Patterns of synchrony in complex networks of adaptively coupled oscillators [original]

Patterns of synchrony in complex networks of

adaptively coupled oscillators

vorgelegt von

M. Sc.

Rico Berner

ORCID: 0000-0003-4821-3366

an der Fakultät II – Mathematik und Naturwissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

- Dr. rer. nat. -

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. Stephan Reitzenstein

Gutachter: Prof. Dr. Dr. h.c. Eckehard Schöll, PhD

Gutachter: PD Dr. Serhiy Yanchuk

Gutachter: Prof. Dr. Alessandro Torcini (Université de Cergy-Pontoise, France)

Tag der wissenschaftlichen Aussprache: 24. September 2020

Berlin 2020

Abstract

Collective phenomena in systems of interconnected dynamical units are omnipresent in nature.

The swarm behavior in flocks of birds or schools of fish, the synchronous flashing of fireflies or

even coherent spiking of neurons in the human brain are just a few examples of collective motion.

Elucidating the mechanisms that give rise to synchronization is crucial in order to understand

biological self-organization. For this sake, the theory of dynamical networks has been successfully

applied over the last decades to boil down the complex dynamics from natural systems to their

essentials. In addition, dynamical networks with adaptive couplings and hence non-constant

network structure appear naturally in real-world systems such as power grid networks, social

networks as well as neuronal networks. In this thesis, we study adaptive networks and their

properties which give rise to the emergence of a variety of synchronization patterns, including

complete and cluster synchronization as well as solitary and chimera-like states. One of the

main fields of application that is investigated in this thesis concerns neuroscience. However,

the methods and approaches developed in this work are not restricted to a specific field of

application.

In the first part of this thesis, we report on the phenomenon of frequency clustering in globally

coupled oscillator network with slow adaptation. As a motivating example, we study a network

of Hodgkin-Huxley neurons with spike timing-dependent plasticity. Here, the clustering leads

to a splitting of a neural population into a few groups synchronized at different frequencies.

We propose a phenomenological model which describes the dynamics of two clusters taking

the adaptive coupling weights into account. Following the successful application of the phe-

nomenological model, we investigate a paradigmatic system of adaptively and globally coupled

phase oscillators inspired by neuronal networks with synaptic plasticity. Our numerical as well

as analytical study of the phase oscillator model allows for a complete description of the mecha-

nism behind the emergence of frequency clustering. Moreover, we unveil the role of individual

clusters for the shape and stability of frequency cluster (multiclusters), which explains the high

level of multistability found in the paradigmatic model.

In the second part, we extend the findings from the first part towards complex network structures.

We observe a variety of partially synchronized states such as phase-locked, multicluster, solitary

and chimera-like states. Solitary states have been observed in a plethora of dynamical systems.

However, the mechanisms behind their emergence were largely unaddressed in the literature.

Here, we show how solitary states emerge in the model of phase oscillators due to the adaptive

feature of the network and classify several bifurcation scenarios in which these states are created

and stabilized. In addition, we investigate the stability of synchronous states for a large class of

complex adaptively coupled oscillator networks. In particular, we generalize the master stability

approach beyond the static network paradigm by taking adaptivity into account. We apply the

new method to an adaptive network of coupled phase oscillators and show how the subtle

interplay between adaptivity and network structure gives rise to the emergence of complex

partial synchronization patterns.

In spite of the lively interest in the topic of adaptive networks, little is known about the interplay

of adaptively coupled groups of networks. Such adaptive multilayer or multiplex networks

appear naturally in neuronal networks. We propose a concept to generate and stabilize diverse

partial synchronization patterns (phase clusters) in adaptive networks. We show that multiplex-

ing induces various stable phase cluster states in a situation where they are not stable or do not

even exist in the single layer. Further, we develop a method for the analysis of Laplacian matrices

of multiplex networks which allows for insight into the spectral structure of these networks

enabling a reduction to the stability problem of single layers. We employ the new method to

provide analytic results for the stability of the multilayer patterns.

Zusammenfassung

Kollektive Phänomene in Systemen von miteinander verbundenen dynamischen Einheiten sind

allgegenwärtig in der Natur. Das Schwarmverhalten in Gruppen von Vögeln oder Fischen,

das synchrone Blinken von Glühwürmchen oder auch das kohärente "Feuern" von Neuronen

im menschlichen Gehirn sind nur einige Beispiele für kollektives Verhalten. Die Erforschung

der Mechanismen, die zur Synchronisation führen, ist entscheidend für das Verständnis von

biologischer Selbstorganisation. Zu diesem Zweck wurde in den letzten Jahrzehnten die The-

orie der dynamischen Netzwerke erfolgreich angewandt, um die komplexen Bewegungen

natürlicher Systeme auf das Wesentliche zu reduzieren. Darüber hinaus kommen dynamische

Netzwerke mit adaptiven Kopplungen und damit nicht konstanter Netzwerkstruktur in realen

Systemen wie Stromnetzwerken, sozialen Netzwerken sowie neuronalen Netzwerken ganz

natürlich vor. In dieser Arbeit untersuchen wir adaptive Netzwerke und deren Eigenschaften,

welche zur Entstehung einer Vielzahl von Synchronisierungsmustern, darunter vollständige

und Cluster-Synchronisierung sowie solitäre und chimärenartige Zustände, beitragen. Eines der

Hauptanwendungsgebiete, das in dieser Arbeit untersucht wird, sind die Neurowissenschaften.

Die in dieser Arbeit entwickelten Methoden und Ansätze sind jedoch nicht auf ein bestimmtes

Anwendungsgebiet beschränkt.

Im ersten Teil dieser Arbeit berichten wir über das Phänomen der Frequenz-Clusterung in global

gekoppelten Oszillator-Netzwerken mit langsamer Adaptivität. Als motivierendes Beispiel un-

tersuchen wir ein Netzwerk von Hodgkin-Huxley-Neuronen mit "spike timing-dependent" Plas-

tizität. Hier führt die Clusterung zur Aufspaltung einer neuronalen Population in einige wenige

Gruppen, die mit unterschiedlichen Frequenzen synchronisiert sind. Wir führen ein phänomenol-

ogisches Modell ein, das die Dynamik von zwei Clustern unter Berücksichtigung der adaptiven

Kopplungsgewichte beschreibt. Nach der erfolgreichen Anwendung des phänomenologischen

Modells untersuchen wir ein paradigmatisches System von adaptiv und global gekoppelten

Phasenoszillatoren, das von neuronalen Netzwerken mit synaptischer Plastizität inspiriert ist.

Unsere numerische sowie analytische Analyse des Phasenoszillatormodells ermöglicht eine

vollständige Beschreibung des Mechanismus hinter der Entstehung von Frequenz-Clustern.

Darüber hinaus stellen wir die tragende Rolle einzelner Cluster für die Form und Stabilität von

Frequenz- Clustern (Multi-Cluster) heraus und erklären damit den hohen Grad an Multistabilität,

der in dem paradigmatischen Modell zu finden ist.

Im zweiten Teil erweitern wir die Erkenntnisse aus dem ersten Teil auf komplexe Netzwerkstruk-

turen. Wir beobachten eine Vielzahl von partiell synchronen Zuständen, wie z.B. phasenstarre,

Multi-Cluster-, solitäre und chimärenartige Zustände. Solitäre Zustände wurden in einer Vielzahl

dynamischer Systeme beobachtet. Die Mechanismen, die hinter ihrer Entstehung stehen, sind in

der Literatur jedoch weitgehend unbehandelt. Hier zeigen wir, wie solitäre Zustände in einem

Model von Phasenoszillatoren aufgrund der adaptiven Eigenschaft des Netzwerks entstehen

und klassifizieren mehrere Bifurkationsszenarien, in denen diese Zustände erzeugt und stabil-

isiert werden. Darüber hinaus untersuchen wir die Stabilität synchroner Zustände für eine große

Klasse komplexer adaptiv gekoppelter Oszillator-Netzwerke. Insbesondere verallgemeinern

wir den Master-Stabilitätsansatz über das statische Netzwerkparadigma hinaus, indem wir

die Adaptivität berücksichtigen. Wir wenden die neue Methode auf ein adaptives Netzwerk

aus gekoppelten Phasenoszillatoren an und zeigen, wie das subtile Zusammenspiel zwischen

Adaptivität und Netzwerkstruktur die Entstehung komplexer partieller Synchronisationsmuster

hervorruft.

Trotz des regen Interesses an adaptiven Netzwerken ist über das Zusammenspiel adaptiv

gekoppelter Netzwerkgruppen wenig bekannt. Solche adaptiven Mehrschicht- oder Multiplex-

Netzwerke kommen in neuronalen Netzwerken ganz natürlich vor. Wir schlagen ein Konzept

zur Erzeugung und Stabilisierung diverser partieller Synchronisationsmuster (Phasen-Cluster)

in adaptiven Netzwerken vor. Wir zeigen, dass das "Multiplexen" verschiedene stabile Phasen-

Cluster-Zustände induzieren kann, selbst wenn diese in den Einschicht-Systemen nicht sta-

bil sind oder gar nicht existieren. Weiterhin entwickeln wir eine Methode zur Analyse von

Laplace-Matrizen von Multiplex-Netzwerken, die einen Einblick in die spektrale Struktur dieser

Netzwerke ermöglicht und damit eine Reduktion des Stabilitätsproblems auf die von Einschicht-

Netzwerken ermöglicht. Wir setzen die neue Methode ein, um analytische Ergebnisse für die

Stabilität der Zustände im Mehrschicht-System zu erhalten.

Publications

[

BER19

]: R. Berner, E. Schöll, and S. Yanchuk: Multiclusters in networks of adaptively coupled

phase oscillators, SIAM J. Appl. Dyn. Syst. 18, 2227 (2019).

[

BER19a

]: R. Berner, J. Fialkowski, D. V. Kasatkin, V. I. Nekorkin, S. Yanchuk, and E. Schöll:

Hierarchical frequency clusters in adaptive networks of phase oscillators, Chaos

, 103134 (2019).

[

ROE19a

]: V. Röhr, R. Berner, E. L. Lameu, O. V. Popovych, and S. Yanchuk: Frequency

cluster formation and slow oscillations in neural populations with plasticity, PLoS ONE

e0225094 (2019).

[

BER19b

]: R. Berner, J. Sawicki, and E. Schöll: Birth and stabilization of phase clusters by

multiplexing of adaptive networks, Phys. Rev. Lett. 124, 088301 (2020).

[

BER20c

]: R. Berner, A. Polanska, E. Schöll, and S. Yanchuk: Solitary states in adaptive nonlocal

oscillator networks (2019), arXiv: 1911.00320.

Table of Contents

Abstract .............................................. ii

Zusammenfassung ....................................... v

Publications ........................................... vii

1 Introduction 1

1.1 Dynamics on complex networks ............................ 1

1.2 Synchronization and collective phenomena ...................... 2

1.3 Dynamics of complex networks ............................. 3

1.4 The role of phase oscillator models for complex dynamical networks ....... 5

1.5 Outline ........................................... 6

2 Fundamentals of adaptive and complex dynamical networks 9

2.1 Complex networks .................................... 9

2.1.1 Networks, subnetworks, and connectivity ................... 9

2.1.2 Special network types .............................. 11

2.1.3 Permutation symmetries in networks ..................... 13

2.2 Dynamics ......................................... 14

2.2.1 Types of coupling ................................. 14

2.2.2 Kuramoto-Sakaguchi type model ........................ 15

2.2.3 Hodgkin-Huxley model with chemical synapses ............... 17

2.3 Adaptive networks in neuroscience ........................... 18

2.3.1 Spike timing-dependent plasticity ....................... 18

2.3.2 Phase difference-dependent plasticity ..................... 19

2.3.3 A network of adaptively coupled phase oscillator .............. 20

2.4 Summary .......................................... 21

I CLUSTER SYNCHRONIZATION IN GLOBALLY COUPLED ADAPTIVE NETWORKS 23

3 Population of Hodgkin-Huxley neurons with spike timing-dependent plasticity 25

3.1

Coupled Hodgkin-Huxley neurons on a network with spike timing-dependent

plasticity .......................................... 26

3.2 Numerical observation of synchrony and frequency clustering ........... 27

3.3 Emergence of two-cluster states ............................. 30

3.4 Phenomenological model with phase difference-dependent plasticity ....... 34

3.4.1 Properties of the model ............................. 36

3.4.2

Comparison of the model and cluster dynamics in Hodgkin-Huxley network

3.4.3 Criteria for the emergence of frequency clusters ............... 37

3.5 Summary .......................................... 39

4 One-cluster states in adaptive networks of coupled phase oscillators 41

4.1 Classification of one-cluster states ............................ 41

4.2 Stability of one-cluster states ............................... 45

4.3 Adaptation rate dependence of one-cluster stability ................. 50

4.4 Double antipodal states ................................. 52

4.5 Summary .......................................... 55

5 Multicluster states in adaptive networks of coupled phase oscillators 57

5.1 Numerical observation of multicluster states ..................... 58

5.1.1 Splay type cluster states ............................. 59

5.1.2 Antipodal type cluster states .......................... 59

5.1.3 Mixed type cluster states ............................. 60

5.2 Splay type multicluster states .............................. 62

5.2.1 Conditions for the emergence of splay type multicluster states ....... 62

5.2.2 Two-cluster states of splay type ......................... 63

5.2.3 Adaptation rate dependence for the emergence of two-cluster states ... 65

5.3 Conditions for the emergence of multicluster states - A generalized approach . . 66

5.4 Antipodal type multicluster states ........................... 69

5.4.1

Asymptotic conditions for the emergence of antipodal type multicluster

states ........................................ 69

5.4.2 Two-cluster states of antipodal type ...................... 71

5.5 Mixed type pseudo-multicluster states ......................... 72

5.5.1

Asymptotic conditions for the emergence of mixed type pseudo-multicluster

states ........................................ 72

5.5.2 Pseudo-two-cluster states of mixed type .................... 73

5.6 Stability of multicluster states .............................. 76

5.6.1 On the stability of multicluster states with evenly sized clusters ...... 77

5.6.2 An effective approach for the stability of multicluster states ........ 78

5.7 Summary .......................................... 79

II INTERPLAY OF ADAPTIVITY AND CONNECTIVITY 81

6 Adaptation on nonlocally coupled ring networks 83

6.1 Multicluster and solitary states ............................. 84

6.1.1 One-cluster states ................................. 85

6.1.2 Multicluster states ................................ 86

6.1.3 Solitary states ................................... 87

6.2 One-cluster states: Local vs. global features ...................... 89

6.2.1 Classification of one-cluster states ....................... 89

6.2.2 Stability of one-cluster states .......................... 91

6.3 The emergence of solitary states ............................. 93

6.4

Adaptive networks with global base topology versus ring base topology: the

differences ......................................... 96

6.5 Summary .......................................... 96

7 Synchronization on adaptive complex network structures 99

7.1 The master stability function for adaptive complex networks ............ 100

7.2 Stability islands in the presence of adaptation ..................... 104

7.3 Stability islands and implications for the emergence of multicluster states .... 107

7.4 Summary .......................................... 109

8 Multilayered adaptive networks 111

8.1 Lifted states in multiplex networks ........................... 112

8.2 Birth and robustness of phase clusters ......................... 114

8.3 Multiplex decomposition ................................. 117

8.4 Stabilizing through multiplexing ............................ 120

8.5 Applications for the multiplex decomposition ..................... 122

8.5.1 The master stability approach for multiplex networks ............ 122

8.5.2 Analytic treatment of diffusive dynamics on multiplex networks ..... 125

8.6 Summary .......................................... 126

9 Conclusion and Outlook 127

APPENDIX 133

A Proof of results from the main text 135

A.1 One-cluster states on globally coupled adaptive networks .............. 135

A.2 Stability of one-cluster states on globally coupled networks ............. 136

A.3 Multicluster states of splay type ............................. 142

A.4 Asymptotic expansions of multicluster states ..................... 142

A.5 From local to global order parameter .......................... 151

A.6 Stability of one-cluster states on nonlocally coupled networks ........... 152

A.7 Stability of lifted one-cluster states ........................... 154

A.8 Example for a complex adjacency matrix ........................ 156

List of Figures 157

List of Tables 163

Acknowledgement 165

Bibliography 167

Introduction1

1.1 Dynamics on complex networks

Complex networks are an ubiquitous paradigm in nature and technology, with a wide field of

applications from physics, chemistry, biology, neuroscience, as well as engineering and socio-

economic systems [

NEW03

]. A lot of work has been devoted to understand the statistical and

topological properties of complex connectivity structures [

ALB02a

COS07

] ranging from ran-

dom [

ERD59

ERD60

] and scale-free networks [

BAR99a

] to small-world structures[

WAT98a

]

and even simplicial complexes that are topological structures used to model many body interac-

tions [GIU16,SIZ19].

The field of complex dynamical networks studies the interplay of dynamics and (static) network

structure [

POR16a

BOC18

]. Powerful methodologies as the Master Stability Function have been

suggested to provide a unified approach to study synchronization of networks with arbitrary

structure [PEC98,CHO09,KYR14,PEC14,WIL14,LEH15b].

The investigation of complex networks and their dynamical features has become a major research

branch in neuroscience. Here, network neuroscience [

BUL09

BAS17

BER19e

] and neuronal

dynamics [

GER14a

BOE17a

] complement each other in order to build up an understanding for

functional and structural properties of the human brain. Within this context, various network

structures equipped with different neuronal models have been studied over the last decades. For

instance, ring-like structures are important motifs in neural networks [

COM03

SPO11

POP11

YAN11

]. Specifically, nonlocally coupled rings where each node is coupled to all nodes within a

certain coupling range, are known to be important systems appearing in many applied problems

and theoretical studies [

PAS95

BRE97

YAN08a

BON09

ZOU09b

HOR09b

PER10c

OME11

KAN13,OME13,YAN15a,SCH16b,KLI17,BUR18,OME18a].

In addition to the nonlocally coupled networks, other coupling structures and motifs such as

modular, scale-free, and small-world structures have been observed in the structural and func-

tional connectivity of human brains [

EGU05

HUM06

BAS06a

MEU10a

WIL12

RIE14

ASH19

The interplay between neuronal dynamics and these structures already revealed important

mechanisms in the functioning of dynamical brain networks [

ZHO06c

ZHO07

BAS17

CHO18

HOE18

WAN19b

RAM19

]. Despite this development and the flourishing interest in network

neuroscience many questions are still unanswered or have not yet been asked [

BAS17

BAS18

Another focus of recent research in network science are multilayer networks, which are sys-

tems interconnected through different types of links [

BOC14

DE13

DE15

KIV14

]. A prominent

example are social networks which can be described as groups of people with different pat-

terns of contacts or interactions between them [

GIR02

AMA17

AMA17a

]. Other applications

are communication, supply, and transportation networks, for instance power grids, subway

networks, or airtraffic networks [

CAR13d

]. In neuroscience, multilayer networks represent for

instance neurons in different areas of the brain, neurons connected either by a chemical link

or by an electrical synapses [

PER14a

MAJ18a

BER19d

], or the modular connectivity structure

21 Introduction

of brain regions [

BEN16

BAT17

VAI18

]. A special case of multilayer networks are multiplex

topologies, where each layer contains the same set of nodes, and only pairwise connections

between corresponding nodes from neighboring layers exist [

ZHA15a

MAK16

SEV16

JAL16

REQ16

GHO16

LEY17a

AND17

FRO18

PIT18

LEY18

GHO18

MIK18

SAW18

SAW18c

SEM18,OME19,RYB19,NIK19,BLA19,JAL19].

1.2 Synchronization and collective phenomena

Collective behavior in networks of coupled oscillatory systems has attracted a lot of attention

over the last decades. Depending on the network and the specific dynamical system, various

synchronization patterns of increasing complexity were explored. Even in simple models of

coupled oscillators, patterns such as complete synchronization [

KUR84

PIK01

], cluster syn-

chronization where the network splits into groups of synchronous elements [

YAN01a

SOR07

BEL11a

DAH12

NIC13

PEC14

GOL16

ZHA20

], and various forms of partial synchronization

patterns have been found. The investigation of synchronization phenomena on complex net-

works [

STR00

PIK01

STR01a

BOC02

STR03

NIS06

ARE08

BAL09

NEK15b

BOC18

MAI18

]

has become fruitful for applications in many areas ranging from physics and chemistry to biology,

neuroscience, physiology, ecology, socio-economic systems, computer science and engineering.

For example, synchronization of neurons is believed to play a crucial role in brain networks under

normal conditions, for instance in the context of cognition and learning [

SIN99

FEL11

], and un-

der pathological conditions such as Parkinson’s disease [

HAM07

], epilepsy [

JIR13

JIR14

ROT14

AND16

], tinnitus [

TAS12

TAS12a

], schizophrenia, or autism, to name a few [

UHL06

]. Another

example is the synchronized flashing in groups of fireflies [

BUC68

]. Also in power grid networks

synchronization is essential for their operation [

ROH12

MOT13a

MEN14

TAH19

]. The control

of dynamics on networks is another important issue with a lot of applications [

SCH07

SCH16

One of the most important partial synchronization patterns are chimera states. As they have

been theoretically predicted [

KUR02a

ABR04

], these states are characterized by a spatial co-

existence of coherence (synchronized domains) and incoherence (desynchronized domains).

Numerous works have been devoted to this phenomenon, theoretically as well as experimentally,

showing that chimera states play a crucial role for the understanding of real-world dynami-

cal systems [

MOT10

OME11

PAN15

YAO16

SCH16a

SCH16b

MAJ18a

OME18a

OME19c

SCH19c

ZAK20

]. Apart from the theoretical importance, complex dynamical scenarios such

as chimera states are found in a wide range of experimental systems, e.g. optical light modu-

lators [

HAG12

] and chemical oscillators [

TIN12

TOT15

TOT18

], mechanical or optomechani-

cal [

MAR13

KAP14

OLM15

PEL20

], electronic or optoelectronic oscillators [

LAR13

LAR15

electrochemical systems [

WIC13

WIC14

SCH14a

], and electronic circuits [

GAM14

ROS14a

Further, it has been shown that chimera states are crucial for the understanding of epileptic

seizures [

AND16

CHO18

] or unihemispheric sleep [

RAT00

RAT16

RAM19

]. Due to their im-

portance, several approaches have been developed to control chimera states and other partial

synchronization patterns [BIC15,OME16,OME18,RUZ19,SAW20].

Beyond chimera states various other types of partial synchronization scenarios have been

found [

PAZ05

ZAK14

MAI14a

ASH15

KLI15

ZHA15a

BI16

]. Solitary [

MAI14a

] or Bellerophon

states [

BI16

] are just two examples of the plethora of synchronization patterns discovered in

1.3 Dynamics of complex networks 3

the transition from incoherence to coherence or as a result of the destabilization of the coherent

state.

Recently, solitary states have been found in diverse dynamical systems. Solitary states are

described as states for which only one single element behaves differently compared with the

behavior of the background group, i.e., the neighboring elements. These kinds of states have

been observed in generalized Kuramoto-Sakaguchi models [

MAI14a

WU18a

TEI19

CHE19b

the Kuramoto model with inertia [

JAR15

JAR18

], models of power grids [

TAH19

HEL20

], the

Stuart-Landau model [

SAT19

], the FitzHugh-Nagumo model [

RYB19a

SCH19a

], systems of

excitable units [

ZAK16b

] as well as in Lozi maps [

RYB17

] and even in experimental setups of

coupled pendula [

KAP14

]. Solitary states are considered as important states in the transition

from coherent to incoherent dynamics [

JAR15

SEM15b

MIK18

]. Despite their appearance in

many well-studied models, the mechanisms of their emergence are less understood. Until now,

only a few works could shed some light on the details behind their formation [

MAI14a

JAR18

SEM18a,TEI19].

1.3 Dynamics of complex networks

Besides the paradigm of static networks the analysis of interconnected dynamical system on

temporally evolving connectivity structures has become a future challenge in nonlinear sci-

ence [

POR19

]. Here, two approaches have to be distinguished. The first which uses a prescribed

temporal evolution of the network structure [

HOL12

HOL15a

], called temporal networks, and

a second where the temporal evolution of the network depends upon on the dynamical state

of the network nodes [

GRO09

], called adaptive networks. In this thesis, we focus on the latter

approach and consider adaptive networks throughout.

With regard to this thesis, one of the main motivations for studying adaptive dynamical networks

comes from the field of neuroscience. Here, a possible mechanisms that causes adaptation, which

may lead to persistent changes in neural connections and relates to learning and memory, is

synaptic plasticity [

HEB49

]. The efficiency of synapses to transmit the electrical potential between

neurons may increase or decrease depending on the mutual neural activity, which results

in short- or long-term potentiation or, respectively, depression of synapses [

BRO88a

BLI93

An example is spike timing-dependent plasticity (STDP) which describes the change of the

synaptic weight as a function of the difference of spiking times between pre- and post-synaptic

neurons [

GER96

MAR97a

BI98

ABB00

BI01

CAP08a

MEI09a

LUE16

]. Thus the network

structure reorganizes adaptively in response to the neuronal dynamics. Similarly, chemical

systems have been reported [JAI01], where the reaction rates adapt dynamically depending on

the variables of the system. Activity-dependent plasticity is also common in epidemics [

GRO06b

]

and in biological or social systems [

GRO08a

]. Many studies are devoted to the mechanism of

plasticity in chemical synapses, see e.g. [

ABB00

MAR11b

], however, it has been shown that also

electrical synapses undergo plastic changes [MER14,PER14a].

A famous example for a plasticity rule is the Hebbian rule which assumes that the modifications

of the synaptic weights are driven by correlations in the firing activity of pre- and post-synaptic

neurons. More specifically, the rule assumes that those connections are potentiated, for which

one neuron contributes to the firing of another [

HEB49

]. Nevertheless, in many publications,

41 Introduction

the Hebbian rule is considered in a narrower sense of closeness between the spiking times: the

smaller the distances between the spikes are the higher is the potentiation of the corresponding

synapse [

HOP96

SEL02

]. In this work, we are dealing with spike-based learning rules rather

than rate-based.

Previous studies of neural networks with STDP (or STDP-like) showed that such networks can

evolve and create various coupling and spiking patterns [

CAT08

MIL13b

MIK14

]. For instance,

the coupling weights can exhibit stable localized spatial structures, which can be interpreted

as receptive fields [

CLO10

]. These structures can be either unidirectionally of bidirectionally

coupled, depending on the plasticity rule or external input properties. The STDP mechanism

plays an important role in temporal coding of information by spikes [

GER96

CLO10

]. On the one

hand, synchronized firing in neural ensembles with STDP can be stabilized through potentiation

of synaptic coupling by stimulation-induced transient synchronization of neurons [

TAS06

TAS12a

POP13

LUE16

]. On the other hand, a desynchronized state can lead to a depression

of synaptic weights [

TAS06

TAS12a

]. Thus, neural networks with plasticity are prone to co-

existence of different stable dynamical and structural states, which may be realized by choosing

appropriate initial conditions or stimulation procedures.

In addition to the localized spatial structures, the emergence of modular and scale-free networks

has been reported [

ITO01a

ITO03

STA10b

GUT11

ASS11

YUA11

AOK12

WIN12

AOK15a

BOT12

BOT14

POP15

CHA17a

] in networks with STDP. This fact underlines the potential

importance of adaptive mechanism in the formation of connectivity structures as they have

been experimentally found in brain networks [

MEU10a

ASH19

]. Furthermore, activity based

adaptive rewiring has even been shown to enhance modularity [RUB09].

Besides direct application to neuroscience, spike timing-dependent plasticity rules have been dis-

cussed for neurocomputing [

HOP99

] and have recently also been implemented into memristive

devices [

DU15

JOH18

]. For early work on electronic switching and S-shaped current-voltage

characteristics see [

SCH87

SHA92

SCH01

]. Memristors and memristor arrays play an important

role in the development of neuromorphic computing [

PIC13

WAL13

IGN15

HAN17

BIR19

]

which is believed to be the future of artificial intelligence. An exhaustive survey on approaches to

neuromorphic computing is presented in Ref. [

SCH17k

] showing the plethora of work devoted

to this questions. The huge common interest of the scientific community in developing brain-

inspired technologies is not least expressed by the flagship project "The human brain project"

funded by the European Union [

MAR12c

AMU19

] or the NIH BRAIN Initiative [

KOR18a

Therefore, there is clearly an urge for the analysis of adaptive neural networks and the develop-

ment of novel methods to understand these systems.

Next to the active interest in adaptive networks with regards to neuroscience, motivation for

systems with adaptive connectivity structure has been found in other dynamical systems from

biology [

PAI14

] and social science [

SAY13

SAY15

AOK16

], in systems that describe swarm-

ing behavior of coupled agents [

IWA10

IWA10a

], or in the dynamics of communication net-

works [

GAV12

]. Furthermore, it has been shown that synchronization in complex networks is

enhanced by an adaptation of the coupling structure [

ZHO06f

ZHU10

]. Due to this, adaptive

network structures have been also successfully applied to control synchronization in complex

and even time-evolving networks [

LEL08

SOR08

WAN08f

LEL09

LEL10

LEL10a

SCH12

SEL12,GUZ13,LEH14,PLO16].

1.4 The role of phase oscillator models for complex dynamical networks 5

1.4 The role of phase oscillator models for complex dynamical

networks

Models of coupled phase oscillators, i.e., models where the dynamical variable is simply given

by an angle, are nowadays a well-known paradigm in order to study collective behavior of

dynamical oscillatory units on complex networks [

ACE05

PIK01

]. In this context, a particularly

important feature is that coupled nonlinear oscillators can be reduced to phase oscillator net-

works in case of weak interactions [

WIN80

HOP97

PIK01

PIE19a

]. The reduction of complex

dynamical systems to networks of coupled phase oscillators is well-known and there exist ex-

haustive reviews highlighting the importance of phase oscillator models [

PIK01

ASH16

PIE19a

Recent studies, in addition, aim to make phase oscillators models even more powerful by gener-

alizing the conditions under which the reduction techniques are valid [

KLI17a

ROS19a

ROS19b

ERM19]

As a representative from the class of phase oscillator models, the Kuramoto model where all

oscillators are coupled in an all-to-all manner, particularly, has attracted a lot of attention due

to its simple form and mathematical tractability [

KUR84

STR00

]. The Kuramoto model has

gained additional popularity due to its application for real-world problems [

STR93

PIK01

STR03

STR05a

ROD16

]. Despite the simple structure of the Kuramoto model many different

dynamical regimes have been observed. In order to get a hand on the dynamics of the Kuramoto

model sophisticated methods have been developed. In particular, sinusoidally and globally

coupled phase oscillators have been shown to be partially integrable. The Watanabe-Strogatz

theory, which was introduced to show the integrability, allows for a reduction of a system with

finite size to only three dimensions and can be applied to even more general classes of phase

oscillator models [

WAT93a

WAT94

STE11b

]. Another approach which has been developed

to understand coupled oscillators in the continuum limit is the Ott-Antonson ansatz [

OTT08

The Ott-Antonson theory allows for a reduction to a two dimensional dynamical system in the

case of an infinite number of oscillators. The theory has been successfully applied to describe

the emergence of chimera states even beyond phase oscillator models [

OME08

ABR08

LAI09

OME13

OME18a

OME19c

]. Remarkably for both reduction techniques, the Watanabe-Strogatz

and the Ott-Antonson theory, the reduced systems possess a clear physical interpretation, and

both approaches are intimately related with each other [

MAR09c

PIK15

]. Just recently, another

very promising reduction technique for finite systems has been introduced which make use of

so-called collective coordinates [HAN18b,SMI19,SMI20].

Beyond the classical Kuramoto model various generalization have been proposed. Starting from

generalization to complex networks [

GOM07

DOE14

], the theory of phase oscillators has been

further developed to study phenomena of phase transitions [

PAZ05a

GOM11a

BOC16

], net-

work symmetries [

NIC13

], the impact of inertia [

ERM91

FIL08a

MAI14a

SCH14m

OLM15a

]

and other forms of frequency adaptation [

TAY10

], delayed coupling [

YEU99a

], or the effect of

time-depending parameters [PET12] to name just a few.

Another generalization which gained a lot of attention over the last years concerns the effects

and phenomena in networks of phase oscillators with adaptive coupling. Various models have

been suggested and studied to gain insights into the interplay of collective dynamics and adap-

tivity [

MAS17c

]. A lot of works are inspired by recent findings in neuroscience and implement

synaptic plasticity into their models [

SEL02

REN07

MAI07

MAS07c

AOK09

NIY09

TAK09

61 Introduction

LI10c

AOK11

GUT11

ASS11

SKA13a

CHA14a

REN14

TIM14

AOK15

HA16a

KAS16a

NEK16

ASL17

KAS17

ASL18

ASL18a

KAS18a

KAR19

]. Despite the rich interest in neurosci-

entific applications, also other forms of adaptation and other applications have been considered.

For example, effects of adaptive rewiring, an adaptive network growth, or evolutionary edge-

snapping are studied in Refs. [

GLE06

LI11a

SCA15

PAP17

DAM19

]. Also the interplay of

adaptivity and complex network structure such as multilayers have been only recently numeri-

cally investigated [MAK16a,KAS18,KAS19].

1.5 Outline

This thesis is devoted to the analysis of partial synchronization patterns in complex oscillator

networks with adaptive coupling strength. This analysis is organized in two parts. After two

introductory chapters, in the first part, the Hodgkin-Huxley neuron model and the phase

oscillator model on a globally coupled network with synaptic plasticity are investigated. Here, we

observe the emergence of multiclusters due to plasticity and underline the importance of phase

oscillator models in order to understand these complex synchronization pattern. By rigorous

mathematical methods, we provide a comprehensive understanding of the cluster structures. In

part II, the paradigmatic model of adaptively coupled phase oscillators, introduced in part I, is

used in order to study the interplay of adaptivity and complex connectivity structures.

The basic formalism and the standard notation which is used through out part I and II is

presented in Chapter 2. Here, we provide graph theoretical preliminaries and introduce all

classes of networks studied in this thesis. Furthermore, a brief introduction into the dynamics

on complex networks is given where we discuss different types of coupling and models. In the

last section, we show how adaptivity is modeled in the context of synaptic plasticity. In addition,

the reduction to phase oscillator models with phase difference-dependent plasticity is given.

Part I starts with Chapter 3. In this chapter, we analyze a population of Hodgkin-Huxley neurons

with spike timing-dependent plasticity. The numerical observation of multiclusters is shown

and the detailed mechanisms of the stability of frequency clusters is explained by using the

simplest case of two clusters. Here, we propose a phenomenological model, which describes the

dynamics of two clusters taking into account the adaptation of the weights. The model is shown

to reflect not only qualitative, but also some basic quantitative properties of the two-cluster

formation. We also determine the set of plasticity functions (update rules), which lead to the

clustering.

In Chapter 4, the existence and stability of one-cluster states is rigorously analyzed in the

phase oscillator model. We provide a full classification of one-cluster states and investigate

the impact of the timescale separation on their stability. Different types are characterized by

different phase relations of the nodes. Finally, we discuss the role of a special type of one-

clusters, called double antipodal states. Although such states are unstable for all parameter

values, they appear as saddles connecting synchronous and splay states. As a result, double

antipodal states are observed during a "meta-stable" transition between the phase-synchronous

and non-phase-synchronous state.

Chapter 5is devoted to the phenomenon of multicluster states which are distinguished by

different frequencies of the individual clusters. The multiclusters are, first, illustrated using

1.5 Outline 7

numerical integration, and then classified. Complementing earlier work, we find a mixed

multicluster state. These new multiclusters consist simultaneously of splay and (in-) anti-phase

synchronous states. It results from this that all one-cluster states can serve as building blocks

for a multicluster. Further, we provide an analytic description of all types of multicluster states

using explicit methods as well as methods from the theory of asymptotic expansions. We show

that multiclusters corresponding to certain fixed frequency ratio appear in continuous families,

and, moreover, multiclusters with different frequency ratios coexist for the same parameter

values. We derive algebraic equations for the frequencies of coexisting multicluster solutions. In

a particular case of two clusters, these equations can be explicitly solved. In the end, the stability

of multicluster states is discussed and connected to the stability of their individual building

blocks, i.e., one-cluster states.

The second part starts with Chapter 6. In this chapter, we analyze the emergence of solitary states

on a nonlocally coupled ring in the presence of plastic coupling weights. For this, numerical

results and a rigorous definition for multicluster and solitary states on complex networks

are presented. We further provide a detailed analysis of one-cluster states and derive explicit

relations between local and global properties. A reduced model for two-clusters is derived

which allows us to unveil the bifurcation scenarios in which solitary states are formed and

(de)stabilized.

Complex networks of adaptively coupled oscillators are studied in Chapter 7. Here, we consider

the stability of synchronous states and derive the master stability function for a large class

of adaptive networks. The new method is applied to a system of adaptively coupled phase

oscillators with complex connectivity structure. We observe the existence of stability islands in

the master stability function and connect these islands to the emergence of multicluster as well

as chimera-like states.

In chapter 8we show that a plethora of novel patterns can be generated by multiplexing adaptive

networks in a multilayer configuration. In particular, partial synchronization patterns like phase

clusters and more complex cluster states which are unstable in the corresponding monoplex

network can be stabilized, or even states which do not exist in the single-layer case for the

parameters chosen, can be born by multiplexing. We elucidate the delicate balance of adaptation

and multiplexing which is a feature of many real-world networks even beyond neuroscience.

The thesis concludes with Chapter 9where the obtained results are summarized and discussed.

Furthermore, an outlook on possible continuations of this work is given.

Fundamentals of adaptive and complex

dynamical networks 2

Throughout this thesis well-established mathematical and physical concepts are used and ex-

tended in order to describe dynamical systems with complex and adaptive coupling structure. In

this chapter, we fix the notation and introduce the basic formalism that is used in the subsequent

considerations.

The chapter is organized as follows. In Section 2.1, we provide graph theoretical preliminaries

and introduce all classes of networks studied in this thesis. Furthermore, a brief introduction

into dynamics on complex networks is given in Section 2.2 where we discuss different types of

coupling and models. In the Section 2.3, we show how adaptivity is modeled in the context of

synaptic plasticity. In addition, the reduction to phase oscillator models with phase difference-

depend plasticity is given.

2.1 Complex networks

This section is devoted to the introduction of graph theoretical concepts for networks and the

description of special classes of networks that will be considered in this thesis. Here, we follow

standard textbooks and literature on graph theory and complex networks [

GOD01

COS07

NEW10].

2.1.1 Networks, subnetworks, and connectivity

In network science the notions graph and network, vertex and node, and edge and link, respec-

tively, are used interchangeably. For the sake of clarity, we use the notions network, node, and

link. In general we define a directed network

as a triple

N=(𝑉

𝐸

Ψ)

of finite sets

𝑉

and

𝐸

, and

Ψ:𝐸→ {(𝑣

𝑤) ∈ 𝑉×𝑉}

. Elements from the sets

𝑉

and

𝐸

are called nodes and links,

respectively. The total number of nodes of a network is denote by

𝑁=|𝑉|

. Usually the range

of the map

𝜓

is restricted to

{(𝑣

𝑤) ∈ 𝑉×𝑉:𝑣≠𝑤}

in order to avoid nodes to be connected to

themselves. However, in this thesis self-coupling is allowed for all further consideration. Note

that for an undirected network

Ψ:𝐸→ {𝑋⊆𝑉:|𝑋| ∈ {

1, 2

}}

which means that

𝜓

maps the

edge

𝑒∈𝐸

𝑋=𝑣

if the mapping

𝜓(𝑒)

refers to self-coupling or it maps the edge to

𝑋={𝑣

𝑤}

if 𝜓(𝑒)refers to an undirected link between node 𝑣and 𝑤(𝑣≠𝑤).

By definition, two links can be mapped through

to the same element of

𝑉×𝑉

. If this is the case,

i.e., there exist

𝑒

𝑒0∈𝐸

such that

𝜓(𝑒)=𝜓(𝑒0)

, the network is called a multinetwork (multigraph

is more commonly used). If on the contrary all links can be uniquely identified with their images

under the map

, the network is called a simple network. This latter type of networks build the

backbone of hundreds of studies on complex networks and forms the basis of the subsequent

chapters. Note that for simple networks with

𝑉={𝑣1

. . .

𝑣𝑁}

every link

𝑒

can be assigned to

its image

Ψ(𝑒)

which allows for the shorthand notation

𝑒𝑖 𝑗

as the link connecting two nodes

starting at node

𝑣𝑗

and ending at node

𝑣𝑖

(

𝑖

𝑗=

. . .

𝑁

). In this case the map

can be omitted

10 2 Fundamentals of adaptive and complex dynamical networks

in the definition of a network. For the sake of simplicity, we refer to simple networks as networks

unless stated differently. Further, we associate an undirected network

N0=(𝑉0

𝐸0)

to a directed

network

N=(𝑉

𝐸)

𝑉0=𝑉

𝐸0={𝑒𝑖 𝑗 :𝑒𝑖 𝑗 ∈𝐸or 𝑒𝑗𝑖 ∈𝐸

𝑖

𝑗=

. . .

𝑁}

. Note that it is also

possible to assign a directed network to an undirected network by introducing an orientation of

the links [GOD01].

Another important notion which will be used in this thesis is the induced subnetwork. A

subnetwork of a network

is given by

N0=(𝑉0

𝐸0

Ψ)

𝑉0⊆𝑉

and

𝐸0⊆𝐸

. In addition, a

subnetwork is denoted by induced if 𝐸0={𝑒𝑖 𝑗 ∈𝐸:𝑣𝑖,𝑣𝑗∈𝑉0}.

The connectivity structure for a simple network is often represented by an

𝑁×𝑁

matrix

𝐴

, called

adjacency matrix, which possesses the entries

𝑎𝑖 𝑗 =(1, if 𝑒𝑖 𝑗 ∈𝐸,

0, otherwise. (2.1)

This algebraic view on networks by representing their structure in form of a matrix is the one

we use throughout this work [

CVE97

GOD01

NEW10

]. This viewpoint allows one to define

the in-degree 𝑑(𝑖)of node 𝑣𝑖by considering the row sum of the adjacency matrix, i.e.,

𝑑(𝑖)=

𝑁

𝑗=1

𝑎𝑖 𝑗 . (2.2)

Further, we define the Laplacian matrix 𝐿of a network as

𝐿=©«

𝑑(1)0··· 0

0.......

.......0

0··· 0𝑑(𝑁)

ª®®®®®®¬

−𝐴. (2.3)

Note that this matrix is a discrete version of the well-known Laplacian operator known from, e.g.,

the theory of waves. In particular, for certain dynamical networks such as nonlocally coupled

rings with diffusive coupling, as they are defined later on in Section 2.2, the relation between

the discrete and the continuum version of the Laplacian can be shown explicitly [

KOU14

ISE15

ISE15a].

𝑣𝑖

𝑣𝑗

walk of length

𝐿∈N

on a network is a sequence

𝑣𝑖

𝑒𝑘1𝑖

𝑣𝑘1

. . .

𝑒𝑗𝑘𝐿−1

𝑣𝑗

for

𝑣𝑖

𝑣𝑗∈𝑉

and

𝑒𝑘𝑚𝑘𝑚−1∈𝐸

for all

𝑚=

. . .

𝐿

where

𝑘0=𝑖

, and

𝑘𝐿=𝑗

. With this, a network is said to be

connected (or weakly connected [

KOR18

]) if for any two nodes

𝑣𝑖

𝑣𝑗∈𝑉

(

𝑣𝑖≠𝑣𝑗

) exists at least

one

𝑣𝑖

𝑣𝑗

walk on the associated undirected network. Note that according to this definition, two

nodes

𝑣𝑖

and

𝑣𝑗

are connected if either

𝑒𝑖 𝑗 ∈𝐸

𝑒𝑗𝑖 ∈𝐸

. We can utilize the adjacency matrix

in order to analyse whether there is a

𝑣𝑖−𝑣𝑗

walk of length

𝐿

between two nodes. For this, we

consider the

𝐿

th power of the adjacency matrix for the associated undirected network and find

that there exists a

𝑣𝑖−𝑣𝑗

walk if

(𝐴𝐿)𝑗𝑖 ≠

0. Therefore, a network is connected if the matrix

Í𝑁

𝑙=1𝐴𝑙

possesses no vanishing entry. The spectrum of the Laplacian matrix, also called graph

spectrum, allows for another way to find whether a network is connected. Note that by definition

any Laplacian matrix possesses a zero eigenvalue which belongs to the eigenvector

(

. . .

, 1

)𝑇

2.1 Complex networks 11

(a) (b) (c)

P= 3

(d)

P= 1

Figure 2.1:

Illustration of different networks with a total number of 12 nodes: (a) globally coupled network, (b)

nonlocally coupled network (coupling range

𝑃=

3), (c) locally coupled network (coupling range

𝑃=

1), and (d)

multiplex network consisting of two layers for which one layer is globally coupled and the other layer has a nonlocally

coupled network structure.

Further, for undirected networks the Laplacian is symmetric and therefore its spectrum consist

of eigenvalues such that 0

=𝜆1(𝐿) ≤ 𝜆2(𝐿) ≤ ··· ≤ 𝜆|𝑉|(𝐿)

. The second smallest eigenvalues is

called algebraic connectivity (sometimes denoted as Fiedler number) and vanishes if and only if

the network is unconnected [FIE73,FIE89].

Lastly, for each network we can define a link weighting

Ξ:𝐸→R

which assigns real numbers

(weights) to each link of the network. According to this map

, the weight matrix

𝜅

with entries

𝜅𝑖 𝑗 =(Ξ(𝑒𝑖 𝑗 ), if 𝑒𝑖 𝑗 ∈𝐸,

0, otherwise. (2.4)

With regards to dynamical networks, the weight matrix is also called coupling matrix and its

entries coupling weights.

In the following we describe several types of networks with special structural properties.

2.1.2 Special network types

Globally coupled network

The simplest network is the globally coupled network (also complete or all-to-all network) which

consist of

𝑁

nodes where each node is connected to every other node. Hence, the entries of the

adjacency are given by

𝑎𝑖 𝑗 =(1, if 𝑖≠𝑗,

0, otherwise, (2.5)

if there is no self-coupling, otherwise

𝑎𝑖 𝑗 =

1 for all

𝑖

𝑗=

. . .

𝑁

. An illustration is provided in

Fig. 2.1(a).

The spectral properties of this type of network are easily determined. The Laplacian for a globally

coupled network possesses, besides the single zero eigenvalue, the

𝑁−

1-times degenerate

eigenvalue 𝑁.

12 2 Fundamentals of adaptive and complex dynamical networks

Circulant networks and nonlocally coupled rings

Another important and already well-studied network structure is represented by circulant

adjacency matrices, i.e., where each row vector

𝒂𝑖=(𝑎𝑖1

. . .

𝑎𝑖𝑁 )

is rotated one element to the

right relative to the preceding row vector [

DAV79

GRA05

]. These adjacency matrices correspond

to a ring network where each element has the same coupling structure. For this type of networks

the spectrum can be determined by using the discrete Fourier transformation. In particular, the

discrete Fourier vectors (𝑙=1, . . . ,𝑁)

𝜻𝑙=1

√𝑁𝑒i2𝜋𝑙

𝑁,. . . ,𝑒i2𝜋(𝑁−1)𝑙

𝑁, 1𝑇(2.6)

are the 𝑁eigenvectors for the 𝑁corresponding eigenvalues

𝜆𝑙=

𝑁

𝑚=1

𝑎1𝑚𝑒i2𝜋𝑚𝑙

𝑁. (2.7)

A particular case of circulant structures are nonlocally coupled ring networks. These networks

are given by

𝑎𝑖 𝑗 =(1, if (𝑖−𝑗)mod 𝑁≤𝑃,

0, otherwise, (2.8)

where

𝑃

denotes the so-called coupling range. Illustrations for these networks are shown in

Fig. 2.1(b,c). In Fig. 2.1(b), we show a nonlocally coupled ring network for

𝑃=

3. Figure 2.1(c)

displays a locally coupled ring network with coupling range

𝑃=

1 where each node is only

coupled to its direct neighbor. By changing the coupling range between

𝑃=

1 and

𝑃=𝑁/

2 (if

𝑁

is even, else

𝑃=(𝑁+

2) the corresponding network structure range from the locally coupled

to the globally coupled network, respectively.

Another intriguing circulant network structure arises in the context of fractal connectivities.

Here, a Cantor construction algorithm has been used to derive iteratively a vector

𝒂

from a base

pattern [HIZ15,OME15,PLO16a,TSI16,ULO16,KRI17,SAW19].

Multilayer and multiplex networks

Multilayer networks are networks where the whole set of nodes is divided into subsets of nodes

which are said to belong together. The induced subnetworks for the individual subsets are

then called layers. From the mathematical point of view, multilayer networks are just networks.

However, the special structure of a multilayer network, i.e., the partition into several subsets,

has recently been considered very important in order to describe the dynamics on real-world

networks. For exhaustive reviews on multilayer networks and their mathematical description we

refer to Refs. [

DOM13

BOC14

KIV14

]. In the following, we discuss some features of multilayer

networks by considering the special class of multiplex networks.

Multiplex networks are a particular kind of multilayer networks. These networks consist of

𝐿∈N

layers of

𝑁

nodes where the connectivity structure within the layer is given by

𝐿

, generally

2.1 Complex networks 13

different, adjacency matrices

𝐴𝜇

(

𝜇=

. . .

𝐿

). These adjacency matrices determine the intra-

layer networks structure. Let

Per(𝑁)

denote the set of all

𝑁×𝑁

permutation matrices [

LIE15

Then, the connectivity between the layers is given by

(𝐿2−𝐿)

matrices

𝐼𝜇𝜈 ∈Per(𝑁)Ðˆ

where

is the zero matrix. Hence, all nodes of one layer are either uniquely identified with all nodes

of another layer via the inter-layer structure

𝐼𝜇𝜈

or there are no connection between the layers.

An illustration of a multiplex networks with two layers is provided in Fig. 2.1(d). Here, the

two layers are bidirectionally connected with

𝐼12 =𝐼21 =I𝑁,𝑁

. Within the layers, however, the

coupling structure is different. One layer possesses a globally coupled while the other layer

possesses a nonlocally coupled ring intra-layer structure.

An algebraic representation of multilayer networks is achieved by multilinear forms [

DOM13

By flattening of these tensor structures, however, a representation via an adjacency matrix

is obtained. Here flattening means that one can relate a finite dimensional tensor space to

another finite dimensional vector space via an isomorphism, see [

KIV14

]. In this context the

representation is called supra-adjacency matrix and takes the following block matrix form

𝐴=©«

𝐴1𝐼12 ··· 𝐼1𝐿

𝐼21 .......

.......𝐼(𝐿−1)𝐿

𝐼𝐿1··· 𝐼𝐿(𝐿−1)𝐴𝐿

ª®®®®®®¬

. (2.9)

Usually the inter-layer structures are assumed to be given by the identity matrix I𝑁,𝑁.

The spectral properties of multiplex networks have been firstly discussed in [

GOM13

SOL13a

Here, results based on perturbation for weighted multiplex networks in case of weak and strong

inter-layer coupling have been derived. As a part of the present thesis, novel results on the

spectrum of multiplex networks are provided in Chapter 8.

2.1.3 Permutation symmetries in networks

Here, we introduce the notion of symmetry for networks. Symmetries are always described

as special transformations leaving certain structures of interest invariant. For this, in case of

networks, permutations of nodes are considered as symmetry transformation. A permutation

of nodes of a network is given by the endomorphism

Π:𝑉→𝑉

𝑣𝑖↦→ 𝑣𝜋(𝑖)

with

𝜋∈𝑆𝑁

and

𝑖=

. . .

𝑁

. By these means, we call such a transformation a symmetry if the following

holds:

𝑒𝑖 𝑗 ∈𝐸

if and only if

𝑒𝜋(𝑖)𝜋(𝑗)∈𝐸

. Transformations with this property, also called

automorphisms of a network, form a group, so-called automorphism group or symmetry group,

under the composition [BOL98]. The symmetry group of a network Nis denoted by Aut(N).

Symmetries appear in many real-world systems [

MAC08

]. Utilizing these symmetry properties

the analysis of the spectral properties [

MAC09a

] of complex networks can be simplified, complex

networks can be reduce to simpler structure [

XIA08

]. Moreover, certain scenarios in the spatial

and temporal dynamics of systems are found to be induced by network symmetry [

FIE88

GOL88a,ASH92b].

14 2 Fundamentals of adaptive and complex dynamical networks

2.2 Dynamics

In dynamical networks all nodes and links of a weighted network

represent dynamical entities

and their weighted interaction with each other, respectively. Formally, a dynamical network of

𝑁

nodes equipped with dynamical

𝑑

-dimensional variables

𝒙𝑖∈C𝑑

(

𝑖=

. . .

𝑁

), as they are

analyzed throughout this thesis, may be written as follows:

𝒙𝑖=𝑓𝑖(x𝑖(𝑡)) −𝜎

𝑁

𝑗=1

𝑎𝑖 𝑗 𝜅𝑖 𝑗𝐺𝑖 𝑗 (𝒙𝑖,𝒙𝑗), (2.10)

where

𝑓𝑖∈𝐶1(C𝑑

C𝑑)

describes the local dynamics of node

𝑖

, the functions

𝐺𝑖 𝑗 ∈𝐶1(C𝑑

C𝑑)

determine the pairwise coupling between the nodes

𝑖

and

𝑗

. Here,

𝐶1(C𝑑

C𝑑)

denotes the set of

differentiable functions

𝑓:C𝑑→C𝑑

with continuous derivative. The basic coupling structure

is given by the matrix entries

𝑎𝑖 𝑗 ∈ {0, 1}

of the

𝑁×𝑁

adjacency matrix

𝐴

. Each existing link is

additionally weighted by the coupling weights

𝜅𝑖 𝑗

. The parameter

𝜎∈R

is the overall coupling

constant.

In the previously described framework of dynamical systems only pairwise interactions are

considered. Note that there exist generalizations in order to describe many-body interaction

by using, for instance, simplicial complexes which are topological generalization of "classi-

cal" networks [

TAN11b

ASH16b

BIC16

SKA19

LEO19

]. In generalizations using simplicial

complexes, interactions between nodes are not only described by links (1-simplex) but also by

higher

𝑛

-dimensional

𝑛

-simplices that can be thought of as the convex hull of

(𝑛+

)

vertices,

see [GIU16] for an introduction. However, these models are beyond the scope of this work.

2.2.1 Types of coupling

In real world systems, single dynamical units are not isolated but interact by in some cases very

complicated interaction mechanism. For the purpose of describing these interactions between

the units, simplified interaction schemes given by the coupling function

𝐺

have been developed.

In the following, we introduce the most important schemes which are used in the subsequent

chapters

In the context of diffusive systems on complex networks, linear diffusive processes were consid-

ered in order to study the dynamics on social as well as transport networks [

BAR11d

GOM13

Moreover, also in neuronal populations the coupling through electrical synapses (gap-junctional

connections) is modeled by a linear diffusive map [

BEN04a

KOP04

CON04

CUR16

REI17a

ALC19]. The described form of coupling can be formally written as

𝐺𝑖 𝑗 (𝒙𝑖,𝒙𝑗)=𝑮𝑖 𝑗 (𝒙𝑖−𝒙𝑗)(2.11)

where the coupling functions becomes

𝑑×𝑑

matrices

𝑮𝑖 𝑗

. Usually the coupling scheme is

assumed to be identical for all nodes, i.e.,

𝑮𝑖 𝑗 =𝑮

for all

𝑖

𝑗=

. . .

𝑁

. In the case of an identical

coupling scheme, the input which the

𝑖

th node receives via the interaction with the other nodes

of the network is given by the sum

𝑠(𝑖)=Í𝑁

𝑗=1𝑎𝑖 𝑗 𝑮(𝒙𝑖−𝒙𝑗)

. Here,

𝑎𝑖 𝑗

are the entries of the

adjacency matrix of the network. Remarkably, this sum can be written as

𝑠(𝑖)=Í𝑁

𝑗=1𝑙𝑖 𝑗 𝑮𝒙 𝑗

where

2.2 Dynamics 15

𝑙𝑖 𝑗

are the entries of the Laplacian matrix

𝐿

, see

(2.3)

. This fact allows for the relation between

dynamical properties and the Laplacian structure of a network. It has been used, for instance, to

introduce the powerful methodology of the master stability function [PEC98,LEH15b] .

Linear coupling is in some sense the simplest type of coupling. More complex forms of diffusive

couplings can be written as

𝐺𝑖 𝑗 (𝒙𝑖,𝒙𝑗)=𝐺𝑖 𝑗 (𝒙𝑖−𝒙𝑗), (2.12)

where

𝐺𝑖 𝑗

are nonlinear functions depending on the difference

𝒙𝑖−𝒙𝑗

. For example, in the

well-known Kuramoto model

𝐺𝑖 𝑗

is given by a sin function [

KUR84

]. Other forms of nonlinear

coupling schemes, not necessarily diffusive, are used to model interactions in various contexts,

see for instance Refs. [

TRU13

PEN19

BAU20

]. Note that despite the nonlinear coupling scheme

the corresponding variational equations around each state are still linear and diffusive. Hence,

stability features depend on the Laplacian structure of the network, as well.

Very elaborate types of coupling schemes are used in models of neurons that exhibit inhibitory

coupling through chemical synapses. Here, the coupling is usually assumed to be nonlinear

with a slow synaptic decay. The coupling through chemical synapses is modeled by a first order

kinetics of the form

𝐺𝑖 𝑗 (𝑉𝑖,𝑉𝑗)=𝑠𝑗· (𝑉𝑖−𝑉𝑟)

¤𝑠𝑗=𝛼𝑠

1+𝑒−(𝑉𝑗−Θ𝑠)/𝜎𝑠(1−𝑠𝑗)−𝑠𝑗/𝜏𝑠

(2.13)

where

𝑉𝑖

is the membrane potential of neuron

𝑖

𝑉𝑟

is the reverse potential,

𝑠𝑗

is the gating

variable describing the synaptic transmission from the

𝑗

th neuron,

𝜏𝑠

is the synaptic decay time

constant. Further, the parameter

𝛼𝑠

Θ𝑠

, and

𝜎𝑠

characterize the sigmoidal part in the dynamics

of the gating variables [

WAN92

WHI98

HAU09

POP13

POP15

]. Depending on which process

is causing the interaction of neurons, other models for chemical coupling can be found similar to

the one described [RIN90,GIL04].

Building on these different types of coupling, we introduce the dynamical network models

which are used in this thesis.

2.2.2 Kuramoto-Sakaguchi type model

A rather simple model, known as Kuramoto-Sakaguchi type model, describes a dynamical

network of 𝑁coupled phase oscillators

𝑑𝜙𝑖

𝑑𝑡 =𝜔𝑖−1

𝑁

𝑗=1

𝑎𝑖 𝑗 𝜅𝑖 𝑗 sin(𝜙𝑖−𝜙𝑗+𝛼), (2.14)

where

𝜙𝑖∈𝑇𝑁

𝑇𝑁

denoting the

𝑁

-torus, represents the phase of the

𝑖

th oscillator (

𝑖=

. . .

𝑁

)

and

𝜔𝑖

its natural frequency. The interaction is the same between each pair of phase oscillators

and given by a sinusoidal coupling kernel. The weighted network is represented by the entries

of the adjacency matrix

𝑎𝑖 𝑗

and the coupling weigths

𝜅𝑖 𝑗

. The parameter

𝛼

can be considered as

a phase-lag of the interaction [

SAK86

]. System

(2.14)

has attracted a lot of attention over the last

16 2 Fundamentals of adaptive and complex dynamical networks

three decades since it is a first choice paradigmatic model for the modeling of synchronization

and partial synchronization scenarios on complex dynamical networks [

KUR84

STR00

ACE05

OME12b,PIK08].

Coherence (phase synchronization) in a network of phase oscillators can be measured by the

so-called Kuramoto-Daido order parameter [

KUR84

DAI92a

DAI94

]. The

𝑛

th order parameter

for the state 𝝓∈T𝑁is defined as

𝑍𝑛(𝝓)=𝑅𝑛(𝝓)𝑒i𝜃𝑛(𝝓)=1

𝑁

𝑗=1

𝑒i𝑛𝜙𝑖, (2.15)

where

𝑛∈N

. The symbols

𝑅𝑛>

0 and

𝜃𝑛∈T1

denote the modulus and the phase of the

complex order parameter, respectively. The number

𝑛

can be also called moment of the order

parameter. Note that the symbol i denotes the imaginary unit to distinguish it from the index

𝑖

In case of full phase synchronization, i.e.,

𝝓=(𝑎

. . .

𝑎)𝑇

for any

𝑎∈𝑇

, the modulus of the order

parameter

𝑅𝑛=

1. More generally, if for any

𝑛∈N

the phase of the

𝑖

th oscillator is given by

𝜙𝑖∈ {𝑎

𝑎+

𝜋/𝑛

𝑎+

𝜋/𝑛

. . .

𝑎+

(𝑛−

)𝜋/𝑛}

for some

𝑎∈𝑇

then

𝑅𝑛=

1. Thus,

𝑅𝑛

is a measure

for a particular type of coherence with respect to a certain discrete phase distribution. Hence,

the classical Kuramoto order parameter

𝑍1

measures in-phase synchrony. If

𝑅𝑛=

0 we consider

the phase distribution as incoherent with respect to the 𝑛th moment.

