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Adaptive stochastic resonance in self-organized small-world neuronal networks with time delay
Accepted Manuscript
Adaptive stochastic resonance in self-organized small-world neuronal networks with time delay Haitao Yu , Xinmeng Guo , Jiang Wang , Chen Liu , Bin Deng , Xile Wei PII: DOI: Reference:
S1007-5704(15)00190-2 10.1016/j.cnsns.2015.05.017 CNSNS 3559
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Received date: Revised date: Accepted date:
3 November 2014 16 April 2015 15 May 2015
Please cite this article as: Haitao Yu , Xinmeng Guo , Jiang Wang , Chen Liu , Bin Deng , Xile Wei , Adaptive stochastic resonance in self-organized small-world neuronal networks with time delay, Communications in Nonlinear Science and Numerical Simulation (2015), doi: 10.1016/j.cnsns.2015.05.017
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ACCEPTED MANUSCRIPT Highlights
Stochastic resonance in small-world neuronal networks is numerically studied.
Spike-time-dependent plasticity may depress the network stochastic resonance.
Effect of stochastic resonance can be either promoted or destroyed by time delay.
The small-world topology can significantly affect the stochastic resonance.
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Adaptive stochastic resonance in self-organized small-world neuronal networks with time delay Haitao Yu*, Xinmeng Guo, Jiang Wang, Chen Liu, Bin Deng, Xile Wei School of Electrical Engineering and Automation, Tianjin University, Tianjin, 300072, P. R. China.
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(*Corresponding author: [email protected], Tel: 86-22-27402293) Abstract: In this paper, adaptive stochastic resonance in time-delayed Newman-Watts small-world neuronal networks is studied, where the strength of synaptic connections between neurons is adaptively modulated by spike-timing-dependent plasticity (STDP). Numerical results show that, in
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the absence of information transmission delay, the phenomenon of stochastic resonance occurs and the efficiency of networked stochastic resonance can be slightly depressed by STDP. Due to the reduction of strong couplings induced by STDP, the larger the adjusting rate of STDP is, the smaller peak value of the resonance response obtains. In addition, the effect of stochastic resonance can be
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either promoted or destroyed by time delay, and multiple stochastic resonances appear intermittently
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at the integer multiples of periods of the subthreshold forcing. Furthermore, it is demonstrated that the networked stochastic resonance can also be dramatically affected by the small-world topology.
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For small and moderate adjusting rate of STDP, fine-tuning of the probability of adding links can significantly enhance the effect of stochastic resonance in adaptive neural network. Additionally,
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there is an optimal probability of adding links by which the noise-induced transmission of weak
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periodic signal peaks and the location of this span depends largely on the time delay and adjusting rate.
Keywords: spike-timing-dependent plasticity, time delay, adaptive stochastic resonance, neuronal network, small-world I.
INTRODUCTION Noise can emerge as a constructive component of nonlinear systems [1-3]. One important
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representative of this fact is stochastic resonance (SR), which occurs when the response of a nonlinear dynamical system to a weak periodic signal is optimized by moderate intensity of random fluctuations [4-7]. Recently, SR has sparked growing interest both in theoretical models of neural systems and in experimental neuroscience [8-10]. It is shown that the ability of sensory neurons to
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process weak input signals can be significantly enhanced by adding noise to the system [11-12]. Additionally, statistical signal processing in coupled neural systems improves with the aid of various levels of stochastic noise via SR [13, 14]. Therefore, the study of stochastic resonance is valuable for
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understanding the signal transmission and information propagation in neuronal networks.
The last decade has growing body of modelling work on stochastic resonance in complex neuronal networks, especially in scale-free networks [15, 16] and small-world neuronal networks
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[17-19]. Perc studied stochastic resonance on weakly paced scale-free networks and indicated that, all the features of the placement of the pacemaker, coupling strength and the inhomogeneous
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structure of scale-free networks play a crucial role in SR [15]. On the other hand, the effect of SR on
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excitable small-world networks can be amplified only for intermediate coupling strengths in excitable networks via pacemaker [18]. The study on stochastic resonance in Newman-Watts
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small-world also demonstrates that fine-tuning of the small-world network structure can largely
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enhance the stochastic resonance in neuronal networks [20]. Time delay, which occurs due to the finite speed of action potentials propagating across neuron
axons and finite reaction times for dendritic and synaptic processing, is an important fundamental feature of the nervous system [21]. In recent years, much attention has been focused on the various dynamical phenomena in time-delayed neural systems [22-24]. For example, Wang and his colleagues have extensively studied the effects of information transmission delays on the
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synchronization transitions in complex neuronal networks [25-27]. It is shown that fine-tuned information transmission delays are vital for assuring optimally synchronized excitatory fronts on complex neuronal networks [25]. Moreover, resonance, as an indispensable part of neural dynamics, has also been widely investigated in delay-induced systems [28-30]. Researchers explored the SR in
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scale-free neuronal networks with transmission delay and found that multiple stochastic resonances can be induced by appropriately tuned delays irrespective to the placing of the subthreshold periodic pacemaker [31].
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However, 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 time-dependent processes. One important form of these
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biological synaptic processes is spike-timing-dependent plasticity (STDP), which modulates the coupling strength adaptively based on the relative timing between pre- and post-synaptic action
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potentials [32, 33]. A series of biological works have confirmed the existence of STDP, which
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commonly occurs at excitatory synapses onto neocortical [34] and hippocampal pyramidal neurons [33], excitatory neurons in auditory brainstem [35], parvalbumin-expressing fast-spiking striatal
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interneurons [36], etc. Experimental researches show that the functional structures in the brain can be
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remapped through STDP, which is prevalently reorganized into both small-world and scale-free networks [37, 38]. More recently, modelling studies on functional role of STDP in neural dynamics have gained increasing interest [39-41]. For example, Lee et al. use a simplified biophysical model of a cortical network with STDP, which provides a mechanism for potentiation and depression depending on input frequency, and suggest that the slow NMDAR current decay helps to regulate the optimal amplitude and duration of the plasticity [42]. In addition, it is reported that, STDP, which
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adaptively modifies strengths of synaptic connections, considerably weakens the synchronization of neuronal activity in small-world networks [43]. Furthermore, coherence resonance and stochastic resonance have also been investigated in self-organized neural network with STDP [44]. It is shown that the selectively refined connectivity modified through STDP can highly enhance the ability of
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neuronal communications and improve the efficiency of signal transmission in the network [45]. In the present work, the pivotal effects of time delay and STDP on stochastic resonance in small-world neuronal networks will be studied. We aim to investigate how the network connections
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evolve during the process refined by STDP and the dependence of SR on it. Furthermore, fundamental roles of information transmission delay and small-world structure in networked stochastic resonance will be discussed as well. The remainder of this paper is organized as follows:
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in Sec. II，a simplified model of time-delayed small-world neuronal network is established and STDP rule is used to modify the strengths of synaptic connections between neurons. We explore the
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evolution of connection synapses via STDP within a noisy background in Sec. III. In Sec. IV, the
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dependence of stochastic resonance on STDP, time delay, as well as small-world topology is systematically studied. Finally, a brief conclusion of this paper is drawn in Sec. V.
