Chaos, Solitons & Fractals 56 (2013) 19–27

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Chaos, Solitons & FractalsNonlinear Science, and Nonequilibrium and Complex Phenomena

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

Review

Cooperative dynamics in neuronal networks

0960-0779/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.chaos.2013.05.003

⇑ Corresponding author at: Department of Dynamics and Control,Beihang University, Beijing 100191, PR China. Tel./fax: +86 10 82332003.

E-mail address: nmqingyun@163.com (Q. Wang).

Qingyun Wang a,b,⇑, Yanhong Zheng a,c, Jun Ma d

a Department of Dynamics and Control, Beihang University, Beijing 100191, PR Chinab State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, PR Chinac School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, PR Chinad Department of Physics, Lanzhou University of Technology, Lanzhou 730050, PR China

a r t i c l e i n f o a b s t r a c t

Article history:

There exist rich cooperative behaviors and their transitions in biological neuronal systemsas some key biological factors are changed. Among all of cooperative behaviors of neuronalsystems, the existing experiments have shown that the spatiotemporal pattern and syn-chronization dynamics are very crucial, which are closely related to normal function anddysfunction of neuronal systems. Based on different neuron models, the recent works havebeen made to explore the mechanisms of pattern formation and synchronization transi-tion. This paper mainly overviews the recent studies of the cooperative dynamics includingthe pattern formation and synchronization transition in biological neuronal networks.Firstly, we review complicated spatiotemporal pattern dynamics of neuronal networks.Secondly, the interesting synchronization transition is reviewed. Finally, conclusion isgiven and we put forward some outlooks of research on the cooperative behaviors in realneuronal networks.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The cerebral cortex is a highly interconnected networkof neurons, in which the activity in any neuron is necessar-ily related to the combined activity of collective neurons.Several empirical studies of structural and functional brainnetworks in humans and other animals have witnessedsmall-world architectures over a wide range of scales inboth space and time [1–3]. The first nervous system to beformally quantified as a small-world network is at themicroscopic scale of the neuronal network of Caenorhabdi-tis elegans [4]. Microscopic neuronal systems in the medialreticular formation of the vertebrate brain have also beenshown to have a small-world architecture [5]. By usingfunctional magnetic resonance imaging, power-law distri-butions were obtained upon linking correlated fMRI voxels[6], and the robustness against simulated lesions of ana-tomic cortical networks has also been found to rely mostly

on the scale-free structure [7]. Cortical areas are arrangedinto modules, which closely follow functional subdivisionsby modality [1,8], and two cortical areas are more likelyconnected if both are involved in the processing of thesame modal information. In addition, some cortical areasare extensively connected (referred as hubs) with projec-tions to areas in all modalities [9–12]. Based on anatomicaldata of cats’ brain, it is found that cortico-cortical networksdisplay a few prominent characteristics such as modularorganization, abundant alternative processing paths, andthe presence of highly connected hubs. These propertiescan support rich dynamical behaviors, facilitating thecapacity of the brain to process and to integrate sensoryinformation [13]. Hence, we can believe that neuronal sys-tems can exhibit the typical features of complex networks,which is formed via interconnection of thousands of syn-apses (chemical synapses and electrical gap junctions)among all neurons.

Cooperative behaviors in biological neuronal networksincluding pattern formation, waves, synchronization, andcoherence have been extensively investigated [14–28]because they are surely related to biological normal

