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Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks
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Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks Qi Wang, Yubing Gong∗ School of Physics and Optoelectronic Engineering, Ludong University, Yantai, Shandong 264025, China
a r t i c l e
i n f o
Article history: Received 11 July 2015 Revised 19 February 2016 Accepted 3 March 2016 Available online xxx Keywords: Neuron Autapse Newman–Watts network Multiple coherence resonance Synchronization transitions
a b s t r a c t In this paper, we study the effect of delayed autaptic self-feedback activity on the spiking temporal coherence and spatial synchronization of Newman–Watts networks of stochastic Hodgkin–Huxley neurons. As autaptic delay is varied, the spiking behaviors of the neurons intermittently become synchronous and non-synchronous, and meanwhile they are ordered and disordered, exhibiting both synchronization transitions (ST) and multiple coherence resonance (MCR). Moreover, the autaptic delays for CR and synchronization are close. When coupling strength or network randomness increases, or when channel noise intensity decreases, MCR is enhanced, but ST becomes strongest at optimal coupling strength, network randomness, or channel noise intensity. These results show that autaptic activity can induce both MCR and ST in the neuronal networks. This implies that autaptic activity can simultaneously enhance or reduce the temporal coherence and synchronization of the neuronal network. These findings could find potential implications for the information processing and transmission in neural systems. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Information transmission delays are inherent to neural systems due to the finite propagation speeds and time lapses occurring in both dendritic and synaptic processing [1]. Physiological experiments have revealed that time delay introduced by chemical and electrical synapses synapses can be up to several tenths of milliseconds in length and comparably short about 0.05 ms, respectively [2,3]. Several decades ago, Van der Loos and Glaser found a special synapse, known as autapse which occurs between the dendrites and axon of the same neuron and connects a neuron to itself, and these self-connections could establish a time-delayed feedback mechanism at the cellular level [4]. Autapses serve as feedback circuits, which are common in the nervous system and have been discovered in a variety of brain areas. Tamas et al. showed anatomically that inhibitory interneurons in visual cortex form approximately 10–30 autapses [5]. Lübke et al. presented that autaptic connections exist in approximately 80% of the cortical pyramidal neurons, including neurons of the human brain [6]. Bacci et al. reported that fast-spiking but not low-threshold spiking interneurons of layer V in neocortical slices exhibit inhibitory autaptic activity [7]. Over the past decade, the effects of autapse on the firing dynamics of neurons have been extensively studied [8–21], for example, Bacci and Huguenard experimentally found that autaptic transmission enhances the precision of spike times of neurons [10]; Popovych et al. showed that time-delayed self-feedback can desynchronize groups of model ∗
Corresponding author. Tel.: +86 5356697550. E-mail address: [email protected] (Y. Gong).
http://dx.doi.org/10.1016/j.apm.2016.03.003 S0307-904X(16)30126-3/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: Q. Wang, Y. Gong, Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.03.003
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neurons [12]; Prager et al. reported a semi-analytical method to study noise induced oscillation with time-delayed feedback [13]; Saada et al. found that an autapse can mediate positive feedback, which maintains persistent activity [16]; Rusin et al. experimentally demonstrated that delayed self-feedback stimulation can engineer the synchronization of action potentials in cultured neurons [18]; Hashemi et al. showed that the spike rate of a neuron depended on the duration of the activity of autapse [19]. Recently, Wang et al. found that delayed autaptic activity switches the spiking activity among quiescent, periodic and chaotic firing patterns in a Hindmarsh–Rose neuron [20], and the firing frequencies and inter-spike interval distribution of the spike train of a neuron shows periodic behaviors as autaptic delay time is increased [21]. They have also reviewed the firing dynamics of an autaptic neuron [22]. Very recently, Yilmaz and Özer have found that autaptic activity can enhanced detection of weak periodic signals in a stochastic Hodgkin–Huxley neuron [23] and pacemaker induced stochastic resonance in a scale-free neuronal network [24], as well as pacemaker mediated propagation of weak rhythmic activity across small-world neuronal networks [25]. Stochastic resonance (SR) is a counterintuitive phenomenon that a suitable level of noise enhances the response of a nonlinear system to external signals, and coherence resonance (CR) is a phenomenon that a suitable level of noise amplifies the intrinsic oscillation signal of nonlinear systems. SR and CR have been extensively studied in various nonlinear systems including neuronal systems over the past decades [26–28]. Synchronization phenomenon often occurs in the spiking activity of coupled neurons, and it is particularly correlated with many physiological mechanisms of normal and pathological brain functions including several neural diseases [29–33]. In the past years, synchronization phenomenon has been widely studied in complex networks including neuronal networks [34,35]. In recent decade, people have found some novel SR and synchronization phenomena in neuronal systems, such as temporal and spatial SR and CR in excitable and neuronal systems [36–42], channel blocking enhanced spiking regularity in clustered neuronal networks [43], and synchronization due to time delay in neuronal networks [44–49]. In particular, multiple SR (MSR) and multiple CR (MCR) (i.e., SR or CR intermittently appear with a varying parameter) [50–52], as well as synchronization transitions (ST) due to time delay [53–57], coupling strength [56–60], and noise [61,62] have been found in various neuronal networks. Very recently, we have observed ST induced by channel noise in delayed Newman–Watts networks of stochastic Hodgkin–Huxley (HH) neurons [63], ST induced by autaptic delay in non-delayed Newman–Watts HH neuron networks [64] and in delayed Newman–Watts