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II. MATHEMATICAL MODEL
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The FitzHugh–Nagumo (FHN) neuron model [44] is used to describe the neuronal dynamics, and the temporal evolution of each unit can be defined as follows:
e
dVi V3 Vi i Wi I ex I isyn i , dt 3
dWi Vi a bW i i, dt where i ( i 1, 2,
(1)
(2)
, N ) is the index of neurons. Vi t represents the fast transmembrane voltage of
ACCEPTED MANUSCRIPT the ith neuron, whereas Wi t is a slow recovery variable. The small value of parameter e ( e 0.08 ), which is the time scale ratio of membrane and recovery variable, guarantees that Vi t evolves much faster than Wi t . I ex stands for the external stimulus current. The Gaussian white noise i with mean 0 satisfies i (t ) j (t ) ij (t t ) .
denotes the intensity of the noisy
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background. The parameters of a and b codetermine the dynamics of a single neuron. In our simulations, the parameter a is constantly chosen as 0.7. Accordingly, the dynamics of a single neuron depends largely on the value of variable b : the neuron is excitable for b 0.45 ; while for
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b 0.45 , it exhibits an oscillatory behavior generating periodic spikes. Considering that the neurons in nervous system are not identical, in our study, bi (the value of parameter b of the ith neuron) is randomly distributed in 0.5,0.75 , so that all neurons within the network are of different excitability.
N
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The synaptic current I isyn is described by:
j 1 j i
gij Cij s j t Vi Vsyn ,
(3)
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I isyn
where Cij is a bidirectional connectivity matrix, if neuron j couples to neuron i , then Cij 1
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and C ji 1 , otherwise Cij C ji 0 and Cii 0 . The type of synapse is subjected to the value of
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reversal potential Vsyn , which is set to be Vsyn 0 as only excitatory synapses are considered in this
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work. The synaptic variable s j takes the forms of: s j V j 1 s j s j ,
V j 0 / 1 e
V j t /Vshp
.
The synaptic recovery function V j can be taken as the Heaviside function.
(4) (5)
is the
information transmission delay in coupled FHN neurons with synaptic connection, which is one of key parameters to be discussed in this work. Vshp 0.05 determines the threshold, above which the presynaptic neuron is able to influence the postsynaptic one. Other parameters 0 and are
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chosen as 0 2 and 1 . The synaptic coupling strength g ij from the j th neuron to the i th one varies through STDP modification function F , which is defined as follows gij gij gij ,
(6)
gij gij F t ,
(7) if t 0 if t 0 ,
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A exp t / + F t A exp t / 0
(8)
if t 0
where t ti t j , ti (or t j ) is marked as the spiking time of the i th (or j th) neuron. The values
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of adjusting rate A and A constrain the maximum amount of synaptic modification. and determine the temporal window over which synaptic strengthening and weakening occur. Experimental investigations suggest that the temporal window for synaptic weakening is roughly the
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same as that for synaptic strengthening [33, 34]. Potentiation is consistently induced when the postsynaptic spike generates within a time window of 20 msec after presynaptic spike, and
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depression is induced conversely. Thus the temporal windows are set to be + = - =20 . According to
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Ref. [39], for the consideration of stable synaptic modification, we use A- /A+ =1.05 . In what follows, for the purpose of exploring the effect of STDP on the adaptive stochastic resonance, the adjusting
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rate A+ is chosen as a main variable of STDP. Hence, based on the previous modelling studies of
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STDP [39, 45], a reasonable parameter space of adjusting rate is set to [0, 0.05]. All synapses considered are initiated as gij gmax / 2 0.05 , where gmax 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. Moreover, as the existence of small-world properties in the networks of the brain has been revealed, the conventional scheme of small-world network is chosen. According to Newman-Watts procedure [46], the neural network, consisting of N 100 neurons, is initiated as a regular ring in
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which each unit is connected to its K 2 nearest neighbors. Thus, the total number of links is N N 1 / 2 . New links are added into the network with probability p , and the number of added
links ne satisfies ne pN N 1 / 2 . If p 0 , the network is a regular ring, while it is globally coupled for p 1 . The small-world network is obtained by an intermediate case of
p
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( 0 p 1 ).The normalized link number p is another one of the main parameters to be investigated in this paper.
III. SYNAPTIC EVOLUTION VIA STDP IN NOISY SMALL-WORLD NEURONAL
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Based on the neural network established above, we check how the coupling strengths evolve during the STDP regulating process in a noisy background. Distributions of synaptic weights arising from STDP for various values of simulation time are presented in Fig. 1. Due to the competition
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among synapses, the fraction of strong ( gij 0.9* gmax ) and weak ( gij 0.1* gmax ) weight expands
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with the enlargement of simulation time, while that of moderate ( 0.1* gmax gij 0.9* g max ) synaptic
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strength declines correspondingly. It is also shown that, the modification of synaptic strength experiences dramatic change in initial simulating stage, while it becomes saturated as the simulation
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time is long enough.
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Fig. 1 Percentage of synapses at three levels: g 0.1* gmax , g 0.9* gmax and the others. Other parameters are: =0.2 , A 0.01 , =0 and p 0.1. In order to obtain a quantitative perspective on the development of neural network through STDP, g ij is averaged over the whole population with synaptic connections and time, and the
g
can be written as [47]: g
1 MT
N
N
i 1 j 1
T
0
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average coupling strength
gij t ,
(9)
where M is the total number of synaptic connections within the network and T is the integration
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time. The dependence of average coupling strength on the noise intensity
for different values of
adjusting rate A is presented in Fig. 2. It is shown that, for each particular value of A ,
g
almost remains the same for small noise intensity, which is so weak that cannot excite neurons to
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spike or only evoke sparse neurons to spike. However, the mean coupling strength slumps as
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continues to increase. This phenomenon is depicted more detailed in Fig. 3(a)-(d), where the final distribution of synaptic weights for different values of
is clearly presented. Obviously, with the
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augmentation of the intensity of noisy background, the fraction of weak synaptic strength increases
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rapidly, while the part of strong coupling strength decreases accordingly. As it is demonstrated above (Fig. 1) that the change of synaptic strength distribution saturates as the simulation time is long
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enough, the obtained result is robust to large simulation time.
Fig. 2 Dependence of
g
on the noise intensity
for different values of adjusting rate A .
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Other Parameters are: =0 and p=0.1 .
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(b)
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(a)
(c)
(d)
Fig. 3 Histogram and distribution of the synaptic matrix for different values of noise intensity
. (a)
0.1 , (b) 0.15 , (c) 0.2 , and (d) 0.25 . Other parameters are: p 0.1, A 0.01 and =0 .
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Fig. 4 STDP modification function for different values of adjusting rate A , where A / A 1.2 .
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The solid lines are the real STDP modification functions for different adjusting rate. And the dash lines are the reverse of the lines in the third quadrant.