A measure that can be used in order to detect frequency synchronization between two oscillators

relies on the average frequency of each phase oscillator

Ω𝑖=lim

𝑇→∞

𝑇(𝜙𝑖(𝑡0+𝑇)−𝜙(𝑡0))(2.16)

and the frequency synchronization measure is given by

Ω𝑖 𝑗 =(1, if Ω𝑖−Ω𝑗=0,

0, otherwise, (2.17)

see e.g. [

ASH15

]. Numerically the limit is approximated by very long average window. In

addition, we use a sufficiently small threshold

𝜛

in order to detect frequency synchronization

numerically, i.e.,

Ω𝑖 𝑗 =

1 if

Ω𝑖−Ω𝑗< 𝜛

. In the subsequenty analysis we used

𝜛=

0.001. Using

the measure Ω𝑖 𝑗 , we may define the following value which we call cluster parameter

𝑅𝐶=1

𝑁2

𝑁

𝑖,𝑗=1

Ω𝑖 𝑗. (2.18)

The cluster parameter measures the following. First, for each frequency cluster, the total number

of links connecting nodes of the same cluster is computed. In other word, how much space is

occupied by a single cluster. Second, all ratios are summed up and normalized by the number of

all possible links. In case of full frequency synchronization, i.e.,

Ω𝑖 𝑗 =

1 for all

𝑖

𝑗=

. . .

𝑁

then

𝑅𝐶=

1 because the single cluster "occupies" the whole network. A similar measure be found

in Ref [

KAS18a

]. In the context of phase oscillator many different measures were introduced

to characterize the dynamical states. For an overview and a comparison of different coherence

measures for complex networks of phase oscillators see [

SCH17m

]; see also [

MEH17a

MEH18

]

2.2 Dynamics 17

where a similar generalized order parameter has been suggested independently.

2.2.3 Hodgkin-Huxley model with chemical synapses

A more realistic model for neuronal dynamics is given by the Hodgkin-Huxley model [

HOD52

GER14a

]. Nowadays there exists a zoo of neuron models describing different features of neural

cells. Due to its rich dynamics the Hodgkin-Huxley model is, however, one of the most prominent

neuron models but also one of the most computationally expensive [

IZH04

]. In this work we are

interested in neurons with tonic spiking dynamics. These kinds of neurons are excitable neurons

which show a periodic spiking pattern in case of a constant synaptic input [IZH04]. A network

𝑁

excitatory Hodgkin-Huxley neurons with inhibitory coupling is described by the following

system [HOD52,HAN93b,POP13,LUE16]

𝐶¤

𝑉𝑖=𝐼𝑖−𝑔𝑁 𝑎𝑚3

𝑖ℎ𝑖(𝑉𝑖−𝐸𝑁 𝑎) −𝑔𝑘𝑛4

𝑖(𝑉𝑖−𝐸𝐾)−𝑔𝐿(𝑉𝑖−𝑉𝑟)− (𝑉𝑖−𝑉𝑟)

𝑁

𝑗=1

𝜅𝑖 𝑗 𝑠𝑗,

¤𝑚𝑖=𝛼𝑚(𝑉𝑖)(1−𝑚𝑖) − 𝛽𝑚(𝑉𝑖)𝑚𝑖,

ℎ𝑖=𝛼ℎ(𝑉𝑖)(1−ℎ𝑖) − 𝛽ℎ(𝑉𝑖)ℎ𝑖,

¤𝑛𝑖=𝛼𝑛(𝑉𝑖)(1−𝑛𝑖) − 𝛽𝑛(𝑉𝑖)𝑛𝑖,

¤𝑠𝑖=5(1−𝑠𝑖)

1+𝑒−𝑉𝑖+3

8−𝑠𝑖.

(2.19)

Here

𝑉𝑖

is the membrane potential of the

𝑖

th neuron with the corresponding equilibrium po-

tentials

𝐸𝑁 𝑎 =

50mV,

𝐸𝐾=−

77mV, and

𝐸𝑙=−

54.4mV.

𝐶=

𝜇𝐹/𝑐𝑚2

. Our choice of the reverse

potential

𝑉𝑟=

20mV corresponds to the excitatory neurons. The three other variables

𝑚

ℎ

, and

𝑛

are gating variables in natrium and potassium, and their dynamics depends on the opening and

closing rates

𝛼𝑚(𝑉)=0.1𝑉+4

1−𝑒(−0.1𝑉−4),

𝛽𝑚(𝑉)=4𝑒(−𝑉−65

18 ),

𝛼ℎ(𝑉)=0.07𝑒(−𝑉−65

20 ),

𝛽ℎ(𝑉)=1

1+𝑒(−0.2𝑉−3.5),

𝛼𝑛(𝑉)=0.01𝑉+0.55

1−𝑒(−0.1𝑉−5.5),

𝛽𝑛(𝑉)=0.125𝑒(−𝑉−65

80 ).

(2.20)

The parameters are

𝑔𝑁 𝑎 =

120mS/cm

𝑔𝐾=

36mS/cm

, and

𝑔𝑙=

0.3mS/cm

. The constant

current

𝐼𝑖

is set to 9

𝜇𝐴/cm2

so that the individual neurons are identical and oscillatory. Note that

we usually assume no autapses and set 𝜅𝑖𝑖 =0.

18 2 Fundamentals of adaptive and complex dynamical networks

2.3 Adaptive networks in neuroscience

A dynamical network is called an adaptive network if its topological structure is non-static

and depends on the state of the dynamical nodes. In this sense, the topological structure is

determined by either the coupling matrix

𝜅(𝑡)

or the adjacency matrix

𝐴(𝑡)

. We note at this point

that adaptive networks are to distinguish from temporal networks which possess a non-static

connectivity structure, as well, but whose temporal dynamics is independent from the state of

the nodes [HOL12,HOL15a].

Adaptive networks appear in many real-world systems [

GRO08a

GRO09

SAY13

]. In the frame-

work of this thesis, the most important example of this class of dynamical systems are given by

(tonically) spiking neurons with synaptic plasticity. Synaptic plasticity, in neural systems, is in-

duced by various different interdependent mechanisms [

FEL12

]. We focus on the plasticity which

is induced by the timing of spikes, so called spike timing-dependent synaptic plasticity (STDP).

In addition, we consider plasticity of chemical synapses only. Note, however, that plastic changes

in the connectivty through electrical synapses have been also reported [

LAN05a

MAT14

]. Note

that in the description of the large-scale structure of the human brain exist completely other mod-

els of plasticity. These models, for instance, assimilate probabilistic forms of neuronal plasticity

and do not rely on the precise timing of the neuron [

GER14a

BRE17a

]. The interplay of all forms

of plasticity plays certainly a crucial role in structure formation, however, the analysis of this in-

terplay is beyond the scope of this thesis. In the following, we briefly introduce the models used

in order to describe STDP. For more details on the modeling and the physiological properties, we

refer the reader to the reviews [ABB00,DAN06,LET07,CAP08a,SJO08,FRO10,MAR11b].

2.3.1 Spike timing-dependent plasticity

The synaptic input current from

𝑗

th presynaptic to

𝑖

th postsynaptic neuron is scaled by the

synaptic weight

𝜅𝑖 𝑗

, see for example

(3.1)

. The coupling weights

𝜅𝑖 𝑗

have been shown to be a

variable entity. Adaptation of

𝜅𝑖 𝑗

due to the spike timings of neurons occurs discontinuously

whenever one of the neurons

𝑖

𝑗

spikes. More specifically, the discontinuous change is formally

given by the following plasticity function

𝜅𝑖 𝑗 →

0, if 𝜅𝑖 𝑗 +𝛿𝑊 (Δ𝑡𝑖 𝑗 )<0,

𝜅𝑖 𝑗 +𝜖𝑊 (Δ𝑡𝑖 𝑗 ), if 0 ≤𝜅𝑖 𝑗 +𝛿𝑊 (Δ𝑡𝑖 𝑗 ) ≤ 𝜅max,

𝜅max, if 𝜅𝑖 𝑗 +𝛿𝑊 (Δ𝑡𝑖 𝑗 )> 𝜅max,

(2.21)

where

Δ𝑡𝑖 𝑗 =𝑡𝑖−𝑡𝑗

is the spike time difference between the postsynaptic and presynaptic neurons;

𝜖 >

0 is a small parameter determining the size of the single update; and

𝜅max >

0 is the maximal

coupling [GER96].

In neuronal systems various forms of spike timing-depending plasticity exist depending for

example on the type of synapse or the type of the postsynaptic neuron [

BI98

]. An overview on

the diversity of plasticity rules has been presented in [

SHU13

]. In the subsequent chapter 3, we

particularly consider the plasticity function [GER96,BEL97a]

𝑊(Δ𝑡𝑖 𝑗)=𝑐𝑝𝑒−|Δ𝑡𝑖 𝑗 |

𝜏𝑝−𝑐𝑑𝑒−|Δ𝑡𝑖 𝑗 |

𝜏𝑑(2.22)

2.3 Adaptive networks in neuroscience 19

with positive parameters 𝑐𝑝,𝜏𝑝,𝑐𝑑, and 𝜏𝑑.

2.3.2 Phase difference-dependent plasticity

Models which show self-sustained limit cycle oscillations are often used to describe excitatory

neurons [

IZH03

IZH04

ASH16

]. In the context of these models a neuron is said to spike, and

thus releases an action potential, if the dynamical variable which describes the membrane

potential reaches its maximum.

It is well-known that networks of such oscillators interacting with each other can be reduced to

dynamical networks of weakly coupled phase oscillators [

HAN93b

HOP96a

PIE19a

]. In this

framework a neuron is said to spike whenever the neuron’s phase equals zero. In order to reduce

the complexity of neuronal networks with STDP it is desirable to establish a reduced model of

coupled phase oscillators accompanied by synaptic plasticity which depends rather on the phase

than the spike time difference. For this model of synaptic plasticity, Lücken et. al. introduced the

notion "phase difference-dependent plasticity" (PDDP) and drew a relation between STDP and

PDDP [LUE16]. Such a PDDP rule is written as

¤𝜅𝑖 𝑗 =𝜖ℎ(𝜙𝑖−𝜙𝑗). (2.23)

The relation between STDP and PDDP is derived as follows. Consider the spiking time of a

postsynaptic

𝑡post

and presynaptic

𝑡pre

. the spiking time difference

Δ𝑡=𝑡post −𝑡pre

is thus always

positive or negative (or zero) if the postsynaptic or the presynaptic neuron spikes (if the neurons

spike synchronously), respectively. Consider now the phase difference

𝜃=𝜙pre −𝜙post

and

assume that the phase difference changes much slower than the phase dynamics. Then by

averaging the synaptic effect induced by a spike of the post and presynaptic neuron and by the

assumption that the chances due to the plasticity are sufficiently small, we get the following

continuous PDDP

ℎ(𝜃)=Ω

2𝜋𝑊+𝜃

Ω+𝑊−𝜃−2𝜋

Ω

where

Ω/

𝜋

is the number of spike per unit of time,

𝑊+=𝑊(Δ𝑡)

for

Δ𝑡 >

0, and

𝑊−=𝑊(Δ𝑡)

for

Δ𝑡≤

0 (

Δ𝑡≤

0 if

𝑊

is discontinuous) [

LUE16

]. Note that this adaptation rule does not guarantee

that the values of the coupling weights are bounded during the course of time. This is why

they have to be forced to stay within a bounded interval

[𝜅min

𝜅max]

by either by a hard-cut as

presented in (2.21) or by a soft-cut given as

¤𝜅𝑖 𝑗 =−2𝜖

(𝜅max −𝜅min)𝜅−𝜅𝜅−𝜅𝜅𝑖 𝑗 −(𝜅max −𝜅min)ℎ(𝜙𝑖−𝜙𝑗)+𝜅max𝜅−𝜅min𝜅(2.24)

where

𝜅=max𝜃∈[0,2𝜋]ℎ(𝜃)

and

𝜅=min𝜃∈[0,2𝜋]ℎ(𝜃)

. This soft-cut rule is obtained by normal-

ization. Another rule using coupling weight dependent scaling of the plasticity function has

been used in [

MAI07

]. In the following, we use a simplified PDDP in order to introduce a

paradigmatic model for adaptively coupled phase oscillators.

20 2 Fundamentals of adaptive and complex dynamical networks

Hebbian-like STDP Anti-Hebbian-like

∆φ/π ∆φ/π∆φ/π

−sin(∆φ)

(a) (b) (c)

Figure 2.2:

The plasticity function

−sin(Δ𝜙+𝛽)

and corresponding plasticity rules. (a)

𝛽=−𝜋

, (b)

𝛽=

0, (c)

𝛽=𝜋

Figure taken from [BER19,BER19a].

2.3.3 A network of adaptively coupled phase oscillator

In the following we describe the model which is in the focus of this thesis. By introducing

a PDDP, we introduced a differential equation which governs the dynamics of the coupling

weights, see Eq.

(2.23)

. In addition, the adaptation function

ℎ

is a 2

𝜋

-periodic that can expanded

in a Fourier series. Consider an adaptive networks of

𝑁

adaptively coupled phase oscillators as

given by

(2.14)

with PDDP of the form

(2.24)

where

ℎ

is approximated by the first Fourier mode

and we restrict the coupling weights to [−1, 1]. Then, the dynamical networks reads

𝑑𝜙𝑖

𝑑𝑡 =𝜔−1

𝑁



𝑗=1

𝜅𝑖 𝑗 sin(𝜙𝑖−𝜙𝑗+𝛼), (2.25)

𝑑𝜅𝑖 𝑗

𝑑𝑡 =−𝜖𝜅𝑖 𝑗 +sin(𝜙𝑖−𝜙𝑗+𝛽), (2.26)

The phase space of the system

(𝜙(𝑡),𝜅(𝑡))∈T𝑁× [−

1, 1

]𝑁2

(𝑁+𝑁2)

-dimensional. System

(2.25)

–

(2.26)

have attracted a lot of attention recently [

REN07

AOK09

AOK11

PIC11a

TIM14

GUS15a

KAS16a

NEK16

KAS17

AVA18

], since it is a first choice paradigmatic model for the

modeling of the dynamics of adaptive networks. In particular, it generalizes the Kuramoto (or

Kuramoto-Sakaguchi) model with fixed 𝜅.

The matrix

𝜅(𝑡)

characterizes the coupling topology of the network at time

𝑡

. Assume that

𝜖

is a small but not vanishing parameter. Then, the dynamical equation

(2.26)

describes the

adaptation of the network topology depending on the dynamics of the network nodes. The

chosen adaptation function in the form

sin(𝜙𝑖−𝜙𝑗+𝛽)

with control parameter

𝛽

can take into

account different plasticity rules that can occur in neuronal networks, see Fig. 2.2. For instance,

for

𝛽=−𝜋/

2, the Hebbian rule is obtained where the coupling

𝜅𝑖 𝑗

is increasing between any

two systems with close phases, i.e.,

𝜙𝑖−𝜙𝑗

close to zero [

HEB49

HOP96

SEL02

AOK15

𝛽=

0 the link

𝜅𝑖 𝑗

will be strengthened if the

𝑗

-th oscillator is advancing the

𝑖

-th. Such a

relationship is typical for spike-timing-dependent plasticity in neuroscience [

CAP08a

MAI07

LUE16,POP13].

Let us mention important properties of the model. Firstly, the parameter

𝜖

1 separates the

time scales of the slow dynamical behavior of the coupling strengths and the fast dynamics of

the oscillatory system [

KUE15

]. Due to the invariance of system

(2.25)

–

(2.26)

with respect to

the phase-shift

𝜙𝑖↦→ 𝜙𝑖+𝜓

for all

𝑖=

. . .

𝑁

and

𝜓∈T1

, the frequency

𝜔

can be set to zero

in the co-rotating coordinate frame

𝜙↦→ 𝜙+𝜔𝑡

. Moreover, one can restrict the consideration of

2.4 Summary 21

the coupling weights to the interval

−

≤𝜅𝑖 𝑗 ≤

1 due to the existence of the attracting region

𝐺:=𝜙𝑖,𝜅𝑖 𝑗 :𝜙𝑖∈T1,|𝜅𝑖 𝑗 | ≤ 1, 𝑖,𝑗=1, . . . ,𝑁[KAS17].

Finally, let us mention the parameter symmetries of the model

(𝛼,𝛽,𝜙𝑖,𝜅𝑖 𝑗 ) ↦→ (−𝛼,𝜋−𝛽,−𝜙𝑖,𝜅𝑖 𝑗 ),

(𝛼,𝛽,𝜙𝑖,𝜅𝑖 𝑗 ) ↦→ (𝛼+𝜋,𝛽+𝜋,𝜙𝑖,−𝜅𝑖 𝑗 ).

As a result of these symmetries one can restrict the analysis to the parameter region

𝛼∈ [

𝜋/

)

and 𝛽∈ [−𝜋,𝜋).

2.4 Summary

In this chapter, we have given the theoretical framework which is used throughout the subse-

quent chapters. References for further reading are provided and the most important results for

different topics are highlighted. We have first introduced notations and concepts from graph

theory in order to have a proper description of weighted networks. Several examples for complex

network structures are presented and some of their topological properties are reviewed. The

notion of symmetry in networks is briefly introduced.

After this, we have used the concept of networks to define systems of coupled dynamical

units. We have described how the links of a network relate to the coupling dynamics of nodes

and what kind of coupling functions can be found in physical applications. Subsequently, a

Kuramoto-Sakaguchi-like and the Hodgkin-Huxley model are introduced. These models are

used throughout this thesis. Particularly, for the phase oscillator model, we have presented

measures which allow for the quantitative description of synchronization in the system.

Extending the concepts from coupled dynamical units on static networks, we have described

how adaptivity is modeled in the context of neuroscience. We have introduced spike timing-

dependent plasticity which is used to describe changes of the synaptic coupling weights de-

pendent on the spike-timing of the neurons. Under the assumption of topically spiking neuron,

neuronal systems with spike-timing dependent plasticity rules can be reduced to phase oscil-

lators models with phase difference-dependent plasticity. We have shown how this is done

and introduced a paradigmatic model for adaptively coupled phase oscillators with synaptic

plasticity.

Part I

CLUSTER SYNCHRONIZATION IN

GLOBALLY COUPLED ADAPTIVE

NETWORKS

Population of Hodgkin-Huxley neurons with

spike timing-dependent plasticity 3

In this chapter we report on the phenomenon of clustering with respect to connectivity and

frequencies in a network of adaptively coupled Hodgkin-Huxley neurons. The spike timing-

dependent plasticity is considered to be symmetric as it was experimentally found for hip-

pocampal synapses [

WIT06

] and derived from asymmetric STDP for an "effective time win-

dow" [

KEP02

]. In this case the observed clusters are bidirectionally coupled [

POP15

]. Splitting

of a neural population to a few clusters that are synchronized at different frequencies could lead

to a slow waxing and waning of the amplitude of the mean field, where the clusters transiently

gather together and move apart as the time evolves [

POP15

]. The frequency of such a modulation

of the mean neural activity could be much smaller than the firing rate of individual neurons and

depends on the differences between the clusters’ frequencies. The emergence of synchronized

clusters could explain the origin of the low-frequency modulation of the power spectral density

of macroscopic brain signals like local field potentials (LFP) or electroencephalographic (EEG)

signals in higher frequency bands, which also correlates with slow oscillations of the blood

oxygen level-dependent (BOLD) signal measured by fMRI [

MAN07

MAG12a

MON08

ALV14

Several other modeling studies have also reported on clustering in the neural populations

with plasticity [

MAI07

CAT08

POP15

]. These clusters have been observed for different models

that ranged from simple phase oscillators to the models of spiking and bursting neurons and

demonstrate stability with respect to heterogeneity of the interacting neurons and random

perturbations [

MAI07

CAT08

POP15

]. In this chapter we provide a simple phenomenological

model and explain a mechanism governed by synaptic plasticity for the stabilization of such

clusters in a neural population.

This chapter includes contents that have been published in [

ROE19a

] and is structured as follows.

In the first Section 3.1 we present the model. The next Section 3.2 shows numerically observed

multiclusters. The detailed mechanisms of the stability of frequency clusters is explained after-

wards in Section 3.3 using the simplest case of two clusters. Then, in Section 3.4, we propose

a phenomenological model which describes the dynamics of two clusters taking into account

the adaptation of the weights. The model is shown to reflect not only qualitative, but also

some basic quantitative properties of the two-cluster formation. We also determine the set of

plasticity functions (update rules), which lead to the clustering. The findings of this chapter are

summarized in Section 3.5.

26 3 Population of Hodgkin-Huxley neurons with spike timing-dependent plasticity

3.1 Coupled Hodgkin-Huxley neurons on a network with spike

timing-dependent plasticity

For the present study we consider

𝑁

coupled Hodgkin-Huxley neurons which are described by

the dynamical equations

𝐶¤

𝑉𝑖=𝐼𝑖−𝑔𝑁 𝑎𝑚3

𝑖ℎ𝑖(𝑉𝑖−𝐸𝑁 𝑎) −𝑔𝑘𝑛4

𝑖(𝑉𝑖−𝐸𝐾)−𝑔𝐿(𝑉𝑖−𝑉𝑟)− (𝑉𝑖−𝑉𝑟)

𝑁

𝑗=1

𝜅𝑖 𝑗 𝑠𝑗,

¤𝑚𝑖=𝛼𝑚(𝑉𝑖)(1−𝑚𝑖) − 𝛽𝑚(𝑉𝑖)𝑚𝑖,

ℎ𝑖=𝛼ℎ(𝑉𝑖)(1−ℎ𝑖) − 𝛽ℎ(𝑉𝑖)ℎ𝑖,

¤𝑛𝑖=𝛼𝑛(𝑉𝑖)(1−𝑛𝑖)−𝛽𝑛(𝑉𝑖)𝑛𝑖,

¤𝑠𝑖=5(1−𝑠𝑖)

1+𝑒−𝑉𝑖+3

8−𝑠𝑖.

(3.1)

Each of the neurons (

𝑖=

. . .

𝑁

) is described by the membrane potential

𝑉𝑖

and the three gating

variables

𝑚𝑖

ℎ𝑖

, and

𝑛𝑖

. The parameters are chosen such that each individual neuron gives rise

to self-sustained oscillatory dynamics. For more details, we refer to

(2.19)

–

(2.20)

in Chapter 2.

The interaction between the neurons is given by the synaptic input current

𝑠𝑗

which each neuron

receives from the 𝑗th neuron. the input current is scaled by the synaptic strength 𝜅𝑖 𝑗 , which we

assume to change due to plasticity. The adaptation of

𝜅𝑖 𝑗

occurs discontinuously whenever one

of the neurons

𝑖

𝑗

spikes. More specifically, the discontinuous change is given by the following

plasticity function, see also (2.21)

𝜅𝑖 𝑗 →

0, if 𝜅𝑖 𝑗 +𝛿𝑊 (Δ𝑡𝑖 𝑗 )<0

𝜅𝑖 𝑗 +𝛿𝑊 (Δ𝑡𝑖 𝑗 ), if 0 ≤𝜅𝑖 𝑗 +𝛿𝑊 (Δ𝑡𝑖 𝑗 ) ≤ 𝜅max

𝜅max, if 𝜅𝑖 𝑗 +𝛿𝑊 (Δ𝑡𝑖 𝑗 )> 𝜅max

(3.2)

where

Δ𝑡𝑖 𝑗 =𝑡𝑖−𝑡𝑗

is the spike time difference between the postsynaptic and presynaptic neurons;

𝛿 >

0 is a small parameter determining the size of the single update;

𝜅max >

0 is the maximal

coupling; and the plasticity function [GER96,MAR97a,BI98,CLO10] is

𝑊(Δ𝑡𝑖 𝑗)=𝑐𝑝𝑒−|Δ𝑡𝑖 𝑗 |

𝜏𝑝−𝑐𝑑𝑒−|Δ𝑡𝑖 𝑗 |

𝜏𝑑(3.3)

with positive parameters 𝑐𝑝,𝜏𝑝,𝑐𝑑, and 𝜏𝑑. We also assume no autapses and set 𝜅𝑖𝑖 =0.

An example of the considered plasticity function

𝑊

used in our simulations is shown in Fig. 3.1.

This is a symmetric function, which corresponds to a potentiation of the coupling weights of the

neurons with highly correlated firing. As we will discuss at the end of this chapter, there is a

family of plasticity functions of similar form that allow for the frequency clustering.

3.2 Numerical observation of synchrony and frequency clustering 27

Figure 3.1: Plasticity function 𝑊(Δ𝑡𝑖 𝑗 )for 𝜏𝑝=2, 𝜏𝑑=5, 𝑐𝑝=2, 𝑐𝑑=1.6. Figure taken from [ROE19a].

3.2 Numerical observation of synchrony and frequency clustering

In order to investigate the dynamics of the network

(3.1)

with plasticity 3.3, we initialize the

neurons and the coupling randomly and integrate the system numerically. For the parameter

values

𝜏𝑝=

𝜏𝑑=

𝑐𝑝=

𝑐𝑑=

1.6, and

𝜅max =

1.5 we observe two phenomena: complete

synchronization and the emergence of frequency clusters hierarchical in size, see Figs. 3.2 and

3.3, respectively.

Figure 3.2(a) shows the initial coupling weights, and Fig. 3.2(c) illustrates the spike times of the

neurons at the beginning of the simulation. One can observe that while the neurons start with an

incoherent spiking, they enhance the coherence already after a few spikes due to the interaction

between them as well as the plasticity. The plasticity potentiates the connections of neurons that

fire together. A complete synchronization is established and the coupling weights increase to

𝜅max

after a transient (Fig. 3.2(b,d)). In a completely synchronized state, the individual neurons

spike simultaneously, hence, the spike time differences Δ𝑡𝑖 𝑗 =0.

The emergence of frequency clusters is shown in Fig. 3.3 for two clusters. The system in Fig. 3.3

possesses the same parameters as in Fig. 3.2, and the difference is just another realization of

random initial conditions. In contrast to the synchronized state, the final state shown in Fig. 3.3(f)

consist of two groups of synchronized neurons. These cluster states also manifest themselves

as two groups of strongly coupled elements in the coupling matrix

𝜅

(Fig. 3.3(c)). The coupling

weights between the neurons from the different groups is very small or zero.

We observe that the largest cluster is formed rather quickly as time evolves, whereas the for-

mation of the small cluster takes much more time. This is illustrated in Fig. 3.4, where the time

courses of the mean coupling within each of the two clusters are shown. The average coupling

within the big cluster reaches its maximum fast (at

𝑡≈

1000, solid curve in Fig. 3.4), whereas the

smaller cluster in Fig. 3.3(c,f) is formed through the merging of transient clusters and finally

establishes at 𝑡≈17000 (dashed curve in Fig. 3.4).

For the states with more clusters, each new formed cluster is significantly smaller than the

previous one, see Fig. 3.5, where three clusters are shown. The spiking period of the cluster

appears to be proportional to its size: the bigger the cluster the larger is the period. Simulation of

the cases with even more clusters becomes computationally expensive due to large transients.

28 3 Population of Hodgkin-Huxley neurons with spike timing-dependent plasticity

(a)

(d)(c)

(b)

Figure 3.2:

Evolution of the coupling matrix

𝜅𝑖 𝑗 (𝑡)

starting from random initial conditions and converging to a

completely synchronous state. Panel (a) shows initial coupling matrix, (b) the coupling matrix after the transient

𝑡=

2000ms. Raster plot of spiking times at the beginning of simulations (c) and after the transient (d). The asymptotic

state (b,d) is a completely synchronized spiking with all coupling weights

𝜅𝑖 𝑗

potentiated to

𝑘max

. Other parameters

𝑁=200, 𝜏𝑝=2, 𝜏𝑑=5, 𝑐𝑝=2, 𝑐𝑑=1.6, and 𝜅max =1.5. Figure modified from [ROE19a].

(a) (b) (c)

(d) (e) (f)

Figure 3.3:

Evolution of the coupling matrix

𝜅𝑖 𝑗 (𝑡)

starting from random initial conditions and converging to

frequency clusters hierarchical in size. Panel (a) shows initial coupling matrix, (b) the coupling matrix after the

transient

𝑡=

5600ms, and (c)

𝑡=

20000ms. (d-f) Corresponding raster plots of spike times. The asymptotic state (c,f) is

a hierarchical cluster state with the coupling weights

𝜅𝑖 𝑗

potentiated to

𝑘max

within each cluster and small or zero

otherwise. Other parameters as in Fig. 3.2. The oscillators are ordered accordingly to their mean frequency. Figure

modified from [ROE19a].

3.2 Numerical observation of synchrony and frequency clustering 29

Figure 3.4:

Formation of individual clusters over time (corresponds to the dynamical scenario in Fig. 3.3). The dashed

and solid curves depict the time course of the mean coupling within the small and big clusters, respectively. Figure

taken from [ROE19a].

Figure 3.5:

Example of a three-cluster state for

𝑁=

500,

𝜏𝑝=

𝜏𝑑=

𝑐𝑝=

𝑐𝑑=

1.6, and

𝜅max =

1.5 with a random

initial distribution of 𝜅𝑖 𝑗 in [0, 0.75]. Figure taken from [ROE19a].

Clustering with independent random input

To investigate the robustness of our findings, we added an

𝛼

-train as additional independent

random input to the membrane potential 𝑉𝑖of every neuron:

𝐼𝑖𝑛𝑝𝑢𝑡

𝑖(𝑡)=𝐼(𝑉𝑟−𝑉𝑖(𝑡)) Õ

𝜏𝑖,𝑘<𝑡

𝛼(𝑡−𝜏𝑖,𝑘)𝑒−𝛼(𝑡−𝜏𝑖,𝑘))(3.4)

The Eq. (3.4) models a postsynaptic potential (PSP) that is received by the neuron at certain ran-

dom times

𝜏𝑖,𝑘

. The inter-spike interval is Gaussian distributed

𝜏𝑖,𝑘+1−𝜏𝑖,𝑘= Δ𝜏𝑖,𝑘∼N(

𝑚𝑠

, 4

𝑚𝑠)

𝛼is set to 24/hΔ𝜏𝑖,𝑘i.

The numerical simulations Fig. 3.6 show that the clustering is still observed under the influence

of random input

𝐼𝑖𝑛𝑝𝑢𝑡

𝑖(𝑡)

of intensity

𝐼

. More specifically, for sufficiently weak perturbations

with

𝐼 <

0.01, all three clusters survive (Fig. 3.6(a)). With increasing the amplitude

𝐼

, the clusters

start to decouple. The smaller clusters are affected first (Fig. 3.6(b-d)), they start desynchronizing

𝐼=

0.01. The biggest cluster keeps shrinking in size while

𝐼

is increased and finally for

𝐼=

0.07

the whole network decouples (Fig. 3.6(e)).

30 3 Population of Hodgkin-Huxley neurons with spike timing-dependent plasticity

(a) (b) (c) (d) (e)

Figure 3.6:

Coupling matrices for

𝑡=

10000ms and different amplitudes of independent random input

𝐼

(see Eq. (3.4)).

(a)

𝐼=

0.005, (b)

𝐼=

0.01, (c)

𝐼=

0.02, (d)

𝐼=

0.05 and (e) I=0.07. All other parameters as in Fig. 3.5. Figure modified

from [ROE19a].

3.3 Emergence of two-cluster states

In this section we numerically show that depending on the relative size of the two clusters, such

two-cluster states can be either dynamically stable or transient leading to complete synchroniza-

tion. In order to investigate the cluster stability, we initialize the system in a two cluster state

with the number of neurons

𝑁𝑠

in the small cluster and

𝑁𝑏=𝑁−𝑁𝑠

in the big cluster. The total

number of neurons is set to

𝑁=

50. The inter-cluster couplings are set to zero initially while the

intra-cluster couplings equals

𝜅max

. All neurons in the same cluster are initialized with the same

initial conditions, so the clusters are fully synchronized at 𝑡=0.

Figure 3.7(a) shows frequency difference of two uncoupled clusters as a function of the size of

the small cluster. The frequency difference demonstrates an almost linear dependence on the

cluster size and decays as the size of the smaller cluster increases. Moreover, we also observe

that clusters with sufficiently different sizes are stable while the clusters of similar sizes, in the

considered case with

𝑁𝑠>

8, are transient, merge into a single cluster and eventually lead to a

stable completely synchronous state, see Fig. 3.8.

Although the threshold of how different the clusters should be in order to be stable is certainly

model dependent, the synchronization of similar clusters is a general property. The merging

of two clusters can be explained qualitatively as follows. Initially uncoupled clusters evolve

(a) (b)

Figure 3.7:

(a) Difference between synchronization frequencies of the two clusters for different size of the smaller clus-

ter

𝑁𝑠

. (b) Time until cluster fusion for different initial size of the smaller cluster

𝑁𝑠

. Figure modified from [

ROE19a

3.3 Emergence of two-cluster states 31

each with their natural frequencies

Ω𝑠

and

Ω𝑏

. If their sizes are different, then

Ω𝑠≠ Ω𝑏

, and

the clusters arrive in-phase periodically with the "beating" frequency

ΔΩ = Ω𝑏−Ω𝑠

. As soon as

such an in-phase episode occurs, the interspike intervals

Δ𝑡𝑖 𝑗

between any two neurons from

different clusters become small and, hence, due to the plasticity rule

𝑊

(see Eq.

(3.2)

and Fig. 3.1)

the inter-cluster coupling weights increase. Moreover, the duration of such an in-phase episode

depends on the frequency difference between the clusters. As a result, for a small frequency

difference

ΔΩ

, the time interval where the clusters are practically in-phase is sufficiently long in

order to potentiate the coupling weights to their maximum value. This unites the two clusters

into one. In contrast, for large

ΔΩ

, such an episode is short, and the inter-cluster coupling remains

small, which keeps the clusters oscillating at different frequencies in a stable manner.

Figure 3.8(a-c) displays an example of two-cluster stable state with

𝑁𝑠=

8. Starting from the

two-cluster state, after

𝑡=

1500ms, the coupling between the clusters increases, see Fig. 3.8(b)

due to the "in-phase episode" when the clusters are synchronous. Afterwards, however, the

inter-cluster coupling weights return to their initial configuration (Fig. 3.8(c)), since the spike

time differences for neurons from different clusters are again far enough apart to cause the

depression of the inter-cluster synapses. Such a process is repeated every time the clusters meet

and is typical for the stable cluster states. A typical case of transient clusters is presented in

Fig. 3.8(d-f) for

𝑁𝑠=

9. The inter-cluster coupling is again potentiated when the clusters meet,

but it does not decrease again, and the clusters merge in a single cluster of a fully coupled and

synchronized regime (Fig. 3.8(f)). The transient time that could be elapsed until the cluster fusion

depends on the cluster size as illustrated in Fig. 3.7(b).

Figures 3.8(g,h) show how the spiking frequency of the clusters change over time. During the

in-phase episode, the cluster with the higher natural spiking rate slows down significantly, while

the slower cluster (with larger number of neurons

𝑁𝑏

) speeds up a little. For a stable cluster state

the cluster frequencies again deviate from each other (Fig. 3.8(g)), whereas all neurons fire with

the same frequency when the clusters unite into one (Fig. 3.8(h)). We found this phenomenon

for different numbers of neurons and different

𝜅max

. Increasing

𝜅max

increases the initial period

difference, but the behavior in general stays the same.

Figure 3.9 shows the dynamics of the mean synaptic activity

𝑆(𝑡)=1

𝑁Í𝑁

𝑖=1𝑠𝑖(𝑡)

of the network in

the case of two stable clusters, which models the dynamics of LFP. During the in-phase episodes

of the two clusters,

𝑆(𝑡)

has a higher amplitude, because both clusters spike synchronously.

The maximum amplitude is generated by maximum synchronization in the network. The low

amplitude of

𝑆(𝑡)

, on the other hand, corresponds to the time intervals when the clusters are

out of phase. In the latter case, the mean synaptic activity shows two peaks, the higher peak

is generated by the larger cluster and the lower by the smaller one, see Fig. 3.9(b). For the

considered case, the synchronized oscillations of individual neurons in the clusters take place at

a time scale of several milliseconds (period

∼

15 ms, Fig. 3.9(b)), see also Fig. 3.8(g,h). The neurons

are tonically spiking. The frequency difference

ΔΩ

between clusters is, however, of the order

of sub-Hz, because the corresponding cluster frequencies are close to each other (Fig. 3.8(g,h)).

Then the modulation of

𝑆(𝑡)

is observed at a much slower timescale of a few seconds, which is

of two orders of magnitude slower than the intrinsic neural firing, see Fig. 3.9(a), as observed in

empirical data of the brain activity [MAN07,MAG12a].

In the following section, a phenomenological model is introduced in order to further investigate

the dynamics of two clusters.

32 3 Population of Hodgkin-Huxley neurons with spike timing-dependent plasticity

(a) (b) (c)

(d) (e) (f)

(g) (h)

Figure 3.8:

Evolution of the coupling matrix for

𝑁=

50 and the number of neurons

𝑁𝑠=

8 (a-c) and

𝑁𝑠=

9 (d-f) in

the small cluster. In panels (a-c) the clusters are stable, while in (d-f) they are merging to one synchronous cluster. (g,

h) Time courses of the spiking synchronization frequencies of small (

𝑁𝑠

neurons) and large (

𝑁𝑏

neurons) clusters

depicted by dashed and solid curves, respectively, for (g)

𝑁𝑠=

8 and

𝑁𝑏=

42 and (h)

𝑁𝑠=

9 and

𝑁𝑏=

41. Parameter

𝜅𝑚𝑎𝑥 =1.0. Figure modified from [ROE19a].

3.3 Emergence of two-cluster states 33

(a) (a)

(b)

Figure 3.9:

Mean synaptic activity

𝑆(𝑡)

of the neural population in the case of stable two cluster state. Panel (a) shows

the dynamics of

𝑆(𝑡)

on the time interval of 12 s, where modulation of the amplitude (blue line) is visible, while the

fast oscillations are not recognized on this timescale. The maximum amplitude corresponds to the two clusters being

synchronised, while the low amplitude corresponds to the clusters being out of phase. Panel (b) shows the zoom of a

small time interval. The modulation takes place on the timescale which is two orders of magnitude larger than the

individual spikes of

𝑆(𝑡)

as well as individual neural spikes in both clusters. Cluster frequencies

𝜔1=

0.065012 kHz

and 𝜔2=0.065416 kHz. The corresponding period of modulation is 𝑇≈2.5𝑠. Figure modified from [ROE19a].

34 3 Population of Hodgkin-Huxley neurons with spike timing-dependent plasticity

3.4 Phenomenological model with phase difference-dependent

plasticity

In this section we introduce a reduced qualitative model for the coupling and phase difference of

two clusters. The model is based on the assumption that oscillators are synchronized identically

within each cluster and the coupling between the clusters is weak. As a result, the interaction

between oscillatory clusters can be described in the framework of two coupled phase oscillators

that are interacting via their phase differences [HOP97,PIK01,GUC75,WIN01]

¤𝜑1=𝜔1−𝐹1(𝜑1−𝜑2), (3.5)

¤𝜑2=𝜔2−𝐹2(𝜑2−𝜑1), (3.6)

where

𝜔1

and

𝜔2

are the natural frequencies of the individual clusters,

𝐹1

and

𝐹2

are effective

interaction functions. For the phase difference 𝜑=𝜑1−𝜑2, this system reads

¤𝜑=𝜔−𝐹(𝜑)(3.7)

where 𝜔=𝜔1−𝜔2is the difference of the natural frequencies, and 𝐹(𝜑)=𝐹1(𝜑)−𝐹2(−𝜑).

Since the clusters are synchronized for a sufficiently small frequency mismatch

𝜔

, the periodic

interaction function

𝐹(𝜑)

must satisfy

𝐹(

0 and

𝐹0(

0. The latter means that there is

a stable equilibrium

𝜑=

0 for small

𝜔

. Aiming at a qualitative insight, we further simplify

the model by assuming that

𝐹(𝜑)=𝜎sin(𝜑+𝛼)

, where

sin 𝜑

can be viewed as a first Fourier

harmonic of the interaction function and

𝜎

as an effective coupling weight. The parameter

𝛼=sin−1(𝜔/𝜎max)

is a constant phase shift assuring that the phase difference of the synchronized

cluster is zero. In fact, for small

𝜔

, this parameter is also small and it does not play important

role in the qualitative behavior of the model apart from a small shift of the synchronized state to

𝜑=0.

Another component of the model is the plasticity-driven changes of the coupling

𝜎

. In order

to derive the equation for

𝜎

, we consider the STDP update in the case of a periodic motion of

the clusters. We assume that the coupling

𝜎

is proportional to an averaged coupling between

the clusters. This is a natural assumption in the case of weakly coupled systems. Let us find

out how the update of the intercluster coupling depends on the phase difference

𝜑

. For a given

phase difference

𝜑

and the frequencies

𝜔1=¯𝜔+𝜔/

𝜔2=¯𝜔−𝜔/

2 (here we introduced the mean

frequency

¯𝜔

), the spiking period of the both clusters can be approximated as

𝑇≈

𝜋/¯𝜔

up to

small terms of order 𝜔, and the distance Δ𝑇between the spikes of two clusters

Δ𝑇=h𝑇𝜑1

2𝜋−𝑇𝜑2

2𝜋imod 𝑇=h𝑇𝜑

2𝜋imod 𝑇≈𝜑mod 2𝜋

¯𝜔.

Since the spike time differences

Δ𝑇

and

𝑇−Δ𝑇

occur recursively, see Fig. 3.10(b), the updates

per unit time sum to the function

𝛿

𝑇(𝑊(𝑇−Δ𝑇)+𝑊(Δ𝑇)) =𝛿¯𝜔

2𝜋𝐺(𝜑), (3.8)

3.4 Phenomenological model with phase difference-dependent plasticity 35

(a) (b)

Figure 3.10:

(a) Update function

𝐺(𝜑)

for

𝜏𝑝=

𝜏𝑑=

𝑐𝑝=

2, and

𝑐𝑑=

1.6. (b) Schematic spiking of two oscillators

with spike time difference Δ𝑇and periods close to 𝑇. Figure modified from [ROE19a].

where

𝐺(𝜑):=𝑊2𝜋−(𝜑mod 2𝜋)

¯𝜔+𝑊𝜑mod 2𝜋

¯𝜔. (3.9)

Since the update of

𝜎

is proportional to the obtained function, and taking into account the

smallness of

𝛿

, this update can be written as

¤𝜎=𝜀𝐺(𝜑)

, where

𝜀

is a small parameter of the

coupling adaptation that controls the scale separation between the fast dynamics of the clusters

and the slow dynamics of the coupling.

Additionally, the coupling strength

𝜎(𝑡)

should be bounded to the interval

[

𝜎max]

by imposing

cut-off conditions. More specifically, the derivative

¤𝜎

is discontinuous at the boundaries

𝜎=

and

𝜎=𝜎max

, i.e.

¤𝜎=max{

𝜀𝐺(𝜑)}

for

𝜎=

0 and

¤𝜎=min{

𝜀𝐺(𝜑)}

for

𝜎=𝜎max

. The

considered cut-off corresponds to "hard" bound conditions [

SON00

]. Another possibility would

be "soft" or "multiplicative" bounds [

RUB01

], when the update is proportional to the distance to

the boundary. We consider here the hard bound, since it corresponds to the hard bound of the

STDP rule for Hodgkin-Huxley system.

The final phenomenological model reads as follows

¤𝜑=𝜔−𝜎sin(𝜑+𝛼), (3.10)

¤𝜎=𝜀·

𝐺(𝜑)for 0 < 𝜎 < 𝜎max,

max{0, 𝐺(𝜑)} for 𝜎=0,

min{0, 𝐺(𝜑)} for 𝜎=𝜎max.

(3.11)

with frequency mismatch 𝜔 > 0 and 𝛼=sin−1(𝜔/𝜎max).

36 3 Population of Hodgkin-Huxley neurons with spike timing-dependent plasticity

(a) (b) (c)

Figure 3.11:

Phase portraits of model (3.10)-(3.11) for (a) monostable regime of complete synchronization; (b) co-

existence of stable synchronized and clustered states; and (c) bifurcation moment of transition between the phase

portraits illustrated in (a) and (b). The basins of attraction of the synchronized regime (point

𝑆

), clustered state (limit

cycle indicated by thick black curve) and the saddle fixed point

(𝜑∗

𝜎∗)

are depicted by gray, blue, and white colors,

respectively. The nullclines of the system and stable and unstable manifolds of the saddle point are indicted by the

thin gray and black curves, respectively. Parameters (a)

𝜔=

0.037 kHz, (b)

𝜔=

0.06 kHz, (c)

𝜔≈

0.455 Hz, and the

other parameters 𝜏𝑝=2, 𝜏𝑑=5, 𝑐𝑝=2, 𝑐𝑑=1.6, and 𝜀=0.08. Figure modified from [ROE19a].

3.4.1 Properties of the model

Phase space of system (3.10)-(3.11) is two dimensional with

(𝜑

𝜎) ∈ 𝑆1×[

𝜎max]

. The nullclines

are given by

𝐺(𝜑)=

0 for

¤𝜎=

0 and

𝜎=𝜔/sin(𝜑+𝛼)

for

¤𝜑=

0 in the internal points of the

phase space. For the parameter values as in Fig. 3.11, the

𝜑

-nullcline corresponds to the two lines

𝜑=𝜑∗≈

0.23 and

𝜑=−𝜑∗

, while the

𝜎

-nullcline to a U-shaped nonlinear curve (grey lines in

Fig. 3.11).

There is just one fixed point

(𝜑∗

𝜎∗=𝜔/sin 𝜑∗)

of saddle-type within the region

𝜎∈ (

𝜎max)

This point is given by the intersection of the nullclines. Figure 3.11 shows this fixed point and

its stable and unstable separatrixes (black lines). An additional fixed point as well as periodic

attractor emerge in system (3.10)-(3.11) due to the non-smoothness at the boundaries. More

specifically, three situations are observed:

(I)

: One globally stable fixed point

𝑆=(

𝜎max)

which corresponds to the fusion of the two

clusters into one. The coupling

𝜎=𝜎max

and the phase difference is zero at the fixed point, see

Fig. 3.11. All orbits are approaching this stable fixed point with time. This corresponding phase

portrait is shown in Fig. 3.11(a).

(II)

: Coexistence of the stable fixed point

𝑆=(

𝜎max)

and a stable periodic orbit, see Fig. 3.11(b).

As in the case (I), the fixed point corresponds to the merging of two clusters. The periodic orbit

corresponds to two simultaneously existing clusters. The clusters possess different frequencies

and, as a result, the phase difference is not bounded and rotate along the circular direction

𝜑

. Part of the periodic orbit is located on the boundary

𝜎=

0, i.e. vanishing inter-cluster

coupling. The coupling

𝜎(𝑡)

increases between

−𝜑∗

and

𝜑∗

and decreases otherwise. In fact,

one can parameterise the coupling

𝜎

by the phase

𝜑

on the periodic attractor. In the case when

𝜎(𝜑∗)< 𝜎∗, the solution returns to the boundary 𝜎=0, moves along it till the orbit reaches the

point (−𝜑∗, 0), and the periodic motion repeats.

(III)

: When

max𝜑𝐺(𝜑)<

0 then there exists globally stable periodic solution

𝜑=𝜔𝑡 +𝜑0

𝜎=

0. In such a case, the fixed point on the boundary disappears. Formally, this corresponds

3.4 Phenomenological model with phase difference-dependent plasticity 37

to an uncoupling between the clusters. However, in the original Hodgkin-Huxley system, this

parameter regime corresponds to complete uncoupling of all oscillators because of the depression

of all synapses.

In fact, the parameter boundary between the cases (I) and (II) is determined by the condition

𝜎(𝜑∗)=𝜎∗

, which can be interpreted geometrically as hitting the point

(−𝜑∗

, 0

)

by the stable

manifold of the saddle equilibrium point, see Fig. 3.11(c). In this special case, the saddle equilib-

rium attracts the whole set of points from the phase space that is below the stable manifolds,

see white area in Fig. 3.11(c). In case (II), the separation between the basins of attraction of the

fixed point and the periodic orbit are given by the saddle equilibrium and its stable manifolds.

A sufficient condition for the case (III) is given by

𝑐𝑑≥𝑐𝑝

and

𝜏𝑑≥𝜏𝑝

. Under these conditions

𝐺(𝜑) ≤ 0 for all 𝜑.

Summarizing, the case (II) corresponds to the situation when clusters are stable and do not

merge into one. For this, initial conditions must belong to the basin of attraction of the periodic

solution (Fig. 3.11(b), blue domain). The analysis of the phenomenological model indicates that

the cluster case always coexists with stable complete synchronization.

3.4.2 Comparison of the model and cluster dynamics in Hodgkin-Huxley network

In order to compare dynamics of the phenomenological model (3.10)-(3.11) and the original

system (3.1)-(3.2), we ran a series of simulations of the Hodgkin-Huxley network for parameter

values that allow for a stable two-cluster solution. The phases of the clusters are calculated

𝜑1,2(𝑡)=

𝜋𝑡−𝑡𝑘

𝑡𝑘+1−𝑡𝑘+

𝜋𝑘 for 𝑡∈ [𝑡𝑘

𝑡𝑘+1)

, where

{𝑡1

, ...,

𝑡𝑛

, ...

}

are spiking times with

𝑡𝑘< 𝑡𝑘+1

[

PIK01

]. Correspondingly, the phase difference is

𝜑𝐻 𝐻 (𝑡)=𝜑1(𝑡)−𝜑2(𝑡)

. The coupling measure

𝜎𝐻 𝐻 is given by the mean inter-cluster coupling.

Extracting the quantities

𝜎𝐻 𝐻

and

𝜑𝐻 𝐻

from the numerically computed solutions of Hodgkin-

Huxley system (3.1)-(3.3) we obtain a two-dimensional projection of the solution to the plane

(𝜑𝐻 𝐻

𝜎𝐻 𝐻 )

, see Fig. 3.12. The discontinuities in the orbits are related to the discrete STDP

updates. Additionally, since the phases

𝜑1,2(𝑡)

can be firstly accessed after the both clusters

fired, some of the area of the phase diagram (see white area in Fig. 3.12) was not accessible.

This "empty" area corresponds to anti-phase initial conditions, which are very sensitive, and,

after each cluster fires, they appear immediately either in the red or blue area. Nevertheless, the

behavior has the same qualitative features as in the phenomenological model, compare Figs. 3.11

and 3.12.

3.4.3 Criteria for the emergence of frequency clusters

Model (3.10)-(3.11) can be used to describe plasticity functions, which lead to multiple clusters.

For this, we investigate numerically the condition

𝜎(𝜑∗)=𝜎∗

. More specifically, system (3.10)-

(3.11) was initialized at the point

(−𝜑∗

, 0

)

and numerically integrated forward in time. If

𝜎(𝜑∗)<

𝜎∗

, the two clusters are stable and do not merge. This procedure can be repeated for different

parameter values.

In order to restrict the set of plasticity parameters, we fix

𝜏𝑝=

2 and

𝜏𝑑=

5 and vary

𝑐𝑝

and

𝑐𝑑

The results of the simulation are shown in Fig. 3.13(a). The white, black and grey parameter

38 3 Population of Hodgkin-Huxley neurons with spike timing-dependent plasticity

Figure 3.12:

Dynamics of the phase difference between the clusters

𝜑𝐻 𝐻

and mean inter-cluster coupling

𝜎𝐻 𝐻

for

the solutions of the Hodgkin-Huxley system (3.1)-(3.3) for different initial conditions.

𝑁=

50 with

𝑁𝑠=

7 neurons in

the small cluster and

𝑁𝑏=

43 in the big one. Red orbits converge to the regime of complete synchronization, and blue

trajectories lead to a stable two-cluster solutions. The nullclines of the phenomenological model are shown in gray.

Other parameters: 𝜏𝑝=2, 𝜏𝑑=5, 𝑐𝑝=2, 𝑐𝑑=1.6, and 𝜅max =1.5. Figure taken from [ROE19a].

areas correspond to the appearance of stable periodic solution of (3.10)-(3.11) (case (II)), globally

stable fixed point (case (I)) and the case (III), respectively.

In order to compare the parameter regions obtained for the phenomenological model (Fig. 3.13(a))

with those for the original Hodgkin-Huxley system, we ran numerical simulations of system

(3.1)-(3.3) with

𝑁=

50 neurons and

𝑁𝑠=

7 neurons in the small cluster. Starting from the two-

cluster state, we monitor the dynamics of the clusters. Figure 3.13(b) shows the results: the white

region corresponds to the case when the clusters survive and stay apart after the simulation

time 3000 ms, black - when the clusters merge into one synchronous group, and grey - when the

clusters split into uncoupled neurons. This behaviour stays qualitatively the same for different

cluster sizes. However, depending on the frequency difference between the clusters, the set of

parameters allowing stable cluster states may change its size.

Comparison of the results for the phenomenological system and the Hodgkin-Huxley system in

the Figs. 3.13(a,b) shows that the phenomenological model provides a reasonable approxima-

tion.

3.5 Summary 39

(a) (b)

Figure 3.13:

Panel (a): system (3.10)-(3.11). White region: stable periodic solution coexisting with a stable fixed point,

case II. Black region: globally stable fixed point, case I. Grey region: globally stable periodic solution with

𝜎=

Panel (b): original system (3.1)-(3.3). White: stable two-clusters (white); black: stable synchrony and no stable clusters;

grey: decoupling of all neurons. Other parameters

𝜏𝑝=

𝜏𝑑=

𝑁=

50,

𝑁𝑠=

7, and

𝜅max =

1. Figure modified

from [ROE19a].

3.5 Summary

In summary, our results show that adaptive neural networks are able to generate self-consistently

dynamics with different frequency bands. In our case, each cluster corresponds to a strongly

connected component with a fixed frequency. Due to a sufficiently large difference of the

cluster sizes and frequencies, the inter-cluster interactions are depreciated, while the intra-

cluster interactions are potentiated. In this study, we have described the mechanisms behind the

formation and stabilization of these clusters. In particular, we have explained why the significant

difference between the cluster sizes is important for the decoupling of the clusters. From a

larger perspective, the decoupling of the clusters in our case is analogous to the decoupling of

timescales in systems with multiple timescales.

In addition, we have presented a two-dimensional phenomenological model which allows for

a detailed study of the clustering mechanisms. Despite of the approximations made by the

derivation, the model coincides surprisingly well with the adaptive Hodgkin-Huxley network.

Using the phenomenological model, we have found parameter regions of the plasticity function,

where stable frequency clustering can be observed.

Clustering behavior also emerges at the brain scale, where synchronized communities of brain

regions constituting large distributed functional networks can intermittently be formed and dis-

solved [

DEC09

PON15

]. Such clustering dynamics can shape the structured spontaneous brain

activity at rest as measured by fMRI. In this study, we have shown that slow oscillations based

on the modulation of synchronized neural activity can already be formed at the resolution level

of a single neural population if adaptive synapses are taken into account. These modulations of

the amplitude of the mean field can be generated in a stable manner, see Fig.3.9 and Ref. [

POP15

40 3 Population of Hodgkin-Huxley neurons with spike timing-dependent plasticity

The mechanism relies on fluctuations of the extent of synchronization of tonically firing neurons.

This is caused by the splitting of the neural population into clusters and the corresponding clus-

ter dynamics. It might contribute to the emergence of slow brain rhythms of electrical (LFP, EEG)

and metabolic (BOLD) brain activity reported by Refs. [MAN07,MAG12a,MON08,ALV14].

However, other mechanisms for generating slow oscillations are possible. The papers [

BAZ02

COM03

] discussed the emergence of slow oscillatory activity (

1Hz) that can be observed in

vivo in the cortex during slow-wave sleep, under anesthesia or in vitro in neural populations.

The suggested mechanism relies on the corresponding modulation of the firing of individual

neurons, and the slow oscillation at the population level was proposed to be the result of

very slow bursting of individual neurons that synchronize across the neural population. In

contrast, the present work shows that the slow oscillations of the population mean field can

also emerge when the firing of individual neurons is not affected. The neurons may tonically

fire at high frequencies. The amplitude of the population mean field then oscillates at much

lower frequencies due to the slow modulation caused by the cluster dynamics. Additionally,

more recent work suggested that slow collective oscillations in neural networks with spike

timing-dependent plasticity are induced by the so-called Sisyphus effect unveiled using a free

energy approach on the temporal behavior of the order parameter.

One-cluster states in adaptive networks of

coupled phase oscillators 4

The findings in Chapter 3suggested that the emergence of frequency clusters in neuronal

populations with synaptic plasticity is well described by a phase oscillator model with phase

difference-dependent plasticity. A model of this kind was introduced in Ref. [

AOK09

KAS17

see also (2.25)–(2.26) in Section 2.3.3, and reads

𝑑𝜙𝑖

𝑑𝑡 =𝜔−1

𝑁

𝑗=1

𝜅𝑖 𝑗 sin(𝜙𝑖−𝜙𝑗+𝛼), (4.1)

𝑑𝜅𝑖 𝑗

𝑑𝑡 =−𝜖𝜅𝑖 𝑗 +sin(𝜙𝑖−𝜙𝑗+𝛽). (4.2)

System

(4.1)

–

(4.2)

has attracted a lot of attention recently [

SEL02

REN07

AOK09

AOK11

PIC11a

TIM14

GUS15a

KAS16a

NEK16

KAS17

AVA18

], since it is a first choice paradigmatic

model for the modeling of the dynamics of globally coupled adaptive networks. In particular, it

generalizes the Kuramoto (or Kuramoto-Sakaguchi) model with fixed

𝜅

. As it is known from

numerical studies performed in Ref. [

KAS17

], the system

(4.1)

–

(4.2)

reaches different cluster

states depending on the initial conditions. One-cluster states are a particular kind of state

characterized by a frequency synchronized motion of all oscillators. In this chapter, we discuss

in detail the nature and properties of these states. We provide necessary and sufficient condition

for their existence as well stability and provide insights in the impact of parameter changes.

This chapter includes contents that have been published in [

BER19

BER19a

]. The structure of

this chapter is as follows. In Section 4.1 we prove the existence of one-cluster states for the

system

(4.1)

–

(4.2)

and provide a complete classification of these states. The stability of one-

clusters is rigorously described in the subsequent Section 4.2. Building on these result, the

impact of the adaptation rate on the stability is explored, Section 4.3. Finally, we discuss the

role of double antipodal states in Section 4.4. All findings of this chapter are summarized in

Section 4.5.

4.1 Classification of one-cluster states

We start with the collective dynamics, where all oscillators are synchronous up to phase shifts, i.e.

𝜙𝑖=𝑠(𝑡)+𝑎𝑖

. It is easy to see from (4.1), that

𝑑𝑠/𝑑𝑡 =const

in this case, and, hence,

𝑠(𝑡)= Ω𝑡

with

some constant frequency

. Moreover, due to the symmetry of system (4.1)–(4.2) with respect to

the phase-shift

𝜙𝑖↦→ 𝜙𝑖−𝑎1

one can consider

𝑎1=

0 without loss of generality. Therefore, we

define the following solutions.

Definition 4.1.1 Phase oscillators 𝜙𝑖(𝑡),𝑖=1, . . . ,𝑁are said to be

(i) in-phase synchronous if 𝜙𝑖(𝑡)=𝑠(𝑡)for all 𝑖;

(ii) antiphase synchronous if 𝜙𝑖(𝑡)=𝑠(𝑡)+𝑎𝑖with 𝑎𝑖∈ {0, 𝜋}and there are 𝑖≠𝑗such that 𝑎𝑖≠𝑎𝑗;

(iii) rotating-waves if 𝜙𝑖(𝑡)=𝑠(𝑡)+(𝑖−1)2𝜋𝑘/𝑁, where 𝑘∈{1, . . . ,𝑁}is the wave number;

(iv) phase-locked if 𝜙𝑖=𝑠(𝑡)+𝑎𝑖with arbitrary 𝑎𝑖∈T1.

42 4 One-cluster states in adaptive networks of coupled phase oscillators

Table 4.1:

The table summarizes the values for

𝑛

th order parameter for the phase-locked solutions introduced in

Definition 4.1.1.

State 𝑛th order parameter

in-phase: 𝜙𝑖= Ω𝑡 𝑅𝑛=1

anti-phase: 𝜙𝑖= Ω𝑡+𝑎𝑖,𝑎𝑖∈{0, 𝜋}𝑅2𝑛=

𝑅2𝑛+1=2𝑁1

𝑁−1

, where

𝑁1

is the num-

ber of 𝑎𝑖=0

rotating-wave: 𝜙𝑖= Ω𝑡+𝑖2𝜋

𝑁𝑘,𝑘≠0, 𝑁

2𝑅𝑛=1 if 𝑛·𝑘

𝑁∈N0and 𝑅𝑛=0

phase-locked: 𝜙𝑖= Ω𝑡+𝑎𝑖,𝑎𝑖∈T1𝑅𝑛=1

𝑁Í𝑁

𝑗=1𝑒i𝑛𝑎𝑗

Note that the following implications hold: rotating-waves with

𝑘=

0 and

𝑘=𝑁/

2 are in-phase

and anti-phase synchronous, respectively. Rotating-waves, in-phase, and antiphase solutions are

phase-locked. The term "rotating-wave" relates to the rotating symmetry of the solutions with

respect to the spatial coordinate given by the index

𝑖

. These solutions are also known as twisted

states [WIL06,GIR12] or splay states [CHO09].

