Effects of time delay and coupling strength on synchronization transitions in excitable homogeneous random network

Commun Nonlinear Sci Numer Simulat 18 (2013) 3509–3516

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Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier .com/locate /cnsns

Effects of time delay and coupling strength on synchronizationtransitions in excitable homogeneous random network

1007-5704/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cnsns.2013.05.008

⇑ Corresponding author at: Department of Physics and Information Technology, Baoji University of Arts and Sciences, Baoji 721007, China.13759756802.

E-mail addresses: [email protected] (Y. Qian), [email protected] (Z. Zhang), [email protected] (Y. Mi).

Yu Qian a,b,c,⇑, Yaru Zhao a, Fei Liu a, Xiaodong Huang d, Zhaoyang Zhang e, Yuanyuan Mi b

a Department of Physics and Information Technology, Baoji University of Arts and Sciences, Baoji 721007, Chinab Center for Systems Biology, Soochow University, Suzhou 215000, Chinac State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, Chinad Department of Physics, South China University of Technology, Guangzhou 510640, Chinae Department of Physics, Beijing Normal University, Beijing 100875, China

a r t i c l e i n f o

Article history:Received 26 December 2012Accepted 7 May 2013Available online 20 May 2013

Keywords:DelaySynchronization transitionsNeuronal networks

a b s t r a c t

In this paper we numerically investigate the effects of time delay and coupling strength onsynchronization transitions in excitable homogeneous random network. Different roles oftime delay and coupling strength have been discovered by synchronization parameter andspace–time plots. Specifically, we have found three distinct parameter regions, i.e., asyn-chronous region (domain I for small time delay), transition region (domain II for moderatetime delay) and synchronous region (domain III for large time delay) as time delay isincreased. The phenomenon of multi-stability is observed in the transition region. Whilecoupling strength can enhance synchronization in the transition region and can reducesynchronization time in the synchronous region. All these results are independence onthe system size.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

In recent three decades, self-organized collective behaviors in coupled oscillators have attracted great attentions in thefield of nonlinear science. Synchronization phenomena are the most typical collective behaviors and have been extensivelystudied due to their popular existences and potential applications in many realistic systems, such as neuronal networks, bio-logical systems, ecological systems, and so on [1,2]. Several kinds of synchronization have been discovered in theoretical re-searches, such as identical synchronization, phase synchronization, lag synchronization and generalized synchronization [3–6]. Experimental studies have shown that synchronous oscillations can emerge in many special areas of the brain, especiallyin the olfactory system or the hippocampal region [7–9], and play important roles in information processing [10]. Synchro-nous oscillations in nervous systems are also regarded as the pathogenesis for epilepsy and tremor in Parkinson’ disease[11,12]. As the concepts of ‘‘small-world’’ and ‘‘scale-free’’ have been proposed [13,14], lots of works have been focusedon the synchronization of complex network to investigate the self-organized collective behaviors determined by the systemdynamics and network topology [15–21].

In reality, time delays in information transmission must be considered for neuronal networks due to the finite propaga-tion velocities in the conduction of signals over a distance. In recent years, lots of works have been taken on the neuronalnetworks with time delay and several amazing phenomena have been discovered [22–32]. For example, Dhamala et al.[22] investigated the enhancement of neural synchrony by time delay. Roxin et al. [23] studied the effect of delays on the

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dynamics of large networks of neurons. Wang et al. [27] observed the time delay induced coherence of spiral waves in noisyHodgkin–Huxley networks. In Refs. [28–30], authors found time delays can cause the transitions between in-phase and anti-phase synchronizations in coupled noisy neurons. Although many contributions have been achieved in delayed neuronalnetworks, more investigations are still needed to be taken in this field for its potential applications.

In the previous studies [33–35], we investigated the mechanisms of self-sustained oscillations in excitable homogeneousrandom networks (EHRNs). The loop structures self-organized in the networks have been discovered by dominant phase-ad-vanced driving method to sustain the oscillations. Now we extend the scope of synchronization transitions induced by timedelay and coupling strength in this field. The remainder of this paper is organized as follows. Section 2 introduces the math-ematical model and the corresponding methods which will be employed in the investigation. In Section 3, we study the ef-fects of time delay on synchronization transitions by synchronization parameter and space–time plots. And the effects ofcoupling strength on synchronization transitions are investigated in Section 4. In Section 5, the previous results are extendedto the EHRNs with different system sizes. Finally, we give the conclusion in the last section.

