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. 2014 Sep 9;9(1):93–101. doi: 10.1007/s11571-014-9310-4

Local and global synchronization transitions induced by time delays in small-world neuronal networks with chemical synapses

Haitao Yu 1, Jiang Wang 1,✉, Jiwei Du 1, Bin Deng 1, Xile Wei 1
PMCID: PMC4454130  PMID: 26052365

Abstract

Effects of time delay on the local and global synchronization in small-world neuronal networks with chemical synapses are investigated in this paper. Numerical results show that, for both excitatory and inhibitory coupling types, the information transmission delay can always induce synchronization transitions of spiking neurons in small-world networks. In particular, regions of in-phase and out-of-phase synchronization of connected neurons emerge intermittently as the synaptic delay increases. For excitatory coupling, all transitions to spiking synchronization occur approximately at integer multiples of the firing period of individual neurons; while for inhibitory coupling, these transitions appear at the odd multiples of the half of the firing period of neurons. More importantly, the local synchronization transition is more profound than the global synchronization transition, depending on the type of coupling synapse. For excitatory synapses, the local in-phase synchronization observed for some values of the delay also occur at a global scale; while for inhibitory ones, this synchronization, observed at the local scale, disappears at a global scale. Furthermore, the small-world structure can also affect the phase synchronization of neuronal networks. It is demonstrated that increasing the rewiring probability can always improve the global synchronization of neuronal activity, but has little effect on the local synchronization of neighboring neurons.

Keywords: Synchronization, Time delay, Neuronal network, Small-world

Introduction

Synchronization in biological neuronal systems has attracted extensive attention over the last few years and a series of meaningful phenomena have been observed (Pikovsky et al. 2001; Acker et al. 2003; Steriade 1997). Many experimental and theoretical works have demonstrated that the synchronization of neuronal activity is of particular important for brain functions (Fries et al. 1997, 2002; Gray 1994). It is shown that well-coordinated global synchronization of neuronal spike trains play a vital role in the neuronal signaling and information processing and transmitting across different areas of brain cortex (Singer 1993; Singer and Gray 1995). Globally synchronized activities involving widespread populations of neurons in neocortex are associated with the execution of complex sensorimotor tasks and have been proposed to participate in the ‘binding’ of sensory attributes during perceptual synthesis (Chawla et al. 2001). On the other hand, recent experiments in monkey visual cortex also demonstrate the synchrony among γ activities (30–90 Hz) in local areas and its important role in perceptual modulation (Eckhorn et al. 2004). Moreover, the phenomenon of zero-lag synchronization between distant cortical areas is reported to rely on local connectivity, such as a specific network motif (Gollo et al. 2014). Motivated by above observations, it is necessary to explore the generation mechanisms underlying the variety of neuronal synchrony.

To elucidate this question, a large amount of studies have been carried out to investigate the temporal coherence and spatial synchrony of different neuronal ensembles. For instance, synchronization and propagation of bursts has been studied in networks of coupled map neurons and the underlying mechanism for this synchronization was explained (Tanaka et al. 2006). Furthermore, the effects of mutual chaotic phase synchronization in ensembles of map-based bursting oscillators have been investigated (Ivanchenko et al. 2004; Batista et al. 2009). More recently, Batista have studied synchronization in clustered networks of bursting neurons and given bounds for the smallest coupling strength needed to achieve global synchronization (Batista et al. 2012). All these studies demonstrate that the interplay between the intrinsic dynamics of the constituent neurons and their complex structure of connectivity strongly affects the synchronization and oscillatory behavior of the resulting network. However, the exact conditions under which can large neuronal ensembles synchronize their spiking or bursting activity are in general not fully understood.

