Ordered chaotic bursting and multiple coherence resonance by time-periodic coupling strength in Newman–Watts neuronal networks Li Wang, Yubing Gong ⇑ , Xiu Lin School of Physics, Ludong University, Yantai 264025, People’s Republic of China article info Article history: Received 8 September 2011 Accepted 1 November 2011 Available online 25 November 2011 abstract In this paper, we study the effect of time-periodic coupling strength (TPCS) on the temporal coherence of the chaotic bursting of Newman–Watts thermosensitive neuron networks. It is found that the chaotic bursting can exhibit coherence resonance and multiple coherence resonance behavior when TPCS amplitude and frequency is varied, respectively. It is also found that TPCS can also enhance the temporal coherence and spatial synchronization of the optimal spatio-temporal bursting in the case of fixed coupling strength. These results show that TPCS can tame the chaotic bursting and can repeatedly enhance the temporal coherence of the chaotic bursting neuronal networks. This implies that TPCS may play a more efficient role for improving the time precision of the information processing in cha- otic bursting neurons. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Information transmission in neural systems is fulfilled by coupled neurons and neuronal network, and the brain functional networks have character of small-world (SW) and scale-free (SF) topology. There are several small-world network models, whereby the Watts–Strogatz model [1] and the Newman–Watts model [2–4] are perhaps the most famous and well-studied ones. In Watts–Strogatz model, the final topology is achieved by rewiring each edge at ran- dom with a probability p, resulting in either a regular (p = 0), disordered (p = 1) or SW networks (0 < p < 1). Sub- sequently, Newman and Watts developed another type of SW network [2–4], in which ‘‘long-range’’ random short- cuts are added between pairs of non-adjacent-vertices cho- sen at random, while maintaining the original edges of the underlying ring, thus the new long-range edges increase the total number of connections from that of the original network. SW networks capture the characteristics of infor- mation transmission in neurons and are often used to de- scribe neuronal networks. Due to importance for the information processing in nervous systems, stochastic res- onance (SR) and coherence resonance (CR) in neuronal sys- tems have been extensively studied in the past years. More importantly, the roles of SR and CR have been broadened in the neuronal firing in recent years. On one hand, some no- vel SR phenomena and their important roles have been found, such as SR in the encoding and processing of infor- mation in the nervous system [5], the amplification of information transfer in excitable systems [6], spatial CR in neuronal media with discrete local dynamics [7], the ef- fects of small-world connectivity on noise-induced tempo- ral and spatial order in neural media [8], SR on excitable SW networks via a pacemaker [9] and weakly paced SF net- works [10] as well as Newman–Watts networks of Hodg- kin–Huxley (HH) neurons with local periodic driving [11], SR-modulated neural synchronization within and between cortical sources [12], SR assisting phasic neurons to encode slow signals [13], and optimal network configuration for maximal CR in excitable systems [14]. For neuronal networks, the coupling among neurons is key to the neuronal firing activity. In the past decades, the roles of coupling in neuronal firing dynamics have been extensively studied, and many phenomena of SR and CR in- duced by coupling type and strength have been found in 0960-0779/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2011.11.001 ⇑ Corresponding author. Tel.: +86 535 6697550. E-mail address: [email protected](Y. Gong). Chaos, Solitons & Fractals 45 (2012) 131–136 Contents lists available at SciVerse ScienceDirect Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
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Chaos, Solitons & Fractals 45 (2012) 131–136
Contents lists available at SciVerse ScienceDirect
Chaos, Solitons & FractalsNonlinear Science, and Nonequilibrium and Complex Phenomena
journal homepage: www.elsevier .com/locate /chaos
Ordered chaotic bursting and multiple coherence resonance bytime-periodic coupling strength in Newman–Watts neuronal networks
Li Wang, Yubing Gong ⇑, Xiu LinSchool of Physics, Ludong University, Yantai 264025, People’s Republic of China
a r t i c l e i n f o a b s t r a c t
Article history:Received 8 September 2011Accepted 1 November 2011Available online 25 November 2011
0960-0779/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.chaos.2011.11.001
In this paper, we study the effect of time-periodic coupling strength (TPCS) on the temporalcoherence of the chaotic bursting of Newman–Watts thermosensitive neuron networks. Itis found that the chaotic bursting can exhibit coherence resonance and multiple coherenceresonance behavior when TPCS amplitude and frequency is varied, respectively. It is alsofound that TPCS can also enhance the temporal coherence and spatial synchronization ofthe optimal spatio-temporal bursting in the case of fixed coupling strength. These resultsshow that TPCS can tame the chaotic bursting and can repeatedly enhance the temporalcoherence of the chaotic bursting neuronal networks. This implies that TPCS may play amore efficient role for improving the time precision of the information processing in cha-otic bursting neurons.