20 Q. Wang et al. / Chaos, Solitons & Fractals 56 (2013) 19–27

functioning and generation of some neural diseases.Among the array of possible cooperative dynamical behav-iors, the spatiotemporal patterns and synchronizationseem very important, through which the efficient process-ing and transmission of information can be conductedacross the nervous system (see e.g., [29,30]). Resultantly,the spatiotemporal patterns and synchronization have be-come currently a hot research topic in theoretical neurosci-ence [31]. Recently, a lot of works have been made toexplore the spatiotemporal patterns and synchronizationby means of theoretical modeling as well as biologicalexperiments [1,25,32–35,37–40]. For examples, thechanges in firing patterns in basal ganglia neurons werehave been explored by means of reasonable models, whichcan uncover the mechanism of pathological network oscil-lations with a regularized pattern of neuronal firing [41–44]. The effects of coupling strength on the spatiotemporalpatterns and synchronization have been studied [32]. It isshown that the network topology have crucial impact onneuronal synchronization [33]. In particular, the synchro-nization on small-world neuronal networks is studied bymeans of various neuron models [25,32–35,37]. Zhouet al. reported synchronization dynamics in the corticalbrain network of the cat, which displays a hierarchicallyclustered organization [35,36]. Wang’s group has analyzedthe phase synchronization in neuronal systems based onphase equations [45–48]. Phase synchronization transitionfrom burst synchronization to spike synchronization wasstudied in two coupled HR neurons with gap junction[49]. Kurths and his cooperators have extensively investi-gated phase synchronization of bursting neurons in thenetworks with different topologies, and properties of sometypical bursting phase synchronization were analyzed assome key parameters vary [50–53].

The information transmission delays are inherent to thenervous system because of the finite speed at which actionpotentials propagate across neuron axons, as well as due totime lapses occurring by both dendritic and synaptic pro-cessing. Recent results have suggested that time delayscan facilitate neural synchronization and lead to manyinteresting and even unexpected pattern formation andsynchronization transition [17,18,54,55], such as zig-zagfronts of excitations, clustering anti-phase synchronizationand in-phase synchronization [17,18]. The studies haveshown that the higher ordered spatiotemporal dynamicsinduced by intermediate delays could be the result of alocking between the period-1 neuronal spiking activityand the delay [56]. Sompolinsky et al. have compared the-oretical results with recent experimental evidence oncoherent oscillatory activity in the cat visual cortex, andstudied the effect of axonal propagation delays on synchro-nization of oscillatory activity [19]. In addition, ion chan-nels are prominent components of the nervous system[57], and the number of working ion channels for a givenmembrane patch is of great importance, particularly forunderstanding the impact of a specific ion channel typeon the neuronal dynamics in terms of intrinsic noise [58–64]. Since Fox and Lu have extended the Hodgkin–Huxleymodel by taking into account the fluctuations of the num-ber of open ion channels around their corresponding meanvalues [65], different effects of channel blocking on the

patterns and synchronization of neuronal systems havebeen studied [66–69]. Moreover, it was shown that byblocking some portion of either potassium or sodium ionchannels, it is possible to either increase or decrease theregularity of the spike firing patterns [70,71].

This review is intended to briefly outline recent ad-vances of both spatiotemporal patters and synchronizationtransition, which are two types of important cooperativedynamics in realistic neuronal systems. The remainder ofthis paper is organized as follows. In Section 2, we presentthe typical spatiotemporal pattern dynamics of neuronalnetworks. Synchronization transition are presented in Sec-tion 3, whereas in the last Section we will summarize theexisting results and give some potential problems.

2. Spatiotemporal pattern dynamics

The spatiotemporal pattern dynamics in neuronal sys-tems is important behaviors that need to be explored forunderstanding neural function. An interesting work in pri-mate cortex has shown that cortical motor control andinformation transfer may be understood with respect tothe spatiotemporal patterns of neuronal oscillatory activity[72]. In human seizures, the existing results have shownthat spatiotemporal patterns had a consistent dynamicalevolution as the seizure episodes initiate, develop, and ter-minate [73]. Oscillatory episodes in the middle layers ofmammalian cortex can display irregular and chaotic spa-tiotemporal wave activity, within which spontaneouslyemerge spiral and plane waves [74]. Thalamocortical net-works, which are responsible for the generation of sleeprhythms, can clearly display very different rhythmic activ-ity during different sleep stages [75]. It has been shownthat by using spike-timing-dependent plasticity, neuronscan adapt to the beginning of a repeating spatio-temporalfiring pattern in their input, and this mechanism can be ex-tended to train recognizers for longer spatio-temporal in-put signals [76]. Spatiotemporal pattern formation intwo-dimensional neural circuits have been observed asthe refractoriness and noise change [77]. Hence, each ofthese neuronal systems can display complicated firing pat-terns such as the correlated rhythmic activity, uncorrelatedspiking and propagating waves.

