Stochastic resonance enhancement of small-world neural networks by hybrid synapses and time delay

Commun Nonlinear Sci Numer Simulat 42 (2017) 532–544

Contents lists available at ScienceDirect

Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns

Research paper

Stochastic resonance enhancement of small-world neural networks by hybrid synapses and time delay Haitao Yu∗, Xinmeng Guo, Jiang Wang School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, PR China

a r t i c l e

i n f o

Article history: Received 9 October 2015 Revised 15 January 2016 Accepted 13 June 2016 Available online 15 June 2016 Keywords: Stochastic resonance Hybrid synapses Time delay Small-world network

a b s t r a c t The synergistic effect of hybrid electrical-chemical synapses and information transmission delay on the stochastic response behavior in small-world neuronal networks is investigated. Numerical results show that, the stochastic response behavior can be regulated by moderate noise intensity to track the rhythm of subthreshold pacemaker, indicating the occurrence of stochastic resonance (SR) in the considered neural system. Inheriting the characteristics of two types of synapses—electrical and chemical ones, neural networks with hybrid electrical-chemical synapses are of great improvement in neuron communication. Particularly, chemical synapses are conducive to increase the network detectability by lowering the resonance noise intensity, while the information is better transmitted through the networks via electrical coupling. Moreover, time delay is able to enhance or destroy the periodic stochastic response behavior intermittently. In the time-delayed small-world neuronal networks, the introduction of electrical synapses can significantly improve the signal detection capability by widening the range of optimal noise intensity for the subthreshold signal, and the efficiency of SR is largely amplified in the case of pure chemical couplings. In addition, the stochastic response behavior is also profoundly influenced by the network topology. Increasing the rewiring probability in pure chemically coupled networks can always enhance the effect of SR, which is slightly influenced by information transmission delay. On the other hand, the capacity of information communication is robust to the network topology within the time-delayed neuronal systems including electrical couplings. © 2016 Elsevier B.V. All rights reserved.

1. Introduction The dynamical behavior of large-scale neuronal networks is closely related to brain functions [1–3]. Neural information processing in the brain is based on the coordinated interactions of large numbers of neurons within different brain areas. Although still a subject of research, the phenomenon of stochastic resonance (SR) provides new insights into the mechanisms of information processing in the brain [4–6]. Stochastic resonance is a process by which the ability of threshold-like systems to detect and transmit weak (periodic) signals can be enhanced by the presence of a certain level of noise [7–9]. Up to now, SR has been observed, quantified, and described in a series of realistic systems, especially in neural systems. It is shown that, SR in the central nervous system of mammalians may account for the higher brain functions, such as human tactile sensation, visual perception, and animal feeding behavior [10–12]. Thus, understanding potential benefits of SR in information processing of nervous systems is of great importance. On the other hand, the past decades have been growing ∗

Corresponding author. E-mail address: [email protected] (H. Yu).

http://dx.doi.org/10.1016/j.cnsns.2016.06.021 1007-5704/© 2016 Elsevier B.V. All rights reserved.

H. Yu et al. / Commun Nonlinear Sci Numer Simulat 42 (2017) 532–544

533

Fig. 1. Spatiotemporal patterns of the hybrid small-world neuronal networks obtained from different noise intensity σ with f = 0.5. (a) σ = 0.015, (b) σ = 0.02, (c) σ = 0.026, (d) σ = 0.03, (e)σ = 0.04.(τ = 0).

Fig. 2. (a) Dependence of Q on the noise intensity σ for different probability of chemical synapse. (b) Dependence of Q on the noise intensityσ and the chemical probability f.

body of modeling work on stochastic resonance in complex neuronal networks, which demonstrates that both the network property and coupling strength have a direct effect on SR in complex neuronal networks [13–18]. Particularly, Perc studied the SR in excitable small-world neuronal networks via a pacemaker, and found that the small-world property is able to enhance the stochastic resonance only for intermediate coupling strengths [19]. In neural systems, neurons contact with each other through two different types of synapses, electrical and chemical ones [20]. For electrical synapses, the coupling acts through gap junction, where the strength depends linearly on the difference between the membrane potentials [21,22]. That is to say, the electrical coupling can work as long as this potential difference exists. While in the chemical case, the synapse is mediated by neurotransmitters and the connection occurs between the dendrites and the axons. The chemical coupling emerges once the presynaptic neuron spikes and its strength decays exponentially afterwards [23,24]. Recently, since the evidences showed that electrical synapses and chemical ones can coexist in the neural system [25–30], the fundamental role of these two types of synapses on network dynamics has sparked increas-

