Effects of spike-time-dependent plasticity on the stochastic resonance of small-world neuronal networks Haitao Yu, Xinmeng Guo, Jiang Wang, Bin Deng, and Xile Wei
Citation: Chaos 24, 033125 (2014); doi: 10.1063/1.4893773 View online: http://dx.doi.org/10.1063/1.4893773 View Table of Contents: http://aip.scitation.org/toc/cha/24/3 Published by the American Institute of Physics
Articles you may be interested in Effects of time delay on the stochastic resonance in small-world neuronal networks Chaos: An Interdisciplinary Journal of Nonlinear Science 23, 013128 (2013); 10.1063/1.4790829 Chaotic phase synchronization in small-world networks of bursting neurons Chaos: An Interdisciplinary Journal of Nonlinear Science 21, 013127 (2011); 10.1063/1.3565027
CHAOS 24, 033125 (2014)
Effects of spike-time-dependent plasticity on the stochastic resonance of small-world neuronal networks Haitao Yu, Xinmeng Guo, Jiang Wang,a) Bin Deng, and Xile Wei School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, People’s Republic of China
(Received 4 June 2014; accepted 12 August 2014; published online 22 August 2014) The phenomenon of stochastic resonance in Newman-Watts small-world neuronal networks is investigated when the strength of synaptic connections between neurons is adaptively adjusted by spike-time-dependent plasticity (STDP). It is shown that irrespective of the synaptic connectivity is fixed or adaptive, the phenomenon of stochastic resonance occurs. The efficiency of network stochastic resonance can be largely enhanced by STDP in the coupling process. Particularly, the resonance for adaptive coupling can reach a much larger value than that for fixed one when the noise intensity is small or intermediate. STDP with dominant depression and small temporal window ratio is more efficient for the transmission of weak external signal in small-world neuronal networks. In addition, we demonstrate that the effect of stochastic resonance can be further improved via fine-tuning of the average coupling strength of the adaptive network. Furthermore, the small-world topology can significantly affect stochastic resonance of excitable neuronal networks. It is found that there exists an optimal probability of adding links by which the C 2014 AIP Publishing LLC. noise-induced transmission of weak periodic signal peaks. V [http://dx.doi.org/10.1063/1.4893773]
The phenomenon of noise-induced stochastic resonance (SR) in complex neuronal networks has been extensively studied in recent years. However, most of the previous studies were devoted to a static description of synaptic connectivity, while in reality the synaptic strength varies as a function of neuromodulation and time-dependent processes. Spike-time-dependent plasticity (STDP) is an important biological process, which can adjust the strength of connections between neurons adaptively based on the relative timing between pre- and postsynaptic action potentials. In this paper, we study the fundamental role of STDP in stochastic resonance of small-world neuronal networks. We aim to investigate how the network connections evolve during the learning process refined by STDP and the dependence of network stochastic resonance on it.
I. INTRODUCTION
Noise plays a constructive role in nonlinear systems.1,2 One typical representative of this effect is SR, which occurs when the response of a nonlinear dynamical system to a weak periodic signal is optimized by moderate intensity of random fluctuations.3–5 Due to its wide applications, the investigation of stochastic resonance has been extended to many different fields, particularly neural systems.6–11 Experimental and theoretical researches have shown that the ability of sensory neurons to process weak input signals can be significantly enhanced by adding noise to the system.12–15 Additionally, it is demonstrated that the signal propagation a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]. Tel.: 86-22-27402293.
1054-1500/2014/24(3)/033125/11/$30.00
in coupled neural systems is largely improved by external noise via stochastic resonance.16–18 Therefore, it is of great importance to study stochastic resonance for understanding the signal transmission and information propagation in neuronal networks. Recently, much attention has been devoted to the stochastic resonance in complex neuronal networks.19–23 One typical topology is small-world networks, which can capture the characteristics of many real-world complex networks.24 By now, a series of empirical investigations have shown the existence of small-world architecture in many structural and functional brain networks of humans and animals.25–28 The small-world architecture exhibits a relative short length path but along with a high clustering coefficient.24 Therefore, it has much dynamical potentials, such as improving the signal propagation speed, computational power, and synchronizability of neural systems.29,30 In recent years, the effects of small-world connectivity on noise-induced temporal and spatial order in neural media have been extensively studied.31–33 Perc found that the temporal order can be greatly enhanced by the introduction of small-world connectivity, whereby the effect increases with the increasing fraction of introduced shortcut links. On the other hand, it is argued that the small-world network topology destroys spatial order due to the lack of a precise internal spatial scale. In addition, the phenomenon of noise-induced stochastic resonance in small-world neuronal networks has been deeply studied. It has been reported that the effect of stochastic resonance in neuronal networks with small-world topology depends largely on the fraction of randomly added shortcuts.34 Moreover, Ozer et al. studied stochastic resonance on small-world networks of Hodgkin-Huxley neurons with local periodic driving and found that the stochastic
24, 033125-1
C 2014 AIP Publishing LLC V
033125-2
Yu et al.
Chaos 24, 033125 (2014)
resonance can also be amplified via fine-tuning of coupling strength among neurons.35 Most of the previous studies of stochastic resonance on complex neuronal networks were devoted to a static description of synaptic connectivity, while in reality the synaptic strength varies as a function of neuromodulation and timedependent processes. STDP is one of those important biological processes, which can regulate the strength of connections between neurons adaptively based on the relative timing between pre- and post-synaptic action potentials.36 Up to now, the existence of STDP has been found in many neocortical layers and brain regions.37–39 Experimental researches show that the functional structures in the brain can be adjusted to both small-world and scale-free networks by synaptic reorganization via STDP.40 Modeling studies also demonstrate that the synaptic modification of STDP can automatically balance synaptic strengths to make post-synaptic firing irregular but more sensitive to pre-synaptic spiking timing.41 Recently, functional dynamics of complex neuronal networks with STDP have gained increasing interests.42–45 For example, Kube et al. studied the synchronous activity in small-world neural networks with adaptive coupling and found that STDP modifies the weights of synaptic connections in such a way that synchronization of neuronal activity is considerably weakened.46 Moreover, coherence resonance and stochastic resonance have been investigated in self-organized neural network via STDP.47 It has been shown that the selectively refined connectivity modified through STDP can highly enhance the ability of neuronal communications and improve the efficiency of signal transmission in the network.48,49 In this paper, we will study the fundamental role of STDP in stochastic resonance of small-world neuronal networks. We aim to investigate how the network connections evolve during the process refined by STDP and the dependence of stochastic resonance on it. In addition, effects of small-world structure on the network stochastic resonance will be discussed as well. The paper is organized as follows: in Sec. II, the neural model of small-world neuronal network is established, where the strength of connections between neurons is varied through STDP rules. The effects of STDP and small-world topology on SR are systematically investigated in Sec. III. In Sec. IV, we further study the evolution of coupling strength via STDP and its benefit on SR. Finally, a brief conclusion of this paper is drawn in Sec. V. II. MATHEMATICAL MODEL
The FitzHugh–Nagumo neuron model47 is employed to describe the neuronal dynamics, and the temporal evolution of each unit can be defined as follows: eV_ i ¼ Vi –Vi3 =3 Wi þ Iex þ Iisyn þ rni ;
(1)
W_ i ¼ Vi þ a bi Wi ;
(2)
where i ¼ 1; 2; ; N. Vi is a fast variable, representing the membrane potential; whereas Wi is a slow recovery variable. The time scale ratio is chosen as e ¼ 0:08, warranting that Vi evolves much faster than Wi . ni is Guasian white noise with mean 0 and variance 1, satisfying hni ðtÞnj ðt0 Þi ¼ dij dðt t0 Þ.
r stands for the intensity of the noisy background. Iex denotes the external weak periodic signal. Parameter a is set to be 0:7. The value of b is crucial for the dynamics of a neuron, i.e., in the absence of noise, when b > 0:45, the neuron is excitable; while for b < 0:45, the system has a stable periodic solution generating periodic spikes. In our study, bi is randomly distributed in ½0:5; 0:75, so that all neurons within the network are of different excitability. The synaptic current Iisyn takes the form of Iisyn ¼
N X
gij Cij sj ðVi Vsyn Þ;
(3)
j¼1ðj6¼iÞ
where Cij is the connective matrix, since all connections considered in this paper are bidirectional, that is to say, if neuron j couples to neuron i, then Cij ¼ 1 and Cji ¼ 1, otherwise Cij ¼ Cji ¼ 0 and Cii ¼ 0. The reversal potential is chosen as Vsyn ¼ 0, guaranteeing that all synapses are excitatory. The synaptic variable sj is defined as s_ j ¼ aðVj Þð1 sj Þ bsj ;
(4)
aðVj Þ ¼ a0 =ð1 þ eVj =Vshp Þ:
(5)
Here, the synaptic recovery function aðVj Þ can be taken as the Heaviside function. When the presynaptic neuron is in the silent state (Vj < 0), sj can be reduced to s_ j ¼ bsj ; otherwise, sj jumps quickly to 1 and acts on the postsynaptic ones. Vshp ¼ 0:05 determines the threshold above which the postsynaptic neuron is affected by a presynaptic one. The parameters a0 and b are set to be a0 ¼ 2 and b ¼ 1. Synaptic conductance gij from the j th neuron to the i th neuron updates through STDP modification function F, which is defined as follows: gij ¼ gij þ Dgij ; Dgij ¼ gij FðDtÞ; 8 > < Aþ exp ðjDtj=sþ Þ FðDtÞ ¼ A exp ðjDtj=s Þ > : 0
(6) (7) if Dt > 0 if Dt < 0
(8)
if Dt ¼ 0;
where Dt ¼ ti tj and ti and tj are the spiking time of the i th and j th neuron, respectively. The maximum amount of synaptic modification is limited by the value of adjusting rate Aþ and A . sþ and s determine the temporal window over which synaptic strengthening and weakening occur. The adjusting rate and temporal window are two main parameters to be discussed in this paper. The adjusting ratio is defined as fA ¼ A =Aþ (fA > 1 is considered to be dominant depression, otherwise dominant potentiation), and the temporal window ratio is fs ¼ sþ =s . Experimental studies suggest that the temporal window for synaptic weakening is roughly the same as that for synaptic strengthening.36,37,50,51 Potentiation is consistently induced when the postsynaptic spike generates within a time window of 20 ms after presynaptic spike, and depression is induced conversely. In this paper, the parameters are initially chosen as fA ¼ 1:05 (Aþ ¼ 0:05) and fs ¼ 1 (sþ ¼ s– ¼ 20). All synapses considered in the network are initiated as gij ¼ gmax =2 ¼ 0:05,
033125-3
Yu et al.