Following experimental studies, theoretical explora-tions have simultaneously been made to investigate thedynamical mechanism of spatiotemporal patterns thathave been observed in realistic neuronal networks. Asmost original and biological Hodgkin and Huxley (HH)neuron, it is the standard mathematical tool in studydynamical behaviors of biologically realistic neurons[78]. By means of HH neuron model, Many works onthe dynamics of spatiotemporal patterns in neuronal sys-tems have been theoretically investigated. For examples,we have found that complex wave propagations can existin a ring HH neuronal network when the coupling andnoise level are changed [79]. Gong et al. have found thatthe most pronounced spatiotemporal pattern in chaoticHH neuronal networks can appear for a connection prob-ability of the optimal random, and the optimal random-ness shifts towards a smaller value as the couplingstrength is increased [80].

Q. Wang et al. / Chaos, Solitons & Fractals 56 (2013) 19–27 21

Interestingly, the delay-induced complexity of spatio-temporal patterns such as irregular and regular fronts ofexcitations has been shown in neuronal networks withcomplex topology [81–84]. As shown in Fig. 1, regularand irregular pattern of membrane potential Vi(t) insmall-world Hodgkin–Huxley (HH) neuronal networkscan be observed as the information transmission delay schanges [81]. Hence, the information transmission delayhas nontrivial effect on the spatiotemporal dynamics ofneuronal networks. For other network topology, similarcomplicated propagations of the spatiotemporal patternscan be still observed [82]. More details, one can review re-cent results in Refs. [81,82].

Spatial patterns were investigated in a square lattice HHneuronal network with nearest diffusive coupling. Resultshave showed that there exist the ordered circular waveswith layered structure in this network at an intermediate

(a) (b) (c)

Fig. 1. Space–time plots of Vi(t) obtained in neuronal networks with sodium chaequalling: (a) 0, (b) 5, (c) 10, (d) 17.5, (e) 22.5, (f) 30. In all panels the system size i� 80 and black depicting 40 values of Vi(t) (the scale is partitioned into ten diffefrom the excitable steady states). For more detailed information, one can see Re

(a) (b)

Fig. 2. Spatial pattern formation out of noise on the diffusively coupled HH neurogiven time t. The noise level r is: (a) 1.2, (b) 1.5 and (c) 2.1, whereas the employethat as noise level is intermediate, the spatial patterns of neuronal networks cdepicting minimal and black maximal values of Vi,j. More detailed explanations

noise level [85]. We also found that the spatiotemporalpatterns of spatially extended neuronal networks withthe delay can exhibit order waves as shown in Fig. 2 asthe noise level is suitable [86]. In particular, we can ob-serve in Fig. 2 that for small and large noise levels, the pre-sented spectra show no particularly expressed spatialfrequency. Only for intermediate r, the spatial structurefunction develops several well-expressed circularly sym-metric rings, indicating the existence of a preferred spatialfrequency induced by additive Gaussian noise. Moreover,another alterative neuron models such as Fitzhugh–Nagu-mo, Morris–Lecar and Rulkov map have also been used tostudy patterns dynamics of neuronal systems, and manyinteresting results have been obtained [87–93], whichcan further help us to understand the work mechanismof realistic neurons. For example, the effects of small-worldconnectivity on noise-induced spatial patterns in neural

(d) (e) (f)

nnel blocking for D = 0.2 with different information transmission delays s,s i = 1,2 . . . ,100 = N. In all panels the color profile is linear, white depictingrent gray levels to enable the color coding of small-amplitude deviationsf. [81].