534

H. Yu et al. / Commun Nonlinear Sci Numer Simulat 42 (2017) 532–544

Fig. 3. Spatiotemporal patterns of the hybrid small-world neuronal networks obtained from different time delay τ with f = 0.5. (a) τ = 0, (b) τ = 350, (c) τ = 700, (d) τ = 1050, (e)τ = 1400, (f) τ = 1750. Noise intensity σ = 0.026.

ing interests [31–35]. For example, in the study of coupled neurons with Gaussian noise, it is demonstrated that chemical coupling is more sensitive than electrical coupling for the local signal input [36]. It is also shown that the topology of hybrid scale-free network can significantly influence the occurrence of SR and an optimal topology is founded for the amplification of the response to the weak input signal [37]. As an extension of SR, the effect of different synapses on the vibrational resonance (VR) in neuron populations is investigated, which found that electrical synaptic coupling is better in signal transmission than chemical coupling [38]. In addition, the researches into synchrony in neuronal networks found that these two types of synapses can perform complementary synchronization roles [39–41]. For instance, both coupling modes of delayed inhibitory and fast electrical synapses play a crucial role in the synchronous behavior of interneuronal networks [42]. Furthermore, due to the finite speed of action potential propagating across neuron axons and time lapses occurring by both dendritic and synaptic processes, information transmission delay is an intrinsic feature of neural systems, which is in the range of several tens of milliseconds [43]. In recent years, a series of studies show that the collective dynamics of neuronal ensembles can be largely affected by time delay [44–47]. Specially, the effects of information transmission delays on the synchronization in complex neuronal networks have been broadly investigated by Wang and his colleagues, who demonstrate that fined-tuned information transmission delays are vital for assuring optimally synchronized excitatory fronts on complex neuronal networks [42,48,49]. Moreover, stochastic resonance, as an indispensable part of neural dynamics, has also been widely investigated in delay-induced systems [50,51]. Yu et al. in their research on SR in small-world with time delay, indicate that, multiple stochastic resonance appears intermittently at integer multiples of the oscillation periods of the pacemaker, and increasing the rewiring probability can largely enhance the efficiency of pacemaker-driven stochastic resonance for small time delays [52]. In the present work, the study of stochastic resonance is extended into the time-delayed hybrid electrical-chemical coupling small-world neuronal networks. We aim to find out the synergistic effect of hybrid synapses and time delay on the phenomenon of SR. Moreover, the influence of network structure on SR in the considered neuronal networks is also explored. The remainder of this paper is governed as follows: in Sec. II, the mathematic model of considered neuronal networks is introduced. In Sec. III, we will discuss the dependence of SR on the parameters, such as hybrid electrical-chemical synapses, transmission delay and network structure. Finally, a brief conclusion and discussion is drawn in Sec. IV.

H. Yu et al. / Commun Nonlinear Sci Numer Simulat 42 (2017) 532–544

535

Fig. 4. Dependence of Q on the time delay τ for different values of noise intensity σ . (a) f = 0, (b) f = 0.5, (c) f = 1.

2. Mathematical model and methods Based on Watts–Strogatz procedure [53], the small-world network, comprising N = 200 neurons, is initiated as a regular ring with K = 6 nearest neighbors. We randomly rewire each edge in the network with the probability p. The small-world network is obtained in the intermediate situation (0 < p < 1), while in the extreme case the network is regular ( p = 0) or completely random ( p = 1). Furthermore, from pure electrical synapses by changing electrical synapses to chemical ones with the probability f, a hybrid synaptic small-world neuronal network is established. For the sake of computational simplification for large neural networks, a two-dimensional Rulkov map is employed to describe the individual neuronal dynamics. The Rulkov map is a two-dimensional iterated map used to model a biological neuron [54, 55]. Compared with conductance-based models, such as Hodgkin–Huxley model [56], the Rulkov map-based model is low-dimension, but can capture the main dynamical features of real neurons [57]. It is often taken as a minimal model for the bursting generation and more efficient for dynamical simulation than a continuous neuronal model. According to Rulkov [54], the temporal evolution of each unit can be described by the following set of discrete equations:

xi ( n + 1 ) =

α 1 + x2i (n )

+ yi (n ) + Iisyn (n ) + σ ξi (n ),

yi ( n + 1 ) = yi ( n ) − β xi ( n ) − γ ,

(1)

(2)

where i = 1, 2, . . . , N. xi (n) is the fast dynamical variable representing the neuronal membrane potential, and yi (n) is the slow variable denoting the slow gating process, which is due to the small values of parameters β = γ = 0.001. n is the discrete time index. The dynamics of a neuron largely depends on the value of parameter α : for α < 2.0, all neurons are excitable; whereas for α > 2.0, these neurons can exhibit spikes or bursts. In this work, we choose α = 1.95 so that all neurons are excitable. Each neuron is initiated from fixed point (x∗ = −1, y∗ = −1 − (α /2 ) = −1.995). ξ i (n) is Gaussian