2. Mathematical model and methods

In this paper, we consider HRN consisting of N excitable nodes. And the B€ar-Eiswirth [36] model is used for local dynam-ics. The evolution of the studied network dynamics influenced by time delay and coupling strength is described by the fol-lowing equations:

duiðtÞdt

¼ �1�

uiðtÞ½uiðtÞ � 1� uiðtÞ �v iðtÞ þ b

a

� �þ D

Xj

Mi;j½ujðt � sÞ � uiðtÞ�; ð1Þ

dv iðtÞdt

¼ f ½uiðtÞ� � v iðtÞ; ð2Þ

where i ¼ 1;2; . . . ;N. The function f ½uiðtÞ� takes the form: f ½uiðtÞ� ¼ 0 for uiðtÞ < 13; f ½uiðtÞ� ¼ 1� 6:75uiðtÞ½uiðtÞ � 1�2 for

13 6 uiðtÞ 6 1; and f ½uiðtÞ� ¼ 1 for uiðtÞ > 1. Here variables uiðtÞ and v iðtÞ describe the activator and the inhibitor of the ithnode, respectively. The small relaxation parameter � represents the time ratio between activator uiðtÞ and inhibitor v iðtÞand a; b are two other parameters of the system. In this paper we fix a ¼ 0:84; b ¼ 0:07 and � ¼ 0:04 to ensure the localdynamics is excitable. Mi;j is the adjacency matrix element and is defined as Mi;j ¼ 1 if node i is connected with node jand Mi;j ¼ 0 otherwise. For HRNs, we assume symmetric couplings and adopt identical degree k ¼ 3 for each node (i.e., eachnode couples to three other nodes, and the bidirectional couplings are chosen randomly). D is the coupling strength of acti-vator u and s is the time delay in information transmission. Our work will focus on time delay s and coupling strength D andwill vary these two parameters in a wide range to investigate the synchronization transitions in EHRNs. While other param-eters will be fixed throughout this paper. The dynamical equations are integrated by the forward Euler integration schemewith time step Dt ¼ 0:001.

In order to quantitatively study the synchronization transitions in EHRNs, the synchronization parameter R, which hasbeen used in the earlier studies [37,38], is imposed, and can be calculated effectively as:

R ¼ h�u2i � h�ui21N

PNi¼1ðhu2

i i � huii2Þ; ð3Þ

where

�u ¼ 1N

XN

i¼1

ui: ð4Þ

From Eq. (3) we can find that the value of R close to zero indicates that the states of the individual excitable nodes aresignificantly different and the whole system oscillates asynchronously at all. While a value of R close to unity indicatesall nodes in the network are in complete synchronization. To avoid the fluctuation induced by network structure and initialcondition, 50 samples with different random network structures and different random initial conditions are employed foreach set of parameter values in the simulation. And we will use

�R ¼ 150

X50

i¼1

Ri ð5Þ

as an order parameter to measure the degree of synchronization induced by time delay and coupling strength in EHRNs.

3. The effects of time delay on synchronization transitions

In this section, we firstly fix coupling strength D ¼ 0:5 and vary s to investigate the effects of time delay on synchroni-zation transitions in EHRNs. System size is set as N ¼ 100, and it will be used in Sections 3 and 4. Fig. 1 displays

Fig. 1. Dependence of synchronization parameters R (50 samples for each s, depicted by black dots) and �R (the average of Rs for 50 samples, depicted by reddots) on time delay s. Three distinct parameter regions, i.e., asynchronous region (domain I for small s), transition region (domain II for moderate s,indicated by gray rectangle) and synchronous region (domain III for large s) are displayed. (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.)

Fig. 2. Space–time plots of u for different time delay s under fixed coupling strength D ¼ 0:5. (a) s ¼ 0:0, (b) s ¼ 1:0, (c) s ¼ 2:4, (d) s ¼ 3:0. The figures areplotted in grayscale from black (lowest value at 0.0) to white (highest value at 1.0). And this grayscale will be used throughout this paper.