It is straightforward to check the values of the

𝑛

th order parameter

(2.15)

for the special solutions

defined above. For instance, for an in-phase synchronous solution it holds

𝑍𝑛(𝝓(𝑡)) =𝑒i𝑛Ω𝑡

and,

hence

𝑅𝑛=

1 for all

𝑛∈N

and

𝑡∈R

. For the other solutions the results are summarized in

Table 4.1. If

𝝓(𝑡)

is a phase-locked solution, the modulus of the

𝑛

th order parameter does not

depend on

Ω𝑡

, and hence it will be often referred to as

𝑅𝑛(a)

, where

a=(𝑎1

. . .

𝑎𝑁)𝑇

is the

phase shift vector.

With the help of the order parameter we introduce the following three types of phase-locked

states.

Definition 4.1.2 Phase oscillators 𝜙𝑖(𝑡),𝑖=1, . . . ,𝑁are said to form a

(i) (Splay cluster) if 𝑅2(𝝓)=0;

(ii) (Antipodal cluster) if 𝑅2(𝝓)=1, i.e., 𝜙𝑖∈{0, 𝜋}for all 𝑖=1, . . . ,𝑁;

(iii) (Double antipodal cluster) if 𝜙𝑖∈{0, 𝜋,𝜓,𝜓+𝜋}for all 𝑖=1, . . . ,𝑁with 𝜓∈ (0, 𝜋).

Note that if

𝝓

is in-phase or anti-phase synchronous,

𝝓

forms an antipodal cluster as well. Rotat-

ing waves from Definition 4.1.1 are also known as splay states or incoherent clusters [

CHO09

For all theses rotating waves, it holds

𝑅1(𝝓)=𝑅2(𝝓)=

0. As it will be shown in Proposition 4.1.1,

system (4.1)–(4.2) generically possesses solutions with

𝑅2(𝝓)=

0 rather than

𝑅1(𝝓)=

0. Both

uniformity criteria are clearly related since

𝑅2(𝝓)=𝑅1(

𝝓)

. We use the notion splay cluster in

a more general sense to stress that the phases are uniformly distributed around the unit circle

with respect to the second moment of the order parameter. The following result describes all

possible phase-locked solutions in the system of adaptively coupled oscillators

(4.1)

–

(4.2)

. We

call these solutions one-cluster solutions, since all oscillators possess the same frequency.

Proposition 4.1.1 System (4.1)–(4.2) possesses the following phase-locked solutions

𝜙𝑖= Ω𝑡+𝑎𝑖, (4.3)

𝜅𝑖 𝑗 =−sin(𝑎𝑖−𝑎𝑗+𝛽),𝑖,𝑗=1, . . . ,𝑁(4.4)

4.1 Classification of one-cluster states 43

Figure 4.1:

Illustration of the three types of one-cluster solutions given by (4.3)–(4.4) for an ensemble of 50 oscillators.

One-cluster solutions (a) of splay type (

𝑅2(a)=

0) for

𝛼=

0.3

𝜋

𝛽=

0.1

𝜋

, (b) of antipodal type (

𝑅2(a)=

1), for

𝛼=

0.2

𝜋

𝛽=−

0.95

𝜋

and (c) of double antipodal type satisfying condition (iii) of Proposition 4.1.1 with

𝑚=

30 for

𝛼=

0.3

𝜋

𝛽=−0.15𝜋. Figure taken from [BER19].

if and only if one of the following three conditions is fulfilled:

(i) the phases 𝑎𝑖form a splay cluster, i.e., 𝑅2(a)=0;

(ii) the phases 𝑎𝑖form an antipodal cluster, i.e., 𝑅2(a)=1;

(iii) the phases

𝑎𝑖

form a double antipodal cluster with

𝑚∈{1, . . . ,𝑁−1}

𝑎𝑖∈{0, 𝜋,𝜓𝑚,𝜓𝑚+𝜋}

𝑖=1, . . . ,𝑁and 𝜓𝑚being the unique modulo 2𝜋solution to the following equation

𝑁−𝑚

𝑚sin(𝜓−𝛼−𝛽)=sin(𝜓+𝛼+𝛽), (4.5)

and the number of phase shifts 𝑎𝑖such that 𝑎𝑖∈ {0, 𝜋}equals to 𝑚.

The corresponding frequencies are given by

Ω = 

cos(𝛼−𝛽)/2if 𝑅2(a)=0,

sin 𝛼sin 𝛽if 𝑅2(a)=1,

2(cos(𝛼−𝛽) −𝑅2(a)cos(𝜓))in case (iii).

(4.6)

The proof of this and other propositions are given in the Appendix A. Note that for the special

cases

𝛼=

𝜋

together with

𝛽=𝛼+𝜋/

𝛼+

𝜋/

2 solutions with

𝑅2(a)∉{0, 1}

were discussed

in [

GUS15a

]. Moreover, similar solutions were found in experimental settings with delay-coupled

chemical oscillators [BLA13].

Note that conditions (i)–(iii) of Proposition 4.1.1 imply that there are three possible types of

one-cluster solutions: splay, antipodal, and double antipodal. We illustrate these solutions in

Figs. 4.1(a–c).

In the following, we focus our study mainly on the splay and antipodal clusters with

𝑅2(a) ∈

44 4 One-cluster states in adaptive networks of coupled phase oscillators

Figure 4.2: Illustration of the family of solutions 𝑆(a) 𝑁=2, (b) 𝑁=3, (c) 𝑁=4. Figure taken from [BER19].

{

0, 1

}

, i.e., the phase-locked solutions given by cases (i) and (ii) of Proposition 4.1.1. The role of

double antipodal states is discussed in the last section of this chapter. We further remark that the

phase-locked solutions

(4.3)

–

(4.4)

are relative equilibria with respect to the phase-shift defined

in Section 2.3.3, i.e., they are equilibria in the co-rotating frame 𝜙↦→ 𝜙+Ω𝑡.

𝑅2(a)=

1, Proposition 4.1.1 implies that

𝑎𝑖

are either 0 or

𝜋

. Therefore, there are 2

𝑁−1

isolated

solutions of this kind. Note that the in-phase synchronous solution is an antipodal one-cluster

solution.

The situation is different for the splay cluster. The relation

𝑅2(a)=

0 gives the

𝑁−

2 parametric

(𝑁 > 2) family

𝑆:=n(𝜙𝑖,𝜅𝑖 𝑗):𝜙𝑖= Ω𝑡+𝑎𝑖,𝜅𝑖 𝑗 =−sin(𝑎𝑖−𝑎𝑗+𝛽),

𝑁

𝑗=1

sin(2𝑎𝑗)=

𝑁

𝑗=1

cos(2𝑎𝑗)=0o, (4.7)

where

Ω = cos(𝛼−𝛽)/

2. Moreover, analogously to [

ASH08

], one can show that

𝑆

is the union of

𝑁−2 dimensional manifolds.

The structure of the solution family

(4.7)

is illustrated in Figs. 4.2(a–c) for

𝑁=

2, 3, 4. Figure 4.2(a)

shows one of the two disjoint one-dimensional subsets of

𝑆

for the case of two adaptively

coupled oscillators modulo common rotation of the phases on the circle. In fact, the oscillators

have to have a phase shift of

𝜋/

2 in order to meet the condition

𝑅2(a)=

0, i.e.,

a=(𝛾,𝛾+𝜋/2)

with

𝛾∈ [

0, 2

𝜋)

. The dimension of

𝑆

for

𝑁=

2 is one. For a system consisting of three or

four phase oscillators the dimension of

𝑆

is either 1 or 2, respectively. For

𝑁=

3, one has

(𝛾,𝛾+𝜋/3, 𝛾+2𝜋/3)

, see Fig. 4.2(b). For the case

𝑁=

4, we have

a=(𝛾

𝛾+𝜉

𝛾+𝜋/

𝛾+𝜉+𝜋/

)

with 𝛾,𝜉∈ [0, 2𝜋), see Fig. 4.2(c).

Note that the set of phases satisfying the condition

𝑅1(a)=

0 was described in [

ASH08

BUR11

ASH16a

]. Our case of splay clusters can be related to this set using the equality

𝑅2(a)=𝑅1(

a)=

Rotating-waves are a particular case of the splay cluster, namely, the following corollary holds.

Corollary 4.1.2 For any 𝑘∈{1, . . . ,𝑁−1},𝑘≠𝑁/2the rotating-wave

𝜙𝑖= Ω𝑡+(𝑖−1)2𝜋𝑘

𝑁,

𝜅𝑖 𝑗 =−sin (𝑖−𝑗)2𝜋𝑘

𝑁+𝛽,

4.2 Stability of one-cluster states 45

with Ω = cos(𝛼−𝛽)/2is a solution of system (4.1)–(4.2).

Let us make a short remark, which allows for a better understanding and interpretation of

the phase-locked solutions given in Proposition 4.1.1. Assume that the phase variables are in

a phase-locked solution

𝜙𝑖= Ω𝑡+𝑎𝑖

. Then, the coupling weights

𝜅𝑖 𝑗

have to satisfy the linear

system

¤𝜅𝑖 𝑗 =−𝜖𝜅𝑖 𝑗 −𝜖sin(𝑎𝑖−𝑎𝑗+𝛽). (4.8)

This system has the unique solution

𝜅𝑖 𝑗 =−sin(𝑎𝑖−𝑎𝑗+𝛽)

which is constant, bounded on

, and asymptotically stable as

𝑡→ ∞

. Therefore, the specific network connectivity

𝜅𝑖 𝑗 =

−sin(𝑎𝑖−𝑎𝑗+𝛽)is associated with a given phase-shift a.

4.2 Stability of one-cluster states

In the previous section the existence of one-cluster were discussed. This section is devoted to the

analysis of their stability.

In order to study the local stability of one-cluster solutions described in Section 4.1, we lin-

earize the system

(4.1)

–

(4.2)

around the solutions

(4.3)

–

(4.4)

. We obtain the following linearized

system

𝑑

𝑑𝑡 𝛿𝜙𝑖=1

𝑁

𝑁−1

𝑚=0

sin(𝑎𝑖−𝑎𝑖+𝑚+𝛽)cos(𝑎𝑖−𝑎𝑖+𝑚+𝛼)(𝛿𝜙𝑖−𝛿𝜙𝑖+𝑚)(4.9)

−1

𝑁

𝑁−1

𝑚=0

sin(𝑎𝑖−𝑎𝑖+𝑚+𝛼)𝛿𝜅𝑖(𝑖+𝑚),

𝑑

𝑑𝑡 𝛿𝜅𝑖(𝑖+𝑚)=−𝜖𝛿𝜅𝑖(𝑖+𝑚)+cos(𝑎𝑖−𝑎𝑖+𝑚+𝛽)(𝛿𝜙𝑖−𝛿𝜙𝑖+𝑚), (4.10)

where we have introduced the new label

𝑚:=𝑗−𝑖

and the convention

𝑖+𝑚=(𝑖+𝑚)mod 𝑁

for convenience. Throughout this paragraph we will make use of the Schur complement [

BOY04

LIE15

] in order to simplify characteristic equations. More precisely, any

𝑚×𝑚

matrix

𝑀

in the

2×2 block form can be written as

𝑀= 𝐴 𝐵

𝐶 𝐷!= I𝑝𝐵𝐷−1

0I𝑞! 𝐴−𝐵𝐷−1𝐶0

0𝐷! I𝑝0

𝐷−1𝐶I𝑞!(4.11)

where

𝐴

is a

𝑝×𝑝

matrix and

𝐷

is an invertible

𝑞×𝑞

matrix. The matrix

𝐴−𝐵𝐷−1𝐶

is called

Schur complement. A simple formula for the determinant of

𝑀

can be derived with the decom-

position (4.11)

det(𝑀)=det(𝐴−𝐵𝐷−1𝐶) ·det(𝐷).

This result is important for the subsequent stability analysis. Note that in the following an

overline indicates the complex conjugate.

46 4 One-cluster states in adaptive networks of coupled phase oscillators

Lemma 4.2.1

Suppose

a𝑘=(

0, 2

𝜋𝑘/𝑁

. . .

(𝑁−

)

𝜋𝑘/𝑁)𝑇

with

𝑘∈ {

. . .

𝑁−

}

and the linear

system around the one-cluster solution

𝝓= Ω𝑡·(

. . .

, 1

)𝑇+a𝑘

is given by (4.9)–(4.10). Then there exist

new coordinates

(𝛿𝜓

𝛿𝜁)

such that the linearized system can be decomposed into

𝑁

linear differential

equations of the form

©«

𝛿¤

𝜓𝑙

𝛿¤

𝜁𝑙0

𝛿¤

𝜁𝑙(𝑁−1)

ª®®®®®¬

=𝐶𝑙©«

𝛿𝜓𝑙

𝛿𝜁𝑙0

𝛿𝜁𝑙(𝑁−1)

ª®®®®®¬

𝑙=0, . . . ,𝑁−1 (4.12)

with

𝐶𝑙:= ˆ

𝜆𝑙𝑏

𝑐𝑙−𝜖I𝑁!,

where, I𝑁is the 𝑁-dimensional identity matrix and

𝜆𝑙=1

2((𝑍1(a𝑙)−1)sin(𝛼−𝛽)−=(𝑍2(a𝑘))cos(𝛼+𝛽) +<(𝑍2(a𝑘))sin(𝛼+𝛽))(4.13)

4𝑍1(a2𝑘−𝑙)i𝑒i(𝛼+𝛽)−𝑍1(a2𝑘+𝑙)i𝑒−i(𝛼+𝛽).

𝑏=1

𝑁sin(−𝛼),. . . , sin((𝑁−1)𝑘2𝜋

𝑁−𝛼), (4.14)

𝑐𝑙=0, cos(𝑘2𝜋

𝑁−𝛽)1−𝑒i𝑙2𝜋

𝑁,. . . , cos((𝑁−1)𝑘2𝜋

𝑁−𝛽)1−𝑒i𝑙(𝑁−1)2𝜋

𝑁𝑇

(4.15)

with any 𝑗∈ {1, . . . ,𝑁}.

Proof.

Due to the cyclic structure in the equations

(4.9)

and

(4.10)

it is possible to decouple them

using a discrete Fourier ansatz [PER10c]

𝛿𝜙𝑗=

𝑁−1

𝑙=0

𝑒i𝑙 𝑗 2𝜋

𝑁𝛿𝜓𝑙,

𝛿𝜅 𝑗(𝑗+𝑚)=

𝑁−1

𝑙=0

𝑒i𝑙 𝑗 2𝜋

𝑁𝛿𝜁𝑙𝑚.

Taking this Fourier ansatz and plugging it into the equations (4.9) and (4.10) we get

𝑁−1

𝑙=0

𝑒i𝑙 𝑗 2𝜋

𝑁¤

𝛿𝜓𝑙=1

𝑁

𝑁−1

𝑚=0

sin(−𝑚𝑘 2𝜋

𝑁+𝛽)cos(−𝑚𝑘 2𝜋

𝑁+𝛼)

𝑁−1

𝑙=0

𝑒i𝑙 𝑗 2𝜋

𝑁1−𝑒i𝑙𝑚 2𝜋

𝑁𝛿𝜓𝑙

−1

𝑁

𝑁−1

𝑚=0

sin(−𝑚𝑘 2𝜋

𝑁+𝛼)

𝑁−1

𝑙=0

𝑒i𝑙 𝑗 2𝜋

𝑁𝛿𝜁𝑙𝑚,

𝑁−1

𝑙=0

𝑒i𝑙 𝑗 2𝜋

𝑁¤

𝛿𝜁𝑙𝑚 =−𝜖

𝑁−1

𝑙=0𝑒i𝑙 𝑗 2𝜋

𝑁𝛿𝜁𝑙𝑚 +cos(−𝑚𝑘 2𝜋

𝑁+𝛽)𝑒i𝑙 𝑗 2𝜋

𝑁1−𝑒i𝑙𝑚 2𝜋

𝑁𝛿𝜓𝑙.

After making use of well known trigonometric identities and using the order parameters defined

4.2 Stability of one-cluster states 47

in (2.15) we find

𝜆𝑙=1

2𝑁

𝑁−1

𝑚=0sin(−4𝜋

𝑁𝑚𝑘 +𝛼+𝛽)−sin(𝛼−𝛽)1−cos(𝑙𝑚 2𝜋

𝑁)−i sin(𝑙𝑚 2𝜋

𝑁)

2((𝑍1(a𝑙)−1)sin(𝛼−𝛽)−=(𝑍2(a𝑘))cos(𝛼+𝛽)+<(𝑍2(a𝑘))sin(𝛼+𝛽))

4𝑍1(a2𝑘−𝑙)i𝑒i(𝛼+𝛽)−𝑍1(a2𝑘+𝑙)i𝑒−i(𝛼+𝛽).

The row and the column vectors

𝑏

and

𝑐𝑙

can directly be read of from the transformed equation

above.

Note that the values

𝜆𝑙

are exactly the eigenvalues for the case where no interaction between

the oscillators and their coupling are assumed or the dynamics of the coupling weights are left

constant. One might expect that due to the slow-fast dynamics of the system

(4.1)

–

(4.2)

a small

perturbation in the coupling weights could be neglected for the analysis of stability [

AOK11

]. In

contrast, we show that the local dynamics of the system around the one-cluster solution depends

on the interplay between phases and couplings.

Proposition 4.2.2

Suppose

a𝑘=(

2𝜋

𝑁𝑘

. . .

(𝑁−

)2𝜋

𝑁𝑘)𝑇

and the linear system around the one-

cluster solution

𝝓= Ω𝑡· (

. . .

, 1

)𝑇+a𝑘

is given by (4.9)–(4.10). Then the Jacobian

𝐽

of this linearized

system possesses the following spectrum

𝜎(𝐽)=−𝜖,(𝜆𝑙;1,2)𝑙=0,...,𝑁−1

with

𝜆𝑙;1,2 =ˆ

𝜆𝑙−𝜖

2±1

2q(ˆ

𝜆𝑙+𝜖)2+4𝜖(𝑏·𝑐)𝑙(4.16)

and ˆ

𝜆𝑙,𝑏𝑙,𝑐𝑙as defined in (4.13), (4.14) and (4.15).

Proof.

Using Lemma 4.2.1 we decompose the linear system (4.9)–(4.10) into the

𝑁

blocks (4.12).

Consider now the characteristic polynomial for the

(𝑁+

)×(𝑁+

)

matrix

𝐶𝑙

and assume that

𝜆𝑙≠−𝜖then by (4.11) we obtain

det(𝜆𝑙I𝑁+1−𝐶𝑙)=det 𝜆𝑙−ˆ

𝜆𝑙−𝑏𝑙

−𝜖𝑐𝑙(𝜖+𝜆𝑙)I𝑁!=(𝜖+𝜆𝑙)𝑁−1(𝜖+𝜆𝑙)(𝜆𝑙−ˆ

𝜆𝑙)−𝜖(𝑏·𝑐)𝑙=0.

Thus for each

𝑙∈0, . . . ,𝑁−1

there are

𝑁−

1 eigenvalues

𝜆𝑙=−𝜖

. For the two remaining

eigenvalues we have to solve the quadratic equation

𝜆2

𝑙+(𝜖−ˆ

𝜆𝑙)𝜆𝑙−𝜖ˆ

𝜆𝑙−𝜖(𝑏·𝑐)𝑙=0. (4.17)

In the case of no weight dynamics or no interdependence between the oscillators and the weights

the eigenvalues would read

𝜆𝑙;1 =ˆ

𝜆𝑙

and

𝜆𝑙;2 =−𝜖

. Therefore, the spectrum would look like

𝜎𝑐={−𝜖

(ˆ

𝜆𝑙)𝑙=0,...,𝑁−1}

with

(𝑁−

)𝑁

-fold multiplicity for the eigenvalue

−𝜖

. In contrast to that,

48 4 One-cluster states in adaptive networks of coupled phase oscillators

we get in general 2

𝑁

eigenvalues that are different from

−𝜖

which stem from the interplay of

phases and coupling weights. We should further mention that

𝜆𝑙(𝛼+𝜋

𝛽−𝜋

2)=(𝑏·𝑐)𝑙(𝛼

𝛽)

With this we write equation (4.17) as

𝜆2

𝑙(𝛼,𝛽)+ 𝜖−ˆ

𝜆𝑙(𝛼,𝛽)𝜆𝑙−𝜖ˆ

𝜆𝑙(𝛼,𝛽)+ ˆ

𝜆𝑙(𝛼−𝜋

2,𝛽+𝜋

2)=0.

The following corollary summarizes the results on the spectrum of the linearized system

(4.9)

–

(4.10).

Corollary 4.2.3

Suppose we have

a𝑘=(

2𝜋

𝑁𝑘

. . .

(𝑁−

)2𝜋

𝑁𝑘)𝑇

and the linear system (4.9)–(4.10)

then

1. (in-phase and anti-phase synchrony) if 𝑘=0or 𝑘=𝑁/2, the spectrum is given by

𝜎(𝐽)=(0)1,(−𝜖)(𝑁−1)𝑁+1,(𝜆1)𝑁−1,(𝜆2)𝑁−1

where 𝜆1and 𝜆2solve 𝜆2+(𝜖−cos(𝛼)sin(𝛽))𝜆−𝜖sin(𝛼+𝛽)=0,

2. (incoherent rotating-wave) if 𝑘≠0, 𝑁/2, 𝑁/4, 3𝑁/4, the spectrum is

𝜎(𝐽)=(0)𝑁−2,(−𝜖)(𝑁−1)𝑁+1,−sin(𝛼−𝛽)

2−𝜖𝑁−3

,(𝜗1)1,(𝜗2)1,𝜗11,𝜗21

where 𝜗1and 𝜗2solve 𝜗2+𝜖+1

2sin(𝛼−𝛽) − 1

4i𝑒i(𝛼+𝛽)𝜗−𝜖

2i𝑒i(𝛼+𝛽)=0,

3. (4-rotating-wave solution) if 𝑘=𝑁/4, 3𝑁/4, the spectrum is

𝜎(𝐽)=(0)𝑁−1,(−𝜖)(𝑁−1)𝑁+1,−sin(𝛼−𝛽)

2−𝜖𝑁−2

,(𝜆1)1,(𝜆2)1

where 𝜆1and 𝜆2solve 𝜆2+(𝜖+sin(𝛼)cos(𝛽))𝜆+𝜖sin(𝛼+𝛽)=0.

Here, the multiplicities for each eigenvalue are given as lower case labels.

As we can see from this corollary there exists always at least one zero eigenvalue. This is due

to the phase-shift symmetry of

(4.1)

–

(4.2)

we already discussed in Section 4.1. The additional

zero eigenvalues for the wave numbers

𝑘≠

𝑁/

2 can be explained with our findings from

Proposition 4.1.1 and Corollary 4.1.2. These linear rotating-wave solutions belong to a

𝑁−

dimensional family of solutions characterized by

𝑅2(a)=

0. Thus, around any point of this

family the linear equation

𝑁

𝑗=1

𝑒𝑖2𝑎𝑗𝛿𝜙𝑗=0 (4.18)

holds for a certain choice of coordinates

𝛿𝜙,𝛿𝜅𝑖 𝑗 

, and hence there are two linearly independent

equations for the infinitesimal perturbations

𝛿𝜙𝑖

. This explains the appearance of

𝑁−

2 zero

eigenvalues. They correspond to the variation along the manifold of solution. In the special case

𝑘=𝑁/

4, 3

𝑁/

4 the two algebraic equations

(4.18)

are linear dependent and we are thus left

with only one linear equation which increases the multiplicity of the zero eigenvalue by one.

The results of Corollary 4.2.3 are presented in Fig. 4.3(a–c) and compared with numerical

4.2 Stability of one-cluster states 49

Figure 4.3:

Stability diagrams for rotating-wave clusters depending on the parameters

𝛼

and

𝛽

are shown. The

regions are colored according to numerical simulation. Blue regions correspond to stable solutions while yellow

regions correspond to unstable solutions. The black dashed lines show to the borders of stability determined by

Corollary 4.2.3. Parameter

𝜖=

0.01 is fixed fo all simulations. (a)

𝑘=

1, (b)

𝑘=𝑁/

2, (c)

𝑘=𝑁/

4. Figure taken

from [BER19].

Figure 4.4:

Stability diagram for splay and antipodal one-cluster solutions depending on the parameters

𝛼

and

𝛽

are shown. The regions are colored according numerical eigenvalues of the Jacobian 4.9–4.10. Blue areas correspond

to stable while yellow areas correspond to unstable regions. Parameter

𝜖=

0.01 is fixed in all simulations. (a) Splay

solution as in Fig. 4.1(a), (b) Anti-phase solution as in Fig. 4.1(b). Figure modified from [BER19].

simulations. The numerical results are obtained by numerical integration of system

(4.1)

–

(4.2)

with

𝑁=

20. The initial conditions for each simulation are set to the one-cluster solution given

in Proposition 4.1.1 with a small perturbation added to each dynamical variable and randomly

chosen from the interval

[−

0.01, 0.01

]

. The numerical integration is stopped after

𝑡=

5000 time

steps. The relative coordinates

Θ𝑖:=𝜙𝑖−𝜙1

for

𝑖=

. . .

𝑁

are introduced in order to compare the

initial phase configuration with the distribution of the phases after numerical integration. A one-

cluster is said to be stable if

𝚯

after numerical integration is closer to the theoretical one-cluster

state than

𝚯

of the initially perturbed phase distribution. Otherwise, the one-cluster solution is

considered as unstable. Closeness is measured by the Euclidean distance. The parameter regions

in the

(𝛼

𝛽)

plane for stable one-cluster solutions are colored blue while the regions for unstable

one-cluster solutions are colored yellow. The black dashed lines correspond to the borders of

stability determined with the results in Corollary 4.2.3. In particular, a state is asymptotically

stable if

<(𝜆)<

0 for all

𝜆∈𝜎(𝐶)

except the zero eigenvalues related to the perturbations along

the solution families. In all three cases the numerical and analytic results agree very well.

In addition to the analysis of rotating-wave solutions, we investigate the stability for the splay

solutions characterized by

𝑅(a)=

0 and the antipodal solutions characterized by

𝑅(a)=

1. For

50 4 One-cluster states in adaptive networks of coupled phase oscillators

this, the stability is calculated semi-analytically by taking the solutions displayed in Fig. 4.1(a–

b), plugging them into the Jacobian matrix given by the linearized equations

(4.9)

–

(4.10)

and

determining the eigenvalues of the Jacobian numerically. The results of this procedure are shown

in Fig. 4.4 together with the borders of stability calculated with Proposition 4.1.1. In comparison

with Fig. 4.3, the analysis yields the same stability regions which are found for the rotating-wave

solutions. The numerical findings indicate that the stability for all splay and antipodal solutions

coincide with the stability of the rotating-waves.

Complementary to the previous stability results, all cases of antipodal, 4-phase, and double

antipodal cluster are analyzed. For these states the analysis is slightly more subtle and given in

the appendix A.2. In the following Proposition, we summarize the findings.

Proposition 4.2.4 The following statements hold true.

The set of eigenvalues of the linearized system

(4.9)

–

(4.10)

around all antipodal states with

𝑎𝑖∈

{0, 𝜋}agrees with the set 𝐿in Prop. 4.2.2 for rotating-wave states with 𝑘=0, 𝑁/2.

The set of eigenvalues to the linearized system

(4.9)

–

(4.10)

around all 4-phase-cluster states with

𝑎𝑖∈ {

𝜋/

𝜋

, 3

𝜋/

}

and

𝑅2(𝒂)=

0agrees with the set

𝐿

in Prop. 4.2.2 for 4-rotating-wave

states.

3. The double antipodal states are unstable for all 𝛼and 𝛽.

As indicated by the numerical analysis in Fig. 4.4, Proposition 4.2.4 shows that in fact in- as well

as anti-phase cluster share the same stability features as all other antipodal clusters. In addition,

also general 4-phase cluster possess the same Lyapunov spectrum as their special 4-rotating

wave cluster.

Remarkably, the third type of one-cluster, namely, the double antipodal state is unstable every-

where in parameter space. In Section 4.4, we describe the role of double antipodal state for the

global structure of the phase space.

4.3 Adaptation rate dependence of one-cluster stability

In the previous sections, the existence and stability of one-cluster solutions are rigorously

described. Building on this, we analyze dependence of the one-cluster states on the adaptation

rate

𝜖

. In contrast to the existence of one-cluster states, the stability of those states depend

crucially on the time-separation parameter.

The diagram in Fig. 4.5 shows the regions of stability for antipodal and rotating-wave one-

cluster states. The diagram is based on the analytic results presented in Corollary 4.2.3 and

Proposition 4.2.4.

In Fig. 4.5 the regions of stability are presented for several values of the time separation parameter

𝜖

. The first case in panel (a) assumes

𝜖=

0, where the network structure is non-adaptive but fixed

to the values given by the one-cluster states, i.e.,

𝜅𝑖 𝑗 =−sin(𝑎𝑖−𝑎𝑗+𝛽)

as given in Section 4.1.

4.3 Adaptation rate dependence of one-cluster stability 51

β/π

rotating wave states (splay)antipodal states

= 0.01

= 0.1

= 0

Asynchronous

behaviour

(b)

α/π

(c)

α/π

(a)

(d)

α/π

= 1

Asynchronous

behaviour

Asynchronous

behaviour

Figure 4.5:

The regions of stability for antipodal and rotating-wave states are presented in (

𝛼

𝛽

) parameter space for

different values of

𝜖

. Coloured and hatched areas refer to stable regions for these states as indicated in the legend.

White areas refer to region where these one-cluster states are unstable. (a)

𝜖=

0; (b)

𝜖=

0.01; (c)

𝜖=

0.1; (d)

𝜖=

Figure taken from [BER19a].

The linearized system in this case is given by

d𝛿𝜙𝑖

d𝑡=−1

2𝑁

𝑁



𝑗=1sin(𝛼−𝛽) − sin(2(𝑎𝑖−𝑎𝑗) + 𝛼+𝛽)𝛿𝜙𝑖−𝛿𝜙 𝑗. (4.19)

For the synchronized or antipodal state, the value 2

𝑎𝑖mod

𝜋

is the same for all

𝑖

. Hence,

the term 2

(𝑎𝑖−𝑎𝑗)

disappears and the linearized system

(4.19)

possesses the same form as

the linearized system for the synchronized state of the Kuramoto-Sakaguchi system [

BUR11

]

with coupling constant

𝜎(𝛽)=−sin(𝛽)

. As it follows from Ref. [

BUR11

], the synchronized as

well as all other antipodal states are stable for

𝜎(𝛽)cos(𝛼)>

0. The region of stability of the

rotating-wave cluster has a more complex shape, see hatched area in Fig 4.5(a). We find large

areas where both types of one-cluster states are stable simultaneously, as well as the regions

where no frequency synchronized state is stable. The results shown in Fig 4.5(a) are in agreement

with Ref. [

AOK11

], where the authors consider the case

𝜖=

0 in order to approximate the limit

case of extremely slow adaptation

𝜖→

0. However, such an approach for studying the stability

of clusters for small adaptation is not correct in general. As Figs. 4.5(b-d) show, the stability of

the network with small adaptation 𝜖 > 0 is different.

The case

𝜖=

0.01 is shown in Fig 4.5(b), where we observe regions for stable antipodal and

rotating-wave states as well. The introduction of a small but non-vanishing adaptation changes

the regions of stability significantly. The diagram in Fig 4.5(b) remains qualitatively the same for

smaller values of

𝜖

. This can be read of from the analytic findings presented in the Corollary 4.2.3

and Proposition 4.2.4. The changes in the stability areas are due to subtle changes in the equation

52 4 One-cluster states in adaptive networks of coupled phase oscillators

which determine the eigenvalue of the corresponding linearized system. In fact, the adaptation

introduces the necessary condition

sin(𝛼+𝛽)<

0 for the stability of antipodal states. Additionally,

for all rotating-wave states of splay type, the necessary condition

sin(𝛼−𝛽)+

𝜖 >

0 is introduced,

see Corollary 4.2.3. This is why, a non-trivial effect of adaptivity on the stability in Figs. 4.5

is observed. In particular, the parameter

𝛽

, which determines the plasticity rule, has now a

non-trivial impact on the stability of antipodal states for any

𝜖 >

0. As it can be seen in Fig. 4.5(b),

one-cluster states of antipodal type are supported by a Hebbian-like adaptation (

𝛽≈ −𝜋/

2) while

splay states are supported by causal rules (

𝛽≈

0). For the asynchronous region, the dynamical

system

(4.1)

–

(4.2)

can exhibit very complex dynamics and show chaotic motion [

KAS16a

]. This

region is supported by an anti-Hebbian-like rule (𝛽≈𝜋/2).

By increasing the parameter

𝜖

, see Fig. 4.5(c,d), two observations can be made. First, the region of

asynchronous dynamical behavior is shrinking. For

𝜖=

1, we find at least one stable one-cluster

state for any choice of the phase lag parameters

𝛼

and

𝛽

. Secondly, the regions where both types

of one-cluster states are stable are shrinking as well. In the limit of instant network adaptation,

i.e.,

𝜖→ ∞

, the stability regions are completely separated. Both types of one-cluster states divide

the whole parameter space into two areas. In this case, the boundaries are described by

𝛼+𝛽=

and

𝛼+𝛽=𝜋

. This division can be seen from the analytic findings presented in Corollary 4.2.3. In

the case of antipodal states, the quadratic equation which determines the Lyapunov coefficients

has negative roots if and only if

𝜖+cos(𝛼)sin(𝛽)>

0 and

sin(𝛼+𝛽)<

0. Here, even for

𝜖 >

the condition

sin(𝛼+𝛽)<

0 is the only remaining one. Similarly, we find

sin(𝛼+𝛽)>

0 as a

condition for the stability of rotating-wave states for 𝜖→ ∞.

4.4 Double antipodal states

Splay and antipodal clusters serve as building blocks for multicluster states. The third type, the

double antipodal clusters, are not of this nature since they appear to be unstable everywhere. As

unstable objects, they can still play an important role for the dynamics. Here we would like to

present an example, where the double antipodal clusters become part of a simple heteroclinic

network. As a result, they can be observed as metastable states in numerics.

As an example, we first analyze the system of

𝑁=

3 adaptively coupled phase oscillators which

is the smallest system with a double antipodal state. According to the definition of a double

antipodal state, laid out in Section 4.1, the phases

𝑎𝑖

of the oscillators

𝜙𝑖

are allowed to take values

from the set

{

𝜋

𝜓

𝜓+𝜋}

where

𝜓

uniquely solves Eq.

(4.5)

for a given

𝑚∈ {

. . .

𝑁}

. Further,

at least one oscillator

𝜙𝑖

with

𝑎𝑖∈ {

𝜋}

and one oscillator

𝜙𝑗

(

𝑗≠𝑖

) with

𝑎𝑗∈ {𝜓

𝜓+𝜋}

are

needed in order to represent one of the two antipodal groups. Note that for the parameters given

in Fig. 4.6 and 𝑁=3, the equation (4.5) yields 𝜓=1.602𝜋if 𝑎1,𝑎2∈ {0, 𝜋}and 𝑎3∈ {𝜓,𝜓+𝜋}.

In Fig. 4.6(a) we present trajectories which initially start close to antipodal clusters. The trajecto-

ries in phase space are represented by the relative coordinates

𝜃12 =𝜙1−𝜙2

and

𝜃13 =𝜙1−𝜙3

In particular, the two configurations with

(𝜃12 =

𝜃13 =

)

and

(𝜃12 =

𝜃13 =𝜋)

are consid-

ered. The coupling weights are initialized according to Eq.

(4.4)

. With the given parameters,

the unstable manifold of the antipodal state is one-dimensional which can be determined via

Corollary 4.2.2. For the numerical simulation, we perturb the antipodal state in such a way that

two distinct orbits close to the unstable manifold are visible. For both configuration the two

4.4 Double antipodal states 53

orbits are displayed in Fig. 4.6(a). It can be observed that after leaving the antipodal state the

trajectories approach the double antipodal states before leaving it towards the direction of a

splay state. With this we numerically find orbits close to "heteroclinic", which connect antipodal,

double antipodal, and splay clusters, see schematic picture on the right in Fig. 4.6(b). The phase

differences

𝜃12

and

𝜃13

at the double antipodal state agree with the solution

𝜓

of Eq.

(4.5)

𝜓+𝜋.

Figure 4.6(b) further justifies our statements on the heteroclinic contours. Here, we see the time

series for the second moment order parameter for all trajectories in Fig. 4.6(b). It can be seen

that in all cases we start at an antipodal cluster (

𝑅2(𝝓)=

1) from which the double antipodal

state (

𝑅2(𝝓) ≈

0.447, theoretical) is quickly approached. The trajectories stay close to the double

antipodal cluster for approximately 2000 time units (shaded area) before leaving the invariant

set towards the splay state (𝑅2(𝝓)=0).

As a second example we analyze a system of

𝑁=

100 adaptively coupled phase oscillators. Here,

we choose two particular antipodal states as initial condition and add a small perturbation to

both. One of the states is chosen as an in-phase synchronous cluster. In both cases, the couplings

weights are initialized in accordance with Eq

(4.4)

. For Figure 4.6(c) we depict the trajectories

which show a clear heteroclinic contour between antipodal, double antipodal, and splay state as

in the example of three phase oscillators. We illustrate the heteroclinic connections and present

𝑅2(𝑡)

for the corresponding trajectories in Fig. 4.6(c). Here, the zoomed view clearly shows

that the trajectories for both initial conditions starting at an antipodal cluster (

𝑅2(𝝓)=

1) again

first approach a double antipodal state (

𝑅2(𝝓) ≈

0.990, theoretical) before leaving it towards

a splay state (

𝑅2(𝝓)=

0). More precisely, each trajectory comes close to a particular double

antipodal state for which only one oscillator has a phase in

{𝜓

𝜓+𝜋}

. Remarkably, these states,

also known as solitary states, have been found in a range of other systems of coupled oscillators,

as well [

MAI14a

]. With the results in Proposition A.2.3 in the Appendix, one can show that this

double antipodal states have a stable manifold with co-dimension one which thus divides the

phase space. Next to this fact, numerical evidence for the existence of heteroclinic connections

between antipodal and double antipodal states as well as between double antipodal and the

family of splay states is provided in Fig. 4.6(c). With this, double antipodal states play an

important role for the organization of the dynamics in system (4.1)–(4.2).

54 4 One-cluster states in adaptive networks of coupled phase oscillators

(b)

R2(φ(t))

(a)

θ/π

(c)

R2(φ(t))

θ12

θ13

time

metastable

double antipodal

cluster

Figure 4.6:

Heteroclinic orbits between several steady states in a system of 3 and 100 adaptively coupled phase

oscillators. (a) The time series for the relative phases

𝜃12

(solid lines) and

𝜃13

(dashed lines) for

𝑁=

3 are shown.

Lines with the same colour correspond to the same trajectories. Panel (b,c) show time series for the second moment

order parameter

𝑅2(𝝓(𝑡))

as well as a schematic illustration of the observed heteroclinic connections (right) for (b)

𝑁=3 and (c) 𝑁=100. Parameter values: 𝜖=0.01, 𝛼=0.4𝜋, and 𝛽=−0.15𝜋. Figure taken from [BER19a].

4.5 Summary 55

4.5 Summary

The numerical analysis in Ref. [

KAS17

] suggested that one-cluster states in system

(4.1)

–

(4.2)

may serve as building blocks for multicluster states. In this chapter, we have exhaustively

analyzed the properties of one-cluster states in a network of adaptively coupled phase oscillators.

In particular, we have found that there are only three types of one-cluster states: splay, antipodal,

and double antipodal. It has been shown that all one-cluster solutions of splay type form an

𝑁−

2 dimensional family and thus give rise to infinitely many solutions the system can achieve.

In order to understand the stability of these cluster states, we have performed a linear stability

analysis. We have provided analytic results for the stability of antipodal as well as of special types

of splay states, namely, rotating-wave states. The stability of these states is rigorously described,

and the impact of all parameters has been shown. Note that due to the

𝑁−

2 dimensional

family of splay type cluster, the rotating-wave states possess

𝑁−

2 neutrally stable directions.

Remarkably, this property of splay states has been also found in networks of pulse-coupled

rotators [CAL09a] and excitable neurons [DIP12].

In this chapter, we have seen that ring-like structures are dynamically associated with rotating-

wave and splay states. Furthermore, ring networks are important motifs in neural networks

[

COM03

SPO11

POP11

] and have been the basis for many interesting dynamical regimes found

for different systems, e.g. [

ABR04

KLI15

JAR18

OME19c

]. Moreover, ring-structures were

recently found in fly brains [

KIM17a

]. In our study, we have presented how ring-like structures

could emerge due to adaptation. In particular, we have shown that causal plasticity rules, see

also Fig. 3.1, as they are found in many experimental studies [

ABB00

BI01

CAP08a

] support

the appearance of ring-like networks.

While the time-scale separation has no influence upon the existence of the one-cluster states

found in this study, it plays an important role for the stability of the one-clusters. The regions

of stability in parameter space have been presented for different choices of the time-scale

separation parameter. The singular limit (

𝜖→

0) and the limit of instantaneous adaptation

have been analyzed. The latter shows that the stability regions of the splay and the antipodal

states divide the whole space into two equally sized regions without intersection. Instantaneous

adaptation cancels multistability of these states. The consideration of the singular limit shows

that it differs from the case of no adaptation. Therefore, even for very slow adaptation, the

oscillatory dynamics alone is not sufficient to describe the stability of the system.

Additionally to the analysis of splay and antipodal clusters, we have proved that the double

antipodal states are unstable in the whole parameter range. In fact, they appear to be saddle-

points in the phase space. We have found that in a system of 3 oscillators, double antipodal states

are transient states in a small heteroclinic network between antipodal and splay states. They

appear to be metastable, i.e., observable for a relatively long time and therefore are physically

important transient states. Moreover, an additional analysis for an ensemble of 100 phase

oscillators has revealed the importance of the double antipodal states for the global dynamics of

the whole system.

Multicluster states in adaptive networks of

coupled phase oscillators 5

In this chapter we investigate the shape and properties of frequency-cluster (multicluster) states

as they have been found in Chapter 3, i.e., where each cluster has a different frequency. Inspired

by the reduced phenomenological model presented in Chapter 3, we have studied a paradigmatic

model of adaptively and globally coupled phase oscillator in Chapter 4which reads

𝑑𝜙𝑖

𝑑𝑡 =𝜔−1

𝑁

𝑗=1

𝜅𝑖 𝑗 sin(𝜙𝑖−𝜙𝑗+𝛼), (5.1)

𝑑𝜅𝑖 𝑗

𝑑𝑡 =−𝜖𝜅𝑖 𝑗 +sin(𝜙𝑖−𝜙𝑗+𝛽), (5.2)

see also Section 2.3.3. In Chapter 4, the properties of one-cluster state have been exhaustively

analyzed. In this chapter we study numerically as well as analytically multicluster states which

we observe by solving the equations

(5.1)

–

(5.2)

From the numerical study in Ref. [

KAS17

we know that certain phase-locked solutions may serve as building blocks for (hierarchical)

multicluster solutions. Here, we show that indeed multiclusters are build-up by one-cluster

states and even may inherit their dynamical features. Multicluster states are formally defined as

follows.

Definition 5.0.1

Phase oscillators

𝜙𝑖(𝑡)

form a

multicluster

if they can be separated into

𝑀

groups of

phase-locked oscillators (clusters), i.e., for all

𝜇∈ {

. . .

𝑀}

the phase oscillators

𝜙𝑖,𝜇

𝑖∈ {

. . .

𝑁𝜇}

from each group 𝜇satisfy 𝜙𝑖,𝜇(𝑡)=𝑠𝜇(𝑡)+𝑎𝑖,𝜇.

The appearance of multiclusters is interesting and nontrivial, since such solutions, in contrast

to one-clusters, are no more relative equilibria of

(5.1)

–

(5.2)

, but are periodic or quasi-periodic

solutions, which appear due to the special structure of the equation and adaptive nature of the

coupling. The oscillators within one cluster possess a synchronized temporal dynamics with

possible phase lags. In a multicluster, the coupling matrix

𝜅

is divided into different blocks

according to the division by clusters:

𝑘𝑖 𝑗,𝜇𝜈

will refer to the coupling weight between the

𝑖

oscillator of cluster 𝜇to the 𝑗th oscillator of cluster 𝜈.

This chapter includes contents that have been published in [

BER19

BER19a

]. The chapter is

organized as follows. First, in Section 5.1, system

(5.1)

–

(5.2)

is numerically analyzed and all

cluster multicluster states are qualitatively described and classified. Subsequently, we derive

analytic expression for all types of multicluster states. The results are given and exhaustively

discussed in Sections 5.2,5.4, and 5.5 for splay, antipodal, and mixed type multicluster states,

respectively. Additionally, the main existence theorem for multicluster states, particularly for

antipodal and mixed type, is given in Section 5.3. In Section 5.6, we analyze numerically as well

as analytically the stability of multicluster states and discuss the hierarchical structure of cluster

sizes. All findings of this chapter are summarized in Section 5.7.

58 5 Multicluster states in adaptive networks of coupled phase oscillators

Figure 5.1:

Three-frequencycluster of splay type at

𝑡=

10000. (a) Coupling weights represented as a graph (left) and

as a coupling matrix (right). In the graph representation, the dynamical nodes are represented by red nodes and

the edges are coloured with respect to the coupling weight. Red and blue refer to positive and negative coupling

weights, respectively. Light and dark colors refer to weak and strong coupling weights, respectively. (b) Distribution

of the phases

𝜙𝑖

for each of the three clusters. Each node represents one oscillator and is coloured with respect to

the cluster to which it belongs. Parameter values:

𝜖=

0.01,

𝛼=

0.3

𝜋

𝛽=

0.23

𝜋

𝜔=

0, and

𝑁=

100. Figure modified

from [BER19a].

5.1 Numerical observation of multicluster states

Figure 5.1 shows a hierarchical multicluster state. The solution was obtained by integrating the

system

(5.1)

–

(5.2)

numerically starting from uniformly distributed random initial conditions. The

self-couplings

𝜅𝑖𝑖

are set to zero in numerical simulations, since they do not influence the relative

dynamics of the system. This is due to the fact that all variables

𝜅𝑖𝑖

converge to

−sin 𝛽

, which

leads to the same constant term

sin(𝛽)sin(𝛼)/𝑁

in the right hand side of Eq. (5.1). The latter term

can be absorbed in the co-rotating coordinate frame. We reorder the oscillators (after sufficiently

long transient time) by first sorting the oscillators with respect to their average frequencies. After

that the oscillators with the same frequency are sorted by their phases. Figure 5.1(a) displays the

coupling matrix (right) of the multicluster state and a representation of the coupling structure as

a network graph (left). The coupling matrix demonstrates a clear splitting into three groups. This

splitting is also visible in the graph representation of the coupling network. The coupling weights

between oscillators of the same group vary in a larger range than between those of different

groups which are generally smaller in magnitude. The splitting into three groups is manifested

in the behaviour of the phase oscillators, as well. We find that the oscillators of the same group

possess the same constant frequency with possible phase lags, Fig.5.1(b). We call the groups of

oscillators (frequency) clusters and the corresponding dynamical states multi(frequency)cluster

states.

In a multicluster, the coupling matrix

𝜅

is divided into different blocks. We observe that the

behavior for each oscillator in a 𝑀-cluster state takes the form

𝜙𝑖,𝜇(𝑡)= Ω𝜇𝑡+𝑎𝑖,𝜇+𝑠𝑖,𝜇(𝑡)𝜇=1, . . . ,𝑀

𝑖=1, . . . ,𝑁𝜇

(5.3)

where

𝑀

is the number of clusters,

𝑁𝜇

is the number of oscillators in the

𝜇

-th cluster,

𝑎𝑖,𝜇∈ [

0, 2

𝜋)

are phase lags, and

Ω𝜇∈R

is the collective frequency of the oscillators in the

𝜇

-th cluster. The

functions 𝑠𝑖,𝜇are bounded.

5.1 Numerical observation of multicluster states 59

The numerical analysis of system

(5.1)

–

(5.2)

shows the appearance of different multicluster

states depending on particular choices of the phase lag parameters

𝛼

and

𝛽

as well as on initial

conditions. Starting from uniformly distributed random initial conditions the system end up in

several states such as multiclusters and chimera-like states [

KAS17

]. Figure 5.2 shows examples

for the three types of multicluster states which appear dynamically in (5.1)–(5.2).

5.1.1 Splay type cluster states

The first type is called splay type multicluster state, see Fig. 5.2(a). The separation into three

clusters is clearly visible in the coupling matrix, as well as a hierarchical structure in the cluster

sizes. Regarding the distribution of the phases, we notice that the oscillators from each group

are almost homogeneously dispersed on the circle. In fact, the phases from each cluster fulfill

the condition

𝑅2(𝝓𝜇)=

0 (

𝜇=

1, 2, 3) with order parameter

(2.15)

. Note that splay states as they

are defined in several other works [NIC92,STR93a,CHO09] share the property 𝑅1(𝝓)=0. This

property can be seen as a measure of incoherence for the oscillator phases, as well. In fact, it

was shown that splay states are part of a whole family of solutions [

BUR11

ASH16a

] given by

exactly

𝑅1(𝝓)=

0. Further,

𝑅2(𝜙)=𝑅1(

𝝓)

relates the two measures of incoherence. These facts

motivate the definition of those clusters with 𝑅2(𝝓)=0 as splay type clusters.

The temporal behavior for all phase oscillators in the splay multicluster state is characterized

by a constant frequency which differs for the different clusters, i.e., according to

(5.3)

𝜙𝑖,𝜇(𝑡)=

Ω𝜇𝑡+𝑎𝑖,𝜇

with

𝑅2(𝒂𝜇)=

0 for all

𝜇=

1, 2, 3 and

𝑖=

. . .

𝑁𝜇

. In addition, the hierarchical cluster

sizes are reflected in the frequencies. Oscillators of a big cluster have a higher frequency than

those of smaller clusters. The coupling weights between the phase oscillators are fixed or change

periodically with time depending on whether the oscillators belong to the same or different

clusters, respectively. Moreover, the amplitude of coupling weights between clusters depends

on the frequency difference of the corresponding clusters. The higher the frequency difference,

the smaller is the amplitude. The periodic behavior of the coupling weights between clusters is

present in all types of multicluster states (Fig. 5.2(a,b,c)).

5.1.2 Antipodal type cluster states

Figure 5.2(b) shows another possible multicluster state. As in Fig. 5.2(a) the clusters are clearly

visible and their oscillators show frequency synchronized temporal behavior. In addition, the

time series for the oscillators show periodic modulations on top of the linear growth. This

additional dynamics is the same for all oscillators of the same cluster, and hence they are still

temporally synchronized. We have

𝜙𝑖,𝜇(𝑡)= Ω𝜇𝑡+𝑎𝑖,𝜇+𝑠𝜇(𝑡)

. In analogy to the coupling weights

between the clusters, the amplitudes of the bounded function

𝑠𝜇(𝑡)

depend on the differences of

the cluster frequencies.

In contrast to the splay states, the phase distribution fulfills

𝑅2(𝒂𝜇)=

1 for all

𝜇=

1, 2, 3, see

Fig 5.2(b), middle panel. Hence, all oscillators of a cluster have either the same phase

𝑎𝜇∈ [

0, 2

𝜋)

or the antipodal phase

𝑎𝜇+𝜋

such that 2

𝑎𝑖,𝜇=

𝑎𝜇

modulo 2

𝜋

for all

𝑖=

. . .

𝑁𝜇

. Therefore,

the clusters represented in Fig 5.2(b) are called antipodal type clusters. Note that with this formal

definition of an antipodal state, in-phase clusters belong to the class of antipodal clusters.

60 5 Multicluster states in adaptive networks of coupled phase oscillators

5.1.3 Mixed type cluster states

The third type of multicluster states combines the previous two types. The 2-cluster state

shown in Fig. 5.2(c) consists of one splay cluster and one antipodal cluster. We call these states

mixed type multicluster. Despite using several different uniformly distributed initial conditions,

we have not found multi cluster by numerical simulations. Note that uniformly distributed

phases are closer to splay states than to antipodal states. In fact, due to the co-stability with

splay type multiclusters, mix type multicluster states are very unlikely to find from uniformly

distributed random initial conditions. In previous studies mix type multiclusters have not even

been mentioned [

KAS19

]. In order to find these particular states, we used specially prepared

initial condition motivated by the results described in Section 5.5.

As we have seen before, the interaction of a cluster with an antipodal cluster induces a modu-

lation

𝑠(𝑡)

additional to the linear growth of the oscillator’s phase. In contrast, the interaction

with a splay cluster does not introduce any modulation. Thus, the temporal dynamics of the

oscillators in the antipodal cluster (

𝜇=

1) have

𝑠𝑖,1(𝑡) ≡

0 while the oscillators in the splay cluster

(

𝜇=

2) show additional bounded modulations

𝑠𝑖,2(𝑡)

, see Fig. 5.2(c). For the oscillators of the

splay cluster we plot the time series of two representatives. We notice the temporal shift in the

dynamics of the two representatives of the splay cluster. The oscillators in the splay cluster

are not completely temporally synchronized. More specifically we have

𝜙𝑖,1(𝑡)= Ω1𝑡+𝑎𝑖,1

with

𝑅2(𝒂1)=1 for 𝑖=1, . . . ,𝑁1and 𝜙𝑖,2(𝑡)= Ω2𝑡+𝑎𝑖,2 +𝑠𝑖,2(𝑡)with 𝑅2(𝒂2)=0 for 𝑖=1, . . . ,𝑁2.

Despite the complexity of the three types of multiclusters states, the structures can be broken

down into simple blocks. In fact, one-cluster states of splay and antipodal type serve as building

blocks in order to create more complex multicluster structures. In the following sections we

provide an in-depth analysis of these blocks.

5.1 Numerical observation of multicluster states 61

index j

κij

index i

(a)

(b)

(c)

φi,µ − hΩµit φ1,µ − hΩµit φ1,µ − hΩµit

Cluster 1

Cluster 2

Cluster 3

time

Cluster 1 (i= 1)

Cluster 2 (i= 1)

Cluster 2 (i= 21)

Figure 5.2:

Three different types of multicluster states at

𝑡=

10000 with

𝑁=

100 and

𝜖=

0.01. For all types, the

coupling matrix (left), distribution of the phases (middle), and time series of representative phase oscillators from

each cluster (right) are presented. In the plot of the phase distribution, each node represents one oscillator and is

colored with respect to the cluster to which it belongs. The time series are shown after subtracting the average linear

growth

𝜙𝑖,𝜇(𝑡) − Ω𝜇𝑡

. The colouring of the time series (shaded for visibility) of a representative phase oscillator

from one cluster is in accordance with the pictures in the middle panel. (a) Splay type 3-cluster for

𝛼=

0.3

𝜋

𝛽=

0.23

𝜋

;

(b) Antipodal type 3-cluster for

𝛼=

0.3

𝜋

𝛽=−

0.53

𝜋

; (c) Mixed type 2-cluster for

𝛼=

0.3

𝜋

𝛽=−

0.4

𝜋

. Figure taken

from [BER19a].

62 5 Multicluster states in adaptive networks of coupled phase oscillators

5.2 Splay type multicluster states

As seen in Fig. 5.2, one-cluster of splay type serve as building blocks for multicluster states. In

the following, we describe this build-up explicitly.

5.2.1 Conditions for the emergence of splay type multicluster states

The multicluster solutions of splay type are composed by the clusters from the continuous family

𝑆

of phase-locked solutions with

𝑅2(a)=

0 and different frequencies. The following proposition

describes them.

Proposition 5.2.1 System 5.1–(5.2)possesses the multicluster solution

𝜙𝑖,𝜇(𝑡)= Ω𝜇𝑡+𝑎𝑖,𝜇,𝑖=1, . . . ,𝑁𝜇

𝜇=1, . . . ,𝑀(5.4)

𝜅𝑖 𝑗,𝜇𝜈 (𝑡)=−𝜌𝜇𝜈 sin(ΔΩ𝜇𝜈𝑡+𝑎𝑖,𝜇−𝑎𝑗,𝜈+𝛽−𝜓𝜇𝜈),𝑗=1, . . . ,𝑁𝜈

𝜈=1, . . . ,𝑀(5.5)

with pairwise different frequencies

Ω𝜇

ΔΩ𝜇𝜈 := Ω𝜇−Ω𝜈

𝜌𝜇𝜈 :=1+ΔΩ𝜇𝜈/𝜖2−1

and

𝜓𝜇𝜈 :=

arctan(ΔΩ𝜇𝜈/𝜖)if and only if

𝑅2(a𝜇)=

0for all

𝜇=

. . .

𝑀

and the frequencies

(Ω1

. . .

Ω𝑀)

solve the following system of equations

Ω𝜇=1

2𝑁

𝑀

𝜈=1

𝜌𝜇𝜈 𝑁𝜈cos(𝛼−𝛽+𝜓𝜇𝜈),𝜇=1, . . . ,𝑀. (5.6)

Note that 𝜌𝜇𝜈 and 𝜓𝜇𝜈 are functions of Ω𝜇−Ω𝜈.

The proof of Proposition 5.2.1 is to be found in the appendix A.3. Similarly to the one-cluster

case, multicluster solutions of splay type give rise to a

(𝑁−

𝑀−

)

-dimensional manifold of

solutions

𝑆𝑀:=n(𝜙𝑖,𝜇,𝜅𝑖 𝑗,𝜇𝜈):(𝜙𝑖,𝜇,𝜅𝑖 𝑗,𝜇𝜈)as in (5.4)-(5.5), 𝑅2(a𝜇)=0 for all 𝜇=1, . . . ,𝑀o.

Let us remark that the collective frequencies

(5.6)

are only defined up to a constant due to the

phase-shift symmetry of system (5.1)–(5.2) while the frequency difference is unaffected. An

example of a 3-cluster solution of splay type is shown in Fig. 5.3. The solution was obtained

by integrating system

(5.1)

–

(5.2)

numerically starting from random initial conditions. After

sufficiently long transient time, the order of the oscillators is given by first sorting the oscillators

with respect to their average frequencies. After that the oscillators with the same frequency are

sorted by their phases. It can be seen from the pictures that the sizes of the three clusters

𝑁𝜇

(

𝜇=

1, 2, 3) possesses a hierarchical structure, i.e.,

𝑁3< 𝑁2< 𝑁1

. The coupling strengths between

oscillators of the same cluster vary in a larger range than between those of different clusters.

The coupling between different clusters scales with

𝜖

since

𝜌𝜇𝜈 =𝜖/ΔΩ𝜇𝜈 +O𝜖/ΔΩ𝜇𝜈 3

and is

5.2 Splay type multicluster states 63

Figure 5.3:

Three-cluster of splay type. (a) Coupling weights at

𝑡=

10000 showing three clusters; (b) Distribution of

the phases within each cluster; space-time raster plot; (c) Average frequency of oscillators; each plateau corresponds

to one cluster; (d) Oscillator phases

𝜙𝑖(𝑡)

at fixed time

𝑡=

10000. Parameter values:

𝜖=

0.01,

𝛼=

0.3

𝜋

𝛽=

0.23

𝜋

, and

𝑁=100. Figure taken from [BER19].

thus close to zero (uncoupled). The oscillators of the same cluster evolve in time with the same

frequencies ¤

𝜙𝑖,𝜇= Ω𝜇,𝑖=1, . . . ,𝑁𝜇.

5.2.2 Two-cluster states of splay type

Let us consider the case of two-clusters in more details. Let

𝜙𝑖,𝜇(𝜇=1, 2)

with

𝑁1

and

𝑁2

being

the numbers of oscillators in cluster 1 and 2, respectively. The following result follows from the

Proposition 5.2.1.

Corollary 5.2.2 Suppose 𝑅2(a𝜇)=0for 𝜇=1, 2, then

𝜙𝑖,1 = Ω1𝑡+𝑎𝑖,1,𝑖=1, . . . ,𝑁1

𝜙𝑖,2 = Ω2𝑡+𝑎𝑖,2,𝑖=1, . . . ,𝑁2

𝜅𝑖 𝑗,𝜇𝜇 =−sin(𝑎𝑖,𝜇−𝑎𝑗,𝜇+𝛽),𝜇=1, 2

𝜅𝑖 𝑗,𝜇𝜈 =−𝜌𝜇𝜈 sin(ΔΩ𝜇𝜈𝑡+𝑎𝑖,𝜇−𝑎𝑗,𝜈+𝛽−𝜓𝜇𝜈),𝜇,𝜈=1, 2; 𝜇≠𝜈

64 5 Multicluster states in adaptive networks of coupled phase oscillators

Figure 5.4:

The figures show all one- and two-cluster solutions of splay type for the system

(5.1)

–

(5.2)

. For this, the

frequency differences

ΔΩ12

are displayed corresponding to the equations (4.6) and (5.7). The dotted lines (black)

indicate unstable solutions while the solid lines (blue) indicate stable solutions. Here, every second solution is plotted

for the sake of visibility. Parameter values: (a)

𝑁=

20,

𝜖=

0.01; (b)

𝑁=

50,

𝜖=

0.01; (c)

𝑁=

50,

𝜖=

0.001; (d)

𝑁=

50,

𝜖=0.1; 𝛼=0.3𝜋is fixed for all panels. Figure taken from [BER19].

is a two-cluster solution of system (5.1)–(5.2)with

(ΔΩ12)1,2 =1

2𝑛1−1

2cos(𝛼−𝛽) ± 1

2s𝑛1−1

22

cos2(𝛼−𝛽)−2𝜖(2𝜖+sin(𝛼−𝛽)), (5.7)

Ω𝜇=1

2𝑛𝜇cos(𝛼−𝛽) + 𝜌𝜇𝜈𝑛𝜈cos(𝛼−𝛽+𝜓𝜇𝜈),(𝜇,𝜈=1, 2; 𝜇≠𝜈), (5.8)

where 𝑛𝜇=𝑁𝜇/𝑁and 𝜓𝜇𝜈,𝜌𝜇𝜈 as in Proposition 5.2.1.

