Chimeras in Multiplex Networks: Interplay of Inter- and Intra-Layer Delays [original]

ORIGINAL RESEARCH

published: 24 April 2019

doi: 10.3389/fams.2019.00019

Frontiers in Applied Mathematics and Statistics | www.frontiersin.org 1April 2019 | Volume 5 | Article 19

Edited by:

Jun Ma,

Lanzhou University of Technology,

China

Reviewed by:

Tanmoy Banerjee,

University of Burdwan, India

Carlo Laing,

College of Sciences, Massey

University, New Zealand

V. K. Chandrasekar,

SASTRA University, India

*Correspondence:

Jakub Sawicki

[email protected]

Specialty section:

This article was submitted to

Dynamical Systems,

a section of the journal

Frontiers in Applied Mathematics and

Statistics

Received: 10 January 2019

Accepted: 28 March 2019

Published: 24 April 2019

Citation:

Sawicki J, Ghosh S, Jalan S and

Zakharova A (2019) Chimeras in

Multiplex Networks: Interplay of Inter-

and Intra-Layer Delays.

Front. Appl. Math. Stat. 5:19.

doi: 10.3389/fams.2019.00019

Chimeras in Multiplex Networks:

Interplay of Inter- and Intra-Layer

Delays

Jakub Sawicki1*, Saptarshi Ghosh 2, Sarika Jalan 2and Anna Zakharova 1

1Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany, 2Complex Systems Lab, Discipline of

Physics, Indian Institute of Technology Indore, Indore, India

Time delay in complex networks with multiple interacting layers gives rise to special

dynamics. We study the scenarios of time delay induced patterns in a three-layer network

of FitzHugh-Nagumo oscillators. The topology of each layer is given by a nonlocally

coupled ring. For appropriate values of the time delay in the couplings between the

nodes, we find chimera states, i.e., hybrid spatio-temporal patterns characterized by

coexisting domains with incoherent and coherent dynamics. In particular, we focus

on the interplay of time delay in the intra-layer and inter-layer coupling term. In the

parameter plane of the two delay times we find regions where chimera states are

observed alternating with coherent dynamics. Moreover, in the presence of time delay

we detect full and relay inter-layer synchronization.

Keywords: chimera states, multiplex networks, FitzHugh-Nagumo oscillator, time delay, relay synchronization

1. INTRODUCTION

During the early eighteenth century, Leonhard Euler published a paper on The Seven Bridges of

Königsberg providing a mathematical background on vertices and edges [1]. Later on, it became

the cornerstone of the field of network science. Network science presents a unique platform to

study various complex real-world systems by analyzing the interactions between its constituent

entities and collectively investigating its behaviors [2–4]. A recent addition to the network science

is the multiplex framework which incorporates multiple types of interactions among nodes by

representing them in different layers [5,6]. For example, the neurons in the brain form different

groups consisting of the same neurons but interacting in different ways (chemical interaction

or electrical synapses) to perform different tasks [7,8]. Multiplex framework divides these

neuronal groups into different layers based on their functionalities [9]. Similarly, transportation

networks, communication networks, social networks and a lot of other real-world networks can be

represented in a multiplex framework to understand their structural and dynamical features in a

better fashion [10].

Recently, various synchronization scenarios have been investigated in multilayer structures,

including remote and relay synchronization [11–14]. Moreover, it has been shown that

multiplexing can be used to control spatio-temporal patterns in networks [15–18]. The advantage

of control schemes based on multiplexing is that they allow to achieve the desired state in a certain

layer without manipulating its parameters, and they can work for weak inter-layer coupling. For

example, it has been found that weak multiplexing can induce coherence resonance [19] as well as

chimera states [18] in neural networks.

Sawicki et al. Chimeras in Multiplex Networks

Chimera state is a peculiar partial synchronization pattern that

refers to a hybrid dynamics where coherence and incoherence

emerge simultaneously in a network of identical oscillators [20–

22]. Since its inception [23,24], chimera state has attracted

massive interest from the nonlinear community for both its

significance in understanding complex spatiotemporal patterns

and its probable applicabilities in various fields, especially in

neuroscience [25]. Here we study the role of the interplay of

intra- and inter-layer time delays for the emergence of chimera

states in a multi(tri)plex network. Time delays represent an

essential factor in real-world networks due to the finite speed of

information propagating through channels connecting the nodes.

They play a crucial role in determining the dynamical behavior

of a complex system [26–32]. Time delays have been shown to

heavily influence the parameter range for which chimera states

appear for both single and multiplex networks [14,15,33–35].