Recent developments in the quantitative analysis of complex networks, mostly revealed by anatomical, electroencephalographic (EEG), and neuroimaging data, demonstrate that many biological neuronal networks present the typical properties of small-world networks, the short path length of which is associated with a high level of local clustering (Watts and Strogatz 1998). Up to now, small-world topology has been found in the brain’s structural and functional systems both at the whole-brain scale of human neuroimaging and at a cellular scale in non-human animals (Bullmore and Sporns 2009; van der Heuvel et al. 2008; Reijneveld et al. 2007). Sporns (Sporns et al. 2000) have described small-world topological properties, and investigated the relationship between topology and complex dynamics, in brain networks. Experimental studies indicate that small-worldness might be a conserved property of brain networks over different species and spatial scales (Bullmore and Sporns 2009; Bassett and Bullmore 2006). Modeling studies show that, compared with regular networks, neuronal systems with small-world connectivity are easier to get synchronization. Increasing the number of shortcut links between long-range units can largely enhance the synchronization degree of small-world networks (Han et al. 2008). Moreover, the significant role of small-world topology on the resonance dynamics of excitable neural media has been explored (Perc 2007; Ozer et al. 2009; Gao et al. 2001).

In biological neural systems, the information transmission delay is ubiquity, the cause of which is mainly attributed to the time needed for the conduction of a nerve impulse from one neuron to the next across a synapse. In synapses with a chemical transmission mechanism the synaptic delay can last from 0.3 to 0.5 ms to several tens of milliseconds (Kandel et al. 1991), determining the dynamics and functions of neuronal networks. It has been demonstrated that the time delay can play many different key roles, such as introducing stable oscillations (Bélair et al. 1996), enhancing neural synchrony (Dhamala et al. 2004; Manju Shrii et al. 2012), and shaping spatiotemporal order (Gong et al. 2012; Roxin et al. 2005; Yang et al. 2012), as well as inducing multiple stochastic resonances (Yu et al. 2013; Wang et al. 2009). Particularly, the effects of time delay on the synchronization among neurons have been extensively studied, and many interesting phenomena have been reported. It is shown that, as the delay increases, regions of irregular and regular propagating excitatory fronts related to the synchronization transitions appear intermittently in excitable small-world and scale-free neuronal networks (Pérez et al. 2011; Wang et al. 2008, 2009, 2010). However, most work has focus on the effect of time delay on the collective dynamics of the whole network; less attention has been given to its impact on the synchronization properties of locally connected neurons, i.e., local synchronization.

In this paper, we extend this subject by studying the effects of time delay on the local and global synchronization in small-world networks of chemically coupled map-based neurons. We consider both excitatory and inhibitory coupling types. We aim to explore the dependence of local and global synchronization transitions in small-world neuronal networks on the information transmission delay and other network parameters. Accordingly, the remainder of this paper is organized as follows. In “Mathematical model and methods”, we describe the mathematical model of the time-delayed small-world neuronal network with chemical synapses. Local and global synchronization transitions induced by the time delay are investigated in “Results”. Finally, a brief conclusion of this paper is given in “Conclusions”.

Mathematical model and methods

In small-world networks, the temporal evolution of each neuron with chemical synapses can be described by a two-dimensional map (Rulkov 2001):

xi(n+1)=α1+xi2(n)+yi(n)-Iisyn(n) 1
yi(n+1)=yi(n)-βxi(n)-γ, 2

where xi(n) and yi(n) are fast and slow dynamical variable of the map, respectively. Small values of parameters β = γ = 0.001 determine a slow temporal evolution of yi(n). The parameter α determines the spiking-bursting activity of the neuron. Here, we first set α = 2.3, such that each individual neuron exhibits spike excitations with the natural period T0 = 850. n is the discrete time index. Isyni(n) is the synaptic current of neuron i, and take the form

Iisyn(n)=gc(xi(n)-v)∑j=1NεijΓ(xi(n)) 3

gc is the coupling strength. v is the synaptic reversal potential, whose value determines the types of synapse: inhibitory or excitatory. For the inhibitory synapses, we choose v = −1.8; whereas for the excitatory synapses, v = 1.8. The matrix ɛ = (ɛij) is an connectivity matrix: ɛij = 1 if neuron i is connected to neuron j,ɛij = 0 otherwise, and ɛii = 0. The delayed synaptic coupling function is modeled by the sigmoidal function

Γ(xi(n))=1(1+exp-λ[xj(n-τ)-Θs]) 4

where Θs is the threshold, above which the postsynaptic neuron is affected by the presynaptic one. Here, we take Θs = −1.0. λ = 30 represents a constant rate for the onset of excitation or inhibition. τ is the information transmission delay among neurons.