Chaos 24, 033125 (2014)
where gmax ¼ 0:1 is the maximum value. Numerical integration of the system is done by the explicit Euler algorithm, with a time step of 0.05. Additionally, based on Newman-Watts procedure,52 the small-world network, comprising of N ¼ 100 neurons, is initiated as an originally regular ring in which each unit is connected to its K ¼ 2 nearest neighbors. Accordingly, the total number of shortcuts is NðN 1Þ=2. New shortcuts are added into the network with probability p, and the number of added shortcuts ne satisfies ne ¼ pNðN 1Þ=2. If p ¼ 0, the network is a regular ring, while it is globally coupled for p ¼ 1. The small-world network is obtained by an intermediate case of p (0 < p < 1). The normalized shortcut number p is another one of the main parameters to be investigated in this paper. III. STOCHASTIC RESONANCE IN SMALL-WORLD NEURONAL NETWORK VIA STDP
In this section, we investigate the effect of STDP on the stochastic resonance in small-world neuronal networks. The external periodic input is chosen as Iex ¼ B sin ðxtÞ, with B ¼ 0:1 and x ¼ 0:2, guaranteeing no spike for all neurons in the absence of random fluctuations. Fourier coefficient Q is utilized to quantitatively characterize the cooperative effect of the weak signal and external noise within the considered network and defined as ðÞ
i ¼ Q sin
iÞ Qðcos
x 2pn
x ¼ 2pn ðiÞ
Q
ð 2pn=x
2Vi ðtÞsinðxtÞdt;
(9)
2Vi ðtÞcosðxtÞdt;
(10)
0
ð 2pn=x 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðiÞ2 ðiÞ2 ¼ Q sin þ Q cos ;
Q¼
N 1X Qð iÞ ; N i¼1
(11) (12)
where Vi is the membrane potential of the i th neuron in the network. n is the number of periods 2p=x covered by the integration time. The final results are obtained by averaging over 20 independent runs. In Fig. 1(a), Q is calculated as a function of the noise intensity r. It is shown that irrespective of the synaptic connectivity is fixed or adapted via STDP, the phenomenon of stochastic resonance arises in small-world neuronal networks, i.e., the temporal coherence between the temporal output series of excitable neurons and the stimulus frequency x of the weak external signal reaches an optimum. Compared with fixed coupling, STDP in the coupling process can largely enhance the efficiency of network stochastic resonance, as the optimal value of Q becomes larger. Moreover, it can be seen that the resonance point shifts rightward with the increase of Aþ . In addition, the dependence of Q on noise intensity r with different s is shown in Fig. 1(b). When the value of s is fixed, the correlation between the output of the system and weak periodic signal achieves best by a moderate noise intensity. With the enlargement of s , the resonance point shifts rightward to a larger noise intensity. However, for even large values of s , for instance, s ¼ 20 and s ¼ 30, the SR effects are nearly coincided. To generalize the above obtained results, we calculate the value of Q in a two-dimensional parameter space. As shown in Fig. 2(a), the dependence of Q on both r and Aþ is exhibited. It can be seen that for each particular value of adjusting rate Aþ , there exists a span of optimal noise intensities by which Q is maximum, and the span of optimal noise intensities r shifts rightward with the augmentation of Aþ . Moreover, the dependence of Q on both noise intensity r and temporal window s is presented in Fig. 2(b). For smaller temporal windows, weaker noise intensity is needed to induce stochastic resonance, and the resonant regions almost remain unchanged for larger temporal windows (s > 20). It can thus be concluded that STDP has a nontrivial impact on the stochastic resonance in small-world neuronal networks.
FIG. 1. Dependence of Q on the noise intensity r for different values of: (a) adjusting rate Aþ , s ¼ 20 and (b) temporal window s , Aþ ¼ 0:05. Other parameters are: p ¼ 0:3, fA ¼ 1:05, and fs ¼ 1. The fixed synaptic strength is set to be gij ¼ 0:05. Irrespective of synaptic connectivity is fixed or adapted via STDP, an intermediate intensity of additive noise is able to optimize the temporal coherence between the spiking activity and the weak subthreshold signal.
033125-4
Yu et al.
Chaos 24, 033125 (2014)
FIG. 2. (a) Contour plot of Q in dependence on the noise intensity r and adjusting rate Aþ , s ¼ 20. (b) Contour plot of Q in dependence on the noise intensity r and temporal window s , Aþ ¼ 0:05. Other parameters are: fA ¼ 1:05, fs ¼ 1, and p ¼ 0:3. In the case with STDP, resonance point shifts rightward as the adjusting rate or temporal window increases.
Considering the influence of small-world network structure on STDP-induced SR of neuronal systems, we plot Q versus r for different values of p, as presented in Fig. 3(a). It can be observed that for each particular value of p, there exists a peak value of Q, indicating the emergence of stochastic resonance. Remarkably, the resonance point shifts leftward to a smaller value of r as the normalized shortcut number p increases. Indeed, with the increase of p, more edges are added into the network which is connected with two neurons. The increased connections between neurons may help mutual excitation and lead to the decrement of resonance noise intensity. Similar results are obtained in Ref. 35. For an overall inspection, we plot the value of Q in the two dimensional space (r, p) in Fig. 3(b). Evidently, the normalized shortcut p has a significant effect on the resonance value of the neural system.
To gain more insight into the effect of STDP on SR in small-world neuronal networks, Q is plotted against p for different values of Aþ and s , respectively (Fig. 4). It can be found that irrespective of the value of Aþ and s , the network stochastic resonance can be maximized by a moderate value of p, i.e., there is an optimal probability of adding links by which the noise-induced transmission of weak periodic signal peaks. Similar results are also obtained by Ozer et al. by studying the stochastic resonance in small-world neuronal networks of Hodgkin-Huxley neurons with a fixed coupling strength,35 which shows that SR phenomenon can be further amplified via fine-tuning of the small-world network structure. In addition, the peak value of Q decreases with Aþ and s . That is to say, large adjusting rate or temporal window of STDP will reduce the efficiency of stochastic resonance due to the network topology. All the results obtained above demonstrate that both
FIG. 3. (a) Dependence of Q on noise intensity r for different values of normalized shortcut p. (b) Contour plot of Q in dependence on the noise intensity r and normalized shortcut p. Other parameters are: Aþ ¼ 0:05, s ¼ 20, fA ¼ 1:05, and fs ¼ 1. The network with more shortcuts will need less noise intensity for the occurrence of SR in small-world neuronal networks with STDP.
033125-5
Yu et al.
Chaos 24, 033125 (2014)
FIG. 4. (a) Dependence of Q on p for different Aþ , s ¼ 20. (b) Dependence of Q on p for different s , Aþ ¼ 0:05. Other parameters are: r ¼ 0:1, fA ¼ 1:05, and fs ¼ 1. There exists an optimal network topology by which the efficiency of SR is maximized, and this efficiency is reduced for large adjusting rate or temporal window.
STDP and connectivity structure have a significant impact on the stochastic resonance in small-world neuronal networks. Adjusting ratio fA and temporal window ratio fs are another two parameters determining the feature of dynamical synaptic coupling in small-world neuronal networks. The dependence of Q on the noise intensity r for different values of fA and fs is presented in Figs. 5(a) and 5(b), respectively. It is shown that the phenomenon of SR indeed occurs for each particular value of fA and fs . With the enlargement of fA , the optimal noise intensity turns larger and the response of neuronal network to external force is enhanced. Similar transition is observed for the decrease of fs . It is thus indicated that STDP with dominant depression and small temporal window ratio (i.e., sþ s ) is more efficient for the transmission of weak external signal in small-world neuronal networks via stochastic resonance.
Finally, the effect of network size on stochastic resonance is presented in Fig. 6. For N ¼ 200 (Fig. 6(a)), similar transitions are obtained compared with the case of N ¼ 100 (presented in Fig. 1(a)). Moreover, as shown in Fig. 6(b), it is evident that increasing the neuron number N, the maximal value of Q will be decreased. It means that the larger is the network size, the less prominent SR effect can be observed. But the enlargement of the neuronal network size has little effect on the optimal intensity of random fluctuations. IV. EVOLUTION OF SYNAPTIC CONNECTIVITY THROUGH STDP AND ITS BENEFIT ON STOCHASTIC RESONANCE
From the observations above, it is shown that STDP has a profound impact on the SR in small-world neuronal
FIG. 5. Dependence of Q on noise intensity r for different values of: (a) adjusting ratio fA , s ¼ 20, and fs ¼ 1; (b) temporal window ratio, fA ¼ 1:05. Other parameters are: p ¼ 0:3, Aþ ¼ 0:05, and r ¼ 0:1 (sþ ¼ 20 for fs 1 and s ¼ 20 for fs > 1). STDP with dominant depression and small temporal window ratio is more efficient for the transmission of weak external signal in small-world neuronal networks via stochastic resonance.