� 2011 Elsevier Ltd. All rights reserved.
1. Introduction
Information transmission in neural systems is fulfilledby coupled neurons and neuronal network, and the brainfunctional networks have character of small-world (SW)and scale-free (SF) topology. There are several small-worldnetwork models, whereby the Watts–Strogatz model [1]and the Newman–Watts model [2–4] are perhaps the mostfamous and well-studied ones. In Watts–Strogatz model,the final topology is achieved by rewiring each edge at ran-dom with a probability p, resulting in either a regular(p = 0), disordered (p = 1) or SW networks (0 < p < 1). Sub-sequently, Newman and Watts developed another type ofSW network [2–4], in which ‘‘long-range’’ random short-cuts are added between pairs of non-adjacent-vertices cho-sen at random, while maintaining the original edges of theunderlying ring, thus the new long-range edges increasethe total number of connections from that of the originalnetwork. SW networks capture the characteristics of infor-mation transmission in neurons and are often used to de-scribe neuronal networks. Due to importance for the
. All rights reserved.
Gong).
information processing in nervous systems, stochastic res-onance (SR) and coherence resonance (CR) in neuronal sys-tems have been extensively studied in the past years. Moreimportantly, the roles of SR and CR have been broadened inthe neuronal firing in recent years. On one hand, some no-vel SR phenomena and their important roles have beenfound, such as SR in the encoding and processing of infor-mation in the nervous system [5], the amplification ofinformation transfer in excitable systems [6], spatial CRin neuronal media with discrete local dynamics [7], the ef-fects of small-world connectivity on noise-induced tempo-ral and spatial order in neural media [8], SR on excitableSW networks via a pacemaker [9] and weakly paced SF net-works [10] as well as Newman–Watts networks of Hodg-kin–Huxley (HH) neurons with local periodic driving [11],SR-modulated neural synchronization within and betweencortical sources [12], SR assisting phasic neurons to encodeslow signals [13], and optimal network configuration formaximal CR in excitable systems [14].