The explicit expressions for the frequencies

ΔΩ12

Ω1,2

and other parameters of the solutions

follow from the system of equations (5.6), which can be solved explicitly leading to (5.7)–(5.8) for

𝑀=

2. For any given relative cluster size

𝑛𝜇

, equations (5.7) and (5.8) provide either two, one, or

no solutions corresponding to the two-cluster solution. Hence, for each fixed set of parameters,

there may be up to 2(𝑁−4)such solutions.

Figure 5.4 shows the frequency differences

ΔΩ12

of these solutions as functions of parameter

𝛽

for different number of oscillations

𝑁

and adaptation parameters

𝜖

. Interestingly, the frequencies

of the solutions depend only on the difference

𝛼−𝛽

, see

(5.7)

–

(5.8)

. By increasing the number

of oscillators

𝑁

in the system, the number of solutions increases accordingly. This can be seen

from Fig. 5.4(a–b) where we increase the number of oscillators from

𝑁=

20 to

𝑁=

50 with

all other parameters fixed. The set of 2-cluster solutions is represented by all

ΔΩ12(𝛽)

for a

given parameter

𝛽

. In accordance with

(5.7)

the number of solutions increases with increasing

5.2 Splay type multicluster states 65

𝑁

. The region of non-existence of the multicluster solutions corresponds to the cases where the

argument beneath the root in

(5.7)

becomes negative. The size of the existence gap depends

furthermore on the choice of the time separation parameter

𝜖

. This can be seen by comparing

Fig. 5.4(b–d) where we vary the value for 𝜖.

5.2.3 Adaptation rate dependence for the emergence of two-cluster states

In this section we show the importance of the time-separation parameter

𝜖

for the appearance of

multicluster states. In particular, we obtain the critical value

𝜖𝑐

above which the multicluster

states cease to exist.

It is quite remarkable that in case of splay-type clusters the multicluster solution can be explicitly

given as it was shown in the Proposition 5.2.1 and Corollary 5.2.2. With this, we can study

directly the role of several parameters for the existence of multicluster states. In Figure 5.4(b-d),

solutions for Eq.

(5.7)

are presented depending on the parameter

𝛽

. The number of oscillators

in the system is chosen as

𝑁=

50. Each line

ΔΩ12(𝛽)

in Fig. 5.4(b-d) represents a frequency

difference of two clusters for which the two-cluster state of splay type exists with fixed relative

number

𝑛1

of oscillators in the first cluster. Note that the number of possible two-cluster states

increases proportionally to the total number of oscillators

𝑁

. Different panels show solutions for

different values of

𝜖

. We note that the existence of those two-cluster states depends only on the

difference of

𝛾:=𝛼−𝛽

, see Eq.

(5.7)

. The necessary condition for the existence of a two-cluster

state reads 𝑛1−1

22

cos2𝛾 > 2𝜖(2𝜖+sin 𝛾). (5.9)

From Eq.

(5.9)

we immediately see that the value of the time separation parameter

𝑒𝑝𝑠𝑖𝑙𝑜𝑛

in system

(5.1)

–

(5.2)

is important for the existence of the multicluster states. This dependence

is in contrast to the findings for one-cluster states. First of all, note that the left hand side of

condition

(5.9)

is positive for

𝛾≠±𝜋/

2. Hence, for all parameters, there is a critical value

𝜖𝑐

such

that there exists no two-cluster state for 𝜖 > 𝜖𝑐. Explicitly, we have

𝜖𝑐=−1

4sin 𝛾+1

2s1

4sin2𝛾+𝑛1−1

22

cos2𝛾, (5.10)

which is illustrated in Fig. 5.5. The figure shows the critical value

𝜖𝑐

depending on the parameter

𝛾

for different values of

𝑛1

. The function possesses a global maximum with

𝜖𝑐=

0.5. This means

that there is a particular requirement on the time separation in order to have two-cluster states of

splay type. Indeed, the adaptation of the network has to be at most half as fast as the dynamics

of the oscillatory system.

Further let us remark that the two-cluster state with equally sized clusters

𝑛1=

0.5 exists only

for 𝛼−𝛽∈ (𝜋, 2𝜋), i.e., 𝜖𝑐=0 for all 𝛼−𝛽∈ [0, 𝜋].

In the subsequent sections we discus that the combination of one-cluster states to a multicluster

state can result in modulated dynamics of the oscillators additional to the linear growth. In fact,

this additional temporal behavior is due to the interaction of the clusters. As we see in Fig. 5.2(a),

66 5 Multicluster states in adaptive networks of coupled phase oscillators

γ/π

c

n1= 0.5

n1= 0.7

n1= 0.9

Figure 5.5:

For the case of two-cluster states of splay type, the critical value

𝜖𝑐

of time-separation parameter

𝜖

plotted as a function of

𝛾=𝛼−𝛽

for different cluster sizes

𝑛1=𝑁1/𝑁

. The function is given explicitly by Eq.

(5.10)

Figure taken from [BER19a].

oscillators interacting with a splay-type cluster will not be forced to perform additional dynamics.

This is the reason why we are able to derive a closed analytic expression for multicluster states

of splay type. It is possible to determine the frequencies explicitly as in Eq.

(5.7)

. Therefore, here

the interaction between cluster causes only changes in the collective frequencies which are small

whenever 𝜖is small.

In contrast to splay clusters, the interaction with antipodal cluster leads to bounded modulation

of the oscillator dynamics besides the constant-frequency motion. The modulations scale with

𝜖

and hence depend on the time-separation parameter. In case of a mixed type two-cluster state,

both interaction phenomena are present. The oscillators in the antipodal cluster interact with the

splay cluster leading to no additional modulation. On the contrary, the phase oscillators of the

splay cluster get additional modulation via the interaction with the antipodal cluster.

In the following Section 5.3, we provide a general result on the asymptotic existence of splay,

antipodal, and mixed type multicluster states. In the Section 5.4 and 5.5, this general result will

be applied to give explicit expression for antipodal as well as mixed type multiclusters.

5.3 Conditions for the emergence of multicluster states - A

generalized approach

In this section we give an analytic description of multicluster solutions in terms of an asymptotic

expansion. We consider therefore the expansion of

𝑟

-th order together with a multi-time scale

ansatz [VER06]

𝜙(𝑟)

𝑖,𝜇(𝜖,𝑡):= Ω(𝑟)

𝜇(𝜏0,. . . ,𝜏𝑟)+𝑎𝑖,𝜇+

𝑟

𝑙=1

𝜖𝑙𝑝𝑖,𝜇;𝑙(𝑡)

𝜅(𝑟)

𝑖 𝑗,𝜇𝜈 (𝜖,𝑡):=

𝑟

𝑙=0

𝜖𝑙𝑘𝑖 𝑗,𝜇𝜈;𝑙(𝑡)

𝜇,𝜈=1, . . . ,𝑀

𝑖,𝑗=1, . . . ,𝑁𝜇

(5.11)

5.3 Conditions for the emergence of multicluster states - A generalized approach 67

where

Ω(𝑟)

𝜇∈𝐶1(R𝑟+1)

is a function depending on the multi-time scales

𝜏𝑙:=𝜖𝑙𝑡

. We show under

under which conditions this expansion describes the time evolution for the system (5.1)–(5.2).

The main result on the asymptotic expansion for (pseudo) multicluster solutions reads as follows.

Proposition 5.3.1

Let

𝑟∈N

. Suppose the system (5.1)– (5.2) possesses a (pseudo) multicluster solu-

tion

(𝜙𝑖,𝜇

𝜅𝑖 𝑗,𝜇𝜈)

with

𝜙𝑖,𝜇(𝜖

𝑡)= Ω𝜇(𝜖)𝑡+𝑎𝑖,𝜇+𝑠𝑖,𝜇(𝜖

𝑡)

where

𝑎𝑖,𝜇∈T1

and the coupling matrix

𝜅𝑖 𝑗,𝜇𝜈 (𝜖

𝑡)

is given as the parametrization of the pullback attractor defined in (A.14). Assume further

that

𝑀1

clusters are of antipodal type (2

𝑎𝑖,𝜇=𝑎𝜇

) and

𝑀2

are of splay type (

𝑅2(a𝜇)=

0). Then, the

𝑟

-th

order asymptotic expansion of 𝜙𝑖,𝜇(𝜖,𝑡)for 𝑡∈𝑂(1/𝜖𝑟)as 𝜖→0is given by

𝜙(𝑟)

𝑖,𝜇(𝜖,𝑡):= Ω(𝑟)

𝜇,0𝑡+𝑎𝑖,𝜇+

𝑟

𝑙=1

𝜖𝑙Ω(𝑟)

𝜇,𝑙𝑡+𝑝𝑖,𝜇;𝑙(𝑡)𝜇,𝜈=1, . . . ,𝑀(5.12)

𝜅(𝑟)

𝑖 𝑗,𝜇𝜈 (𝜖,𝑡):=

𝑟

𝑙=0

𝜖𝑙𝑘𝑖 𝑗,𝜇𝜈;𝑙(𝑡),𝑖=1, . . . ,𝑁𝜇

𝑗=1, . . . ,𝑁𝜈

(5.13)

where

(i) all coefficients of the expansion can be found inductively;

(ii) the first order approximation can be written as

𝜙(1)

𝑖,𝜇=©«

𝑛𝜇

2(cos(𝛼−𝛽) −cos(𝛼+𝛽))−𝜖

𝑀

𝜈=𝑀1+1

𝑛𝜈

2ΔΩ(1)

𝜇𝜈

sin(𝛼−𝛽)ª®¬𝑡+𝑎𝑖,𝜇+𝜖 𝑝𝜇;1

for 𝜇=1, . . . ,𝑀1, and

𝜙(1)

𝑖,𝜇=©«

𝑛𝜇

2cos(𝛼−𝛽) −𝜖

𝑀

𝜈=𝑀1+1

𝜈≠𝜇

𝑛𝜈

2ΔΩ(1)

𝜇𝜈

sin(𝛼−𝛽)ª®®®¬

𝑡+𝑎𝑖,𝜇+𝜖 𝑝𝑖,𝜇;1(𝑡)

for 𝜇=𝑀1+1, . . . ,𝑀with

𝑝𝜇;1 =−

𝑀1

𝜈=1

𝜈≠𝜇

𝑛𝜈

4ΔΩ(1)

𝜇𝜈 2cos(2ΔΩ(1)

𝜇𝜈 𝑡+𝑎𝜇−𝑎𝜈+𝛼+𝛽)𝜇=1, . . . ,𝑀1

𝑝𝑖,𝜇;1 =−

𝑀1

𝜈=1

𝑛𝜈

4ΔΩ(1)

𝜇𝜈 2cos(2ΔΩ(1)

𝜇𝜈 𝑡+2𝑎𝑖,𝜇−𝑎𝜈+𝛼+𝛽);𝜇=𝑀1+1, . . . ,𝑀

the coupling weights are given by

𝜅(1)

𝑖 𝑗,𝜇𝜇 =−sin(𝑎𝑖,𝜇−𝑎𝑗,𝜇+𝛽),

𝜅(1)

𝑖 𝑗,𝜇𝜈 =𝜖

ΔΩ(1)

𝜇𝜈

cos(ΔΩ(1)

𝜇𝜈 𝑡+𝑎𝑖,𝜇−𝑎𝑗,𝜈+𝛽);

68 5 Multicluster states in adaptive networks of coupled phase oscillators

and the cluster frequencies Ω(1)

𝜇solve the system of equations

Ω(1)

𝜇=©«

Ω𝜇;0 −𝜖

𝑀

𝜈=1

𝜈≠𝜇

𝑛𝜈

2ΔΩ(1)

𝜇𝜈

sin(𝛼−𝛽)ª®®¬

with

Ω𝜇;0 =𝑛𝜇sin(𝛼)sin(𝛽)𝜇=1, . . . ,𝑀1

Ω𝜇;0 =𝑛𝜇

2cos(𝛼−𝛽).𝜇=𝑀1+1, . . . ,𝑀

where 𝜇=1, . . . ,𝑀,𝑖,𝑗=1, . . . ,𝑁𝜇and ΔΩ(1)

𝜇𝜈 := Ω(1)

𝜇−Ω(1)

𝜈.

The proof makes use of several lemmas and is presented in the appendix A.4. Overall, we aim

to describe the following particular form for the dynamical behaviour of the phase oscillators.

The phases of the oscillators

𝜙𝑖,𝜇

form a pseudo multicluster, c.f. definition 5.5.1. Further, the

bounded modulations for the phases of each oscillator are given as Taylor expansions in

𝜖

with

periodic coefficients that can be expressed as Fourier sums with even modes. The strategy for

the proof of the main result is as follows.

Assume that the phases of the oscillators are given as finite Taylor sums in

𝜖

with periodic

coefficients, which are represented as finite Fourier sums. With this, the equations

(5.2)

can be explicitly solved. Introducing the pull-back attractor provides us with a unique

expression for the asymptotic solutions (

𝑡→ ∞

). An explicit form for the expansion in

𝜖

these solutions of the coupling weights 𝜅is given in Lemma A.4.2.

The solutions of the coupling weights depend on the Fourier modes of the periodic

expansion coefficients of the oscillators. An statement on their explicit dependence is

provided by Lemma A.4.3. More specifically, the expansion coefficients of the couplings

weights only consist of even modes whenever the expansion coefficients for the phases of

the oscillators consist only of odd modes.

The expressions for the coupling weights in Lemma A.4.2 are used to derive the explicit

form for the phase oscillators. More specifically, the expansions coefficients are derived

such that they satisfy the equations (5.1). Higher order terms which contribute to a linear

growth are absorbed in an expansion for the oscillator frequencies.

Finally, we find an iterative scheme to determine all expansion coefficients of the phases

and coupling weights up to any order. Moreover, it is shown that the coefficients provided

by the iterative scheme are consistent with the assumption on the expansion coefficients

given in the beginning of the proof.

In the following this general result is applied to antipodal and mixed multicluster in order to

characterize their properties. Note that if we apply the expansion result to splay type multicluster,

i.e.

𝑀1=

0, the expressions would agree with the Taylor expansion in

𝜖

of the explicit formulas

presented in Proposition 5.2.1.

5.4 Antipodal type multicluster states 69

5.4 Antipodal type multicluster states

In this section we apply the results presented in Proposition 5.2.1 to antipodal type multicluster

states.

5.4.1 Asymptotic conditions for the emergence of antipodal type multicluster states

In the case when the oscillators are phase synchronized or in an antiphase relation within

each cluster, the situation is different to what was described for the splay type multiclusters.

Particularly, the linear growth of the oscillator phases within each cluster is modulated by

periodic or quasi-periodic terms of order

𝜖

. That is, the clusters possess the form

𝜙𝑖,𝜇(𝑡)= Ω𝜇𝑡+

𝑎𝑖,𝜇+𝜖 𝑝𝜇(𝑡

𝜖)

. Here, we give important necessary conditions for the existence of such solutions

and their asymptotic expansion in

𝜖

. Additionally, we provide numerical results showing these

solutions. In particular, we present a system of equations for the cluster frequencies Ω𝜇.

Proposition 5.4.1

Suppose 2

𝑎𝑖,𝜇=𝑎𝜇mod

𝜋

for all

𝜇=

. . .

𝑀

and

𝑖=

. . .

𝑁𝜇

. If system

(5.1)–(5.2) possesses antipodal multicluster solution

(𝜙𝑖,𝜇

𝜅𝑖 𝑗,𝜇𝜈)

then its first asymptotic expansion in

𝜖

is given by

𝜙(1)

𝑖,𝜇= Ω(1)

𝜇𝑡+𝑎𝑖,𝜇−𝜖

𝑀

𝜈=1

𝜈≠𝜇

𝑛𝜈

4ΔΩ(1)

𝜇𝜈 2cos(2ΔΩ(1)

𝜇𝜈 𝑡+𝑎𝜇−𝑎𝜈+𝛼+𝛽),

𝜅(1)

𝑖 𝑗,𝜇𝜇 =−sin(𝑎𝑖,𝜇−𝑎𝑗,𝜇+𝛽),

𝜅(1)

𝑖 𝑗,𝜇𝜈 =𝜖

ΔΩ(1)

𝜇𝜈

cos(ΔΩ(1)

𝜇𝜈 𝑡+𝑎𝑖,𝜇−𝑎𝑗,𝜈+𝛽),𝜇≠𝜈

with the cluster frequencies

Ω(1)

𝜇

up to first order in

𝜖

whenever the following implicit equation can be

solved

Ω(1)

𝜇=©«

𝑛𝜇sin(𝛼)sin(𝛽)−𝜖

𝑀

𝜈=1

𝜈≠𝜇

𝑛𝜈

2ΔΩ(1)

𝜇𝜈

sin(𝛼−𝛽)ª®®¬

. (5.14)

Here, 𝜇=1, . . . ,𝑀,𝑖,𝑗=1, . . . ,𝑁𝜇and ΔΩ𝜇𝜈 := Ω(1)

𝜇−Ω(1)

𝜈.

This first order perturbation reveals a nonlinear modulation

𝑝𝜇

, which is periodic or quasi-

periodic with the frequencies

ΔΩ(1)

𝜇𝜈

given by the differences in the frequencies of the clusters.

Figure 5.6 shows the numerically obtained 3-cluster solution of antipodal type. The dynamics of

system

(5.1)

–

(5.2)

is shown after a sufficiently long transient so that it represents dynamically

stable solution (more on stability in Section 5.6). One can clearly observe three clusters in the

coupling matrix. Similarly to the previous multicluster cases, we first sort the oscillators with

respect to their average frequency and subsequently by their phases. In contrast to the splayed

distribution of the phases described in Section 5.2, the oscillators within the clusters additionally

form two groups, in which the phases differ by 𝜋.

70 5 Multicluster states in adaptive networks of coupled phase oscillators

Figure 5.6:

Three-cluster of antipodal type. (a) Coupling weights at

𝑡=

10000 showing three clusters; (b) Distribution

of the phases within each cluster; space-time raster plot; (c) Average frequency of oscillators; each plateau corresponds

to one cluster; (d) Oscillator phases

𝜙𝑖(𝑡)

at fixed time

𝑡=

10000. Parameter values:

𝜖=

0.01,

𝛼=

0.3

𝜋

𝛽=−

0.53

𝜋

, and

𝑁=100. Figure taken from [BER19].

time ¯ω

φi,µ(t)− hΩµit

S(¯ω)

(a) (b)

Cluster 1

Cluster 2

Cluster 3

Figure 5.7:

For 3-cluster solution from Fig. 5.6, panel (a) shows time series of an oscillator from one of the clusters

after subtracting the average linear growth

𝜙𝜇,𝑖(𝑡)−hΩ𝜇i𝑡

. The black dashed lines show the corresponding analytic

results from the asymptotic expansion in Proposition 5.4.1. (b) Power spectrum of the time series given in (a). Figure

modified from [BER19].

5.4 Antipodal type multicluster states 71

In order to observe the modulation of the cluster frequencies, time series for three representa-

tive oscillators from each cluster are shown in Fig. 5.7(a). The averaged linear growth of the

phases due to

hΩ𝜇i𝑡

has been subtracted to show the modulation. Such small but non-vanishing

oscillations do not exist in the case of splay type multiclusters. In addition, the black dashed

lines show the modulation given by Proposition 5.4.1 confirming that the asymptotic expansion

describes the whole temporal behaviour very well. Furthermore, Proposition 5.4.1 implies that

the amplitudes of the modulations are proportional to

𝑛𝜈/(ΔΩ(1)

𝜇𝜈 )2

. Thus, if the difference in the

frequencies is high the amplitude is small and vice versa. This relation is also reflected by the

power spectrum, see Fig. 5.7(b). Figure 5.7(b) confirms that the frequencies of the modulation

oscillations correspond to the differences of the average frequencies.

5.4.2 Two-cluster states of antipodal type

Let us consider the case of two clusters in more details. Let

𝑁1

and

𝑁2=𝑁−𝑁1

be the numbers

of oscillators in group 1 and 2, respectively. The following result follows from Proposition 5.4.1.

Corollary 5.4.2

Suppose 2

𝑎𝑖,𝜇=𝑎𝜇

for all

𝜇=

1, 2 and

𝑖=

. . .

𝑁𝜇

. If system (5.1)–(5.2) possesses an

antipodal multicluster solution (𝜙𝑖,𝜇,𝜅𝑖 𝑗,𝜇𝜈)then its first order asymptotic expansion in 𝜖is given by

𝜙(1)

𝑖,1 = Ω(1)

1𝑡+𝑎𝑖,1 −𝜖𝑛2

4ΔΩ(1)

12 2cos(2ΔΩ(1)

12 𝑡+𝑎1−𝑎2+𝛼+𝛽),

𝜙(1)

𝑖,2 = Ω(1)

2𝑡+𝑎𝑖,2 −𝜖𝑛1

4ΔΩ(1)

12 2cos(2ΔΩ(1)

12 𝑡+𝑎1−𝑎2−𝛼−𝛽),

𝜅(1)

𝑖 𝑗,𝜇𝜇 =−sin(𝑎𝑖,𝜇−𝑎𝑗,𝜇+𝛽),

𝜅(1)

𝑖 𝑗,𝜇𝜈 =𝜖

ΔΩ(1)

𝜇𝜈

cos(ΔΩ(1)

𝜇𝜈 𝑡+𝑎𝑖,𝜇−𝑎𝑗,𝜈+𝛽),

with

Ω(1)

𝜇= 𝑛𝜇sin(𝛼)sin(𝛽)−𝜖𝑛𝜈

2ΔΩ(1)

𝜇𝜈

sin(𝛼−𝛽)!

for 𝜇=1, 2,𝜈≠𝜇,𝑖,𝑗=1, . . . ,𝑁𝜇,ΔΩ(1)

𝜇𝜈 := Ω(1)

𝜇−Ω(1)

𝜈,

ΔΩ(1)

12 1,2

=𝑛1−1

2sin(𝛼)sin(𝛽)±s𝑛1−1

22

sin2𝛼sin2𝛽−𝜖

2sin(𝛼−𝛽). (5.15)

This result follows directly from Proposition 5.4.1. It shows, in particular, that the system of

equations (5.14) can be solved explicitly by (5.15) in case of two clusters.

Similarly to the splay multiclusters, for any fixed set of parameters and each

𝑛1

, equation (5.15)

can lead to two antipodal multiclusters with two different frequency differences. Hence, a

large number of antipodal two-clusters can coexist for the same parameter values. Figure 5.8

illustrates such a coexistence, where we present the one-cluster solutions given by (4.6) and

72 5 Multicluster states in adaptive networks of coupled phase oscillators

∆Ω12

(a)

(b)

β/π β/π

Figure 5.8:

Two-cluster solutions (upper panels) and one-cluster solutions (lower panels) of antipodal type given

by the asymptotic expansion in Corollary 5.4.2 and Proposition 4.1.1, respectively. For this, the difference of the

frequencies

ΔΩ(1)

is displayed corresponding to (5.15) and (4.6). The dotted lines (black) indicate unstable solutions

while the solid lines (blue) indicate stable solutions. Here, every second solution is plotted for the sake of visibility.

The insets show a blow-up of the interval

[−𝜖

𝜖]

. Parameter values: (a)

𝑁=

20,

𝜖=

0.01; (b)

𝑁=

50,

𝜖=

0.01;

𝛼=

0.3

𝜋

is fixed for all panels. Figure taken from [BER19].

the solutions to the equation (5.15). Blue solid lines represent those solutions for which the

asymptotic expansion led to an existing and stable two-cluster solutions of antipodal type. Note

further that for two-cluster solutions of antipodal type, the asymptotic expansion presented in

Proposition 5.4.1 turns into a formal expansion whenever

|ΔΩ|> 𝜖

, i.e.,

𝜖

is not assumed to be

infinitesimal (𝜖→0). The interval [−𝜖,𝜖]is therefore highlighted in Fig. 5.8.

5.5 Mixed type pseudo-multicluster states

In this section we apply the results presented in Proposition 5.2.1 to mixed type multicluster

states. In order to this the notion of multiclusters has to be slightly adjusted which leads to the

definition of pseudo multicluster states.

5.5.1 Asymptotic conditions for the emergence of mixed type pseudo-multicluster

states

We have seen how clusters are described consisting of oscillator groups of splay type (Section 5.2)

as well as clusters consisting of oscillator groups with in- and anti-phase relation (Section 5.4). It

is therefore reasonable to ask for multicluster solutions that consist of both of these types. In

order to describe these solutions we have to loosen the definition of a multicluster solution.

Definition 5.5.1

Phase oscillators

𝜙𝑖(𝑡)

form a

pseudo multicluster

if they can be separated into

𝑀

groups such that for all

𝜇∈ {

. . .

𝑀}

the phase oscillators

𝜙𝑖,𝜇

𝑖∈ {

. . .

𝑁𝜇}

, from each group

𝜇

satisfy 𝜙𝑖,𝜇(𝑡)= Ω𝜇𝑡+𝑠𝑖,𝜇(𝑡)with bounded functions 𝑠𝑖,𝜇.

Note that every multicluster solution is by definition already a pseudo multicluster solution.

5.5 Mixed type pseudo-multicluster states 73

Proposition 5.5.1

Suppose 2

𝑎𝑖,𝜇=𝑎𝜇

for all

𝜇=

. . .

𝑀1

, and

𝑅2(a𝜇)=

0for all

𝜇=𝑀1+

. . .

𝑀

𝑖=

. . .

𝑁𝜇

where

𝑀1

is the number of antipodal type clusters. The mixed pseudo multicluster solutions

of (5.1)–(5.2) with

𝜙𝑖,𝜇(𝑡)= Ω𝜇(𝜖)𝑡+𝑠𝑖,𝜇(𝑡)+𝑎𝑖,𝜇

possess the following first order asymptotic expansion

in 𝜖

𝜙(1)

𝑖,𝜇= Ω(1)

𝜇𝑡+𝑎𝑖,𝜇+𝜖 𝑝𝑖,𝜇;1(𝑡),

𝜅(1)

𝑖 𝑗,𝜇𝜇 =−sin(𝑎𝑖,𝜇−𝑎𝑗,𝜇+𝛽),

𝜅(1)

𝑖 𝑗,𝜇𝜈 =𝜖

ΔΩ(1)

𝜇𝜈

cos(ΔΩ(1)

𝜇𝜈 𝑡+𝑎𝑖,𝜇−𝑎𝑗,𝜈+𝛽),

with

𝑝𝑖,𝜇;1(𝑡)=𝑝𝜇;1 =−

𝑀1

𝜈=1

𝜈≠𝜇

𝑛𝜈

4ΔΩ(1)

𝜇𝜈 2cos(2ΔΩ(1)

𝜇𝜈 𝑡+𝑎𝜇−𝑎𝜈+𝛼+𝛽)

for 𝜇=1, . . . ,𝑀1,

𝑝𝑖,𝜇;1(𝑡)=−

𝑀

𝜈=1

𝜈≠𝜇

𝑛𝜈

4ΔΩ(1)

𝜇𝜈 2cos(2ΔΩ(1)

𝜇𝜈 𝑡+𝑎𝑖,𝜇−𝑎𝜈+𝛼+𝛽)

for

𝜇=𝑀1+

. . .

𝑀

, and the cluster frequencies

Ω(1)

𝜇

up to second order in

𝜖

whenever the following

system of equations can be solved

Ω(1)

𝜇=©«

Ω𝜇;0 −𝜖

𝑀

𝜈=1

𝜈≠𝜇

𝑛𝜈

2ΔΩ(1)

𝜇𝜈

sin(𝛼−𝛽)ª®®¬

(5.16)

with

Ω𝜇;0 =𝑛𝜇sin(𝛼)sin(𝛽)𝜇=1, . . . ,𝑀1

Ω𝜇;0 =𝑛𝜇

2cos(𝛼−𝛽).𝜇=𝑀1+1, . . . ,𝑀

Here, 𝜇=1, . . . ,𝑀,𝑖,𝑗=1, . . . ,𝑁𝜇and ΔΩ(1)

𝜇𝜈 := Ω(1)

𝜇−Ω(1)

𝜈.

5.5.2 Pseudo-two-cluster states of mixed type

As in the previous sections, we are going to show that equation (5.16) possesses solutions. For

this, we consider the case of two clusters 𝜙𝑖,𝜇(𝜇=1, 2).

Corollary 5.5.2

Suppose 2

𝑎𝑖,1 =𝑎1

for all

𝑖=

. . .

𝑁1

and

𝑅(a2)=

0. The mixed pseudo multiclusters

74 5 Multicluster states in adaptive networks of coupled phase oscillators

of system (5.1)–(5.2) possess the following first order asymptotic expansion in 𝜖

𝜙(1)

𝑖,1 = Ω(1)

1𝑡+𝑎𝑖,1,

𝜙(1)

𝑖,2 = Ω(1)

2𝑡+𝑎𝑖,2 −𝜖𝑛1

4ΔΩ(1)

12 2cos(2ΔΩ(1)

12 𝑡+𝑎1−𝑎𝑖,2 −𝛼−𝛽),

𝜅(1)

𝑖 𝑗,𝜇𝜇 =−sin(𝑎𝑖,𝜇−𝑎𝑗,𝜇+𝛽),

𝜅(1)

𝑖 𝑗,𝜇𝜈 =𝜖

ΔΩ(1)

𝜇𝜈

cos(ΔΩ(1)

𝜇𝜈 𝑡+𝑎𝑖,𝜇−𝑎𝑗,𝜈+𝛽),

where

Ω(1)

𝜇= Ω𝜇;0 −𝜖𝑛𝜈

2ΔΩ(1)

𝜇𝜈

sin(𝛼−𝛽)!,

Ω1;0 =𝑛1sin(𝛼)sin(𝛽),

Ω2;0 =𝑛2

2cos(𝛼−𝛽),

ΔΩ(1)

12 1,2

2𝑛1−1

2cos(𝛼−𝛽) − 𝑛1

2cos(𝛼+𝛽)

±1

2s𝑛1−1

2cos(𝛼−𝛽) − 𝑛1

2cos(𝛼+𝛽)2

−2𝜖sin(𝛼−𝛽)

(5.17)

for 𝜇=1, 2,𝜈≠𝜇and 𝑖,𝑗=1, . . . ,𝑁𝜇.

Illustration of the mixed 2-clusters is shown in Fig. 5.9.

Moreover, we performed a Fourier analysis of the temporal behaviour of the oscillators, see

Fig. 5.10. First, it can be observed that the oscillators representing the second cluster (

𝑖=

10, 30) show the same evolution in time but with a phase lag due to the spatial dependency

described above. In order to show the agreement with the asymptotic expansion presented in

Corollary 5.5.2, the analytic results are displayed with black dashed lines. Furthermore, the

power spectrum shows a prominent peak at 2

hΔΩi12

for both oscillators of the second cluster

and a flat curve for the representative of the first cluster. These numerical results are in complete

agreement with the analytic findings.

Analogously, to the antipodal two-clusters, for any fixed set of parameters and each

𝑛1

, equa-

tion (5.17) can lead to two multiclusters of mixed type with two different frequency differences.

Hence, a large number of those clusters can coexist for the same parameter values. Figure 5.11

illustrates such a coexistence, where we present the solutions to the equation (5.17). Again,

blue solid lines represent those solutions for which the asymptotic expansion led to an existing

and stable two-cluster solutions of mixed type. Additionally, Figure 5.11 shows the one-cluster

solutions of splay and antipodal type (in both cases

ΔΩ12 =

0) together with their common

regions of stability. As in the case of two-clusters of antipodal type, the asymptotic expansion

presented in Proposition 5.4.1 turns into a formal expansion whenever

|ΔΩ|> 𝜖

. The interval

[−𝜖,𝜖]is therefore highlighted in Fig. 5.11.

5.5 Mixed type pseudo-multicluster states 75

Figure 5.9:

2-Cluster solution of mixed type. (a) Coupling weights at

𝑡=

10000 showing two clusters, (b) Distribution

of the phases within each cluster, space-time representation. (c) Average frequency of each oscillator, (d) Oscillator

phases

𝜙𝑖

for fixed time

𝑡=

10000. Parameter values:

𝜖=

0.01,

𝛼=

0.3

𝜋

𝛽=−

0.4

𝜋

𝑁=

100. Figure taken from [

BER19

time ¯ω

φi,µ(t)− hΩµit

S(¯ω)

(a) (b)

Cluster 1

Cluster 2 (i= 10)

Cluster 2 (i= 30)

Figure 5.10:

For mixed type 2-cluster solution from Fig. 5.9, panel (a) shows time series of an oscillator from one of the

clusters after subtracting the average linear growth

𝜙𝜇,𝑖(𝑡)−hΩ𝜇i𝑡

. The black dashed lines show the corresponding

analytic results from the asymptotic expansion in Proposition 5.5.1. (b) Power spectrum of the time series given in (a).

Figure moified from [BER19].

76 5 Multicluster states in adaptive networks of coupled phase oscillators

Figure 5.11:

Two-cluster solutions of mixed type (upper panels) and one-cluster solutions (lower panels) of either

splay or antipodal type given by the asymptotic expansion in Corollary 5.5.2 and Proposition 4.1.1, respectively. For

this, the difference of the frequencies

ΔΩ(1)

is displayed corresponding to (5.17) and (4.6). The dotted lines (black)

indicate unstable solutions while the solid lines (blue) indicate stable solutions. Here, every second solution is plotted

for the sake of visibility. The insets show a blow-up of the interval

[−𝜖

𝜖]

. Parameter values: (a)

𝑁=

20,

𝜖=

0.01; (b)

𝑁=50, 𝜖=0.01; 𝛼=0.3𝜋is fixed for all panels. Figure taken from [BER19].

5.6 Stability of multicluster states

In Section 5.2 we discussed multicluster solutions of splay type and showed under which

condition they exist. The solutions for two-cluster solutions of splay type and their stability

are presented in Fig. 5.4. In Fig. 5.4(b) the solution for the case of 50 oscillators is shown. The

solid lines (blue) correspond to solutions that are stable. It can be seen that whenever a 2-cluster

solution is stable the one-cluster solution (with ΔΩ12 =0) is also stable.

A more detailed validation of this statement is presented in Fig. 5.12, where we show the stability

regions of both one- and two-cluster solutions in the

(𝛼

𝛽)

plane. The stability for each type of

cluster solution is determined numerically. The numerical approach was already introduced in

Section 4.2. For the two-cluster solutions the norm for the phase configuration is calculated in

the relative coordinates given by

Θ𝑖,𝜇=𝜙𝑖,𝜇−𝜙1,𝜇

with

𝜇=

1, 2. Additionally, we calculated the

maximal value of all inter-cluster connections and compared it to the theoretical maximum given

𝜌12

. If after numerical integration the maximal inter-cluster coupling is bigger than

𝜌12 +

0.01,

the two-cluster is considered as unstable. Here, region where the both types of solutions are

stable are colored in dark blue. Regions of only stable one-cluster solutions are colored in light

blue. Since two-cluster solutions do not exist for certain values of

𝛼

and

𝛽

, we can find a light

blue stripe in the middle of Fig. 5.12. Further, we have not found any configuration of

𝛼

and

𝛽

for which two-cluster solutions are stable and one-cluster are not. This supports the claim

that the stability of a one-cluster solution is necessary condition for the stability of a two-cluster

solution. This can be explained by the fact that for the stability of the multiclusters, it is necessary

that its one-cluster components are each stable with respect to the perturbations that disturb

the structure of just one cluster (see similarly in [

LUE12a

]). A more rigorous formulation of this

issue is beyond the scope of this thesis.

Figure 5.4(b) further provides us with information about the stability of two-cluster solutions

depending on the ratio between cluster sizes. First, due to (5.7) there exist two branches of

5.6 Stability of multicluster states 77

β/π

α/π

Figure 5.12:

Stability diagram for the one-cluster and two-cluster solution of the splay type depending on the

parameters

𝛼

and

𝛽

. Yellow region corresponds to the instability of both solutions, dark blue to the stability of both

solutions, and light-blue to the stability of only the one-cluster solution. Parameter

𝜖=

0.01 is fixed in all simulations.

Figure taken from [BER19].

two-cluster solution of splay type. Only solutions with higher frequency difference are stable

which can be seen in the inset of Fig. 5.4(b). For an increasing number of oscillators in the second

cluster of relative size

𝑛2=

−𝑛1

the stability changes. Above a certain value of

𝑛2

both branches

are unstable. This observation explains why only multicluster solutions with unequal as well as

hierarchical cluster sizes were found in simulations, see Fig. (5.3) and [KAS17].

5.6.1 On the stability of multicluster states with evenly sized clusters

In the following, we give a necessary condition for the stability of all one-cluster states of splay

type complementing the result on rotating-wave states given in Proposition 4.2.2. This extension

enables us to determine the stability of evenly sized multiclusters of splay type.

In general all splay one-cluster states have the property

𝑅2(𝒂)=

0 for the phase given by the

vector

𝒂

. Therefore, the splay states form

𝑁−

2 dimensional family of solution. Hence, around

each splay states there are

𝑁−

2 neutral variational directions

(𝛿𝝓,𝛿𝜅)𝑇

which are determined by

the condition Í𝑁

𝑗=1𝑒i2𝑎𝑗𝛿𝜙 𝑗=0. Note, 𝛿𝜅𝑖 𝑗 =−cos(𝑎𝑖−𝑎𝑗+𝛽)𝛿𝜙𝑖−𝜙𝑗in neutral direction.

Proposition 5.6.1

Consider an asymptotically stable one-cluster state of splay type. Then,

𝜖+sin(𝛼−

𝛽)/2>0.

Proof.

For some notation we refer the reader to appendix A.2. Due to the block form of the lin-

earised equation

(A.6)

and the Schur decomposition

(4.11)

, any eigenvalue comes with a second.

We have already seen this in Lemma A.2.1 and Proposition A.2.3. Variation along the neutral

direction gives

𝑁−

2 times the eigenvalue 0. Suppose we have

𝛿𝝓

such that

Í𝑁

𝑗=1𝑒i2𝑎𝑗𝛿𝜙 𝑗=

and 𝛿𝜅𝑖 𝑗 =−cos(𝑎𝑖−𝑎𝑗+𝛽)𝛿𝜙𝑖−𝜙𝑗. Applying Schur decomposition (4.11), we get

(𝑀−𝜆I)𝑁2+𝑁 𝛿𝝓

𝛿𝜅!= I𝑁−(𝜖+𝜆)𝐵

0I𝑁2! (𝐴−𝜆I𝑁)+ 1

𝜖+𝜆𝐵𝐶 0

0−(𝜖+𝜆)! 𝛿𝝓

−1

𝜖+𝜆𝐶𝛿𝝓+𝛿𝜅!=0.

(5.18)

78 5 Multicluster states in adaptive networks of coupled phase oscillators

With this, we have to find

𝜆

such that the last equality in

(5.18)

is fulfilled. This is equivalent

to solving

((𝐴−𝜆I𝑁)(𝜖+𝜆) +𝐵𝐶)𝛿𝝓=

0 of which in general only

𝑁−

2 equations are linearly

independent. The equivalence can be seen by multiplying

𝜖+𝜆

from both sides and keeping

in mind that

𝛿𝜅

is already determined by

𝛿𝝓

. Using the definition of

𝛿𝝓

the matrices

𝐴

and

𝐵𝐶

can be effectively reduced in such a way that they are independent of the actual values for the

phases 𝑎𝑗. In fact,

𝑎𝑖 𝑗 =(−𝑁−1

2𝑁sin(𝛼−𝛽)𝑖=𝑗

2𝑁sin(𝛼−𝛽),𝑖≠𝑗

(𝑏𝑐)𝑖 𝑗 =(𝜖𝑁−1

2𝑁sin(𝛼−𝛽),𝑖=𝑗

−𝜖

2𝑁sin(𝛼−𝛽).𝑖≠𝑗

In turn, this gives

((𝐴−𝜆I𝑁)(𝜖+𝜆) +𝐵𝐶)

a circulant structure which can be used to diagonalise

the matrix, in analogy to Proposition A.2.3. For circulant matrices we immediately know the

eigenvalues. They are

𝜇𝑙=−𝜆2− 𝑁−1

2𝑁sin(𝛼−𝛽) − 1

2𝑁sin(𝛼−𝛽) 𝑁−1

𝑘=0

𝑒i2𝜋𝑘𝑙/𝑁−1!+𝜖!𝜆

with

𝑙=

. . .

𝑁−

1 and

det ((𝐴−𝜆I𝑁)(𝜖+𝜆) +𝐵𝐶)=𝜇0(𝜆)···𝜇𝑁−1(𝜆)

. Remember we have in

general

𝑁−

2 independent equations. Thus, solving

𝜇𝑙(𝜆)=

0 for

𝜆

results in

𝑁−

2 eigenvalues

𝜆=

0, 1 eigenvalue

𝜆=−𝜖

and

𝑁−

3 eigenvalues

𝜆=−𝜖−sin(𝛼−𝛽)/

2. Note that for 4-phase-

cluster states, as considered in Corollary A.2.5, the number of independent equations is

𝑁−

This is due to the fact that in this case the equations for the imaginary and real part from

Í𝑁

𝑗=1𝑒i2𝑎𝑗𝛿𝜙 𝑗=0 agree.

Finally, we note why the equally-sized splay-clusters are not found to be stable. Indeed, from

Eq.

(5.7)

we know that 2

𝜖+sin(𝛼−𝛽)<

0 is a necessary condition to have such equally-sized

(

𝑛1=

2) clusters. However, any one-cluster splay state is unstable for 2

𝜖+sin(𝛼−𝛽)<

0 by

Proposition 5.6.1.

5.6.2 An effective approach for the stability of multicluster states

As mentioned above, the stability of one-cluster states is important for the stability of multicluster

states. For two weakly coupled clusters, the stability of one-clusters serves as a necessary

condition for the stability of the two-cluster state. In Figure 5.4 the possible two-clusters of splay

type are plotted. We notice that for

𝜖=

0.001 a stable two-cluster state exists for almost every

relative cluster size 𝑛1, while this is not true for 𝜖=0.01 and even more so for 𝜖=0.1.

Another observation from Figs. 5.4(b,c) is that the possible

𝛽

-values where the two-cluster states

can be stable mainly correspond to the

𝛽

values where the one-cluster state is stable. This is true

for small values of

𝜖

, however, a careful inspection of Fig. 5.4(d) for the case of larger

𝜖

, here

𝜖=

0.1, shows that some two-cluster states appear to be stable for a parameter region where

the corresponding one-cluster state is unstable. This can be explained as follows. According

(5.1)

, in the case of one-cluster states, the inter-cluster interactions are summed over all

𝑁

5.7 Summary 79

oscillators of the whole system. Additionally, the interactions are scaled with the factor 1

/𝑁

Therefore, the total interaction scales with 1. For two-cluster states, the inter-cluster interactions

for each individual cluster are only a sum over the

𝑁𝜇

(

𝜇=

1, 2) oscillators whereas the scaling

remains 1

/𝑁

. Hence, the total inter-cluster interaction scales with

𝑛𝜇=𝑁𝜇/𝑁

, the relative size

of the cluster. Therefore, the effective oscillatory system, when neglecting the interaction to the

other cluster, reads

d𝜙𝑖,𝜇

d𝑡=−𝑛𝜇

𝑁𝜇

𝑗=1

𝜅𝑖 𝑗,𝜇𝜇 sin(𝜙𝑖,𝜇−𝜙𝑗,𝜇+𝛼),

d𝜅𝑖 𝑗,𝜇𝜇

d𝑡=−𝜖𝜅𝑖 𝑗,𝜇𝜇 +sin(𝜙𝑖,𝜇−𝜙𝑗,𝜇+𝛽).

This system is equivalent to

(5.1)

–

(5.2)

with

𝑁𝜇

oscillators by rescaling

𝜖↦→ 𝜖/𝑛𝜇

. Thus, the

stability of the inter-cluster system has to be evaluated with respect to the rescaled effective

parameter

𝜖eff :=𝜖/𝑛𝜇

. Since

𝑛𝜇<

1 for

𝜇=

1, 2, we have

𝜖eff > 𝜖

. As we have discussed, the

stability for the one-cluster changes with increasing

𝜖

. With this, the influence of the cluster size

as well as the slight boundary shift in the regions of stability, see Fig. 5.4, can be explained.

5.7 Summary

In this chapter, we have focused on multicluster solutions in a network of adaptively coupled

phase oscillators. The multicluster states are composed of several one-clusters with distinct

frequencies. Starting from random initial conditions, our numerical simulations show two

different types of states. These are the splay and the antipodal type multicluster states. A third

mixed type multicluster state is found by specially prepared initial conditions. For all these

states the collective motion of oscillators, the shape of the network, and the interaction between

the frequency clusters is presented in detail. It turns out that the oscillators are able to form

groups of strongly connected units. The interaction between the groups is weak compared to the

interaction within the groups. The numerical analysis of multicluster states reveals the building

blocks for these states.

While the one-cluster solutions are relative equilibria of our system due to the phase-shift

symmetry, the multicluster solutions contain components with different frequencies, and, hence,

they cannot be reduced to an equilibrium by transforming into another co-rotating frame. As a

result, the study of multiclusters is more involved. However, to our surprise, we have still been

able to find an explicit form of multiclusters with the components of the splay type. Remarkably,

in addition to its ring-like spatial structure that dynamically emerges, the network behaves in

such a case (quasi-)periodically in time such that the whole solution can be interpreted as a

spatial-temporal wave.

The analysis of multicluster solutions of antipodal type is more subtle due to the modulation of

the frequency. More specifically, we look at multiclusters with bounded frequency modulation.

For these types of multiclusters, we derive an asymptotic expansion in the parameter

𝜖

that gives

explicit existence conditions. In addition, we have shown the existence of mixed multiclusters,

which consist of clusters of splay type and clusters of antipodal type. For the mixed multiclusters,

the temporal behavior within one cluster has been shown to be slightly non-identical, namely,

80 5 Multicluster states in adaptive networks of coupled phase oscillators

the oscillators possess the same averaged frequency, but they still can have a bounded quasi-

periodically modulated phase difference.

For the splay clusters we analytically show the existence of two-cluster states. Remarkably, while

the existence of the one-cluster states does not depend on the time-scale separation parameter

𝜖

, the multicluster states crucially depend on the time-scale separation. In fact, we provide an

analysis showing that there exists a critical value for the time-scale separation. Moreover, we

show that in the case of two-cluster states of splay type the adaptation of the coupling weights

must be at most half as fast as the dynamics of the oscillators. This fact is of crucial importance

for comparing dynamical scenarios induced by short-term or long-term plasticity [FRO16].

The stability of two-cluster states is analyzed numerically and presented for different values of

the time-scale separation parameter. By assuming weakly interacting clusters, we describe the

stability of the two-cluster with the help of the analysis of one-cluster states. The main messages

from this analysis are as follows: there is a high degree of coexistence of stable multiclusters

that can be reached from different initial conditions; in particular, a certain amount of imbalance

in the number of oscillators within the clusters is needed to achieve stability. This explains the

appearance of only hierarchical structures in numerical simulations. In fact, the simulations

indicate that there are no stable two-cluster states with clusters of the same size. We provide a

rigorous argument to understand this property of the system.

Moreover, the findings on multicluster solutions as they are reported in this chapter are in

very good agreement with previous results on adaptive neural networks [

POP15

CHA17a

Here, stable multicluster solutions of coherently spiking neurons with weak but time-dependent

inter-cluster coupling are reported. With this work we shed some light on these generic time-

dependent network patterns.

Part II

INTERPLAY OF ADAPTIVITY AND

CONNECTIVITY

Adaptation on nonlocally coupled ring

networks 6

In this chapter, we go beyond the model of globally coupled oscillators which were in the focus

of the Chapters 3–5. In particular, here, we study the interplay of adaptivity with a non-locally

coupled ring structure. Ring-like structures are important motifs in neural networks [

COM03

SPO11

POP11

YAN11

]. Specifically, nonlocally coupled rings where each node is coupled to

all nodes within a certain coupling range, are known to be important for systems appearing

in many applied problems and theoretical studies [

PAS95

BRE97

YAN08a

BON09

ZOU09b

HOR09b,PER10c,OME11,KAN13,OME13,YAN15a,KLI17,BUR18,OME18a].

In the subsequent sections, we consider the following system of

𝑁

adaptively coupled identical

phase oscillators [AOK09,AOK11,NEK16,KAS17]

d𝜙𝑖

d𝑡=1−1

Í𝑁

𝑗=1𝑎𝑖 𝑗

𝑁

𝑗=1

𝑎𝑖 𝑗 𝜅𝑖 𝑗 sin(𝜙𝑖−𝜙𝑗+𝛼)(6.1)

d𝜅𝑖 𝑗

d𝑡=−𝜖𝜅𝑖 𝑗 +sin(𝜙𝑖−𝜙𝑗+𝛽)(6.2)

where

𝑖

𝑗=

. . .

𝑁

𝜙𝑖∈𝑆1

are the phases of the oscillators,

𝑎𝑖 𝑗 ∈ {

0, 1

}

are the entries of the

adjacency matrix

𝐴

determining the base topology,

𝜅𝑖 𝑗 ∈ [−

1, 1

]

are slowly changing adaptive

coupling strengths, 0

< 𝜖 

1 is the rate of the adaptation, and

𝛼

𝛽

are coupling and adaptation

phase lags. Note that the natural oscillation frequency has been normalized to 1 by the rotating

coordinate frame.

The base topology given by the adjacency matrix

𝐴

determines the structure of the network,

on which the adaptation takes place. Equation (6.2) for the adaptation is used only for "active"

weights

𝜅𝑖 𝑗

corresponding to

𝑎𝑖 𝑗 =

1. Similarly, the sum in (6.1) goes over these links. Here we

consider the topology of a nonlocally coupled ring given by

𝑎𝑖 𝑗 =(1 for 0 <(𝑖−𝑗)mod 𝑁≤𝑃,

0 otherwise. (6.3)

This means that any two oscillators are coupled if their indices

𝑖

and

𝑗

are separated at most by the

coupling radius

𝑃

. The coupling Eq. (6.3) defines a nonlocal ring structure with coupling range

𝑃

to each side and two special limiting cases: local ring for

𝑃=

1 and globally coupled network for

𝑃=𝑁/

2 (if

𝑁

is even, else

𝑃=(𝑁+

2). The matrix of the form (6.3) is circulant [

GRA06

] and

has constant row sum, i.e.,

Í𝑁

𝑗=1𝑎𝑖 𝑗 =

𝑃

for all

𝑖=

. . .

𝑁

. Due to the same symmetries as they

are present in

(2.25)

–

(2.26)

, the analysis can be restricted to the parameter regions

𝛼∈ [

𝜋/

)

and 𝛽∈ [−𝜋,𝜋).

Two measures of coherence are used in this chapter: first, the

𝑛

th moment of the

𝑖

th (

𝑖=

. . .

𝑁

)

84 6 Adaptation on nonlocally coupled ring networks

complex local order parameter as given by

𝑍(𝑛)

𝑖(𝝓):=1

2𝑃

𝑁

𝑗=1

𝑎𝑖 𝑗 𝑒i𝑛𝜙𝑗=𝑅(𝑛)

𝑖(𝝓)𝑒i𝜗(𝑛)

𝑖(𝝓)(6.4)

where

𝝓=(𝜙1

. . .

𝜙𝑁)𝑇

𝑅(𝑛)

𝑖(𝝓)

is the

𝑛

th local order parameter and

𝜗(𝑛)

𝑖(𝝓)

the

𝑛

th local

mean-phase; second, the complex (global) order parameter

𝑍(𝑛)(𝝓):=1

𝑁

𝑗=1

𝑒i𝑛𝜙𝑗=𝑅(𝑛)(𝝓)𝑒i𝜗(𝑛)(𝝓)(6.5)

where

𝑅(𝑛)(𝝓)

is the

𝑛

th (global) order parameter and

𝜗(𝑛)(𝝓)

the

𝑛

th (global) mean-phase, see

also Section 2.2. Both measures are used throughout the chapter to characterize asymptotic states

of (6.1)–(6.2).

This chapter includes contents that have been published in [

]. The chapter is organized as

follows. Numerical results and a rigorous definition for multicluster and solitary states on

complex networks are presented in Section 6.1. In Section 6.2 we provide a more detailed

analysis of one-cluster states. Here, relations between local and global properties are derived.

The salient role of rotating-wave clusters are underlined and the crucial dependence of their

stability on the coupling range and the wavenumber are rigorously described. Some of the proofs

are presented in the App. A.5 and A.6. After this, we focus on the analysis of solitary states

in Sec. 6.3. A reduced model for two-clusters is derived and a variety of bifurcations in which

solitary states are born and stabilized are presented. Section 6.5 summarizes our findings.

6.1 Multicluster and solitary states

This section is devoted to the numerical analysis of system

(6.1)

–

(6.2)

and the description of

several dynamical states which occur for this system. More specifically, we report one-cluster,

multicluster, and solitary states. While this section describes the states in a phenomenological

fashion, more rigorous results are presented in the subsequent Sections 6.2–6.3.

Note that the observed one-cluster and multicluster states are similar to those reported in Chap-

ters 4and 5for the all-to-all coupling base topology. However, there are important differences

due to the ring structure of our system, which will be discussed in detail.

For the numerical simulations in this section a system of

𝑁=

100 oscillators with coupling

range

𝑃=

20 is studied. The value of

𝜖

is set to 0.01 and the system parameter

𝛼

and

𝛽

are

varied in the ranges

[

𝜋/

]

and

[−𝜋

𝜋]

, respectively. All results are obtained starting from

uniformly distributed random initial conditions and a simulation time of

𝑡=

20000. Several

types of synchronization patterns are found in the numerical simulations, depending on the

values of 𝛼and 𝛽.

6.1 Multicluster and solitary states 85

(a)

(c)

100

0 1.00.5−0.5−1.0

(d)

(b)

2π

φi

2π

φi

20 40 60 80 100 i

20 40 60 80 100

Figure 6.1:

Illustration for two types of one-cluster states. The panels (a,c) show the asymptotic coupling matrices

and (b,d) snapshots of the phases at a fixed time. Results for the one-cluster states of antipodal type are presented in

(a,b) where

𝛼=

0.19

𝜋

𝛽=−

0.66

𝜋

and of splay type in (c,d) where

𝛼=

0.35

𝜋

𝛽=

0.01

𝜋

. Parameters:

𝑁=

100,

𝑃=

20,

𝜖=0.01. Figure modified from [?].

6.1.1 One-cluster states

A one-cluster is defined as a frequency synchronized state

𝜙𝑖= Ω𝑡+𝜒𝑖,𝑖=1, . . . ,𝑁

with a collective frequency

Ω∈R

and individual phase shifts

𝜒𝑖∈ [

0, 2

𝜋)

, see Section 4.1.

The two types of one-cluster states found in the numerical simulations are either of antipodal

or splay type whose asymptotic configurations are displayed in Fig. 6.1(a,b) and Fig. 6.1(c,d),

respectively. The antipodal and splay-type clusters have been introduced previously in Chapter 4.

In the antipodal cluster, all phases

𝜙𝑖

are either in-phase or in anti-phase, i.e.,

𝜒𝑖∈ {𝜒

𝜒+𝜋}

with

𝜒∈ [

0, 2

𝜋)

and hence

𝑅(2)(𝝓)=

1. In the splay cluster the phases are distributed across

the interval

[

0, 2

𝜋)

such that the global second order parameter, as defined in equation

(6.5)

vanishes, i.e.,

𝑅(2)(𝝓)=

0. In Fig. 6.1(a,c) the coupling structures corresponding to the two types

of one-clusters are displayed. Note that the coupling weights are solely described by the phase

differences of the oscillators and are given by

𝜅𝑖 𝑗 =−sin(𝜒𝑖−𝜒𝑗+𝛽).

The one-cluster states, which exist in our ring case and the all-to-all base topology case from

Chapter 4have the same representation except for the fact that some of the coupling weights are

absent in the case of the ring, see empty entries in Fig. 6.1(a,c).

86 6 Adaptation on nonlocally coupled ring networks

Figure 6.2:

Schematic figure illustrating the definition of multicluster and subnetworks induced by groups of nodes

with the same average frequency. The full network (left) consists of

𝑁=

20 nodes and has a nonlocal ring structure

with

𝑃=

4. The colors of the nodes indicate their average frequencies. Clusters are shown by the equally colored

nodes that form connected sub-networks. Even though the two blue groups I and II possess the same averaged

frequencies, they form two different clusters, since they are not connected. Figure taken from [BER20c].

6.1.2 Multicluster states

As described in Chapter 5, one-clusters can serve as building blocks for multi-frequency clustered

states where the phase dynamics and the coupling matrix

𝜅

are divided into different groups;

𝜅𝑖 𝑗,𝜇𝜈

refers to the coupling weight for the connection from the

𝑖

th oscillator of the

𝜇

th cluster to

the

𝑗

th oscillator of the

𝜈

th cluster. Analogously,

𝜙𝑖,𝜇

denotes the

𝑖

th phase oscillator in the

𝜇

cluster. The temporal behavior for each oscillator in an

𝑀

-cluster state takes the form, see also

Definition 5.0.1

𝜙𝑖,𝜇(𝑡)= Ω𝜇𝑡+𝜒𝑖,𝜇+𝑠𝑖,𝜇(𝑡)𝜇=1, . . . ,𝑀

𝑖=1, . . . ,𝑁𝜇

where

𝑀

is the number of clusters,

𝑁𝜇

is the number of oscillators in the

𝜇

th cluster,

𝜒𝑖,𝜇∈ [

0, 2

𝜋)

are phase lags, and

Ω𝜇∈R

is the collective frequency of the oscillators in the

𝜇

th cluster. The

functions 𝑠𝑖,𝜇(𝑡)are assumed to be bounded.

Both types of one-clusters give rise to multi-frequency cluster states, see Fig. 6.1, similarly

to the case of all-to-all base coupling. However, in case of more complex network structures

the definition of a frequency-cluster has to be refined to account for the connectedness of the

individual building blocks. Therefore, a multi-frequency-cluster (shortly: multicluster) consists of

groups of frequency synchronized oscillators for which the subnetwork (or subgraph), induced by the

individual groups of nodes, is connected. Here, we say a network is connected if there is directed

path from each node to every other node of the network. In case of a directed graph, we require

the property of weak connectedness for the induced subgraph for the cluster. For an introduction

to the terminology we refer the reader to [KOR18].

Let us illustrate the above definition. Consider a nonlocal ring network of

𝑁=

20 oscillators

with coupling range

𝑃=

4 as presented in Fig. 6.2. Suppose that for each node of the network

we have a certain average frequency which is indicated by the color. In Fig. 6.2, we have three

different average frequencies denoted by the green, blue, and red colors. The individual clusters

6.1 Multicluster and solitary states 87

are given by the connected subnetworks induced by equally colored nodes. Note that even

though the blue nodes have the same average frequency, they are forming two different clusters

(I,II) corresponding to two connected components. Note that the induced subnetworks are

not necessarily regular even if the base topology is regular, see for instance the red or green

subnetworks.

Figure 6.3 shows two-cluster states of antipodal (Figs. 6.3(a-c)) and splay type (Figs. 6.3(d-f)). In

both cases, Figs. 6.3(c,f) show that there are two distinct groups of oscillators with different aver-

aged frequencies (green, red). It is easily verified that these groups form connected subnetworks

and, hence, they form a two-cluster state. Due to the frequency difference between the individual

clusters, the groups of oscillators decouple effectively. This can be seen in Figs. 6.3(a,d) where

only oscillators of the same cluster are strongly coupled compared to the coupling between the

clusters given by the respective coupling weights 𝜅𝑖 𝑗,𝜇𝜇 and 𝜅𝑖 𝑗,𝜇𝜈 ≈0 (𝜇≠𝜈).

Snapshots of the phase distributions are presented in Figs 6.3(b,e). Figure 6.3(b) is showing

the phase distribution of an antipodal cluster. In contrast to the case of global coupling, see

Chapters 4and 5, the phases do not possess the exact antipodal property anymore. In fact,

the scattering of the antipodal phase distribution is caused by the structure of the induced

subnetwork which is not regular anymore for the in-degree of each node, i.e., the subnetwork

does not have constant row sum, see e.g. Fig. 6.2. Note that in case of a global base topology,

all induced subnetworks are global. The phase snapshot for the splay multicluster is displayed

in Fig. 6.3(e). Here, as in the case with global base structure, the phase distribution possess the

property that 𝑅(2)(𝝓)=0.

Another, more complex, antipodal five-cluster is presented in Fig. 6.3(g-i). In Fig. 6.3(i), we ob-

serve three groups, a big one (green) and two smaller ones (red, blue), with different frequencies.

Moreover, in accordance with the definition of a frequency cluster and the illustration in Fig. 6.2,

the blue and the red groups possesses two connected components (I,II) each. This fact implies

the presence of five individual frequency clusters in Fig. 6.3(g-i). Remarkably, the red clusters I

and II as well as the blue clusters I and II are of the same size. This observation is in contrast to

the hierarchical structures discovered and analyzed in [

KAS17

] and Chapter 5. Hence, in the

case of the ring base structure, the evenly sized clusters can appear, which was not possible in

the case of global base structure [KAS17] and identical oscillators, see also Chapter 5.