536

H. Yu et al. / Commun Nonlinear Sci Numer Simulat 42 (2017) 532–544

Fig. 5. Dependence of Q on the noise intensity σ and the time delay τ for (a) f = 0, (b) f = 0.5, (c) f = 1.

noise with zero mean and intensity σ . In neural systems, noise is universal, mainly inherently generated from the random processes in ion channels and synapses. Markov chains provide realistic models for these stochastic processes [58,59]. It is demonstrated that the channel noise shows a Poisson distribution and its effect depends on the number of ion channels [60,61]. As channel number turns larger, the channel noise can be simply modeled as Gaussian noise. The synaptic current syn syn syn syn Ii (n ) comprises two parts: the electrical and chemical ones, i.e., Ii (n ) = Ii,e (n ) + Ii,c (n ). In the case of electrical coupling, syn Ii,e ( n ) = ge

N 





Ce (i, j ) x j (n − τ ) − xi (n ) ,

(3)

j=1, j=i

where ge stands for the electrical coupling strength. Ce is the electrical connectivity matrix: if neuron i couples to neuron j via an electrical synapse, then Ce (i, j ) = Ce ( j, i ) = 1, otherwise Ce (i, j ) = Ce ( j, i ) = 0, and Ce (i, i ) = 0. τ is the transmission delay between connected neurons, which is one of the main parameters discussed below. While for the chemical case, syn Ii,c ( n ) = gc ( xi ( n ) − v )

N 





Cc (i, j ) x j (n ) ,

(4)

j=1, j=i

where gc is the chemical coupling strength. v denotes the reversal potential associated with chemical synapses. Its value determines the types of chemical synapses. The synapse will be excitatory if v is higher and inhibitory if v is lower than the range of neuronal membrane potential xi, n . In the Rulkov map-based model, the range of membrane potential is between -1.5 to 0. Here, we set v = 0.5 guaranteeing that all the chemical synapses are excitatory. Cc is the chemical coupling matrix: if neuron i is coupled to neuron j through a chemical synapse, then Cc (i, j ) = 1, otherwise Cc (i, j ) = 0, and Cc (i, i ) = 0.

H. Yu et al. / Commun Nonlinear Sci Numer Simulat 42 (2017) 532–544

537

Additionally, the chemical synaptic coupling function is defined as follows [49]

       x j (n ) = 1/ 1 + exp −λ x j (n − τ ) − s ,

(5)

where the parameter λ = 30 represents the constant rate for the onset of excitation. s = −1 is regarded as the threshold. The pre-synaptic neuron is able to affect the post-synaptic one once the membrane potential of presynaptic one exceeds the threshold. To quantitatively characterize the response of the small-world neuronal network, the average membrane potential X (n ) = (1/N ) N i=1 xi (n ) is utilized as the main output of the neuronal population. The correlation of the average membrane potential X(n) with the frequency of the pacemaker ω = 2π /t is computed via the Fourier coefficients according to [19]

Qsin =

Tt 2  X (n ) sin (ωn ), Tt

(6)

Tt 2  X (n ) cos (ωn ), Tt

(7)

2 + Q2 , Qsin cos

(8)