For neuronal networks, the coupling among neurons iskey to the neuronal firing activity. In the past decades,the roles of coupling in neuronal firing dynamics have beenextensively studied, and many phenomena of SR and CR in-duced by coupling type and strength have been found in
132 L. Wang et al. / Chaos, Solitons & Fractals 45 (2012) 131–136
neurons [15–21]. For example, Hu and Zhou demonstratedthat coupling can greatly enhance the noise-inducedcoherence in two-way coupled non-identical FHN system[15]. Zhou et al. studied the effect of CR in a heterogeneousarray of two-way coupled FitzHugh–Nagumo (FHN)neurons [19]. Wang et al. stated that the effect of CR canbe enhanced greatly by the coupling regardless of the spa-tio-temporal order of the system in globally coupled HHneurons [21]. These studies have shown that the responseof neurons could be improved by coupling in two-way cou-pled, one-way coupled or globally coupled neurons. How-ever, coupling strength in these studies has always beenassumed constant. It is known that neurons are coupledto each other via synapses and form neural networks.Experimental studies have shown that synapses have a fea-ture of plasticity (i.e., the ability of the connection betweentwo neurons to change in strength in response to either useor disuse of transmission over synaptic pathways), andsynaptic plasticity is spiking-timing-dependent [22–25].Consequently, the coupling strength of neurons is adaptiveand will change with time nonlinearly. The coupling ofneurons comprises electrical (gap-junction) and chemicalsynaptic coupling. Gap-junction coupling represents thedirect information transmission between two neurons,and the coupling strength may symbolize the magnitudeof transmitted information. Studies have already shownthat synapses are plastic, and the coupling among neuronsis changing so that the neurons can instantly adjust theirfiring behavior and achieve new synchronization. There-fore, the neural networks with either gap-junction orchemical synapses are plastic, and the coupling of neuronsshould be time-varying. On the other hand, finite couplingstrength requires time-varying coupling strength shouldchange nonlinearly but not monotonically, which makestime-periodic coupling strength (TPCS) allowable. Very re-cently, Birzu et al. have studied the effect of TPCS on thefiring dynamics of a globally coupled array of Fitzhugh–Na-gumo oscillators and have observed rich oscillatory andresonant behavior with the frequency of coupling strength[26].
In this paper, we study the influence of TPCS on the cha-otic bursting of Newman–Watts modified HH (MHH) neu-ron networks, aiming to investigate how TPCS affects thetemporal coherence of the chaotic bursting activity of theneurons. We show that the TPCS can tame the chaoticbursting by inducing CR and multiple CR (MCR). In addi-tion, TPCS can also enhance the optimal spatio-temporalbursting in the case of fixed coupling strength.
The MHH model [27–30] is temperature-dependent andcan exhibit distinct dynamical behaviors with changingtemperature. Bifurcation analysis showed that the modelbehaves like regular spikes (T < 7.31 �C), chaotic bursts(7.31 �C 6 T 6 10 �C), and regular bursts (T > 10 �C) [31].Here, we set T = 8.2 �C such that each neuron initiallyexhibits chaotic bursting.
According to Newman–Watts network topology [2–4],the Newman–Watts MHH neuron network comprising
N = 60 identical chaotic neurons starts with one-dimen-sional regular ring of neurons, each neuron being connectedto two nearest neighbors, and links are randomly added be-tween non-nearest vertices. In the limit case that all neu-rons are coupled to each other, the network containsN(N � 1)/2 edges. Using M to denote the number of addedshortcuts, the fraction of shortcuts is given by p = M/[N(N � 1)/2], which is used to characterize the randomnessof the network. Note that for a given p there are a lot of net-work realizations.
The membrane potential of each neuron of the networkis given by
g = 0.012 lA, k = 0.17. ni(t) is Gaussian white noise with zeromean and auto-correlation function hni(t)nj(t0)i = dijd(t � t0);P
jeij½Vj � Vi� is coupling term, Vi,j is the membrane potentialof the i or jth neuron at time t, 1 6 (i, j) 6 N, and the summa-tion takes over all neurons; eij is a coupling between the neu-rons i and j, which is determined by the coupling pattern ofthe system and is identical for any two neurons, i.e., eij = e, ifthe neurons i and j are connected; e = 0 otherwise. Here, weassume TPCS e has the form as [26]:
e ¼ e0½1þ cosðxtÞ�: ð2Þ
Other parameter values are listed in Table 1. More inter-pretations of these parameters can be found in Ref. [31].
The temporal regularity of bursting is quantitativelycharacterized by characteristic correlation time s [32],which is based on the normalized auto-correlation func-tion c(sd) defined as
cðsdÞ ¼ h�VðtÞ�Vðt þ sdÞi=h�V2i; ð3Þ
where V(t) is the membrane potential of the neuron at timet, sd is time delay, �VðtÞ ¼ VðtÞ � hVðtÞi, and the averaging is
Fig. 3. Contour plot of s as functions of x and p at e0 = 0.002. Threeislands are clearly present, which displays the global structure of MCRinduced by frequency x and CR induced by network randomness p.
