Chimeras in physics and biology : Synchronization and desynchronization of rhythms [original]

Nova Acta Leopoldina NF Nr. 425, 67–95 (2020), doi:10.26164/leopoldina_10_00275

Chimeras in Physics and Biology:

Synchronization and Desynchronization of Rhythms

Eckehard Schöll (Berlin)

Das Video zum Vortrag ist verfügbar unter https://doi.org/10.26164/leopoldina_10_00292.

Eckehard Schöll

68 Nova Acta Leopoldina NF Nr. 425, 67–95 (2020)

Abstract

Rhythms influence our life in various ways, e.g., through heart beat and respiration, oscillating brain currents, life

cycles and seasons, clocks and metronomes, pulsating lasers, transmission of data packets, and many others. The

physics of complex nonlinear systems has developed methods to describe and analyze periodic oscillations and

their synchronization in complex networks, which are composed of many components. Synchronized oscillations

as well as completely asynchronous chaotic oscillations play a major role in many networks in nature and technol-

ogy. For instance, the synchronous firing of all neurons in the brain represents a pathological state, like in epilepsy

or Parkinson’s disease, and should be suppressed, as well as the synchronous mechanical vibration of bridges. On

the other hand, synchronization is desirable for the stable operation of power grids or in encrypted communication

with chaotic signals. In networks composed of identical components, intriguing hybrid states (“chimeras”) may

form spontaneously, which consist of spatially coexisting synchronized and desynchronized domains, i.e., seemingly

incongruous parts. This might be of relevance in inducing and terminating epileptic seizures, or in unihemispheric

sleep which is found in certain migratory birds and mammals, or in cascading failures of the power grid.

Zusammenfassung

Rhythmen prägen unser Leben auf vielfältige Weise, z. B. durch Herzschlag und Atmung, oszillierende Gehirnströ-

me, Lebenszyklen und Jahreszeiten, Uhren und Metronome, pulsierende Laser, Übertragung von Datenpaketen, und

vieles andere. Die Physik komplexer nichtlinearer Systeme hat Methoden entwickelt, wie periodische Schwingungen

und deren Synchronisation in komplexen Netzwerken, die aus vielen Bestandteilen zusammengesetzt sind, beschrie-

ben und analysiert werden können. Synchronisierte Oszillationen, aber auch völlig desynchronisierte, chaotische

Oszillationen spielen eine große Rolle in vielen Netzwerken in Natur und Technik. Beispielsweise ist das synchroni-

sierte Feuern aller Neuronen im Gehirn ein pathologischer Zustand, etwa bei Epilepsie oder Parkinson-Erkrankung,

und sollte unterdrückt werden, wie auch synchrone mechanische Schwingungen von Brücken. Andererseits ist die

Synchronisation erwünscht beim stabilen Betrieb von Stromnetzen oder bei der verschlüsselten Kommunikation mit

chaotischen Signalen. In Netzwerken aus identischen Komponenten können sich überraschenderweise auch spontan

Hybrid-Zustände („Schimären“) bilden, die aus räumlich koexistierenden synchronisierten und desynchronisierten

Bereichen bestehen, welche scheinbar nicht zusammen passen. Diese könnten relevant sein bei der Auslösung oder

Beendigung epileptischer Anfälle, oder beim halbseitigen Schlaf einer Gehirnhälfte, der bei bestimmten Zugvögeln

oder Säugetieren auftritt, oder beim kaskadenartigen Zusammenbruch des Stromnetzes.

Chimeras in Physics and Biology: Synchronization and Desynchronization of Rhythms

Nova Acta Leopoldina NF Nr. 425, 67–95 (2020) 69

1. Synchronization of Rhythms in Complex Networks

Synchronization phenomena in nonlinear dynamical systems (Haken 1983, Pikovsky et

al. 2001, Mosekilde et al. 2002, Haken 2008, Balanov et al. 2009, Schöll et al. 2016,

Boccaletti et al. 2018) are of great importance in many areas ranging from physics and

chemistry to biology, neuroscience, socio-economic systems, and engineering. Probably the

first example was given by Christiaan Huygens (1629 –1695), who observed that while two

individual pendulum clocks show slightly deviating times, they spontaneously synchronize

at exactly the same frequency if they are weakly coupled via a wooden beam (Fig. 1). Even

if individual systems, e.g., semiconductor lasers, exhibit chaotic dynamics, they may sponta-

neously synchronize their chaotic time series if coupled (Soriano et al. 2013).

Fig. 1 Christiaan Huygens (1671) and his sketch of the two coupled pendulum clocks which he observed to synchro-

nize spontaneously. (After Huygens 1932.)

Eckehard Schöll

70 Nova Acta Leopoldina NF Nr. 425, 67–95 (2020)

Synchronization of rhythms is also observed generally if more than two oscillating elements

are coupled in a network, even for large and complex networks. Complex networks are a

ubiquitous paradigm in nature and technology, and a central issue in nonlinear science with

applications to different fields ranging from natural to technological and socio-economic

systems (Fig. 2). The interplay of nonlinear dynamics, network topology, naturally arising

delays, and random fluctuations results in a plethora of spatio-temporal patterns (Schöll et

al. 2016).