033125-6
Yu et al.
Chaos 24, 033125 (2014)
FIG. 6. (a) Dependence of Q on noise intensity r for different values of adjusting rate, N ¼ 200. (b) Dependence of Q on noise intensity r for different values of network size N, (Aþ ¼ 0:05). Other parameters are: p ¼ 0:3, s ¼ 20, fA ¼ 1:05, and fs ¼ 1.
networks, which may arise from the dynamical synaptic coupling modulated by STDP. In order to investigate how coupling strengths evolve during the STDP regulating process, gij is averaged over the whole population and time with the absence of subthreshold external signal, and the average coupling strength hgi can be written as53 hgi ¼
N X N 1 X N 2 T i¼1 j¼1
ðT
gij ðtÞ;
(13)
0
where T is the integration time. In Fig. 7(a), the average coupling strength hgi is plotted against the noise intensity r for different values of adjusting rate Aþ . For each particular value of Aþ , Aþ ¼ 0:05, for instance, it is observed that hgi slumps with the noise intensity r. This phenomenon is presented more detailed in Figs. 8(a)–8(c), where the temporal processes of synaptic modification for different r are clearly shown. Obviously, with the increase of noise intensity, the fraction of strong (gij 0:9 gmax ) and moderate (0:1 gmax < gij < 0:9 gmax ) synaptic strength decreases, while the part of weak synaptic strength (gij 0:1 gmax ) increases rapidly during the evolution. Moreover, as crucial parameters of STDP, the adjusting rate Aþ and A determine the growth and decline speed of the coupling strength. As mentioned above, the value of A is a little stronger than Aþ (we choose fA ¼ 1:05 here), which means that the decrement of synaptic strength is larger than the increment for the same spiking interval. Hence, the larger the value of Aþ is, the smaller the average coupling strength hgi is. Indeed, the fraction of moderate (0:1 gmax < gij < 0:9 gmax ) synaptic strength declines correspondingly with the enlargement of Aþ , comparing the results in Figs. 8(d)–8(f). Furthermore, Fig. 7(b) depicts the variation of the average coupling strength hgi with respect to the noise intensity r for different temporal windows s . It is demonstrated that the neuronal system with high levels of temporal window can obtain a strong synaptic coupling hgi. However, when the temporal window
s > 20, synaptic modification differs little with s increase. Indeed, the ability for correlated spiking to introduce synaptic potentiation or depression decreases rapidly as the interval of spike timing increases,36 which is a possible explanation for the saturation of mean coupling strength as temporal window widening. In turn, the saturation of coupling strength may account for the near coincidence of the SR effects between s ¼ 20 and s ¼ 30. In addition, the effect of network topology on the average coupling strength hgi is also investigated. Figs. 7(c) and 7(d) plot the dependence of the average coupling strength on the normalized shortcut p for different values Aþ and s , respectively. As a major parameter of the small-world neuronal network, p is the normalized number of shortcuts added into the network. Obviously, without STDP, the average coupling strength increases linearly with p. Larger p means more shortcuts added into the neuronal network. When STDP is introduced into the coupling process, hgi is also an increasing function of normalized shortcut p, but the increment is weaker than fixed one and depends largely on the features of STDP, i.e., the value of Aþ and s . More specifically, the value of hgi increases more rapidly for high values of s , while decreases with the enlargement of Aþ . In STDP rules, the parameters of adjusting ratio fA and temporal window ratio fs can also affect the dynamical coupling strength of small-world neuronal networks. Fig. 9(a) presents the dependence of hgi on fA for different values of Aþ . It is shown that the average coupling strength declines with the adjusting ratio fA , and larger Aþ results in a smaller value of hgi. Moreover, we explore the variation of average coupling strength against temporal window ratio fs . Here, we consider both cases of fs > 1 and fs < 1. As presented in Fig. 9(b), for each particular value of adjusting rate Aþ , the average coupling strength is enlarged by the increase of the temporal window ratio fs . In conclusion, all the features of noise intensity, network structure, and STDP have a profound impact on the average coupling strength of neural systems.
033125-7
Yu et al.
Chaos 24, 033125 (2014)
FIG. 7. Dependence of hgi on the noise intensity r for different values of: (a) adjusting rate Aþ , s ¼ 20 and (b) temporal window s , Aþ ¼ 0:05, and p ¼ 0:3. Dependence of hgi on the normalized shortcut p for different values of: (c) adjusting rate Aþ , s ¼ 20 and (d) temporal window s , Aþ ¼ 0:05, and r ¼ 0:15. Other parameters are: fA ¼ 1:05 and fs ¼ 1. The fixed synaptic strength is set to be gij ¼ 0:05. The average coupling strength is a decreasing function of noise intensity and adjusting rate, while it monotonously increases with the normalized shortcut and temporal window.
The results obtained above demonstrate that all the adjustments of STDP parameters finally result in the change of average coupling strength of the neuronal network. To analyze the effect of mean coupling strength on SR, variations of Q with respect to hgi (which is calculated from Eq. (13) by varying the parameter Aþ from Aþ ¼ 103 to 100:5 ) for different values of noise intensity r are exhibited in Fig. 10. It is shown that for r 0:08, Q passes through a peak when hgi is increased, indicating the occurrence of stochastic resonance due to the average coupling strength. However, when the noise intensity is increased to r 0:12, the effectiveness of coupling is overwhelmed by large intensity of noise, and the response of the neural system to external signal decreases monotonically. It thus can be concluded that the phenomenon of stochastic resonance can be further enhanced via fine-tuning of the average coupling strength of small-world networks. An optimal average coupling strength modulated by STDP exists to maximize the dynamical response of the neural system to the external subthreshold signal.
To quantitatively characterize the difference effect of fixed coupling and adaptive coupling adjusted via STDP on the stochastic resonance of the neural system, we calculate Q as a function of fixed coupling strength g and the average coupling strength hgi for different noise intensity r. (g is the average value of fixed synaptic coupling strength of the small-world neuronal network.) Results are presented in Figs. 11 and 12. For s ¼ 20, it is evident that when the noise intensity is small and moderate, there is a peak for Q by an optimal value of hgi, where the phenomenon of stochastic resonance happens (Figs. 12(a) and 12(b)). Remarkably, for r ¼ 0:08 (Fig. 11(b)), the resonance of the adaptive coupling can reach a much larger value than that of fixed coupling. This phenomenon may be interpreted as that a fraction of synapses with large strength (g 0:9 gmax ) emerge in STDP modification process (all synapses are initiated as g ¼ 0:5 gmax ), which will enhance the excitability of neuronal network. This may be the explanation of the spike-time-dependent plasticity-improved networked
033125-8
Yu et al.
Chaos 24, 033125 (2014)
FIG. 8. [(a)–(c)] Percentage of synapses at three levels: g 0:1 gmax , g 0:9 gmax , and the others for r ¼ 0:08, r ¼ 0:15, and r ¼ 0:2, respectively, Aþ ¼ 0:05, s ¼ 20, fA ¼ 1:05, and fs ¼ 1. [(d)–(f)] Percentage of synapses at three levels: g 0:1 gmax , g 0:9 gmax , and the others for Aþ ¼ 0:01, Aþ ¼ 0:05, and Aþ ¼ 0:1, respectively, r ¼ 0:06, s ¼ 20, fA ¼ 1:05, and fs ¼ 1. Other parameter: p ¼ 0:3. All coupling strengths are initiated as gij ¼ gmax =2, and during the evolution of time, a part of synapses with weak (gij 0:1 gmax ) and strong (gij 0:9 gmax ) coupling strength emerges, and the fraction of each strength level depends largely on r and Aþ .
stochastic resonance. Similar results can also be obtained for small temporal window s ¼ 5, but both with weak and moderate noise intensity (Figs. 12(a) and 12(b)). While for a
large noise intensity, the resonance effect of STDP is not obvious, as shown in Figs. 11(c) and 12(c). The reason may be that the noise is too large to overwhelm the weak periodic
033125-9
Yu et al.
Chaos 24, 033125 (2014)
FIG. 9. (a) Dependence of hgi on the adjusting ratio fA for different Aþ , s ¼ 20, and fs ¼ 1. (b) Dependence of hgi on the temporal window ratio fs for different Aþ , fA ¼ 1:05. (sþ ¼ 20 for fs 1 and s ¼ 20 for fs > 1). Other parameters are: r ¼ 0:1 and p ¼ 0:3. Large adjusting ratio and small temporal window ratio result in small average coupling strength.
FIG. 10. Dependence of Q on the average coupling strength hgi for different values of r. s ¼ 20, fA ¼ 1:05, fs ¼ 1, p ¼ 0:3, and Aþ ¼ 103 –100:5 . Fine-tuning of the average coupling strength can further enhance the stochastic resonance in small-world neuronal networks.
signal. Thus, we can get a conclusion that STDP in the coupling process can largely enhance the efficiency of stochastic resonance in small-world neuronal networks, which depends greatly on the temporal window of STDP. Up to now, a series of biological works have demonstrated the existence of STDP, which commonly occurs at excitatory synapses onto neocortical54,55 and hippocampal pyramidal neurons,36,56 excitatory neurons in auditory brainstem,57 parvalbumin-expressing fast-spiking striatal interneurons,58 etc. Moreover, stochastic-resonance-type effects have been demonstrated in a range of sensory systems. These include crayfish mechanoreceptors,59 shark multimodal sensory cells,15 cricket cercal sensory neurons,13 and human muscle spindles.60 It has been shown that the ability of sensory neurons to transmit and process weak input signals can be significantly enhanced via SR. As is well known, the coupling strengths between neurons play a crucial role by the stochastic resonance of the network.35,61 However, the fundamental role of STDP in stochastic resonance has been uncovered. In this work, we further suggest that the selective refinement is a possible explanation when the network is adaptive through STDP. Compared to fixed coupling,
FIG. 11. Dependence of Q on the fixed coupling strength g and the average coupling strength hgi. (a) r ¼ 0:03, (b) r ¼ 0:08, and (c) r ¼ 0:15. Other parameters are: s ¼ 20 and p ¼ 0:3. The phenomenon of SR due to mean coupling strength occurs for small and moderate noise intensity. For large temporal window, the resonance of the adaptive coupling can reach a much larger value than that of fixed coupling within an intermediate noisy background.