n=1

Qcos =

Q=



n=1

where T = 70 is the number of periods of the pacemaker used in the simulation. The final results are obtained by averaged over 20 independent runs. The Fourier coefficient is proportional to the square of the spectral power amplification, which is usually used as a measure for stochastic resonance [62]. In this work, we use Fourier coefficient as a stochastic resonance factor as in Refs. [13–15,19,37], since we are more interested in the transfer of information encoded in the frequency ω. The maximum of Q shows the optimal correlation between input signal and neural firing. In fact, although Fourier analysis is linear, the method is often used in data analysis of neural systems. For example, Fourier analysis is taken as a standard procedure for computing the power spectrum of EEG [63]. 3. Results In this section, the synergistic effect of hybrid electrical-chemical synapses and information transmission delay on stochastic resonance in small-world neuronal networks is investigated. Within the obtained network, the subthreshold periodic signal, which is a pulse train with the oscillation period of t =700, is imposed on a random neuron as a pacemaker. The width and amplitude of each pulse is 50 and 0.0015, respectively. The rewiring probability of the neural network is initially set to be p = 0.1. The coupling strength of electrical and chemical synapses is assumed to be equaled, i.e., ge = gc = 0.003, and the probability of chemical synapse f is initiated as f = 0.5. The spatiotemporal patterns of spiking activity in hybrid small-world neuronal networks for different noise intensity σ are presented in Fig. 1. It is shown that, at low noise levels, the neural system spikes sparsely to respond to the subthreshold input and few signals are detected. When the noise intensity is intermediate, the outreach of the considered network is ordered and follows the rhythm of the pacemaker. However, as the noise level keeps increasing, this visible regularity is broken by the noise. This phenomenon indicates that appropriate level of random fluctuation can regulate the stochastic response behavior of small-world neuronal networks and improve the ability of weak signal detection. In order to obtain a quantitative inspect on the fundamental role of noise in the networked dynamics, we calculate Fourier coefficient Q on the dependence of noise intensity (Fig. 2(a)). Irrespective of probability of chemical synapse f, a peak value of Q emerges when the noise intensity is moderate, indicating the existence of stochastic resonance in small-world neuronal networks with hybrid electrical-chemical synapses. Obviously, the peak point Qmax decreases and slightly moves to smaller noise intensity as more chemical synapses are added into the network. It is suggested that electrical synaptic coupling is better in signal transmission, while chemical coupling is more sensitive in signal detection. This phenomenon is presented detailed in the two-dimensional space (σ , f) for an overall insight into the combined effect of noise intensity and synapse type on SR (Fig. 2(b)). The different mechanisms between electrical and chemical coupling may be a possible interpretation for the transitions above. For electrically coupled neurons, the coupling functions through membrane potential all the time; whereas chemically coupled neurons connect to each other only when pre-synaptic one fires. In the case of chemical synapses, due to the selective coupling, neurons are given more chance to detect and response to the weak signal with the help of stochastic noise. While for electrical synapses, the enhanced correlations among neurons may promote the information transmission through the networks. As mentioned in a large number of studies that, information transmission delay is an inherent factor in the nervous system, which can significantly affect the dynamical behavior of neural systems. Therefore, we extend our investigation into the time-delayed hybrid neuronal networks. From the space-time plots of the considered network (Fig. 3), the information transmission delay indeed can largely affect the stochastic response behavior by enhancing or destroying the ordered periodic response of neuronal excitation intermittently. To quantize the effect of time delay, we show the dependence of Q on τ for different noise intensity in Fig. 4. Multiple stochastic resonances apparently occur due to the change of time delay. When

538

H. Yu et al. / Commun Nonlinear Sci Numer Simulat 42 (2017) 532–544

Fig. 6. Dependence of Q on the noise intensityσ for different values of time delay τ . (a) f = 0, (b) f = 0.5, (c) f = 1.

the network is only composed of electrical synapses, all the resonance points mainly locate at the identical time delay for different σ . However, as the chemical synaptic couplings are added into the network, the peaks move rightward with the enlargement of random fluctuation. And this difference is amplified by large f. In addition, the combined effect of noise and time delay is examined. There exists some narrow-banded regions with high values of Q, confirming the emergence of multiple stochastic resonances in the time-delayed hybrid small-world neuronal networks (shown in Fig. 5). For pure electrically coupled network (Fig. 5(a)), the banded resonance regions are horizontal for small and moderate noise intensity, whereas these regions slightly incline upward with the further increase of random disturbance. On the contrary, the resonance regions for pure chemical coupling only locate near intermediate noise intensity without plateau, as depicted in Fig. 5(c). Interestingly, the hybrid scheme synthesizes the dynamic features of both synaptic couplings, for the transition is similar with pure electrical ones but the slope is much steeper than it. Time delay not only evokes the multiple stochastic resonances in the considered networks, but also influences the single SR. Particularly, the phenomenon of stochastic resonance indeed exists for particular values of time delay (τ = 0, τ = 700 and τ = 1400, presented in Fig. 6). The spans of noise intensities that warrant relatively high values of Q largely extend as τ increases. Compared with these cases with different fraction of chemical synapses, it can be seen that, the extension of optimal noise intensities induced by time delay is more significant in the networks with electrical synapses, while the variation of this span is less obvious in the networks only consisting of chemical coupling. It is indicated that, with the aid of time delay, the signal detection capability of small-world neural networks is largely improved by the introduction of electrical synapses. On the other hand, in the time-delayed neural systems, the chemical synaptic coupling can considerably promote the transmission of periodic signal, for the increment of peak value of Q in Fig. 6(c) is more profound than that presented in Fig. 6(a) and (b). The network topology is also one of the critical factors influencing the stochastic response behavior of neural systems. We first inspect the effect of network structure on SR without information transmission delay. It is shown that, the peak value of Q increases with the enlargement of p (presented in Fig. 7), indicating that the increase of rewiring probability