Fig. 2 Complex networks exist in many diverse fields: (A) brain, (B) internet, (C), (D) power grids (panel D shows

the high-voltage power grid of Germany [Taher et al. 2019]), (E) social network.

There exist many applications of synchronization in various nonlinear systems, and so-

metimes synchrony is desirable, whereas sometimes it may be undesirable. Synchronization

of lasers with chaotic dynamics, for instance, may lead to new secure communication sche-

mes (Cuomo and Oppenheim 1993, Boccaletti et al. 2002, Argyris et al. 2005, Kanter et

al. 2008). The London Millenium Bridge was opened in the year 2000, but had to be closed on

the same day, after pedestrians experienced an alarming lateral swaying motion, and it took

almost two years while modifications and repairs were made to keep the bridge stable. The

instability was explained by a simple network model leading to spontaneous collective crowd

synchronization at a critical density of the pedestrians streaming onto the bridge (Strogatz

et al. 2005). Synchronization of power grids to the nominal frequency of 50 Hz is essential

A B

Chimeras in Physics and Biology: Synchronization and Desynchronization of Rhythms

Nova Acta Leopoldina NF Nr. 425, 67–95 (2020) 71

for their stable operation (Motter et al. 2013, Tumash et al. 2019, Taher et al. 2019). The

synchronization of neurons (Vicente et al. 2008) is believed to play a crucial role in the brain

under normal conditions, for instance in the context of memory, cognition and learning (Sin-

ger 1999), and under pathological conditions such as Parkinson’s disease (Tass et al. 1998) or

epilepsy (Andrzejak et al. 2006, Jiruska et al. 2013, Jirsa et al. 2014). Fireflies are known

to synchronize their flashing (Buck and Buck 1968). Especially in biology, rhythms and bio-

logical clocks play an important role and have given the name to a whole discipline, namely

chronobiology (Peschke 2011), see also the Abstracts by Jessica Grahn, Steve Kay, and

Russell Foster, in this Meeting. Examples are the circadian rhythms in single-cell organisms,

plants, fruit flies, mammals, and humans; the high relevance of chronobiology is documented

by the award of the 2017 Nobel Prize for Physiology or Medicine to Jeffrey C. Hall, Michael

Rosbash, and Michael W. Young for their discoveries of molecular mechanisms controlling

the circadian rhythm.

In many realistic dynamical networks time delay effects are a key issue (Just et al. 2010,

Flunkert et al. 2013, Schöll 2013). For example, the finite propagation time of light bet-

ween coupled semiconductor lasers (Wünsche et al. 2005, Carr et al. 2006, Erzgräber

et al. 2006, Fischer et al. 2006, D’Huys et al. 2008, Soriano et al. 2013) significantly in-

fluences the dynamics on networks. Similar effects occur in neuronal (Rossoni et al. 2005,

Masoller et al. 2008, Schöll et al. 2009) and biological gene expression networks (Tiana

and Jensen 2013) due to signal processing and propagation delays. Time delay has two com-

plementary, counterintuitive and almost contradicting facets. On the one hand, delay is able

to induce instabilities, generate new dynamical behavior, e.g. periodic and quasiperiodic time

evolution, multistability and chaotic motion. On the other hand, delay can suppress instabi-

lities, stabilize unstable stationary or periodic states and may control deterministic chaos.

Both facets open up efficient methods of designing and controlling nonlinear dynamics by

time-delayed feedback (Schöll and Schuster 2008, Sun and Ding 2013).

There exist different forms of synchronization, i.e., complete or isochronous (zerolag)

synchronization, generalized synchronization (where the oscillations of the individual ele-

ments of the network are not identical, but functionally related), phase synchronization (whe-

re only the phases but not the amplitudes of the oscillations are synchronized), cluster or

group synchronization (Sorrentino and Ott 2007, Dahms et al. 2012, Williams et al. 2013,

Taylor et al. 2011, Nkomo et al. 2013) (where within each cluster all elements are comple-

tely synchronized, but between the clusters there is a phase lag), and many other forms. Some

progress has been made in generalizing this work, for instance, towards adaptive networks

(Lehnert et al. 2014, Kasatkin et al. 2017, Berner et al. 2019a, b, 2020) (where the strength

of the links is adapted dynamically), inhomogeneous local dynamics (Sorrentino and Pe-

cora 2016) and heterogeneous delay times (Cakan et al., 2014), distributed (Kyrychko et

al. 2014, Wille et al. 2014), state-dependent, or time-varying delays (Gjurchinovski et al.

2014). In general, the stability of synchronization in delay-coupled networks of oscillators

depends in a complicated way on the local dynamics of the nodes and the coupling topology.

However, for large coupling delays synchronizability relates in a simple way to the spectral

properties of the network topology, characterized by the eigenvalue spectrum of the coupling

matrix. The master stability function (Pecora and Carroll 1998) used to determine the

stability of synchronous solutions has a universal structure in the limit of large delay: it is ro-

tationally symmetric and monotonically increasing. This allows for a universal classification

of networks with respect to synchronization properties (Flunkert et al. 2010). For smaller

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