HH neuron networks [65]. So far, however, the effect of autaptic delay on the synchronization of stochastic HH neuron networks has not yet been studied. Particularly, it is not clear if autaptic delay can induce MCR or both MCR and ST in stochastic HH neuron networks. In this paper, we study the effect of autaptic self-feedback activity on the temporal coherence and spatial synchronization of Newman–Watts stochastic HH neuron networks. We aim to investigate if autaptic activity can induce MCR and ST in the neuronal network. We first study how autaptic delay induces MCR and ST, and then explore the effects of coupling strength, network randomness and channel noise on the MCR and ST. We find that the neurons exhibit both MCR and ST as autaptic delay is varied, and CR and synchronization appear almost at the same autaptic delays. When coupling strength or network randomness increases, MCR is enhanced and ST becomes strongest at optimal coupling strength or network randomness. Similarly, when channel noise intensity decreases, MCR is enhanced and ST becomes strongest at optimal channel noise intensity. Finally, mechanisms are briefly discussed, and conclusion is given. 2. Model and equations According to Hodgkin and Huxley’s work [66], the dynamics of the membrane potentials of a HH neuron with a large number of ion channels can be described by deterministic equation:
C
dV = −gNa m3 h(V − VNa ) − gK n4 (V − VK ) − gL (V − VL ) + Iaut , dt
(1)
where C = 1 μF cm−2 is the membrane capacitance, and VNa = 50 mV, VK = −77 mV, VL = −54.4 mV are the reversal potentials for the sodium, potassium, and leakage currents, respectively, and gK = 36 mS cm−2 and gK = 120 mS cm−2 gL = 0.3 mS cm−2 are maximal conductances for potassium, sodium, and leakage currents, respectively. However, for a HH neuron with a small number of ion channels, there is fluctuation (noise) arising from stochastic opening–closing of the channel gates, and the dynamics of the neuron should be described by stochastic HH model. The stochastic gating variables m, h, and n obey the following Langevin equations [67,68]:
dm = αm (V )(1 − m ) − βm (V )m + ξm (t ), dt dh = αh (V )(1 − h ) − βh (V )h + ξh (t ), dt dn = αn (V )(1 − n ) − βn (V )n + ξn (t ), dt
(2a) (2b) (2c)
with the experimentally determined voltage-dependent opening and closing transition rates:
αm (V ) =
0.1(V + 40 ) , 1 − exp [−(V + 40 )/10]
(3a)
Please cite this article as: Q. Wang, Y. Gong, Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.03.003
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3
βm (V ) = 4 exp [−(V + 65)/18],
(3b)
αh (V ) = 0.07 exp [−(V + 65)/20],
(3c)
βh (V ) = {1 + exp [−(V + 35)/10]}−1 ,
(3d)
0.01(V + 55 ) , 1 − exp [−(V + 55 )/10]
αn (V ) =
(3e)
βn (V ) = 0.125 exp [−(V + 65)/80],
(3f) = 0 and ξx (t )ξx (t ) = Dx δ (t − t ). Dx = m,h,n represent channel
where ξx = m,n,h (t ) are Gaussian white noises with ξx (t ) noise intensities, which are inversely proportional to the total number of sodium or potassium channels in the membrane patch:
Dm =
2 NNa
αm (V )βm (V ) , αm (V ) + βm (V )
(4a)
Dh =
2 NNa
αh (V )βh (V ) , αh (V ) + βh (V )
(4b)
Dn =
2 NK
αn (V )βn (V ) . αn (V ) + βn (V )
(4c)
With assumption of homogeneous ion channel densities ρNa = 60 μm−2 and ρK = 18 μm−2 , we have ion channel numbers NNa = ρNa S and NK = ρK S, where S is membrane patch size. As S decreases, the fluctuation of the number of open ion channels increases and hence channel noise intensity increases. Iaut is autaptic current. Here we use electrical autaptic current in diffusive coupling form [17,20]:
Iaut = gaut [V (t − τ ) − V (t )],
(5)
and chemical autaptic current in nonlinear coupling form [20,69]:
Iaut = −gaut [V (t ) − Vsyn ]S(t − τ ), S(t − τ ) = 1/{1 + exp [−k(V (t − τ ) − θ )]},
(6)
where gaut is autaptic coupling strength, V (t − τ ) is the action potential at earlier time t − τ , τ (in unit of ms) is autaptic delayed time. We assume all neurons have equal gaut and τ . Other parameter values are: Vsyn = 2 mV, k = 8,θ = −0.25. According to Newman–Watts topology [70], the present neuronal network comprising of N = 60 identical HH neurons starts with an originally regular ring in which each unit is connected to its two nearest neighbors, and then links are randomly added with probability p (network randomness) between non-nearest vertices. When all neurons are coupled with each other, the network contains N (N − 1 )/2 edges. The number of added random shortcuts M satisfies M = p × N (N − 1 )/2. If p = 0, the network is a regular ring, while it is globally coupled for p = 1. The Newman–Watts small-world network occurs for 0 < p < 1. Note that there are a lot of network realizations for a given p. In the presence of autaptic current, the dynamics of Newman–Watts stochastic HH neuron networks can be written as:
dVi (t ) = −gNa mi 3 hi (Vi − VNa ) − gK ni 4 (Vi − VK ) − GL (Vi − VL ) + εi j V j (t ) − Vi (t ) + Iauti , dt N
C
(7)
j=1
d xi = αxi (Vi )(1 − xi ) − βxi (Vi )xi + ξxi (t ), dt
(x = m, h, n ),
(8) N
where Vi (t ) and V j (t ) (1 ≤ i, j ≤ N ) are the action potentials of neurons i and j at time t , j=1 εi j [V j (t ) − Vi (t )] is electrical coupling term, and εi j is coupling strength between neurons i and j; if there is coupling, εi j = ε , otherwise, εi j = 0. The temporal coherence of the spikes of a neuron on the network can be characterized by the inverse of coefficient of variationλ, which is defined as [71]: N 1 λ= λi N i=1
with
λi =
k Ti
2 , 2 (Tik ) − Tik
(9)
η where Tik = tk − tk−1 is interspike time interval, tk is the time of the kth spike in the spike train;Tik = η1 k=1 (tk − tk−1 ) 2 2 η 2 and (Tik ) = η1 k=1 (tk − tk−1 ) are mean and mean-squared values of Tik , respectively, and (Tik ) − Tik 2 is the variance of interspike intervals. λi measures the degree of spiking regularity of each neuron; λ is the average of λi overall neurons, which represents average spiking regularity of each neuron. A larger λ denotes higher average spiking regularity and temporal coherence of each neuron. The detection threshold for the occurrence of a spike isVi (t ) = 0. Please cite this article as: Q. Wang, Y. Gong, Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.03.003
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Fig. 1. Spatiotemporal patterns of the membrane potentials for different autaptic delay τ at p = 0.1,ε = 0.12,gaut = 0.15,S = 8 μm2 . As τ increases, the neurons intermittently become ordered and synchronous where τ = 18, 32, 54 and disordered and non-synchronous where τ = 10, 28, 50, exhibiting both MCR and ST.