While several examples for one- and multicluster states have been described above, Fig. 6.4

shows that these states are observable in a wide range in the

(𝛼

𝛽)

parameter space. The diagram

in Fig. 6.4 is produced by running simulations of

(6.1)

–

(6.2)

from random initial conditions. In

case a one-cluster or multicluster is found, the region is colored or hatched, respectively, in

accordance with the legend in Fig. 6.4. We further used a continuation method in the

(𝛼

𝛽)

parameter space to show the full extent where the various types of multiclusters can be observed.

In Section 6.2.1 and 6.2.2, we provide a more rigorous description for the existence and stability

properties of the one-clusters.

6.1.3 Solitary states

For systems with global base coupling, the clusters in the multicluster states were found to be

hierarchical in nature, i.e. the clusters varied in size significantly [

KAS17

], see also Chapter 5. As

88 6 Adaptation on nonlocally coupled ring networks

Figure 6.3:

Illustration of the different types of multicluster states. The panels (a,d,g) show the coupling matrix, (b,e,h)

phase snapshots and (c,f,i) average frequencies. (a-c): antipodal two-cluster for

𝛼=

0.23

𝜋

𝛽=−

0.56

𝜋

; (d-f): splay

two-cluster for

𝛼=

0.19

𝜋

𝛽=−

0.45

𝜋

; (g-i): antipodal five-cluster (I, II denote the two connected components of the

red and the blue clusters) for

𝛼=

0.3

𝜋

𝛽=−

0.53

𝜋

. Parameters:

𝑁=

100,

𝑃=

20,

𝜖=

0.01. Figure taken from [

BER20c

Figure 6.4:

Map of regimes for one- and multicluster states of antipodal and splay type in

(𝛼

𝛽)

parameter space.

Parameters:

𝑁=

100,

𝑃=

20,

𝜖=

0.01. The horizontal black line at

𝛼=

0.1 shows the location for the parameter

𝛽

where the emergence of solitary states is analyzed, see Fig. 6.7 in Sec. 6.3. Figure taken from [BER20c].

6.2 One-cluster states: Local vs. global features 89

Figure 6.5:

Illustration of solitary states. The panels (a,d) show coupling matrix, (b,e) phase snapshots, and (c,f)

average frequencies. (a-c): single solitary state for

𝛼=

0.1

𝜋

𝛽=−

0.3

𝜋

; (d-f): three uncoupled solitary states for

𝛼=0.15𝜋,𝛽=−0.41𝜋. Parameters: 𝑁=100, 𝑃=20, 𝜖=0.01. Figure modified from [BER20c].

we have mentioned above, in the nonlocal base coupling case, multicluster states with one large

and many smaller, similar in size, clusters have been observed. Figure 6.5 shows a particular

example of this phenomenon, called solitary states, where either one single oscillator (upper

panels) or three single oscillators (lower panels) decouple from a large cluster. The solitary

states are particular examples of multiclusters with a large group of frequency synchronized

oscillators (background cluster) and individual solitary nodes with different frequency, i.e.,

clusters consisting of only one oscillator. These special kind of states, for which we provide an

analysis of their emergence in Sec. 6.3, are of particular interest as they are found in various

dynamical systems [

MAI14a

ASH15

SEM15b

WOJ16

PRE16

MAI17

JAR18

TEI19

TAH19

6.2 One-cluster states: Local vs. global features

6.2.1 Classification of one-cluster states

In this section we study the antipodal and splay one-cluster states in more details. Due to the

𝑆1

symmetry of system (6.1)-(6.2), the following phase-locked solutions appear generically

𝜙𝑖(𝑡)= Ω𝑡+𝜒𝑖,𝑖=1, . . . ,𝑁, (6.6)

where

𝜒𝑖∈ [

0, 2

𝜋)

are fixed phase lags and

the cluster frequency. It is clear that such solutions

describe a one-cluster state since the frequencies are the same. By substituting

(6.6)

into

(6.1)

–

90 6 Adaptation on nonlocally coupled ring networks

(6.2), we obtain

𝜅𝑖 𝑗 =−sin(𝜒𝑖−𝜒𝑗+𝛽), (6.7)

Ω = 1

2cos(𝛼−𝛽) − 1

4𝑃

𝑖+𝑃

𝑗=𝑖−𝑃

cos(2𝜒𝑖−2𝜒𝑗+𝛼+𝛽). (6.8)

The equation (6.8) implies that the one-cluster state exists only if the following expression is

independent of the index 𝑖

2𝑃

𝑖+𝑃

𝑗=𝑖−𝑃

cos(2𝜒𝑖−2𝜒𝑗+𝛼+𝛽)=<𝑅(2)

𝑖𝑒i(𝜗(2)

𝑖−2𝜒𝑖−𝛼−𝛽). (6.9)

Equation (6.9) allows for the distinction of two types of distributions of the phase-lags for which

it is independent of 𝑖. We call a cluster of

(i)

Antipodal type, if

𝜒𝑖∈ {

𝜋}

for all

𝑖=

. . .

𝑁

. In this case, the sum in (6.9) equals

cos(𝛼+𝛽)

and of

(ii)

Local splay type, if

𝜒𝑖∈ [

0, 2

𝜋)

are such that the second order parameter

𝑍(2)

𝑖=𝑅(2)

𝑖(𝝌)𝑒i𝜗(2)

𝑖

satisfies

𝑅(2)

𝑖(𝝌)=𝑅(2)

𝑐(𝝌)(6.10)

𝜗(2)

𝑖=2𝜒𝑖+𝜒0(6.11)

with 0 ≤𝑅(2)

𝑐(𝝌)<1 and 𝜒0∈ [0, 2𝜋)independent on 𝑖.

Note that the additional degree of freedom for clusters of local splay type, i.e.

𝜒0∈ [

0, 2

𝜋)

arbitrary, is due to the

𝑆1

symmetry. The constant

𝜒0

can be set to 0 without loss of generality.

The both types of phase distribution lead to one-cluster states for the system

(6.1)

–

(6.2)

with

frequency

Ω = 

sin 𝛼sin 𝛽antipodal type,

2cos(𝛼−𝛽) −𝑅(2)

𝑐cos(𝛼+𝛽)local splay type. (6.12)

In the context of globally coupled base topologies, antipodal and splay type phase distribution

have been extensively discussed [

ASH08

] where the (global) splay clusters, also called fuzzy

clusters [

MAI14a

], are defined by the global condition

𝑍(2)(𝝌)=

0, see also Chapter 4. Remark-

ably, if a phase distribution is of the local splay type (as described by (6.10)-(6.11)) then it is of

splay type as well, see Appendix A.5 for more details. The converse is not true in general. Hence,

the class of local splay clusters is "smaller" than the class of global splay clusters. In addition,

local splay clusters do not necessarily form families of solutions. According to the definition of

local splay cluster, generically

𝑁

complex algebraic equations have to be solved for

𝑁

unknown

phase-lags

𝜒𝑖

. Therefore, the set of equations for the phase-lags is overdetermined and the set

of local splay states might be empty. However, it is not the case due to the symmetries of the

system and the base coupling structure. The symmetry of the nonlocal ring structure allows for

constructing explicit, symmetric examples for the clusters of local splay type. These are clusters

of the rotating-wave type.

6.2 One-cluster states: Local vs. global features 91

(ii’)

The clusters are of rotating-wave type, if

𝜒𝑖=𝑖𝑘 2𝜋

𝑁

, where

𝑘=

. . .

𝑁

is the wavenumber. In

the literature, the notion "splay state" is often restricted to this definition.

Let us show that the rotating wave clusters (ii’) are the local splay states (ii). For this we write

the phase distribution as 𝝌𝑘=(2𝜋𝑘/𝑁,. . . 2𝜋𝑘 (𝑁−1)/𝑁, 0)𝑇. Then, we have

𝑍(𝑛)

𝑖(𝝌𝑘)=1

2𝑃

𝑖+𝑃

𝑗=𝑖−𝑃

𝑒i𝑛𝑘 𝑗 2𝜋

𝑁=𝑒i𝑛𝑘𝑖 2𝜋

𝑁𝑅(𝑛)

𝑁(𝝌𝑘), (6.13)

where

𝑅(𝑛)

𝑁(𝝌𝑘)=1

𝑃©«

𝑃

𝑗=1

cos(𝑛𝑘 𝑗 2𝜋

𝑁)ª®¬. (6.14)

we conclude that all rotating-wave states with

𝑘≠

𝑁/

2 are local splay states and thus solutions

(6.1)

–

(6.2)

. Rotating-wave clusters with

𝑘=

𝑁/

2 are of antipodal type. The

𝑛

th moment local

order parameter

𝑍(𝑛)

𝑖(𝝌𝑘)=𝑅(𝑛)

𝑁(𝝌𝑘)

is constant for all

𝑖=

. . .

𝑁

and its value depends on

the wavenumber 𝑘.

Note further that

𝑍(𝑛)

𝑖(𝝌𝑘)=𝑍(𝑛𝑘)

𝑖(𝝌1)

, which connects the moment of the order parameters

with the wavenumber of the rotating-wave states. For globally coupled base structures, the

rotating-wave states are found to be very important in describing the main features of antipodal

and global splay type clusters such as stability. The next section is devoted to the description of

the stability condition for rotating-wave states.

6.2.2 Stability of one-cluster states

In the following, the stability of one-clusters is analyzed. In order to study the local stability of

one-cluster solutions described in Sec. 6.2.1, we linearize the system of differential equations

(6.1)–(6.2) around the phase-locked states

𝜙𝑖(𝑡)= Ω𝑡+𝑎𝑖,

𝜅𝑖 𝑗 =−sin(𝑎𝑖−𝑎𝑗+𝛽).

These solutions are equilibria relative to the

𝑆1

symmetry [

GOL88a

], therefore the linearization

around such solutions leads to a linear system with constant coefficients, despite the time

dependency of

𝜙(𝑡)

. Practically, one can first move to the co-rotating coordinate system by

introducing the new variable

𝜙(𝑡) −Ω𝑡

and then linearize around the equilibrium in the new

coordinates. As a result, we obtain the following linearized system for the perturbations

𝛿𝜙𝑖

and

𝛿𝜅𝑖 𝑗 :

𝑑

𝑑𝑡 𝛿𝜙𝑖=1

4𝑃

𝑖+𝑃

𝑗=𝑖−𝑃sin(𝛽−𝛼) +sin(2(𝑎𝑖−𝑎𝑗)+𝛼+𝛽)𝛿𝜙𝑖−𝛿𝜙𝑗

−1

2𝑃

𝑖+𝑃

𝑗=𝑖−𝑃

sin(𝑎𝑖−𝑎𝑗+𝛼)𝛿𝜅𝑖 𝑗 ,

(6.15)

92 6 Adaptation on nonlocally coupled ring networks

Figure 6.6:

Stability of one-cluster states for different wavenumbers

𝑘

and coupling ranges

𝑃

. Regions of stability for

the one-cluster states are colored in blue, while instability in yellow. The borders of stability (black dashed lines) are

obtained from the eigenvalues

(6.17)

. Parameters are as follows: (a)

𝑃=

10,

𝑘=

1; (b)

𝑃=

10,

𝑘=

4; (c)

𝑃=

10,

𝑘=

25;

(d)

𝑃=

𝑘=

1; (e)

𝑃=

20,

𝑘=

1; and (f)

𝑃=

25,

𝑘=

1. The other parameters are

𝑁=

50 and

𝜖=

0.01. Figure modified

from [BER20c].

and

𝑑

𝑑𝑡 𝛿𝜅𝑖 𝑗 =−𝜖𝑎𝑖 𝑗 𝛿𝜅𝑖 𝑗 +cos(𝑎𝑖−𝑎𝑗+𝛽)𝛿𝜙𝑖−𝛿𝜙 𝑗. (6.16)

System (6.15)-(6.16) is a

(𝑁+𝑁2)

-dimensional linear system of ordinary differential equations,

which can be written in the form

𝒙0=𝑳𝒙

𝒙∈R𝑁+𝑁2

, and the stability of which is determined

by the eigenvalues of the matrix

𝑳

. For all antipodal and rotating-wave states the stability

analysis can be done explicitly. However, the calculations are quite lengthy (see Appendix A.6).

Summarizing the results of these calculations, the spectrum

𝑆

of the eigenvalues, corresponding

to the rotating-wave one-clusters, is given by

𝑆=n0, −𝜖,𝜆𝑙,1𝑁

𝑙=1,𝜆𝑙,2𝑁

𝑙=1o. (6.17)

Here 𝜆𝑙,1 and 𝜆𝑙,2 are the solutions of the quadratic equation

𝜆2

𝑙−𝜆𝑙

2h𝐿(𝛼,𝛽,𝑙,𝑘)+(𝑅(𝑙)

𝑁(𝝌1) −1)sin(𝛼−𝛽) −2𝜖i−𝜖 𝐿(𝛼,𝛽,𝑙,𝑘)=0, (6.18)

where the complex function

𝐿

as defined in Eq.

(A.27)

in Appendix A.6,

𝑘

is the wavenumber.

In Figure 6.6 we show the stability of rotating-wave one-clusters in an ensemble of

𝑁=

oscillators. The figures demonstrate regions of stability in the

(𝛼

𝛽)

parameter plane for different

wavenumber

𝑘

and coupling range

𝑃

. The stability is obtained by numerical simulations as

well as by the Lyapunov spectrum Eq.

(6.17)

. The borders of stability, as they are provided by

the analytical results, are displayed with a dashed black line. Numerically the stability was

computed as follows: (i) the theoretical shape of the one-cluster given by

(6.6)

–

(6.7)

is used as

initial conditions with a small random perturbation in the range of

[−

0.01, 0.01

]

; (ii) then we solve

the system numerically for

𝑡=

20000 time units; (iii) compute the euclidean norm between the

initially perturbed state and the theoretical one as well as between the final state after

𝑡=

20000

and the theoretical one; (iv) in case the second norm is smaller than the first, meaning that the

trajectory approaches the theoretical one-cluster state, we consider the one-cluster state as stable

and color the corresponding region in blue, otherwise the state is considered as unstable and the

corresponding region is colored in yellow.

The diagrams in the first row of Fig. 6.6 show the influence of the wavenumber on the stability of

6.3 The emergence of solitary states 93

one-clusters. Here, the coupling range is fixed to

𝑃=

10. For adaptively coupled phase oscillators

with a global base topology, it has been shown that the stability of rotating-waves of local splay

type, i.e.,

𝑘≠

𝑁/

2, does not depend on the wavenumber, see Chapter 4. However. we observe

that in case of a nonlocal base structure the shape of the stability regions crucially depends on

the wavenumber

𝑘

, see Fig. 6.6(a,b,c). Moreover, there are no common regions of stability for

the one-cluster states in Fig. 6.6(b,c), i.e., the region of stability in both figures have an empty

intersection.

The diagrams in Fig. 6.6(d-f) exemplify the influence of the coupling range on the stability of

the rotating-wave cluster with

𝑘=

1. We see that in comparison with Fig. 6.6(a), the regions

of stability change significantly. Note further that for

𝑃=

25 the stability regions resemble the

results known for the globally coupled base topology, see Chapter 4.

In contrast to the local splay type clusters, the stability regions for the antipodal one-cluster

states are the same which can be derived from the following. For antipodal states, the quadratic

equation (6.18) simplifies to

𝜆2

𝑙−𝜆𝑙

2h(1−𝑅(𝑙)

𝑁(𝝌1))sin(𝛽)cos(𝛼) −2𝜖i−𝜖(1−𝑅(𝑙)

𝑁(𝝌1))sin(𝛼+𝛽)=0. (6.19)

since 𝐿(𝛼,𝛽,𝑙,𝑘)=sin(𝛼+𝛽)(1−𝑅(𝑙)

𝑁(𝝌1)) for 𝑘=0, 𝑁/2.

The regions of stability in Fig. 6.6(c) for a nonlocal coupling structure are in agreement with

the regions of stability found for a global coupling structure, see Chapter 4. However, note

that Eq.

(6.19)

differs analytically from the expression found in Corr. 4.2.3. The similarity in

the stability regions is only due to the small value of

𝜖

and the differences would be more

pronounced in the presence of larger

𝜖

. Note further, that in case of

𝑃=𝑁/

2, which is equivalent

to global coupling, the local order parameter

𝑅(𝑙)

is either 1 for

𝑙=

0 and 0 otherwise. This agrees

with the findings in Chapter 4.

Our stability analysis shows how strongly the ring network structure and confined coupling

range alter the stability properties of the clusters. Since the analysis in Appendix A.6 is not

restricted to nonlocal coupling structures, it provides the analytic tools to study the influence of

more general complex base topologies on the stability of rotating-wave states.

6.3 The emergence of solitary states

In this section, we unveil the mechanism behind the formation of solitary states as they are

illustrated in Fig. 6.5. As solitary state we define the state where all oscillators in the system

are frequency synchronized except one single oscillator, or several oscillators which do not

share their local neighborhood. The majority cluster of the synchronized oscillators is also called

background cluster.

Let us restrict ourselves to the analysis of a solitary cluster interacting with an in-phase syn-

chronous cluster (see

(6.6)

and

(6.7)

with

𝜒𝑖=

0). Imposing the assumptions, we end-up with

the following 4-dimensional model where

𝜙

and

𝜓

describe the dynamics of the solitary cluster

94 6 Adaptation on nonlocally coupled ring networks

Figure 6.7:

Phase portraits for two-dimensional system

(6.20)

–

(6.21)

. The graphics show the two classes of asymptotic

states that are equilibria (colored nodes) and periodic solutions (colored lines). The stability properties of the

individual asymptotic states are indicated by the coloring where the blue refers to stable and the red (dashed) to

unstable states. In addition, several trajectories are plotted in black including those close to the stable and unstable

manifold of the equilibria. The nullclines are displayed as gray lines. For the different panels parameter

𝛽

is varied as

shown in Fig. 6.4: (a)

𝛽=−

0.601

𝜋

; (b)

𝛽=−

0.599

𝜋

; (c)

𝛽=−

0.58

𝜋

; (d)

𝛽=−

0.5515

𝜋

; (e)

𝛽=−

0.5

𝜋

; (f)

𝛽=−

0.08

𝜋

; (g)

𝛽=−0.0563𝜋; and (h) 𝛽=0.05𝜋. The other parameters are 𝛼=0.1𝜋and 𝜖=0.01. Figure modified from [BER20c].

and the background in-phase synchronized cluster, respectively, which are dynamically coupled

through 𝜅1and 𝜅2:

𝜙=1−𝑁−1

𝑁𝜅1sin(𝜙−𝜓+𝛼),

𝜓=1+𝑁−2

𝑁sin 𝛼sin 𝛽−1

𝑁𝜅2sin(𝜙−𝜓−𝛼),

¤𝜅1=−𝜖(𝜅1+sin(𝜙−𝜓+𝛽)),

¤𝜅2=−𝜖(𝜅2−sin(𝜙−𝜓−𝛽)).

The latter equations can be simplified by introducing the phase difference

𝜃=𝜙−𝜓

as well as by

considering a large ensemble of oscillators (𝑁→ ∞). We obtain the following two-dimensional

system for the dynamics of two clusters, one of which is solitary, in the large ensemble limit:

𝜃=−sin 𝛼sin 𝛽−𝜅sin(𝜃+𝛼), (6.20)

¤𝜅=−𝜖(𝜅+sin(𝜃+𝛽)), (6.21)

where we denote 𝜅=𝜅1.

In the following, we study the structure of the phase space of

(6.20)

–

(6.21)

for fixed

𝛼=

0.1

𝜋

and

different values of parameter

𝛽

. Several bifurcation scenarios are discovered which give rise to

the birth and stability changes of the solitary states. We observe how the stable solitary state

emerges in a subcritical pitchfork bifurcation of periodic orbits and disappears in a homoclinic

bifurcation with increasing

𝛽

. Note that solitary states are given by periodic solutions, where

6.3 The emergence of solitary states 95

the phase difference

𝜃(𝑡)

rotates. Equilibria of this system describe one-cluster states. Figure 6.7

shows several characteristic phase portraits of

(6.20)

–

(6.21)

illustrating the bifurcation scenarios

with increasing 𝛽(from (a) to (h)), see also Fig. 6.4.

In Fig. 6.7(a), we observe four equilibria which correspond to certain one-cluster solution. The

stable equilibria at

𝜃=

0 and

𝜃=𝜋

correspond to in-phase synchronous and antipodal where

𝑎1=𝜋

and

𝑎𝑖≠1=

0, respectively. The other two saddle equilibria correspond to the special

class of double antipodal states, see Chapter 4, and describe therefore phase-cluster similar

to those described in [

MAI14

TEI19

]. While these equilibria can be stable for the reduced

system

(6.20)

–

(6.21)

, they are always unstable for

(6.1)

–

(6.2)

in case of global coupling, see

Prop. 4.2.4. Additionally, to these four equilibria we find an unstable periodic orbit which

corresponds to an unstable solitary state.

With increasing

𝛽

, we observe a subcritical pitchfork bifurcation of periodic orbits at

𝛼+𝛽=−𝜋/

in which the unstable periodic orbit is stabilized and two additional periodic orbits are created.

Figure 6.7(b) shows the phase portrait directly after the pitchfork bifurcation. Therefore, we

conclude that there exist three solitary states, two of which are unstable and one stable. It is

worth to remark that the stability for the reduced system is only necessary but not sufficient to

be a stable asymptotic state for the network (6.1)–(6.2).

By increasing

𝛽

even further the basin of attraction of the stable periodic orbit increases and

its boundaries are given by the unstable periodic orbits, see Fig. 6.7(c). For

𝛽=−

0.5515

𝜋

, the

trajectories of the unstable solitary states merge with the equilibria and become homoclinic

orbits of the saddle equilibria (Fig. 6.7(d)). The phase portrait after this homoclinic bifurcation is

shown in Fig. 6.7(e).

After the homoclinic bifurcation, with increasing

𝛽

, the equilibria are moving towards each

other in phase space and exchange their stability in a transcritical bifurcation. This can seen

analytically by considering the determining equations for the equilibria (𝜃,𝜅)of (6.20)–(6.21)

0=cos(𝛼+𝛽) −cos(2𝜃+𝛼+𝛽), (6.22)

𝜅=−sin(𝜃+𝛽).

In general, equation

(6.22)

possesses two solutions for 2

𝜃

. At

𝛼+𝛽=

𝜋

, however, these two

solutions coincide which describes the point of the transcritical bifurcation. The stability of

the equilibria can be further computed by considering the two-dimensional system linearized

around the equilibria. In a more general setup this has be done Sec. 6.2.2. Figure 6.7(f) displays the

phase portrait after the transcritical bifurcation. Remember that although the double antipodal

clusters are stable for the reduced system, they are always unstable for the full system, see

Prop. 4.2.4.

In Figure 6.7(g,h) another homoclinic bifurcation is presented in which the stable solitary state

becomes a homoclinic orbit of the in-phase and antipodal cluster. The phase portrait close to

the homoclinic bifurcation is presented in Fig. 6.7(h). After the homoclinic bifurcation the phase

space is divided into the basins of attraction of the two double antipodal states and no more

solitary state exist.

96 6 Adaptation on nonlocally coupled ring networks

6.4 Adaptive networks with global base topology versus ring base

topology: the differences

Adaptive networks of coupled phase oscillators have been extensively studied on an all-to-all

base structure [

AOK09

AOK11

KAS17

], see also Chapters 4and 5. Here, we extend previous

work towards more complex base topologies by considering a nonlocal ring base topology on

which adaptation takes place. In this section we briefly summarize the main differences resulting

from the different base topologies.

For the global base topology, all links between the nodes with the same frequency become active

leading to the all-to-all structures within each cluster. Therefore, strongly connected components

can be equivalently described by the frequency synchronization of nodes. In contrast to this,

for ring networks (also more complex base structures), the frequency synchronization does

not necessarily imply connectivity, see Fig. 6.2. As a result, we have adapted the definition of

the frequency cluster on a complex base topology as a connected subnetwork with frequency

synchronized oscillators.

Another effect induced by the ring base topology concerns the hierarchical ordering of cluster

sizes. For global base structures is has been found that a sufficiently large difference of the

cluster sizes is necessary for the appearance of multicluster states. In case of a ring structure,

the hierarchy is not necessary anymore. In figure 6.3(g-i), we present a five-cluster states that

possesses two clusters each of size 7 and additionally two clusters each of size 2. For solitary

states this nonhierarchical clustering implies that on a ring structure there can be several solitary

nodes (Fig. 6.5(d-f)). A simple explanation for the appearance of the clusters of a similar size is

based on the fact that such clusters can be uncoupled in the base coupling structure and, hence,

not synchronized. In contrast, in networks with the global base structure, similar clusters tend to

be synchronized and merge into one larger cluster.

Regarding the stability of rotating-wave states another striking difference between global and

ring base topology is observed. Here the differences are twofold. For a global base topology

rotating-waves constitute a

𝑁−

2-dimensional family of solutions with the same collective

frequency. On a nonlocal ring, this invariant family is not present anymore and all rotating-

wave are different from each other including their frequencies. The same holds true for their

stability. While the stability features of all rotating-waves agree on global structures, the stability

properties depend crucially on the wavenumber (see Fig. 6.6(a-c)).

6.5 Summary

In summary, a model of adaptively coupled identical phase oscillators on a nonlocal ring has been

studied. Various frequency synchronized states are observed including one-cluster, multicluster,

and solitary states. Those states are similar to those found for a global base topology [

KAS17

also Chapters 4and 5. However, to account for the complex base topology, we have introduced

a new definition of one-clusters by means of connected induced subnetworks. This definition

allows furthermore to distinguish between multicluster and solitary states in a more strict way

than it was done before.

6.5 Summary 97

Since one-cluster states form building blocks for multicluster states, Chapter 5, we have first

investigated the existence and stability properties of one-cluster states. Here, we have introduced

a novel type of phase-locked states for complex networks, namely local splay states, and have

shown that this class of states is nonempty for any nonlocal ring base topology. In particular, we

have proved that rotating-wave as well as antipodal states are always phase-locked solutions.

Compared with the case of a global base topology, the different clusters of local splay type on a

nonlocal ring structure can possess different collective frequencies. In addition, we have proved

that local splay cluster are always global splay cluster. This statement relates, therefore, local

with global (with respect to "spatial" structures in the network) properties of solutions.

The stability features of rotating-wave states have been studied numerically and analytically. The

comparison of both approaches results in a very good agreement. Due to the analytic findings

for rotating-wave states on a nonlocal ring, we are able to describe their stability depending on

the coupling range

𝑃

and the wavenumber

𝑘

. The limiting case of global coupling, i.e.

𝑃=𝑁/

is shown to be in agreement with the results presented in Chapter 4.

An interesting feature of the system’s behavior are solitary states. They have been previously

found to emerge in the Kuramoto-Sakaguchi model with inertia [

JAR18

]. In this chapter, we

show that solitary states are born in a homoclinic bifurcation and can be (de)stabilized in a

pitchfork bifurcation of periodic orbits. In order to show this, a two-dimensional effective model

is derived governing the dynamics of solitary states. In contrast to the Kuramoto-Sakaguchi

model with inertia, we observe a much more complicated bifurcation behavior. In particular,

three different solitary states are created due to two individual homoclinic bifurcations. Two

of these three solitary states, however, are unstable and bifurcate together with stable solitary

states in a subcritical pitchfork bifurcation of periodic orbits.

Our results highlight the delicate interplay between adaptivity and complexity of the network

structure. Since this interplay has been rarely investigated from the mathematical viewpoint, so

far, this work raises many questions for future research which could be conducted for different

network structures beyond nonlocal rings, other dynamical models for the local dynamics,

nonidentical units or different adaptation rules.

Synchronization on adaptive complex network

structures 7

In this chapter, we extend the master stability approach to complex dynamical network of

𝑁

diffusively [

PEC98

KEA12

LEH15b

] and adaptively coupled oscillators. Moreover, we apply

the result to the paradigmatic model of adaptively coupled phase oscillators and provide a novel

description for the emergence of multicluster due to the presence of stability islands.

In the following, we consider a network of

𝑁

diffusively [

PEC98

KEA12

LEH15b

] and adap-

tively coupled oscillators of the form

𝒙𝑖=𝑓(𝒙𝑖(𝑡)) −𝜎

𝑁

𝑗=1

𝑎𝑖 𝑗 𝜅𝑖 𝑗𝐺(𝒙𝑖−𝒙𝑗), (7.1)

¤𝜅𝑖 𝑗 =−𝜖𝜅𝑖 𝑗 +𝑎𝑖 𝑗 𝜌𝐻(𝒙𝑖−𝒙𝑗), (7.2)

where

𝒙𝑖

is a

𝑑

-dimensional state vector of the

𝑖

th node with

𝒙𝑖∈C𝑑

𝑓∈𝐶1(C𝑑

C𝑑)

describes

the local dynamics of node

𝑖

, the function

𝐺∈𝐶1(C𝑑

C𝑑)

determines the coupling between

nodes. The coupling between the nodes is weighted by the dynamical variables

𝜅𝑖 𝑗 ∈R

which

are adapted according to the function

𝐻∈𝐶1(C𝑑

. The parameter

𝜎∈R

is the coupling

constant and

𝜌∈R

is called the adaptation strength. The basic coupling structure is given by the

matrix entries

𝑎𝑖 𝑗 ∈ {0, 1}

of the

𝑁×𝑁

adjacency matrix

𝐴

where a constant row sum

𝑟=Í𝑁

𝑗=1𝑎𝑖 𝑗

for all 𝑖=1, . . . ,𝑁is assumed.

Let

𝒔(𝑡)

be the synchronized state, meaning that

𝒙𝑖=𝒔(𝑡)

for all

𝑖=

. . .

𝑁

. Hence the fully

synchronized solution for the equations

(7.1)

–

(7.2)

is given by the solution of the following

equations

𝒔=𝑓(𝒔)+𝑟𝜎𝜌𝐻(0)𝐺(0), (7.3)

𝜅sync =(−𝜌𝐻(0), if 𝑎𝑖 𝑗 =1,

0, otherwise. (7.4)

Note that the synchronous solution depends on the row sum if neither

𝐺(

)≠

0 nor

𝐻(

)≠

By scaling 𝜎with 1/𝑟this dependency can be omitted.

The chapter is structured as follows. In Section 7.1, we establish the master stability approach

for complex adaptive networks of coupled oscillators and prove the main Proposition 7.1.3.

This result is then applied to a complex network of adaptively coupled phase oscillators for

which the master stability function is completely described in Sec. 7.2. Here, we further show

the emergence of stability islands. In Section 7.3, we use the master stability functions to show

how multicluster states as well as chimera-like states emergence due to the stability islands. The

results of this chapter are summarized in Section 7.4.

100 7 Synchronization on adaptive complex network structures

7.1 The master stability function for adaptive complex networks

In Ref. [

PEC98

], the master stability function for dynamical systems of the form

(7.1)

was

introduced. Here, the master stability function is defined as the largest Lyapunov exponent

as a function of a complex parameter, the master function parameter, for which eventually

the eigenvalues of the Laplacian matrix will be inserted. Once the master stability function is

determined, the stability of the synchronous states can be directly deduced for any topology.

In the following we follow and extend the approach of Refs. [

PEC98

LEH15b

] and present a

master stability function which takes adaptation of the coupling weights into account. For this

we have to consider the variational equations along the synchronous solution. For convenience

we define the following notation

𝒂𝑖=(𝑎𝑖1,. . . ,𝑎𝑖𝑁 ), diag(𝒂𝑖)=©«

𝑎𝑖1

...

𝑎𝑖𝑁 ª®®®¬

and the 𝑁×𝑁2,𝑁2×𝑁, and 𝑁2×𝑁matrices

𝐵=©«

𝒂1

...

𝒂𝑁ª®®®¬

,𝐶=𝐵𝑇−D, 𝐷=©«

diag(𝒂1)

diag(𝒂𝑁)ª®®®¬

respectively.

Then, the variational equations for system

(7.1)

–

(7.2)

along the synchronous solution

𝒔(𝑡)

can be

written in the following vectorized form

𝝃

𝝌!= 𝑆−𝜎𝐵 ⊗𝐺(0)

−𝜖 𝜌𝐶 ⊗𝐷𝐻(0) −𝜖I𝑁2! 𝝃

𝝌!, (7.5)

where

𝑆=I𝑁⊗𝐷 𝑓 (𝒔)+𝜎𝜌𝐻(

)(𝐿(𝐴) ⊗ 𝐷𝐺(0))

𝝃=𝒙−I𝑁⊗𝒔

, and

𝝌=𝜿−𝜅sync

with Kronecker

product denoted by

⊗

. Further,

𝒙∈C𝑁·𝑑

is the system state vector where all individual nodal

states are stacked on each other ordered by the node index, i.e.,

𝒙=(𝒙1

. . .

𝒙𝑁)𝑇

. Analogously,

𝜿=(𝜅11

. . .

𝜅1𝑁

. . .

𝜅𝑁 𝑁 )

. Further, the Laplacian

𝐿(𝐴)

for the corresponding adjacency matrix

𝐴is given by

𝐿(𝐴)=D(𝐴)− 𝐴,

where

D(𝐴)=©«

Í𝑁

𝑗=1𝑎1𝑗

...Í𝑁

𝑗=1𝑎𝑁 𝑗 ª®®®¬

, (7.6)

see (2.3).

7.1 The master stability function for adaptive complex networks 101

In the following Lemma, we summarize a few simple relations which are used to simply

calculation, later on.

Lemma 7.1.1 Let 𝐵,𝐶, and 𝐷given as above. Then the following relations hold true:

1. 𝐵·𝐵𝑇=𝑟I𝑁,

2. 𝐵·𝐷=𝐴,

3. 𝐵·𝐶=𝐿(𝐴),

4. 𝐶𝑇𝐶=𝐿(𝐴)+𝐿(𝐴𝑇).

Proof. The results are proved by direct calculation.

𝐵·𝐵𝑇=©«

𝒂1

...

𝒂𝑁ª®®®¬©«

𝒂𝑇

1...

𝒂𝑇

𝑁ª®®®¬

=©«

Í𝑁

𝑗=1𝑎1𝑗𝑎1𝑗

...Í𝑁

𝑗=1𝑎𝑁 𝑗 𝑎𝑁 𝑗 ª®®®¬

𝐵·𝐷=©«

𝒂1

...

𝒂𝑁ª®®®¬©«

diag(𝒂1)

diag(𝒂𝑁)ª®®®¬

=©«

𝒂1diag(𝒂1)

𝒂𝑁diag(𝒂𝑁)ª®®®¬

=©«

𝑎11𝑎11 ··· 𝑎1𝑁𝑎1𝑁

.....

𝑎𝑁1𝑎𝑁1··· 𝑎𝑁 𝑁 𝑎𝑁 𝑁 ª®®®¬

3. Using the results from 1. and 2., we get

𝐵·𝐶=𝐵·𝐵𝑇−𝐵·𝐷=𝐿(𝐴).

4. By applying 1. and 2., we find

𝐶𝑇·𝐶=𝐵−𝐷𝑇·𝐵𝑇−𝐷=𝑟I𝑁−𝐴−(𝐵·𝐷)𝑇+

𝑁

𝑗=1

diag(𝒂𝑗)2

=𝑟I𝑁−𝐴−𝐴𝑇+D(𝐴𝑇).

It was shown in Chapter 4and 6that due to the structure of the variational equations, there

are locally always degrees of freedom corresponding to the

𝑁2−𝑁

eigenvalues

𝜆=−𝜖

. In

order to reduce the equations

(7.5)

, we are studying the corresponding eigenspace. Consider the

equation 𝑆−𝜖I𝑁 𝑑 −𝜎𝐵 ⊗𝐺(0)

−𝜖 𝜌𝐶 ⊗𝐷𝐻(0)0! 𝝃

𝝌!=0.

From

𝐶⊗𝐷𝐻(

)𝝃=

0, it can be immediately deduced that

𝝃𝑖=¯

𝝃∈C𝑑

for all

𝑖=

. . .

𝑁

Assuming that

𝝃=

0 would lead to

𝐵𝝌=

0 as the defining equation for the eigenvectors. In fact,

the following results show that

𝐵𝝌=

0 suffices to find

𝑁2−𝑁

many linearly independent vectors

spanning the eigenspace. The eigenvectors themselves are additionally independent of time.

102 7 Synchronization on adaptive complex network structures

Lemma 7.1.2

Let

𝝌=(𝝌𝑇

. . .

𝝌𝑇

𝑁)𝑇

with

𝝌𝑇

𝑖=((𝝌𝑖)1

. . .

(𝝌𝑖)𝑁)

for all

𝑖∈ {

. . .

𝑁}

, then the

following two relations hold true

1. ker(𝐵)=n𝝌∈R𝑁2:Í𝑁

𝑗=1𝑎𝑖 𝑗 (𝝌𝑖)𝑗=0∀𝑖∈ {1, . . . ,𝑁}o

2. ker(𝐷𝑇)=n𝝌∈R𝑁2:Í𝑁

𝑗=1𝑎𝑗𝑖 (𝝌𝑗)𝑖=0∀𝑖∈ {1, . . . ,𝑁}o.

Further, dim(ker(𝐵)) =dim(ker(𝐷𝑇)) =𝑁2−𝑁.

Proof. 1. We find by direct calculation

𝐵𝝌=©«

𝒂1𝝌1

𝒂𝑁𝝌𝑁ª®®®¬

=©«

Í𝑁

𝑗=1𝑎1𝑗(𝝌1)𝑗

Í𝑁

𝑗=1𝑎𝑁 𝑗 (𝝌𝑁)𝑗ª®®®¬

𝝌𝑇𝐷=𝝌𝑇

1,. . . ,𝝌𝑇

𝑁©«

diag(𝒂1)

diag(𝒂𝑁)ª®®®¬

𝑁

𝑗=1

𝝌𝑇

𝑗𝒂𝑗=

𝑁

𝑗=1(𝝌𝑗)1𝑎𝑗1,. . . ,(𝝌𝑗)𝑁𝑎𝑗 𝑁 

It is easy to verify that

rank(𝐵)=rank(𝐷𝑇)=𝑁

and

dim(ker(𝐵)) =dim(ker(𝐷𝑇)) =𝑁2−𝑁

With this result we are able to find a proper transformation in order to determine the master

stability function for adaptively coupled dynamical systems.

Proposition 7.1.3

Let

(7.1)

–

(7.2)

possess a synchronous state given by

(7.3)

–

(7.4)

. Further, let

(7.5)

the variational equations around this synchronous solution and assume that the Laplacian matrix

𝐿(𝐴)

is diagonalizable. Then, the synchronous solution is stable if and only if for all eigenvalues

𝜇∈C

of the

Laplacian matrix the largest Lyapunov exponent of the following differential equations is smaller than

zero.

d𝜁

d𝑡=(𝐷 𝑓 (𝒔)+𝜎𝜌𝜇𝐻(0)𝐷𝐺(0))𝜁−𝜎𝐺(0)𝜅(7.7)

d𝜅

d𝑡=−𝜖(𝜌𝜇D𝐻(0)𝜁+𝜅). (7.8)

Here, 𝜁∈C𝑑and 𝜅∈C.

Proof.

Due to Lemma 7.1.2 there are

𝑁2−𝑁

independent vectors

𝒘𝑙

(

𝑙=

. . .

𝑁2−𝑁

) spanning

the kernel of

𝐵

. Using the Gram-Schmidt procedure we find an ortho-normal basis for

ker(𝐵)=

span{𝒗1

. . .

𝒗𝑁2−𝑁}

. With this we define the

𝑁2×(𝑁2−𝑁)

matrix

𝑄=𝒗1,. . . ,𝒗𝑁2−𝑁

. Consider

now the (𝑁2+𝑁𝑑) × (𝑁2+𝑁𝑑)matrix

𝑅= I𝑁 𝑑 0 0

0(1/𝑟)𝐵𝑇𝑄!

7.1 The master stability function for adaptive complex networks 103

with left inverse

𝑅−1=©«

I𝑁 𝑑 0

0𝐵

0𝑄𝑇ª®®¬

i.e.,

𝑅−1𝑅=I𝑁2+𝑁 𝑑

. Introduce the new coordinates given by

𝑅 ˆ

𝝃

ˆ𝝌!= 𝝃

𝝌!

for which the varia-

tional equations (7.5) read

d𝑡 ˆ

𝝃

ˆ𝝌!=𝑅−1 𝑆−𝜎𝐵 ⊗𝐺(0)

−𝜖 𝜌𝐶 ⊗𝐷𝐻(0) −𝜖I𝑁2!𝑅 ˆ

𝝃

ˆ𝝌!.

We further observe

𝑅−1 𝑆−𝜎𝐵 ⊗𝐺(0)

−𝜖 𝜌𝐶 ⊗𝐷𝐻(0) −𝜖I𝑁2!𝑅=𝑅−1 𝑆−𝜎I𝑁⊗𝐺(0)0

−𝜖 𝜌𝐶 ⊗𝐷𝐻(0) −𝜖/𝑟𝐵𝑇−𝜖𝑄!

=©«

𝑆−𝜎I𝑁⊗𝐺(0)0

−𝜖 𝜌𝐿(𝐴) ⊗ 𝐷𝐻(0) −𝜖I𝑁0

−𝜖 𝜌𝑄𝑇𝐶⊗𝐷𝐻(0)0−𝜖I𝑁2−𝑁ª®®¬

These equations yield that there are the 𝑁𝑑 +𝑁many coupled differential equations left

d𝑡 ˆ

𝝃

˜𝝌!= 𝑆−𝜎I𝑁⊗𝐺(0)

−𝜖 𝜌𝐿(𝐴) ⊗ 𝐷𝐻(0) −𝜖I𝑁! ˆ

𝝃

˜𝝌!(7.9)

with ˜𝝌=ˆ𝝌1that determine the stability for the synchronous state and 𝑁2−𝑁equations

d𝑡¯𝝌=−𝜖 𝜌𝑄𝑇𝐶⊗𝐷𝐻(0)0−𝜖I𝑁2−𝑁©«

𝝃

˜𝝌

¯𝝌ª®®¬

with

¯𝝌=(ˆ𝝌𝑇

. . .

ˆ𝝌𝑇

𝑁)𝑇

which are unidirectionally coupled to the variables

𝝃

and

˜𝝌

and

can be solved explicitly once the latter once are known. By assumption, there is a unitary

matrix

𝐷𝐿(𝐴)=𝑈𝐻𝐿(𝐴)𝑈

where

𝐷𝐿(𝐴)

is the diagonalization of the Laplacian matrix

𝐿(𝐴)

Transforming the differential equations (7.9) by using the unitary transformation 𝑈, we get

d𝑡 𝜻

𝜿!= I𝑁⊗𝐷 𝑓 (𝒔)+𝜎𝜌𝐻(0)𝐷𝐿(𝐴)⊗𝐷𝐺(0)−𝜎I𝑁⊗𝐺(0)

−𝜖 𝜌𝐷𝐿(𝐴)⊗𝐷𝐻(0) −𝜖I𝑁! 𝜻

𝜿!

where 𝑈⊗I𝑑0

0𝑈! ˆ

𝝃

˜𝝌!= 𝜻

𝜿!.

By using the result in Proposition 7.1.3, the master stability function can be determined for any

diffusive dynamical system on a adaptive network. Note that Eq.

(7.8)

is explicitly solvable. the

104 7 Synchronization on adaptive complex network structures

solutions reads

𝜅=𝜅0𝑒−𝜖(𝑡−𝑡0)−𝜖 𝜌𝜇D𝐻(0)∫𝑡

𝑡0

𝑒−𝜖(𝑡−𝑡0)𝜁(𝑡0)d𝑡0.

Here, the first term vanishes as

𝑡→ ∞

. Hence, we can neglect it in order to study the asymptotic

dynamics of (7.7)–(7.8) which are rewritten in integro-differential form (as 𝑡→ ∞)

d𝜁

d𝑡=D𝑓(𝒔)𝜁+˜𝜇𝐻(0)D𝐺(0)𝜁+𝜖D𝐻(0)𝐺(0)∫𝑡

𝑡0

𝑒−𝜖(𝑡−𝑡0)𝜁(𝑡0)d𝑡0. (7.10)

with master function parameter

˜𝜇=𝜌𝜎𝜇

. We apply this result in the subsequent sections and

connect it to our understanding of the stability of multicluster states.

7.2 Stability islands in the presence of adaptation

In this section, we apply the general result on the master stability function for complex adaptive

networks, as it was derived in the previous section. For this, we use a phase oscillator model.

Note, however, that the master stability approach presented above is not restricted to this kind

of networks.

In analogy with Chapter 6, we consider the following adaptive network of

𝑁

coupled phase

oscillators

d𝜙𝑖

d𝑡=1−𝜎

𝑁

𝑗=1

𝑎𝑖 𝑗 𝜅𝑖 𝑗 sin(𝜙𝑖−𝜙𝑗+𝛼)(7.11)

d𝜅𝑖 𝑗

d𝑡=−𝜖𝜅𝑖 𝑗 +𝑎𝑖 𝑗 sin(𝜙𝑖−𝜙𝑗+𝛽). (7.12)

Let us further consider all networks with constant row-sum, i.e.,

𝑟=Í𝑁

𝑗=1𝑎𝑖 𝑗

for all

𝑖=

. . .

𝑁

Then the synchronous solution of (7.11)–(7.12) is given by

𝜙sync =(1−𝑟𝜎 sin(𝛼))𝑡(7.13)

𝜅sync =−sin(𝛽). (7.14)

Note that this sort of solutions have been already exhaustively analyzed for the case of a globally

coupled network in Chapter 4. The synchronous state

(7.13)

–

(7.14)

is an equilibrium relative to

the shift symmetry of

(7.11)

–

(7.12)

. Thus, the synchronous state is independent of the value

𝑟

In order to analyze the stability of the synchronous states

(7.13)

–

(7.14)

, we have to linearize the

equations

(7.11)

–

(7.12)

around these states. Using the result in Proposition 7.1.3, the stability of

the synchronous solution is governed by the two dimensional set of equations

d𝑡 𝜁

𝜅!= 𝜇𝜎cos(𝛼)sin(𝛽) −𝜎sin(𝛼)

−𝜖 𝜇 cos(𝛽) −𝜖! 𝜁

𝜅!,

where

𝜇∈C

is an eigenvalue of the Laplacian matrix

𝐿

corresponding to the base network

described by the adjacency matrix

𝐴

. The characteristic polynomial in

𝜆

of the latter system is of

7.2 Stability islands in the presence of adaptation 105

degree two and reads

𝜆2+(𝜖−𝜎𝜇 cos(𝛼)sin(𝛽))𝜆−𝜖𝜎𝜇 sin(𝛼+𝛽)=0. (7.15)

Note that for

𝜇=

0 the eigenvalues are

𝜆=−𝜖

and

𝜆=

0 which corresponds to the shift

symmetry. With this, the master stability function for the equations

(7.11)

–

(7.12)

is given by

max(<(𝜆1(˜𝜇)))

<(𝜆2(˜𝜇)))

where

𝜆1,2

solve Eq.

(7.15)

for a given set of parameters

𝛼

𝛽

and

𝜖

Here, the master function parameter is given by ˜𝜇=𝜎𝜇.

Corollary 7.2.1

Let

(7.13)

–

(7.14)

be the synchronous solution for the equations

(7.11)

–

(7.12)

with

𝜎 >

0. Further, assume the Laplacian matrix

𝐿(𝐴)

is symmetric, i.e., the network is undirected. Then the

synchronous solution is stable if the parameters 𝛼and 𝛽are from following set

{−𝜋 < 𝛼 +𝛽 < 0}Ùn−𝜋

2< 𝛼 < 𝜋

2,−𝜋 < 𝛽 < 0oÙ𝜋

2< 𝛼 < 3𝜋

2, 0 < 𝛽 < 𝜋 mod 2𝜋.

(7.16)

Proof.

Since

𝐿(𝐴)

is symmetric it is diagonalizable and has only positive real eigenvalues. As

shown above, we are allowed to use Proposition 7.1.3. From equation

(7.15)

, we find that the

eigenvalues

𝜆

are strictly negative if and only if

sin(𝛼+𝛽)<

0 and

(𝜖−𝜎𝜇 cos(𝛼)sin(𝛽)) >

The first of the latter inequalities implies the neccessary condition for stability

−𝜋 < 𝛼 +𝛽 <

The other inequality yields

𝜖

𝜎𝜇 >cos(𝛼)sin(𝛽).

Since 𝜖,𝜇,𝜎 > 0, the latter inequality implies the statement of the corollary.

In corollary 7.2.1, a relation between stability and the parameters

𝛼

and

𝛽

is established for

undirected networks. In case of directed networks, the Laplacian eigenvalue might be complex,

and hence the stability cannot be immediately deduced from the Eq.

(7.15)

. Figure 7.1 displays

the master stability function determined for different values for the parameter of

𝛽

. Here, we

fix

𝛼

and

𝜖

. The blue colored areas correspond to master function parameter

˜𝜇=𝜎𝜇

, where

𝜇

an eigenvalues o the Laplacian, that would lead to stable dynamics. By changing the control

parameter 𝛽various shapes of the stable regions are visible.

We, first, observe that the master stability function in all cases is symmetric with respect to

the real axis (

=( ˜𝜇)=

0). This symmetry stems from the fact that if

𝜆1,2

solve

(7.15)

for

𝜇

then

𝜆1,2

solve

(7.15)

for

𝜇

where the overline indicates the complex conjugate. Secondly, form some

parameter, e.g. Fig. 7.1(a,d,e), almost the whole half-space from the left and right of the imaginary

axis belong to the stable regime. Note that in case of no adaptation the stability of the synchronous

solution is solely described by the

cos(𝛼)

which is then scaled by

˜𝜇sin(𝛽)

. As a third observation,

we find parameters where almost all regions of the

˜𝜇

parameter space correspond to unstable

dynamics, e.g. Fig. 7.1(b,c,f). However, there exist small islands, i.e., bounded regions in

˜𝜇

parameter space, that correspond to stable dynamics. To understand the emergence of these

islands, we further anylze the boundary that separates the stable (

<(𝜆1,2)>

0) from the unstable

region (<(𝜆1,2)>0).

106 7 Synchronization on adaptive complex network structures

Figure 7.1:

The figure shows the master stability function for the equations

(7.11)

–

(7.12)

. Regions belonging to

negative Lyapunov exponents are colored blue. The one dimensional curve where at least one eigenvalue of

(7.15)

has zero real part is given as a black dotted line. Parameter: (a)

𝛽=−

0.8

𝜋

, (b)

𝛽=−

0.2

𝜋

, (c)

𝛽=−

0.02

𝜋

, (d)

𝛽=

0.05

𝜋

(e) 𝛽=0.1𝜋, and (f) 𝛽=0.98𝜋. The other parameters are 𝛼=0.3𝜋and 𝜖=0.01.

In order to describe the boundaries between regions in

˜𝜇

parameter space that would lead to

stable local dynamics or regions that would lead to unstable local dynamics, we consider

𝜆=

𝛾

Plugging this into the equation

(7.15)

we obtain the following parameterized expression for the

boundary

˜𝜇(˜𝛾)=𝜖−sin(𝛼)cos(𝛽)˜𝛾2+isin(𝛼+𝛽)˜𝛾+cos(𝛼)sin(𝛽)˜𝛾3)

sin2(𝛼+𝛽)+cos2(𝛼)sin2(𝛽)˜𝛾2

where we use

˜𝛾=𝛾/𝜖

. The curves given by the latter parametrization of the boundary are

displayed in Fig. 7.1 as dotted lines and agree very well with the actual borders. We further

notice that the emergence of the stability islands is a direct consequence of the presence of

adaptation. This can be seen by using

𝜆=

𝛾

and Eq.

(7.15)

where we put

𝜖=

0. This ansatz

results in the simple linear boundary equation

˜𝜇=

𝛾/(cos(𝛼)sin(𝛽))

which does not give rise to

stability islands. Due to the symmetry of the master stability function, a necessary condition to

observe find a stability island is that the curve

˜𝜇(˜𝛾)

possesses two crossings with the real axis.

From Eq. 7.15, we thus deduce the following condition

Corollary 7.2.2 The master stability function of (7.11)–(7.12)possesses stability islands if and only if

sin(𝛼+𝛽)

cos(𝛼)sin(𝛽)<0. (7.17)

The presence of stability islands is a very intriguing and unexpected effect introduced by adap-

tation. In the following section the impact of stability islands on the formation of multicluster is

described.

7.3 Stability islands and implications for the emergence of multicluster states 107

Figure 7.2:

Results for an adiabatic continuation in the coupling constant

𝜎

of the full synchronous solution

𝜙𝑖=

and

𝜅𝑖 𝑗 =−sin(𝛽)

for a globally coupled network of

𝑁=

200 oscillators

(7.11)

–

(7.12)

. The cluster parameter

𝑅𝐶

for

different values of

𝜎

is presented. As an inset, for the three values (a)

𝜎=

0.002, (b)

𝜎=

0.006, and (c)

𝜎=

0.025,

the master stability function, the phases

𝜙𝑖

of the final state, and the frequencies of the oscillators

Ω𝑖

are plotted.

The oscillators are sorted as in Fig 5.2. If more than or equal to ten (for numerical convenience) oscillators have the

same frequency (coherent groups) all nodes of this group are plotted as circles and with a respective color. All other

oscillators are plotted as an asterisk. The master function parameter

˜𝜇=𝜎𝜇

for the subnetworks induced by the

coherent groups are plotted together with the master stability function. The colors of each coherent group agree in all

three plots. Parameters: 𝛼=0.49𝜋,𝛽=0.88𝜋,𝜖=0.01.

7.3 Stability islands and implications for the emergence of

multicluster states

In this section, we explore the relation between the stability islands and the formation of

multiclusters. In order to do this, we perform the following numerical analysis. We choose

parameters

𝛼

and

𝛽

such that there exists a stability island. Then we prepare the initial condition

to be an in-phase synchronous state, i.e.,

𝜙𝑖=

0 and

𝜅𝑖 𝑗 =−sin(𝛽)

and integrate numerically

the system

(7.11)

–

(7.12)

with

𝑁=

200 oscillators for

𝑡=

30000. Thus, we perform an adiabatic

continuation starting from

𝜎=

0.001 and ending at

𝜎=

0.03. Note that

𝜎=

/𝑁=

0.005 lies

within this range. In order to illustrate the impact of the stability island on the dynamics, we

perform this analysis for a globally coupled as well as for a complex network. The complex

network which we have chosen for the subsequent analysis is connected, directed and has row

sum 𝑟=50. An illustration of the adjacency matrix can be found in the Appendix A.8.

Figure 7.2 shows the cluster parameter

𝑅𝐶

, see

(2.18)

in Chapter 2, for different values of the

coupling constant

𝜎

. Note that

𝑅𝐶=

1 refers to full in-phase synchrony of the oscillators. We

observe that for small

𝜎

the synchronous solution is stable, see Fig. 7.2(a). For the sake of

108 7 Synchronization on adaptive complex network structures

simplicity, the coupling matrix

𝜅𝑖 𝑗

is not displayed for all examples Fig. 7.2(a–c). Here, the

stability of the synchronous state is directly implied by the master stability function. We note

that all Laplacian eigenvalues

𝜇

of a globally coupled network are given by

𝜇=

0 and

𝜇=𝑁

. In

Fig. 7.2(a), all master function parameters 𝜎𝜇 lie within the stability islands.

By increasing the coupling constant, the master function parameter gets pushed out of the

regions of stability and the synchronous states becomes unstable. For intermediate values of

𝜎

the emergence of multiclusters with hierachical structure in the cluster size is observed, see

also Chapter 5. In Fig. 7.2(b) a multicluster states is shown with three clusters. For numerical

convenience, we only count groups of oscillators with more than or equal to 10 members as a

cluster. Note that for the systems

(7.11)

–

(7.12)

in-phase synchronous and antipodal clusters have

the same properties. In Chapter 5, the role of the hierarchical structure of the cluster sizes have

been discussed. We have found that due to the frequency difference the coupling between the

clusters vanish on average. Hence we argued that the stability is effectively described by the

stability of the one-clusters of which the multicluster consists, see Section 5.6. Here, we follow

this argument and consider the subnetworks induced by the groups of nodes (oscillators) with

the frequency. In the case of global coupling each of the subnetworks is globally coupled, as well.

The individual master function parameter for the induced subnetworks of the three clusters are

plotted together with the master stability function. For all three subnetworks the master function

parameters lie in the stable region and thus the corresponding one-clusters are stable and hence

the multicluster. This formation of a multicluster is remarkable and is well explained by the

master stability function. In addition it is in very good agreement with the numerical analysis in

Ref. [KAS17].

Increasing the coupling constant further shows the emergence of incoherence. In Figure 7.2(c), we

show the coexistence of a coherent and an incoherent cluster. These states, also called chimera-

like states, have been numerically analyzed in Refs. [

KAS17

KAS18

KAS18a

]. Surprisingly,

also for these states, the stability of the coherent cluster is determined by the master function

parameter corresponding to the subnetwork induced by the coherent nodes. This fact underlines,

once more, the necessity of the building block approach outlined in Chapters 4and 5.

In the following we show that the results obtained for the global network translate to set-ups

with complex network structure. In Figure 7.3, again, the cluster parameter

𝑅𝐶

for different

values of the coupling constant

𝜎

is presented. As in the case of global coupling, we observe that

the synchronous state is stable for small values of the coupling constant. After the destabilization

of the synchronous state, the emergence of a multicluster is shown, see Fig. 7.3(b). In contrast to

the globally coupled case, the coherent groups do not necessarily induce subnetworks with a

constant row sum. Hence, clusters with slightly perturbed phase distribution occur. However,

the stability of the single clusters is again well described by the master stability function since all

master function parameters for the individual clusters lie within the stability region. The same

holds true for the state presented in Fig. 7.3(c). Here, we observe the coexistence of an incoherent

and a coherent cluster. The stability of the latter is given by the master stability function.

7.4 Summary 109

Figure 7.3:

Results for an adiabatic continuation in the coupling constant

𝜎

of the full synchronous solution

𝜙𝑖=

0 and

𝜅𝑖 𝑗 =−sin(𝛽)

for a complex network (see Appendix A.8) of

𝑁=

200 oscillators

(7.11)

–

(7.12)

. The cluster parameter

𝑅𝐶

for different values of

𝜎

is presented. As an inset, for the three values (a)

𝜎=

0.003, (b)

𝜎=

0.007, and (c)

𝜎=

0.019,

the master stability function, the phases

𝜙𝑖

of the final state, and the frequencies of the oscillators

Ω𝑖

are plotted. If

more than or equal to ten ()for numerical convenience) oscillators have the same frequency (coherent groups) all

nodes of this group are plotted as circles and with a respective color. All other oscillators are plotted as an asterisk.

The master function parameter

˜𝜇=𝜎𝜇

for the subnetworks induced by the coherent groups are plotted together

with the master stability function. The colors of each coherent group agree in all three plots. Parameters:

𝛼=

0.49

𝜋

𝛽=0.88𝜋,𝜖=0.01.

7.4 Summary

In this chapter, we have studied the stability of the synchronous solution for a complex network

of adaptively coupled oscillators. For this, the well-known master stability approach is general-

ized to networks with adaptive and hence time evolving coupling weights. The master stability

approach is already well established and was generalized for various applications, e.g., to un-

derstand the stability of cluster synchronous states [

DAH12

PEC14

SOR16a

], to study systems

with single or distributed delays [

SOR07

FLU10b

DAH11b

HEI11

KYR14

WIL14

LEH15b

or even to allow for discontinuous dynamical systems [

LAD13

COO16

]. In addition, there

are several works where synchronization on networks with time evolving have been inves-

tigated [

BEL04a

ZHO06f

KOH14

SOR08

]. So far, however, all generalizations of the master

stability approach were either introduced for a static topological structure or the interdependence

between node and coupling dynamics was not taken into account.

In the case of an adaptive network the topological structure is non-constant in time and depends

on the state of the network nodes. In turn, the nodal dynamics depends on the coupling structure

given by the coupling adaptive weights. This subtle interplay changes the theoretical approach

entirely that has to be taken in order to derive a master stability function for adaptive networks.

110 7 Synchronization on adaptive complex network structures

In this chapter, we have developed a master stability approach and thus provided a novel

extension towards adaptive networks that can be regarded as non-standard. Studying network

synchronization by adapting the coupling weights [

YU12

LEL10

LEH14

HOE16

] is only one

field of application where our new approach could be applied.

The master stability approach has been further applied to the paradigmatic model of adaptively

coupled phase oscillators. We have illustrated several forms for the master stability function

with respect to different adaptation rules, i.e., different values for the control parameter

𝛽

Remarkably, the emergence of bounded regions that would lead to stable synchronous dynamics

has been observed in the master stability function. These bounded regions are called stability

islands. We have described the structure and formation of stability islands via a cubic curve and

derived an existence criterion. Additionally, we have analytically provided sufficient conditions

for the system parameters which imply the stability of the synchronous state on a undirected

background network structure.

In addition, the implications of the stability islands have been further explored. Using an

adiabatic continuation of the synchronous state with respect to the coupling constant

𝜎

, we have

obtained that the presence of a stability islands gives rise to the emergence of multicluster states

and chimera-like states. The stable existence of certain multiclusters and chimera-like states

have been shown numerically and analytically explained using the master stability function.

Thus, the findings do not only complement the results obtained in Chapters 5and 6. They also

underline the subtle interplay between cluster size and network structure for the stability of

complex synchronization patterns.