Fig. 2. Dependence of s on x at e0 = 0.002 and p = 0.12. As x increases, spasses through three peaks, demonstrating the occurrence of MCR.
L. Wang et al. / Chaos, Solitons & Fractals 45 (2012) 131–136 133
taken over the time. In the present case of limited and dis-crete sampling with N data points, the characteristic corre-lation time is given by
s ¼ 1NDt
XN
k¼1
c2ðskÞDt; ð4Þ
where sk = kDt with Dt being the sampling time, and NDtbeing the length of the time series.
Numerical integrations of Eqs. (1) and (2) are carriedout using explicit Euler method with time step of0.001 ms. Periodic boundary conditions are employedand the parameter values for all neurons are assumedidentical except for distinct initial values of the potentialVi0 and the noise terms ni(t) for each neuron.
3. Results and discussion
Our previous study showed that the bursting temporalcoherence and synchronization of the present Newman–Watts neuronal network is closely dependent on the net-work randomness p. The bursting achieves CR state at anoptimal value of p, and the bursting synchronization in-creases with increasing p and arrives at complete synchro-nization state when p = 1 [32]. Here, we focus on the effectof TPCS on the temporal coherence of bursting for fixed pvalues.
3.1. MCR by TPCS frequency x
To address the effect of TPCS frequency x, we fix TPCSamplitude at e0 = 0.002. We have studied the evolution ofbursting behavior with changing x under different p val-ues. In Fig. 1, we display the space–time plots of the burst-ing for different values of x at p = 0.12 and the bursting forfixed coupling strength e = 0.002 for comparison. It is seenthat the bursting regularity intermittently becomes muchbetter at x = 0.0025 and 0.0055 and a little better atx = 0.0085 when x is increased, which illustrates theoccurrence of MCR. Fig. 2 shows the corresponding quanti-tative characterization of the phenomenon by the variationof characteristic correlation time s as a function of x. It is
Fig. 1. Space–time plots of neuronal bursting for different values of frequency xdisordered. As x increases, the bursting intermittently becomes ordered at x =
seen that, as x is increased, s passes through two higherpeaks at x = 0.0025 and 0.0055 and one lower peak at
at e0 = 0.002 and p = 0.12. For fixed coupling strength, the bursting is too0.0025, 0.0055 and weakly ordered at x = 0.0085.
Fig. 4. Space–time plots of bursting for different amplitudes e0 at x = 0.0025 and p = 0.14. There is an optimal value of e0 at which the bursting becomes themost ordered.
134 L. Wang et al. / Chaos, Solitons & Fractals 45 (2012) 131–136
x = 0.0085. When x is further increased up to x > 0.01, CRphenomenon no longer emerges. This quantitatively dem-onstrates the presence of MCR induced by TPCS frequency.
We have also studied the cases for other values of p. Tomake an overall inspection, we plot the dependence of s inFig. 3 as functions of both x and p. Clearly, three islands liein the range of 0.05 < p < 0.25 and 0.002 < x < 0.01, whichprovides a global view of x-induced MCR for different pvalues.
3.2. CR by TPCS amplitude e0
To address the effect of TPCS amplitude e0, we fix TPCSfrequency at x = 0.0025. Fig. 4 displays the space–timeplots for different at p = 0.14. As e0 is increased, the burst-ing becomes increasingly regular in time and achieves thebest performance at e0 = 0.0025, but then the bursting reg-ularity is reduced as e0 is further increased. This demon-
Fig. 5. Contour plot of s in dependence on p and e0 at x = 0.0025. A longand narrow island appears, indicating the occurrence of CR by e0. As p isincreased, the CR moves to smaller coupling strength amplitude.
strates the occurrence of the phenomenon of CR inducedby e0. In Fig. 5, the dependence of s is plotted.