The spatial synchronization of the neurons on the network is characterized by standard deviation σ defined as [71]:
σ = [σ (t )] with σ (t ) =
1 N
N
i=1 Vi
(t )2 −
1 N N
N−1
i=1 Vi
2 (t )
,
(10)
2 1 N 2 where N1 N i=1 Vi (t ) − ( N i=1 Vi (t ) ) is the variance of the membrane potentials of all neurons at each time t; · denotes the average over time and [·] the average over different network realizations for each set of parameter values. Larger σ (t ) represents larger deviation between the neurons, and smaller σ (t ) shows higher synchronization. Numerical integrations of Eqs. (7) and (8) are carried out using explicit Euler algorithm with time step of t = 0.001 ms. Periodic boundary conditions are used and the parameter values for all neurons are identical except for the noise terms ξxi (t ). 3. Results and discussion 3.1. MCR and ST induced by autaptic delay Throughout this paper, we set p = 0.1, ε = 0.12, gaut = 0.15 and S = 8 μm2 unless stated otherwise. Using Iaut described by Eq. (6), we first study the effect of chemical autaptic delay on the temporal coherence and synchronization of the spiking activity of the neuronal networks. In Fig. 1, the spatiotemporal patterns of the membrane potentials of the neurons for different values of autaptic delay τ are displayed. It is shown that the both the regularity and the synchrony of the spiking behaviors of the neurons change for different τ values. As τ increases, they intermittently become non-synchronous at τ = 10, 28, 50 and synchronous at τ = 18, 32, 54, and meanwhile they are also disordered at τ = 10, 28, 50 and ordered at τ = 18, 32, 54. This shows that the neurons can exhibit both ST and MCR as autaptic delay is varied, and the autaptic delays for the occurrence of CR and synchronization are very close. These phenomena can be quantified by the inverse of coefficient of variation λ described by Eq. (9) and standard deviation σ described by Eq. (10). We plot λ and σ as a function of τ in Fig. 2(a) and (b), respectively (red curves). Fig. 2(a) shows that λ passes through a few peaks at τ = 18, 32, 54 and valleys at τ = 10, 28, 50 as τ increases, which quantitatively characterize the occurrence of MCR. Fig. 2(b) shows that σ undergoes a few peaks at τ = 10, 28, 50 and valleys at τ = 18, 32, 54, which quantitatively characterize the occurrence of ST. The results in Fig. 2(a) and (b) clearly quantify the presence of MCR and ST induced by autaptic delay. Meanwhile, they show that the autaptic delays for the occurrence of CR and synchronization are close. We have also obtained results for other autaptic strengths gaut . MCR and ST for different gaut values are displayed in Fig. 2(a) and (b), respectively. It is shown that the amplitudes of λ peaks become higher when gaut increases, but the amplitudes of σ peaks increase and become highest at around gaut = 0.5, and then they decrease for gaut > 0.8. Please cite this article as: Q. Wang, Y. Gong, Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.03.003
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Fig. 2. Dependence of (a)λ and (b)σ on τ for different autaptic strength gaut at p = 0.1, ε = 0.12,S = 8 μm2 . As τ increases, λ and σ pass through a few peaks, quantifying the occurrence of MCR and ST. As gaut increases, the amplitudes of λ peaks increase, but there is optimal gaut by which the amplitudes of σ peaks become highest. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Contour plot of (a)λ and (b)σ as functions of τ and gaut at p = 0.1,ε = 0.12, S = 8 μm2 . As gaut increases, λ islands become more and more pronounced, but there is optimal gaut by which σ islands become most pronounced.
To get a global view, we have studied for more other autaptic strengths gaut . All results are displayed via the contour plot of λ and σ as functions of gaut and τ in Fig. 3. It is seen from Fig. 3(a) that λ passes through a few islands as τ increases. However, as gaut increases, λ islands become more and more pronounced, From Fig. 3(b), it is seen that σ passes through a few islands as τ increases, and the islands are enhanced and become most pronounced at around gaut = 0.45. This shows that MCR is enhanced when autaptic strength increases, while there is optimal autaptic strength by which ST becomes strongest. Please cite this article as: Q. Wang, Y. Gong, Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.03.003
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Fig. 4. Dependence of (a)λ and (b)σ on τ for different coupling strengthε atgaut = 0.15, p = 0.1, S = 8 μm2 . As ε increases, the amplitudes of λ peaks always increase, but the amplitudes of σ peaks become highest at aroundε = 0.04.
