Spike-timing-dependent plasticity enhanced synchronization transitions induced by autapses in adaptive Newman-Watts neuronal networks

Accepted Manuscript Title: Spike-timing-dependent plasticity enhanced synchronization transitions induced by autapses in adaptive Newman-Watts neuronal networks Author: Yubing Gong Baoying Wang Huijuan Xie PII: DOI: Reference:

S0303-2647(16)30233-7 http://dx.doi.org/doi:10.1016/j.biosystems.2016.09.006 BIO 3704

To appear in:

BioSystems

Received date: Revised date: Accepted date:

5-5-2016 19-7-2016 21-9-2016

Please cite this article as: Gong, Yubing, Wang, Baoying, Xie, Huijuan, Spiketiming-dependent plasticity enhanced synchronization transitions induced by autapses in adaptive Newman-Watts neuronal networks.BioSystems http://dx.doi.org/10.1016/j.biosystems.2016.09.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Spike-timing-dependent plasticity enhanced synchronization transitions induced by autapses in adaptive Newman–Watts neuronal networks Yubing Gong1, Baoying Wang2, Huijuan Xie1 1. School of Physics and Optoelectronic Engineering, Ludong University, Yantai, Shandong 264025, China

2. Library, Ludong University, Yantai, Shandong 264025, China

Abstract: In this paper, we numerically study the effect of spike-timing-dependent plasticity (STDP) on synchronization transitions induced by autaptic activity in adaptive Newman–Watts Hodgkin–Huxley neuron networks. It is found that synchronization transitions induced by autaptic delay vary with the adjusting rate Ap of STDP and become strongest at a certain Ap value, and the Ap value increases when network randomness or network size increases. It is also found that the synchronization transitions induced by autaptic delay become strongest at a certain network randomness and network size, and the values increase and related synchronization transitions are enhanced when Ap increases. These results show that there is optimal STDP that can enhance the synchronization transitions induced by autaptic delay in the adaptive neuronal networks. These findings provide a new insight into the roles of STDP and autapses for the information transmission in neural systems. Keywords: Neuron; autapse; adaptive Newman–Watts network; spike-timing-dependent plasticity; synchronization transition.



Corresponding author. E-mail address: [email protected] (Y. Gong) 1

1. Introduction Synchronization is an important phenomenon occurring in many realistic systems of biology, ecology, and so on (Arenas et al., 2008; Suykens and Osipov, 2008). In neural systems, synchronization is correlated with many physiological mechanisms of normal and pathological brain functions (Gray and Singer, 1989; Bazhenov et al., 2001; Mehta et al., 2001) and particularly with several neurological diseases such as epilepsy and tremor in Parkinson’s disease (Levy et al., 2000; Mormann et al., 2003). In neural systems, information transmission between neurons occurs at electrical and chemical synapses, and information transmission delays are inherent due to the finite propagation speeds and time delays occurring by both dendritic and synaptic processing. Physiological experiments have revealed that the transmission delays introduced by chemical and electrical synapses are several tenths of milliseconds and 0.05 ms in length, respectively (Mann, 1981; Izhikevich, 2006). Several decades ago, Van der Loos and Glaser found a unique synapse, known as autapse. They pointed out that autapse occurs between 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 (Van der Loos and Glaser, 1972.). In addition, there are many studies reporting the possible existence of autapses in different brain regions (Tamás et al., 1997; Lübke et al., 1996; Bacci et al., 2003). In recent decade, the roles of autapses in the firing dynamics of neurons have been extensively studied. It is found that autaptic activity can enhance the precision of spike times of neurons (Bacci and Huguenard, 2006), engineer the synchronization of action potentials in cultured neurons (Rusin et al., 2011), induce rich firing patterns in a Hindmarsh–Rose model neuron (Wang et al., 2014), enhance pacemaker-induced