033125-10
Yu et al.
Chaos 24, 033125 (2014)
FIG. 12. Dependence of Q on the fixed coupling strength g and the average coupling strength hgi. (a) r ¼ 0:03, (b) r ¼ 0:08, and (c) r ¼ 0:15. Other parameters are: s ¼ 5 and p ¼ 0:3. For small temporal window, the resonance of the adaptive coupling can reach a much larger value within both weak and intermediate noisy background.
the adaptive coupling is inhomogeneous, i.e., a fraction of synapses are pushed to strong coupling, whereas a part of weak coupling emerges. This heterogeneity of synaptic coupling can selectively excite neurons in the network to respond to the subthreshold forcing with spikes. It was shown that this selective refinement of synaptic process can enhance weak signal detection by inducing stochastic resonance on neuronal networks. V. CONCLUSIONS
In this paper, we study stochastic resonance in NewmanWatts small-world neuronal networks with spike-time-dependent plasticity. The effects of fixed and adaptive couplings on stochastic resonance of the neural system are also discussed. The obtained numerical results show that an intermediate intensity of additive noise is able to optimize the temporal response of the neural system to the subthreshold periodic signal, where the phenomenon of stochastic resonance occurs. It is demonstrated that STDP in coupling process can largely enhance the efficiency of network stochastic resonance. Particularly, the resonance for adaptive coupling can reach a much larger value than that for fixed coupling when the noise intensity is small or intermediate. STDP with dominant depression and small temporal window ratio is more efficient for the transmission of weak external signal in small-world neuronal networks. In addition, the effect of stochastic resonance can be further improved via fine-tuning of the average coupling strength of small-world networks. There exists an optimal average coupling strength modulated by STDP, which can always induce dynamical resonance of the neural system. More importantly, it is found that the smallworld topology has a significant impact on the stochastic resonance in neuronal networks. There is an optimal probability of adding links, warranting the largest peak value of system resonance. In a word, both STDP and connectivity structure play an important role in the stochastic resonance on smallworld neuronal networks, determining the ability to enhance the transmission of weak periodic signal. ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (Grant Nos. 61072012 and 61302002)
and Tianjin Research Program of Application Foundation and Advanced Technology (No. 14JCQNJC01200). 1
P. H€anggi and R. Bartussek, in Nonlinear Physics of Complex Systems, edited by J. Parisi, S. C. M€ uller, and W. Zimmermman (Springer, New York, 1999). 2 M. Perc, Phys. Rev. E 72, 016207 (2005). 3 R. Benzi, A. Sutera, and A. Vulpiani, J. Phys. A: Math. Gen. 14, L453 (1981). 4 L. Gammaitoni, P. H€anggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998). 5 P. Jung and G. Mayer-Kress, Phys. Rev. Lett. 74, 2130 (1995). 6 S. Fauve and F. Heslot, Phys. Lett. A 97, 5 (1983). 7 B. McNamara, K. Wiesenfeld, and R. Roy, Phys. Rev. Lett. 60, 2626 (1988). 8 R. Bates, O. Blyuss, and A. Zaikin, Phys. Rev. E 89, 032716 (2014). 9 J. Paulsson and M. Ehrenberg, Phys. Rev. Lett. 84, 5447 (2000). 10 G. Mato, Phys. Rev. E 58, 876 (1998). 11 B. J. Gluckman, T. I. Netoff, E. J. Neel, W. L. Ditto, M. L. Spano, and S. J. Schiff, Phys. Rev. Lett. 77, 4098 (1996). 12 J. K. Douglass, L. Wilkens, E. Pantazelou, and F. Moss, Nature (London) 365, 337 (1993). 13 J. E. Levin and J. P. Miller, Nature (London) 380, 165 (1996). 14 A. Logtin, A. Bulsara, and F. Moss, Phys. Rev. Lett. 67, 656 (1991). 15 H. A. Braun, H. Wissing, K. Schafer, and M. C. Hirsch, Nature (London) 367, 270 (1994). 16 J. F. Lindner, S. Chandramouli, A. R. Bulsara, M. Locher, and W. L. Ditto, Phys. Rev. Lett. 81, 5048 (1998). 17 X. Li, J. Wang, and W. Hu, Phys. Rev. E 76, 041902 (2007). 18 N. G. Stocks and R. Mannella, Phys. Rev. E 64, 030902 (2001). 19 M. Perc, Phys. Rev. E 78, 036105 (2008). 20 Q. Y. Wang, M. Perc, Z. S. Duan, and G. R. Chen, Chaos 19, 023112 (2009). 21 M. Perc and M. Gosak, New J. Phys. 10, 053008 (2008). 22 Q. Y. Wang, H. H. Zhang, and G. R. Chen, Chaos 22, 043123 (2012). 23 E. Yilmaz, M. Uzuntarla, M. Ozer, and M. Perc, Physica A 392, 5735 (2013). 24 D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998). 25 E. Bullmore and O. Sporns, Nat. Rev. Neurosci. 10, 186 (2009). 26 S. Achard, R. Salvador, B. Whitcher, J. Suckling, and E. Bullmore, J. Neurosci. 26, 63 (2006). 27 O. Sporns, D. Chialvo, M. Kaiser, and C. C. Hilgetag, Trends Cognit. Sci. 8, 418 (2004). 28 D. S. Bassett and E. T. Bullmore, Neuroscientist 12, 512 (2006). 29 Q. Y. Wang, Z. S. Duan, M. Perc, and G. R. Chen, Europhys. Lett. 83, 50008 (2008). 30 I. D. Steenbuck, M. Butz, M. Ruiter, and A. van Ooyen, BMC Neurosci. 12, P357 (2011). 31 M. Perc, Chaos, Solitons Fractals 31, 280 (2007). 32 X. J. Sun, M. Perc, Q. S. Lu, and J. Kurths, Chaos 18, 023102 (2008). 33 M. Perc, New J. Phys. 7, 252 (2005). 34 Y. Gong, M. Wang, Z. Zhou, and H. Xin, ChemPhysChem 6, 1042 (2005). 35 M. Ozer, M. Perc, and M. Uzuntarla, Phys. Lett. A 373, 964 (2009). 36 G. Bi and M. Poo, J. Neurosci. 18, 10464 (1998).
033125-11 37
Yu et al.
L. I. Zhang, H. W. Tao, C. E. Holt, W. A. Harris, and M. M. Poo, Nature 395, 37 (1998). 38 D. E. Feldman, Neuron 27, 45 (2000). 39 D. Debanne, B. H. Gahwiler, and S. M. Thompson, J. Physiol. 507, 237 (1998). 40 C. W. Shin and S. Kim, Phys. Rev. E 74, 045101 (2006). 41 S. Song, K. D. Miller, and L. F. Abbott, Nat. Neurosci. 3, 919 (2000). 42 K. Mikkelsen, A. Imparota, and A. Torcini, Phys. Rev. Lett. 110, 208101 (2013). 43 M. Bayati and A. Valizadeh, Phys. Rev. E 86, 011925 (2012). 44 C. B. Picallo and H. Riecke, Phys. Rev. E 83, 036206 (2011). 45 Q. S. Ren, K. M. Kolwankar, A. Samal, and J. Jost, Phys. Rev. E 86, 056103 (2012). 46 K. Kube, A. Herzog, B. Michaelis, A. D. de Lima, and T. Voigt, Neurocomputing 71, 1694 (2008). 47 X. Li, J. Zhang, and M. Small, Chaos 19, 013126 (2009). 48 Q. S. Ren, Z. Q. Zhang, and J. Y. Zhao, Phys. Rev. E 85, 022901 (2012). 49 X. Li and M. Small, Chaos 22, 023104 (2012).