3.2. Effects of coupling strength, network randomness and channel noise on MCR and ST induced by autaptic delay In this section, we study the effects of coupling strength, network randomness and channel noise intensity on ST and MCR induced by autaptic delay. First, we study the effect of coupling strength ε on MCR and ST induced by autaptic delay. In Fig. 4, we plot λ and σ as a function of τ for different ε values at gaut = 0.15, p = 0.1, S = 8 μm2 . Fig. 4(a) shows that the amplitudes of λ peaks increase as ε increases, but, λ peaks decrease more rapidly with the increase of τ for a largerε . For instance, for ε = 0.12 the amplitudes of λ peaks gradually decrease with increasing τ , and for ε = 0.3 λ peaks almost disappear for τ > 40. This shows that MCR is enhanced when coupling strength increases. However, for bigger coupling strengths, MCR is weakened as autaptic delays increases. Fig. 4(b) shows that σ passes through a few peaks with the increase of τ . As ε increases, the amplitudes of σ peaks increase and become highest at around ε = 0.04 and then decrease again. This shows that ST induced by autaptic delay is strongest when coupling strength is optimal. Secondly, we study the effect of network randomness p on MCR and ST induced by autaptic delay. In Fig. 5(a) and (b), λ and σ are plotted as a function of τ for different p values, respectively. It is shown that λ passes through a few peaks as τ increases, and the amplitudes of λ peaks increase with the increase of p. However, for p > 0.05, the amplitudes of λ peaks decrease as τ increases. This shows that MCR induced by autaptic delay enhances with the increase of network randomness. However, it is weakened with the increase of autaptic delay for a larger number of random shortcuts. Fig. 5(b) shows that there are a few σ peaks with the increase of τ , and the amplitudes of σ peaks become highest at around p = 0.01, and then they decrease again. This shows that there is optimal network randomness by which ST induced by autaptic delay becomes strongest. Finally, we study the effect of channel noise intensity on MCR and ST induced by autaptic delay. In Fig. 6, λ and σ are plotted as a function of τ for different S values. Fig. 6(a) shows that the amplitudes of λ peaks increase as S increases (channel noise intensity decreases). However, for a bigger S value the amplitudes of λ peaks decrease more rapidly with the increase of τ . For instance, for S = 8 μm2 the amplitudes of λ peaks begin to decrease with the increase of τ , and for S = 15 μm2 the amplitudes of λ peaks decrease more rapidly for τ > 40. This shows that MCR induced by autaptic delay is enhanced as channel noise intensity decreases, but, for a smaller channel noise intensity MCR is weakened more rapidly as autaptic delay increases. Fig. 6(b) shows that the amplitudes of σ peaks increase and become highest at around S = 8 μm2 , and then they decrease again. This shows that there is optimal channel noise intensity by which ST induced by autaptic delay becomes strongest. Using Iaut described in Eq. (5), we have also studied ST induced by electrical autaptic current and obtained qualitatively similar results. We now briefly discuss the mechanisms for the above results. In Ref. [71], the bifurcation analysis showed that time delay can induce a sudden jump in the frequency (phase) of the oscillation and makes the oscillators change from higher to low synchronization state, or vice versa. It is concluded that time delay can cause phase slips and induce ST [54,72]. Similarly, autaptic self-feedback delay can also induce ST by introducing phase slips in the spiking activity of a neuron itself. On the other hand, Wang et al showed that MSR induced by time delay in neuronal networks is attributed to the locking between Please cite this article as: Q. Wang, Y. Gong, Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.03.003
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Fig. 5. Dependence of (a)λ and (b)σ on τ for different network randomness p at gaut = 0.15,ε = 0.12, S = 8 μm2 . As p increases, the amplitudes of λ peaks increase, but the amplitudes of σ peaks become highest at around p = 0.01.
Fig. 6. Plots of (a)λ and (b)σ against τ for different patch sizes S atgaut = 0.15, p = 0.1, ε = 0.12. As S increases, the amplitudes of λ peaks increase, but the amplitudes of σ peaks become highest at around S = 8 μm2 .
the delay length and the global resonant oscillation period of individual neurons [50]. From Fig. 1, it is seen that the resonant oscillation period is T = 18–20 ms, and the spiking behaviors of the neurons become ordered at around τ = 18, 32, 54, which are very close to integer multiple of the resonant oscillation period. This shows that MCR induced by autaptic delay is also attributed to the locking between the delay length and the resonant oscillation period. On the other hand, as autaptic strength increases, the self-feedback coupling of the action potentials of a neuron is enhanced, and hence MCR induced by autaptic delay is enhanced as well. However, for too big self-feedback coupling, it is difficult for autaptic delay to change the synchrony state of the spiking behaviors of a neuron, and thus ST is weakened. Therefore, there exists intermediate and optimal autaptic strength by which ST induced by autaptic delay become strongest. The effects of coupling strength, network randomness and channel noise can be explained as follows. When coupling strength or network randomness increases, or when channel noise intensity decreases, the neurons are coupled to each other more strongly, and thus MCR is enhanced. It is shown that the synchronization of the Newman–Watts neuronal network Please cite this article as: Q. Wang, Y. Gong, Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.03.003
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increases as network randomness increases [71]. For too small network randomness, the neurons are in low synchronization, while, for too big network randomness, the neurons are highly synchronized. In these two cases, it is difficult for autaptic self-feedback activity to change the synchronization states, and thus ST induced by autaptic delay is weak. When network randomness is intermediate, the neurons are moderately synchronized, and autaptic delay can change the synchronization state more effectively, and thus ST induced by autaptic delay is strong. In particular, when network randomness is optimal, ST induced by autaptic delay becomes strongest. The mechanisms for the effects of coupling strength and channel noise intensity on ST by autaptic delay are similar. It should be noted that, in our previous work [63], we studied the effect of channel noise on the synchronization of Newman–Watts HH neuron networks with synaptic time delays, and ST induced by channel noise are observed. While, in the present work, we have studied the effect of autaptic delay on the synchronization of Newman–Watts networks of stochastic HH neurons, and both MCR and ST induced by autaptic delay are observed. Thus, the model and the results obtained in the present work are completely different from those in previous works. 