Multilayered adaptive networks8

In this chapter, we show that a plethora of novel patterns can be generated by multiplexing

adaptive networks. In particular, partial synchronization patterns like phase clusters and more

complex cluster states which are unstable in the corresponding monoplex network can be sta-

bilized, or even states which do not exist in the single-layer case for the parameters chosen,

can be born by multiplexing. Thus our aim is to provide fundamental insight into the com-

bined action of adaptivity and multiplex topologies. Hereby we elucidate the delicate balance

of adaptation and multiplexing which is a feature of many real-world networks even beyond

neuroscience [

GRO06b

SHA13b

WAR14

KLI16a

]. As local dynamics we use the paradigmatic

Kuramoto phase oscillator model, which is a simple generic model and has been successfully

applied in the modeling of synchronization phenomena in a wide range of natural and techno-

logical systems [BOC18].

A general multiplex network with

𝐿

layers each consisting of

𝑁

identical adaptively coupled

phase oscillators is described by

𝜙𝜇

𝑖=𝜔−1

𝑁

𝑗=1

𝜅𝜇

𝑖 𝑗 sin(𝜙𝜇

𝑖−𝜙𝜇

𝑗+𝛼𝜇𝜇)−

𝐿

𝜈=1,𝜈≠𝜇

𝜎𝜇𝜈 sin(𝜙𝜇

𝑖−𝜙𝜈

𝑖+𝛼𝜇𝜈),

¤𝜅𝜇

𝑖 𝑗 =−𝜖𝜅𝜇

𝑖 𝑗 +sin(𝜙𝜇

𝑖−𝜙𝜇

𝑗+𝛽𝜇),

(8.1)

where

𝜙𝜇

𝑖∈ [

0, 2

𝜋)

represents the phase of the

𝑖

th oscillator (

𝑖=

. . .

𝑁

) in the

𝜇

th layer

(

𝜇=

. . .

𝐿

), and

𝜔

is the natural frequency. The interaction between the phase oscillators within

each layer is described by the coupling matrix elements

𝜅𝜇

𝑖 𝑗 ∈ [−

1, 1

]

. The intra-layer coupling

weights

𝜅𝜇

𝑖 𝑗

are determined adaptively, whereas the inter-layer coupling weights

𝜎𝜇𝜈 ≥

0 are

fixed. The parameters

𝛼𝜇𝜈

are the phase lags of the interaction [

SAK86

]. The adaptation rate

< 𝜖 

1 is assumed to be a small parameter separating the time scales of the slow dynamics of

the coupling weights and the fast dynamics of the oscillatory system. The adaptation function in

each layer is chosen as in the Chapters 4and 5.

Let us note important properties of the model. First,

𝜔

can be set to zero without loss of

generality due to the shift-symmetry of Eq.

(8.1)

, i.e., considering the co-rotating frame

𝜙→𝜙+

𝜔𝑡

. Moreover, due to the existence of the attracting region

𝐺≡n𝜙𝜇

𝑖,𝜅𝜇

𝑖 𝑗 :𝜙𝜇

𝑖∈ (0, 2𝜋],|𝜅𝜇

𝑖 𝑗 | ≤ 1,

𝑖,𝑗=1, . . . ,𝑁,𝜇=1, . . . ,𝐿}

, one can restrict the range of the coupling weights to the interval

−1≤𝜅𝑖 𝑗 ≤1 [KAS17]. Finally, based on the parameter symmetries of the model

(𝜶,𝜷,𝝓,𝜿) ↦→ (−𝜶,𝜋−𝜷,−𝝓,𝜿),

(𝛼𝜇𝜇,𝛽𝜇,𝜙𝜇

𝑖,𝜅𝜇

𝑖 𝑗) ↦→ (𝛼𝜇𝜇 +𝜋,𝛽𝜇+𝜋,𝜙𝜇

𝑖,−𝜅𝜇

𝑖 𝑗),

where

𝜶

𝜷

𝝓

𝜿

abbreviate the whole set of variables and parameters, it is sufficient to analyze

the system within the parameter region

𝛼11 ∈ [

𝜋/

)

𝛼𝜇𝜇 ∈ [

𝜋)

(

𝜇≠

1),

𝛼𝜇𝜈 ∈ [

0, 2

𝜋)

(

𝜇≠𝜈

)

and 𝛽𝜇∈ [−𝜋,𝜋).

112 8 Multilayered adaptive networks

This chapter includes contents that have been published in [BER19b]. The chapter is organized

as follows. In Section 8.1, we introduce the notion of lifted states and show how states from

one-layer can be used to find states for multiplex systems. Additionally, the existence of new

states induced by the muliplex structure is shown in Section 8.2. We describe the properties of

these states and study numerically their robustness against heterogeneity ind the local dynamics.

In order to study the stability of lifted states, we develop the new methodology of the multiplex

decomposition in Section 8.3. Subsequently in Section 8.4, the method is applied to lifted states

to prove the stabilizing features of the multiplex set-up. In Section 8.5, we provide a brief

outlook on further applications of the multiplex decomposition. All findings are summarized in

Section 8.6.

8.1 Lifted states in multiplex networks

In Chapter 4, we exhaustively described the properties of one-cluster in case of one layer and

provided rigorous existence and stability results. In this section, we lift these finding to the case

of two layers.

Let us now consider one-cluster states in multiplex structures. In general, one-cluster states are

given as relative equilibria

𝜙𝜇

𝑖= Ω𝑡+𝑎𝜇

𝑖,

𝜅𝜇

𝑖 𝑗 =−sin(𝑎𝜇

𝑖−𝑎𝜇

𝑗+𝛽𝜇),(8.2)

with collective frequency

and relative phases

𝑎𝜇

𝑖∈ [

0, 2

𝜋)

. In order to connect one-cluster

states of the single layer case to one-cluster states of the multiple layer case, we introduce the

notion of lifted one-cluster states.

Definition 8.1.1

Let

(𝝓1

. . .

𝝓𝐿

𝜅1

. . .

𝜅𝐿)

be a one-cluster state

(8.2)

solving the equations

(8.1)

Then

(𝝓𝜇

𝜅𝜇)

is called a lifted one-cluster state if for each layer

𝜇=

. . .

𝐿

the state

(𝝓𝜇

𝜅𝜇)

is a

monoplex one-cluster, i.e.,

(𝝓𝜇(𝑡)

𝜅𝜇(𝑡))

solves

(6.1)

–

(6.2)

and hence the distribution of phases

𝒂𝜇

are of

splay, antipodal, or double antipodal type.

In the following we show that in duplex systems (

𝐿=

2) the phase difference of oscillators

between the layers Δ𝑎≡𝑎1

𝑖−𝑎2

𝑖takes only two values and solves

ΔΩ = 𝜎12 sin(Δ𝑎+𝛼12)+𝜎21 sin(Δ𝑎−𝛼12), (8.3)

where

ΔΩ ≡Ω(𝛼11

𝛽1) −Ω(𝛼22

𝛽2)

is given in

(4.6)

for the three different one-cluster states

(splay, antipodal, double antipodal).

To show this, suppose we have two one-cluster states where each is of either splay, antipodal, or

double antipodal type which form a duplex one-cluster

(8.2)

for

𝐿=

2 and

𝜙𝜇

𝑖= Ω(𝛼𝜇𝜇

𝛽𝜇)𝑡+

𝜔𝜇𝑡+𝑎𝜇

𝑖

(

𝜇=

1, 2), where

Ω(𝛼𝜇𝜇

𝛽𝜇)

is given by

(4.6)

and the coupling weights are given by

𝜅𝜇

𝑖 𝑗 =−sin(𝑎𝜇

𝑖−𝑎𝜇

𝑗+𝛽𝜇)

. We verify by directly inserting that

𝜙𝜇

𝑖

and

𝜅𝜇

𝑖 𝑗

solve Eq.

(2.26)

. For the

8.1 Lifted states in multiplex networks 113

given ansatz, Eq. (8.1) reads

Ω(𝛼11,𝛽1)+𝜔1=1

2cos(𝛼11 −𝛽1) − 1

2<𝑒−i(2𝑎1

𝑖+𝛼11+𝛽1)𝑍2(a1)

−𝜎12 sin(ΔΩ𝑡+Δ𝜔𝑡 +𝑎1

𝑖−𝑎2

𝑖+𝛼12),

and

Ω(𝛼22,𝛽2)+𝜔2=1

2cos(𝛼22 −𝛽2) − 1

2<𝑒−i(2𝑎1

𝑖+𝛼22+𝛽2)𝑍2(a2)

+𝜎21 sin(ΔΩ𝑡+Δ𝜔𝑡 +𝑎1

𝑖−𝑎2

𝑖−𝛼21).

where

ΔΩ = Ω(𝛼11

𝛽1)−Ω(𝛼22

𝛽2)

and

Δ𝜔=𝜔1−𝜔2

, respectively. Thus,

𝜙=(𝝓1

𝝓2

𝜅1

𝜅2)

is a

duplex one-cluster if

ΔΩ +Δ𝜔=0

which is equivalent to

ΔΩ = 𝜎12 sin(𝑎1

𝑖−𝑎2

𝑖+𝛼12)+𝜎21 sin(𝑎1

𝑖−𝑎2

𝑖−𝛼21)

for all

𝑖=

. . .

𝑁

. Note that

ΔΩ

is not necessarily zero even if the phase-lag parameters for both

layers agree. They can still differ in the type of one-cluster state. The former equation can be

written as

ΔΩ

𝐶=sin(𝑎1

𝑖−𝑎2

𝑖+𝜈)(8.4)

with

sin(𝜈)=1

𝐶𝜎12 sin(𝛼12)−𝜎21 sin(𝛼21), (8.5)

cos(𝜈)=1

𝐶𝜎12 cos(𝛼12)+𝜎21 cos(𝛼21),

where

𝐶=q(𝜎12)2+(𝜎21)2+2𝜎12𝜎21 cos(𝛼12 +𝛼21).

Whenever (𝜎12)2+(𝜎21)2+2𝜎12𝜎21 cos(𝛼12 +𝛼21) ≥ 0 and

(𝜎12)2+(𝜎21)2+2𝜎12𝜎21 cos(𝛼12 +𝛼21) ≥ ΔΩ2, (8.6)

Eq. (8.4) has the two solutions

𝑎1

𝑖−𝑎2

𝑖=arcsin(ΔΩ/𝐶) −𝜈

and

𝑎1

𝑖−𝑎2

𝑖=𝜋−arcsin(ΔΩ/𝐶) −𝜈

Considering the inverse function

arcsin : [−

1, 1

] → [−𝜋/

𝜋/

]

applied to Eq. (8.5) determines

𝜈

to be either

𝜈0

𝜋−𝜈0

, where

𝜈0:=arcsin(sin(𝜈))

and

sin(𝜈)

as given in (8.5). The second

equation for cos(𝜈)then fixes 𝜈to take one of the values.

The condition

(8.6)

is a relation between all parameters of the system which has to be fulfilled for

the existence of duplex relative equilibria. Note that for any given inter-layer coupling

𝜎12 ≠

and

𝛼12 +𝛼21 ≠±𝜋/

2 or

𝜋/

2 there exists a minimum coupling weight

𝜎21 <∞

such that the

114 8 Multilayered adaptive networks

Layer 1 Layer 2 Layer 1 Layer 2

∆a

ψ1ψ2

10 30 50 10 30 50 10 30 50 10 30 50

j j

φµ

(a)

2π

(b)

Figure 8.1:

Different duplex states of Eq.

(4.3)

(

𝐿=

2) for an ensemble of 50 oscillators in each layer with color-coded

coupling weights

𝜅𝜇

𝑖 𝑗

(upper panels, color code as in Fig.1), phases

𝜙𝜇

𝑗

(lower panels): Duplex one-cluster states (a) of

lifted splay type (

𝑅2(a𝜇)=

0) for

𝛼12/21 =

0.3

𝜋

𝜎12/21 =

0.07; (b) of lifted antipodal type (

𝑅2(a𝜇)=

1) for

𝛼12 =

0.3

𝜋

𝛼21 =

0.75

𝜋

𝜎12/21 =

0.62; (c) of double antidodal type (not a lifted state) for

𝛼12/21 =

0.05

𝜋

𝜎12/21 =

0.28; (d) of

lifted splay type for

𝛼12 =

0.3

𝜋

𝛼21 =

0.4

𝜋

𝜎12/21 =

0.8, and

𝜖=

0.01. In the lower panels phase differences between

the two layers are indicated by

Δ𝑎≡𝑎1

𝑖−𝑎2

𝑖

, and between the two new antipodal states (c) by

𝜓1

𝜓2

. Figure taken

from [BER19b].

lifted one-clusters exist. In case of unidirectional coupling, i.e.,

𝜎12 =

0, the condition gives the

minimum weight 𝜎21 ≥ΔΩ.

In Fig. 8.1 different duplex one-cluster states are presented that are observed by numerical

simulations. Panels (a),(b),(d) in Fig. 8.1 display lifted states of splay, antipodal, and splay type,

respectively. The phase distributions in both layers are the same but shifted by the constant

value

Δ𝑎

in agreement with the above equation. In contrast to the lifted states, Fig. 8.1(c) shows

another possible one-cluster for the duplex network. Due to the interaction of the two layers we

can find a phase distribution which is of double antipodal type in each layer but not a lifted state.

This means that these states are born by the duplex set-up. Moreover, in contrast to the other

examples the phase distribution between the layers does not agree,

𝜓1≠𝜓2

. For the monoplex

case, it has been shown that double antipodal states are unstable for any set of parameters, see

Chapter 4. Hence, finding stable double antipodal states which interact through the duplex

structure is unexpected.

8.2 Birth and robustness of phase clusters

For more insight into the birth of phase-locked states by multiplexing, Fig. 8.2 displays the emer-

gence of double antipodal states in a parameter regime where they do not exist in single-layer

networks. They are characterized by the second moment order parameter

𝑅2

. It is remarkable

that the new double antipodal state can be found for a wide range of the inter-layer coupling

8.2 Birth and robustness of phase clusters 115

φµ

2π

105

0.8

0.6

0.4

0.2

00 0.511.5

πψ1ψ2

hR2(φ2)i

hR2(φ1)i

σc

Figure 8.2:

Birth of double antipodal state in a duplex network (

𝑁=

12) for a wide range of inter-layer coupling

strength

𝜎=𝜎12 =𝜎21

. The solid lines are the temporal averages for the second moment order parameter

𝑅2

the individual layers (layer 1: black, layer 2: red). The error bars for

𝜎 < 𝜎𝑐

denote the standard deviation of the

temporal evolution of

𝑅2

. The dashed horizontal lines represent the unique values of

𝑅2

for the double antipodal

state in a monoplex network. The plot was obtained by adiabatic continuation of a duplex double antipodal state (see

inset) in both directions starting from

𝜎=

0.5. Parameters:

𝛼11/22 =

0.3

𝜋

𝛼12/21 =

0.05,

𝛽1=

0.1

𝜋

𝛽2=−

0.95

𝜋

, and

𝜖=0.01. Figure taken from [BER19b].

strength larger than a certain critical value

𝜎𝑐

, and is clearly different from those of the monoplex.

Below the critical value

𝜎𝑐

, the double antipodal states are no longer stable, and more complex

temporal dynamics occurs which causes temporal changes in

𝑅2

. This leads to non-vanishing

temporal variance indicated by the error bars in Fig. 8.2.

Robustness of the phase clusters for inhomogeneous natural frequencies

In the main text, we investigate a system of identical oscillators. There the existence of particular

phase cluster states of double-antipodal type is demonstrated in Figs. 8.1 and 8.2. In order to

show that these states are also present in a system of heterogeneous phase oscillators, we modify

the equations (8.1)–(2.26) as follows:

𝑑𝜙𝜇

𝑖

𝑑𝑡 =𝜔𝜇

𝑖−1

𝑁



𝑗=1

𝜅𝜇

𝑖 𝑗 sin(𝜙𝜇

𝑖−𝜙𝜇

𝑗+𝛼𝜇𝜇) −

𝐿



𝜈=1,𝜈≠𝜇

𝜎𝜇𝜈 sin(𝜙𝜇

𝑖−𝜙𝜈

𝑖+𝛼𝜇𝜈), (8.7)

𝑑𝜅𝜇

𝑖 𝑗

𝑑𝑡 =−𝜖𝜅𝜇

𝑖 𝑗 +sin(𝜙𝜇

𝑖−𝜙𝜈

𝑗+𝛽𝜇). (8.8)

For the numerical analysis of system

(8.7)

–

(8.8)

, we consider randomly uniformly distributed

natural frequencies on the interval

[−Δ𝜔

Δ𝜔]

. To check for the robustness of the phase cluster

presented in the inset of Fig. 8.2, the following steps are performed. We fix a random realization

of a uniform distribution. For any inter-layer coupling strength

𝜎=𝜎𝜇𝜈

we take the final state

from the simulation with

𝜔𝜇

𝑖=𝜔𝜈

𝑖=

0 (i.e., those obtained from Fig. 8.2) as initial condition.

We perform an adiabatic continuation of the state by running the simulation for

𝑡=

5000 and

increasing the width of the distribution

Δ𝜔

with a stepsize of 0.01. This is done until

Δ𝜔

reaches

0.5. The continuation is performed for each value of

𝜎

and for 10 different realizations of the

uniform distribution. Afterwards, we first check whether the final state has still the same form as

the one presented in Fig. 8.2. For this, we calculate the second moment order parameter for both

states in each layer individually, determine the difference of the order parameters for both layers,

116 8 Multilayered adaptive networks

πψ1

ψ2

πψ1

ψ2

φµ

2π

∆ω

(a) (b)

10 105 5 10 105 5

Figure 8.3:

The figure shows the range

Δ𝜔

vs where duplex one-cluster states in general (gray) and of the form

presented in Fig. 3 of the main text (red) can be found. For this the system

(8.7)

(8.8)

is integrated numerically for 10

different random uniform distributions of the natural frequencies in the interval

[−Δ𝜔

Δ𝜔]

. The results are obtained

by adiabatic continuation starting with the phase clusters found for

Δ𝜔=

0 (see Fig. 3 of the main text). Duplex

one-cluster states of double antipodal type with (a)

𝜎=

0.5,

Δ =

0.02 and (b)

𝜎=

0.5,

Δ =

0.07 are shown as insets.

Parameters: 𝛼11/22 =0.3𝜋,𝛼12/21 =0.05, 𝛽1=0.1𝜋,𝛽2=−0.95𝜋,𝜖=0.01, and 𝑁=12. Figure taken from [BER19b].

and set the upper limit to 0.01. States with a difference of less than the limit are considered to

possess the same form. Secondly, we check whether the final state is still a phase-locked state, i.e,

all oscillators are frequency-synchronized. The range and the boundaries up to which the final

state is still a duplex one-cluster state with or without the form from Fig. 8.2 are presented in

Fig. 8.3. For the boundaries and the range the mean value over the 10 realizations is determined

and the error bars indicate the standard deviation.

It is clearly visible that duplex one-cluster states of double-antipodal type are still present for

a considerable range of heterogeneity

Δ𝜔

of the natural frequencies. In the inset Fig. 8.3(a)

we present a duplex one-cluster double-antipodal state of the same form as shown in Fig. 8.2.

The phases are distorted slightly due to the frequency distribution but the double-antipodal

configuration is still clearly visible. The inset Fig. 8.3(b) shows another one-cluster state of

double-antipodal type. However, due to the frequency mismatch the phase distribution becomes

different compared with the one presented in Fig. 8.2.

A similar result, as it is shown in Fig. 8.3, is obtained if we consider a Gaussian instead of an

uniform distribution of frequencies, see Fig. 8.4.

8.3 Multiplex decomposition 117

φµ

2π

10 105 5

ψ1

ψ2

Figure 8.4:

The figure shows the standard deviation

𝜌

where duplex one-cluster states in general (gray) and of the

form presented in Fig. 3 of the main text (red) can be found. For this the system

(8.7)

(8.8)

is integrated numerically

for 10 different random normal distributions of the natural frequencies with standard deviation

𝜌

and zero mean. The

results are obtained by adiabatic continuation starting with the phase clusters found for

𝜌=

0 (see Fig. 3 of the main

text). A duplex one-cluster state of double antipodal type with

𝜎=

0.5,

Δ =

0.02 is shown as an inset. Parameters:

𝛼11/22 =0.3𝜋,𝛼12/21 =0.05, 𝛽1=0.1𝜋,𝛽2=−0.95𝜋,𝜖=0.01, and 𝑁=12. Figure taken from [BER19b].

8.3 Multiplex decomposition

In this section, we provide important tools and theorems to find the spectrum of multiplex

networks. These results are then subsequently used to analyze the stability of lifted one-cluster

states.

Let us start with a general result on the determinant of block matrices.

Theorem 8.3.1 Let Rbe a commutative subring of C𝑁×𝑁and let 𝑀∈R𝐿×𝐿. Then,

detC𝑀=detC(detR𝑀).

The proof can be found in Refs. [

SIL00

SOT17

]. This rather abstract result allows for a very

nice decomposition for pairwise commuting matrices and yields a useful tool to study the local

dynamics in multiplex systems.

Proposition 8.3.2

Let

𝑀∈C𝑁×𝑁

be a unitary diagonalizable matrix with

𝑀=𝑈𝐷𝑀𝑈𝐻

where

𝑈

𝑈𝐻

and

𝐷𝑀

are a unitary, its adjoint and a diagonal matrix, respectively. Let further

D𝑀

be the set of

simultaneously diagonalizable matrices to

𝑀

, i.e., the set of all matrices which commute pairwise with

𝑀

Then,

det 

𝐴11 · · · 𝐴1𝐿

.....

𝐴𝐿1· · · 𝐴𝐿𝐿



=det 

𝜎∈𝑆𝐿

sgn(𝜎)

𝐿



𝜇=1

𝐷𝐴𝜇,𝜎(𝜇)(8.9)

118 8 Multilayered adaptive networks

where 𝐴𝜇𝜈 ∈D𝑀for 𝜇,𝜈=1, . . . ,𝐿and 𝑆𝐿is the set of all permutations of the numbers 1, . . . ,𝐿.

Proof.

Consider any

𝐴

𝐵∈D𝑀

, then they are simultaneously diagonalizable with

𝑀

and hence

𝐴=𝐷𝐴𝑈𝐻

and

𝐵=𝑈𝐷𝐵𝑈𝐻

with the same

𝑈

. Thus, all

𝐴𝜇𝜈

can be diagonalized with the same

𝑈. Since 𝑈is unitary,i.e. (det𝑈)2=1, we find

det ©«

𝐴11 ··· 𝐴1𝐿

.....

𝐴𝐿1··· 𝐴𝐿𝐿ª®®®¬

=det ©«

𝐷𝐴11 ··· 𝐷𝐴1𝐿

.....

𝐷𝐴𝐿1··· 𝐷𝐴𝐿𝐿 ª®®®¬

by applying the block diagonal matrices

diag(𝑈

···

𝑈)

and

diag(𝑈𝐻

···

𝑈𝐻)

from the left and

right, respectively. The set of diagonal matrices with usual matrix multiplication and addition

form a commutative subring of

C𝑁×𝑁

. Applying Theorem 8.3.1 and using the well-known

determinant representation of Leibniz, the expression (8.9) follows.

Remark 8.3.1

The set

D𝑀

consists of all matrices which commute with

𝑀

and all the other

elements of D𝑀. In particular, the identity matrix I𝑁∈D𝑀for any 𝑀∈C𝑁×𝑁.

In the following, we apply the last result to a duplex and triplex system and connect the local

dynamics on the one-layer network to the multiplex case. We specify our consideration by

defining two special multiplex systems.

Definition 8.3.1

Suppose

𝐴

𝐵

𝐶∈C𝑁×𝑁

and

𝑚𝑖 𝑗 ∈C

(

𝑖

𝑗=

. . .

, 3). Then, the 2

𝑁×

𝑁

block

matrix

𝑀(2)= 𝐴 𝑚12I

𝑚21I𝐵!(8.10)

and the 3𝑁×3𝑁block matrix

𝑀(3)=©«

𝐴 𝑚12I𝑚13I

𝑚21I𝐵 𝑚23I

𝑚31I𝑚32I𝐶ª®®¬

(8.11)

are called (complex) duplex and triplex network, respectively.

Suppose we know how to diagonalize the individual layer topologies. The next result shows

how the eigenvalues of the individual layers are connected to eigenvalues of the multiplex

system. This will be done for the duplex and triplex network. For the proof of our following

statement, we provide two different ways.

The first approach makes use of the Schur decomposition [

BOY04

LIE15

], see

(4.11)

in Chapter 4,

which will be used, later on, in order to derive the characteristic equations. An extension of the

first approach to any number of layers in the network can be found by induction but is very

technical, see [

KOV99

POW11a

SOT17

]. The second approach uses Proposition 8.3.2 which

allows for a straightforward extension to any number of layers in a multiplex network.

8.3 Multiplex decomposition 119

Proposition 8.3.3

Suppose

𝐴

𝐵

𝐶∈C𝑁×𝑁

, they commute pairwise, and are diagonalizable with

diagonal matrices

𝐷𝐴

𝐷𝐵

𝐷𝐶

and unitary matrix

𝑈

. Then, the eigenvalues

𝜇

for the multiplex networks

𝑀(2)and 𝑀(3)can be found by solving the 𝑁quadratic

𝜇2−((𝑑𝐴)𝑖+(𝑑𝐵)𝑖)𝜇+(𝑑𝐴)𝑖(𝑑𝐵)𝑖−𝑚12𝑚21 =0 (8.12)

and cubic polynomial equations

𝜇3+𝑎2,𝑖𝜇2+𝑎1,𝑖𝜇+𝑎0,𝑖=0, (8.13)

respectively, with

𝑎2,𝑖=−((𝑑𝐴)𝑖+(𝑑𝐵)𝑖+(𝑑𝐶)𝑖)

𝑎1,𝑖=(𝑑𝐴)𝑖(𝑑𝐵)𝑖+(𝑑𝐴)𝑖(𝑑𝐶)𝑖+ (𝑑𝐵)𝑖(𝑑𝐷)𝑖

−𝑚12𝑚21 −𝑚13𝑚31 −𝑚23𝑚32

𝑎0,𝑖=𝑚12𝑚21(𝑑𝐶)𝑖+𝑚13𝑚31(𝑑𝐵)𝑖+𝑚23𝑚32(𝑑𝐴)𝑖

−(𝑑𝐴)𝑖(𝑑𝐵)𝑖(𝑑𝐶)𝑖−𝑚12𝑚23𝑚31 −𝑚13𝑚32𝑚21

and (𝑑𝐴)𝑖,(𝑑𝐵)𝑖, and (𝑑𝐶)𝑖being the respective diagonal elements of 𝐷𝐴,𝐷𝐵, and 𝐷𝐶.

Proof.

Since

𝐴

𝐵

𝐶

are diagonalizable and commute, Proposition 8.3.2 can be applied to both

matrices

𝑀(2)

𝑀(3)

. Anyhow, for the matrix

𝑀(2)

we will provide another proof using Schur’s

decomposition.

The determinant is an antisymmetric multilinear form. Thus, we can write

det 𝑀(2)−𝜇I2𝑁=det 𝐴−𝜇I𝑁𝑚12 ·I𝑁

𝑚21 ·I𝑁𝐵−𝜇I𝑁𝑚!=(−1)𝑁det 𝑚12 ·I𝑁𝐴−𝜇I𝑁

𝐵−𝜇I𝑁𝑚21 ·I𝑁.!

By assumption

𝐴

and

𝐵

are both diagonalizable with respect to the unitary transformation

matrix 𝑈, and so are 𝐴−𝜇Iand 𝐵−𝜇I. This allows us to write

det 𝑚12 ·I𝑁𝐴−𝜇I𝑁

𝐵−𝜇I𝑁𝑚21 ·I𝑁.!=det 𝑚12I𝑁𝐷𝐴−𝜇I𝑁

𝐷𝐵−𝜇I𝑁𝑚21I𝑁!

by applying the block diagonal matrices

diag(𝑈

···

𝑈)

and

diag(𝑈𝐻

···

𝑈𝐻)

from the left and

right, respectively. Now, using Schur’s decomposition (4.11) the determinant can written as

det 𝑚12I𝑁𝐷𝐴−𝜇I𝑁

𝐷𝐵−𝜇I𝑁𝑚21I𝑁!=𝑛𝑁det 𝑚−1

𝑛(𝐷𝐴−𝜇I𝑁) (𝐷𝐵−𝜇I𝑁)

=det (𝑚12𝑚21I𝑁−(𝐷𝐴−𝜇I𝑁) (𝐷𝐵−𝜇I𝑁)).

The last expression together with det 𝑀(2)−𝜇I2𝑁=0 yields the 𝑁quadratic equations (8.12).

Using that

(𝐴−𝜇I)

(𝐵−𝜇I)

(𝐶−𝜇I)

commute pairwise, Proposition 8.3.2 can be applied. We

120 8 Multilayered adaptive networks

find

det 𝑀(3)−𝜇I3𝑁=det ((𝐷𝐴−𝜇I𝑁)[(𝐷𝐵−𝜇I𝑁)(𝐷𝐶−𝜇I𝑁)−𝑚23𝑚32I𝑁]

−𝑚21 [𝑚12(𝐷𝐶−𝜇I𝑁)−𝑚13𝑚32I𝑁]+𝑚31 [𝑚12𝑚23I𝑁−𝑚13(𝐷𝐵−𝜇I𝑁)])

The last expression together with det 𝑀(3)−𝜇I3𝑁=0 yields the 𝑁cubic equations (8.13).

Let us briefly discuss some special cases for both the duplex and triplex network. Consider a

duplex network with master and slave layer, i.e., either

𝑚12 =

0 or

𝑚21 =

0. Then, the quadratic

equations (8.12) yield

(𝜇−(𝑑𝐴)𝑖) (𝜇− (𝑑𝐵)𝑖)=0. (8.14)

As shown in Proposition 8.3.2, the eigenvalues for special triplex networks can be found by

solving cubic equations. For the solution even closed forms exist. Despite this, the explicit form

of the solutions is rather tedious, in general. However, if we consider

𝐴=𝐵=𝐶

and a ring-like

inter-layer connection between the networks, i.e.,

𝑚12 =𝑚23 =𝑚31 =

0, then equation

(8.13)

has

the following solutions for all 𝑗=1, . . . ,𝑁

𝜇1=−(𝑑𝐴)𝑗+(𝑚13𝑚32𝑚21)1/3,

𝜇2=−(𝑑𝐴)𝑗+1

2i(i+√3)(𝑚13𝑚32𝑚21)1/3,

𝜇3=−(𝑑𝐴)𝑗−1

2(i+√3)(𝑚13𝑚32𝑚21)1/3,

where i denotes the imaginary unit. In analogy to equation

(8.14)

, a decoupling for the eigen-

values can be found. Consider three pairwise commuting matrices

𝐴

𝐵

𝐶

, and the structure

between the layers is a directed chain, i.e., 𝑚12 =𝑚13 =𝑚31 =𝑚23 =0 , then

(𝜇−(𝑑𝐴)𝑖) (𝜇− (𝑑𝐵)𝑖) (𝜇−(𝑑𝐶)𝑖)=0. (8.15)

In the following we apply the results of this section to the stability analysis of lifted duplex

one-clsuter states. In the last Section 8.5 of this chapter, further two different applications of the

multiplex decomposition are briefly discussed.

8.4 Stabilizing through multiplexing

In the following we show how the dynamics in a neighborhood of a single layer one-cluster

state can be lifted to the dynamics in a neighborhood of lifted multiplex one-cluster state, i.e., we

investigate and relate their local stability features. Everything is exemplified for antipodal type

states but can be generalized to lifted splay and lifted double antipodal states in a straightforward

manner. To study the dynamics around the one-cluster states described by Eq.

(8.2)

, we linearize

8.4 Stabilizing through multiplexing 121

Eq. (8.1) around these states:

𝛿𝜙𝜇

𝑖=1

𝑁

𝑗=1sin(Δ𝑎+𝛽𝜇)cos(Δ𝑎+𝛼𝜇𝜇)Δ𝜇𝜇

𝑖 𝑗 𝛿𝜙

−sin(Δ𝑎+𝛼𝜇𝜇)𝛿𝜅𝜇

𝑖 𝑗 −

𝑀

𝜈=1

𝜎𝜇𝜈 cos(Δ𝑎+𝛼𝜇𝜈)Δ𝜇𝜈

𝑖 𝑗 𝛿𝜙,

𝛿𝜅𝜇

𝑖 𝑗 =−𝜖𝛿𝜅𝜇

𝑖 𝑗 +cos(Δ𝑎+𝛽𝜇)Δ𝜇𝜇

𝑖 𝑗 𝛿𝜙(8.16)

where Δ𝜇𝜈

𝑖 𝑗 𝛿𝜙 ≡𝛿𝜙𝜇

𝑖−𝛿𝜙𝜈

𝑗.

Consider a duplex antipodal one-cluster state Eq.

(8.2)

with

𝑎1

𝑖∈ {

𝜋}

and

𝑎2

𝑖=𝑎1

𝑖−Δ𝑎

Eq.

(8.16)

can be brought to the duplex form in Definition

(8.3.1)

and possesses the following set

of eigenvalues

Duplex ={−𝜖,𝜆𝑖,1,𝜆𝑖,2,𝜆𝑖,3,𝜆𝑖,4𝑖=1,...,𝑁}

where 𝜆𝑖,1,...,4 solve the following 𝑁quartic equations

(𝜆+𝜖)2𝑚1𝑚2−h𝜆−𝜌1

𝑖,1·𝜆−𝜌1

𝑖,2+𝑚1(𝜆+𝜖)ih𝜆−𝜌2

𝑖,1𝜆−𝜌2

𝑖,2+𝑚2(𝜆+𝜖)i=0, (8.17)

with

𝑚1=𝜎12 cos(Δ𝑎+𝛼12)

𝑚2=𝜎21 cos(Δ𝑎−𝛼21)

and the eigenvalues

𝜌𝜇

𝑖,1,2 ≡𝜌𝑖,1,2(𝛼𝜇𝜇

𝛽𝜇)

for the monoplex system, see Corollary 4.2.3. Using the real parts of the eigenvalues of this

analysis for the duplex antipodal clusters, the stability for duplex antipodal states is found. All

details and the proof of (8.17) is provided in the appendix A.7.

We have seen that the stability analysis of the duplex system can be reduced to that of the

monoplex case. Thus, we are now able to analyze the stabilizing and destabilizing features

of a duplex network numerically and analytically. To illustrate the effect of multiplexing, the

interaction between two clusters of antipodal type is presented in Fig. 8.5. The stability of these

states is determined by integrating Eq.

(4.3)

numerically starting with a slightly perturbed lifted

antipodal state. The states are stable if the numerical trajectory is approaching the lifted antipodal

state. Otherwise, the state is considered as unstable. The black contour lines in Fig. 8.5 show the

borders of the stability regions in dependence of the coupling strength

𝜎21

, as calculated from

the Lyapunov exponents.The borders are in remarkable agreement with the numerical results.

We investigate two different situations in Fig. 8.5: In both panels the parameters for the first layer

𝛼11

𝛽1

are chosen such that the antipodal state is stable without inter-layer coupling. The stability

of the duplex antipodal states is displayed in the

(𝛼22

𝛽2)

parameter plane for several values of

the inter-layer coupling

𝜎21

. To compare the effects of the duplex network with the mono-layer

case, the stability regions for monoplex antipodals states are displayed, as well, as red hatched

areas. They are markedly different. In Figure 8.5(a), the two layers are connected unidirectionally

(

𝜎12 =

0). It can be seen that with increasing inter-layer coupling weight

𝜎21

the region of stability

for the lifted antipodal state also grows. Already for small values of the inter-layer couplings

𝜎21

, a stabilizing effect of the duplex network can be noticed. For

𝜎=

0.1 there exist already

regions for which the duplex antipodal state is stable but the corresponding monoplex state

would not be stable. The opposite effect is found as well where the duplex network destabilizes

a lifted state. Figure 8.5(b) shows the results for two layers with bidirectional coupling. In this

122 8 Multilayered adaptive networks

σ21

= 0.3σ

= 0

/π β

/π

(a) (b)

Figure 8.5:

Regions of stability (blue) and instability (white) of the lifted antipodal state in the

(𝛼22

𝛽2)

parameter

plane for different values of interlayer coupling (indicated by different blue shading)

𝜎21

, where regions of stronger

coupling

𝜎21

(lighter blue) include such of weaker

𝜎21

(darker blue). Stability regions for single-layer antipodal

clusters are indicated by red hatched areas. The inter-layer coupling is considered as (a) unidirectional (

𝜎12 =

and (b) bidirectional (

𝜎12 =𝜎21

). Parameters:

𝛼11 =

0.2

𝜋

𝛽1=−

0.8

𝜋

𝛼12 =

𝛼21 =

0.3

𝜋

, and

𝜖=

0.01. Figure taken

from [BER19b].

case, the duplex structure can have stabilizing and destabilizing effects, as well. Further, for

the bidirectional coupling we also notice a growth of the stability region with increasing

𝜎21

similar to the unidirectional case. However, the regions of stability grow at different rates in

dependence on

𝜎21

and non-monotonically with respect to the parameters

𝛼22

𝛽2

. Comparing

the size of the stability region for both cases, one can see that for small values of 𝜎21 the region

for bidirectional coupling is larger. In turn, for higher inter-layer coupling, the regions for the

unidirectional case are larger. It is worth noting that in Fig. 8.5 the stability regions for smaller

values of 𝜎21 are always contained in the region for larger values of 𝜎21.

8.5 Applications for the multiplex decomposition

In the previous section, we have shown how the multiplex decomposition can be applied to

understand the stability features of lifted states. Here, we provide two additional perspective

where the newly derived decomposition can be used to generalize existing results and make

others analytically accessible.

8.5.1 The master stability approach for multiplex networks

In Ref. [

TAN19

], the master stability function for dynamical systems on multiplex networks was

introduced. In their article, the authors considered diffusive systems similar to

(2.10)

. Consider

now the synchronous solution which solves



s=𝑓(s)

. The master stability function is then

derived from the following variational equations



𝝃=I𝑁 𝐿 ⊗𝐷 𝑓 (s) − 𝜎Lintra ⊗𝐻−𝜌Linter ⊗𝐺𝝃, (8.18)

where

𝐻

and

𝐺

are linear coupling functions. Here,

𝝃=x−I𝐿⊗I𝑁⊗s

and

x∈C𝐿·𝑁·𝑑

is the

system state vector where all individual nodal states are stacked on each other ordered by the

8.5 Applications for the multiplex decomposition 123

layer and node index. Further, the intra-layer Laplacian is defined is defined as

Lintra =

𝐿

𝑙=1

𝐿𝑙=©«

𝐿1

...

𝐿𝐿ª®®®¬

where

𝐿𝑘=©«

Í𝑁

𝑗=1𝑎𝑘

1𝑗

...Í𝑁

𝑗=1𝑎𝑘

𝑁 𝑗 ª®®®¬−𝐴𝑘.

The inter-layer Laplacian is defined as Linter =𝐿𝐼⊗I𝑁where

𝐿𝐼=©«Í𝐿

𝑙=1𝑚1𝑙

...Í𝐿

𝑙=1𝑚𝐿𝑙ª®®®¬−𝑀,

where

𝑚𝑘𝑙

with

𝑚𝑙𝑙 =

0 are the entries of the

𝐿×𝐿

matrix

𝑀

. All further necessary details on the

model equations can be found in Ref. [TAN19].

In Ref [

TAN19

], it was shown that if

Lintra

and

Linter

commute, a master stability equation for

system (8.18) can be found which reads

y=[𝐷 𝑓 (s)−𝛼𝐻 −𝛽𝐺]y,

where

𝛼=𝜎𝜆

𝛽=𝜌𝜇

𝜆

and

𝜇

are the (complex) eigenvalues of

Lintra

and

Linter

, respectively.

Note that if 𝐻=𝐺the master stability equation can be reduced to

y=[𝐷 𝑓 (s)−𝛾𝐻]y, (8.19)

with

𝛾=𝛼+𝛽

and

y∈C𝑑

. Equation

(8.19)

is called the master stability equation for the

composite system where a single supra-Laplacian matrix

𝜎Lintra +𝜌Linter

described the network

topology [DOM13,KIV14].

Direct evaluation shows that

Lintra,Linter=

0 is equivalent to

𝑚𝑘𝑙 𝐿𝑙−𝑚𝑙𝑘 𝐿𝑘=

0 for all

𝑙

𝑘=

. . .

𝐿

. Thus, to require

Lintra,Linter=

0 yields a linear dependence between the

individual layer topologies. Using Proposition 8.3.1 for composite system, the master stability

function can be used under much milder conditions.

Proposition 8.5.1

Let us consider the multiplex dynamical system

(8.18)

, with linear function

𝐻=𝐺

Suppose that further all

𝐿𝑙

𝑙=

. . .

𝐿

, commute pairwise. Then, the master stability equation is given

y=[𝐷 𝑓 (s)− 𝜇𝐻]y, (8.20)

and

𝜇=𝜇(𝜆1

𝑖

. . .

𝜆𝑙

𝑖)

being a non-linear function, mapping the

𝑖

th eigenvalues

𝜆𝑙

𝑖

(

𝑖=

. . .

𝑁

) of the

layer topologies

𝐿𝑙

, to the master function parameter

𝜇

(8.20)

. The non linear mapping is given as the

124 8 Multilayered adaptive networks

formal solution to the 𝐿th order polynomial equation

det Õ

𝜎∈𝑆𝐿"sgn(𝜎)

𝐿

𝑙=1

𝐷Lsupra

𝑙𝜎(𝑙)−𝜇I𝑁𝛿𝑙 𝜎 (𝑙)#!=0, (8.21)

where

Lsupra =𝜎Lintra +𝜌Linter

is a

𝐿×𝐿

block matrix divided into

𝑁×𝑁

matrices which we individu-

ally refer to with Lsupra

𝑘𝑙 ,𝑘,𝑙=1, . . . ,𝐿, and 𝛿𝑘𝑙 is the Kronecker symbol.

Proof.

Using

𝐻=𝐺

, the variation equation for

(8.18)

on the synchronous solution

s(𝑡)

is given

𝝃=I𝑁 𝐿 ⊗𝐷 𝑓 (s)−Lsupra ⊗𝐻𝝃,

By assumption all

𝐿𝑙

𝑙=

. . .

𝐿

, commute pairwise and

Linter

is a

𝐿×𝐿

block matrix consisting

𝑁×𝑁

identity matrices multiplied by scalar. Hence, Proposition 8.3.2 can be applied to

(Lsupra −𝜇I𝐿·𝑁)in order to diagonalize Lsupra.

With this Proposition, we have a very powerful tool in order to investigate not only the influence

of the multiplex structure network on the stability of the synchronous state but also the impact

of different layer topologies which are not necessarily linearly dependent. As an example, we

consider a duplex systems with [𝐿1,𝐿2]=0 and

Lsupra = 𝜎𝐿1+𝜌𝑚12I−𝜌𝑚12I

−𝜌𝑚21I𝜎𝐿2+𝜌𝑚21I!.

Knowing the eigenvalues for

𝐿1

and

𝐿2

, the master function parameter

𝜇

is determined using

Eq.

(8.12)

. The matrices

𝐿1

and

𝐿2

are Laplacian matrices which have at least one zero eigen-

value, corresponding to the so-called Goldstone mode

1=(

. . . 𝑁 −times ···

, 1

)𝑇

. As a result

there are two parameters

𝜇=

0 and

𝜇=𝜌(𝑚12 +𝑚21)

. The first value corresponds to the Gold-

stone mode

ˆ

1, ˆ

1𝑇

. The second parameter is exclusively induced by the duplex structure and

completely independent from the individual layer topologies. Further, we find that in case of

𝑚12 =−𝑚21

another zero parameter

𝜇

exists which corresponds to the eigenmode

ˆ

1, −ˆ

1𝑇

. The

other eigenvalues of the supra-Laplacian matrix

Lsupra

are given by the non-linear mappings

𝜇(𝜆1,𝜆2)=𝜎(𝜆1+𝜆2)+𝜌(𝑚12 +𝑚21)

2±1

2r𝜎(𝜆1−𝜆2)+𝜌(𝑚12 −𝑚21)2+4𝜌2𝑚12𝑚21 (8.22)

for which we have formally solved equation

(8.12)

with respect to

𝜇

. Let us consider two special

cases.

First, we assume that there is no connection from the second to the first layer. We have a master-

slave set-up which means that

𝑚12 =

0. With this, the master function parameter is

𝜇(𝜆1

𝜆2)=𝜎𝜆1

and

𝜇(𝜆1

𝜆2)=𝜎𝜆2+𝜌𝑚21

. Remarkably, in this set-up the stability of a synchronous state in a

duplex network is reduced to the pure one-layer system. The stability in the duplex system is

determined by the spectrum of the individual layer topologies where only in the second layer

the spectrum is shifted due to the interaction.

8.5 Applications for the multiplex decomposition 125

The second case starts from the consideration in [

TAN19

]. In particular, we consider

Lintra,Linter=

0 which leads to a pairwise linear dependence of all individual layer topologies, and hence

𝜆𝑙

𝑖=𝜆𝑖

for all

𝑙=

. . .

𝐿

and

𝑖=

. . .

𝑁

with

𝜆𝑙

𝑖∈C

. Taking this into account, the equation

for the master function parameter yields

𝜇(𝜆1

𝜆2)=𝜎𝜆1+𝜌𝑚12 +𝑚21

and

𝜇(𝜆1

𝜆2)=𝜎𝜆1

In order to see that the master function parameter agrees with the set for the parameter

𝛾

(8.19)

, we determine the eigenvalues of

Lintra

and

Linter

individually. Since

𝐿(1)

and

𝐿(2)

are

linearly dependent, the set of eigenvalues for

Lintra

consists of the eigenvalues of

𝐿1

with dou-

ble multiplicity. Using Proposition 8.3.3, we find that

Linter

has eigenvalues 0 and

(𝑚12 +𝑚21)

each with multiplicity

𝑁

. Since both Laplacian matrices commute, there exists a common set of

eigenvectors. We find

𝛾=𝜌𝜆(1)

for the eigenvector

(

1, 1

)𝑇⊗𝑣

and

𝛾=𝜌𝜆(1)+𝜎(𝑚12 +𝑚21)

for

the eigenvector

(𝑚12

−𝑚21)𝑇⊗𝑣

where

𝐿1𝑣=𝜆1𝑣

. Here, again everything boils down to a pure

one-layer set up with an additional shift.

8.5.2 Analytic treatment of diffusive dynamics on multiplex networks

In the previous section we considered the dynamics of linear systems as they are given by the

variational equation

(8.18)

. In the context of diffusive systems on complex networks, recently

linear diffusive processes were considered in order to study the dynamics on social as well

as transport networks [

BAR11d

GOM13

]. Compared with equation in [

GOM13

], the authors

investigate a duplex system (

𝐿=

2) with

𝑓=

𝜎=𝐷𝑘

(

𝑘=

1, 2),

𝐻

𝐺=I

, and

𝜌𝑚𝑘𝑙 =𝐷𝑥

Additionally, they considere weighted networks for which instead of

𝑎𝑘

𝑖 𝑗 ∈ {

0, 1

}

coupling

weights

𝜅𝑘

𝑖 𝑗

(

𝑘=

1, 2) were taken. Note that the former results on multiplex matrices still hold

true for weighted connections.

Let us assume that the super-Laplacian matrix is given by

Lsupra = 𝐷1𝐿1+𝐷𝑥I−𝐷𝑥I

−𝐷𝑥I𝐷2𝐿2+𝐷𝑥I!.

with

[𝐿1

𝐿2]=

0. Due to the structure of the supra-Laplacian we are allowed to apply Proposi-

tion 8.5.1. In accordance with [

GOM13

] and our former findings in Section 8.5.1, we have the

two eigenvalues

𝜇=

0 and

𝜇=

𝐷𝑥

corresponding to the Goldstone mode

(ˆ

1)𝑇

and the vector

(ˆ

−ˆ

1)𝑇

, respectively. The other eigenvalues are given as solution to the equations

(8.12)

and

read

𝜇𝑖=𝐷1𝜆1

𝑖+𝐷2𝜆2

𝑖+2𝐷𝑥

2±1

2r𝐷1𝜆1

𝑖−𝐷2𝜆2

𝑖2+4(𝐷𝑥)2,

where

𝜆1

𝑖

and

𝜆2

𝑖

are the eigenvalues of the Laplacian matrices

𝐿1

and

𝐿2

, respectively. With this

we derived a complete analytic expression for the spectral properties of the diffusive system

which was considered in [GOM13] without using any perturbation method.

126 8 Multilayered adaptive networks

8.6 Summary

In summary, we have proposed a concept to induce diverse partial synchronization patterns

(phase clusters) in adaptively coupled phase oscillator networks. While adaptive networks have

recently attracted a lot of attention in the fields of neuroscience and social sciences, biology,

engineering, and other disciplines, and multilayer networks are a paradigm for real-world

complex networks, little has been known about the interplay of multilayer structures and

adaptivity. We have aimed to fill this gap within a rigorous framework of theoretical analysis

and computer simulations. We have shown that multiplexing in a multi-layer with symmetry

can induce various stable phase cluster states like splay states, antipodal states, and double

antipodal states, in a situation where they are not stable or do not even exist in the single layer.

Further, we have developed a novel method for analysis of Laplacian matrices of duplex net-

works which allows for insight into the spectral structure of these networks, and can easily be

generalized to more than two layers. This new approach of multiplex decomposition has a broad

range of applications to physical, biological, socio-economic, and technological systems, ranging

from plasticity in neurodynamics or the dynamics of linear diffusive systems [

GOM13

SOL13a

]

to generalizations of the master stability approach [

PEC98

TAN19

] for adaptive networks. We

have used the multiplex decomposition to provide analytic results for the stability of lifted states

in the multilayer system. As local dynamics we have used the paradigmatic Kuramoto phase

oscillator model, supplemented by adaptivity of the link strengths with a phase lag parameter

which can model a whole range of adaptivity rules from Hebbian via spike-timing dependent

plasticity to anti-Hebbian.

Conclusion and Outlook9

Part I of this thesis was devoted to the analysis of cluster states in populations of globally

coupled neurons with synaptic plasticity. Here, we used mathematical as well as numerical tools

in order to understand the mechanism behind the emergence of frequency clustering.

In Chapter 3, we showed that adaptive neural networks are able to generate self-consistently

dynamics with different frequency bands. Here, each cluster corresponds to a strongly connected

component with a fixed frequency. Due to a sufficiently large difference of the cluster sizes and

frequencies, the inter-cluster interactions are suppressed, while the intra-cluster interactions are

enhanced. In this study, we described the mechanisms behind the formation and stabilization

of these clusters. In particular, we explained why the significant difference between the cluster

sizes is important for the decoupling of the clusters. From a broader perspective, the decoupling

of the clusters in our case is analogous to the decoupling of timescales in systems with multiple

timescales.

In addition, we presented a two-dimensional phenomenological model which allows for a de-

tailed study of the clustering mechanisms. Despite of the approximations made by the derivation,

the model coincides surprisingly well with the adaptive Hodgkin-Huxley network. Using the

phenomenological model, we found parameter regions of the plasticity function, where stable

frequency clustering can be observed.

Clustering behavior also emerges at the brain scale, where synchronized communities of brain

regions constituting large distributed functional networks can intermittently be formed and

dissolved [

DEC09

PON15

]. Such clustering dynamics can shape the structured spontaneous

brain activity at rest as measured by fMRI. In this study, we showed that slow oscillations

based on the modulation of synchronized neural activity can already be formed at the reso-

lution level of a single neural population if adaptive synapses are taken into account. These

modulations of the amplitude of the mean field can be generated in a stable manner, see Fig.3.9

and Ref. [

POP15

]. The mechanism relies on fluctuations of the extent of synchronization of

tonically firing neurons that are neurons which fire periodically provided they receive a constant

synaptic input. The fluctuations are caused by the splitting of the neural population into clus-

ters and the corresponding cluster dynamics. Our findings might contribute to the emergence

of slow brain rhythms of electrical (LFP, EEG) and metabolic (BOLD) brain activity reported

by [MAN07,MAG12a,MON08,ALV14].

However, other mechanisms for generating slow oscillations are possible. The papers [

BAZ02

COM03

] discussed the emergence of slow oscillatory activity (

1Hz) that can be observed in

vivo in the cortex during slow-wave sleep, under anesthesia or in vitro in neural populations.

The suggested mechanism relies on the corresponding modulation of the firing of individual

neurons, and the slow oscillation at the population level was proposed to be the result of very

slow bursting of individual neurons that synchronize across the neural population. In contrast,

the presented work shows that the slow oscillations of the population mean field can also

emerge when the firing of individual neurons is not affected. The neurons may tonically fire

128 9 Conclusion and Outlook

at high frequencies. The amplitude of the population mean field then oscillates at much lower

frequencies due to the slow modulation caused by the cluster dynamics.

Additionally, we would like to mention that the observed frequency clustering resembles phe-

nomenologically the weak chimera states [

ASH15

BIC16b

] where clusters with different fre-

quencies are formed in symmetrically coupled oscillators without adaptation. However the

properties and mechanisms of the appearance of such clusters are different from those presented

here, which are essentially based on the slow adaptation.

In Chapter 3.4 the phenomenological phase oscillator model was shown to be a very powerful

tool to predict the clustering dynamics of neuronal population with synaptic plasticity. It is well-

known that various models of weakly coupled oscillatory systems can be reduced to coupled

phase oscillators. For this, we implemented a simplified phase oscillator model in Chapters 4–5

which is able to describe the slow adaptive change of the network depending on the oscillatory

states. The slow adaptation is controlled by a time-scale separation parameter.

In this framework, we developed the analytical approach of building blocks for multicluster

states. We showed that for the case of phase oscillators the multiclusters are composed of certain

one-cluster states (building blocks) that are themselves phase-locked solutions of the adaptive

model. Moreover, we provided rigorous conditions under which multiclusters emerge and can

be found stable. For the analysis we divided the problem into two main step. Firstly, we analyzed

the single one-cluster states. Secondly, the multicluster states were analytically described and

were shown to inherit important properties from their building blocks.

In Chapter 4, we exhaustively described the one-cluster states in an adaptive network of glob-

ally coupled phase oscillators. In particular, the following three types of one-cluster (relative

equilibria) are possible building blocks for multicluster states: splay, antipodal, and double

antipodal. In order to understand the stability of these states, we performed a linear stability

analysis for the relative equilibria. The stability of these states was rigorously described, and the

impact of all parameters was shown. We showed that there exists an

𝑁−

2 dimensional family

of splay type cluster states which gives rise to

𝑁−

2 neutrally stable directions. Remarkably,

this property of splay states was also found in networks of pulse-coupled rotators [

CAL09a

]

and excitable neurons [

DIP12

]. We further proved that the double antipodal states are unstable

in the whole parameter range. They appear to be saddle-points in the phase space. While the

time-scale separation has no influence upon their existence, it plays an important role for the

stability the relative equilibria. The regions of stability in parameter space were presented for

different choices of the time-scale separation parameter. The singular limit (

𝜖→

0) and the limit

of instantaneous adaptation were analyzed. The latter shows that the stability region of the

splay and the antipodal states divide the whole space into two equally sized regions without

intersection. Instantaneous adaptation cancels multistability of these states. The consideration of

the singular limit showed that it differs from the case of no adaptation. Therefore, even for very

slow adaptation, the oscillatory dynamics alone is not sufficient to describe the stability of the

system.

Subsequently, the role of double antipodal states was discussed. We found that in a system of 3

oscillators these states are transient states in a small heteroclinic network between antipodal and

splay states. They appear to be metastable, i.e. observable for a relative long time and therefore

are physically important transient states. Moreover an additional analysis for an ensemble of 100

129

phase oscillators revealed the importance of the double antipodal states for the global dynamics

of the whole system.

In Chapter 5, we described the appearance of several different frequency cluster states. Starting

from random initial conditions, our numerical simulations showed two different types of states.

These are the splay and the antipodal type multicluster states. A third mixed type multicluster

state was found by using well prepared initial condition. For all these states the collective motion

of oscillators, the shape of the network, and the interaction between the frequency clusters

was presented in detail. It turned out that the oscillators are able to form groups of strongly

connected units. The interaction between the groups is weak compared to the interaction within

the groups.

While the one-cluster are relative equilibria of the model due to the phase-shift symmetry, the

multicluster states contain components with different frequencies, and, hence, they cannot

be reduced to an equilibrium by transforming into another co-rotating frame. As a result,

the study of multiclusters is more involved. However, to our surprise, we were still able to

find an explicit form of multiclusters with the components of the splay type. Remarkably, in

addition to its ring-like spatial structure that dynamically emerges, the network behaves in

such a case (quasi-)periodically in time such that the whole solution can be interpreted as a

spatio-temporal wave. Moreover, while the existence of the one-cluster states does not depend

on the time-scale separation parameter, the multicluster states crucially depend on the time-

scale separation. In fact, we provided an analysis showing that there exists a critical value for

the time-scale separation. Moreover, we showed that in the case of two-cluster states of splay

type the adaptation of the coupling weights must be at most half as fast as the dynamics of

the oscillators. For the splay clusters we analytically provided an explicit existence criterion.

This fact is of crucial importance for comparing dynamical scenarios induced by short-term or

long-term plasticity [FRO16].

In contrast to the splay type multicluster, the analysis of multicluster states of antipodal type is

more subtle due to the modulation of the frequency. More specifically, we looked at multiclusters

with bounded frequency modulation. For these types of multiclusters, we derived an asymptotic

expansion in the parameter

𝜖

that gave explicit existence conditions. In addition, we showed

analytically the existence of mixed multiclusters, which consist of clusters of splay type and

clusters of antipodal type. For the mixed multiclusters, the temporal behavior within one cluster

was shown to be slightly non-identical, namely, the oscillators possess the same averaged

frequency, but they still can have a bounded quasi-periodically modulated phase difference.

The stability of two-cluster states was analyzed numerically and was presented for different

values of the time-scale separation parameter. By assuming weakly interacting clusters, we

described the stability of the two-cluster with the help of the analysis of one-cluster states. The

simulations showed that there are no stable two-cluster states with clusters of the same size. We

provided an argument to understand this property of the system.

Moreover, the findings on multicluster solutions as they are reported in this work are in very good

agreement with previous results on adaptive neural networks [

POP15

CHA17a

]. Here, stable

multicluster states of coherently spiking neurons with weak but time-dependent inter-cluster

coupling were reported. With this work we shed some light on these generic time-dependent

network patterns.

130 9 Conclusion and Outlook

In Part II, we extended the findings of the first part towards more complex connectivity struc-

tures.

In Chapter 6, a model of adaptively coupled identical phase oscillators on a nonlocal ring

was studied. Various frequency synchronized states were observed including one-cluster, mul-

ticluster, and solitary states. Those states are similar to those found for a global base topol-

ogy [

KAS17

KAS18a

]. However, to account for the complex base topology, we introduced a new

definition of one-cluster states by means of connected induced subnetworks. This definition

allows furthermore to distinguish between multicluster and solitary states in a more rigorous

way than it had been done before.

Since one-cluster states form building blocks for multicluster states, see Part I of this thesis, we

first investigated the existence and stability properties of one-cluster states. Here, we introduced

a novel type of phase-locked states for complex networks, namely local splay states, and showed

that this class of states is nonempty for any nonlocal ring base topology. In particular, we proved

that rotating-wave as well as antipodal states are always phase-locked solutions. Compared

with the case of a global base topology, the different clusters of local splay type on a nonlocal

ring structure can possess different collective frequencies. In addition, we proved that local splay

cluster are always global splay cluster. This statement relates, therefore, local with global (with

respect to "spatial" structures in the network) properties of solutions.

The stability features of rotating-wave states were studied numerically and analytically. The

comparison of both approaches resulted in a very good agreement. Due to the analytic findings

for rotating-wave states on a nonlocal ring, we were able to describe their stability depending on

the coupling range

𝑃

and the wavenumber

𝑘

. The limiting case of global coupling, i.e.

𝑃=𝑁/

was shown to be in agreement with the results presented in Chapter 4.

An interesting feature of the system’s behavior are solitary states. They had been previously

found to emerge in the Kuramoto-Sakaguchi model with inertia [

JAR18

]. In this article, we

showed that solitary states are born in a homoclinic bifurcation and can be (de)stabilized in a

pitchfork bifurcation of periodic orbits. In order to show this, a two-dimensional effective model

was derived governing the dynamics of solitary states. In contrast to the Kuramoto-Sakaguchi

model with inertia, we observed a much more complicated bifurcation behavior. In particular,

three different solitary states are created due to two individual homoclinic bifurcations. Two of

these three solitary states, however, are unstable and bifurcate together with the stable solitary

states in a subcritical pitchfork bifurcation of periodic orbits.

Our results highlight the delicate interplay between adaptivity and complexity of the network

structure. Since this interplay has been rarely investigated from the mathematical viewpoint, so

far, this work raises many questions for future research which could be conducted for different

network structures beyond nonlocal rings, other dynamical models for the local dynamics,

nonidentical units or different adaptation rules.

In Chapter 7, we studied the stability of the synchronous state for a complex network of adap-

tively coupled oscillators. For this, the well-known master stability approach was generalized

for networks with adaptive and hence time evolving coupling weights. The master stability

approach is already well established and was generalized for various applications, e.g., to

understand the stability of cluster synchronous states [

DAH12

PEC14

SOR16a

] or to study

systems with single or distributed delays [

SOR07

CHO09

FLU10b

DAH11b

HEI11

KYR14

131

WIL14

LEH15b

]. Despite these advances, the novel extension towards adaptive networks can

be regarded as non-standard. So far all generalizations were introduced for a static topological

structure. However, in the case of an adaptive network the topological structure is non-constant

in time which changes the theoretical approach entirely.