Chaos 24, 033125 (2014) 50
R. G€ utig, R. Aharonov, S. Rotter, and H. Sompolinsky, J. Neurosci. 23, 3697 (2003). H. Markram, J. Lubke, M. Frotscher, and B. Sakmann, Science 275, 213 (1997). 52 M. E. J. Newman and D. J. Watts, Phys. Lett. A 263, 341 (1999). 53 D. Wu, S. Q. Zhu, X. Q. Luo, and L. Wu, Phys. Rev. E 84, 021102 (2011). 54 D. E. Feldman and M. Brecht, Science 310, 810 (2005). 55 T. Nevian and B. Sakmann, J. Neurosci. 26, 11001 (2006). 56 M. Nishiyama, K. Hong, K. Mikoshiba, M. M. Poo, and K. Kato, Nature 408, 584 (2000). 57 T. Tzounopoulos, M. E. Rubio, J. E. Keen, and L. O. Trussell, Neuron 54, 291 (2007). 58 E. Fino, V. Paille, J. M. Deniau, and L. Venance, Neuroscience 160, 744 (2009). 59 K. Wiesenfeld and F. Moss, Nature 373, 33 (1995). 60 P. Cordo, J. T. Inglis, S. Verschueren, J. J. Collins, D. M. Merfeld, S. Rosenblum, S. Buckley, and F. Moss, Nature 383, 769 (1996). 61 M. Perc, Phys. Rev. E 76, 066203 (2007). 51
Citation: Chaos 24, 033125 (2014); doi: 10.1063/1.4893773 View online: http://dx.doi.org/10.1063/1.4893773 View Table of Contents: http://aip.scitation.org/toc/cha/24/3 Published by the American Institute of Physics
Articles you may be interested in Effects of time delay on the stochastic resonance in small-world neuronal networks Chaos: An Interdisciplinary Journal of Nonlinear Science 23, 013128 (2013); 10.1063/1.4790829 Chaotic phase synchronization in small-world networks of bursting neurons Chaos: An Interdisciplinary Journal of Nonlinear Science 21, 013127 (2011); 10.1063/1.3565027
CHAOS 24, 033125 (2014)
Effects of spike-time-dependent plasticity on the stochastic resonance of small-world neuronal networks Haitao Yu, Xinmeng Guo, Jiang Wang,a) Bin Deng, and Xile Wei School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, People’s Republic of China
(Received 4 June 2014; accepted 12 August 2014; published online 22 August 2014) The phenomenon of stochastic resonance in Newman-Watts small-world neuronal networks is investigated when the strength of synaptic connections between neurons is adaptively adjusted by spike-time-dependent plasticity (STDP). It is shown that irrespective of the synaptic connectivity is fixed or adaptive, the phenomenon of stochastic resonance occurs. The efficiency of network stochastic resonance can be largely enhanced by STDP in the coupling process. Particularly, the resonance for adaptive coupling can reach a much larger value than that for fixed one when the noise intensity is small or intermediate. STDP with dominant depression and small temporal window ratio is more efficient for the transmission of weak external signal in small-world neuronal networks. In addition, we demonstrate that the effect of stochastic resonance can be further improved via fine-tuning of the average coupling strength of the adaptive network. Furthermore, the small-world topology can significantly affect stochastic resonance of excitable neuronal networks. It is found that there exists an optimal probability of adding links by which the C 2014 AIP Publishing LLC. noise-induced transmission of weak periodic signal peaks. V [http://dx.doi.org/10.1063/1.4893773]
The phenomenon of noise-induced stochastic resonance (SR) in complex neuronal networks has been extensively studied in recent years. However, most of the previous studies were devoted to a static description of synaptic connectivity, while in reality the synaptic strength varies as a function of neuromodulation and time-dependent processes. Spike-time-dependent plasticity (STDP) is an important biological process, which can adjust the strength of connections between neurons adaptively based on the relative timing between pre- and postsynaptic action potentials. In this paper, we study the fundamental role of STDP in stochastic resonance of small-world neuronal networks. We aim to investigate how the network connections evolve during the learning process refined by STDP and the dependence of network stochastic resonance on it.
I. INTRODUCTION
Noise plays a constructive role in nonlinear systems.1,2 One typical representative of this effect is SR, which occurs when the response of a nonlinear dynamical system to a weak periodic signal is optimized by moderate intensity of random fluctuations.3–5 Due to its wide applications, the investigation of stochastic resonance has been extended to many different fields, particularly neural systems.6–11 Experimental and theoretical researches have shown that the ability of sensory neurons to process weak input signals can be significantly enhanced by adding noise to the system.12–15 Additionally, it is demonstrated that the signal propagation a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]. Tel.: 86-22-27402293.
1054-1500/2014/24(3)/033125/11/$30.00
in coupled neural systems is largely improved by external noise via stochastic resonance.16–18 Therefore, it is of great importance to study stochastic resonance for understanding the signal transmission and information propagation in neuronal networks. Recently, much attention has been devoted to the stochastic resonance in complex neuronal networks.19–23 One typical topology is small-world networks, which can capture the characteristics of many real-world complex networks.24 By now, a series of empirical investigations have shown the existence of small-world architecture in many structural and functional brain networks of humans and animals.25–28 The small-world architecture exhibits a relative short length path but along with a high clustering coefficient.24 Therefore, it has much dynamical potentials, such as improving the signal propagation speed, computational power, and synchronizability of neural systems.29,30 In recent years, the effects of small-world connectivity on noise-induced temporal and spatial order in neural media have been extensively studied.31–33 Perc found that the temporal order can be greatly enhanced by the introduction of small-world connectivity, whereby the effect increases with the increasing fraction of introduced shortcut links. On the other hand, it is argued that the small-world network topology destroys spatial order due to the lack of a precise internal spatial scale. In addition, the phenomenon of noise-induced stochastic resonance in small-world neuronal networks has been deeply studied. It has been reported that the effect of stochastic resonance in neuronal networks with small-world topology depends largely on the fraction of randomly added shortcuts.34 Moreover, Ozer et al. studied stochastic resonance on small-world networks of Hodgkin-Huxley neurons with local periodic driving and found that the stochastic
24, 033125-1
C 2014 AIP Publishing LLC V
033125-2
Yu et al.
Chaos 24, 033125 (2014)
resonance can also be amplified via fine-tuning of coupling strength among neurons.35 Most of the previous studies of stochastic resonance on complex neuronal networks were devoted to a static description of synaptic connectivity, while in reality the synaptic strength varies as a function of neuromodulation and timedependent processes. STDP is one of those important biological processes, which can regulate the strength of connections between neurons adaptively based on the relative timing between pre- and post-synaptic action potentials.36 Up to now, the existence of STDP has been found in many neocortical layers and brain regions.37–39 Experimental researches show that the functional structures in the brain can be adjusted to both small-world and scale-free networks by synaptic reorganization via STDP.40 Modeling studies also demonstrate that the synaptic modification of STDP can automatically balance synaptic strengths to make post-synaptic firing irregular but more sensitive to pre-synaptic spiking timing.41 Recently, functional dynamics of complex neuronal networks with STDP have gained increasing interests.42–45 For example, Kube et al. studied the synchronous activity in small-world neural networks with adaptive coupling and found that STDP modifies the weights of synaptic connections in such a way that synchronization of neuronal activity is considerably weakened.46 Moreover, coherence resonance and stochastic resonance have been investigated in self-organized neural network via STDP.47 It has been shown that the selectively refined connectivity modified through STDP can highly enhance the ability of neuronal communications and improve the efficiency of signal transmission in the network.48,49 In this paper, we will study the fundamental role of STDP in stochastic resonance of small-world neuronal networks. We aim to investigate how the network connections evolve during the process refined by STDP and the dependence of stochastic resonance on it. In addition, effects of small-world structure on the network stochastic resonance will be discussed as well. The paper is organized as follows: in Sec. II, the neural model of small-world neuronal network is established, where the strength of connections between neurons is varied through STDP rules. The effects of STDP and small-world topology on SR are systematically investigated in Sec. III. In Sec. IV, we further study the evolution of coupling strength via STDP and its benefit on SR. Finally, a brief conclusion of this paper is drawn in Sec. V. II. MATHEMATICAL MODEL
The FitzHugh–Nagumo neuron model47 is employed to describe the neuronal dynamics, and the temporal evolution of each unit can be defined as follows: eV_ i ¼ Vi –Vi3 =3 Wi þ Iex þ Iisyn þ rni ;
(1)
W_ i ¼ Vi þ a bi Wi ;
(2)
where i ¼ 1; 2; ; N. Vi is a fast variable, representing the membrane potential; whereas Wi is a slow recovery variable. The time scale ratio is chosen as e ¼ 0:08, warranting that Vi evolves much faster than Wi . ni is Guasian white noise with mean 0 and variance 1, satisfying hni ðtÞnj ðt0 Þi ¼ dij dðt t0 Þ.
r stands for the intensity of the noisy background. Iex denotes the external weak periodic signal. Parameter a is set to be 0:7. The value of b is crucial for the dynamics of a neuron, i.e., in the absence of noise, when b > 0:45, the neuron is excitable; while for b < 0:45, the system has a stable periodic solution generating periodic spikes. In our study, bi is randomly distributed in ½0:5; 0:75, so that all neurons within the network are of different excitability. The synaptic current Iisyn takes the form of Iisyn ¼
N X
gij Cij sj ðVi Vsyn Þ;
(3)
j¼1ðj6¼iÞ
where Cij is the connective matrix, since all connections considered in this paper are bidirectional, that is to say, if neuron j couples to neuron i, then Cij ¼ 1 and Cji ¼ 1, otherwise Cij ¼ Cji ¼ 0 and Cii ¼ 0. The reversal potential is chosen as Vsyn ¼ 0, guaranteeing that all synapses are excitatory. The synaptic variable sj is defined as s_ j ¼ aðVj Þð1 sj Þ bsj ;
(4)
aðVj Þ ¼ a0 =ð1 þ eVj =Vshp Þ:
(5)
Here, the synaptic recovery function aðVj Þ can be taken as the Heaviside function. When the presynaptic neuron is in the silent state (Vj < 0), sj can be reduced to s_ j ¼ bsj ; otherwise, sj jumps quickly to 1 and acts on the postsynaptic ones. Vshp ¼ 0:05 determines the threshold above which the postsynaptic neuron is affected by a presynaptic one. The parameters a0 and b are set to be a0 ¼ 2 and b ¼ 1. Synaptic conductance gij from the j th neuron to the i th neuron updates through STDP modification function F, which is defined as follows: gij ¼ gij þ Dgij ; Dgij ¼ gij FðDtÞ; 8 > < Aþ exp ðjDtj=sþ Þ FðDtÞ ¼ A exp ðjDtj=s Þ > : 0
(6) (7) if Dt > 0 if Dt < 0
(8)
if Dt ¼ 0;
where Dt ¼ ti tj and ti and tj are the spiking time of the i th and j th neuron, respectively. The maximum amount of synaptic modification is limited by the value of adjusting rate Aþ and A . sþ and s determine the temporal window over which synaptic strengthening and weakening occur. The adjusting rate and temporal window are two main parameters to be discussed in this paper. The adjusting ratio is defined as fA ¼ A =Aþ (fA > 1 is considered to be dominant depression, otherwise dominant potentiation), and the temporal window ratio is fs ¼ sþ =s . Experimental studies suggest that the temporal window for synaptic weakening is roughly the same as that for synaptic strengthening.36,37,50,51 Potentiation is consistently induced when the postsynaptic spike generates within a time window of 20 ms after presynaptic spike, and depression is induced conversely. In this paper, the parameters are initially chosen as fA ¼ 1:05 (Aþ ¼ 0:05) and fs ¼ 1 (sþ ¼ s– ¼ 20). All synapses considered in the network are initiated as gij ¼ gmax =2 ¼ 0:05,
033125-3
Yu et al.