The small-world network can be obtained by starting from a ring-like network with regular connectivity containing N = 200 vertices, each one connecting to its K = 6 nearest neighbors. Each link is then randomly rewired with probability p. By increasing the probability p, the architecture of the network is tuned between two extremes: regular (p = 0) and random (p = 1) networks. For 0 < p < 1, the resulting network have small-world properties, characterized by a distinctive combination of short characteristic path length, comparable with that of a random network, and high clustering coefficient, just like a regular network (Watts and Strogatz 1998). The time delay τ and rewiring probability p are two main parameters to be investigated in this paper.

In order to quantitatively characterize the synchronization degree of spiking neurons on small-world networks, we introduce the phase of neuron i as:

ϕin=2πk+2πn-nknk+1-nk,(nk<n<nk+1) 5

where nk is the time of the kth firing of neuron i and nk+1 − nk is the inter-spike interval. Thus, the neuronal phase ϕi(n) increases linearly between the firing moments nk and nk+1, and gains a 2π growth over each inter-spike interval.

To measure the phase synchronization between neuron i and the set of its neighbors v(i), we calculate the quantity (Pérez et al. 2011):

sin=1ki∑j∈visin2ϕin-ϕjn2, 6

where ki represents the degree of neuron i, i.e., the number of neurons connected to neuron i. Then, the local phase synchronization of the network is measured by the order parameter Sloc, which is defined as

Sloc=1t∑n=1t1N∑i=1Nsin, 7

where t is the period of numerical integration.To measure the global phase synchronization, we first calculate the synchronization degree of neuron i with the rest of the network as (Pérez et al. 2011)

si′n=1N∑j=1Nsin2ϕin-ϕjn2, 8

Then, by averaging over all units, we obtain a global order parameter Sglob, described as

Sglob=1t∑n=1t1N∑i=1Nsi′n. 9

It is evident that the order parameters Sloc and Sglob are zero if the phases of the neurons are equal, i.e., in-phase synchronization, and unity if they differ by π, termed as anti-phase synchronization. When the phases of the neurons are randomly distributed, i.e., out-of-phase synchronization, the order parameters take an intermediate value of 0.5.

Results

In what follows, we present the effects of information transmission delay τ and rewiring probability p on the synchronization of small-world neuronal networks with chemical coupling. We first consider the network with excitatory coupling and the rewiring probability p = 0.1. Results presented in Fig. 1 illustrates the spatiotemporal dynamics of neurons that are evoked by different values of τ. As one can see, when τ = 0, τ = 850 and τ = 1,700, the neuronal spiking is clearly synchronous, while this synchronized state disappears for τ = 400, τ = 1,200. For inhibitory coupling, similar transitions of spiking synchronization can also occur for appropriate time delays (not shown here). It can thus be concluded that the information transmission delay plays a pivotal role in the synchronization of neuronal networks by either enhancing or destroying the synchrony of neuronal activity.

Fig. 1.

Fig. 1

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Spatiotemporal plots obtained for different time delays. a τ = 0, b τ = 400, c τ = 850, d τ = 1,200, e τ = 1,700. The color coding is linear, white depicting −2 and black depicting 0.1 values of x(n). Other parameter values are g c = 0.01 and p = 0.1

In Fig. 2a, we plot the local and global synchronization coefficients Slocand Sglob versus the time delay τ for excitatory coupling. Numerical results show that, as the delay increases, three minima of Sloc and Sglob approximately appear at τ = 850, τ = 1,700, and τ = 2,550. It is proved that the delay-induced transitions to synchronized neuronal activity indeed appear intermittently as the delay increases. Importantly, all these synchronization transitions roughly occur at integer multiples of the spiking period of individual neurons. For inhibitory connections, similar synchronization transitions also occur, but at the odd multiples of the half of the firing period of neurons (Fig. 2b). This difference is mainly result from phase slips induced by the time delay. Indeed, the delay-induced multiple synchronization states of neuronal activity are due to an intricate interplay between the inherent dynamics of constituting neurons and the locking between the delay time and the oscillatory period of neuronal spiking (Pérez et al. 2011; Wang et al. 2008, 2009, 2010). On the other hand, for both coupling types the local synchronization transition is more profound than the global synchronization transition, as the value of Sglob is no larger than 0.5. That is to say, the small-world neuronal network cannot get overall anti-phase synchronization, which can only be achieved by locally connected neurons with appropriate time delays.