The master stability approach was further applied to the paradigmatic model of adaptively

coupled phase oscillators. We illustrated several forms for the master stability function with

respect to different adaptation rules, i.e., different values for the control parameter

𝛽

. Remark-

ably, the emergence of bounded regions that would lead to stable synchronous dynamics was

observed in the master stability function. These bounded regions were called stability islands.

We described the structure and formation of stability islands via a cubic curve and derived an

existence criterion. Additionally, we analytically provided sufficient conditions for the system

parameters which imply the stability of the synchronous state on a undirected background

network structure.

Afterwards, the implications of the stability islands were further explored. Using an adiabatic

continuation of the synchronous state with respect to the coupling constant

𝜎

, we obtained

that the presence of a stability island gives rise to the emergence of multicluster states and

chimera-like states. The stable existence of certain multiclusters and chimera-like states were

shown numerically and analytically explained using the master stability function. Thus, the

findings do not only complement the results obtained in Chapters 5and 6. They also underline

the subtle interplay between cluster size and network structure for the stability of complex

synchronization patterns.

We showed how the generalized master stability approach is used to understand the mechanisms

behind the formation of frequency clusters. Another field of application could be the analysis

of synchronous states in real-world neuronal systems with synaptic plasticity. Such systems

have been discussed recently in order to study pathological diseases such as tinnitus [

TAS12a

However, the connectivity structure of the neuronal system was predefined and fixed in the

study [

TAS12a

]. Our master stability approach for adaptive networks is not restricted to a partic-

ular complex network. Thus, the new method could be used to introduce a generalized approach

in order to understand unwanted neural synchronization in complex neuronal networks with

synaptic plasticity.

In Chapter 8, we proposed a concept to induce diverse partial synchronization patterns (phase

clusters) in adaptively coupled multilayer phase oscillator networks. While adaptive networks

have recently attracted a lot of attention in the fields of neuro- and social sciences, biology,

engineering, and other disciplines, and multilayer networks are a paradigm for real-world

complex networks, little had been known about the interplay of multilayer structures and

adaptivity. We aimed to fill this gap within a rigorous framework of theoretical analysis and

computer simulations. We showed that multiplexing in a multilayer with symmetry can induce

various stable phase cluster states like splay states, antipodal states, and double antipodal states,

in a situation where they are not stable or do not even exist in the single layer. Further, we

developed a novel method for the analysis of Laplacian matrices of duplex networks which

allows for insight into the spectral structure of these networks, and can easily be generalized

to more than two layers. This new approach of multiplex decomposition has a broad range of

applications to physical, biological, socio-economic, and technological systems, ranging from

plasticity in neurodynamics or the dynamics of linear diffusive systems [

GOM13

SOL13a

] to

132 9 Conclusion and Outlook

generalizations of the master stability approach [

PEC98

TAN19

] for adaptive networks. We

have used the multiplex decomposition to provide analytic results for the stability of lifted

states in the multilayer system. As local dynamics we used the paradigmatic Kuramoto phase

oscillator model, supplemented by adaptivity of the link strengths with a phase lag parameter

which can model a whole range of adaptivity rules from Hebbian via spike timing-dependent

plasticity to anti-Hebbian.

While several applications of the multiplex decomposition were presented in this thesis, many

other possible fields of application remain untouched. An intriguing opportunity to apply

our new methodology arises in the control of power grid networks. For these particular real-

world networks, it is of great importance to retain synchronization when single nodes or lines

fail [

WIT16

]. Multiplexing with a control layer could be used to establish an efficient way to sta-

bilize power grids against topological changes [

TOT20

]. A control layer approach might also be

used to prevent the power grids from the cascading of line failures due to overloads [

SCH18i

As we saw in this thesis, adaptive networks of phase oscillators give rise to a plethora of

dynamical scenarios. These scenarios include complete and cluster synchronization, frequency

clustering, solitary states, and chimera-like states. The reason behind this variety of states is

the adaptivity of the coupling weights. This flexibility in the coupling structure allows each

individual oscillator to adapt indirectly its own frequency and thus leads to the formation

of different frequency clusters. Frequency adaptation is also a property of Kuramoto phase

oscillators with inertia that is a key model in the theory of power grids. The introduction of

inertia enables the oscillators to adapt their frequencies directly. Remarkably, frequency clustered

states are also widely observed in networks of Kuramoto phase oscillators with inertia [

OLM15a

JAR18

MEH18

TUM18

TAH19

TUM19a

HEL20

TOT20

]. Frequency adaptation is present but

implemented differently in both systems. Therefore, an open and very intriguing question arises

as to whether it is possible to relate the models of phase oscillators on adaptive networks to the

models of phase oscillators with inertia.

APPENDIX

Proof of results from the main textA

A.1 One-cluster states on globally coupled adaptive networks

Here we provide a proof of Proposition 4.1.1. We first need a preliminary lemma.

Lemma A.1.1

For a phase-locked solution

𝝓(𝑡)

𝑅2(𝝓(𝑡)) =

1for all

𝑡

if and only if

𝝓(𝑡)

is either an

in-phase or an anti-phase synchronous solution.

Proof.

As follows from table 4.1,

𝑅2(a)=1

𝑁Í𝑁

𝑗=1𝑒i2𝑎𝑗=

1 for all in-phase and anti-phase

solutions. Let us show the opposite. If

𝑅2(a)=

1, then

𝑒i2𝑎𝑗=𝑒i2𝑎1=

1 for all

𝑗

, since

Í𝑁

𝑗=1𝑒i2𝑎𝑗≤

Í𝑁

𝑗=1𝑒i2𝑎𝑗=𝑁

. Hence,

𝑎𝑗∈{0, 𝜋}

. The latter means that the phase-locked solution is either

in-phase, if all 𝑎𝑗have the same values, or anti-phase otherwise.

Now we present the proof of Proposition 4.1.1.

Proof. Substituting (4.3)–(4.4) into (2.25)–(2.26) we obtain ¤𝜅𝑖 𝑗 =0 and

𝜙𝑖(𝑡)=Ω= 1

𝑁

𝑗=1

sin(𝑎𝑖−𝑎𝑗+𝛽)sin(𝑎𝑖−𝑎𝑗+𝛼)

2cos(𝛼−𝛽) − 1

2<𝑒−i(2𝑎𝑖+𝛼+𝛽)𝑍2(𝑎). (A.1)

Therefore (4.3)–(4.4) are solutions if and only if the expression on the right-hand side of the

equation (A.1) is independent of the oscillators index

𝑖=

. . .

𝑁

. In particular, for any choice of

𝑎𝑖

the complex second order parameter is either zero or can be written as

𝑍2(a)=𝑅2(a)𝑒−i𝛾

. Thus,

according to (A.1)

𝑎𝑖

has to be such that

𝑅2(𝒂)=

0 or

cos(

𝑎𝑖+𝛼+𝛽+𝛾)

is independent of

𝑖

. For

any

𝛼

𝛽

and

𝛾

the latter requirement is equivalent with 2

𝑎𝑖∈{0, −2(𝛼+𝛽+𝛾)}

. Here, we made

use of the phase-shift symmetry of (2.25)–(2.26) by setting 2

𝑎1=

0. Due to the definition of the

complex order parameter the value for

𝛾

depends on the choice of the phase lags

𝑎𝑖

. Assuming

that one fraction

𝑞1=𝑄1/𝑁

of the oscillators have 2

𝑎𝑖=

0 with

𝑄1∈ {

. . .

𝑁−

}

and the other

fraction of oscillators 𝑞2=1−𝑞1have 2𝑎𝑖=−2(𝛼+𝛽+𝛾)one obtains

𝑞1+𝑞2𝑒−i2(𝛼+𝛽)𝑒−i2𝛾=𝑅2(a)𝑒−i𝛾.

which is equivalent to the equations

𝑞1cos(𝛾) +𝑞2cos(𝛾+2𝜗)=𝑅2(a),

𝑞1sin(𝛾) −𝑞2sin(𝛾+2𝜗)=0. (A.2)

with

𝜗=𝛼+𝛽

. Here, the first equation gives the value for the second order parameter while

the second equation determines

𝛾

. A special solution can be given if we set

𝑞2=

0, equivalently

136 A Proof of results from the main text

𝑞1=

1 . Then

𝛾=

0 and

𝛾=𝜋

would solve the equation above and thus 2

𝑎𝑖=

0 for all

𝑖=

. . .

𝑁

Note that, for both values of

𝛾

the value for the complex second order parameter coincide

𝑍2(a)=

1. This solution corresponds to

𝑅2(a)=

1. In any other case the last equation can be

written in the form

sin(𝛾−𝜈)=

0 which has two solutions

𝛾=𝜈

𝜈+𝜋

. Both solution would

coincide while determining 2𝑎𝑖. Writing (A.2) as

𝐶((𝑞1−(1−𝑞1)cos(2𝜗))sin(𝛾)−(1−𝑞1)sin(2𝜗)cos(𝛾))=0

where the normalization constant 𝐶is defined as

𝐶=q(𝑞1−(1−𝑞1)cos(2𝜗))2+(1−𝑞1)2sin2(2𝜗),

yields the equations

sin(𝜈)=sin(2𝜗)

r𝑞1

1−𝑞12+1−2𝑞1

1−𝑞1cos(2𝜗)

, (A.3)

cos(𝜈)=𝑞1− (1−𝑞1)cos(2𝜗)

q(1−𝑞1)2+𝑞2

1−2𝑞1(1−𝑞1)cos(2𝜗)

Therefore, considering the inverse function

arcsin : [−

1, 1

] → [−𝜋/

𝜋/

]

applied to (A.3)

determines

𝜈

to be either

𝜈0

𝜋−𝜈0

, where

𝜈0:=arcsin(sin(𝜈))

and

sin(𝜈)

as given in (A.3). The

second equation for

cos(𝜈)

then fixes

𝜈

to take one of the values. Thus,

𝛾

exists and is unique for

every

𝑞1∈ [

0, 1

)

. The Proposition is proved by taking into account that for a finite number

𝑁

oscillators 𝑞1takes values in the range from 1/𝑁to (𝑁−1)/𝑁and defining 𝛾:=−𝜓−𝜗.

A.2 Stability of one-cluster states on globally coupled networks

In order to study the local stability of one-cluster solutions described in Sec. 4.1, we linearise

the system of differential equations

(2.25)

–

(2.26)

around the phase locked states described by

𝜙𝑖= Ω𝑡+𝑎𝑖and 𝜅𝑖 𝑗 =−sin(𝑎𝑖−𝑎𝑗+𝛽). We obtain the following linearised system

𝑑

𝑑𝑡 𝛿𝜙𝑖=1

2𝑁

𝑁

𝑗=1

sin(𝛽−𝛼)𝛿𝜙𝑖−𝛿𝜙 𝑗+1

2𝑁

𝑁

𝑗=1

cos(2(𝑎𝑖−𝑎𝑗) +𝛼+𝛽)𝛿𝜙𝑖−𝛿𝜙 𝑗

−1

𝑁

𝑗=1

sin(𝑎𝑖−𝑎𝑗+𝛼)𝛿𝜅𝑖 𝑗 , (A.4)

and

𝑑

𝑑𝑡 𝛿𝜅𝑖 𝑗 =−𝜖𝛿𝜅𝑖 𝑗 +cos(𝑎𝑖−𝑎𝑗+𝛽)𝛿𝜙𝑖−𝛿𝜙𝑗. (A.5)

A.2 Stability of one-cluster states on globally coupled networks 137

Note that this set of equations can be brought into the following block form

d𝑡 𝛿𝝓

𝛿𝜅!= 𝐴 𝐵

𝐶−𝜖I𝑁2! 𝛿𝝓

𝛿𝜅!(A.6)

where

(𝛿𝝓)𝑇=(𝛿𝜙1,. . . ,𝛿𝜙𝑁)

(𝛿𝜅)𝑇=(𝛿𝜅11,. . . ,𝛿𝜅1𝑁,𝛿𝜅21,. . . ,𝛿𝜅𝑁 𝑁 )

𝐵=𝐵1··· 𝐵𝑁

𝐶=©«

𝐶1

𝐶𝑁ª®®®¬

, and

𝐴

𝐵𝑛

𝐶𝑛

are

𝑁×𝑁

matrices with

𝑛=

. . .

𝑁

. The elements of the block matrices

read

𝑎𝑖 𝑗 =

−1

2sin(𝛼−𝛽) − 1

𝑁sin(𝛽)cos(𝛼)

2𝑁

𝑁

𝑘=1

sin(2(𝑎𝑖−𝑎𝑘) +𝛼+𝛽),

𝑖=𝑗

2𝑁sin(𝛼−𝛽) −sin(2(𝑎𝑖−𝑎𝑗)+𝛼+𝛽),𝑖≠𝑗

𝑏𝑖 𝑗;𝑛=(−1

𝑁sin(𝑎𝑛−𝑎𝑗+𝛼),𝑖=𝑛

0, otherwise

𝑐𝑖 𝑗;𝑛=

0, 𝑗=𝑛,𝑖=𝑗

−𝜖cos(𝑎𝑛−𝑎𝑖+𝛽),𝑗=𝑛,𝑖≠𝑗

𝜖cos(𝑎𝑛−𝑎𝑖+𝛽),𝑗≠𝑛,𝑖=𝑗

0, otherwise

Throughout this section we will make use of the Schur complement [

BOY04

LIE15

] in order to

simplify characteristic equations as it has been introduced Eq.

(4.11)

in Chapter 4. In particular,

the determinant of the

𝑝×𝑝

matrix

𝑀= 𝐴 𝐵

𝐶 𝐷!

can be simplified to the following expression

by using (4.11)

det(𝑀)=det(𝐴−𝐵𝐷−1𝐶) ·det(𝐷).

This result is important for the subsequent stability analysis. So far, we have found the Lyapunov

coefficients for the rotating-wave states, see Corollary 4.2.3. The following two Lemmata are

needed to describe the stability of all antipodal, 4-phase cluster, and double antipodal states as

well.

Lemma A.2.1 Suppose 𝑀is a block square matrix of the form

𝑀= 𝐴 𝑚1ˆ

1𝑝,𝑞

𝑚2ˆ

1𝑞,𝑝𝐵!

where

𝐴

is a circulant

𝑝×𝑝

matrix,

𝐵

is a circulant

𝑞×𝑞

matrix,

1𝑝,𝑞

𝑝×𝑞

where all entries are 1

138 A Proof of results from the main text

and 𝑚1,𝑚2∈R. Then the eigenvector-eigenvalue pairs are given by

𝜆0

𝑘,. . . ,𝜆𝑝−1

𝑘, 0, . . . , 0𝑇,𝜇𝑘=

𝑝−1

𝑙=0

𝑎1(1+𝑙)𝜆𝑙

𝑘(A.7)

0, . . . , 0, 𝜌0

𝑙,. . . ,𝜌𝑞−1

𝑙𝑇,𝜈𝑙=

𝑞−1

𝑙=0

𝑏1(1+𝑙)𝜌𝑙

𝑙(A.8)

(1, . . . , 1, 𝑎1,. . . 𝑎1)𝑇, ¯𝜇=𝜇0+𝑚1𝑞𝑎1(A.9)

(1, . . . , 1, 𝑎2,. . . ,𝑎2)𝑇, ¯𝜈=𝜇0+𝑚1𝑞𝑎2(A.10)

with

𝜆𝑘=𝑒i2𝜋

𝑝𝑘

and

𝜌𝑙=𝑒i2𝜋

𝑞𝑙

for

𝑘=

. . .

𝑝−

1and

𝑙=

. . .

𝑞−

1, respectively, and

𝑎1

and

𝑎2

solve the equation

𝑎2+𝜇0−𝜈0

𝑚1𝑞𝑎−𝑚2𝑝

𝑚1𝑞=0

with 𝜇0=Í𝑝

𝑗=1𝑎1𝑗and 𝜈0=Í𝑞

𝑗=1𝑏1𝑗.

Proof. We can prove the Lemma by direct calculation and find

𝑀𝜆0

𝑘,. . . ,𝜆𝑝−1

𝑘, 0, . . . , 0𝑇

=𝐴𝜆0

𝑘,. . . ,𝜆𝑝−1

𝑘𝑇

𝑚2Í𝑝−1

𝑙=0𝜆𝑙

𝑘

=𝜇𝑘𝜆0

𝑘,. . . ,𝜆𝑝−1

𝑘, 0, . . . , 0𝑇.

Here, we use that

𝐴

is a circulant matrix and that

Í𝑝−1

𝑙=0𝜆𝑙

𝑘=

0 for all

𝑘=

. . .

𝑝−

1. Analogous

arguments hold for

(A.8)

. The last two eigenvector-eigenvalue pairs

(A.9)

–

(A.10)

can be obtained

𝑀(1, . . . , 1, 𝑎,. . . ,𝑎)𝑇= 𝐴(1, . . . , 1)𝑇+𝑚1𝑞𝑎 (1, . . . , 1)𝑇

𝑚2𝑝

𝑎(1, . . . , 1)𝑇+𝑎𝐵 (1, . . . , 1)𝑇!

= 𝜇0+𝑚1𝑞𝑎

𝑚2𝑝

𝑎+𝜈0!(1, . . . , 1, 𝑎,. . . ,𝑎)𝑇

which solves the eigenvalue problem if 𝑎is chosen to be either 𝑎1or 𝑎2.

Lemma A.2.2

Suppose we have a phase locked state with phases

𝑎𝑖∈ [

0, 2

𝜋)

. Then, the solution for the

characteristic equations corresponding to the linearised system

(4.9)

–

(4.10)

are given by

𝜆=−𝜖

with

multiplicity 𝑁2−𝑁and by the solution of the following set of equations

det ((𝐴−𝜆I𝑁) (𝜖+𝜆)+𝐵𝐶)=0.

Proof.

Applying Schur’s decomposition

(4.11)

to the linearised system in the block form

(A.6)

yields the result.

Proposition A.2.3

Suppose we have a state with phases

𝑎𝑖∈ {

𝜋

𝜓

𝜓+𝜋}

where

𝑖=

. . .

𝑁

. Further

set

𝑞1=𝑄1/𝑁

and

𝑞2=𝑄2/𝑁

, where

𝑄1

and

𝑄2

denote the numbers of phases which are either 0

A.2 Stability of one-cluster states on globally coupled networks 139

𝜋

and

𝜓

𝜓+𝜋

, respectively. Then, the linear system (4.9)–(4.10) possesses the following set

𝐿

eigenvalues

𝐿=(0)1,(−𝜖)(𝑁−1)𝑁+1,(𝜆1)𝑁1−1,(𝜆2)𝑁1−1,(𝜗1)𝑁2−1,(𝜗2)𝑁2−1,(𝜌1)1,(𝜌2)1

where 𝜆1and 𝜆2solve

𝜆2+1

2(sin(𝛼−𝛽) −𝑞1sin(𝛼+𝛽)−𝑞2sin(−2𝜓+𝛼+𝛽)+2𝜖)𝜆

−𝜖𝑞1sin(𝛼+𝛽)−𝜖𝑞2sin(−2𝜓+𝛼+𝛽)=0,

𝜗1and 𝜗2solve

𝜗2+1

2(sin(𝛼−𝛽) −𝑞1sin(2𝜓+𝛼+𝛽)−𝑞2sin(𝛼+𝛽)+2𝜖)𝜗

−𝜖𝑞1sin(2𝜓+𝛼+𝛽)−𝜖𝑞2sin(𝛼+𝛽)=0,

as well as 𝜌1and 𝜌2solve

𝜌2+1

2(sin(𝛼−𝛽) −𝑞1sin(2𝜓+𝛼+𝛽)−𝑞2sin(−2𝜓+𝛼+𝛽)+2𝜖)𝜌

−𝜖𝑞1sin(2𝜓+𝛼+𝛽)−𝜖𝑞2sin(−2𝜓+𝛼+𝛽)=0.

The multiplicities for each eigenvalue are given as subscripts.

Proof.

For an arbitrary solution of the form

𝜙𝑖= Ω𝑡+𝑎𝑖

we consider the linearized system (4.9)–

(4.10) in the block form

(A.6)

and apply Lemma A.2.2. The elements of the second term

𝐷:=𝐵𝐶

of Schur’s complement are then

𝑑𝑖 𝑗 =−𝜖

2𝑁sin(𝛼−𝛽) +sin(2(𝑎𝑖−𝑎𝑗)+𝛼+𝛽)

if 𝑖≠𝑗and

𝑑𝑖𝑖 =𝜖©«

2sin(𝛼−𝛽) − 1

𝑁sin(𝛼)cos(𝛽)+ 1

2𝑁

𝑁

𝑗=1

sin(2(𝑎𝑖−𝑎𝑗) +𝛼+𝛽)ª®¬.

Defining the matrix 𝑀:=(𝐴−𝜆I𝑁) (𝜖+𝜆)+𝐷we get

𝑚𝑖 𝑗 =(−𝜆2+(𝑎𝑖𝑖 −𝜖)𝜆+𝜖𝑎𝑖𝑖 +𝑑𝑖𝑖 𝑖=𝑗

𝜆𝑎𝑖 𝑗 +𝜖𝑎𝑖 𝑗 +𝑑𝑖 𝑗.𝑖≠𝑗

Using the assumption for the phases 𝑎𝑖, then one group of oscillators (group 𝐼) have 𝑎𝑖∈ {0, 𝜋}

and the remaining

𝑄2

oscillators (group

𝐼𝐼

) have

𝑎𝑖={𝜓

𝜓+𝜋}

. Putting this into the definition

140 A Proof of results from the main text

of 𝑚𝑖 𝑗 , we find that the whole square matrix 𝑀can be written as

𝑀=

𝑄1×𝑄1

z }| {

𝑚𝐼¯𝑚··· ¯𝑚

¯𝑚.......

.......¯𝑚

¯𝑚··· ¯𝑚 𝑚𝐼

𝑚1ˆ

1𝑄1,𝑄2

𝑚2ˆ

1𝑄2,𝑄1

𝑚𝐼 𝐼 ¯𝑚··· ¯𝑚

¯𝑚.......

.......¯𝑚

¯𝑚··· ¯𝑚 𝑚𝐼 𝐼

| {z }

𝑄2×𝑄2

ª®®®®®®®®®®®®®®®®®®®®®®¬

where

𝑚1

¯𝑚

𝑚𝐼

, and

𝑚𝐼 𝐼

are real values which depend on all the system parameters

𝛼

𝛽

𝜖

and

additionally on

𝜓

and

𝜆

. Note that all diagonal blocks are circulant matrices. The determinant is

invariant under basis transformations which is why we diagonalize the matrix

𝑀

and therewith

derive equations for the values

𝜆

. In order to do so, we look for the eigenvalues of

𝑀

determined

by the characteristic equation

det (𝑀−𝜇I𝑁)=0.

Due to the structure of

𝑀

we can apply Lemma A.2.1 and find the following set of eigenvalues

𝜇𝑘=−𝜆2−1

2(sin(𝛼−𝛽) −𝑞1sin(𝛼+𝛽)−𝑞2sin(−2𝜓+𝛼+𝛽)+2𝜖)𝜆

+𝜖𝑞1sin(𝛼+𝛽)+𝜖𝑞2sin(−2𝜓+𝛼+𝛽)

for

𝑘=

. . .

𝑄1−

1. Analogously, we obtain the equations for

𝜈𝑘

(

𝑘=

. . .

𝑄2−

1) where

𝑚𝐼

is substituted with

𝑚𝐼 𝐼

. The two other eigenvalue are given by

¯𝜇=𝜇0+𝑚1𝑄2𝑎1

and

¯𝜈=

𝜇0+𝑚1𝑄2𝑎2, respectively, where

𝜇0=𝑚𝐼+(𝑄1−1)¯𝑚

and 𝑎1,2 are given by

𝑎2+(𝑚𝐼−𝑚𝐼 𝐼 )+(𝑄1−𝑄2)¯𝑚

𝑚1𝑄2

𝑎−𝑚2𝑄1

𝑚1𝑄2

=0.

Considering the row sums of

𝑀

we find that all agree with

−𝜆2−𝜖𝜆

and therefore

¯𝜇=−𝜆2−𝜖𝜆

Resulting from this 𝑎1=−𝜆2+𝜖𝜆 +𝜇0/𝑚1𝑄2=1. Hence,

𝑎2=(𝑚𝐼 𝐼 −𝑚𝐼)+(𝑄2−𝑄1)¯𝑚

𝑚1𝑄2−1

A.2 Stability of one-cluster states on globally coupled networks 141

and we find

¯𝜈=𝑚𝐼 𝐼 +(𝑄2−1)¯𝑚−𝑚1𝑄2

=−𝜆2−1

2(sin(𝛼−𝛽) −𝑞1sin(2𝜓+𝛼+𝛽)−𝑞2sin(−2𝜓+𝛼+𝛽)+2𝜖)𝜆

+𝜖𝑞1sin(2𝜓+𝛼+𝛽)+𝜖𝑞2sin(−2𝜓+𝛼+𝛽)

After diagonalizing the matrix 𝑀the determinant can be easily written as

det(𝑀)=¯𝜇·𝜇1····· 𝜇𝑁1−1·¯𝜈·𝜈1·····𝜈𝑁2−1.

Therewith, finding

𝜆0𝑠

such that at least one of the eigenvalues of

𝑀

vanishes solves the initial

eigenvalue problem.

We will now sum up the results with the following corollaries

Corollary A.2.4

The set of eigenvalues of the linearised system (4.9)–(4.10) around all antipodal states

with 𝑎𝑖∈ {0, 𝜋}agrees with the set 𝐿in Prop. 4.2.2 for rotating-wave states with 𝑘=0, 𝑁/2.

Proof. Put 𝑄2=0 in Prop. A.2.3, then there is only the equation for 𝜆left.

Corollary A.2.5

The set of eigenvalues to the linearised system (4.9)–(4.10) around all 4-phase-cluster

states with

𝑎𝑖∈ {

𝜋/

𝜋

, 3

𝜋/

}

and

𝑅2(𝒂)=

0agrees with the set

𝐿

in Prop. 4.2.2 for 4-rotating-wave

states.

Proof.

The requirement

𝑅2(𝒂)=

0 yields

𝑄1=𝑄2

. The statement of this proposition follows by

using Prop. A.2.3.

Corollary A.2.6 For all 𝛼and 𝛽the double antipodal states are unstable.

Proof.

Suppose the polynomial equation

𝑝(𝑥)=𝑥2+𝑎𝑥 +𝑏=

0. This equation has two negative

roots if and only if

𝑏 >

0 and

𝑎 >

0 meaning that

𝑝(

0 and the vertex of the parabola is at

𝑥 <

0, respectively. In order to have stable double antipodal states these two conditions have to be

met by all three equations for

𝜆

𝜗

and

𝜌

in Proposition A.2.3. From the condition on the existence

of double antipodal states

(4.5)

we find

𝑞1sin(

𝜓+𝛼+𝛽) +𝑞2sin(−

𝜓+𝛼+𝛽)=−sin(𝛼+𝛽)

With this assumption on the quadratic equation and the latter equation, we find the following

two necessary conditions for the stability of double antipodal states, (1)

𝑞1sin(

𝜓+𝛼+𝛽) +

𝑞2sin(𝛼+𝛽)>

0 and (2)

𝑞1sin(

𝜓+𝛼+𝛽) +𝑞2sin(𝛼+𝛽)<

0. The two condition cannot be

equally fulfilled.

142 A Proof of results from the main text

A.3 Multicluster states of splay type

Proof of Proposition 5.2.1.We prove by direct substitution. Plugging (5.4) and (5.5) into (2.26) the

identity is obtained. Further, substituting (5.4) and (5.5) into (2.26) we obtain

Ω𝜇=1

2𝑁

𝑀

𝜈=1

𝜌𝜇𝜈

𝑁𝜇

𝑗=1cos(𝛼−𝛽+𝜓𝜇𝜈)−cos(2(ΔΩ𝜇𝜈𝑡+𝑎𝑖,𝜇−𝑎𝑗,𝜈) +𝛼+𝛽−𝜓𝜇𝜈)

𝑀

𝜈=1

𝜌𝜇𝜈 𝑛𝜈

2cos(𝛼−𝛽+𝜓𝜇𝜈)− 1

2<𝑒−i(2ΔΩ𝜇𝜈𝑡+2𝑎𝑖,𝜇+𝛼+𝛽−𝜓𝜇𝜈 )𝑍2(a𝜈).

If 𝑍2(a𝜇)=0 for all 𝜇=1, . . . ,𝑀then

Ω𝜇=1

𝑁

𝑀

𝜈=1

𝑁𝜇

𝑗=1

𝜌𝜇𝜈 cos(𝛼−𝛽+𝜓𝜇𝜈)

which agrees with the system (5.6) for the frequencies

Ω𝜇

. On the contrary, assume that the

multicluster phase-locked solution (5.4) and (5.5) solve the equation (2.25). Analog to Propo-

sition 4.1.1

Í𝑀

𝜈=1𝜌𝜇𝜈<𝑒−i(2ΔΩ𝜇𝜈 𝑡+2𝑎𝑖,𝜇+𝛼+𝛽−𝜓𝜇𝜈)𝑍2(a𝜈)

has to be independent of the oscillator

index

𝑖=

. . .

𝑁𝜇

and

𝑡∈R

for all

𝜇=

. . .

𝑀

. Take any

𝜇=

. . .

𝑀

and suppose

𝑍2(a𝜈)≠

for 𝜈∈𝐴with 𝐴⊆ {1, . . . ,𝑀}. Then

𝑀

𝜈=1

𝜌𝜇𝜈<𝑒−i(2ΔΩ𝜇𝜈 𝑡+2𝑎𝑖,𝜇+𝛼+𝛽−𝜓𝜇𝜈)𝑍2(a𝜈)=

𝜈∈𝐴

𝜌𝜇𝜈 𝑅2(a𝜈)cos(2ΔΩ𝜇𝜈𝑡+2𝑎𝑖,𝜇+𝛼+𝛽−𝜓𝜇𝜈 +𝛾𝜈)(A.11)

where 𝛾𝜈is defined as in Proposition 4.1.1, see type 2 or 3. For fixed 𝜇all frequency differences

ΔΩ𝜇𝜈

differ due to the assumption that the frequencies

Ω𝜇

are all pairwise different. This is why

only terms with

ΔΩ𝜇𝜈

and

ΔΩ𝜇𝜈0=−ΔΩ𝜇𝜈

(

𝜈

𝜈0∈𝐴

) are candidates to compensate each other in

the right hand side of (A.11) to give a constant value for

Ω𝜇

. Therefore, the number of clusters

with

𝑍2(a𝜈)≠

0 excluding the

𝜇

-th cluster which is under consideration has to be even, i.e.,

|𝐴\{𝜇}|

even for all

𝜇=

. . .

𝑀

. This already yields that

|𝐴|

odd and

𝐴={

. . .

𝑀}

. Consider

now

𝜇

such that

Ω𝜇=min𝜈∈1,...,𝑀Ω𝜈

. Then for every other

𝜈∈ {

. . .

𝑀}

with

ΔΩ𝜇𝜈 <

0 there

has to be

𝜈0∈ {

. . .

𝑀}

so that

−ΔΩ𝜇𝜈 = ΔΩ𝜇𝜈0= Ω𝜇−Ω𝜈0

. Hence,

Ω𝜈0<Ω𝜇

which contradicts

that

Ω𝜇=min𝜈∈1,...,𝑀Ω𝜈

. Therefore, for this choice of

𝜇∈1, . . . ,𝑀

the expression in (A.11)

cannot be constant contradicting the assumption made in the beginning.

A.4 Asymptotic expansions of multicluster states

In this section we give an analytic description of multicluster solutions in terms of an asymptotic

expansion. We consider therefore the expansion of

𝑟

-th order together with a multi-time scale

A.4 Asymptotic expansions of multicluster states 143

ansatz [VER06] as given in 5.11

𝜙(𝑟)

𝑖,𝜇(𝜖,𝑡):= Ω(𝑟)

𝜇(𝜏0,. . . ,𝜏𝑟)+𝑎𝑖,𝜇+

𝑟

𝑙=1

𝜖𝑙𝑝𝑖,𝜇;𝑙(𝑡)

𝜅(𝑟)

𝑖 𝑗,𝜇𝜈 (𝜖,𝑡):=

𝑟

𝑙=0

𝜖𝑙𝑘𝑖 𝑗,𝜇𝜈;𝑙(𝑡)

𝜇,𝜈=1, . . . ,𝑀

𝑖,𝑗=1, . . . ,𝑁𝜇

where

Ω(𝑟)

𝜇∈𝐶1(R𝑟+1)

is a function depending on the multi-time scales

𝜏𝑙:=𝜖𝑙𝑡

. We show

under under which conditions this expansion describes the time evolution for the system

(2.25)–(2.26).

The section is organized as follows. We first introduce some notations and prove some technical

lemmas that will help us to prove the main result in Proposition 5.3.1. For ease of notation, for

the remainder of the section indices are used as follows. Small Latin letters

𝑖

𝑗

in the subscript

are oscillator indices while Greek letters

𝜇

𝜈

represent cluster indices. These are separated by a

comma. Two further indices, separated by semicolon, are the coefficient index and the mode

index, respectively. The indices in superscript are either powers or the order for the expansion

which are then written in parenthesis.

The following definition is introduced to handle the order of approximation.

Definition A.4.1

Let

𝑓:R×R→R

and

𝑔:R×R→R

real functions. We define the following

notations:

1. 𝑓(𝜖

𝑡) ∈ 𝑂(𝑔(𝜖

𝑡))

𝜖→

0 on the interval

𝐼⊆R

if for any

𝑡∈𝐼

there exist

𝐶(𝑡)>

0 and

𝜖0(𝑡)>0 such that |𝑓(𝜖,𝑡)| < 𝐶(𝑡)|𝑔(𝜖,𝑡)| for all 𝜖 < 𝜖0(𝑡),

2. 𝑓(𝜖

𝑡) ∈ 𝑜(𝑔(𝜖

𝑡))

𝜖→

0 on the interval

𝐼⊆R

if for any

𝑡∈𝐼

and all

𝐶 >

0 there exist

𝜖0(𝑡)>0 such that |𝑓(𝜖,𝑡)| < 𝐶|𝑔(𝜖,𝑡)| for all 𝜖 < 𝜖0(𝑡).

Remark A.4.1

If the constants

𝐶

and

𝜖0

can be chosen independently of

𝑡∈𝐼

we say that

𝑓(𝜖,𝑡) ∈ 𝑂(𝑔(𝜖,𝑡)) (or 𝑓(𝜖,𝑡) ∈ 𝑜(𝑔(𝜖,𝑡))) as 𝜖→0 uniformly on 𝐼.

In order to find an expressions for the asymptotic expansion of the coupling weights

𝜅

, we

use the concept of the pullback attractor. It is defined as an nonempty, compact and invariant

set and well known from the theory of nonautonomous dynamical systems. For our purposes,

suppose we know the functions

𝜙𝑖(𝑡)

for all

𝑖=

. . .

𝑁

. Then, the differential equations

(2.26)

nonautonomous and can be solved explicitly by

𝜅𝑖 𝑗 (𝑡):=𝜅𝑖 𝑗,0𝑒−𝜖(𝑡−𝑡0)−𝜖∫𝑡

𝑡0

𝑒−𝜖(𝑡−𝑡0)sin(𝜙𝑖(𝑡0) −𝜙𝑗(𝑡0) +𝛽)d𝑡0(A.12)

with 𝜅𝑖 𝑗,0 ∈ [−1, 1]for all 𝑖,𝑗=1, . . . ,𝑁. For this, the pullback attractor Ais given by the set

A:=Ø

𝑡∈R{(𝜅(𝑡))∗}(A.13)

144 A Proof of results from the main text

where

(𝜅𝑖 𝑗 (𝑡))∗:=−lim

𝑡0→−∞ 𝜖∫𝑡

𝑡0

𝑒−𝜖(𝑡−𝑡0)sin(𝜙𝑖(𝑡0) −𝜙𝑗(𝑡0) +𝛽)d𝑡0(A.14)

for all

𝑖

𝑗∈ {

. . .

𝑁}

. Note that

(A.14)

is the unique bounded solution of

(2.26)

, see Theorem

IV.1.1 of [

HAL80

]. We remark the following important properties. For given functions

𝜙𝑖

the

equations

(2.26)

possess the compact absorbing set

𝐺:={𝜅𝑖 𝑗 :𝜅𝑖 𝑗 ∈ [−

1, 1

]

𝑖

𝑗=

. . .

𝑁}

. Hence,

the pullback attractor exists, c.f. Theorem 3.18 in [

KLO11

], and is unique due to Proposition 3.8

[

KLO11

]. Moreover,

(𝜅(𝑡))∗

is a solution for the nonautonomous system which can be shown

by direct computation. We call

𝜅(𝑡))∗

the parametrization of the pullback attractor and use it

in order to find an analytic expression for the (pseudo-)multicluster states. Note that we have

already seen such parametrizations explicitly in

(4.4)

and

(5.5)

. For more details regarding

nonautonomous systems and the pullback attractor we refer the reader to [RAS06a,KLO11].

We use the following notations for the sake of brevity.

M:={m=(𝑚1,. . . ,𝑚𝑀):𝑚1,. . . ,𝑚𝑀∈Z},

𝑐m:=𝑐𝑚1,...,𝑚𝑀,

ΔΩ(m):=

𝑀

𝜇=1

𝑚𝜇Ω𝜇.

Furthermore, we say that two elements

n∈M

are equivalent

m∼n

if and only if

ΔΩ(m)=

ΔΩ(n)

. The corresponding quotient space is denoted by

M:=M/∼

. If

Ω𝜇

is considered as

frequencies the equivalence relation factors out all resonant linear combinations of those. Let

us further define

M(𝑓)

as the set of all

(𝑚1

. . .

𝑚𝑀)

such that the function

𝑓

can be written as

𝑓=Ím∈M(𝑓)𝑐m𝑒iΔΩ(m)𝑡

for some

𝑐m∈C

. Finally, we introduce the shorthand notion

(𝑚𝜇𝑛𝜈):=

(

. . .

, 0,

𝑚𝜇

, 0,

. . .

, 0,

𝑚𝜈

, 0,

. . .

, 0

)

with

𝑚𝜇=𝑚

and

𝑚𝜈=𝑛

for further convenience if only

frequencies of two distinguished clusters are considered.

The proof of Proposition 5.3.1 makes use of several lemmas and is presented at the end of this

section. Overall, we aim to describe the following particular form for the dynamical behavior

of the phase oscillators. The phases of the oscillators

𝜙𝑖,𝜇

form a pseudo multicluster, see

Definition 5.5.1. Further, the bounded modulations for the phases of each oscillator are given

as Taylor expansions in

𝜖

with periodic coefficients that can be expressed as Fourier sums with

even modes.

To determining the asymptotic expansion explicitly, derivatives of composed function have to

be carried out. The following Lemma provides us with a general form.

Lemma A.4.1 Suppose we have 𝑛-times differentiable real functions 𝑓and 𝑔. Let

𝑇𝑛:={(𝑘1,. . . ,𝑘𝑛): 1𝑘1+2𝑘2+···+𝑛𝑘𝑛=𝑛,𝑘1,. . . ,𝑘𝑛∈N0}

denote the partitions of

𝑛

. The composition

(𝑓◦𝑔)

is also

𝑛

-times differentiable and the

𝑛

th derivative can

A.4 Asymptotic expansions of multicluster states 145

be written as

𝐷𝑛

𝑥(𝑓◦𝑔)(𝑥0)=Õ

(𝑘1,... ,𝑘𝑛)∈𝑇𝑛

𝑛!

𝑘1!· ··· ·𝑘𝑛!𝐷𝑘1+...+𝑘𝑛

𝑥𝑓)◦𝑔(𝑥0)

𝑛

𝑚=1𝐷𝑚𝑔

𝑚!𝑘𝑚

(𝑥0). (A.15)

Proof. See [ABR88, pp. 95–96].

This expression for the 𝑛th-derivative is also known as the Faà di Bruno formula.

Lemma A.4.2 Suppose the phase oscillators behave as

𝜙𝑖,𝜇(𝜖,𝑡):= Ω𝜇𝑡+𝑎𝑖,𝜇+

𝑟

𝑙=1

𝜖𝑙𝑝𝑖,𝜇;𝑙(𝑡),𝜇=1, . . . ,𝑀

𝑖=1, . . . ,𝑁𝜇

with

Ω𝜇∈R

𝑎𝑖,𝜇∈T1

and there are

𝑚𝑖,𝜇;𝑙∈N0

such that the bounded functions

𝑝𝑖,𝜇;𝑙:R→R

are

given by the finite multi-Fourier series

𝑝𝑖,𝜇;𝑙(𝑡)=Õ

m∈˜

M(𝑝𝑖,𝜇;𝑙)

𝑐𝑖,𝜇;𝑙;m𝑒iΔΩ(m)𝑡

with |˜

M(𝑝𝑖,𝜇;𝑙)| =𝑚𝑖,𝜇;𝑙.

Then, for

𝑠≤𝑟

there exist functions

𝑘𝑖 𝑗,𝜇𝜈;𝑙(𝑡)

such that the asymptotic expansion of the pull-back

attractor 𝜅𝑖 𝑗,𝜇𝜈 ∗defined in (A.14) can be written as

𝜅𝑖 𝑗,𝜇𝜈 ∗=

𝑠

𝑙=0

𝜖𝑙𝑘𝑖 𝑗,𝜇𝜈;𝑙(𝑡)+ ˆ

𝑅𝑖 𝑗,𝜇𝜈 (𝜖,𝑡)

𝜇,𝜈=1, . . . ,𝑀

𝑖=1, . . . ,𝑁𝜇

𝑗=1, . . . ,𝑁𝜈

, (A.16)

where

𝑅𝑖 𝑗,𝜇𝜈 (𝜖

𝑡) ∈ 𝑜(𝜖𝑠)

uniformly on

𝜖→

0and

𝜅(𝑠)

𝑖 𝑗,𝜇𝜈 (𝑡):=Í𝑠

𝑙=0𝜖𝑙𝑘𝑖 𝑗,𝜇𝜈;𝑙(𝑡)

. The

𝜅(𝑠)

𝑖 𝑗,𝜇𝜈 (𝑡)

called the

𝑠

-th-order asymptotic approximation of

𝜅𝑖 𝑗,𝜇𝜈 ∗

. Further, all

𝑘𝑖 𝑗,𝜇𝜈;𝑙

can also be written as a

Fourier sum.

Proof.

For fixed functions

𝜙𝑖,𝜇(𝑡)

the nonautonomous systems corresponding to the differential

equation (2.26) possess the following parametrization of the pullback attractor

(𝜅𝑖 𝑗,𝜇𝜈 (𝑡))∗:=−lim

𝑡0→−∞ 𝜖∫𝑡

𝑡0

𝑒−𝜖(𝑡−𝑡0)sin(𝜙𝑖,𝜇(𝑡0) −𝜙𝑗,𝜈(𝑡0) +𝛽)d𝑡0.

Using (A.15) with

𝑓=sin(𝜙𝑖,𝜇−𝜙𝑗,𝜈+𝛽)

and

𝑔= ΔΩ𝜇𝜈𝑡+𝑎𝑖 𝑗,𝜇𝜈 +𝛽+Í𝑟

𝑙=1𝜖𝑙𝑝𝑖 𝑗,𝜇𝜈;𝑙

we can

perform a Taylor expansion of

𝑓(𝜖

𝑡)=sin(𝜙𝑖,𝜇−𝜙𝑗,𝜈+𝛽)

around

𝜖=

0. Due to Theorem 2.4.15

in [ABR88, pp. 93–94] we get

𝑠

𝑙=0

𝜖𝑙𝑟𝑖 𝑗,𝜇𝜈;𝑙(𝛽,𝑡)+𝜖𝑠𝑅𝑖 𝑗,𝜇𝜈 (𝜖,𝑡):=sin(ΔΩ𝜇𝜈𝑡+𝑎𝑖 𝑗,𝜇𝜈 +𝛽+

𝑟

𝑙=1

𝜖𝑙𝑝𝑖 𝑗,𝜇𝜈;𝑙(𝑡))

=sin(ΔΩ𝜇𝜈𝑡+𝑎𝑖 𝑗,𝜇𝜈 +𝛽)+𝜖cos(ΔΩ𝜇𝜈𝑡+𝑎𝑖 𝑗,𝜇𝜈 +𝛽)𝑝𝑖 𝑗,𝜇𝜈;1 +···+𝜖𝑠𝑅𝑖 𝑗,𝜇𝜈 (𝜖,𝑡)

146 A Proof of results from the main text

where the abbreviations

𝑝𝑖 𝑗,𝜇𝜈;𝑙:=𝑝𝑖,𝜇;𝑙−𝑝𝑗,𝜈;𝑙

and

𝑎𝑖 𝑗,𝜇𝜈 :=𝑎𝑖,𝜇−𝑎𝑗,𝜈

are used. Here,

𝑅𝑖 𝑗,𝜇𝜈

denotes the remainder of the Taylor expansion. The remainder

𝑅(𝜖

𝑡) →

0 and

𝑅𝑖 𝑗,𝜇𝜈 (𝜖

𝑡) ∈ 𝑜(𝜖)

for all 𝑡∈Ras 𝜖→0. We get

𝜅𝑖 𝑗,𝜇𝜈 (𝑡)∗=−

𝑠

𝑙=0

𝜖𝑙+1∫𝑡

−∞

𝑒−𝜖(𝑡−𝑡0)𝑟𝑖 𝑗,𝜇𝜈;𝑙(𝛽,𝑡0)d𝑡0+𝜖𝑠+1∫𝑡

𝑡0

𝑒−𝜖(𝑡−𝑡0)𝑅𝑖 𝑗,𝜇𝜈 (𝜖,𝑡0)d𝑡0. (A.17)

In order to derive the expansion for

𝜅𝑖 𝑗,𝜇𝜈 ∗

the integrals of the formula above have to be

investigated. Faà di Bruno’s formula (A.15) provides us with

𝑟𝑖 𝑗,𝜇𝜈;𝑙(𝛽,𝑡):=Õ

(𝑘1,... ,𝑘𝑙)∈𝑇𝑙𝐷𝑘1+...+𝑘𝑙sin)(ΔΩ𝜇𝜈𝑡+𝑎𝑖 𝑗,𝜇𝜈 +𝛽)

𝑘1!· ··· ·𝑘𝑙!

𝑙

𝑚=1𝑝𝑖 𝑗,𝜇𝜈;𝑚𝑘𝑚

. (A.18)

First, we conclude that the Taylor coefficients

𝑟𝑖 𝑗,𝜇𝜈;𝑙(𝛽)

can also be written in a (finite) multi-

Fourier sum

𝑟𝑖 𝑗,𝜇𝜈;𝑙(𝛽,𝑡)=Õ

m∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;𝑙(𝛽))

𝑑𝑖 𝑗,𝜇𝜈;𝑙;m(𝛽)𝑒iΔΩ(m)𝑡. (A.19)

Second,

𝑑𝑖 𝑗,𝜇𝜈;𝑙;0≠

0 if

Î𝑙

𝑚=1𝑝𝑖 𝑗,𝜇𝜈,𝑚𝑘𝑚

possess a non vanishing term for

𝑒iΔΩ𝜇𝜈𝑡

. With this, we

are able to calculate the integrals in (A.17) and get

∫𝑡

−∞

𝑒−𝜖(𝑡−𝑡0)𝑟𝑖 𝑗,𝜇𝜈;𝑙(𝛽,𝑡)=Õ

m∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;𝑙(𝛽))

𝑑𝑖 𝑗,𝜇𝜈;𝑙;m(𝛽)∫𝑡

−∞

𝑒−𝜖(𝑡−𝑡0)𝑒iΔΩ(m)𝑡0d𝑡0

=Õ

m∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;𝑙(𝛽)) 1

𝜖+iΔΩ(m)𝑑𝑖 𝑗,𝜇𝜈;𝑙;m(𝛽)𝑒iΔΩ(m)𝑡.

The last term in the equation (A.17) is in

𝑜(𝜖𝑠)

which can be seen as follows. Since

𝑅𝑖 𝑗,𝜇𝜈 (𝜖

𝑡) ∈

𝑜(𝜖)

𝜖→

0 for all

𝐶 >

0 there exist an

𝜖0(𝑡)>

0 such that

𝑅𝑖 𝑗,𝜇𝜈 (𝜖,𝑡)< 𝐶𝜖

for all

𝜖 < 𝜖0(𝑡)

Due to the boundedness of all 𝑝𝑖,𝜇;𝑙the remainder 𝑅𝑖 𝑗,𝜇𝜈 (𝜖,𝑡)is also bounded by some positive

number ˜

𝐶. Thus, 𝑅𝑖 𝑗,𝜇𝜈 (𝜖,𝑡) ∈ 𝑜(𝜖)uniformly on Ras 𝜖→0 , hence for all 𝐶 > 0

∫𝑡

𝑡0

𝑒−𝜖(𝑡−𝑡0)𝑅(𝜖,𝑡0)d𝑡0≤𝐶1−𝑒𝜖(𝑡0−𝑡).

Finally, we end up with

𝜅𝑖 𝑗,𝜇𝜈 ∗(𝑡)=−

𝑠

𝑙=0

𝜖𝑙Õ

m∈˜

M(𝑟𝛽

𝑖 𝑗,𝜇𝜈;𝑙)

𝑑𝛽

𝑖 𝑗,𝜇𝜈;𝑙;m

1+iΔΩ(m)

𝜖

𝑒iΔΩ(m)𝑡+˜

𝑅𝑖 𝑗,𝜇𝜈 (𝜖,𝑡)(A.20)

where

𝑅𝑖 𝑗,𝜇𝜈 :=𝜖𝑠+1∫𝑡

𝑡0𝑒−𝜖(𝑡−𝑡0)𝑅(𝜖,𝑡0)d𝑡0∗∈𝑜(𝜖𝑠)

uniformly on

𝜖→

0. By considering

the Laurent series

1+iΔΩ(m)

𝜖

=−∞

𝑛=1

𝑖𝑛𝜖

ΔΩ(m)𝑛

A.4 Asymptotic expansions of multicluster states 147

which converges whenever

𝜖 < |ΔΩ(m)|

, the coefficients of the expansion

𝜅𝑖 𝑗,𝜇𝜈 (𝑠)

∗=

Í𝑠

𝑙=0𝜖𝑙𝑘𝑖 𝑗,𝜇𝜈;𝑙(𝑡)are given by

𝜅𝑖 𝑗,𝜇𝜈;0 =−𝑑𝑖 𝑗,𝜇𝜈;0;0(𝛽),

𝜅𝑖 𝑗,𝜇𝜈;𝑙>0=−𝑑𝑖 𝑗,𝜇𝜈;𝑙;0(𝛽)+

𝑙−1

𝑛=0Õ

m∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;𝑛(𝛽))/{0}

𝑖𝑙−𝑛𝑑𝑖 𝑗,𝜇𝜈;𝑛;m(𝛽)

(ΔΩ(m))𝑙−𝑛𝑒iΔΩ(m)𝑡.

Note that,

𝜖

can always be chosen such that

𝜖 < |ΔΩ(m)|

, since we consider the asymptotic

limit

𝜖→

0. The coefficients are determined via comparing the terms of both sides of the

equation (A.20) with respect to their order in

𝜖

. In the case

𝜇≠𝜈

we get

𝜅𝑖 𝑗,𝜇𝜈;0 =

0. All terms of

order 𝑂(𝜖𝑠+1)and ˜

𝑅𝑖 𝑗,𝜇𝜈 (𝜖,𝑡)are summarized in ˆ

𝑅𝑖 𝑗,𝜇𝜈 (𝜖,𝑡) ∈ 𝑜(𝜖𝑠)as 𝜖→0.

Remark A.4.2

Without considering the asymptotic limit

𝜖→

0, the

𝑠

-th order formal expansion

for

𝜅𝑖 𝑗,𝜇𝜈

would have the form as it is given in Lemma A.4.2 under the condition that

ΔΩ(m)> 𝜖

for all m∈Ð𝑠−1

𝑙=1˜

M(𝑟𝛽

𝑖 𝑗,𝜇𝜈;𝑙).

Lemma A.4.3

Suppose everything is given as in Lemma A.4.2. Then, if for all

𝜇∈1, . . . ,𝑀

𝑖∈

{

. . .

𝑁𝜇}

and

𝑙∈1, . . . ,𝑟

𝑝𝑖,𝜇;𝑙(𝑡)

can be written completely in terms of even modes, i.e., all

𝑚1

. . .

𝑚𝑀

are even for

(𝑚1

. . .

𝑚𝑀) ∈ ˜

𝑴(𝑝𝑖,𝜇;𝑙)

, then for all

(𝑛1

. . .

𝑛𝑀) ∈ ˜

𝑴(𝜅𝑠

𝑖 𝑗,𝜇𝜈 (𝑡))

holds:

𝑛𝜆

are even for 𝜆≠𝜇,𝜈and odd otherwise.

Proof.

It is fairly easy to verify that if

𝑝𝑖,𝜇;𝑙(𝑡)

can be completely written in terms of even modes

for all

𝑖

𝜇

𝑙

, so can

𝑝𝑖 𝑗,𝜇𝜈;𝑙(𝑡)

and moreover, the product

𝑝𝑖 𝑗,𝜇𝜈;𝑙·𝑝𝑖 𝑗,𝜇𝜈;𝑚

. According to (A.18)

𝑟𝑖 𝑗,𝜇𝜈;𝑙

consists only of terms of the form

𝑒±iΔΩ𝜇𝜈𝑡·𝑒iΔΩ(𝑚)𝑡

and hence

𝑚𝜆

are even for every

𝜆≠𝜇

𝜈

and odd otherwise for all

(𝑚1

. . .

𝑚𝑀) ∈ ˜

M(𝑟𝑖 𝑗,𝜇𝜈;𝑙)

. Since integration by time (A.17)

does not make any changes in the modes, the same holds for 𝑘𝑖 𝑗,𝜇𝜈;𝑙(𝑡)and hence 𝜅𝑠

𝑖 𝑗,𝜇𝜈 (𝑡).

Now, we have everything which is needed to proof the main result in Proposition 5.3.1.

Proof.

Note, whenever we write

ΔΩ(m)

ΔΩ(𝑟)(m)

is meant. We omit the superscript for the sake

of readability. (i) Combing the formal time derivative of the first equation in (5.11), the system

equations (2.25)–(2.26) and Lemma A.4.2 we get

𝜙(𝑟)

𝑖,𝜇=

𝑟

𝑙=0

𝜖𝑙𝜕Ω(𝑟)

𝜇

𝜕𝜏𝑙+

𝑟

𝑙=1

𝜖𝑙¤𝑝𝑖,𝜇;𝑙(𝑡)=−1

𝑁

𝑀

𝜈=1

𝑁𝜈

𝑗=1

𝑟

𝑙=0

𝑟

𝑛=0

𝜖𝑙+𝑛𝑘𝑖 𝑗,𝜇𝜈;𝑙(𝑡)𝑟𝑖 𝑗,𝜇𝜈,𝑛(𝛼,𝑡). (A.21)

Assume that 𝑝𝑖,𝜇;𝑙(𝑡)=Ím∈˜

M(𝑝𝑖,𝜇;𝑙)𝑐𝑖,𝜇;𝑙;m𝑒iΔΩ(m)𝑡with |˜

M(𝑝𝑖,𝜇;𝑙)| ∈ N. We get

𝜕Ω(𝑟)

𝜇

𝜕𝑡 =1

𝑁

𝑀

𝜈=1

𝑁𝜈

𝑗=1Õ

m∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;0 (𝛼))

𝑑𝑖 𝑗,𝜇𝜈;0;m(𝛼)𝑑𝑖 𝑗,𝜇𝜈;0;0(𝛽)𝑒iΔΩ(m)𝑡,

𝜕Ω(𝑟)

𝜇

𝜕𝜏𝑙+ ¤𝑝𝑖,𝜇;𝑙(𝑡)=−1

𝑁

𝑀

𝜈=1

𝑁𝜈

𝑗=1

𝑙

𝑚=0

𝑘𝑖 𝑗,𝜇𝜈;𝑚𝑟𝑖 𝑗,𝜇𝜈,𝑙−𝑚(𝛼)(A.22)

148 A Proof of results from the main text

by comparing both sides of the equation (A.21) with respect to the order of

𝜖

. Due to Lemma A.4.2

𝜕Ω(𝑟)

𝜇(𝜏0,. . . ,𝜏𝑚)

𝜕𝑡 =1

𝑁

𝑁𝜇

𝑗=1

𝑑𝑖 𝑗,𝜇𝜇;0;0(𝛼)𝑑𝑖 𝑗,𝜇𝜇;0;0(𝛽)=:Ω𝜇,0 ∈R.

This equation can be solved by

Ω𝜇= Ω𝜇,0𝑡+˜

Ω𝜇,0(𝜏1

. . .

𝜏𝑚)

. Due to our assumptions the right

hand side of equation (A.22) can be written as

−

𝑀

𝜈=1

𝑁𝜈

𝑗=1

𝑙

𝑚=0

𝑘𝑖 𝑗,𝜇𝜈;𝑚𝑟𝑖 𝑗,𝜇𝜈,𝑙−𝑚(𝛼)=

𝑀

𝜈=1

𝑁𝜈

𝑗=1

𝑙

𝑚=0𝑑𝑖 𝑗,𝜇𝜈;𝑚;0(𝛽)−

𝑚−1

𝑛=0Õ

m∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;𝑛(𝛽))/{0}

𝑖𝑚−𝑛𝑑𝑖 𝑗,𝜇𝜈;𝑛;m(𝛽)

(ΔΩ(m))𝑚−𝑛𝑒iΔΩ(m)𝑡×

𝑑𝑖 𝑗,𝜇𝜈;𝑙−𝑚;0(𝛼)+ Õ

m∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;𝑙−𝑚(𝛼))/{0}

𝑑𝑖 𝑗,𝜇𝜈;𝑙−𝑚;m(𝛼)𝑒iΔΩ(m)𝑡.

By using 1.) and 2.) of Lemma A.4.2 we find

𝜕Ω(𝑟)

𝜇

𝜕𝜏𝑙

𝑁

𝑀

𝜈=1

𝑁𝜈

𝑗=1

𝑙

𝑚=0𝑑𝑖 𝑗,𝜇𝜈;𝑚;0(𝛽)𝑑𝑖 𝑗,𝜇𝜈;𝑙−𝑚;0(𝛼)

−

𝑚−1

𝑛=0Õ

m∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;𝑛(𝛽))/{0}

𝑖𝑚−𝑛𝑑𝑖 𝑗,𝜇𝜈;𝑛;m(𝛽)𝑑𝑖 𝑗,𝜇𝜈;𝑙−𝑚;−m(𝛼)

(ΔΩ(m))𝑚−𝑛,

¤𝑝𝑖,𝜇;𝑙=1

𝑁

𝑀

𝜈=1

𝑁𝜈

𝑗=1

𝑙

m∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;𝑙−𝑚(𝛽))/{0}

𝑑𝑖 𝑗,𝜇𝜈;𝑙−𝑚;m(𝛼)𝑒iΔΩ(m)𝑡ª®¬

𝑚−1

𝑛=0Õ

m∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;𝑛(𝛽))/{0}

𝑖𝑚−𝑛𝑑𝑖 𝑗,𝜇𝜈;𝑛;m(𝛽)

(ΔΩ(m))𝑚−𝑛𝑒iΔΩ(m)𝑡ª®¬

−

𝑚−1

𝑛=0Õ

m∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;𝑛(𝛽))/{0}

n∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;𝑙−𝑚(𝛼))/{0,−m}

𝑖𝑚−𝑛𝑑𝑖 𝑗,𝜇𝜈;𝑛;m(𝛽)𝑑𝑖 𝑗,𝜇𝜈;𝑙−𝑚;n(𝛼)

(ΔΩ(m))𝑚−𝑛𝑒iΔΩ(m+n)𝑡

Note that we use the multi-time scale function

Ω(𝑟)

𝜇

to deal with all terms of the expansion

describing a linear growth. All the other terms are considered to determine the behaviour of

𝑝𝑖,𝜇;𝑙

. With this ansatz we are able to maintain the boundedness of

𝑝𝑖,𝜇;𝑙

while letting

Ω(𝑟)

𝜇

alone

describing unbounded behaviour in

𝑡∈R

. Note further that

Ω(𝑟)

𝜇

can be directly computed if all

functions 𝑝𝑖,𝜇;𝑘≤𝑙(𝑡)are known. Thus, we finally end up with

Ω(𝑟)

𝜇=

𝑟

𝑙=0

𝜖𝑙Ω𝜇,𝑙𝑡.

A.4 Asymptotic expansions of multicluster states 149

We assume now that for all

𝑖

𝜇

and

𝑙 >

𝑝𝑖,𝜇;𝑙

can be written completely in terms of even

modes, c.f., Lemma A.4.3. In particular,

(𝜇𝜈)∉𝑀(𝑝𝑖,𝜇;𝑙)

. Thus,

𝑑𝑖 𝑗,𝜇𝜈;𝑙;0(𝛽)=

0 by (A.18) for all

𝜇,𝜈=1, . . . ,𝑀,𝑖=1, . . . ,𝑁𝜇,𝑗=1, . . . ,𝑁𝜈and 𝑙≥1 and

¤𝑝𝑖,𝜇;𝑙=−1

𝑁

𝑀

𝜈=1

𝑁𝜈

𝑗=1

𝑙,𝑚−1

𝑚=1

𝑛=0Õ

m∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;𝑛(𝛽))/{0}

n∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;𝑙−𝑚(𝛼))/{0,−m}

𝑖𝑚−𝑛𝑑𝑖 𝑗,𝜇𝜈;𝑛;m(𝛽)𝑑𝑖 𝑗,𝜇𝜈;𝑙−𝑚;n(𝛼)

(ΔΩ(m))𝑚−𝑛𝑒iΔΩ(m+n)𝑡.