Moreover, recently a scheme has been proposed for engineering

chimera states using suitably placed heterogeneous delays [36].

In the present work we demonstrate that just the variation

of delay values in the intra- and inter-layer edges can lead

to various dynamical states and allows for control of spatio-

temporal patterns.

2. THE MODEL

We study a multiplex network with three layers (triplex) as shown

in Figure 1. Every single layer represents a ring of Nidentical

FitzHugh-Nagumo (FHN) oscillators with non-local (intra-layer)

topology. The outer layers i=1 and i=3 are coupled with the

middle layer i=2, so that it acts as a relay layer. There is no direct

connection between layers 1 and 3 (so-called ordinal coupling).

FIGURE 1 | Triplex network with ordinal coupling: The middle layer i=2 (red)

acts as relay layer between the two outer layers i=1, 3 (blue). Black dots

indicate the nodes, solid lines represent the intra-layer connections, and

dashed lines denote the inter-layer links. The intra-layer coupling is

characterized by the strength σiand time delay τi, and the inter-layer coupling

is characterized by the strength σij and time delay τij. For example, in layer 3

the intra-layer coupling strength is given by σ3and the intra-layer time delay is

τ3. Similarly, for the layers 1 and 2 the inter-layer coupling strength is given by

σ12 and the inter-layer time delay is τ12.

Our system is described by the following equations:

˙xi

k(t)=F(xi

k(t)) +σi

2Ri

k+Ri

l=k−Ri

H[xi

l(t−τi)−xi

k(t)]

j=1

σijH[xj

k(t−τij)−xi

k(t)], (1)

where x=(u,v)T∈R2,k∈ {1, ..., N},i∈ {1, ..., 3}with all

indices modulo N, describe the set of activator (u) and inhibitor

(v) variables. The intra-layer delay time is τiand the inter-layer

delay time is τij. The coupling radius in layer iis given by Ri. The

local dynamics of each oscillator is given by

F(x)=ε−1(u−u3

3−v)

u+a, (2)

where ε=0.05 is the parameter characterizing the time

scale separation. The FHN oscillator exhibits either oscillatory

(|a|<1) or excitable (|a|>1) behavior depending on the

threshold parameter a. In this work we focus on the oscillatory

regime (a=0.5). The parameter σistands for the coupling

FIGURE 2 | Dynamical regimes in the parameter plane of intra-layer coupling

delay τi≡τ1=τ2=τ3and inter-layer coupling delay τij ≡τ12 =τ23: “salt &

pepper” states (green islands) occur in the region of coherent states (blue

region) as traveling waves, cluster or synchronized states. At the border

between these two regimes chimera states can be found (red color). We

distinguish between the different regions on the one hand, by analyzing the

mean phase velocity and a snapshot of variables uk, on the other hand, by

means of the Laplacian distance measure [39]. The boundaries of these

regions are fitted linearly after the (τi,τij) plane has been sampled in steps

1τi=0.05 and 1τij =0.1. For all simulations of Equation (1) random initial

conditions are taken. Parameters are chosen as ε=0.05, a=0.5, σi=0.2,

σij =0.05, N=500, Ri=170, φ=π

2−0.1, and i,j=1, 2, 3.

Frontiers in Applied Mathematics and Statistics | www.frontiersin.org 2April 2019 | Volume 5 | Article 19

Sawicki et al. Chimeras in Multiplex Networks

FIGURE 3 | Dynamics in all three layers for different values of the inter-layer delay time τij:(A) Chimera state for τij =0.4, (B) “salt & pepper” state for τij =0.9, (C)

coherent state (cluster state) for τij =1.7. The intra-layer delay time is fixed at τi=0.8. The left column displays snapshots of variables ui

kfor the layers i=1, 2, 3,

while the right column illustrates the mean phase velocity profile ωk(dark blue) for the individual layers and the local inter-layer synchronization error Eij

k(light yellow).

Other parameters as in Figure 2.

strength inside the layer (intra-layer coupling), and σij is the

inter-layer coupling. For an ordinal inter-layer coupling with

constant row sum we set σ12 =σ23, which yields the inter-layer

coupling matrix

σ=



0σ12 0

σ12

20σ23

0σ23 0



(3)

The connections between the nodes are given by the diffusive

coupling with the following coupling matrix

H=ε−1cos φ ε−1sin φ

−sin φcos φ(4)

and coupling phase φ=π

2−0.1 [37]. This coupling

configuration (i.e., predominantly activator-inhibitor cross-

coupling) is similar to a phase-lag of approximately π/2 in

the Kuramoto model that ensures the occurrence of chimera

states [37].