Moreover, as a main parameter of STDP, the adjusting rate A can also significantly affect the
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synaptic modification. As mentioned above, the value of A is a little stronger than A (i.e.,
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A / A 1.05 ), which means the decrement of synaptic strength is larger than the increment for the same spike interval. For a better understanding, Fig. 4 exemplifies the STDP modification function
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for different adjusting rate (we amplify the value of A and A / A 1.2 to observe significant
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difference). It is shown that, for each particular value of interspike interval t , the decline of modification is larger than the increase. And this difference is enlarged with the increasing of A .
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Consequently, the larger the value of A is, the smaller the average coupling strength
g
is (as
shown in Fig. 2). From the obtained observations, a conclusion can be drawn that enhancing the effects of noise and STDP have a negative impact on the refinement of average synaptic strength in small-world neuronal networks.
IV. STOCHASTIC RESONANCE IN TIME-DELAYED SMALL-WORLD NEURONAL
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NETWORKS VIA STDP In this section, we focus our concentration on how the stochastic resonance is affected by STDP. In addition, the fundamental role of time delay in stochastic resonance within small-world neuronal networks is also discussed. Initially, we simultaneously introduce the external periodic signal and
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STDP modification into the time-delayed small-world neuronal networks. For the investigation of SR, the external periodic signal is set to be I ex B sin t , with B 0.1 and 0.2 , guaranteeing no spike for all neurons with the absence of random disturbances. To quantitatively characterize the
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accordance between the weak subthreshold signal and the temporal output of the considered network, Fourier coefficient is used, which is calculated as: i Qsin
2 n / 2Vi t sin t dt , 2 n 0
2 n / 2Vi t cos t dt , 2 n 0
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i Qcos
Q Qsin Qcos ,
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i
Q
i
2
i
(10)
(11)
2
1 N i Q , N i 1
(12) (13)
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where Vi t is the membrane potential of i th neuron in the network. n is the number of periods
2 / covered by the integration time. Since the Fourier coefficient is in proportion to the square
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of the spectral power amplification, which is frequently used as a measure for stochastic resonance. The maximum of Q indicates the best correlation between input signal and output firing. The final results are obtained by averaging over 20 independent runs to eliminate the randomness.
(a)
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(b)
for different values of adjusting rate A . (b) and adjusting rate A . Other
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Fig. 5 (a) Dependence of Q on the noise intensity
Contour plot of Q in dependence on the noise intensity parameters are: p 0.1 and 0 .
We first inspect the effect of STDP on the stochastic resonance in small-world neuronal
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networks without time delay. As shown in Fig. 5(a), for each particular value of adjusting rate A ,
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as the increase of noise intensity, the value of Q first ascends, then decreases after reaching a peak,
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which means the correlation between the temporal output series of excitable neurons and the stimulus frequency of the weak external signal reaches an optimum. This phenomenon indicates
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the existence of stochastic resonance in small-world neuronal networks with STDP. Particularly, the
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optimal value of noise intensity moves to a much larger one and the maxima of SR curve Qmax decays with the enlargement of adjusting rate. This observation maybe results from the attenuation of strong coupling modulated by STDP. When A turns larger, the average coupling strength of the neuronal network drops, particularly the strong synapses decrease and weak synapses increase accordingly, which will reduce the excitability of neuronal network and impair the efficiency of signal transmission and information propagation via SR in neuronal networks. To generalize the
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obtained results, we calculate the value of Q in the two-dimensional parameter space ( , A ). Indeed, as exhibited in Fig. 5(b), irrespective of the value of A , there exists a span of optimal noise intensities by which Q is maximum, indicating the occurrence of stochastic resonance in small-world neuronal networks. In addition, the span shifts rightward slightly and Qmax declines as
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A becomes larger, which corresponds to the results obtained from Fig. 4(a). Thus, all the observations above indicate that, the phenomenon of SR indeed arises in the small-world neural
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networks with STDP, while the efficiency of SR is slightly weakened by the augmentation of STDP.
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(a)
Fig. 6 (a) Contour plot of Q in dependence on the noise intensity
(b)
and time-delay . Other
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parameters are: p 0.1 and A 0.01 . (b) Contour plot of Q in dependence on the adjusting rate
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A and time-delay . Other parameters are: p 0.1 and 0.1 . Fig. 6(a) plots the function of Q depending on the noise intensity and time delay. Evidently,
there exist some tongue-like regions with high level of resonant response. It demonstrates that multiple stochastic resonances as well emerge in the self-organized time-delayed small-world neuronal networks, as regularity and disorder appear intermittently with the increment of
. All
these resonance regions are roughly located at the integer multiples of the forcing periods of external
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sinusoidal signal, that is to say, the SR frequency is a function of delay and that there are several ‘sweet spot’ delays that generate stronger SR. In addition, the effect of STDP on SR in time-delayed small-world neuronal networks is investigated. Contour plot of Q in dependence on the adjusting rate A and time delay
is presented in Fig. 6(b). Some narrow-banded regions with high values
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of Q appear, and the maximum value of Q decays as the adjusting rate increases. This phenomenon thus indicates that, the effects of STDP on SR only give expression in the resonance amplitude and the peak value of Q is a decreasing function of adjusting rate, while it has little
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impact on the periodicity of networked system due to time delay.
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(a)
(b)
Fig. 7 (a) Dependence of Q on the noise intensity
for different values of normalized link p .
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(b) Dependence of Q on normalized link p for different values of the noise intensity
. Other
parameters are: A 0.01 and =0 . In the following, the impact of network structure on stochastic resonance is explored as well. Fig. 7(a) shows the dependence of Q on
for different values of p without time delay. It is
obvious that, the phenomenon of stochastic resonance occurs for each particular value of p , and the optimal noise intensity for SR reduces as the normalized link increases. Indeed, with the augment of
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p , more links are added into the network to connect with originally separated neurons. The increased connections between neurons may help mutual excitation and lead to the decrement of resonance noise intensity. To better characterize the influence of small-world topology on the stochastic resonance with STDP, the dependence of Q on normalized link p for different values
is exhibited in Fig. 7(b). Evidently, when the noise intensity is small,
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of the noise intensity
=0.05 for instance, the correlation between ensemble activity and subthreshold force keeps improving as more link added into the network. However, a peak emerges along with p for
. It is thus indicated that, for small levels of random fluctuations,
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moderate and large values of
increasing the normalized link can largely enhance the efficiency of the SR in small-world neuronal networks, while that efficiency can be optimized by an appropriate value of normalized link when the intensity of noise is moderate and large. In addition, comparing with several resonance points
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presented in Fig. 7(b), we can notice that the optimal value of p shifts towards lower values with
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the increasing of noise intensity. This may be explained by that, the higher level of random
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background would excite more neurons to spike, and fewer connections are needed to increase the
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mutual communication between neurons.