H. Yu et al. / Commun Nonlinear Sci Numer Simulat 42 (2017) 532–544

539

Fig. 7. Dependence of Q on the noise intensity σ for different values of rewiring probability p. (a) f = 0, (b) f = 0.5, (c) f = 1.

can always promote periodic signal transmission among neuronal networks. On the other hand, as p increases, the optimal resonance noise intensity for chemically coupled networks declines more significantly than the other two cases (shown in Fig. 7(a) and (b)). That is to say, the detectability of nervous system with chemical synapses is considerably amplified by random rewiring probability. In summary, increasing the rewiring probability can significantly enhance the detection and transmission of subthreshold input. The above suggests that the topology of neural systems also has an effect on stochastic resonance. Indeed, Fig. 8 reveals a substantial influence of rewiring probability p and confirms the synergistic effect of hybrid synapses and time delay in the networks. When τ = 0 (Fig. 8(a)), the rewiring probability can facilitate the propagation of weak period signal, and this enhancement is amplified by adding chemical synapses into neuronal networks. However, this transition changes significantly in the time-delayed small-world neuronal networks. As shown in Fig. 8(b) and (c), due to the interaction between hybrid synapses and time delay, it has a strong robustness of network structure in the time-delayed hybrid neural systems, as the network with electrical coupling can maintain a relative steady high level of Q along with p. On the other hand, compared with the curves of f = 1 for different values of τ , it is found that, the tendency respected to rewiring probability varies slightly with time delay. A conclusion may be drawn that, increasing rewiring probability can always enhance the efficiency of stochastic resonance in the pure chemically coupled networks, and this effect is less sensitive to information transmission delay. Additionally, the capacity of information propagation is robust to the network topology within the time-delayed neuronal systems including electrical couplings. To further check the combined effect of hybrid synapses and time delay on stochastic resonance, we calculate the dependence of Q on both noise intensity σ and rewiring probability p in Fig. 9. Irrespective of the chemical probability, a banded region with high value of Q emerges, indicating that the phenomenon of SR occurs for each particular value of p when the time delay is identical to integer multiples of the oscillation periods of the pacemaker. As shown in Fig. 9(a)–(f), when neurons within the network couple with each other through electrical synapses, the spans of noise intensities that

540

H. Yu et al. / Commun Nonlinear Sci Numer Simulat 42 (2017) 532–544

Fig. 8. Dependence of Q on the rewiring probability p for different probability of chemical synapse f. (a) τ = 0, (b) τ = 700, (c) τ = 1400.

warrant relatively high values of Q largely extend as τ increases. Compared with the top and middle panel of Fig. 9, for each particular value of time delay, it is obvious that the extension of optimal noise region is reduced as more chemical synapses are added into the network. Furthermore, the enlargement of optimal noise intensity is less evident in the network only coupled through chemical synapses, which confirms again that the effect of chemical synapses on networked SR is less sensitive to time delay. Finally, Fig. 10 depicts the dependence of Q on the rewiring probability and fraction of chemical synapses for different time delay, respectively. Irrespective of the time delay, there exists a tongue-like region with high value of Q, warranting that the networked response can be optimized by appropriate value of f. It is indicated that, inheriting the characteristics of two types of synapses-electrical and chemical ones, the hybrid neuronal networks have much superiority in signal processing over pure ones. Interestingly, this optimization only occurs for intermediate and large random rewiring probability in the case of τ = 0. On the contrary, when the time delay is an integral number of pacemaker driven period (i.e., τ = 700 and τ = 1400), the collective response of neuronal networks is able to be maximized by moderate value of f for each particular value of p. A main conclusion is therefore that neural networks with hybrid electrical-chemical synapses are of great improvement in signal processing, and this improvement is extended to a wide range of small-world topology by appropriate delay in the information transmission. 4. Conclusion and discussion Recently, effects of neuron property on the neuronal dynamics are focus issues in neuroscience. A series of works have investigated the effect of synapse types, time delay and noise on the dynamic behavior of neuronal systems, including stochastic resonance [13–19,37], vibrational resonance [38] and synchronization [39–42,45–49]. In the previous studies of SR (and VR) in neural systems, only one of those features is to be considered. For example, a bunch of reports demonstrate the fundamental role of time delay in neuronal dynamics by introducing multiple resonances [13,52]. It is also indicated

H. Yu et al. / Commun Nonlinear Sci Numer Simulat 42 (2017) 532–544

541

Fig. 9. Dependence of Q on the rewiring probability p and the noise intensity σ for different probability of chemical synapse f. (a) τ = 0, f = 0; (b) τ = 700, f = 0; (c) τ = 1400, f = 0; (d) τ = 0, f = 0.5; (e) τ = 700, f = 0.5; (f) τ = 1400, f = 0.5; (g) τ = 0, f = 1; (h) τ = 700, f = 1; (i) τ = 1400, f = 1.