Fig. 2.

Fig. 2

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Dependence of the local and global synchronization parameter S loc and S glob on the time delay τ for a excitatory coupling and b inhibitory coupling. Other parameter values are g c = 0.01 and p = 0.1. (Color figure online)

In order to generalize the above obtained results, we calculate the value of the local and global synchronization parameters in a two-dimensional parameter space: time delay τ versus coupling strength gc (Fig. 3). At a local level, for both excitatory and inhibitory coupling, there exists some narrow-banded regions with small values of Sloc ∼ 0, where in-phase synchronization of neuronal activity is realized, as shown in Fig. 3a and c. For excitatory coupling, these regimes are approximately located at τ ≈ mT0; while for inhibitory connections, they appear for τ ≈ (2 m + 1)T0/2. However, parameters outside these tongue-like regions will unfortunately impair the local synchrony, leading to out-of-phase (Sloc ∼ 0.5) even anti-phase synchronization (Sloc ∼ 1). In the latter case, neighboring neurons fire with a phase difference of π between them. This phenomenon occurs at τ = (2 m + 1)T0/2 for excitatory coupling, while happens at τ = mT0 for inhibitory connections. In conclusion, all these observations demonstrate that the neuronal activity enters and exits the in-phase synchronous regions in an intermittent fashion as the delay increases. The different is the delay times at which these synchronization transitions occur.

Fig. 3.

Fig. 3

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Contour plots of the synchronization parameter S loc[S glob] in dependence on the time delay τ and the coupling strength g c for a, b excitatory coupling and c, d inhibitory coupling

At a global scale, the phase synchronization depends on the type of synapses. For excitatory synapses, the local in-phase synchronization observed for some values of the delay also occur at a global scale (Fig. 3b). While for inhibitory ones, this synchronization, observed at the local scale, disappears at a global scale (Fig. 3d). It is indicated that global in-phase synchronization cannot be achieved in small-world neuronal networks with inhibitory chemical coupling. However, in previous studies of global synchronization in neuronal networks with electrical synapses, it is shown that the neural networks with purely inhibitory synapses can even get synchronization for some particular values of time delay (Wang and Chen 2011). The reason for this difference may be the stronger synchronization ability of electrical synapses compared with chemical ones. Chemical coupling acts only when the presynaptic neurons is spiking, whereas electrical coupling connects neurons at all times. Hence electrical synapses can make the dynamics of neurons stay much closer during the non-spiking epoch, leading to the more profound synchronization (Yu et al. 2013). Furthermore, the coupling strength between neurons strongly affects the synchronous spiking transition in small-world neuronal networks. As shown in Fig. 4, with the increase of coupling strength, the synchronization degree of the ensemble is enhanced for both excitatory and inhibitory coupling types. It is thus indicated that increasing the coupling intensity can always induces the transition from non-synchronization to synchronization.

Fig. 4.

Fig. 4

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Dependence of the local and global synchronization parameter S loc[S glob] on the time delay τ for different coupling strength g c: a, b excitatory coupling and c, d inhibitory coupling. (Color figure online)

To gain more insights into the dependence of synchronous spiking transition in small-world neuronal networks on the rewiring probability, we plot the variation of Sloc and Sglob with respect to p, as shown in Fig. 5. It can be seen that, for each kind of synapses, Sglob decreases monotonously and finally reach a constant with the increase of p. It is indicated that the nonsynchronous neuronal activity can transit to regular synchronization and finally saturate as p approaches the random network limit. To make an overall inspection, the dependence of Sglob on both p and τ is presented in Fig. 6b and d. It is shown that the small-world neuronal network can transit to in-phase synchronization as p increases. From Fig. 5, we can also see that Sloc changes much less profoundly as p is varied, thus implying no clear effect of small-world topology on the local synchronization of the neuronal network. Results of Sloc versus both p and τ are shown in Fig. 6a and c. It is evident that Sloc undergoes no significant changes as p increases, which fully support our above assessment. In sum, global synchronization transitions on small-world neuronal networks can also be induced by the variation of rewiring probability p.