(c)

nal network. All panels depict values of Vi,j on a 128 � 128 square grid at ad delay is the same in all three panels, equaling s = 0.08. It is clearly shownan be orderly exhibited. Gray scale coloring in all panels is linear, whitecan be found in Ref. [85].

22 Q. Wang et al. / Chaos, Solitons & Fractals 56 (2013) 19–27

media have also been investigated [87–91], and manyinteresting behaviors of collective dynamics have been dis-cussed. Volman et al. have investigated the possible causesunderlying the appearance of structured spatiotemporalpatterns in the activity of neuronal networks by function-follow-form [92].

The problem about spiral wave in the network of neu-rons has become attractive in experimental and numericalstudies. Some experimental evidences confirmed that thespiral wave in disinhibited mammalian neocortex doesplay an active role in regulating the membranes of neuronslike a pacemaker [94–96], and the development of voltage-sensitive dyes and fast optical imaging techniques give us apractical tool for measuring spatiotemporal patterns ofpopulation neuronal activity in the neocortex [97]. Pres-ently, the mechanism of the spiral waves has been ex-plored extensively in different systems. In particular, Heet al. [98] studied the formation of spiral wave in an inho-mogeneous medium with small-world connections. Thedynamics of spiral wave in the regular networks of Hind-marsh–Rose neurons was ever discussed in brief [99–103]. Furthermore, Ma et al. reported that the emergenceof spiral wave in network of neurons is associated to theunderlying defects due to the blocking of ion channels[104–108]. The dynamics of spiral wave in the regularand/or small-world network can be changed by somebifurcation parameters, thus the effect of white noise, col-ored noise, channel noise and blocking in ion channels onthe spiral wave has been investigated extensively [105–109]. As an illustrative example, it is shown in Fig. 3 thattypical ordered wave in the networks of neurons can van-ish as the connection probability of small-world networksis suitably chosen [108]. In addition, Wang et al. observedthe time delay-induced coherence of spiral waves in noisyHodgkin–Huxley neuronal networks [110].

Fig. 3. Developed spatiotemporal pattern in a small-world network of Hodgt = 1200 time units, where 0.001 6 p 6 p0 < 1. (a) p0 = 0.03; (b) p0 = 0.17; (c) p0 =connection. For the detailed description, one can see Ref. [108].

3. Synchronization transition dynamics

Many experiments have shown that the firing transitionbetween synchronization and desynchronization canextensively exist in biological neuronal systems. For exam-ple, the work using tissue slice preparations, animal mod-els and in humans with Parkinson’s disease hasdemonstrated abnormally synchronized oscillatory activityat multiple levels of the basal ganglia-cortical loop in Par-kinson’s neuronal systems, and excessive synchronizationcorrelates with the motor [111]. Experiments have alsodemonstrated that the patterns of neuronal activity withinthe basal ganglia change between a normal and a patholog-ical synchronization state. However, the cause of the tran-sition from normal to pathological activity is stillunknown. It has been established that this transitionmaybe correlate with a loss of the neurotransmitter dopa-mine within the basal ganglia [112]. In addition, it has beenevidenced that the resting brain spends most of the timenear the critical point of phase transition and exhibits ava-lanches of activity ruled by the same dynamical and statis-tical properties [113–115]. Nonetheless, the phaserelationships of the pyloric rhythm of the stomatogastricganglion (STG) of the crab, Cancer borealis, are remarkablyinvariant for a certain range of temperature. However, asthe isolated STG preparations are exposed to more extremetemperature ranges, behaviors of their networks becomenonrhythmic, or ‘‘crashed’’, in a reversible fashion [116–119].