the different effect of synapse type on SR and VR, for electrical synaptic coupling it indeed promotes the signal propagation while chemical one enhance the detectability of external input [38,64]. However, more importantly, most of the investigations above ignore the synergistic effect of time delay and hybrid synapses. Therefore, we introduce the time delay into the hybrid complex neuronal networks to investigate the fundamental role of hybrid synapses and information transmission delay in stochastic resonance. Numerical results show that, the stochastic response behavior can be modulated by an appropriate level of random noise to keep accordance with rhythm of external pacemaker, indicating the occurrence of stochastic resonance in the hybrid small-world neuronal networks. It is also suggested that, due to the selective connection of chemical synapses, neurons are given more chance to spike, which will lead to the increment of system detectability by lowering the noise intensity for the emergence of SR. While for electrical coupling, the enhanced correlations among neurons are able to promote the information transmission through the networks. Moreover, it is also shown that the information transmission delay can largely effect on the stochastic response behavior by enhancing or destroying the ordered periodic response of neuronal excitation intermittently. The obtained results above are agreed with the previous studies [13,52]. Additionally, when considering the synergistic effect of time delay and hybrid synapses, some intriguing phenomena occur. In the time-delayed small-world neuronal networks, the introduction of electrical synapses can significantly improve the signal detection capability by widening the range of optimal noise intensity for stochastic resonance, and the efficiency of SR is evidently amplified in the case with pure chemical couplings. Moreover, the network topology is also one of critical factors influencing the synergistic effect of hybrid synapses and time delay on stochastic resonance of neural systems. Inheriting the characteristics of two types of synapses—electrical and chemical ones, neural networks with hybrid electricalchemical synapses are of great improvement in neuron communication, and this improvement is extended to a wide range of small-world topology by appropriate delay in the information transmission. Particularly, the efficiency of SR can always

542

H. Yu et al. / Commun Nonlinear Sci Numer Simulat 42 (2017) 532–544

Fig. 10. Dependence of Q on the rewiring probability p and the probability of chemical synapsef for different values of time delay τ . (a) τ = 0, (b) τ = 700, (c) τ = 1400.

be enhanced by increasing rewiring probability in pure chemically coupled networks, and it is less sensitive to information transmission delay. However, the capacity of information communication is robust to the network topology within the time-delayed neuronal systems including electrical couplings. The observations presented above demonstrate that, all the features of hybrid synapses, information transmission delay and network topology play critically important roles in the stochastic resonance within the small-world neuronal networks, determining the collective response behavior of neural ensemble. Since the electrical and chemical synapses can coexist in real neuronal networks, and noise and time delay are ubiquitous in nervous system, the noise-induced dynamic behaviors presented in this paper may have some biological significance. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 61302002 and 61374182), Tianjin Research Program of Application Foundation and Advanced Technology (Grant No. 14JCQNJC01200), and Tangshan Technology Research and Development Program (Grant No. 14130223B). References [1] Rabinovich MI, Varona P. Robust transient dynamics and brain functions. Front Comput Neurosci 2011;5:24. [2] Bressler SL, Menon V. Large-scale brain networks in cognition: emerging methods and principles. Trends Cogn Sci 2010;14:277–90. [3] Reches A, Laufer I, Ziv K, Cukierman G, McEvoy K, Ettinger M. Network dynamics predict improvement in working memory performance following donepezil administration in healthy young adults. NeuroImage 2014;88:228–41. [4] Moss F, Ward LM, Sannita WG. Stochastic resonance and sensory information processing: a tutorial and review of application. Clin Neurophysiol 2004;115:267–81. [5] Winterer G, Ziller M, Dorn H, Frick K, Mulert C, Dahhan N. Cortical activation, signal-to-noise ratio and stochastic resonance during information processing. Clin Neurophysiol 1999;110:1193–203. [6] Kawaguchi M, Mino H, Durand DM. Stochastic resonance can enhance information transmission in neural networks. IEEE Trans Biomed Eng 2011;58:1950–8.