Chaos 24, 033125 (2014)
where gmax ¼ 0:1 is the maximum value. Numerical integration of the system is done by the explicit Euler algorithm, with a time step of 0.05. Additionally, based on Newman-Watts procedure,52 the small-world network, comprising of N ¼ 100 neurons, is initiated as an originally regular ring in which each unit is connected to its K ¼ 2 nearest neighbors. Accordingly, the total number of shortcuts is NðN 1Þ=2. New shortcuts are added into the network with probability p, and the number of added shortcuts ne satisfies ne ¼ pNðN 1Þ=2. If p ¼ 0, the network is a regular ring, while it is globally coupled for p ¼ 1. The small-world network is obtained by an intermediate case of p (0 < p < 1). The normalized shortcut number p is another one of the main parameters to be investigated in this paper. III. STOCHASTIC RESONANCE IN SMALL-WORLD NEURONAL NETWORK VIA STDP
In this section, we investigate the effect of STDP on the stochastic resonance in small-world neuronal networks. The external periodic input is chosen as Iex ¼ B sin ðxtÞ, with B ¼ 0:1 and x ¼ 0:2, guaranteeing no spike for all neurons in the absence of random fluctuations. Fourier coefficient Q is utilized to quantitatively characterize the cooperative effect of the weak signal and external noise within the considered network and defined as ðÞ
i ¼ Q sin
iÞ Qðcos
x 2pn
x ¼ 2pn ðiÞ
Q
ð 2pn=x
2Vi ðtÞsinðxtÞdt;
(9)
2Vi ðtÞcosðxtÞdt;
(10)
0
ð 2pn=x 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðiÞ2 ðiÞ2 ¼ Q sin þ Q cos ;
Q¼
N 1X Qð iÞ ; N i¼1
(11) (12)
where Vi is the membrane potential of the i th neuron in the network. n is the number of periods 2p=x covered by the integration time. The final results are obtained by averaging over 20 independent runs. In Fig. 1(a), Q is calculated as a function of the noise intensity r. It is shown that irrespective of the synaptic connectivity is fixed or adapted via STDP, the phenomenon of stochastic resonance arises in small-world neuronal networks, i.e., the temporal coherence between the temporal output series of excitable neurons and the stimulus frequency x of the weak external signal reaches an optimum. Compared with fixed coupling, STDP in the coupling process can largely enhance the efficiency of network stochastic resonance, as the optimal value of Q becomes larger. Moreover, it can be seen that the resonance point shifts rightward with the increase of Aþ . In addition, the dependence of Q on noise intensity r with different s is shown in Fig. 1(b). When the value of s is fixed, the correlation between the output of the system and weak periodic signal achieves best by a moderate noise intensity. With the enlargement of s , the resonance point shifts rightward to a larger noise intensity. However, for even large values of s , for instance, s ¼ 20 and s ¼ 30, the SR effects are nearly coincided. To generalize the above obtained results, we calculate the value of Q in a two-dimensional parameter space. As shown in Fig. 2(a), the dependence of Q on both r and Aþ is exhibited. It can be seen that for each particular value of adjusting rate Aþ , there exists a span of optimal noise intensities by which Q is maximum, and the span of optimal noise intensities r shifts rightward with the augmentation of Aþ . Moreover, the dependence of Q on both noise intensity r and temporal window s is presented in Fig. 2(b). For smaller temporal windows, weaker noise intensity is needed to induce stochastic resonance, and the resonant regions almost remain unchanged for larger temporal windows (s > 20). It can thus be concluded that STDP has a nontrivial impact on the stochastic resonance in small-world neuronal networks.
FIG. 1. Dependence of Q on the noise intensity r for different values of: (a) adjusting rate Aþ , s ¼ 20 and (b) temporal window s , Aþ ¼ 0:05. Other parameters are: p ¼ 0:3, fA ¼ 1:05, and fs ¼ 1. The fixed synaptic strength is set to be gij ¼ 0:05. Irrespective of synaptic connectivity is fixed or adapted via STDP, an intermediate intensity of additive noise is able to optimize the temporal coherence between the spiking activity and the weak subthreshold signal.
033125-4
Yu et al.
Chaos 24, 033125 (2014)
FIG. 2. (a) Contour plot of Q in dependence on the noise intensity r and adjusting rate Aþ , s ¼ 20. (b) Contour plot of Q in dependence on the noise intensity r and temporal window s , Aþ ¼ 0:05. Other parameters are: fA ¼ 1:05, fs ¼ 1, and p ¼ 0:3. In the case with STDP, resonance point shifts rightward as the adjusting rate or temporal window increases.
Considering the influence of small-world network structure on STDP-induced SR of neuronal systems, we plot Q versus r for different values of p, as presented in Fig. 3(a). It can be observed that for each particular value of p, there exists a peak value of Q, indicating the emergence of stochastic resonance. Remarkably, the resonance point shifts leftward to a smaller value of r as the normalized shortcut number p increases. Indeed, with the increase of p, more edges are added into the network which is connected with two neurons. The increased connections between neurons may help mutual excitation and lead to the decrement of resonance noise intensity. Similar results are obtained in Ref. 35. For an overall inspection, we plot the value of Q in the two dimensional space (r, p) in Fig. 3(b). Evidently, the normalized shortcut p has a significant effect on the resonance value of the neural system.
To gain more insight into the effect of STDP on SR in small-world neuronal networks, Q is plotted against p for different values of Aþ and s , respectively (Fig. 4). It can be found that irrespective of the value of Aþ and s , the network stochastic resonance can be maximized by a moderate value of p, i.e., there is an optimal probability of adding links by which the noise-induced transmission of weak periodic signal peaks. Similar results are also obtained by Ozer et al. by studying the stochastic resonance in small-world neuronal networks of Hodgkin-Huxley neurons with a fixed coupling strength,35 which shows that SR phenomenon can be further amplified via fine-tuning of the small-world network structure. In addition, the peak value of Q decreases with Aþ and s . That is to say, large adjusting rate or temporal window of STDP will reduce the efficiency of stochastic resonance due to the network topology. All the results obtained above demonstrate that both
FIG. 3. (a) Dependence of Q on noise intensity r for different values of normalized shortcut p. (b) Contour plot of Q in dependence on the noise intensity r and normalized shortcut p. Other parameters are: Aþ ¼ 0:05, s ¼ 20, fA ¼ 1:05, and fs ¼ 1. The network with more shortcuts will need less noise intensity for the occurrence of SR in small-world neuronal networks with STDP.
033125-5
Yu et al.
Chaos 24, 033125 (2014)
FIG. 4. (a) Dependence of Q on p for different Aþ , s ¼ 20. (b) Dependence of Q on p for different s , Aþ ¼ 0:05. Other parameters are: r ¼ 0:1, fA ¼ 1:05, and fs ¼ 1. There exists an optimal network topology by which the efficiency of SR is maximized, and this efficiency is reduced for large adjusting rate or temporal window.
STDP and connectivity structure have a significant impact on the stochastic resonance in small-world neuronal networks. Adjusting ratio fA and temporal window ratio fs are another two parameters determining the feature of dynamical synaptic coupling in small-world neuronal networks. The dependence of Q on the noise intensity r for different values of fA and fs is presented in Figs. 5(a) and 5(b), respectively. It is shown that the phenomenon of SR indeed occurs for each particular value of fA and fs . With the enlargement of fA , the optimal noise intensity turns larger and the response of neuronal network to external force is enhanced. Similar transition is observed for the decrease of fs . It is thus indicated that STDP with dominant depression and small temporal window ratio (i.e., sþ s ) is more efficient for the transmission of weak external signal in small-world neuronal networks via stochastic resonance.
Finally, the effect of network size on stochastic resonance is presented in Fig. 6. For N ¼ 200 (Fig. 6(a)), similar transitions are obtained compared with the case of N ¼ 100 (presented in Fig. 1(a)). Moreover, as shown in Fig. 6(b), it is evident that increasing the neuron number N, the maximal value of Q will be decreased. It means that the larger is the network size, the less prominent SR effect can be observed. But the enlargement of the neuronal network size has little effect on the optimal intensity of random fluctuations. IV. EVOLUTION OF SYNAPTIC CONNECTIVITY THROUGH STDP AND ITS BENEFIT ON STOCHASTIC RESONANCE
From the observations above, it is shown that STDP has a profound impact on the SR in small-world neuronal
FIG. 5. Dependence of Q on noise intensity r for different values of: (a) adjusting ratio fA , s ¼ 20, and fs ¼ 1; (b) temporal window ratio, fA ¼ 1:05. Other parameters are: p ¼ 0:3, Aþ ¼ 0:05, and r ¼ 0:1 (sþ ¼ 20 for fs 1 and s ¼ 20 for fs > 1). STDP with dominant depression and small temporal window ratio is more efficient for the transmission of weak external signal in small-world neuronal networks via stochastic resonance.