Hence, we get an equation to determine the value of

𝑝𝑖,𝜇;𝑙

inductively. Due to Lemma A.4.3,

we know that if all

𝑝𝑖,𝜇;𝑙

can be written in terms of even modes then

𝑚𝜇

and

𝑚𝜈

are odd for

all

(𝑚1

. . .

𝑚𝑀) ∈ ˜

M(𝑟𝑖 𝑗,𝜇𝜈;𝑙(𝛼))

. Therefore

𝑝𝑖,𝜇;𝑙

can be written in terms of even modes. This

is consistent with our assumption that

𝑝𝑖,𝜇;𝑙

can be written in terms of even modes. Consider

further

¤𝑝𝑖,𝜇;1 =−1

𝑁

𝑀

𝜈=1

𝑁𝜈

𝑗=1Õ

m∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;0 (𝛽))/{0}

n∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;0 (𝛼))/{0,−m}

𝑖𝑑𝑖 𝑗,𝜇𝜈;0;m(𝛽)𝑑𝑖 𝑗,𝜇𝜈;0;n(𝛼)

(ΔΩ(m))𝑚−𝑛𝑒iΔΩ(m+n)𝑡.

The expression for 𝑝𝑖,𝜇,1 can be found by integration

𝑝𝑖,𝜇;1(𝑡)=−1

𝑁

𝑀

𝜈=1

𝑁𝜈

𝑗=1Õ

m∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;0 (𝛽))/{0}

n∈˜

M(𝑟𝑖 𝑗,𝜇𝜈;0 (𝛼))/{0,−m}

𝑑𝑖 𝑗,𝜇𝜈;0;m(𝛽)𝑑𝑖 𝑗,𝜇𝜈;0;n(𝛼)

(ΔΩ(m))ΔΩ(m+n)𝑒iΔΩ(m+n)𝑡.

Since

𝑝𝑖,𝜇;1

can be written as a Fourier sum the same holds true for all

𝑝𝑖,𝜇,𝑙

by induction. This is

consistent with our assumption for

𝑝𝑖,𝜇,𝑙

in the beginning of this proof. The expressions

𝑘𝑖 𝑗,𝜇𝜈;𝑙

follow from Lemma A.4.2. Furthermore, analog to Lemma A.4.2, from Theorem 2.4.15 in [

ABR88

pp. 93–94] we conclude

𝜅𝑖 𝑗,𝜇𝜈 (𝜖

𝑡) − 𝜅𝑟

𝑖 𝑗,𝜇𝜈 (𝜖

𝑡) ∈ 𝑂(𝜖𝑟)

and

𝜙𝑖,𝜇(𝜖

𝑡) − 𝜙(𝑟)

𝑖,𝜇(𝜖

𝑡) ∈ 𝑂(𝜖𝑟)

for

𝑡∈𝑂(1/𝜖𝑟)as 𝜖→0.

(ii) To achieve this result we apply now (i) which allows for iteratively determining the function

appearing in the asymptotic expansion.

0-th order: For the expansion of the sine-function from equation (A.18) we find

𝑟𝑖 𝑗,𝜇𝜇;0(𝛽)=

sin(𝑎𝑖 𝑗,𝜇𝜇 +𝛽)=𝑑𝑖 𝑗,𝜇𝜇;0;0(𝛽)and

𝑟𝑖 𝑗,𝜇𝜈;0(𝛽)=sin(ΔΩ(1)

𝜇𝜈 𝑡+𝑎𝑖 𝑗,𝜇𝜈 +𝛽)=𝑑𝑖 𝑗,𝜇𝜈;0;(𝜇𝜈)(𝛽)𝑒iΔΩ(1)

𝜇𝜈 𝑡+𝑐.𝑐.

where

𝑑𝑖 𝑗,𝜇𝜈;0;(𝜇𝜈)(𝛽):=(

)𝑒i(𝑎𝑖 𝑗,𝜇𝜈+𝛽)

and

𝑐

. stands for complex conjugated. Hence, for the

coupling matrix we find

𝜅𝑖 𝑗,𝜇𝜇;0 =−sin(𝑎𝑖 𝑗,𝜇𝜇 +𝛽).

Depending on the cluster the zero-th order approximation for the frequencies read

Ω𝜇,0 =1

𝑁

𝑁𝜇

𝑗=1

sin(𝑎𝑖 𝑗,𝜇𝜇 +𝛽)sin(𝑎𝑖 𝑗,𝜇𝜇 +𝛽)=𝑛𝜇

2(cos(𝛼−𝛽) −cos(𝛼+𝛽))

150 A Proof of results from the main text

for all 𝜇=1, . . . ,𝑀1and analogously Ω𝜇,0 =𝑛𝜇

2cos(𝛼−𝛽)for all 𝜇=𝑀1+1, . . . ,𝑀.

1-th order: Since we know the 0-th order expansion we are able to calculate the next order. We

get

Ω𝜇,1 =−1

𝑁

𝑀1

𝜈=1

𝜈≠𝜇

𝑁𝜈

𝑗=1Õ

m∈{(𝜇𝜈),−(𝜇𝜈)}

𝑖𝑑𝑖 𝑗,𝜇𝜈;0;m(𝛽)𝑑𝑖 𝑗,𝜇𝜈;0;−m(𝛼)

ΔΩ(1)(m)

2𝑁

𝑀

𝜈=1

𝜈≠𝜇

𝑁𝜈

𝑗=1

2i 𝑒i(𝑎𝑖 𝑗,𝜇𝜈+𝛽)𝑒−i(𝑎𝑖 𝑗,𝜇𝜈 +𝛼)

ΔΩ(1)

𝜇𝜈 −𝑒−i(𝑎𝑖 𝑗,𝜇𝜈 +𝛽)𝑒i(𝑎𝑖 𝑗,𝜇𝜈+𝛼)

ΔΩ(1)

𝜇𝜈 !

2𝑁

𝑀

𝜈=1

𝜈≠𝜇

𝑁𝜈

𝑗=1

2i 𝑒i(𝛽−𝛼)

ΔΩ(1)

𝜇𝜈 −𝑒−i(𝛽−𝛼)

ΔΩ(1)

𝜇𝜈 !=−

𝑀

𝜈=1

𝜈≠𝜇

𝑛𝜈

2ΔΩ(1)

𝜇𝜈

sin(𝛼−𝛽).

For all 𝜇=1, . . . ,𝑀1we get

¤𝑝𝜇;1 =−1

𝑁

𝑀1

𝜈=1

𝜈≠𝜇

𝑁𝜈

𝑗=1Õ

m∈{(𝜇𝜈),−(𝜇𝜈)} 𝑖𝑑𝑖 𝑗,𝜇𝜈;0;m(𝛽)𝑑𝑖 𝑗,𝜇𝜈;0;m(𝛼)

ΔΩ(1)(m)𝑒i2ΔΩ(1)(m)𝑡

𝑀

𝜈=𝑀1+1

𝑁𝜈

𝑗=1Õ

m∈{(𝜇𝜈),−(𝜇𝜈)} 𝑖𝑑𝑖 𝑗,𝜇𝜈;0;m(𝛽)𝑑𝑖 𝑗,𝜇𝜈;0;m(𝛼)

ΔΩ(1)(m)𝑒i2ΔΩ(1)(m)𝑡

=−1

𝑀

𝜈=1

𝜈≠𝜇

𝑛𝜈

2i 𝑒i(𝑎𝜇−𝑎𝜈+𝛼+𝛽)

ΔΩ(1)

𝜇𝜈 !𝑒i2ΔΩ(1)

𝜇𝜈 𝑡−𝑛𝜈

2i 𝑒−i(𝑎𝜇−𝑎𝜈+𝛼+𝛽)

ΔΩ(1)

𝜇𝜈 !𝑒−i2ΔΩ(1)

𝜇𝜈 𝑡

and for all 𝜇=𝑀1+1, . . . ,𝑀we get

¤𝑝𝑖,𝜇;1 =−1

𝑁

𝑀1

𝜈=1

𝑁𝜈

𝑗=1Õ

m∈{(𝜇𝜈),−(𝜇𝜈)} 𝑖𝑑𝑖 𝑗,𝜇𝜈;0;m(𝛽)𝑑𝑖 𝑗,𝜇𝜈;0;m(𝛼)

ΔΩ(1)(m)𝑒i2ΔΩ(1)(m)𝑡

𝑀

𝜈=𝑀1+1

𝜈≠𝜇

𝑁𝜈

𝑗=1Õ

m∈{(𝜇𝜈),−(𝜇𝜈)} 𝑖𝑑𝑖 𝑗,𝜇𝜈;0;m(𝛽)𝑑𝑖 𝑗,𝜇𝜈;0;m(𝛼)

ΔΩ(1)(m)𝑒i2ΔΩ(1)(m)𝑡

=−1

𝑀

𝜈=1

𝑛𝜈

2i 𝑒i(𝑎𝑖,𝜇−𝑎𝜈+𝛼+𝛽)

ΔΩ(1)

𝜇𝜈 !𝑒i2ΔΩ(1)

𝜇𝜈 𝑡−𝑛𝜈

2i 𝑒−i(𝑎𝑖,𝜇−𝑎𝜈+𝛼+𝛽)

ΔΩ(1)

𝜇𝜈 !𝑒−i2ΔΩ(1)

𝜇𝜈 𝑡.

Thus, solving this fairly easy differential equation the following expression is obtained for

𝜇=1, . . . ,𝑀1

𝑝𝜇;1 =1

𝑀

𝜈=1

𝜈≠𝜇

𝑛𝜈

2©«

𝑒i(𝑎𝜇−𝑎𝜈+𝛼+𝛽)

ΔΩ(1)

𝜇𝜈 2ª®®¬

𝑒i2ΔΩ(1)

𝜇𝜈 𝑡+𝑛𝜈

2©«

𝑒−i(𝑎𝜇−𝑎𝜈+𝛼+𝛽)

ΔΩ(1)

𝜇𝜈 2ª®®¬

𝑒−i2ΔΩ(1)

𝜇𝜈 𝑡.

Analogously we find the expression for

𝑝𝑖,𝜇;1

with

𝜇=𝑀1+

. . .

𝑀

. For the coupling matrix

A.5 From local to global order parameter 151

we get

𝜅𝑖 𝑗,𝜇𝜈;1 =Õ

m∈{(𝜇𝜈),−(𝜇𝜈)} 𝑖𝑑𝑖 𝑗,𝜇𝜈;0;m(𝛽)

ΔΩ(1)(m)𝑒iΔΩ(1)(m)𝑡

=𝑒i(𝑎𝑖 𝑗,𝜇𝜈+𝛽)

2ΔΩ(1)

𝜇𝜈

𝑒iΔΩ(1)

𝜇𝜈 𝑡+𝑒−i(𝑎𝑖 𝑗,𝜇𝜈+𝛽)

2ΔΩ(1)

𝜇𝜈

𝑒−iΔΩ(1)

𝜇𝜈 𝑡.

A.5 From local to global order parameter

From expression

(6.9)

we derived two types of one-cluster solution namely antipodal or local

splay states. In the following we derive a remarkable relation between local and global properties

on a nonlocal ring which is: If a cluster is of local splay type, the cluster is also of global splay

type.

In order to show this, we rewrite the sum

Í𝑁

𝑖=1𝑍(2)

𝑖

in two ways. Firstly, using the pure definition

of the local order parameter (6.4):

𝑁

𝑖=1

𝑍(2)

𝑖(𝝌)=1

2𝑃

𝑁

𝑖,𝑗=1

𝑎𝑖 𝑗 𝑒i2𝜒𝑗=

𝑁

𝑗=1

𝑒i2𝜒𝑗=𝑁 𝑍 (2)(𝝌). (A.23)

Secondly, the sum can be rewritten using the definition of a local splay type cluster with

𝑅(2)

𝑖(𝝌)=𝑅(2)

𝑐(𝝌)and 2𝜒𝑖=𝜗(2)

𝑖. Then

𝑁

𝑖=1

𝑍(2)

𝑖(𝝌)=

𝑁

𝑖=1

𝑅(2)

𝑐𝑒i𝜗(2)

𝑖=𝑅(2)

𝑐(𝝌)

𝑁

𝑖=1

𝑒i2𝜒𝑖=𝑁 𝑅(2)

𝑐(𝝌)𝑍(2)(𝝌)(A.24)

By equating (A.23) and (A.24) we obtain:

(1−𝑅(2)

𝑐(𝝌))𝑍(2)(𝝌)=0

The latter equation yields

𝑅(2)(𝝌)=

0 for all local splay type clusters, since

𝑅(2)

𝑐(𝝌)<

1 by

definition.

152 A Proof of results from the main text

A.6 Stability of one-cluster states on nonlocally coupled networks

First note that the set of equations (6.15)–(6.16) can be brought into the following block form

d𝑡 𝛿𝝓

𝛿𝜅!= 𝑀 𝐵

𝐶−𝜖I𝑁2! 𝛿𝝓

𝛿𝜅!

where

(𝛿𝝓)𝑇=(𝛿𝜙1,. . . ,𝛿𝜙𝑁)

(𝛿𝜅)𝑇=(𝛿𝜅11,. . . ,𝛿𝜅1𝑁,𝛿𝜅21,. . . ,𝛿𝜅𝑁 𝑁 )

𝐵=𝐵1··· 𝐵𝑁

𝐶=©«

𝐶1

𝐶𝑁ª®®®¬

, and

𝑀

𝐵𝑛

𝐶𝑛

are

𝑁×𝑁

matrices with

𝑛=

. . .

𝑁

. The elements of the block

matrices read

𝑚𝑖 𝑗 =

−𝜎

2sin(𝛼−𝛽)

𝑁

𝑘=1

𝑎𝑖𝑘 −𝜎𝑎𝑖𝑖 sin(𝛽)cos(𝛼)+ 𝜎

𝑁

𝑘=1

𝑎𝑖𝑘 sin(2(𝑎𝑖−𝑎𝑘)+𝛼+𝛽),𝑖=𝑗

𝜎𝑎𝑖 𝑗

2sin(𝛼−𝛽) −sin(2(𝑎𝑖−𝑎𝑗)+𝛼+𝛽),𝑖≠𝑗

=

−𝜎

2sin(𝛼−𝛽) +=(𝑒−i(2𝑎𝑖+𝛼+𝛽)𝑍(2)

𝑖(𝒂))𝑁

𝑘=1

𝑎𝑖𝑘 −𝜎𝑎𝑖𝑖 sin(𝛽)cos(𝛼)𝑖=𝑗

𝜎𝑎𝑖 𝑗

2sin(𝛼−𝛽) −sin(2(𝑎𝑖−𝑎𝑗)+𝛼+𝛽),𝑖≠𝑗

𝑏𝑖 𝑗;𝑛=(−𝜎𝑎𝑛 𝑗 sin(𝑎𝑛−𝑎𝑗+𝛼),𝑖=𝑛

0, otherwise

𝑐𝑖 𝑗;𝑛=

0, 𝑗=𝑛,𝑖=𝑗

−𝜖𝑎𝑛𝑖 cos(𝑎𝑛−𝑎𝑖+𝛽),𝑗=𝑛,𝑖≠𝑗

𝜖𝑎𝑛𝑖 cos(𝑎𝑛−𝑎𝑖+𝛽),𝑗≠𝑛,𝑖=𝑗

0, otherwise

where

𝜎=

𝑃

. Note that we use the Schur decomposition

(4.11)

throughout this stability

analysis. Suppose we have a phase locked state with phases

𝑎𝑖∈ [

0, 2

𝜋)

. Then, the solution for

the characteristic equations corresponding to the linearized system

(6.15)

–

(6.16)

are given by

𝜆=−𝜖

with multiplicity

𝑁2−𝑁

and by the solution of the following set of equations, see also

Lemma A.2.2,

det ((𝑀−𝜆I𝑁) (𝜖+𝜆)+𝐵𝐶)=0. (A.25)

The second term 𝐷:=𝐵𝐶 of the Schur complement are element wise given by

𝑑𝑖 𝑗 =−𝜖𝜎

2𝑎𝑖 𝑗 sin(𝛼−𝛽) +sin(2(𝑎𝑖−𝑎𝑗) +𝛼+𝛽)

if 𝑖≠𝑗and

𝑑𝑖𝑖 =𝜖𝜎

2sin(𝛼−𝛽) −=(𝑒−i(2𝑎𝑖+𝛼+𝛽)𝑍(2𝑘)

𝑖(𝒂))𝑁

𝑘=1

𝑎𝑖𝑘 −𝜖𝜎 sin(𝛼)cos(𝛽)

A.6 Stability of one-cluster states on nonlocally coupled networks 153

Consider that

𝑎𝑖(𝑘)=𝑖𝑘 2𝜋

𝑁

and the base topology has constant row sum

𝜌∈N

, i.e.

Í𝑁

𝑘=1𝑎𝑖𝑘 =𝜌

then the matrix in

(A.25)

becomes circulant. For a ring structure, as considered in this thesis,

we have

𝜌=

𝑃

. Hence it can be diagonalized using the

𝑁

eigenvectors

𝜁𝑙=exp(

𝜋𝑙/𝑁)=

exp(i𝑎𝑙(1)) and the eigenvalues 𝜇𝑙(𝜆)are

𝜇𝑙=((𝑚𝑁 𝑁 −𝜆) (𝜖+𝜆)+𝑑𝑁 𝑁 ) +

𝑁−1

𝑗=1(𝑚𝑁 𝑗 (𝜖+𝜆)+𝑑𝑁 𝑗 )𝜁𝑗

𝑙

=−𝜆2+𝜌𝜎

2𝜆i

2𝑒i(𝛼+𝛽)𝑍(𝑙−2𝑘)

𝑁−𝑍(2𝑘)

𝑁−𝑒−i(𝛼+𝛽)𝑍(2𝑘+𝑙)

𝑁−𝑍(2𝑘)

𝑁

+(𝑍(𝑙)

𝑁−1)sin(𝛼−𝛽) − 2𝜖

𝜌𝜎 +𝜖𝜎𝜌 i

2𝑒i(𝛼+𝛽)𝑍(𝑙−2𝑘)

𝑁−𝑍(2𝑘)

𝑁−𝑒−i(𝛼+𝛽)𝑍(2𝑘+𝑙)

𝑁−𝑍(2𝑘)

𝑁.

Here, we use the shorthand form

𝑍(𝑛)

𝑁=𝑍(𝑛)

𝑁(

)

. If any of the

𝜇𝑙

vanishes, the determinant

(A.25)

vanishes, as well. Thus,

𝜇𝑙(𝜆)=

0 is giving the quadratic equation which determine

the Lyapunov spectrum of the linearized system

(6.15)

–

(6.16)

around a phase-locked solution

and with a base topology, given by the adjacency matrix

𝐴

, having constant in-degree. Note that

with the result in Sec. 6.2.1, the complex order parameter can be further simplified

𝑍(𝑛)

𝑁=𝑅(𝑛)

𝑁=1

𝑃©«

𝑃

𝑗=1

cos(𝑛 𝑗 2𝜋

𝑁)ª®¬. (A.26)

Using the latter equation and well-known trigonometric equations, the following relations are

derived:

𝑍(𝑙−2𝑘)

𝑁−𝑍(2𝑘)

𝑁=1

𝑃©«

𝑃

𝑗=1cos((𝑙−2𝑘)𝑗2𝜋

𝑁)−cos(2𝑘 𝑗 2𝜋

𝑁)ª®¬,

=−2

𝑃©«

𝑃

𝑗=1

sin(𝑙 𝑗 𝜋

𝑁)sin((𝑙−4𝑘)𝑗𝜋

𝑁)ª®¬,

=−2

𝑃©«

𝑃

𝑗=1

sin(𝑙 𝑗 𝜋

𝑁)sin(𝑙 𝑗 𝜋

𝑁)cos(4𝑘 𝑗 𝜋

𝑁)−cos(𝑙 𝑗 𝜋

𝑁)sin(4𝑘 𝑗 𝜋

𝑁)ª®¬,

and analogously

𝑍(𝑙+2𝑘)

𝑁−𝑍(2𝑘)

𝑁=−2

𝑃©«

𝑃

𝑗=1

sin(𝑙 𝑗 𝜋

𝑁)sin(𝑙 𝑗 𝜋

𝑁)cos(4𝑘 𝑗 𝜋

𝑁)+cos(𝑙 𝑗 𝜋

𝑁)sin(4𝑘 𝑗 𝜋

𝑁)ª®¬.

154 A Proof of results from the main text

Combining these relations, we find

𝐿(𝛼,𝛽,𝑙,𝑘)=i

2𝑒i(𝛼+𝛽)𝑍(𝑙−2𝑘)

𝑁−𝑍(2𝑘)

𝑁−𝑒−i(𝛼+𝛽)𝑍(2𝑘+𝑙)

𝑁−𝑍(2𝑘)

𝑁

=2 sin(𝛼+𝛽)

𝑃

𝑗=1

sin2(𝑙 𝑗 𝜋

𝑁)cos(4𝑘 𝑗 𝜋

𝑁)

+i2 cos(𝛼+𝛽)

𝑃

𝑗=1

sin(𝑙 𝑗 𝜋

𝑁)cos(𝑙 𝑗 𝜋

𝑁)sin(4𝑘 𝑗 𝜋

𝑁). (A.27)

A.7 Stability of lifted one-cluster states

For an arbitrary duplex equilibrium of the form

𝜙𝜇

𝑖= Ω𝑡+𝑎𝜇

𝑖

with

𝑎1

𝑘=(

2𝜋

𝑁𝑘

. . .

(𝑁−

)2𝜋

𝑁𝑘)𝑇

and

𝑎2

𝑘=a1

𝑘−Δ𝑎

we start with the linearized system (4) of the main text. This can also be written

in the block matrix form

©«

𝛿𝜙1

𝛿𝜙2

𝛿𝜅1

𝛿𝜅2ª®®®®¬

=©«

𝐴1𝑚1I𝑁𝐵10

𝑚2I𝑁𝐴20𝐵2

𝐶10−𝜖I𝑁20

0𝐶20−𝜖I𝑁2

ª®®®®¬©«

𝛿𝜙1

𝛿𝜙2

𝛿𝜅1

𝛿𝜅2ª®®®®¬

with

(𝛿𝜙𝜇,𝛿𝜅𝜇)𝑇=𝛿𝜙𝜇

1,. . . ,𝛿𝜙𝜇

𝑁,𝛿𝜅𝜇

11,. . . ,𝛿𝜅𝜇

1𝑁,𝛿𝜅𝜇

21,. . . ,𝛿𝜅𝜇

𝑁 𝑁 𝑇

, the matrices

𝐴𝜇

𝐵𝜇

, and

𝐶𝜇

follow from system (4) of the main text, and

𝑚1

𝑚2∈R

. With the help of Schur’s decomposition

the characteristic equation for the linearized system takes the form

(𝜆+𝜖)2(𝑁2−𝑁)=0

det (𝜆I𝑁−𝐴1)(𝜆+𝜖)−𝐵1𝐶1−(𝜆+𝜖)𝑚1I𝑁

−(𝜆+𝜖)𝑚2I𝑁(𝜆I𝑁−𝐴2)(𝜆+𝜖)−𝐵2𝐶2!=0. (A.28)

The second equation has the block matrix form which is required from Proposition 8.3.3. All

blocks can be diagonalized and commute since they all possess a cyclic structure; compare

Lemma 4.2.1. Thus, we are allowed to apply Proposition 8.3.3 which we use in order to diagonal-

ize the matrix in Eq.

(A.28)

. For the diagonalized matrix we find the following equations for the

diagonal elements 𝜇𝑖

(𝜆+𝜖)2𝑚1𝑚2−(𝑝1

𝑖(𝜆;𝛼11,𝛽1,𝛼12,𝜎12)−𝜇𝑖)(𝑝2

𝑖(𝜆;𝛼22,𝛽2,𝛼21,𝜎21)−𝜇𝑖)=0 (A.29)

where

𝑖=

. . .

𝑁

𝑝𝜇

𝑖(𝜆

;

𝛼𝜇𝜇

𝛽𝜇)

is a second order polynomial in

𝜆

which depends continuously

𝛼

and

𝛽

as well as functionally on the type of the one-cluster state. For every

𝑖∈ {

. . .

𝑁}

these equations will give us two eigenvalues

𝜇𝑖,1

and

𝜇𝑖,2

for the matrix in Eq.

(A.28)

depending

on 𝜆and the system parameters. Thus, we can write Eq. (A.29) as

𝜇𝑖−𝜇𝑖,1(𝜆;𝜶,𝜷,𝝈)𝜇𝑖−𝜇𝑖,2(𝜆;𝜶,𝜷,𝝈)=0

A.7 Stability of lifted one-cluster states 155

where

𝜶

𝜷

𝝈

represent all system parameter chosen for (2.25)–(2.26). In order to find the eigen-

value

𝜆

of the linearized system

(A.28)

one of the eigenvalues

𝜇

has to vanish. This means that

we have to find 𝜆such that Eq. (A.29) equals

𝜇𝑖𝜇𝑖−𝜇𝑖,2(𝜆;𝜶,𝜷,𝝈)=0

which is equivalent to finding 𝜆such that the following quartic equation is solved

𝑝1

𝑖(𝜆;𝛼11,𝛽1,𝛼12,𝜎12)𝑝2

𝑖(𝜆;𝛼22,𝛽2,𝛼21,𝜎21)− (𝜆+𝜖)2𝑚1𝑚2=0. (A.30)

Note that here the diagonal elements of

𝐴1

are slightly different from those in Prop. 4.2.2 but they

do not affect the result, i.e., the diagonal element equals

𝜌𝑖(𝛼11

𝛽1)−𝑚1(𝛼12)

. The same holds

true for

𝐴2

. Thus, with the two possible eigenvalues

𝜌𝑖,1,2(𝛼𝜇𝜇

𝛽𝜇)

for the monoplex system

from Corr. 4.2.3 one finds the following quartic equation which give the Lyapunov exponents

for the lifted duplex one-cluster

h𝜆−𝜌𝑖,1(𝛼11,𝛽1)·𝜆−𝜌𝑖,2(𝛼11,𝛽1)+𝑚1(𝜆+𝜖)i×

h𝜆−𝜌𝑖,1(𝛼22,𝛽2)𝜆−𝜌𝑖,2(𝛼22,𝛽2)+𝑚2(𝜆+𝜖)i−(𝜆+𝜖)2𝑚1𝑚2=0. (A.31)

In case of an antipodal monoplex one-cluster given by Eq. (2) of the main text with

𝑎𝑖∈ {

𝜋}

then the set of Lyapunov exponents Sfor Eq. (4)in the main text is given by

In the case of a duplex antipodal one-cluster state given by Eq. (2) of the main text with

𝑎1

𝑖∈ {

𝜋}

and 𝑎2

𝑖=𝑎1

𝑖−Δ𝑎, Eq. (3) possesses the following set of eigenvalues

Duplex ={−𝜖,𝜆𝑖,1,𝜆𝑖,2,𝜆𝑖,3,𝜆𝑖,4𝑖=1,...,𝑁}

where 𝜆𝑖,1,...,4 solve the following 𝑁quartic equations

(𝜆+𝜖)2𝑚1𝑚2−h𝜆−𝜌1

𝑖,1·𝜆−𝜌1

𝑖,2+𝑚1(𝜆+𝜖)ih𝜆−𝜌2

𝑖,1𝜆−𝜌2

𝑖,2+𝑚2(𝜆+𝜖)i=0,

with

𝑚1=𝜎12 cos(Δ𝑎+𝛼12)

𝑚2=𝜎21 cos(Δ𝑎−𝛼21)

and the eigenvalues

𝜌𝜇

𝑖,1,2 ≡𝜌𝑖,1,2(𝛼𝜇𝜇

𝛽𝜇)

for the monoplex system, see Corollary 4.2.3.

156 A Proof of results from the main text

A.8 Example for a complex adjacency matrix

index j

index i

Figure A.1:

Adjacency matrix of a connected, directed random network of

𝑁=

200 nodes with constant row sum

𝑟=

50. The illustration shows the adjacency matrix where black and white refer to whether a link between two nodes

exist or not, respectively.

The adjacency matrix displayed in Fig. A.1 is obtained by the following procedure. For each

node

𝑖

of the

𝑁

nodes,

𝑟

links are randomly (with uniform distribution) picked from the set

consisting of all links from a node

𝑗≠𝑖

to node

𝑖

. This procedure results in a random directed

network with

𝑁

nodes and constant row sum (in-degree)

𝑟

. After the procedure we test if the

resulting network is connected [KOR18], see also 2.1.

List of Figures

2.1

Illustration of different networks with a total number of 12 nodes: (a) globally coupled

network, (b) nonlocally coupled network (coupling range

𝑃=

3), (c) locally coupled

network (coupling range

𝑃=

1), and (d) multiplex network consisting of two layers

for which one layer is globally coupled and the other layer has a nonlocally coupled

network structure. ....................................... 11

2.2

The plasticity function

−sin(Δ𝜙+𝛽)

and corresponding plasticity rules. (a)

𝛽=−𝜋

(b) 𝛽=0, (c) 𝛽=𝜋

2. Figure taken from [BER19,BER19a]. ................. 20

3.1

Plasticity function

𝑊(Δ𝑡𝑖 𝑗 )

for

𝜏𝑝=

𝜏𝑑=

𝑐𝑝=

𝑐𝑑=

1.6. Figure taken from [

ROE19a

3.2

Evolution of the coupling matrix

𝜅𝑖 𝑗 (𝑡)

starting from random initial conditions and

converging to a completely synchronous state. Panel (a) shows initial coupling matrix,

(b) the coupling matrix after the transient

𝑡=

2000ms. Raster plot of spiking times

at the beginning of simulations (c) and after the transient (d). The asymptotic state

(b,d) is a completely synchronized spiking with all coupling weights

𝜅𝑖 𝑗

potentiated

𝑘max

. Other parameters

𝑁=

200,

𝜏𝑝=

𝜏𝑑=

𝑐𝑝=

𝑐𝑑=

1.6, and

𝜅max =

1.5.

Figure modified from [ROE19a]. ............................... 28

3.3

Evolution of the coupling matrix

𝜅𝑖 𝑗 (𝑡)

starting from random initial conditions and

converging to frequency clusters hierarchical in size. Panel (a) shows initial coupling

matrix, (b) the coupling matrix after the transient

𝑡=

5600ms, and (c)

𝑡=

20000ms. (d-f)

Corresponding raster plots of spike times. The asymptotic state (c,f) is a hierarchical

cluster state with the coupling weights

𝜅𝑖 𝑗

potentiated to

𝑘max

within each cluster and

small or zero otherwise. Other parameters as in Fig. 3.2. The oscillators are ordered

accordingly to their mean frequency. Figure modified from [ROE19a]. ......... 28

3.4

Formation of individual clusters over time (corresponds to the dynamical scenario in

Fig. 3.3). The dashed and solid curves depict the time course of the mean coupling

within the small and big clusters, respectively. Figure taken from [ROE19a]. ..... 29

3.5

Example of a three-cluster state for

𝑁=

500,

𝜏𝑝=

𝜏𝑑=

𝑐𝑝=

𝑐𝑑=

1.6,

and

𝜅max =

1.5 with a random initial distribution of

𝜅𝑖 𝑗

[

0, 0.75

]

. Figure taken

from [ROE19a]. ........................................ 29

3.6

Coupling matrices for

𝑡=

10000ms and different amplitudes of independent random

input

𝐼

(see Eq. (3.4)). (a)

𝐼=

0.005, (b)

𝐼=

0.01, (c)

𝐼=

0.02, (d)

𝐼=

0.05 and (e) I=0.07.

All other parameters as in Fig. 3.5. Figure modified from [ROE19a]. .......... 30

3.7

(a) Difference between synchronization frequencies of the two clusters for different

size of the smaller cluster

𝑁𝑠

. (b) Time until cluster fusion for different initial size of

the smaller cluster 𝑁𝑠. Figure modified from [ROE19a]. ................. 30

3.8

Evolution of the coupling matrix for

𝑁=

50 and the number of neurons

𝑁𝑠=

8 (a-c)

and

𝑁𝑠=

9 (d-f) in the small cluster. In panels (a-c) the clusters are stable, while in

(d-f) they are merging to one synchronous cluster. (g, h) Time courses of the spiking

synchronization frequencies of small (

𝑁𝑠

neurons) and large (

𝑁𝑏

neurons) clusters

depicted by dashed and solid curves, respectively, for (g)

𝑁𝑠=

8 and

𝑁𝑏=

42 and (h)

𝑁𝑠=9 and 𝑁𝑏=41. Parameter 𝜅𝑚𝑎𝑥 =1.0. Figure modified from [ROE19a]. ...... 32

3.9

Mean synaptic activity

𝑆(𝑡)

of the neural population in the case of stable two clus-

ter state. Panel (a) shows the dynamics of

𝑆(𝑡)

on the time interval of 12 s, where

modulation of the amplitude (blue line) is visible, while the fast oscillations are not

recognized on this timescale. The maximum amplitude corresponds to the two clus-

ters being synchronised, while the low amplitude corresponds to the clusters being

out of phase. Panel (b) shows the zoom of a small time interval. The modulation takes

place on the timescale which is two orders of magnitude larger than the individual

spikes of

𝑆(𝑡)

as well as individual neural spikes in both clusters. Cluster frequencies

𝜔1=

0.065012 kHz and

𝜔2=

0.065416 kHz. The corresponding period of modulation

is 𝑇≈2.5𝑠. Figure modified from [ROE19a]. ........................ 33

3.10

(a) Update function

𝐺(𝜑)

for

𝜏𝑝=

𝜏𝑑=

𝑐𝑝=

2, and

𝑐𝑑=

1.6. (b) Schematic spiking

of two oscillators with spike time difference

Δ𝑇

and periods close to

𝑇

. Figure modified

from [ROE19a]. ........................................ 35

3.11

Phase portraits of model (3.10)-(3.11) for (a) monostable regime of complete syn-

chronization; (b) co-existence of stable synchronized and clustered states; and (c)

bifurcation moment of transition between the phase portraits illustrated in (a) and (b).

The basins of attraction of the synchronized regime (point

𝑆

), clustered state (limit

cycle indicated by thick black curve) and the saddle fixed point

(𝜑∗

𝜎∗)

are depicted

by gray, blue, and white colors, respectively. The nullclines of the system and stable

and unstable manifolds of the saddle point are indicted by the thin gray and black

curves, respectively. Parameters (a)

𝜔=

0.037 kHz, (b)

𝜔=

0.06 kHz, (c)

𝜔≈

0.455 Hz,

and the other parameters

𝜏𝑝=

𝜏𝑑=

𝑐𝑝=

𝑐𝑑=

1.6, and

𝜀=

0.08. Figure modified

from [ROE19a]. ........................................ 36

3.12

Dynamics of the phase difference between the clusters

𝜑𝐻 𝐻

and mean inter-cluster

coupling

𝜎𝐻 𝐻

for the solutions of the Hodgkin-Huxley system (3.1)-(3.3) for different

initial conditions.

𝑁=

50 with

𝑁𝑠=

7 neurons in the small cluster and

𝑁𝑏=

43 in the

big one. Red orbits converge to the regime of complete synchronization, and blue

trajectories lead to a stable two-cluster solutions. The nullclines of the phenomenolog-

ical model are shown in gray. Other parameters:

𝜏𝑝=

𝜏𝑑=

𝑐𝑝=

𝑐𝑑=

1.6, and

𝜅max =1.5. Figure taken from [ROE19a]. .......................... 38

3.13

Panel (a): system (3.10)-(3.11). White region: stable periodic solution coexisting with a

stable fixed point, case II. Black region: globally stable fixed point, case I. Grey region:

globally stable periodic solution with

𝜎=

0. Panel (b): original system (3.1)-(3.3).

White: stable two-clusters (white); black: stable synchrony and no stable clusters;

grey: decoupling of all neurons. Other parameters

𝜏𝑝=

𝜏𝑑=

𝑁=

50,

𝑁𝑠=

7, and

𝜅max =1. Figure modified from [ROE19a]. ......................... 39

4.1

Illustration of the three types of one-cluster solutions given by (4.3)–(4.4) for an

ensemble of 50 oscillators. One-cluster solutions (a) of splay type (

𝑅2(a)=

0) for

𝛼=

0.3

𝜋

𝛽=

0.1

𝜋

, (b) of antipodal type (

𝑅2(a)=

1), for

𝛼=

0.2

𝜋

𝛽=−

0.95

𝜋

and (c) of

double antipodal type satisfying condition (iii) of Proposition 4.1.1 with

𝑚=

30 for

𝛼=0.3𝜋,𝛽=−0.15𝜋. Figure taken from [BER19]. ..................... 43

4.2

Illustration of the family of solutions

𝑆

(a)

𝑁=

2, (b)

𝑁=

3, (c)

𝑁=

4. Figure taken

from [BER19]. .......................................... 44

4.3

Stability diagrams for rotating-wave clusters depending on the parameters

𝛼

and

𝛽

are shown. The regions are colored according to numerical simulation. Blue regions

correspond to stable solutions while yellow regions correspond to unstable solutions.

The black dashed lines show to the borders of stability determined by Corollary 4.2.3.

Parameter

𝜖=

0.01 is fixed fo all simulations. (a)

𝑘=

1, (b)

𝑘=𝑁/

2, (c)

𝑘=𝑁/

4. Figure

taken from [BER19]. ...................................... 49

4.4

Stability diagram for splay and antipodal one-cluster solutions depending on the pa-

rameters

𝛼

and

𝛽

are shown. The regions are colored according numerical eigenvalues

of the Jacobian 4.9–4.10. Blue areas correspond to stable while yellow areas correspond

to unstable regions. Parameter

𝜖=

0.01 is fixed in all simulations. (a) Splay solution as

in Fig. 4.1(a), (b) Anti-phase solution as in Fig. 4.1(b). Figure modified from [BER19]. 49

4.5

The regions of stability for antipodal and rotating-wave states are presented in (

𝛼

𝛽

)

parameter space for different values of

𝜖

. Coloured and hatched areas refer to stable

regions for these states as indicated in the legend. White areas refer to region where

these one-cluster states are unstable. (a)

𝜖=

0; (b)

𝜖=

0.01; (c)

𝜖=

0.1; (d)

𝜖=

1. Figure

taken from [BER19a]. ..................................... 51

4.6

Heteroclinic orbits between several steady states in a system of 3 and 100 adaptively

coupled phase oscillators. (a) The time series for the relative phases

𝜃12

(solid lines)

and

𝜃13

(dashed lines) for

𝑁=

3 are shown. Lines with the same colour correspond

to the same trajectories. Panel (b,c) show time series for the second moment order

parameter

𝑅2(𝝓(𝑡))

as well as a schematic illustration of the observed heteroclinic

connections (right) for (b)

𝑁=

3 and (c)

𝑁=

100. Parameter values:

𝜖=

0.01,

𝛼=

0.4

𝜋

and 𝛽=−0.15𝜋. Figure taken from [BER19a]. ........................ 54

5.1

Three-frequencycluster of splay type at

𝑡=

10000. (a) Coupling weights represented

as a graph (left) and as a coupling matrix (right). In the graph representation, the

dynamical nodes are represented by red nodes and the edges are coloured with

respect to the coupling weight. Red and blue refer to positive and negative coupling

weights, respectively. Light and dark colors refer to weak and strong coupling weights,

respectively. (b) Distribution of the phases

𝜙𝑖

for each of the three clusters. Each node

represents one oscillator and is coloured with respect to the cluster to which it belongs.

Parameter values:

𝜖=

0.01,

𝛼=

0.3

𝜋

𝛽=

0.23

𝜋

𝜔=

0, and

𝑁=

100. Figure modified

from [BER19a]. ......................................... 58

5.2

Three different types of multicluster states at

𝑡=

10000 with

𝑁=

100 and

𝜖=

0.01.

For all types, the coupling matrix (left), distribution of the phases (middle), and time

series of representative phase oscillators from each cluster (right) are presented. In the

plot of the phase distribution, each node represents one oscillator and is colored with

respect to the cluster to which it belongs. The time series are shown after subtracting

the average linear growth

𝜙𝑖,𝜇(𝑡)−hΩ𝜇i𝑡

. The colouring of the time series (shaded for

visibility) of a representative phase oscillator from one cluster is in accordance with

the pictures in the middle panel. (a) Splay type 3-cluster for

𝛼=

0.3

𝜋

𝛽=

0.23

𝜋

; (b)

Antipodal type 3-cluster for

𝛼=

0.3

𝜋

𝛽=−

0.53

𝜋

; (c) Mixed type 2-cluster for

𝛼=

0.3

𝜋

𝛽=−0.4𝜋. Figure taken from [BER19a]. ........................... 61

5.3

Three-cluster of splay type. (a) Coupling weights at

𝑡=

10000 showing three clusters;

(b) Distribution of the phases within each cluster; space-time raster plot; (c) Average

frequency of oscillators; each plateau corresponds to one cluster; (d) Oscillator phases

𝜙𝑖(𝑡)

at fixed time

𝑡=

10000. Parameter values:

𝜖=

0.01,

𝛼=

0.3

𝜋

𝛽=

0.23

𝜋

, and

𝑁=100. Figure taken from [BER19]. ............................ 63

5.4

The figures show all one- and two-cluster solutions of splay type for the system

(5.1)

–

(5.2)

. For this, the frequency differences

ΔΩ12

are displayed corresponding to

the equations (4.6) and (5.7). The dotted lines (black) indicate unstable solutions while

the solid lines (blue) indicate stable solutions. Here, every second solution is plotted

for the sake of visibility. Parameter values: (a)

𝑁=

20,

𝜖=

0.01; (b)

𝑁=

50,

𝜖=

0.01;

(c)

𝑁=

50,

𝜖=

0.001; (d)

𝑁=

50,

𝜖=

0.1;

𝛼=

0.3

𝜋

is fixed for all panels. Figure taken

from [BER19]. .......................................... 64

5.5

For the case of two-cluster states of splay type, the critical value

𝜖𝑐

of time-separation

parameter

𝜖

is plotted as a function of

𝛾=𝛼−𝛽

for different cluster sizes

𝑛1=𝑁1/𝑁

The function is given explicitly by Eq. (5.10). Figure taken from [BER19a]. ....... 66

5.6

Three-cluster of antipodal type. (a) Coupling weights at

𝑡=

10000 showing three

clusters; (b) Distribution of the phases within each cluster; space-time raster plot;

Oscillator phases

𝜙𝑖(𝑡)

at fixed time

𝑡=

10000. Parameter values:

𝜖=

0.01,

𝛼=

0.3

𝜋

𝛽=−0.53𝜋, and 𝑁=100. Figure taken from [BER19]. ................... 70

5.7

For 3-cluster solution from Fig. 5.6, panel (a) shows time series of an oscillator from

one of the clusters after subtracting the average linear growth

𝜙𝜇,𝑖(𝑡)−hΩ𝜇i𝑡

. The black

dashed lines show the corresponding analytic results from the asymptotic expansion

in Proposition 5.4.1. (b) Power spectrum of the time series given in (a). Figure modified

from [BER19]. .......................................... 70

5.8

Two-cluster solutions (upper panels) and one-cluster solutions (lower panels) of

antipodal type given by the asymptotic expansion in Corollary 5.4.2 and Proposi-

tion 4.1.1, respectively. For this, the difference of the frequencies

ΔΩ(1)

is displayed

corresponding to (5.15) and (4.6). The dotted lines (black) indicate unstable solutions

while the solid lines (blue) indicate stable solutions. Here, every second solution is

plotted for the sake of visibility. The insets show a blow-up of the interval

[−𝜖

𝜖]

Parameter values: (a)

𝑁=

20,

𝜖=

0.01; (b)

𝑁=

50,

𝜖=

0.01;

𝛼=

0.3

𝜋

is fixed for all

panels. Figure taken from [BER19]. ............................. 72

5.9

2-Cluster solution of mixed type. (a) Coupling weights at

𝑡=

10000 showing two

clusters, (b) Distribution of the phases within each cluster, space-time representation.

𝜙𝑖

for fixed time

𝑡=

10000. Parameter values:

𝜖=

0.01,

𝛼=

0.3

𝜋

𝛽=−

0.4

𝜋

𝑁=

100. Figure taken

from [BER19]. ......................................... 75

5.10

For mixed type 2-cluster solution from Fig. 5.9, panel (a) shows time series of an oscilla-

tor from one of the clusters after subtracting the average linear growth

𝜙𝜇,𝑖(𝑡)−hΩ𝜇i𝑡

The black dashed lines show the corresponding analytic results from the asymptotic

expansion in Proposition 5.5.1. (b) Power spectrum of the time series given in (a).

Figure moified from [BER19]. ................................ 75

5.11

Two-cluster solutions of mixed type (upper panels) and one-cluster solutions (lower

panels) of either splay or antipodal type given by the asymptotic expansion in Corol-

lary 5.5.2 and Proposition 4.1.1, respectively. For this, the difference of the frequencies

ΔΩ(1)

is displayed corresponding to (5.17) and (4.6). The dotted lines (black) indicate

unstable solutions while the solid lines (blue) indicate stable solutions. Here, every

second solution is plotted for the sake of visibility. The insets show a blow-up of the

interval

[−𝜖

𝜖]

. Parameter values: (a)

𝑁=

20,

𝜖=

0.01; (b)

𝑁=

50,

𝜖=

0.01;

𝛼=

0.3

𝜋

fixed for all panels. Figure taken from [BER19]. ...................... 76

5.12

Stability diagram for the one-cluster and two-cluster solution of the splay type de-

pending on the parameters

𝛼

and

𝛽

. Yellow region corresponds to the instability

of both solutions, dark blue to the stability of both solutions, and light-blue to the

stability of only the one-cluster solution. Parameter

𝜖=

0.01 is fixed in all simulations.

Figure taken from [BER19]. .................................. 77

6.1

Illustration for two types of one-cluster states. The panels (a,c) show the asymptotic

coupling matrices and (b,d) snapshots of the phases at a fixed time. Results for the

one-cluster states of antipodal type are presented in (a,b) where

𝛼=

0.19

𝜋

𝛽=−

0.66

𝜋

and of splay type in (c,d) where

𝛼=

0.35

𝜋

𝛽=

0.01

𝜋

. Parameters:

𝑁=

100,

𝑃=

20,

𝜖=0.01. Figure modified from [?]. .............................. 85

6.2

Schematic figure illustrating the definition of multicluster and subnetworks induced

by groups of nodes with the same average frequency. The full network (left) consists

𝑁=

20 nodes and has a nonlocal ring structure with

𝑃=

4. The colors of the nodes

indicate their average frequencies. Clusters are shown by the equally colored nodes

that form connected sub-networks. Even though the two blue groups I and II possess

the same averaged frequencies, they form two different clusters, since they are not

connected. Figure taken from [BER20c]. .......................... 86

6.3

Illustration of the different types of multicluster states. The panels (a,d,g) show

the coupling matrix, (b,e,h) phase snapshots and (c,f,i) average frequencies. (a-c):

antipodal two-cluster for

𝛼=

0.23

𝜋

𝛽=−

0.56

𝜋

; (d-f): splay two-cluster for

𝛼=

0.19

𝜋

𝛽=−

0.45

𝜋

; (g-i): antipodal five-cluster (I, II denote the two connected components of

the red and the blue clusters) for

𝛼=

0.3

𝜋

𝛽=−

0.53

𝜋

. Parameters:

𝑁=

100,

𝑃=

20,

𝜖=0.01. Figure taken from [BER20c]. ............................ 88

6.4 Map of regimes for one- and multicluster states of antipodal and splay type in (𝛼,𝛽)

parameter space. Parameters:

𝑁=

100,

𝑃=

20,

𝜖=

0.01. The horizontal black line at

𝛼=

0.1 shows the location for the parameter

𝛽

where the emergence of solitary states

is analyzed, see Fig. 6.7 in Sec. 6.3. Figure taken from [BER20c]. ............. 88

6.5

Illustration of solitary states. The panels (a,d) show coupling matrix, (b,e) phase

snapshots, and (c,f) average frequencies. (a-c): single solitary state for

𝛼=

0.1

𝜋

𝛽=−

0.3

𝜋

; (d-f): three uncoupled solitary states for

𝛼=

0.15

𝜋

𝛽=−

0.41

𝜋

. Parameters:

𝑁=100, 𝑃=20, 𝜖=0.01. Figure modified from [BER20c]. ................ 89

6.6

Stability of one-cluster states for different wavenumbers

𝑘

and coupling ranges

𝑃

Regions of stability for the one-cluster states are colored in blue, while instability in

yellow. The borders of stability (black dashed lines) are obtained from the eigenvalues

(6.17)

. Parameters are as follows: (a)

𝑃=

10,

𝑘=

1; (b)

𝑃=

10,

𝑘=

4; (c)

𝑃=

10,

𝑘=

25;

(d)

𝑃=

𝑘=

1; (e)

𝑃=

20,

𝑘=

1; and (f)

𝑃=

25,

𝑘=

1. The other parameters are

𝑁=50 and 𝜖=0.01. Figure modified from [BER20c]. ................... 92

6.7

Phase portraits for two-dimensional system

(6.20)

–

(6.21)

. The graphics show the

two classes of asymptotic states that are equilibria (colored nodes) and periodic

solutions (colored lines). The stability properties of the individual asymptotic states

are indicated by the coloring where the blue refers to stable and the red (dashed) to

unstable states. In addition, several trajectories are plotted in black including those

close to the stable and unstable manifold of the equilibria. The nullclines are displayed

as gray lines. For the different panels parameter

𝛽

is varied as shown in Fig. 6.4:

(a)

𝛽=−

0.601

𝜋

; (b)

𝛽=−

0.599

𝜋

; (c)

𝛽=−

0.58

𝜋

; (d)

𝛽=−

0.5515

𝜋

; (e)

𝛽=−

0.5

𝜋

; (f)

𝛽=−

0.08

𝜋

; (g)

𝛽=−

0.0563

𝜋

; and (h)

𝛽=

0.05

𝜋

. The other parameters are

𝛼=

0.1

𝜋

and

𝜖=0.01. Figure modified from [BER20c]. .......................... 94

7.1

The figure shows the master stability function for the equations

(7.11)

–

(7.12)

. Regions

belonging to negative Lyapunov exponents are colored blue. The one dimensional

curve where at least one eigenvalue of

(7.15)

has zero real part is given as a black

dotted line. Parameter: (a)

𝛽=−

0.8

𝜋

, (b)

𝛽=−

0.2

𝜋

, (c)

𝛽=−

0.02

𝜋

, (d)

𝛽=

0.05

𝜋

, (e)

𝛽=0.1𝜋, and (f) 𝛽=0.98𝜋. The other parameters are 𝛼=0.3𝜋and 𝜖=0.01. ...... 106

7.2

Results for an adiabatic continuation in the coupling constant

𝜎

of the full synchronous

solution

𝜙𝑖=

0 and

𝜅𝑖 𝑗 =−sin(𝛽)

for a globally coupled network of

𝑁=

200 oscil-

lators

(7.11)

–

(7.12)

. The cluster parameter

𝑅𝐶

for different values of

𝜎

is presented.

As an inset, for the three values (a)

𝜎=

0.002, (b)

𝜎=

0.006, and (c)

𝜎=

0.025, the

master stability function, the phases

𝜙𝑖

of the final state, and the frequencies of the

oscillators

Ω𝑖

are plotted. The oscillators are sorted as in Fig 5.2. If more than or equal

to ten (for numerical convenience) oscillators have the same frequency (coherent

groups) all nodes of this group are plotted as circles and with a respective color. All

other oscillators are plotted as an asterisk. The master function parameter

˜𝜇=𝜎𝜇

for the subnetworks induced by the coherent groups are plotted together with the

master stability function. The colors of each coherent group agree in all three plots.

Parameters: 𝛼=0.49𝜋,𝛽=0.88𝜋,𝜖=0.01. ......................... 107

7.3

Results for an adiabatic continuation in the coupling constant

𝜎

of the full synchronous

solution

𝜙𝑖=

0 and

𝜅𝑖 𝑗 =−sin(𝛽)

for a complex network (see Appendix A.8) of

𝑁=

200 oscillators

(7.11)

–

(7.12)

. The cluster parameter

𝑅𝐶

for different values of

𝜎

is presented. As an inset, for the three values (a)

𝜎=

0.003, (b)

𝜎=

0.007, and

(c)

𝜎=

0.019, the master stability function, the phases

𝜙𝑖

of the final state, and the

frequencies of the oscillators

Ω𝑖

are plotted. If more than or equal to ten ()for numerical

convenience) oscillators have the same frequency (coherent groups) all nodes of this

group are plotted as circles and with a respective color. All other oscillators are plotted

as an asterisk. The master function parameter

˜𝜇=𝜎𝜇

for the subnetworks induced

by the coherent groups are plotted together with the master stability function. The

colors of each coherent group agree in all three plots. Parameters:

𝛼=

0.49

𝜋

𝛽=

0.88

𝜋

𝜖=0.01. ............................................. 109

8.1

Different duplex states of Eq.

(4.3)

(

𝐿=

2) for an ensemble of 50 oscillators in each

layer with color-coded coupling weights

𝜅𝜇

𝑖 𝑗

(upper panels, color code as in Fig.1),

phases

𝜙𝜇

𝑗

(lower panels): Duplex one-cluster states (a) of lifted splay type (

𝑅2(a𝜇)=

for

𝛼12/21 =

0.3

𝜋

𝜎12/21 =

0.07; (b) of lifted antipodal type (

𝑅2(a𝜇)=

1) for

𝛼12 =

0.3

𝜋

𝛼21 =

0.75

𝜋

𝜎12/21 =

0.62; (c) of double antidodal type (not a lifted state) for

𝛼12/21 =

0.05

𝜋

𝜎12/21 =

0.28; (d) of lifted splay type for

𝛼12 =

0.3

𝜋

𝛼21 =

0.4

𝜋

𝜎12/21 =

0.8, and

𝜖=

0.01. In the lower panels phase differences between the two layers are indicated

Δ𝑎≡𝑎1

𝑖−𝑎2

𝑖

, and between the two new antipodal states (c) by

𝜓1

𝜓2

. Figure taken

from [BER19b]. ......................................... 114

8.2

Birth of double antipodal state in a duplex network (

𝑁=

12) for a wide range of

inter-layer coupling strength

𝜎=𝜎12 =𝜎21

. The solid lines are the temporal averages

for the second moment order parameter

𝑅2

of the individual layers (layer 1: black,

layer 2: red). The error bars for

𝜎 < 𝜎𝑐

denote the standard deviation of the temporal

evolution of

𝑅2

. The dashed horizontal lines represent the unique values of

𝑅2

for the

double antipodal state in a monoplex network. The plot was obtained by adiabatic

continuation of a duplex double antipodal state (see inset) in both directions starting

from

𝜎=

0.5. Parameters:

𝛼11/22 =

0.3

𝜋

𝛼12/21 =

0.05,

𝛽1=

0.1

𝜋

𝛽2=−

0.95

𝜋

, and

𝜖=0.01. Figure taken from [BER19b]. ............................ 115

8.3

The figure shows the range

Δ𝜔

vs where duplex one-cluster states in general (gray)

and of the form presented in Fig. 3 of the main text (red) can be found. For this

the system

(8.7)

(8.8)

is integrated numerically for 10 different random uniform

distributions of the natural frequencies in the interval

[−Δ𝜔

Δ𝜔]

. The results are

obtained by adiabatic continuation starting with the phase clusters found for Δ𝜔=0

(see Fig. 3 of the main text). Duplex one-cluster states of double antipodal type with

(a)

𝜎=

0.5,

Δ =

0.02 and (b)

𝜎=

0.5,

Δ =

0.07 are shown as insets. Parameters:

𝛼11/22 =

0.3

𝜋

𝛼12/21 =

0.05,

𝛽1=

0.1

𝜋

𝛽2=−

0.95

𝜋

𝜖=

0.01, and

𝑁=

12. Figure taken

from [BER19b]. ......................................... 116

8.4

The figure shows the standard deviation

𝜌

where duplex one-cluster states in general

(gray) and of the form presented in Fig. 3 of the main text (red) can be found. For

this the system

(8.7)

(8.8)

is integrated numerically for 10 different random normal

distributions of the natural frequencies with standard deviation

𝜌

and zero mean.

The results are obtained by adiabatic continuation starting with the phase clusters

found for

𝜌=

0 (see Fig. 3 of the main text). A duplex one-cluster state of double

antipodal type with

𝜎=

0.5,

Δ =

0.02 is shown as an inset. Parameters:

𝛼11/22 =

0.3

𝜋

𝛼12/21 =0.05, 𝛽1=0.1𝜋,𝛽2=−0.95𝜋,𝜖=0.01, and 𝑁=12. Figure taken from [BER19b]. 117

8.5

Regions of stability (blue) and instability (white) of the lifted antipodal state in the

(𝛼22

𝛽2)

parameter plane for different values of interlayer coupling (indicated by

different blue shading)

𝜎21

, where regions of stronger coupling

𝜎21

(lighter blue)

include such of weaker 𝜎21 (darker blue). Stability regions for single-layer antipodal

clusters are indicated by red hatched areas. The inter-layer coupling is considered as

(a) unidirectional (

𝜎12 =

0) and (b) bidirectional (

𝜎12 =𝜎21

). Parameters:

𝛼11 =

0.2

𝜋

𝛽1=−0.8𝜋,𝛼12 =0, 𝛼21 =0.3𝜋, and 𝜖=0.01. Figure taken from [BER19b]. ....... 122

A.1

Adjacency matrix of a connected, directed random network of

𝑁=

200 nodes with

constant row sum

𝑟=

50. The illustration shows the adjacency matrix where black

and white refer to whether a link between two nodes exist or not, respectively. . . . . 156

List of Tables

4.1

The table summarizes the values for

𝑛

th order parameter for the phase-locked solu-

tions introduced in Definition 4.1.1. ............................. 42

Acknowledgement

First of all, I would like to thank Prof. Dr. Dr. h.c. Eckehard Schöll, PhD and PD Dr. Serhiy

Yanchuk for introducing me to the exciting field of adaptive networks and the theory of non-

linear dynamics. I also thank for their supervision, valuable help, and support by suggestions

and questions regarding my research. I appreciate their encouragement to frequent visits of

international conferences which enabled me to profit from the exchange of knowledge within

the research community from a very early stage on.

I am very much in debt to Dr. Jakub Sawicki who provided guidance during the research and

writing of this thesis. Furthermore, I would like to thank Prof. Dr. Vladimir Nekorkin and Dr.

Dmitry Kasatkin for the fruitful collaboration which resulted in some of the results presented in

Chapter 4and 5. Likewise, I am grateful for the collaboration with PD Dr. Oleksandr Popovych. I

owe thanks to all the current, former, and visiting members of the Schöll group and the Yanchuk

group for the wonderful working atmosphere. In particular, I thank Dr. Iryna Omelchenko,

Prof. Dr. Anna Zakharova, Phd, Prof. Dr. Yuri Maistrenko, Dr. Stefan Ruschel, Dr. Simona Olmi,

Dr. Dmitry Puzyrev, Dr. Ewandson Lameu, Dr. Vander Freitas, Jan Fialkowski, Giulia Ruzzene,

Florian Stelzer, Nour Eldine Hanbali, Narges Chinichian, Michael Lindner, Deniz Nikitin, Maria

Masoliver, Franziska Beckschulte, Danila Semenov, my students Jacopo Zurbuch, Sören Nagel,

Philippe Lehmann, Simon Vock, Max Thiele, Amy Searle, Fenja Drauschke, Johanna Czech, Lucas

Kluge, Moritz Gerster, Vera Röhr, and my RISE students Bricker Ostler and Alicja Polanska for

exciting discussions and fruitful collaborations. Additionally, I owe thanks to Prof. Dr. Sarika

Jalan and Dr. Saptarshi Gosh for insightful discussions. Further, I would like to express my

thanks to Andrea Schulze who helped more than once with her tremendous knowledge on the

universities’ bureaucracy to pass administrative challenges and to Peter Orlowski who could

always help me when things went wrong with the IT.

I would like to thank Prof. Dr. Alessandro Torcini for preparing the third assessment of this

thesis and Prof. Dr. Dieter Breitschwerdt for chairing the defense of this thesis.

This work was supported in the framework of the DFG-RSF (Deutsche Forschungsgemeinschaft-

Russian Science Foundation) project: "Complex dynamical networks: effects of heterogeneous,

adaptive and time delayed couplings" under SCHO 307/15-1 and YA 225/3-1. Further, I would

like to thank the DAAD (Deutscher Akademischer Austauschdienst) which provided funding

within the RISE program to host two students for a summer internship at the AG Schöll. I

also thank all members of the IRTG (International Research Training Group) 1740 and the SFB

(Collorative Research Center) 910 for supporting me to take part in their exciting events and

conferences which were a fruitful ground to develop new ideas and get into contact with a lot of

inspiring people.

Last but certainly not least I thank my family for their constant support during the whole time of

my studies and in particular I thank my dear girlfriend Veronika for encouraging me to finalize

this thesis, for supporting me when things went slowly or got stuck and for taking me out of my

scientific bubble sometimes.

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