4. Conclusion In summary, we have numerically studied the effect of delayed autaptic self-feedback activity on the temporal coherence and spatial synchronization of Newman–Watts network of stochastic HH neurons. It is found that the neurons can exhibit both MCR and ST as autaptic delay is varied, and CR and synchronization occur almost at the same autaptic delays. When autaptic strength increases, MCR is enhanced, but ST becomes strongest at optimal autaptic strength. When coupling strength or network randomness increases, MCR is enhanced, but ST becomes strongest when it is optimal. Similarly, when channel noise intensity decreases, MCR is enhanced, but there is optimal channel noise level by which ST becomes strongest. Mechanisms for the results are briefly discussed. These results show that delayed autaptic self-feedback activity can induce both MCR and ST in the neuronal networks. This implies that sutaptic delay can simultaneously enhance or reduce the temporal coherence and spatial synchronization of the spiking activity of the neurons. These findings could find potential implications for the information processing and transmission in neural systems. Acknowledgment This work was supported by the Natural Science Foundation of Shandong Province of China (ZR2012AM013). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks Qi Wang, Yubing Gong∗ School of Physics and Optoelectronic Engineering, Ludong University, Yantai, Shandong 264025, China
a r t i c l e
i n f o
Article history: Received 11 July 2015 Revised 19 February 2016 Accepted 3 March 2016 Available online xxx Keywords: Neuron Autapse Newman–Watts network Multiple coherence resonance Synchronization transitions
a b s t r a c t In this paper, we study the effect of delayed autaptic self-feedback activity on the spiking temporal coherence and spatial synchronization of Newman–Watts networks of stochastic Hodgkin–Huxley neurons. As autaptic delay is varied, the spiking behaviors of the neurons intermittently become synchronous and non-synchronous, and meanwhile they are ordered and disordered, exhibiting both synchronization transitions (ST) and multiple coherence resonance (MCR). Moreover, the autaptic delays for CR and synchronization are close. When coupling strength or network randomness increases, or when channel noise intensity decreases, MCR is enhanced, but ST becomes strongest at optimal coupling strength, network randomness, or channel noise intensity. These results show that autaptic activity can induce both MCR and ST in the neuronal networks. This implies that autaptic activity can simultaneously enhance or reduce the temporal coherence and synchronization of the neuronal network. These findings could find potential implications for the information processing and transmission in neural systems. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Information transmission delays are inherent to neural systems due to the finite propagation speeds and time lapses occurring in both dendritic and synaptic processing [1]. Physiological experiments have revealed that time delay introduced by chemical and electrical synapses synapses can be up to several tenths of milliseconds in length and comparably short about 0.05 ms, respectively [2,3]. Several decades ago, Van der Loos and Glaser found a special synapse, known as autapse which occurs between the dendrites and axon of the same neuron and connects a neuron to itself, and these self-connections could establish a time-delayed feedback mechanism at the cellular level [4]. Autapses serve as feedback circuits, which are common in the nervous system and have been discovered in a variety of brain areas. Tamas et al. showed anatomically that inhibitory interneurons in visual cortex form approximately 10–30 autapses [5]. Lübke et al. presented that autaptic connections exist in approximately 80% of the cortical pyramidal neurons, including neurons of the human brain [6]. Bacci et al. reported that fast-spiking but not low-threshold spiking interneurons of layer V in neocortical slices exhibit inhibitory autaptic activity [7]. Over the past decade, the effects of autapse on the firing dynamics of neurons have been extensively studied [8–21], for example, Bacci and Huguenard experimentally found that autaptic transmission enhances the precision of spike times of neurons [10]; Popovych et al. showed that time-delayed self-feedback can desynchronize groups of model ∗
Corresponding author. Tel.: +86 5356697550. E-mail address: [email protected] (Y. Gong).
http://dx.doi.org/10.1016/j.apm.2016.03.003 S0307-904X(16)30126-3/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: Q. Wang, Y. Gong, Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.03.003
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neurons [12]; Prager et al. reported a semi-analytical method to study noise induced oscillation with time-delayed feedback [13]; Saada et al. found that an autapse can mediate positive feedback, which maintains persistent activity [16]; Rusin et al. experimentally demonstrated that delayed self-feedback stimulation can engineer the synchronization of action potentials in cultured neurons [18]; Hashemi et al. showed that the spike rate of a neuron depended on the duration of the activity of autapse [19]. Recently, Wang et al. found that delayed autaptic activity switches the spiking activity among quiescent, periodic and chaotic firing patterns in a Hindmarsh–Rose neuron [20], and the firing frequencies and inter-spike interval distribution of the spike train of a neuron shows periodic behaviors as autaptic delay time is increased [21]. They have also reviewed the firing dynamics of an autaptic neuron [22]. Very recently, Yilmaz and Özer have found that autaptic activity can enhanced detection of weak periodic signals in a stochastic Hodgkin–Huxley neuron [23] and pacemaker induced stochastic resonance in a scale-free neuronal network [24], as well as pacemaker mediated propagation of weak rhythmic activity across small-world neuronal networks [25]. Stochastic resonance (SR) is a counterintuitive phenomenon that a suitable level of noise enhances the response of a nonlinear system to external signals, and coherence resonance (CR) is a phenomenon that a suitable level of noise amplifies the intrinsic oscillation signal of nonlinear systems. SR and CR have been extensively studied in various nonlinear systems including neuronal systems over the past decades [26–28]. Synchronization phenomenon often occurs in the spiking activity of coupled neurons, and it is particularly correlated with many physiological mechanisms of normal and pathological brain functions including several neural diseases [29–33]. In the past years, synchronization phenomenon has been widely studied in complex networks including neuronal networks [34,35]. In recent decade, people have found some novel SR and synchronization phenomena in neuronal systems, such as temporal and spatial SR and CR in excitable and neuronal systems [36–42], channel blocking enhanced spiking regularity in clustered neuronal networks [43], and synchronization due to time