033125-6
Yu et al.
Chaos 24, 033125 (2014)
FIG. 6. (a) Dependence of Q on noise intensity r for different values of adjusting rate, N ¼ 200. (b) Dependence of Q on noise intensity r for different values of network size N, (Aþ ¼ 0:05). Other parameters are: p ¼ 0:3, s ¼ 20, fA ¼ 1:05, and fs ¼ 1.
networks, which may arise from the dynamical synaptic coupling modulated by STDP. In order to investigate how coupling strengths evolve during the STDP regulating process, gij is averaged over the whole population and time with the absence of subthreshold external signal, and the average coupling strength hgi can be written as53 hgi ¼
N X N 1 X N 2 T i¼1 j¼1
ðT
gij ðtÞ;
(13)
0
where T is the integration time. In Fig. 7(a), the average coupling strength hgi is plotted against the noise intensity r for different values of adjusting rate Aþ . For each particular value of Aþ , Aþ ¼ 0:05, for instance, it is observed that hgi slumps with the noise intensity r. This phenomenon is presented more detailed in Figs. 8(a)–8(c), where the temporal processes of synaptic modification for different r are clearly shown. Obviously, with the increase of noise intensity, the fraction of strong (gij 0:9 gmax ) and moderate (0:1 gmax < gij < 0:9 gmax ) synaptic strength decreases, while the part of weak synaptic strength (gij 0:1 gmax ) increases rapidly during the evolution. Moreover, as crucial parameters of STDP, the adjusting rate Aþ and A determine the growth and decline speed of the coupling strength. As mentioned above, the value of A is a little stronger than Aþ (we choose fA ¼ 1:05 here), which means that the decrement of synaptic strength is larger than the increment for the same spiking interval. Hence, the larger the value of Aþ is, the smaller the average coupling strength hgi is. Indeed, the fraction of moderate (0:1 gmax < gij < 0:9 gmax ) synaptic strength declines correspondingly with the enlargement of Aþ , comparing the results in Figs. 8(d)–8(f). Furthermore, Fig. 7(b) depicts the variation of the average coupling strength hgi with respect to the noise intensity r for different temporal windows s . It is demonstrated that the neuronal system with high levels of temporal window can obtain a strong synaptic coupling hgi. However, when the temporal window
s > 20, synaptic modification differs little with s increase. Indeed, the ability for correlated spiking to introduce synaptic potentiation or depression decreases rapidly as the interval of spike timing increases,36 which is a possible explanation for the saturation of mean coupling strength as temporal window widening. In turn, the saturation of coupling strength may account for the near coincidence of the SR effects between s ¼ 20 and s ¼ 30. In addition, the effect of network topology on the average coupling strength hgi is also investigated. Figs. 7(c) and 7(d) plot the dependence of the average coupling strength on the normalized shortcut p for different values Aþ and s , respectively. As a major parameter of the small-world neuronal network, p is the normalized number of shortcuts added into the network. Obviously, without STDP, the average coupling strength increases linearly with p. Larger p means more shortcuts added into the neuronal network. When STDP is introduced into the coupling process, hgi is also an increasing function of normalized shortcut p, but the increment is weaker than fixed one and depends largely on the features of STDP, i.e., the value of Aþ and s . More specifically, the value of hgi increases more rapidly for high values of s , while decreases with the enlargement of Aþ . In STDP rules, the parameters of adjusting ratio fA and temporal window ratio fs can also affect the dynamical coupling strength of small-world neuronal networks. Fig. 9(a) presents the dependence of hgi on fA for different values of Aþ . It is shown that the average coupling strength declines with the adjusting ratio fA , and larger Aþ results in a smaller value of hgi. Moreover, we explore the variation of average coupling strength against temporal window ratio fs . Here, we consider both cases of fs > 1 and fs < 1. As presented in Fig. 9(b), for each particular value of adjusting rate Aþ , the average coupling strength is enlarged by the increase of the temporal window ratio fs . In conclusion, all the features of noise intensity, network structure, and STDP have a profound impact on the average coupling strength of neural systems.
033125-7
Yu et al.
Chaos 24, 033125 (2014)
FIG. 7. Dependence of hgi on the noise intensity r for different values of: (a) adjusting rate Aþ , s ¼ 20 and (b) temporal window s , Aþ ¼ 0:05, and p ¼ 0:3. Dependence of hgi on the normalized shortcut p for different values of: (c) adjusting rate Aþ , s ¼ 20 and (d) temporal window s , Aþ ¼ 0:05, and r ¼ 0:15. Other parameters are: fA ¼ 1:05 and fs ¼ 1. The fixed synaptic strength is set to be gij ¼ 0:05. The average coupling strength is a decreasing function of noise intensity and adjusting rate, while it monotonously increases with the normalized shortcut and temporal window.
The results obtained above demonstrate that all the adjustments of STDP parameters finally result in the change of average coupling strength of the neuronal network. To analyze the effect of mean coupling strength on SR, variations of Q with respect to hgi (which is calculated from Eq. (13) by varying the parameter Aþ from Aþ ¼ 103 to 100:5 ) for different values of noise intensity r are exhibited in Fig. 10. It is shown that for r 0:08, Q passes through a peak when hgi is increased, indicating the occurrence of stochastic resonance due to the average coupling strength. However, when the noise intensity is increased to r 0:12, the effectiveness of coupling is overwhelmed by large intensity of noise, and the response of the neural system to external signal decreases monotonically. It thus can be concluded that the phenomenon of stochastic resonance can be further enhanced via fine-tuning of the average coupling strength of small-world networks. An optimal average coupling strength modulated by STDP exists to maximize the dynamical response of the neural system to the external subthreshold signal.
To quantitatively characterize the difference effect of fixed coupling and adaptive coupling adjusted via STDP on the stochastic resonance of the neural system, we calculate Q as a function of fixed coupling strength g and the average coupling strength hgi for different noise intensity r. (g is the average value of fixed synaptic coupling strength of the small-world neuronal network.) Results are presented in Figs. 11 and 12. For s ¼ 20, it is evident that when the noise intensity is small and moderate, there is a peak for Q by an optimal value of hgi, where the phenomenon of stochastic resonance happens (Figs. 12(a) and 12(b)). Remarkably, for r ¼ 0:08 (Fig. 11(b)), the resonance of the adaptive coupling can reach a much larger value than that of fixed coupling. This phenomenon may be interpreted as that a fraction of synapses with large strength (g 0:9 gmax ) emerge in STDP modification process (all synapses are initiated as g ¼ 0:5 gmax ), which will enhance the excitability of neuronal network. This may be the explanation of the spike-time-dependent plasticity-improved networked
033125-8
Yu et al.
Chaos 24, 033125 (2014)
FIG. 8. [(a)–(c)] Percentage of synapses at three levels: g 0:1 gmax , g 0:9 gmax , and the others for r ¼ 0:08, r ¼ 0:15, and r ¼ 0:2, respectively, Aþ ¼ 0:05, s ¼ 20, fA ¼ 1:05, and fs ¼ 1. [(d)–(f)] Percentage of synapses at three levels: g 0:1 gmax , g 0:9 gmax , and the others for Aþ ¼ 0:01, Aþ ¼ 0:05, and Aþ ¼ 0:1, respectively, r ¼ 0:06, s ¼ 20, fA ¼ 1:05, and fs ¼ 1. Other parameter: p ¼ 0:3. All coupling strengths are initiated as gij ¼ gmax =2, and during the evolution of time, a part of synapses with weak (gij 0:1 gmax ) and strong (gij 0:9 gmax ) coupling strength emerges, and the fraction of each strength level depends largely on r and Aþ .
stochastic resonance. Similar results can also be obtained for small temporal window s ¼ 5, but both with weak and moderate noise intensity (Figs. 12(a) and 12(b)). While for a
large noise intensity, the resonance effect of STDP is not obvious, as shown in Figs. 11(c) and 12(c). The reason may be that the noise is too large to overwhelm the weak periodic
033125-9
Yu et al.
Chaos 24, 033125 (2014)
FIG. 9. (a) Dependence of hgi on the adjusting ratio fA for different Aþ , s ¼ 20, and fs ¼ 1. (b) Dependence of hgi on the temporal window ratio fs for different Aþ , fA ¼ 1:05. (sþ ¼ 20 for fs 1 and s ¼ 20 for fs > 1). Other parameters are: r ¼ 0:1 and p ¼ 0:3. Large adjusting ratio and small temporal window ratio result in small average coupling strength.
FIG. 10. Dependence of Q on the average coupling strength hgi for different values of r. s ¼ 20, fA ¼ 1:05, fs ¼ 1, p ¼ 0:3, and Aþ ¼ 103 –100:5 . Fine-tuning of the average coupling strength can further enhance the stochastic resonance in small-world neuronal networks.
signal. Thus, we can get a conclusion that STDP in the coupling process can largely enhance the efficiency of stochastic resonance in small-world neuronal networks, which depends greatly on the temporal window of STDP. Up to now, a series of biological works have demonstrated the existence of STDP, which commonly occurs at excitatory synapses onto neocortical54,55 and hippocampal pyramidal neurons,36,56 excitatory neurons in auditory brainstem,57 parvalbumin-expressing fast-spiking striatal interneurons,58 etc. Moreover, stochastic-resonance-type effects have been demonstrated in a range of sensory systems. These include crayfish mechanoreceptors,59 shark multimodal sensory cells,15 cricket cercal sensory neurons,13 and human muscle spindles.60 It has been shown that the ability of sensory neurons to transmit and process weak input signals can be significantly enhanced via SR. As is well known, the coupling strengths between neurons play a crucial role by the stochastic resonance of the network.35,61 However, the fundamental role of STDP in stochastic resonance has been uncovered. In this work, we further suggest that the selective refinement is a possible explanation when the network is adaptive through STDP. Compared to fixed coupling,
FIG. 11. Dependence of Q on the fixed coupling strength g and the average coupling strength hgi. (a) r ¼ 0:03, (b) r ¼ 0:08, and (c) r ¼ 0:15. Other parameters are: s ¼ 20 and p ¼ 0:3. The phenomenon of SR due to mean coupling strength occurs for small and moderate noise intensity. For large temporal window, the resonance of the adaptive coupling can reach a much larger value than that of fixed coupling within an intermediate noisy background.