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(b)
(c)
and normalized link p . (a)
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Fig. 8 Contour plot of Q in dependence on the noise intensity
0 , (b) 7.5 and (c) 15.5 . Other parameter is: A 0.01 . For an overall view, we plot the function of Q on both
and p . As shown in Fig. 8(a), the
spans of noise intensities for the occurrence of SR shift leftward as p increases. In addition, an
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elliptical region of high values of Q indicates that the response of neural system can be maximized
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by some appropriate values of p . In order to gain more insight into the stochastic resonance in delay-induced small-world neuronal networks, we also calculate the values of Q in the
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two-dimensional space ( , p ) for 7.5 and 15.5 in Fig. 8, respectively. In accordance with
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the case without delay, the optimal value of
moves to a smaller one as more links introduced into
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the neural networks. Moreover, there exist several regions of optimal value of p that maximizes the response of the neural system to the periodic force and these optimal regions are affected profoundly by time delay. That is to say, the optimal region varies in the delay-induced small-world neuronal networks with STDP. From the observations above, it can be indicated that all the features of
, p and govern the optimal resonance region in small-world neuronal networks with
STDP.
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Fig. 9 Dependence of Q on the normalized link p for different values of adjusting rate A . Other
(a)
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parameters are: 0.15 and 0 .
(b)
(c)
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Fig. 10 Contour plot of Q in dependence on the normalized link p and adjusting rate A for (a)
0 , (b) 7.5 , and (c) 15.5 . Other parameter is: 0.15 . We turn to explore the impact of STDP on the networked stochastic resonance with small-world topology. First, this effect is studied with the absence of time delay in Fig. 9. It is shown that, for
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small and moderate values of adjusting rate, the stochastic resonance due to normalized link still presents and the resonance points move to a larger value of p as A augments. Whereas the value of Q maintains growing when the adjusting rate is large, A 0.05 for example. This interesting
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observation results from the decay of average coupling strength modulated by STDP. As the analysis mentioned above, the larger the adjusting rate is, the smaller the mean synaptic weight gets, which makes the neural system require more connections between neurons to performance best to keep
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pace with the external periodic signal. In addition, Fig. 10(a) is a generalization of Fig. 9. Despite the observations conformed to Fig. 9, an intriguing finding is exhibited in Fig. 10(a) that there is a span
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of optimal adjusting rate maximizing the entire population activity to meet the frequency of stimulus signal. Furthermore, we extend the investigation to the delay-induced small-world nervous system.
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Fig. 10(b) and (c) present the contour plot of Q in dependence on p and A for 7.5 and
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15.5 , respectively. For each particular value of time delay, similar transition is exhibited. While
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the area of optimal normalized link for time-delayed neuronal network narrows comparing with that without information transmission delay.
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(a)
(b)
for (a)
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Fig. 11 Contour plot of Q in dependence on the normalized link p and time delay
A 0.01 and (b) A 0.05 . Other parameter is: 0.15 .
The effect of time delay on the stochastic resonance due to network structure is probed in Fig. 11. Contour plot of Q in the two-dimensional space ( p , ) for A 0.01 is exhibited in Fig. 11(a).
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Indeed, the region with high values of Q roughly locates at the integer multiples of the forcing
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periods of the stimulus signal. Smaller time delay (close to 0 ) needs larger value of p to
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obtain the maximal value of Q with the comparison of 7.5 and 15.5 . On the other hand, values of Q outside the regions of multiple integers depress the stochastic resonance significantly.
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However, when the adjusting rate A reaches to 0.05 , the value of Q only keeps gaining with p
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at the integer multiples of the external periods, which is consist with the result presented in Fig. 9. It thus can be concluded that, both features of STDP and time delay can significantly affect the networked stochastic resonance, codetermining the optimal response of neural system.
Fig. 12 Dependence of Q on the noise intensity
for different values of network size N . Other
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parameters are: p 0.1, A 0.01 and 0 .
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The neuronal network size is also a variable that influences the response of the nonlinear system. Hence, the impact of network size on the stochastic resonance in the self-organized small-world
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neuronal network is investigated. As presented in Fig. 12, the phenomenon of SR occurs for each
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particular value of network size. Moreover, the optimal noise intensity shifts leftward to a smaller one for larger network size. It is also demonstrated that, with the enlargement of network size, the
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the weak signal.
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coefficient Q increase, indicating enhancement of the correlation between the network response and
(a)
(b)
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Fig. 13 (a) Dependence of Q on the noise intensity Dependence of Q on the noise intensity
for different values of e , a 0.7 . (b)
for different values of a , e 0.08 . Other parameters
are: p 0.1, A 0.01 and 0 . As a core component of neuronal networks, single neuron and its properties play a fundamental
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role in networked dynamics. Therefore, we investigate the effect of model parameters of single neuron on the collective resonance response within adaptive small-world neuronal networks. As shown in Fig. 13(a), with the enlargement of time ratio e , the resonance point moves rightward to a
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larger noise intensity and the maximal Q decreases accordingly, indicating that large value of e reduces the ability of neuronal network in weak signal detection and transmission. The obtained results attribute to that single neuron spikes more and more difficultly with the increase of e (presented detailed in Fig. 14, where direct current I =0.09 is induced to evoke neuron to spike).
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However, the phenomenon of stochastic resonance vanish when e is large enough, as the resonance
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response only increases monotonously with noise intensity. Moreover, the transition of stochastic
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resonance for different values of a is similar to the case of time ratio e (as shown in Fig. 13(b) and Fig. 15). A conclusion can be drawn that the properties of single neuron are of great importance
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to the networked resonance response.
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Fig. 14 Neuronal firing sequence obtained for different values of time ratio e . (a) e=0.07 , (b)
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e=0.08 , (c) e=0.09 , (d) e=0.1 and (e) e=0.15 . Other parameters are: a=0.7 , b=0.5 , I =0.09 .
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Fig. 15 Neuronal firing sequence obtained for different values of time ratio a . (a) a=0.6 , (b)
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a=0.65 , (c) a=0.7 , (d) a=0.75 and (e) a=0.8 . Other parameters are: e=0.08 , b=0.5 , I =0.09 .
V. CONCLUSIONS In this paper we study adaptive stochastic resonance in time-delayed Newman-Watts small-world neuronal networks self-organized via spike-timing-dependent plasticity. The obtained numerical results show that, with the absence of transmission delay, the temporal response of the neural system
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to the external periodic signal can be optimized by an intermediate intensity of additive noise, which implies the occurrence of stochastic resonance. It is also indicated that, STDP in coupling process can slightly depress the efficiency of network stochastic resonance, as the system with larger adjusting rate can reach a much lower peak value near the resonance noise intensity. This observation
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is accounted for that, large adjusting rate, which can result in the increase of the fraction of synapses with small strength, will impair the excitability of the considered neural network and consequently adverse to the communication among neurons. In addition, the effect of stochastic resonance can be
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either promoted or destroyed by time delay. Delay-induced multiple stochastic resonance appear intermittently at the integer multiples of period of the subthreshold forcing. Furthermore, it is demonstrated that the small-world topology plays a constructive role in the stochastic resonance in
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neural systems. For small and moderate adjusting rate of STDP, fine-tuning of the probability of adding links can significantly enhance the effect of stochastic resonance in adaptive neural network.