delay in neuronal networks [44–49]. In particular, multiple SR (MSR) and multiple CR (MCR) (i.e., SR or CR intermittently appear with a varying parameter) [50–52], as well as synchronization transitions (ST) due to time delay [53–57], coupling strength [56–60], and noise [61,62] have been found in various neuronal networks. Very recently, we have observed ST induced by channel noise in delayed Newman–Watts networks of stochastic Hodgkin–Huxley (HH) neurons [63], ST induced by autaptic delay in non-delayed Newman–Watts HH neuron networks [64] and in delayed Newman–Watts HH neuron networks [65]. So far, however, the effect of autaptic delay on the synchronization of stochastic HH neuron networks has not yet been studied. Particularly, it is not clear if autaptic delay can induce MCR or both MCR and ST in stochastic HH neuron networks. In this paper, we study the effect of autaptic self-feedback activity on the temporal coherence and spatial synchronization of Newman–Watts stochastic HH neuron networks. We aim to investigate if autaptic activity can induce MCR and ST in the neuronal network. We first study how autaptic delay induces MCR and ST, and then explore the effects of coupling strength, network randomness and channel noise on the MCR and ST. We find that the neurons exhibit both MCR and ST as autaptic delay is varied, and CR and synchronization appear almost at the same autaptic delays. When coupling strength or network randomness increases, MCR is enhanced and ST becomes strongest at optimal coupling strength or network randomness. Similarly, when channel noise intensity decreases, MCR is enhanced and ST becomes strongest at optimal channel noise intensity. Finally, mechanisms are briefly discussed, and conclusion is given. 2. Model and equations According to Hodgkin and Huxley’s work [66], the dynamics of the membrane potentials of a HH neuron with a large number of ion channels can be described by deterministic equation:
C
dV = −gNa m3 h(V − VNa ) − gK n4 (V − VK ) − gL (V − VL ) + Iaut , dt
(1)
where C = 1 μF cm−2 is the membrane capacitance, and VNa = 50 mV, VK = −77 mV, VL = −54.4 mV are the reversal potentials for the sodium, potassium, and leakage currents, respectively, and gK = 36 mS cm−2 and gK = 120 mS cm−2 gL = 0.3 mS cm−2 are maximal conductances for potassium, sodium, and leakage currents, respectively. However, for a HH neuron with a small number of ion channels, there is fluctuation (noise) arising from stochastic opening–closing of the channel gates, and the dynamics of the neuron should be described by stochastic HH model. The stochastic gating variables m, h, and n obey the following Langevin equations [67,68]:
dm = αm (V )(1 − m ) − βm (V )m + ξm (t ), dt dh = αh (V )(1 − h ) − βh (V )h + ξh (t ), dt dn = αn (V )(1 − n ) − βn (V )n + ξn (t ), dt
(2a) (2b) (2c)
with the experimentally determined voltage-dependent opening and closing transition rates:
αm (V ) =
0.1(V + 40 ) , 1 − exp [−(V + 40 )/10]
(3a)
Please cite this article as: Q. Wang, Y. Gong, Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.03.003
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3
βm (V ) = 4 exp [−(V + 65)/18],
(3b)
αh (V ) = 0.07 exp [−(V + 65)/20],
(3c)
βh (V ) = {1 + exp [−(V + 35)/10]}−1 ,
(3d)
0.01(V + 55 ) , 1 − exp [−(V + 55 )/10]
αn (V ) =
(3e)
βn (V ) = 0.125 exp [−(V + 65)/80],
(3f) = 0 and ξx (t )ξx (t ) = Dx δ (t − t ). Dx = m,h,n represent channel
where ξx = m,n,h (t ) are Gaussian white noises with ξx (t ) noise intensities, which are inversely proportional to the total number of sodium or potassium channels in the membrane patch:
Dm =
2 NNa
αm (V )βm (V ) , αm (V ) + βm (V )
(4a)
Dh =
2 NNa
αh (V )βh (V ) , αh (V ) + βh (V )
(4b)
Dn =
2 NK
αn (V )βn (V ) . αn (V ) + βn (V )
(4c)
With assumption of homogeneous ion channel densities ρNa = 60 μm−2 and ρK = 18 μm−2 , we have ion channel numbers NNa = ρNa S and NK = ρK S, where S is membrane patch size. As S decreases, the fluctuation of the number of open ion channels increases and hence channel noise intensity increases. Iaut is autaptic current. Here we use electrical autaptic current in diffusive coupling form [17,20]:
Iaut = gaut [V (t − τ ) − V (t )],
(5)
and chemical autaptic current in nonlinear coupling form [20,69]:
Iaut = −gaut [V (t ) − Vsyn ]S(t − τ ), S(t − τ ) = 1/{1 + exp [−k(V (t − τ ) − θ )]},
(6)
where gaut is autaptic coupling strength, V (t − τ ) is the action potential at earlier time t − τ , τ (in unit of ms) is autaptic delayed time. We assume all neurons have equal gaut and τ . Other parameter values are: Vsyn = 2 mV, k = 8,θ = −0.25. According to Newman–Watts topology [70], the present neuronal network comprising of N = 60 identical HH neurons starts with an originally regular ring in which each unit is connected to its two nearest neighbors, and then links are randomly added with probability p (network randomness) between non-nearest vertices. When all neurons are coupled with each other, the network contains N (N − 1 )/2 edges. The number of added random shortcuts M satisfies M = p × N (N − 1 )/2. If p = 0, the network is a regular ring, while it is globally coupled for p = 1. The Newman–Watts small-world network occurs for 0 < p < 1. Note that there are a lot of network realizations for a given p. In the presence of autaptic current, the dynamics of Newman–Watts stochastic HH neuron networks can be written as:
dVi (t ) = −gNa mi 3 hi (Vi − VNa ) − gK ni 4 (Vi − VK ) − GL (Vi − VL ) + εi j V j (t ) − Vi (t ) + Iauti , dt N
C
(7)
j=1
d xi = αxi (Vi )(1 − xi ) − βxi (Vi )xi + ξxi (t ), dt
(x = m, h, n ),
(8) N
where Vi (t ) and V j (t ) (1 ≤ i, j ≤ N ) are the action potentials of neurons i and j at time t , j=1 εi j [V j (t ) − Vi (t )] is electrical coupling term, and εi j is coupling strength between neurons i and j; if there is coupling, εi j = ε , otherwise, εi j = 0. The temporal coherence of the spikes of a neuron on the network can be characterized by the inverse of coefficient of variationλ, which is defined as [71]: N 1 λ= λi N i=1
with
λi =
k Ti
2 , 2 (Tik ) − Tik
(9)
η where Tik = tk − tk−1 is interspike time interval, tk is the time of the kth spike in the spike train;Tik = η1 k=1 (tk − tk−1 ) 2 2 η 2 and (Tik ) = η1 k=1 (tk − tk−1 ) are mean and mean-squared values of Tik , respectively, and (Tik ) − Tik 2 is the variance of interspike intervals. λi measures the degree of spiking regularity of each neuron; λ is the average of λi overall neurons, which represents average spiking regularity of each neuron. A larger λ denotes higher average spiking regularity and temporal coherence of each neuron. The detection threshold for the occurrence of a spike isVi (t ) = 0. Please cite this article as: Q. Wang, Y. Gong, Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.03.003
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Fig. 1. Spatiotemporal patterns of the membrane potentials for different autaptic delay τ at p = 0.1,ε = 0.12,gaut = 0.15,S = 8 μm2 . As τ increases, the neurons intermittently become ordered and synchronous where τ = 18, 32, 54 and disordered and non-synchronous where τ = 10, 28, 50, exhibiting both MCR and ST.