033125-10
Yu et al.
Chaos 24, 033125 (2014)
FIG. 12. Dependence of Q on the fixed coupling strength g and the average coupling strength hgi. (a) r ¼ 0:03, (b) r ¼ 0:08, and (c) r ¼ 0:15. Other parameters are: s ¼ 5 and p ¼ 0:3. For small temporal window, the resonance of the adaptive coupling can reach a much larger value within both weak and intermediate noisy background.
the adaptive coupling is inhomogeneous, i.e., a fraction of synapses are pushed to strong coupling, whereas a part of weak coupling emerges. This heterogeneity of synaptic coupling can selectively excite neurons in the network to respond to the subthreshold forcing with spikes. It was shown that this selective refinement of synaptic process can enhance weak signal detection by inducing stochastic resonance on neuronal networks. V. CONCLUSIONS
In this paper, we study stochastic resonance in NewmanWatts small-world neuronal networks with spike-time-dependent plasticity. The effects of fixed and adaptive couplings on stochastic resonance of the neural system are also discussed. The obtained numerical results show that an intermediate intensity of additive noise is able to optimize the temporal response of the neural system to the subthreshold periodic signal, where the phenomenon of stochastic resonance occurs. It is demonstrated that STDP in coupling process can largely enhance the efficiency of network stochastic resonance. Particularly, the resonance for adaptive coupling can reach a much larger value than that for fixed coupling when the noise intensity is small or intermediate. STDP with dominant depression and small temporal window ratio is more efficient for the transmission of weak external signal in small-world neuronal networks. In addition, the effect of stochastic resonance can be further improved via fine-tuning of the average coupling strength of small-world networks. There exists an optimal average coupling strength modulated by STDP, which can always induce dynamical resonance of the neural system. More importantly, it is found that the smallworld topology has a significant impact on the stochastic resonance in neuronal networks. There is an optimal probability of adding links, warranting the largest peak value of system resonance. In a word, both STDP and connectivity structure play an important role in the stochastic resonance on smallworld neuronal networks, determining the ability to enhance the transmission of weak periodic signal. ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (Grant Nos. 61072012 and 61302002)
and Tianjin Research Program of Application Foundation and Advanced Technology (No. 14JCQNJC01200). 1
P. H€anggi and R. Bartussek, in Nonlinear Physics of Complex Systems, edited by J. Parisi, S. C. M€ uller, and W. Zimmermman (Springer, New York, 1999). 2 M. Perc, Phys. Rev. E 72, 016207 (2005). 3 R. Benzi, A. Sutera, and A. Vulpiani, J. Phys. A: Math. Gen. 14, L453 (1981). 4 L. Gammaitoni, P. H€anggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998). 5 P. Jung and G. Mayer-Kress, Phys. Rev. Lett. 74, 2130 (1995). 6 S. Fauve and F. Heslot, Phys. Lett. A 97, 5 (1983). 7 B. McNamara, K. Wiesenfeld, and R. Roy, Phys. Rev. Lett. 60, 2626 (1988). 8 R. Bates, O. Blyuss, and A. Zaikin, Phys. Rev. E 89, 032716 (2014). 9 J. Paulsson and M. Ehrenberg, Phys. Rev. Lett. 84, 5447 (2000). 10 G. Mato, Phys. Rev. E 58, 876 (1998). 11 B. J. Gluckman, T. I. Netoff, E. J. Neel, W. L. Ditto, M. L. Spano, and S. J. Schiff, Phys. Rev. Lett. 77, 4098 (1996). 12 J. K. Douglass, L. Wilkens, E. Pantazelou, and F. Moss, Nature (London) 365, 337 (1993). 13 J. E. Levin and J. P. Miller, Nature (London) 380, 165 (1996). 14 A. Logtin, A. Bulsara, and F. Moss, Phys. Rev. Lett. 67, 656 (1991). 15 H. A. Braun, H. Wissing, K. Schafer, and M. C. Hirsch, Nature (London) 367, 270 (1994). 16 J. F. Lindner, S. Chandramouli, A. R. Bulsara, M. Locher, and W. L. Ditto, Phys. Rev. Lett. 81, 5048 (1998). 17 X. Li, J. Wang, and W. Hu, Phys. Rev. E 76, 041902 (2007). 18 N. G. Stocks and R. Mannella, Phys. Rev. E 64, 030902 (2001). 19 M. Perc, Phys. Rev. E 78, 036105 (2008). 20 Q. Y. Wang, M. Perc, Z. S. Duan, and G. R. Chen, Chaos 19, 023112 (2009). 21 M. Perc and M. Gosak, New J. Phys. 10, 053008 (2008). 22 Q. Y. Wang, H. H. Zhang, and G. R. Chen, Chaos 22, 043123 (2012). 23 E. Yilmaz, M. Uzuntarla, M. Ozer, and M. Perc, Physica A 392, 5735 (2013). 24 D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998). 25 E. Bullmore and O. Sporns, Nat. Rev. Neurosci. 10, 186 (2009). 26 S. Achard, R. Salvador, B. Whitcher, J. Suckling, and E. Bullmore, J. Neurosci. 26, 63 (2006). 27 O. Sporns, D. Chialvo, M. Kaiser, and C. C. Hilgetag, Trends Cognit. Sci. 8, 418 (2004). 28 D. S. Bassett and E. T. Bullmore, Neuroscientist 12, 512 (2006). 29 Q. Y. Wang, Z. S. Duan, M. Perc, and G. R. Chen, Europhys. Lett. 83, 50008 (2008). 30 I. D. Steenbuck, M. Butz, M. Ruiter, and A. van Ooyen, BMC Neurosci. 12, P357 (2011). 31 M. Perc, Chaos, Solitons Fractals 31, 280 (2007). 32 X. J. Sun, M. Perc, Q. S. Lu, and J. Kurths, Chaos 18, 023102 (2008). 33 M. Perc, New J. Phys. 7, 252 (2005). 34 Y. Gong, M. Wang, Z. Zhou, and H. Xin, ChemPhysChem 6, 1042 (2005). 35 M. Ozer, M. Perc, and M. Uzuntarla, Phys. Lett. A 373, 964 (2009). 36 G. Bi and M. Poo, J. Neurosci. 18, 10464 (1998).
033125-11 37
Yu et al.
L. I. Zhang, H. W. Tao, C. E. Holt, W. A. Harris, and M. M. Poo, Nature 395, 37 (1998). 38 D. E. Feldman, Neuron 27, 45 (2000). 39 D. Debanne, B. H. Gahwiler, and S. M. Thompson, J. Physiol. 507, 237 (1998). 40 C. W. Shin and S. Kim, Phys. Rev. E 74, 045101 (2006). 41 S. Song, K. D. Miller, and L. F. Abbott, Nat. Neurosci. 3, 919 (2000). 42 K. Mikkelsen, A. Imparota, and A. Torcini, Phys. Rev. Lett. 110, 208101 (2013). 43 M. Bayati and A. Valizadeh, Phys. Rev. E 86, 011925 (2012). 44 C. B. Picallo and H. Riecke, Phys. Rev. E 83, 036206 (2011). 45 Q. S. Ren, K. M. Kolwankar, A. Samal, and J. Jost, Phys. Rev. E 86, 056103 (2012). 46 K. Kube, A. Herzog, B. Michaelis, A. D. de Lima, and T. Voigt, Neurocomputing 71, 1694 (2008). 47 X. Li, J. Zhang, and M. Small, Chaos 19, 013126 (2009). 48 Q. S. Ren, Z. Q. Zhang, and J. Y. Zhao, Phys. Rev. E 85, 022901 (2012). 49 X. Li and M. Small, Chaos 22, 023104 (2012).
Chaos 24, 033125 (2014) 50
R. G€ utig, R. Aharonov, S. Rotter, and H. Sompolinsky, J. Neurosci. 23, 3697 (2003). H. Markram, J. Lubke, M. Frotscher, and B. Sakmann, Science 275, 213 (1997). 52 M. E. J. Newman and D. J. Watts, Phys. Lett. A 263, 341 (1999). 53 D. Wu, S. Q. Zhu, X. Q. Luo, and L. Wu, Phys. Rev. E 84, 021102 (2011). 54 D. E. Feldman and M. Brecht, Science 310, 810 (2005). 55 T. Nevian and B. Sakmann, J. Neurosci. 26, 11001 (2006). 56 M. Nishiyama, K. Hong, K. Mikoshiba, M. M. Poo, and K. Kato, Nature 408, 584 (2000). 57 T. Tzounopoulos, M. E. Rubio, J. E. Keen, and L. O. Trussell, Neuron 54, 291 (2007). 58 E. Fino, V. Paille, J. M. Deniau, and L. Venance, Neuroscience 160, 744 (2009). 59 K. Wiesenfeld and F. Moss, Nature 373, 33 (1995). 60 P. Cordo, J. T. Inglis, S. Verschueren, J. J. Collins, D. M. Merfeld, S. Rosenblum, S. Buckley, and F. Moss, Nature 383, 769 (1996). 61 M. Perc, Phys. Rev. E 76, 066203 (2007). 51