H. Yu et al. / Commun Nonlinear Sci Numer Simulat 42 (2017) 532–544 [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59]

543

Benzi R, Sutera A, Vulpiani A. The mechanism of stochastic resonance. J Phys A 1981;14:L453–7. Gammaitoni L, Hänggi P, Jung P, Marchesoni F. Stochastic resonance. Rev Mod Phys 1998;70:223–87. Lindner B, Garcia-Ojalvo J, Neiman A, Schimansky-Geier L. Effects of noise in excitable systems. Phys Rep 2004;392:321–424. Lugo E, Doti R, Faubert J. Ubiquitous cross modal stochastic resonance in humans: auditory noise facilitates tactile, visual and proprioceptive sensations. PLoS One 2008;3:e2860. Richardson KA, Imhoff TT, Grigg P, Collins JJ. Using electrical noise to enhance the ability of humans to detect subthreshold mechanical cutaneous stimuli. Chaos 1998;8:599–603. Mcdonnell MD, Ward LM. The benefits of noise in neural systems: bridging theory and experiment. Nat Rev Neurosci 2011;12:415–26. Wang Q, Perc M, Duan Z, Chen G. Delay-induced multiple stochastic resonances on scale-free neuronal networks. Chaos 2009;19:023112. Ozer M, Perc M, Uzuntarla M. Stochastic resonance on Newman-Watts networks of Hodgkin-Huxley neurons with local periodic driving. Phys Lett A 2009;373:964–8. Perc M, Gosak M. Pacemaker-driven stochastic resonance on diffusive and complex networks of bistable oscillators. New J Phys 20 08;10:0530 08. Yu H, Guo X, Wang J, Deng B, Wei X. Effects of spike-time-dependent plasticity on the stochastic resonance of small-world neuronal networks. Chaos 2014;24:033125. Bates R, Blyuss O, Zaikin A. Stochastic resonance in an intracellular genetic perceptron. Phys Rev E 2014;89:032716. Balenzuela P, Rue P, Boccaletti S, Garcia-Ojalvo J. Collective stochastic coherence and synchronizability in weighted scale-free networks. New J Phys 2014;16:013036. Perc M. Stochastic resonance on excitable small-world networks via a pacemaker. Phys Rev E 2007;76:066203. Sudhof TC, Malenka RC. Understanding synapses: past, present, and future. Neuron 2008;60:469–76. Hestrin S. The strength of electrical synapses. Science 2011;334:315–16. Connors BW, Long MA. Electrical synapses in the mammalian brain. Annu Rev Neurosci 2004;27:393–418. Anderson PA. Physiology of a bidirectional, excitatory, chemical synapse. J Neurophysiol 1985;53:821–35. Greengard P. The neurobiology of slow synaptic transmission. Science 2001;294:1024–30. Taugner R, Sonnhof U, Richter DW, Schiller A. Mixed (chemical and electrical) synapses on frog spinal motoneurons. Cell Tissue Res 1978;193:41–59. Galarreta M, Hestrin S. Electrical and chemical synapses among parvalbumin fast-spiking GABAergic interneurons in adult mouse neocortex. Proc Natl Acad Sci USA 2002;99:12438–43. Friedman D, Strowbridge BW. Both electrical and chemical synapses mediate fast network oscillations in the olfactory bulb. J Neurophysiol 2003;89:2601–10. Fukuda T, Kosaka T. The dual network of GABAergic interneurons linked by both chemical and electrical synapses: a possible infrastructure of the cerebral cortex. Neurosci Res 20 0 0;38:123–30. Pereda AE. Electrical synapses and their functional interactions with chemical synapses. Nat Rev Neurosci 2014;15:250–63. Hamzei-Sichani F, Davidson KGV, Yasumura T, Janssen WGM, Wearne SL, Hof PR, . Mixed electrical-chemical synapses in adult rat hippocampus are primarily glutamatergic and coupled by connexin-36. Front Neuroanat 2012;6:13. Viana RL, Borges FS, Iarosz KC, Batista AM, Lopes SR, Caldas IL. Dynamic range in a neuron network with electrical and chemical synapses. Commun Nonlinear Sci Numer Simul 2014;19:164–72. Baptista MS, Kakmeni FMM, Grebogi C. Combined effect of chemical and electrical synapses in Hindmarsh-Rose neural networks on synchronization and the rate of information. Phys Rev E 2010;82:036203. Mamiya A, Manor Y, Nadim F. Short-term dynamics of a mixed chemical and electrical synapse in a rhythmic network. J Neurosci 2003;23:9557–64. Rela L, Szczupak L. Coactivation of motoneurons regulated by a network combing electrical and chemical synapses. J Neurosci 2003;23:682–92. Kanamaru T, Aihara K. Stochastic synchrony of chaos in a pulse-coupled neural network with both chemical and electrical synapses among inhibitory neurons. Neural Comput 2008;20:1951–72. Li X, Wang J, Hu W. Effects of chemical