The spatial synchronization of the neurons on the network is characterized by standard deviation σ defined as [71]:
σ = [σ (t )] with σ (t ) =
1 N
N
i=1 Vi
(t )2 −
1 N N
N−1
i=1 Vi
2 (t )
,
(10)
2 1 N 2 where N1 N i=1 Vi (t ) − ( N i=1 Vi (t ) ) is the variance of the membrane potentials of all neurons at each time t; · denotes the average over time and [·] the average over different network realizations for each set of parameter values. Larger σ (t ) represents larger deviation between the neurons, and smaller σ (t ) shows higher synchronization. Numerical integrations of Eqs. (7) and (8) are carried out using explicit Euler algorithm with time step of t = 0.001 ms. Periodic boundary conditions are used and the parameter values for all neurons are identical except for the noise terms ξxi (t ). 3. Results and discussion 3.1. MCR and ST induced by autaptic delay Throughout this paper, we set p = 0.1, ε = 0.12, gaut = 0.15 and S = 8 μm2 unless stated otherwise. Using Iaut described by Eq. (6), we first study the effect of chemical autaptic delay on the temporal coherence and synchronization of the spiking activity of the neuronal networks. In Fig. 1, the spatiotemporal patterns of the membrane potentials of the neurons for different values of autaptic delay τ are displayed. It is shown that the both the regularity and the synchrony of the spiking behaviors of the neurons change for different τ values. As τ increases, they intermittently become non-synchronous at τ = 10, 28, 50 and synchronous at τ = 18, 32, 54, and meanwhile they are also disordered at τ = 10, 28, 50 and ordered at τ = 18, 32, 54. This shows that the neurons can exhibit both ST and MCR as autaptic delay is varied, and the autaptic delays for the occurrence of CR and synchronization are very close. These phenomena can be quantified by the inverse of coefficient of variation λ described by Eq. (9) and standard deviation σ described by Eq. (10). We plot λ and σ as a function of τ in Fig. 2(a) and (b), respectively (red curves). Fig. 2(a) shows that λ passes through a few peaks at τ = 18, 32, 54 and valleys at τ = 10, 28, 50 as τ increases, which quantitatively characterize the occurrence of MCR. Fig. 2(b) shows that σ undergoes a few peaks at τ = 10, 28, 50 and valleys at τ = 18, 32, 54, which quantitatively characterize the occurrence of ST. The results in Fig. 2(a) and (b) clearly quantify the presence of MCR and ST induced by autaptic delay. Meanwhile, they show that the autaptic delays for the occurrence of CR and synchronization are close. We have also obtained results for other autaptic strengths gaut . MCR and ST for different gaut values are displayed in Fig. 2(a) and (b), respectively. It is shown that the amplitudes of λ peaks become higher when gaut increases, but the amplitudes of σ peaks increase and become highest at around gaut = 0.5, and then they decrease for gaut > 0.8. Please cite this article as: Q. Wang, Y. Gong, Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.03.003
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Fig. 2. Dependence of (a)λ and (b)σ on τ for different autaptic strength gaut at p = 0.1, ε = 0.12,S = 8 μm2 . As τ increases, λ and σ pass through a few peaks, quantifying the occurrence of MCR and ST. As gaut increases, the amplitudes of λ peaks increase, but there is optimal gaut by which the amplitudes of σ peaks become highest. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Contour plot of (a)λ and (b)σ as functions of τ and gaut at p = 0.1,ε = 0.12, S = 8 μm2 . As gaut increases, λ islands become more and more pronounced, but there is optimal gaut by which σ islands become most pronounced.
To get a global view, we have studied for more other autaptic strengths gaut . All results are displayed via the contour plot of λ and σ as functions of gaut and τ in Fig. 3. It is seen from Fig. 3(a) that λ passes through a few islands as τ increases. However, as gaut increases, λ islands become more and more pronounced, From Fig. 3(b), it is seen that σ passes through a few islands as τ increases, and the islands are enhanced and become most pronounced at around gaut = 0.45. This shows that MCR is enhanced when autaptic strength increases, while there is optimal autaptic strength by which ST becomes strongest. Please cite this article as: Q. Wang, Y. Gong, Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.03.003
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Fig. 4. Dependence of (a)λ and (b)σ on τ for different coupling strengthε atgaut = 0.15, p = 0.1, S = 8 μm2 . As ε increases, the amplitudes of λ peaks always increase, but the amplitudes of σ peaks become highest at aroundε = 0.04.