Fig. 5.

Fig. 5

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Dependence of the local and global synchronization parameter S loc and S glob on the rewiring probability p for a excitatory coupling (τ = 850) and b inhibitory coupling (τ = 450). (Color figure online)

Fig. 6.

Fig. 6

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Contour plots of the synchronization parameter S loc[S glob] in dependence on the time delay τ and the rewiring probability p for a, b excitatory coupling and c, d inhibitory coupling

Finally, the local and global synchronization is investigated in the bursting small-world neuronal network. Here α is set to be 4.1 so that each neuron in the network bursts with the natural period T0 = 350. In Fig. 7, we plot the synchronization coefficients Sloc and Sglob versus the time delay τ for excitatory and inhibitory coupling, respectively. It is shown that, for both coupling types, the information transmission delay can induce synchronization transitions of bursting neurons in small-world networks: the synchronous state emerges at τ ≈ mT0 for excitatory coupling and τ ≈ (2 m + 1)T0/2 for inhibitory coupling. The observed results are similar with that obtained in spiking neuron networks. In addition, the variations of Sloc and Sglob with respect to p are presented in Fig. 8. It is shown that, with the increase of p, Sglob decreases monotonously and finally reach a constant, while Sloc increases much less profoundly. It is demonstrated that increasing the rewiring probability can always enhance the global synchronization of neuronal activity, but has little effect on the local synchronization of neighboring neurons, which is in accord with the results shown in Fig. 5. From the results above, it can be concluded that the transitions of the local and global synchronization also occur in the bursting small-world neuronal network with time delay.

Fig. 7.

Fig. 7

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Dependence of the local and global synchronization parameter S loc and S glob on the time delay τ for a excitatory coupling and b inhibitory coupling. Other parameter values are α = 4.1, g c = 0.01 and p = 0.1. (Color figure online)

Fig. 8.

Fig. 8

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Dependence of the local and global synchronization parameter S loc and S glob on the rewiring probability p for a excitatory coupling (τ = 350) and b inhibitory coupling (τ = 800). (Color figure online)

Conclusions

In this paper we have studied the dependence of local and global synchronization transitions in small-world neuronal networks with chemical synapses on the information transmission delay and the wiring probability. The obtained numerical results show that, for both excitatory and inhibitory coupling types, the synaptic time delay can always induce synchronization transitions in small-world neuronal networks. It can either enhance or destroy the synchrony of neuronal activity. In particular, regions of in-phase and out-of-phase synchronization of connected neurons appear intermittently as the delay increases. In addition, for excitatory coupling, all these synchronization transitions occur approximately at integer multiples of the firing period of individual neurons; while for inhibitory coupling, they appear for the odd multiples of the half of the firing period of neurons. More importantly, the local synchronization transition is more profound than the global synchronization transition, depending on the type of synapses. For excitatory synapses, the local in-phase synchronization observed for some values of the delay also occur at a global scale; while for inhibitory ones, this synchronization, observed at the local scale, disappears at a global scale. Furthermore, it is found that the small-world topology can also affect phase synchronization transitions of neuronal system. Irrespectively of the coupling type, increasing the rewiring probability can always enhance the global synchronization of neuronal activity, but has little effect on the local synchronization of neighboring neurons. In conclusion, the information transmission delay and rewiring probability can perform significant but different roles in the synchronization of small-world neuronal networks.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 61302002, 61172009, and 61372010) and Tianjin Research Program of Application Foundation and Advanced Technology (No. 14JCQNJC01200).

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