synapses on the enhancement of signal propagation in coupled neurons near the canard regime. Phys Rev E 2007;76:041902. Yilmaz E, Uzuntarla M, Ozer M, Perc M. Stochastic resonance in hybrid scale-free neuronal networks. Physica A 2013;392:5735–41. Yu H, Wang J, Sun J, Yu H. Effects of hybrid synapses on the vibrational resonance in small-world neuronal networks. Chaos 2012;22:033105. Yu H, Wang J, Liu C, Deng B, Wei X. Delay-induced synchronization transitions in small-world neuronal networks with hybrid electrical and chemical synapses. Physica A 2013;392:5473–80. Kopell N, Ermentrout B. Chemical and electrical synapses perform complementary roles in the synchronization of interneuronal networks. Proc Natl Acad Sci USA 2004;101:15482–7. Liu C, Wang J, Yu H, Deng B, Wei X, Tsang KM. Impact of delays on the synchronization transitions of modular neuronal networks with hybrid synapses. Chaos 2013;23:033121. Guo D, Wang Q, Perc M. Complex synchronous behavior in interneuronal networks with delayed inhibitory and fast electrical synapses. Phys Rev E 2012;85:061905. Swadlow HA. Physiological properties of individual cerebral axons studied in vivo for as long as one year. J Neurophysiol 1985;54:1346–62. Wu H, Hou Z, Xin H. Delay-enhanced spatiotemporal order in coupled neuronal systems. Chaos 2010;20:043140. Dhamala M, Jirsa VK, Ding MZ. Enhancement of neural synchrony by time delay. Phys Rev Lett 2004;92:074104. Roxin A, Brunel N, Hansel D. Role of delays in shaping spatiotemporal dynamics of neuronal activity in large networks. Phys Rev Lett 2005;94:238103. Perez T, Garcia GC, Eguiluz VM, Vicente R, Pipa G, Mirasso C. Effect of the topology and delayed interactions in neuronal networks synchronization. PloS One 2011;6:e19900. Wang Q, Perc M, Duan Z, Chen G. Synchronization transitions on scale-free neuronal networks due to finite information transmission delays. Phys Rev E 2009;80:023206. Wang Q, Chen G. Delay-induced intermittent transition of synchronization in neuronal networks with hybrid synapses. Chaos 2011;21:013123. Liu C, Wang J, Yu H, Deng B, Tsang KM, Chan WL. The effects of time delay on the stochastic resonance in feed-forward-loop neuronal network motifs. Commun Nonlinear Sci Numer Simul 2014;19:1088–96. Luo X, Wu D, Zhu S. Stochastic resonance in a time-delayed bistable system with colored coupling between noise terms. Int J Mod Phys B 2012;26:1250149. Yu H, Wang J, Du J, Deng B, Wei X, Liu C. Effects of time delay and random rewiring on the stochastic resonance in excitable small-world neuronal networks. Phys Rev E 2013;87:052917. Watts DJ, Strogatz SH. Collective dynamics of ’small-world’ networks. Nature 1998;393:440–2. Rulkov NF. Regularization of synchronized chaotic bursts. Phys Rev Lett 2001;86:183–6. Rulkov NF. Modeling of spiking-bursting neural behavior using two-dimensional map. Phys Rev E 2002;65:041922. Hodgkin A, Huxley A. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 1952;117:500–44. Rulkov NF, Timofeev I, Bazhenov M. Oscillations in large-scale cortical networks: Map-based model. J Comput Neurosci 2004;17:203–23. Groff JR, DeRemigio H, Smith GD. Markov chain models of ion channels and calcium release sites. In: Laing CL, Lord GJ, editors. Stochastic methods in neuroscience. New York: Oxford: Oxford University Press; 2010. p. 29–64. Destexhe A, Mainen ZF, Sejnowski TJ. Kinetic models of synaptic transmission. In: Koch C, Segev I, editors. Methods in neuronal modeling. Cambridge: The MIT Press; 1998. p. 1–25.

544

H. Yu et al. / Commun Nonlinear Sci Numer Simulat 42 (2017) 532–544

[60] Goldwyn JH, Imennov NS, Famulare M, Shea-Brown E. Stochastic differential equation models for ion channel noise in Hodgkin-Huxley neurons. Phys Rev E 2011;83:041908. [61] Schneidman E, Freedman B, Segev I. Ion channel stochasticity may be critical in determining the reliability and precision of spike timing. Neural Comput 1998;10:1679–703. [62] Guo D, Li C. Stochastic and coherence resonance in feed-forward-loop neuronal network motifs. Phys Rev E 2009;79:051921. [63] Tong S, Thakor NV. Quantitative EEG analysis methods and clinical applications. London: Artech House; 2009. [64] Deng B, Wang J, Wei X. Effect of chemical synapse on vibrational resonance in coupled neurons. Chaos 2009;19:013117.