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Additionally, there is an optimal probability of adding links by which the noise-induced transmission
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of weak periodic signal peaks and the span of optimal normalized link for small-world neuronal networks shrinks with the increase of time delay. Finally, the networked stochastic resonance can
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also be influenced by single neuron properties. In a word, all the features of STDP, time delay, and
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connectivity structure functions together in the stochastic resonance in small-world 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).
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Adaptive stochastic resonance in self-organized small-world neuronal networks with time delay Haitao Yu , Xinmeng Guo , Jiang Wang , Chen Liu , Bin Deng , Xile Wei PII: DOI: Reference:
S1007-5704(15)00190-2 10.1016/j.cnsns.2015.05.017 CNSNS 3559
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Received date: Revised date: Accepted date:
3 November 2014 16 April 2015 15 May 2015
Please cite this article as: Haitao Yu , Xinmeng Guo , Jiang Wang , Chen Liu , Bin Deng , Xile Wei , Adaptive stochastic resonance in self-organized small-world neuronal networks with time delay, Communications in Nonlinear Science and Numerical Simulation (2015), doi: 10.1016/j.cnsns.2015.05.017
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Stochastic resonance in small-world neuronal networks is numerically studied.
Spike-time-dependent plasticity may depress the network stochastic resonance.
Effect of stochastic resonance can be either promoted or destroyed by time delay.
The small-world topology can significantly affect the stochastic resonance.
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Adaptive stochastic resonance in self-organized small-world neuronal networks with time delay Haitao Yu*, Xinmeng Guo, Jiang Wang, Chen Liu, Bin Deng, Xile Wei School of Electrical Engineering and Automation, Tianjin University, Tianjin, 300072, P. R. China.
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(*Corresponding author: [email protected], Tel: 86-22-27402293) Abstract: In this paper, adaptive stochastic resonance in time-delayed Newman-Watts small-world neuronal networks is studied, where the strength of synaptic connections between neurons is adaptively modulated by spike-timing-dependent plasticity (STDP). Numerical results show that, in
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the absence of information transmission delay, the phenomenon of stochastic resonance occurs and the efficiency of networked stochastic resonance can be slightly depressed by STDP. Due to the reduction of strong couplings induced by STDP, the larger the adjusting rate of STDP is, the smaller peak value of the resonance response obtains. In addition, the effect of stochastic resonance can be
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either promoted or destroyed by time delay, and multiple stochastic resonances appear intermittently
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at the integer multiples of periods of the subthreshold forcing. Furthermore, it is demonstrated that the networked stochastic resonance can also be dramatically affected by the small-world topology.
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For small and moderate adjusting rate of STDP, fine-tuning of the probability of adding links can significantly enhance the effect of stochastic resonance in adaptive neural network. Additionally,
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there is an optimal probability of adding links by which the noise-induced transmission of weak
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periodic signal peaks and the location of this span depends largely on the time delay and adjusting rate.
Keywords: spike-timing-dependent plasticity, time delay, adaptive stochastic resonance, neuronal network, small-world I.
INTRODUCTION Noise can emerge as a constructive component of nonlinear systems [1-3]. One important
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representative of this fact is stochastic resonance (SR), which occurs when the response of a nonlinear dynamical system to a weak periodic signal is optimized by moderate intensity of random fluctuations [4-7]. Recently, SR has sparked growing interest both in theoretical models of neural systems and in experimental neuroscience [8-10]. It is shown that the ability of sensory neurons to
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process weak input signals can be significantly enhanced by adding noise to the system [11-12]. Additionally, statistical signal processing in coupled neural systems improves with the aid of various levels of stochastic noise via SR [13, 14]. Therefore, the study of stochastic resonance is valuable for
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understanding the signal transmission and information propagation in neuronal networks.
The last decade has growing body of modelling work on stochastic resonance in complex neuronal networks, especially in scale-free networks [15, 16] and small-world neuronal networks
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[17-19]. Perc studied stochastic resonance on weakly paced scale-free networks and indicated that, all the features of the placement of the pacemaker, coupling strength and the inhomogeneous
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structure of scale-free networks play a crucial role in SR [15]. On the other hand, the effect of SR on
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excitable small-world networks can be amplified only for intermediate coupling strengths in excitable networks via pacemaker [18]. The study on stochastic resonance in Newman-Watts
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small-world also demonstrates that fine-tuning of the small-world network structure can largely
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enhance the stochastic resonance in neuronal networks [20]. Time delay, which occurs due to the finite speed of action potentials propagating across neuron
axons and finite reaction times for dendritic and synaptic processing, is an important fundamental feature of the nervous system [21]. In recent years, much attention has been focused on the various dynamical phenomena in time-delayed neural systems [22-24]. For example, Wang and his colleagues have extensively studied the effects of information transmission delays on the
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synchronization transitions in complex neuronal networks [25-27]. It is shown that fine-tuned information transmission delays are vital for assuring optimally synchronized excitatory fronts on complex neuronal networks [25]. Moreover, resonance, as an indispensable part of neural dynamics, has also been widely investigated in delay-induced systems [28-30]. Researchers explored the SR in
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scale-free neuronal networks with transmission delay and found that multiple stochastic resonances can be induced by appropriately tuned delays irrespective to the placing of the subthreshold periodic pacemaker [31].
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However, 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 time-dependent processes. One important form of these
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biological synaptic processes is spike-timing-dependent plasticity (STDP), which modulates the coupling strength adaptively based on the relative timing between pre- and post-synaptic action
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potentials [32, 33]. A series of biological works have confirmed the existence of STDP, which
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commonly occurs at excitatory synapses onto neocortical [34] and hippocampal pyramidal neurons [33], excitatory neurons in auditory brainstem [35], parvalbumin-expressing fast-spiking striatal
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interneurons [36], etc. Experimental researches show that the functional structures in the brain can be
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remapped through STDP, which is prevalently reorganized into both small-world and scale-free networks [37, 38]. More recently, modelling studies on functional role of STDP in neural dynamics have gained increasing interest [39-41]. For example, Lee et al. use a simplified biophysical model of a cortical network with STDP, which provides a mechanism for potentiation and depression depending on input frequency, and suggest that the slow NMDAR current decay helps to regulate the optimal amplitude and duration of the plasticity [42]. In addition, it is reported that, STDP, which
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adaptively modifies strengths of synaptic connections, considerably weakens the synchronization of neuronal activity in small-world networks [43]. Furthermore, coherence resonance and stochastic resonance have also been investigated in self-organized neural network with STDP [44]. It is shown that the selectively refined connectivity modified through STDP can highly enhance the ability of
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neuronal communications and improve the efficiency of signal transmission in the network [45]. In the present work, the pivotal effects of time delay and STDP on stochastic resonance in small-world neuronal networks will be studied. We aim to investigate how the network connections
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evolve during the process refined by STDP and the dependence of SR on it. Furthermore, fundamental roles of information transmission delay and small-world structure in networked stochastic resonance will be discussed as well. The remainder of this paper is organized as follows:
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in Sec. II，a simplified model of time-delayed small-world neuronal network is established and STDP rule is used to modify the strengths of synaptic connections between neurons. We explore the
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evolution of connection synapses via STDP within a noisy background in Sec. III. In Sec. IV, the
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dependence of stochastic resonance on STDP, time delay, as well as small-world topology is systematically studied. Finally, a brief conclusion of this paper is drawn in Sec. V.