3.2. Effects of coupling strength, network randomness and channel noise on MCR and ST induced by autaptic delay In this section, we study the effects of coupling strength, network randomness and channel noise intensity on ST and MCR induced by autaptic delay. First, we study the effect of coupling strength ε on MCR and ST induced by autaptic delay. In Fig. 4, we plot λ and σ as a function of τ for different ε values at gaut = 0.15, p = 0.1, S = 8 μm2 . Fig. 4(a) shows that the amplitudes of λ peaks increase as ε increases, but, λ peaks decrease more rapidly with the increase of τ for a largerε . For instance, for ε = 0.12 the amplitudes of λ peaks gradually decrease with increasing τ , and for ε = 0.3 λ peaks almost disappear for τ > 40. This shows that MCR is enhanced when coupling strength increases. However, for bigger coupling strengths, MCR is weakened as autaptic delays increases. Fig. 4(b) shows that σ passes through a few peaks with the increase of τ . As ε increases, the amplitudes of σ peaks increase and become highest at around ε = 0.04 and then decrease again. This shows that ST induced by autaptic delay is strongest when coupling strength is optimal. Secondly, we study the effect of network randomness p on MCR and ST induced by autaptic delay. In Fig. 5(a) and (b), λ and σ are plotted as a function of τ for different p values, respectively. It is shown that λ passes through a few peaks as τ increases, and the amplitudes of λ peaks increase with the increase of p. However, for p > 0.05, the amplitudes of λ peaks decrease as τ increases. This shows that MCR induced by autaptic delay enhances with the increase of network randomness. However, it is weakened with the increase of autaptic delay for a larger number of random shortcuts. Fig. 5(b) shows that there are a few σ peaks with the increase of τ , and the amplitudes of σ peaks become highest at around p = 0.01, and then they decrease again. This shows that there is optimal network randomness by which ST induced by autaptic delay becomes strongest. Finally, we study the effect of channel noise intensity on MCR and ST induced by autaptic delay. In Fig. 6, λ and σ are plotted as a function of τ for different S values. Fig. 6(a) shows that the amplitudes of λ peaks increase as S increases (channel noise intensity decreases). However, for a bigger S value the amplitudes of λ peaks decrease more rapidly with the increase of τ . For instance, for S = 8 μm2 the amplitudes of λ peaks begin to decrease with the increase of τ , and for S = 15 μm2 the amplitudes of λ peaks decrease more rapidly for τ > 40. This shows that MCR induced by autaptic delay is enhanced as channel noise intensity decreases, but, for a smaller channel noise intensity MCR is weakened more rapidly as autaptic delay increases. Fig. 6(b) shows that the amplitudes of σ peaks increase and become highest at around S = 8 μm2 , and then they decrease again. This shows that there is optimal channel noise intensity by which ST induced by autaptic delay becomes strongest. Using Iaut described in Eq. (5), we have also studied ST induced by electrical autaptic current and obtained qualitatively similar results. We now briefly discuss the mechanisms for the above results. In Ref. [71], the bifurcation analysis showed that time delay can induce a sudden jump in the frequency (phase) of the oscillation and makes the oscillators change from higher to low synchronization state, or vice versa. It is concluded that time delay can cause phase slips and induce ST [54,72]. Similarly, autaptic self-feedback delay can also induce ST by introducing phase slips in the spiking activity of a neuron itself. On the other hand, Wang et al showed that MSR induced by time delay in neuronal networks is attributed to the locking between Please cite this article as: Q. Wang, Y. Gong, Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.03.003
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Fig. 5. Dependence of (a)λ and (b)σ on τ for different network randomness p at gaut = 0.15,ε = 0.12, S = 8 μm2 . As p increases, the amplitudes of λ peaks increase, but the amplitudes of σ peaks become highest at around p = 0.01.
Fig. 6. Plots of (a)λ and (b)σ against τ for different patch sizes S atgaut = 0.15, p = 0.1, ε = 0.12. As S increases, the amplitudes of λ peaks increase, but the amplitudes of σ peaks become highest at around S = 8 μm2 .
the delay length and the global resonant oscillation period of individual neurons [50]. From Fig. 1, it is seen that the resonant oscillation period is T = 18–20 ms, and the spiking behaviors of the neurons become ordered at around τ = 18, 32, 54, which are very close to integer multiple of the resonant oscillation period. This shows that MCR induced by autaptic delay is also attributed to the locking between the delay length and the resonant oscillation period. On the other hand, as autaptic strength increases, the self-feedback coupling of the action potentials of a neuron is enhanced, and hence MCR induced by autaptic delay is enhanced as well. However, for too big self-feedback coupling, it is difficult for autaptic delay to change the synchrony state of the spiking behaviors of a neuron, and thus ST is weakened. Therefore, there exists intermediate and optimal autaptic strength by which ST induced by autaptic delay become strongest. The effects of coupling strength, network randomness and channel noise can be explained as follows. When coupling strength or network randomness increases, or when channel noise intensity decreases, the neurons are coupled to each other more strongly, and thus MCR is enhanced. It is shown that the synchronization of the Newman–Watts neuronal network Please cite this article as: Q. Wang, Y. Gong, Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.03.003
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increases as network randomness increases [71]. For too small network randomness, the neurons are in low synchronization, while, for too big network randomness, the neurons are highly synchronized. In these two cases, it is difficult for autaptic self-feedback activity to change the synchronization states, and thus ST induced by autaptic delay is weak. When network randomness is intermediate, the neurons are moderately synchronized, and autaptic delay can change the synchronization state more effectively, and thus ST induced by autaptic delay is strong. In particular, when network randomness is optimal, ST induced by autaptic delay becomes strongest. The mechanisms for the effects of coupling strength and channel noise intensity on ST by autaptic delay are similar. It should be noted that, in our previous work [63], we studied the effect of channel noise on the synchronization of Newman–Watts HH neuron networks with synaptic time delays, and ST induced by channel noise are observed. While, in the present work, we have studied the effect of autaptic delay on the synchronization of Newman–Watts networks of stochastic HH neurons, and both MCR and ST induced by autaptic delay are observed. Thus, the model and the results obtained in the present work are completely different from those in previous works. 4. Conclusion In summary, we have numerically studied the effect of delayed autaptic self-feedback activity on the temporal coherence and spatial synchronization of Newman–Watts network of stochastic HH neurons. It is found that the neurons can exhibit both MCR and ST as autaptic delay is varied, and CR and synchronization occur almost at the same autaptic delays. When autaptic strength increases, MCR is enhanced, but ST becomes strongest at optimal autaptic strength. When coupling strength or network randomness increases, MCR is enhanced, but ST becomes strongest when it is optimal. Similarly, when channel noise intensity decreases, MCR is enhanced, but there is optimal channel noise level by which ST becomes strongest. Mechanisms for the results are briefly discussed. These results show that delayed autaptic self-feedback activity can induce both MCR and ST in the neuronal networks. This implies that sutaptic delay can simultaneously enhance or reduce the temporal coherence and spatial synchronization of the spiking activity of the neurons. These findings could find potential implications for the information processing and transmission in neural systems. Acknowledgment This work was supported by the Natural Science Foundation of Shandong Province of China (ZR2012AM013). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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Please cite this article as: Q. Wang, Y. Gong, Multiple coherence resonance and synchronization transitions induced by autaptic delay in Newman–Watts neuron networks, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.03.003