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synchronization parameters R (50 samples for each s, depicted by black dots) and �R (the average of Rs for 50 samples, de-picted by red dots) for different time delay s. As s ¼ 0:0, i.e., there is no time delay in information transmission, the synchro-nization parameters R and �R are all around 0.20. It indicates that the self-sustained oscillations in EHRNs without time delayis a kind of weak synchronization state. As reported in Refs. [33–35], loop structures can self-organize in the network and akind of target wave like patterns can emerge to sustain the oscillations without any external pacemakers. The nodes in everysame wave front, i.e., located in the same distance to the target center (see Fig. 2(a)–(c) in Ref [34] for detailed description),will oscillate synchronously. That’s why we can observe weak synchronous oscillation without time delay. And a typical spa-tiotemporal pattern in EHRNs for s ¼ 0:0 is shown in Fig. 2(a). Some nodes in the network execute synchronous oscillationand a weak synchronization state of the whole system can be detected. As time delay s is increased, synchronization param-eters R and �R decrease simultaneously. This is caused by the coexistence of loop structure and delayed feedback. The dynam-ics of every excitable node is governed by these two effects, and the whole network exhibits complex and disorderedbehavior. Fig. 2(b) exhibits the firing pattern for s ¼ 1:0. Most of the nodes in the network oscillate asynchronously and

Fig. 3. Space–time plots of u for different random initial conditions under fixed network structure for time delay s ¼ 2:4. Distinct synchronous states, i.e.the phenomenon of multi-stability, have been observed in the transition region: (a) asynchronous state, (b) weak synchronization and (c) completesynchronization.

Fig. 4. (a)–(c) Space–time plots of u for different coupling strength D under fixed time delay s ¼ 2:4 (transition region) with same network structure andsame initial condition: (a) asynchronous state for D ¼ 0:4, (b) weak synchronization for D ¼ 0:6, (c) complete synchronization for D ¼ 0:8 and (d)dependence of synchronization parameter �R on coupling strength D.

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the spatiotemporal pattern becomes irregular. For parameter region s 6 2:0 (domain I in Fig. 1), asynchronous states inducedby time delay in EHRNs can be obtained. When s is in the parameter region of ð2:2;2:8Þ, synchronous performance of thewhole network improves remarkably. And a weak synchronization state for s ¼ 2:4 is revealed in Fig. 2(c). The excitatoryfronts are more ordered both in time and space. We call this parameter region as the transition region (domain II in

Fig. 5. (a)–(c) Space–time plots of u for different coupling strength D under fixed time delay s ¼ 4:0 (synchronous region) with same network structure andsame initial condition: (a) D ¼ 0:3, (b) D ¼ 0:4, (c) D ¼ 0:8, (d) the time series �u of (a) (black line), (b) (red dashed) and (c) (green dotted line). The blue lineshows the criterion for complete synchronization (�uc ¼ 0:95 is used in this paper) and tci (i ¼ 1;2;3) indicates the time that is needed for synchronous statesof (a)–(c). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 1, indicated by gray rectangle). For large time delay, i.e. s P 3:0 (domain III in Fig. 1), synchronization parameters R and �Rincrease rapidly and close to one approximately. It indicates that large information transmission delay can induce completesynchronization in EHRNs. A perfectly synchronous spatiotemporal pattern for s ¼ 3:0 is shown in Fig. 2(d). All nodes in thenetwork fire simultaneously and damp to their rest state together.

Here we should notice that the phenomenon of multi-stability can be observed in the transition region. Fig. 3(a)–(c) re-veal three distinct spatiotemporal firing patterns realized from different random initial conditions under fixed networkstructure for time delay s ¼ 2:4. Asynchronous state (shown by Fig. 3(a)), weak synchronization (shown by Fig. 3(b)) andcomplete synchronization (shown by Fig. 3(c)) have been obtained for same time delay in the transition region. We speculatethe multi-stability in the transition region is induced by the sensitivity of the system to the initial conditions. For most initialconditions (45 samples), weak synchronization can be obtained. While for other few samples, the network performs asyn-chronous oscillations (four samples) or complete synchronization (only one sample).

4. The effects of coupling strength on synchronization transitions

From Section 3, we have found that time delays in neuronal networks are very important for synchronization transitions.Now the effects of coupling strength on synchronization transitions in EHRNs will be discussed in detail in this part. There-fore, time delay is set to be a constant. Here two samples of s in distinct parameter regions are selected as examples (i.e.s ¼ 2:4 in transition region and s ¼ 4:0 in synchronous region). Fig. 4(a)–(c) display the spatiotemporal plots for differentcoupling strength under fixed time delay s ¼ 2:4. Same network structure and same random initial condition are used inFig. 4(a)–(c) to avoid the multi-stability appeared in the transition region. Fig. 4(a) shows irregular firing pattern forD ¼ 0:4. Asynchronous oscillation emerges in the network for small coupling strength. As coupling strength is increased,the firing pattern becomes more ordered. Weak synchronization can realize for moderate coupling strength (see Fig. 4(b)for D ¼ 0:6). While for large coupling strength, the whole network can achieve complete synchronization quickly (seeFig. 4(c) for D ¼ 0:8).