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II. MATHEMATICAL MODEL
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The FitzHugh–Nagumo (FHN) neuron model [44] is used to describe the neuronal dynamics, and the temporal evolution of each unit can be defined as follows:
e
dVi V3 Vi i Wi I ex I isyn i , dt 3
dWi Vi a bW i i, dt where i ( i 1, 2,
(1)
(2)
, N ) is the index of neurons. Vi t represents the fast transmembrane voltage of
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denotes the intensity of the noisy
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background. The parameters of a and b codetermine the dynamics of a single neuron. In our simulations, the parameter a is constantly chosen as 0.7. Accordingly, the dynamics of a single neuron depends largely on the value of variable b : the neuron is excitable for b 0.45 ; while for
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b 0.45 , it exhibits an oscillatory behavior generating periodic spikes. Considering that the neurons in nervous system are not identical, in our study, bi (the value of parameter b of the ith neuron) is randomly distributed in 0.5,0.75 , so that all neurons within the network are of different excitability.
N
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The synaptic current I isyn is described by:
j 1 j i
gij Cij s j t Vi Vsyn ,
(3)
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I isyn
where Cij is a bidirectional connectivity matrix, if neuron j couples to neuron i , then Cij 1
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and C ji 1 , otherwise Cij C ji 0 and Cii 0 . The type of synapse is subjected to the value of
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reversal potential Vsyn , which is set to be Vsyn 0 as only excitatory synapses are considered in this
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work. The synaptic variable s j takes the forms of: s j V j 1 s j s j ,
V j 0 / 1 e
V j t /Vshp
.
The synaptic recovery function V j can be taken as the Heaviside function.
(4) (5)
is the
information transmission delay in coupled FHN neurons with synaptic connection, which is one of key parameters to be discussed in this work. Vshp 0.05 determines the threshold, above which the presynaptic neuron is able to influence the postsynaptic one. Other parameters 0 and are
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chosen as 0 2 and 1 . The synaptic coupling strength g ij from the j th neuron to the i th one varies through STDP modification function F , which is defined as follows gij gij gij ,
(6)
gij gij F t ,
(7) if t 0 if t 0 ,
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A exp t / + F t A exp t / 0
(8)
if t 0
where t ti t j , ti (or t j ) is marked as the spiking time of the i th (or j th) neuron. The values
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of adjusting rate A and A constrain the maximum amount of synaptic modification. and determine the temporal window over which synaptic strengthening and weakening occur. Experimental investigations suggest that the temporal window for synaptic weakening is roughly the
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same as that for synaptic strengthening [33, 34]. Potentiation is consistently induced when the postsynaptic spike generates within a time window of 20 msec after presynaptic spike, and
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depression is induced conversely. Thus the temporal windows are set to be + = - =20 . According to
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Ref. [39], for the consideration of stable synaptic modification, we use A- /A+ =1.05 . In what follows, for the purpose of exploring the effect of STDP on the adaptive stochastic resonance, the adjusting
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rate A+ is chosen as a main variable of STDP. Hence, based on the previous modelling studies of
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STDP [39, 45], a reasonable parameter space of adjusting rate is set to [0, 0.05]. All synapses considered are initiated as gij gmax / 2 0.05 , where gmax 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. Moreover, as the existence of small-world properties in the networks of the brain has been revealed, the conventional scheme of small-world network is chosen. According to Newman-Watts procedure [46], the neural network, consisting of N 100 neurons, is initiated as a regular ring in
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which each unit is connected to its K 2 nearest neighbors. Thus, the total number of links is N N 1 / 2 . New links are added into the network with probability p , and the number of added
links ne satisfies ne pN N 1 / 2 . If p 0 , the network is a regular ring, while it is globally coupled for p 1 . The small-world network is obtained by an intermediate case of
p
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( 0 p 1 ).The normalized link number p is another one of the main parameters to be investigated in this paper.
III. SYNAPTIC EVOLUTION VIA STDP IN NOISY SMALL-WORLD NEURONAL
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NETOWRKS
Based on the neural network established above, we check how the coupling strengths evolve during the STDP regulating process in a noisy background. Distributions of synaptic weights arising from STDP for various values of simulation time are presented in Fig. 1. Due to the competition
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among synapses, the fraction of strong ( gij 0.9* gmax ) and weak ( gij 0.1* gmax ) weight expands
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with the enlargement of simulation time, while that of moderate ( 0.1* gmax gij 0.9* g max ) synaptic
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strength declines correspondingly. It is also shown that, the modification of synaptic strength experiences dramatic change in initial simulating stage, while it becomes saturated as the simulation
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time is long enough.
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Fig. 1 Percentage of synapses at three levels: g 0.1* gmax , g 0.9* gmax and the others. Other parameters are: =0.2 , A 0.01 , =0 and p 0.1. In order to obtain a quantitative perspective on the development of neural network through STDP, g ij is averaged over the whole population with synaptic connections and time, and the
g
can be written as [47]: g
1 MT
N
N
i 1 j 1
T
0
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average coupling strength
gij t ,
(9)
where M is the total number of synaptic connections within the network and T is the integration
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time. The dependence of average coupling strength on the noise intensity
for different values of
adjusting rate A is presented in Fig. 2. It is shown that, for each particular value of A ,
g
almost remains the same for small noise intensity, which is so weak that cannot excite neurons to
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spike or only evoke sparse neurons to spike. However, the mean coupling strength slumps as
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continues to increase. This phenomenon is depicted more detailed in Fig. 3(a)-(d), where the final distribution of synaptic weights for different values of
is clearly presented. Obviously, with the
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augmentation of the intensity of noisy background, the fraction of weak synaptic strength increases
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rapidly, while the part of strong coupling strength decreases accordingly. As it is demonstrated above (Fig. 1) that the change of synaptic strength distribution saturates as the simulation time is long
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enough, the obtained result is robust to large simulation time.
Fig. 2 Dependence of
g
on the noise intensity
for different values of adjusting rate A .
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Other Parameters are: =0 and p=0.1 .
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(b)
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(a)
(c)
(d)
Fig. 3 Histogram and distribution of the synaptic matrix for different values of noise intensity
. (a)
0.1 , (b) 0.15 , (c) 0.2 , and (d) 0.25 . Other parameters are: p 0.1, A 0.01 and =0 .
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Fig. 4 STDP modification function for different values of adjusting rate A , where A / A 1.2 .
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The solid lines are the real STDP modification functions for different adjusting rate. And the dash lines are the reverse of the lines in the third quadrant.
Moreover, as a main parameter of STDP, the adjusting rate A can also significantly affect the
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synaptic modification. As mentioned above, the value of A is a little stronger than A (i.e.,
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A / A 1.05 ), which means the decrement of synaptic strength is larger than the increment for the same spike interval. For a better understanding, Fig. 4 exemplifies the STDP modification function
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for different adjusting rate (we amplify the value of A and A / A 1.2 to observe significant
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difference). It is shown that, for each particular value of interspike interval t , the decline of modification is larger than the increase. And this difference is enlarged with the increasing of A .