3. INTERPLAY OF TIME DELAYS

Chimera states are spatio-temporal patterns where incoherent

and coherent domains coexist in space. For certain values of

coupling strength σiand coupling radius Rione can detect

them in the ith layer [37]. Recently, the phenomenon of

relay synchronization of chimera states has been studied in a

three-layer network of FHN oscillators [38]. In more detail,

for varying the coupling delay and strength in the inter-layer

connections relay synchronization of chimera states in the outer

network layers has been reported. For appropriate parameters

the so-called “double” chimeras are possible where the coherent

parts of the chimera states are synchronized, whereas the

incoherent parts remain desynchronized. Additionally, the

transitions between different synchronization scenarios have

been studied. Moreover, time delay in the inter-layer coupling

has been shown to be a powerful tool for controlling various

partial synchronization patterns in the three-layer network [14].

Therefore, by multiplexing and introducing inter-layer time

delays it is possible to destroy or induce chimera states. Here we

study the interplay of inter- and intra-layer time delay.

To provide an overview of the patterns observed in the

network, we calculate the map of regimes in the parameter

plane of intra-layer delay time τiand inter-layer delay time τij

(Figure 2). The dominating region is the one corresponding to

coherent states (blue region in Figure 2). On the one hand, we

Frontiers in Applied Mathematics and Statistics | www.frontiersin.org 3April 2019 | Volume 5 | Article 19

Sawicki et al. Chimeras in Multiplex Networks

FIGURE 4 | Dynamics in all three layers for different values of the intra-layer delay time τi:(A) Chimera state for τi=2.8, (B) “salt & pepper” state for τi=2.6, (C)

coherent state (traveling wave) for τi=2.4. The inter-layer delay time is fixed at τij =2.6. The left column displays snapshots of variables ui

kfor the layers i=1, 2, 3,

while the right column illustrates the mean phase velocity profile ωk(dark blue) for the individual layers and the local inter-layer synchronization error Eij

k(light yellow).

Other parameters as in Figure 2.

detect the in-phase synchronization regime (see Figure 3C), on

the other hand, we also observe a region of coherent traveling

waves (see Figure 4C). By varying the delay times we can not

only switch between these states, but also adjust the speed of

traveling waves. In addition, we can observe salt and pepper states

(green region in Figure 2), where all nodes oscillate with the same

phase velocity but they are distributed between states with phase

lag πincoherently [40] (see Figures 3B,4B). The reason for

this are strong variations on very short length scales, so that the

dynamical patterns have arbitrarily short wavelengths. Besides

these two patterns characterized by the same mean phase velocity

for all the nodes in the network, we also observe chimera states

(red region in Figure 2), where the oscillators within each layer

show a characteristic arc-shaped mean phase velocity profile.

The mean phase velocities of the oscillators are given by ωk=

2πSk/1T,k=1, ..., N, where Skdenotes the number of complete

rotations performed by the kth oscillator during the time 1T.

The value 1T=10, 000 is fixed throughout the paper. Because

of the delay, the system often exhibits traveling structures.

Therefore, the classical arc-shaped profile is transformed into a

wider, conical one [41] (see Figures 3A,4A (right column)). We

distinguish between the different regions on the one hand, by

analyzing the mean phase velocity and a snapshot of variables uk,

on the other hand, by means of the Laplacian distance measure

[39]. The latter identifies strong local curvature in an otherwise

smooth spatial profile of xkby calculating the discrete Laplacian

k(xk+1−xk)−(xk−xk−1)kfor each k. In case of coherent dynamics

we obtain low values (≈0) for all k, in case of salt and pepper

states we get high values (>2). Chimera states show low and high

values for the coherent and incoherent domains, respectively. By

taking the mean over all k, we can distinguish between coherent,

chimera and “salt & pepper” states. For the numerical integration

an Euler integration method is used with a step size 1t=0.0025

and a transient time ttrans =2, 000. All simulations are evaluated

then after 1T=10, 000 as mentioned above.