In dependence on both p and e0. It is seen that an islandis present, and it moves to smaller e0 when p is increased.This shows that for each value of p there is CR phenomenonwith the increase of e0, and the value of e0 for CR decreaseswhen p is increased. The explanation to this phenomenonis that increasing p (i.e., increasing random connectionsamong the neurons) strengthens the coupling of the neu-rons, which makes the neurons achieve CR performancemore easily in case of weak coupling strength.
To get a global view of the results with x and e0, we plotthe dependence of s as functions of both x and e0 in Fig. 6.Clearly, three islands lie in a range of 0.001 < e < 0.004 and0.002 < x < 0.01. On can see that s passes through three is-lands as x is varied, and passes one island as e0 is varied.This clearly demonstrates the presence of MCR with xand CR with e0.
Fig. 6. Contour plot of s as functions of x and e0 at p = 0.12. There arethree islands with varying x and one island with varying e0, indicatingthe occurrence of MCR with x and CR with e0.
L. Wang et al. / Chaos, Solitons & Fractals 45 (2012) 131–136 135
All the above results show that the chaotic bursting ofthe neuronal networks can exhibit MCR and CR when TPCSfrequency and amplitude are appropriate, respectively.TPCS can tame the chaotic bursting by inducing CR andcan repeatedly tame the chaotic bursting by inducingMCR. This implies TPCS may play a more efficient role thanfixed coupling strength for enhancing the temporal coher-ence of the chaotic bursting of the neuronal networks.
In addition, we have also examined the influence of TPCSon the optimal bursting behavior with p for fixed couplingstrength. Our study showed that, when TPCS is proper, themost temporal coherent bursting becomes more synchro-nized and the synchronized bursting becomes more regular(not shown). This means that TPCS can also enhance the opti-mal spatio-temporal bursting for fixed coupling strength.
The phenomena of CR and MCR by TPCS can be explainedin brief as follows. On one hand, TPCS functions as an externaldriving signal. When TPCS amplitude or frequency is appro-priate, the bursting may employ TPCS’s driving energy toachieve the most regular performance, exhibiting CR behav-ior. This mechanism is similar to that for SR phenomenon. Onthe other hand, the phenomenon of MCR occurs when TPCSfrequency is equal to or multiple of the intrinsic burstingfrequency, which shows that this phenomenon is due tothe locking between TPCS and the predominant burstingfrequency of the individual neurons on the network.
Recently, multiple SR and MCR induced by time delay inneuronal networks have attracted increasing attention[33,34]. The results presented in this paper shows thatMCR may also occur due to TPCS in neuronal networks.Very recently, Gosak et al. have studied the effect of con-stant coupling strength on the SR in a random network ofbistable oscillators, and they found that increasing cou-pling strength can increase the noise intensity for SR occur-rence [35]. The present results reveal the effect of timeperiodic coupling strength on the CR in a similar randomnetwork of chaotic bursting neurons.
4. Conclusion
In summary, we have studied the effect of TPCS on thetemporal coherence of the chaotic bursting of Newman–Watts MHH neuron networks. It is found that the chaoticbursting can exhibit CR behavior when TPCS amplitude isappropriate, and can intermittently exhibit MCR behaviorwith the increase of TPCS frequency. It is also found thatthe temporal coherence and spatial synchronization of theoptimal spatio-temporal bursting for fixed coupling strengthcan be enhanced by TPCS. These results show that TPCS cantame the chaotic bursting repeatedly, which implies thatTPCS may play a more efficient role than fixed couplingstrength for improving the time precision of the informationprocessing in chaotic bursting neuronal networks.
Acknowledgments
The authors are grateful to the anonymous reviewersfor valuable comments and suggestions. This work wassupported by the Natural Science Foundation of ShandongProvince of China (ZR2009AM016).
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