In recent years, the synchronization transition ofneuronal systems has been investigated in detail.Theoretical investigations have shown that the increasingcoupling strength can induce two different transitions tosynchronized states, which are associated with burstsand spikes [121]. It has been reported that there exist a

kin–Huxley neurons with varying long-range connection probability at0.39; (d) p0 = 0.4. p0 is the upper threshold for probability of long-range

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maximum and a minimum of the critical coupling intensityfor synchronization transition in diluted neural networks[120]. We have reported that the coupled neurons withcoexisting attractors can exhibit different types of synchro-nization transitions between spiking and bursting as thecoupling strength increases [122], and synchronizationtransition can be identified by the bifurcation diagram.For two coupled bursters, it was found that the same typeof coupled bursters may have different synchronizationtransition paths from that of two different types of coupledbursters [123]. The coupled cells can burst synchronouslyfor both weak electrical and chemical coupling when anisolated cell exhibits tonic spiking and different types ofbursting can appear as the coupling strength changes[124]. In a neuronal network coupled by excitatory syn-apses, the transition of regular bursting and spiking syn-chronization can be exhibited as the coupling strength ischanged [79]. Transition to two types of burst synchroniza-tion has been studied on a diffusively coupled network ofHindmash–Rose bursting neurons [125]. Furthermore, thetransitions of burst synchronization are explored in a neu-ronal network consisting of subnetworks, and it is foundthat two types of burst synchronization transitions canbe induced not only by the variations of intra-and inter-coupling strengths but also by changing the probabilityof random links between different subnetworks and thenumber of subnetworks [126]. In globally coupled ensem-bles of identical neurons, it is found that the process from aspontaneous synchronization transition, to the entireensemble breaking down into a number of coherent clus-ters, and then to complete mutual synchronization can oc-cur as the coupling strength increases [127]. Synchronizedbursts and loss of synchrony was analyzed among hetero-geneous conditional oscillators [128].

Very recently, the delay induced complicatedtransitions of synchronization have been observed inneuronal networks, whose nodes are composed of HH,

(a) (b)

Fig. 4. (a) Dependence of the synchronization parameter r on the informationContour plots of the synchronization parameter r in dependence on the informaobvious that intermittent synchronization can be induced by the delay, and as thtransition is more profound. More detailed explanations can be found in Ref. [8

Hindmarsh–Rose, Fitzhugh–Nagumo, Morris–Lecar andRulkov map neuron. Interesting results have shown thatthe delay-coupling can induce in-phase and anti-phasesynchronization in noisy Hodgkin–Huxley neurons, andthe coupled neurons exhibit noise-induced phase-flipbifurcations [129]. Zheng and Wang have studied thetime-delay effect on the bursting of the synchronized stateof coupled Hindmarsh–Rose neurons by using the methodof stability switch, and they have explained the mechanismof the transition from bursting oscillation to relaxationoscillation and to chaotic bursting [130]. The behavior ofthe phase synchronization in the networks with non-uni-form delays is different from the networks with distributeddelays, where the phase synchrony decreases as distrib-uted delays are introduced to the networks [131]. Theinterplay between the noise, the time-delay and theexcitable character of the neuronal dynamics has beenshown to be necessary and sufficient for the occurrenceof the synchronization clusters [132]. The study of Rulkovneuronal networks with information transmission delayin a small-world and scale free topology with additivenoise and delay has shown the existence of intermittentsynchronization transitions as the delay is varied[17,133,134]. We have reported different impacts of thedelays and rewiring probability on the synchronizationtransitions of small-world neuronal networks with twotypes of coupling [93]. We also found that the delay caninduce the period adding bursting synchronizationtransition in the Macaque cortical network [135]. Adhikariinvestigated the delay-induced phase-transition in coupledbursting neurons including Hindmarsh–Rose and Leech–Heart interneuron model [136]. Synchronization transitionbetween spike and burst was investigated [137]. Resultsshow that burst synchrony is easier to achieve than spikesynchrony. By increasing the inhibitory synaptic delay, atransition from regular to mixed oscillatory patterns at acritical value can be observed [138]. We also found that

delay s for different probability of sodium ion channel blockings XNa. (b)tion delay s and the probability of sodium ion channel blockings XNa. It ise probability of sodium ion channel blockings increases, synchronization

1].