A quantitative study is carried out to investigate the effects of coupling strength on synchronization transitions in thetransition region. The dependence of synchronization parameter �R on coupling strength D under fixed time delay s ¼ 2:4

Fig. 6. Dependence of synchronization parameter �R on time delay s and coupling strength D.

Fig. 7. Dependence of synchronization parameter �R on time delay s with different system size N under fixed coupling strength D ¼ 0:5.

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is exhibited in Fig. 4(d). �R increases monotonously as we increase the coupling strength. And the enhancement of neuronalnetwork synchronization ability induced by coupling strength in transition region can be revealed by Fig. 4(d) obviously.

Here we investigate the effects of coupling strength on synchronization transitions in synchronous region. The firing pat-terns for different coupling strength under fixed time delay s ¼ 4:0 with same network structure and same random initialcondition are shown in Fig. 5(a)–(c). And Fig. 5(d) reveals the time series �u of figures (a) (black line for D ¼ 0:3), (b) (reddashed for D ¼ 0:4) and (c) (green dotted line for D ¼ 0:8). In this paper we use �uc ¼ 0:95 as the criterion to judge whethercomplete synchronization is achieved in the network (shown by blue line in Fig. 5(d)). And the labels tci (i ¼ 1;2;3) indicatethe time when synchronous states of (a), (b) and (c) is achieved. As we can see, the synchronization time which is needed forthe whole network to realize complete synchronization shortens when we increase the coupling strength.

To have a overall inspection of time delay and coupling strength on synchronization transitions in EHRNs, the contourplot of the synchronization parameter �R in the plane ðs;DÞ is revealed in Fig. 6. Three distinct parameter regions of time de-lay, i.e., asynchronous region (domain I for small s), transition region (domain II for moderate s, indicated by blue rectangle)and synchronous region (domain III for large s) are exposed clearly. And the promotion of network synchronization inducedby coupling strength in transition region is also revealed explicitly.

5. Complex networks with different system sizes

Till now we have studied the effects of time delay and coupling strength on synchronization transitions in EHRNs withsystem size N ¼ 100. In order to test the universality of our findings, we examine the previous results for other system sizes.Figs. 7 and 8 reveal the effects of time delay and coupling strength on synchronization transitions for different network sizeN. Time delay induced synchronization transitions can be observed for other system sizes (see Fig. 7 for N ¼ 50 (blacksquare), 200 (green triangle1), and 300 (purple star)). The three distinct parameter regions can emerge for same time delay

1 For interpretation of color in Fig. 7, the reader is referred to the web version of this article.

Fig. 8. Dependence of synchronization parameter �R on coupling strength D with different system size N under fixed time delay s ¼ 2:4.

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s with different N. While Fig. 8 displays the dependence of synchronization parameter �R on coupling strength D with differentsystem size N in the transition region. The enhancement of network synchronization ability induced by coupling strength canalso be discovered for other system sizes. All the results revealed by Figs. 7 and 8 for different networks sizes are almost thesame. Therefore, we can conclude that the effects of time delay and coupling strength on synchronization transitions foundin the present paper are independence on the system size.

6. Conclusion

In conclusion, we have systematically investigated the effects of time delay and coupling strength on synchronizationtransitions in excitable homogeneous random network by synchronization parameter and space–time firing patterns. Wehave found that, under fixed coupling strength, asynchronous region, transition region and synchronous region can emergefor small, moderate and large time delay, respectively. The multi-stability can be observed for specific time delays in thetransition region. While two kinds of impacts of coupling strength on synchronization transitions under fixed time delayare discovered. In particularly, coupling strength can enhance synchronization ability of the whole network in transition re-gion and still can shorten synchronization time in synchronous region. All these results about time delay and couplingstrength on synchronization transitions are examined for other large and small systems and have been found to be indepen-dence on the system size.

Self-sustained oscillations and synchronization transitions in excitable homogeneous random network are very impor-tant issues in wide practical fields, especially in biological systems. Meanwhile time delay and coupling strength are twoimportant parameters to decide the characteristics of some specific physiological functions of neural systems. Therefore, asystematical investigation of the synchronization transitions induced by time delay and coupling strength is expected tobe useful both for theoretical understandings and practical applications. We do hope that our results may have a useful im-pact in these fields.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11105003 and 11205062).

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