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Consequently, the larger the value of A is, the smaller the average coupling strength
g
is (as
shown in Fig. 2). From the obtained observations, a conclusion can be drawn that enhancing the effects of noise and STDP have a negative impact on the refinement of average synaptic strength in small-world neuronal networks.
IV. STOCHASTIC RESONANCE IN TIME-DELAYED SMALL-WORLD NEURONAL
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NETWORKS VIA STDP In this section, we focus our concentration on how the stochastic resonance is affected by STDP. In addition, the fundamental role of time delay in stochastic resonance within small-world neuronal networks is also discussed. Initially, we simultaneously introduce the external periodic signal and
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STDP modification into the time-delayed small-world neuronal networks. For the investigation of SR, the external periodic signal is set to be I ex B sin t , with B 0.1 and 0.2 , guaranteeing no spike for all neurons with the absence of random disturbances. To quantitatively characterize the
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accordance between the weak subthreshold signal and the temporal output of the considered network, Fourier coefficient is used, which is calculated as: i Qsin
2 n / 2Vi t sin t dt , 2 n 0
2 n / 2Vi t cos t dt , 2 n 0
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i Qcos
Q Qsin Qcos ,
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i
Q
i
2
i
(10)
(11)
2
1 N i Q , N i 1
(12) (13)
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where Vi t is the membrane potential of i th neuron in the network. n is the number of periods
2 / covered by the integration time. Since the Fourier coefficient is in proportion to the square
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of the spectral power amplification, which is frequently used as a measure for stochastic resonance. The maximum of Q indicates the best correlation between input signal and output firing. The final results are obtained by averaging over 20 independent runs to eliminate the randomness.
(a)
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(b)
for different values of adjusting rate A . (b) and adjusting rate A . Other
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Fig. 5 (a) Dependence of Q on the noise intensity
Contour plot of Q in dependence on the noise intensity parameters are: p 0.1 and 0 .
We first inspect the effect of STDP on the stochastic resonance in small-world neuronal
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networks without time delay. As shown in Fig. 5(a), for each particular value of adjusting rate A ,
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as the increase of noise intensity, the value of Q first ascends, then decreases after reaching a peak,
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which means the correlation between the temporal output series of excitable neurons and the stimulus frequency of the weak external signal reaches an optimum. This phenomenon indicates
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the existence of stochastic resonance in small-world neuronal networks with STDP. Particularly, the
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optimal value of noise intensity moves to a much larger one and the maxima of SR curve Qmax decays with the enlargement of adjusting rate. This observation maybe results from the attenuation of strong coupling modulated by STDP. When A turns larger, the average coupling strength of the neuronal network drops, particularly the strong synapses decrease and weak synapses increase accordingly, which will reduce the excitability of neuronal network and impair the efficiency of signal transmission and information propagation via SR in neuronal networks. To generalize the
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obtained results, we calculate the value of Q in the two-dimensional parameter space ( , A ). Indeed, as exhibited in Fig. 5(b), irrespective of the value of A , there exists a span of optimal noise intensities by which Q is maximum, indicating the occurrence of stochastic resonance in small-world neuronal networks. In addition, the span shifts rightward slightly and Qmax declines as
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A becomes larger, which corresponds to the results obtained from Fig. 4(a). Thus, all the observations above indicate that, the phenomenon of SR indeed arises in the small-world neural
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networks with STDP, while the efficiency of SR is slightly weakened by the augmentation of STDP.
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(a)
Fig. 6 (a) Contour plot of Q in dependence on the noise intensity
(b)
and time-delay . Other
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parameters are: p 0.1 and A 0.01 . (b) Contour plot of Q in dependence on the adjusting rate
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A and time-delay . Other parameters are: p 0.1 and 0.1 . Fig. 6(a) plots the function of Q depending on the noise intensity and time delay. Evidently,
there exist some tongue-like regions with high level of resonant response. It demonstrates that multiple stochastic resonances as well emerge in the self-organized time-delayed small-world neuronal networks, as regularity and disorder appear intermittently with the increment of
. All
these resonance regions are roughly located at the integer multiples of the forcing periods of external
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sinusoidal signal, that is to say, the SR frequency is a function of delay and that there are several ‘sweet spot’ delays that generate stronger SR. In addition, the effect of STDP on SR in time-delayed small-world neuronal networks is investigated. Contour plot of Q in dependence on the adjusting rate A and time delay
is presented in Fig. 6(b). Some narrow-banded regions with high values
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of Q appear, and the maximum value of Q decays as the adjusting rate increases. This phenomenon thus indicates that, the effects of STDP on SR only give expression in the resonance amplitude and the peak value of Q is a decreasing function of adjusting rate, while it has little
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impact on the periodicity of networked system due to time delay.
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(a)
(b)
Fig. 7 (a) Dependence of Q on the noise intensity
for different values of normalized link p .
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(b) Dependence of Q on normalized link p for different values of the noise intensity
. Other
parameters are: A 0.01 and =0 . In the following, the impact of network structure on stochastic resonance is explored as well. Fig. 7(a) shows the dependence of Q on
for different values of p without time delay. It is
obvious that, the phenomenon of stochastic resonance occurs for each particular value of p , and the optimal noise intensity for SR reduces as the normalized link increases. Indeed, with the augment of
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p , more links are added into the network to connect with originally separated neurons. The increased connections between neurons may help mutual excitation and lead to the decrement of resonance noise intensity. To better characterize the influence of small-world topology on the stochastic resonance with STDP, the dependence of Q on normalized link p for different values
is exhibited in Fig. 7(b). Evidently, when the noise intensity is small,
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of the noise intensity
=0.05 for instance, the correlation between ensemble activity and subthreshold force keeps improving as more link added into the network. However, a peak emerges along with p for
. It is thus indicated that, for small levels of random fluctuations,
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moderate and large values of
increasing the normalized link can largely enhance the efficiency of the SR in small-world neuronal networks, while that efficiency can be optimized by an appropriate value of normalized link when the intensity of noise is moderate and large. In addition, comparing with several resonance points
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presented in Fig. 7(b), we can notice that the optimal value of p shifts towards lower values with
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the increasing of noise intensity. This may be explained by that, the higher level of random
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background would excite more neurons to spike, and fewer connections are needed to increase the
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mutual communication between neurons.
(a)
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(b)
(c)
and normalized link p . (a)
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Fig. 8 Contour plot of Q in dependence on the noise intensity
0 , (b) 7.5 and (c) 15.5 . Other parameter is: A 0.01 . For an overall view, we plot the function of Q on both
and p . As shown in Fig. 8(a), the
spans of noise intensities for the occurrence of SR shift leftward as p increases. In addition, an
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elliptical region of high values of Q indicates that the response of neural system can be maximized
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by some appropriate values of p . In order to gain more insight into the stochastic resonance in delay-induced small-world neuronal networks, we also calculate the values of Q in the
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two-dimensional space ( , p ) for 7.5 and 15.5 in Fig. 8, respectively. In accordance with
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the case without delay, the optimal value of
moves to a smaller one as more links introduced into
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the neural networks. Moreover, there exist several regions of optimal value of p that maximizes the response of the neural system to the periodic force and these optimal regions are affected profoundly by time delay. That is to say, the optimal region varies in the delay-induced small-world neuronal networks with STDP. From the observations above, it can be indicated that all the features of
, p and govern the optimal resonance region in small-world neuronal networks with
STDP.