In many systems with time delays resonance effects can be

expected if the delay time is an integer or half-integer multiple

of the period in the uncoupled system [42–44]. Regarding the

inter-layer delay time τij we can observe this effect for half-

integer multiples of the period T=2.3 of a uncoupled FHN

oscillator (see Figure 2 for τi≈2.5). Concerning the intra-layer

delay time τiwe find a resonance effect in the case of integer

multiple of the delay. For greater values of the delay times τi

and τij the dynamical regions are becoming curved (see green

islands in Figure 2 at τi≈2.5). This can be explained by the

fact that branches of periodic solutions, which are reappearing for

integer multiples of the intrinsic period, are becoming stretched

with increasing delay time [41]. In comparison to the almost

vertical shape at τi≈0.5, the green islands at τi≈2.5 are

rotated clockwise by approximate π/8. The consequence is an

overlapping of the delay islands for small intra-layer delay time

τi, whereas for larger delays the islands become separated.

Frontiers in Applied Mathematics and Statistics | www.frontiersin.org 4April 2019 | Volume 5 | Article 19

Sawicki et al. Chimeras in Multiplex Networks

4. RELAY SYNCHRONIZATION

Networks with multiple layers demonstrate remote

synchronization of distant layers via a relay layer. Regarding the

inter-layer synchronization, two synchronization mechanisms

are conceivable in a triplex network:

•full inter-layer synchronization when synchronization is

observed between all the layers and

•relay inter-layer synchronization when synchronization occurs

exclusively between the two outer layers.

A useful measure for synchronization between two layers i,jis

given by the global inter-layer synchronization error Eij [14,45]:

Eij =lim

T→∞

NT ZT

k=1



xj

k(t)−xi

k(t)



dt, (5)

where k·kdenotes the Euclidean norm. One can distinguish

between the two synchronization mechanisms by measuring the

global inter-layer synchronization error between the first and

second layer E12 and between the first and third layer E13: In the

case of full inter-layer synchronization E12 =0 as well as E13 =0,

while in the case of relay inter-layer synchronization E12 6= 0 and

E13 =0.

To provide more insight into the synchronizability of patterns

between two layers i,j, the local inter-layer synchronization error

in dependence of every single node kcan be used [14]:

Eij

k=lim

T→∞

TZT

0



xj

k(t)−xi

k(t)



dt. (6)

The local inter-layer synchronization error is convenient for

detecting the synchronized nodes between two layers. In

Figures 3,4(right column) Eij

kis plotted (light yellow) together

with the mean phase velocity (blue): Depending on the delay

times τiand τij we can find full inter-layer synchronization

(see Figure 3C) as well as relay inter-layer synchronization (see

Figures 3A,B,4A,C). These synchronization scenarios can be

found for both coherent and incoherent dynamics. An additional

effect is the partial relay synchronization scenario in Figure 4B:

In all three layers “salt & pepper” dynamics can be observed. The

nodes in the outer layers are almost all synchronized, but a small

part of them destroys the relay synchronization. On the other

hand a few oscillators in the relay layer are synchronized with

the outer layers.

5. CONCLUSION

In conclusion, we have studied chimera states in a three-

layer network of FitzHugh-Nagumo oscillators, where each

layer has a nonlocal coupling topology. Focusing on the role

of time delays in the coupling terms and their influence

on chimera states, we have performed a numerical study

of complex spatio-temporal patterns in the network. In the

parameter plane of the intra-layer τiand the inter-layer τij

time delay, we have determined the regions where chimera

patterns occur, alternating with regimes of coherent states. A

proper choice of time delay allows to achieve the desired state

of the network: chimera state or coherent pattern, full or relay

inter-layer synchronization.

Combining the delayed interactions with the multiplex

framework considered in this work can provide additional

insight into the formation of the complex spatio-temporal

patterns in real-world systems. Specifically, in brain networks

where EEG patterns are recently reported to display chimera-

like behavior at the onset of a seizure [46–48]. Inducing

the chimera states by tuning the inter- and intra-layer

delay values provides us with a powerful tool to control

chimera states.

AUTHOR CONTRIBUTIONS

JS did the numerical simulations and the theoretical

analysis. AZ supervised the study. All authors designed

the study and contributed to the preparation of the

manuscript. All the authors have read and approved the

final manuscript.

ACKNOWLEDGMENTS

This work was supported by the Deutsche

Forschungsgemeinschaft (DFG, German Research

Foundation)—Projektnummer—163436311—SFB 910 and

by the German Academic Exchange Service (DAAD) and the

Department of Science and Technology of India (DST) within

the PPP project (INT/FRG/DAAD/P-06/2018).

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Frontiers in Applied Mathematics and Statistics | www.frontiersin.org 6April 2019 | Volume 5 | Article 19