(a) (b)

Fig. 5. (a) Dependence of the synchronization parameter r on the information delay s for different the probability of potassium ion channel blockings XK.(b) Contour plots of the synchronization parameter r in dependence on the information delay s and the probability of potassium ion channel blockings XK. Itis obvious that intermittent synchronization can be induced by the delay, and as the probability of potassium ion channel blockings increases,synchronization transition is more profound. More detailed explanations can be found in Ref. [81].

24 Q. Wang et al. / Chaos, Solitons & Fractals 56 (2013) 19–27

low levels of reliability tend to destroy synchronizationand, moreover, that interneuronal networks with longinhibitory synaptic delays require a minimal level of reli-ability for the mixed oscillatory pattern to be maintained.Furthermore, Gong et al. have explored the effect oftime-periodic coupling strength on the synchronizationtransition of neuronal networks, and found that time-peri-odic coupling strength may play a more efficient role thanfixed coupling strength for enhancing the temporal coher-ence [139–141].

For the extending results, we have studied the impactsof the ion channel blocking and information transmissiondelays on synchronization transition in small-world HHneuronal networks. It is found that, as the delay increases,neurons within the network with ion channel blocking canexhibit transitions from zig–zag fronts to a nearlyclustering anti-phase synchronization and further toregular in-phase synchronization, and the transition canappear alternatively. Moreover, it is shown that the ionchannel blockings of sodium and potassium have differentimpacts on synchronization. By a measure factor as givenin Refs. [81,82], the process of synchronization transitioncan be exhibited as shown in Figs. 4 and 5, where we cansee the intermittent synchronization transition of neuronalnetworks irrespectively of the probability of ion channelblockings. We also found similar transition behaviors inscale free HH neuronal networks with ion channel blockingand delay [82]. For more details, one can see references[81,82].

4. Conclusion and future works

Nervous systems are anatomically and functionallyconnected as complex networks at macro and micro scales.Currently, we have confirmed that neuronal systems canexhibit rich nonlinear dynamical behaviors such asbifurcation, pattern formation, synchronization, and

coherence. These are closely correlated to normal functionof neuronal systems. In this review, we have outlined therecent studies of spatiotemporal pattern dynamics andsynchronization transition dynamics of neuronal systems.In particular, we briefly clarified the advance of the patternformation and the evolutions of spatiotemporal patterns assome key biological parameters are considered. In addi-tion, the complex dynamics of synchronization transitionwas reviewed, where we can observe various transitionsof neuronal synchronization.

Most of works about pattern dynamics and synchroni-zation transition we have reviewed, maybe be some pow-erful theories for the neuroscientist. With this review, wehave tried to convey rich phenomenon of the patterndynamics and synchronization transition of neuronal sys-tems. Currently, some theories have been developed to ex-plore the generation mechanism of those dynamics.However, up to now, the key mechanism behind the manyphenomena of spatiotemporal pattern dynamics and syn-chronization transition remains unclear and is an openproblem to be explored.

In the future, we should develop efficient methods tofurther study the dynamical mechanism of patterns andsynchronization transition of neuronal systems. Especially,the bifurcation mechanism of different transitions shouldbe explored in detail. Combining theoretical results withrealistic neuronal networks, we will study their patternsand synchronization dynamics to well understand theworking essence of biological neurons. We hope that ourreview can may give some helps on investigating richdynamics of neuronal networks.

Acknowledgments

This research was supported by the National ScienceFoundation of China (Grant Nos. 11172017, 11102041),the Research Fund for the Doctoral Program of Higher

Q. Wang et al. / Chaos, Solitons & Fractals 56 (2013) 19–27 25

Education (No. 20121102110014), and State Key Labora-tory of Mechanical System and Vibration (Grant No.MSV-2013-03).

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