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Fig. 9 Dependence of Q on the normalized link p for different values of adjusting rate A . Other
(a)
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parameters are: 0.15 and 0 .
(b)
(c)
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Fig. 10 Contour plot of Q in dependence on the normalized link p and adjusting rate A for (a)
0 , (b) 7.5 , and (c) 15.5 . Other parameter is: 0.15 . We turn to explore the impact of STDP on the networked stochastic resonance with small-world topology. First, this effect is studied with the absence of time delay in Fig. 9. It is shown that, for
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small and moderate values of adjusting rate, the stochastic resonance due to normalized link still presents and the resonance points move to a larger value of p as A augments. Whereas the value of Q maintains growing when the adjusting rate is large, A 0.05 for example. This interesting
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observation results from the decay of average coupling strength modulated by STDP. As the analysis mentioned above, the larger the adjusting rate is, the smaller the mean synaptic weight gets, which makes the neural system require more connections between neurons to performance best to keep
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pace with the external periodic signal. In addition, Fig. 10(a) is a generalization of Fig. 9. Despite the observations conformed to Fig. 9, an intriguing finding is exhibited in Fig. 10(a) that there is a span
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of optimal adjusting rate maximizing the entire population activity to meet the frequency of stimulus signal. Furthermore, we extend the investigation to the delay-induced small-world nervous system.
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Fig. 10(b) and (c) present the contour plot of Q in dependence on p and A for 7.5 and
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15.5 , respectively. For each particular value of time delay, similar transition is exhibited. While
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the area of optimal normalized link for time-delayed neuronal network narrows comparing with that without information transmission delay.
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(a)
(b)
for (a)
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Fig. 11 Contour plot of Q in dependence on the normalized link p and time delay
A 0.01 and (b) A 0.05 . Other parameter is: 0.15 .
The effect of time delay on the stochastic resonance due to network structure is probed in Fig. 11. Contour plot of Q in the two-dimensional space ( p , ) for A 0.01 is exhibited in Fig. 11(a).
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Indeed, the region with high values of Q roughly locates at the integer multiples of the forcing
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periods of the stimulus signal. Smaller time delay (close to 0 ) needs larger value of p to
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obtain the maximal value of Q with the comparison of 7.5 and 15.5 . On the other hand, values of Q outside the regions of multiple integers depress the stochastic resonance significantly.
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However, when the adjusting rate A reaches to 0.05 , the value of Q only keeps gaining with p
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at the integer multiples of the external periods, which is consist with the result presented in Fig. 9. It thus can be concluded that, both features of STDP and time delay can significantly affect the networked stochastic resonance, codetermining the optimal response of neural system.
Fig. 12 Dependence of Q on the noise intensity
for different values of network size N . Other
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parameters are: p 0.1, A 0.01 and 0 .
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The neuronal network size is also a variable that influences the response of the nonlinear system. Hence, the impact of network size on the stochastic resonance in the self-organized small-world
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neuronal network is investigated. As presented in Fig. 12, the phenomenon of SR occurs for each
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particular value of network size. Moreover, the optimal noise intensity shifts leftward to a smaller one for larger network size. It is also demonstrated that, with the enlargement of network size, the
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the weak signal.
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coefficient Q increase, indicating enhancement of the correlation between the network response and
(a)
(b)
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Fig. 13 (a) Dependence of Q on the noise intensity Dependence of Q on the noise intensity
for different values of e , a 0.7 . (b)
for different values of a , e 0.08 . Other parameters
are: p 0.1, A 0.01 and 0 . As a core component of neuronal networks, single neuron and its properties play a fundamental
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role in networked dynamics. Therefore, we investigate the effect of model parameters of single neuron on the collective resonance response within adaptive small-world neuronal networks. As shown in Fig. 13(a), with the enlargement of time ratio e , the resonance point moves rightward to a
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larger noise intensity and the maximal Q decreases accordingly, indicating that large value of e reduces the ability of neuronal network in weak signal detection and transmission. The obtained results attribute to that single neuron spikes more and more difficultly with the increase of e (presented detailed in Fig. 14, where direct current I =0.09 is induced to evoke neuron to spike).
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However, the phenomenon of stochastic resonance vanish when e is large enough, as the resonance
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response only increases monotonously with noise intensity. Moreover, the transition of stochastic
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resonance for different values of a is similar to the case of time ratio e (as shown in Fig. 13(b) and Fig. 15). A conclusion can be drawn that the properties of single neuron are of great importance
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to the networked resonance response.
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Fig. 14 Neuronal firing sequence obtained for different values of time ratio e . (a) e=0.07 , (b)
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e=0.08 , (c) e=0.09 , (d) e=0.1 and (e) e=0.15 . Other parameters are: a=0.7 , b=0.5 , I =0.09 .
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Fig. 15 Neuronal firing sequence obtained for different values of time ratio a . (a) a=0.6 , (b)
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a=0.65 , (c) a=0.7 , (d) a=0.75 and (e) a=0.8 . Other parameters are: e=0.08 , b=0.5 , I =0.09 .
V. CONCLUSIONS In this paper we study adaptive stochastic resonance in time-delayed Newman-Watts small-world neuronal networks self-organized via spike-timing-dependent plasticity. The obtained numerical results show that, with the absence of transmission delay, the temporal response of the neural system
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to the external periodic signal can be optimized by an intermediate intensity of additive noise, which implies the occurrence of stochastic resonance. It is also indicated that, STDP in coupling process can slightly depress the efficiency of network stochastic resonance, as the system with larger adjusting rate can reach a much lower peak value near the resonance noise intensity. This observation
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is accounted for that, large adjusting rate, which can result in the increase of the fraction of synapses with small strength, will impair the excitability of the considered neural network and consequently adverse to the communication among neurons. In addition, the effect of stochastic resonance can be
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either promoted or destroyed by time delay. Delay-induced multiple stochastic resonance appear intermittently at the integer multiples of period of the subthreshold forcing. Furthermore, it is demonstrated that the small-world topology plays a constructive role in the stochastic resonance in
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neural systems. For small and moderate adjusting rate of STDP, fine-tuning of the probability of adding links can significantly enhance the effect of stochastic resonance in adaptive neural network.
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Additionally, there is an optimal probability of adding links by which the noise-induced transmission
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of weak periodic signal peaks and the span of optimal normalized link for small-world neuronal networks shrinks with the increase of time delay. Finally, the networked stochastic resonance can
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also be influenced by single neuron properties. In a word, all the features of STDP, time delay, and
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connectivity structure functions together in the stochastic resonance in small-world 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).
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