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stochastic resonance in a scale-free neuronal network (Yilmaz, et al., 2016a) and propagation of weak rhythmic activity across small-world neuronal networks (Yilmaz, et al., 2016b), and regulate the firing of interneurons (Guo, et al., 2016a, 2016b). Recently, a phenomenon of synchronization transitions has attracted increasing attention, and synchronization transitions induced by time delay (Wang et al., 2008; 2009a; 2009b; 2011; Gong et al., 2011; Guo et al., 2012; Qian et al., 2013; Wu et al., 2013), coupling strength (Xu et al., 2013; Sun et al., 2011), noise (Wu et al., 2014; 2015a; Wang et al., 2015a), and even autapses (Wu et al., 2015b; Wang et al., 2015b) have been observed in various neuronal networks. However, these studies are devoted to a static description of synaptic connections. In reality, neural networks are adaptive due to synaptic plasticity, and synaptic strength varies as a function of neuromodulation and time-dependent processes (Markram et al., 1997; Bi et al., 1998; Feldman and Brecht, 2005) One representative of this biological effect is spike-timing-dependent plasticity (STDP), which modulates coupling strength adaptively based on the relative timing between pre- and post-synaptic action potentials (Markram et al., 1997; Bi et al., 1998). A series of biological works have confirmed the existence of STDP in excitatory synapses onto neocortical (Feldman and Brecht, 2005) and hippocampal pyramidal neurons (Bi et al., 1998), excitatory neurons in auditory brainstem (Tzounopoulos et al., 2007), parvalbumin-expressing fast- spiking striatal interneurons (Fino et al., 2009.), etc. It is suggested that synaptic plasticity may account for learning and memory (Abbot and Nelson, 2007; Kim and Linden, 2007). In the past years, the roles of STDP in the synchronization of neuronal population have been intensively investigated (Ruan and Zhao, 2009; Nowotny et al., 2003; Zhigulin et al., 2003; Karbowski and Ermentrout, 2012; Perez and Uchida, 2011; Kube

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et al., 2008; Mikkelsen et al., 2013;Yu et al., 2015). It was found that STDP modifies the weights of synaptic connections in such a way that synchronization of neuronal activity is considerably weakened (Kube et al., 2008); STDP induces persistent irregular oscillations between strongly and weakly synchronized states (Mikkelsen et al., 2013); STDP can largely depress the temporal coherence and spatial synchrony induced by external noise and random shortcuts in Newman–Watts neuronal networks (Yu et al., 2015). So far, however, there are few studies on synchronization transitions in adaptive neuronal networks and, particularly, there is no study on synchronization transitions induced by autapses in adaptive neuronal networks with STDP. In this paper, we study how autapses induce synchronization transitions in adaptive Newman–Watts neuronal networks with STDP, focusing on how STDP influences the synchronization transitions. We first present synchronization transitions induced by autaptic delay when the adjusting rate of STDP is fixed, and then study the effect of STDP by investigating how the synchronization transitions vary when the adjusting rate of STDP is varied. We also study how the effect of network randomness and network size on synchronization transitions varies when the adjusting rate of STDP is varied. Finally, mechanism is briefly discussed and conclusion is given. 2. Model and equations Here Hodgkin–Huxley (HH) neuron model (Hodgkin and Huxley, 1952) and Newman–Watts (NW) networks are used. According to Newman–Watts (NW) topology (Newman and Watts, 1999), the present network comprising of N identical Hodgkin–Huxley neurons starts with a regular ring, each neuron having two nearest neighbors, and then links

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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 is M  p N(N 1) 2 . If p = 0, the network is a regular ring; for p = 1, the network is a globally coupled random network; for 0 < p < 1, a Newman–Watts small-world network occurs. Note that there are a lot of network realizations for a given p. In the presence of autapses, the dynamics of adaptive NW HH neuronal networks can be written as: C

dVi (t )  gNa mi3hi (Vi  VNa )  gK ni 4 (Vi  VK )  GL (Vi  VL )  I auti  Iisyn  i (t ) dt

(1)

where C  1 μFcm-2 is the membrane capacitance; gK  36 , gNa  120 , and GL  0.3 mS cm-2 are the maximal conductance of potassium, sodium, and leakage currents, respectively;

VK  77 mV , VNa  50 mV , and VL  54.4 mV represent corresponding reversal potentials. i  t  is Gaussian white noises with zero mean i (t )  0 and auto-correlation functions i (t ) j (t)  Dij (t  t) , noise intensity D =2.0. Gating variables m, h and n governing the stochastic dynamics of sodium and potassium channels obey the following equations: dxi   xi Vi 1  xi   xi Vi  xi , dt

(x = m, h, n)

(2)

with opening and closing rates:

m (Vi )  i

0.1Vi  40 , mi (Vi )  4exp  Vi  65 18 , 1  exp  Vi  40 10

h (Vi )  0.07exp  Vi  65 20 , h (Vi )  1  exp  Vi  35 10 , 1

i

i

n (Vi )  i

0.01Vi  55 , ni (Vi )  0.125exp  Vi  65 80 , 1  exp  Vi  55 10

Iauti is autaptic current. Here we use chemical autaptic current (Burić et al., 2008; Wang 5

et al., 2014):

Iauti  gaut Vi (t ) Vsyn  Si t   ,





Si t    1 1  exp k Vi (t  )    ,

(3)

and electrical autaptic current [Li et al., 2010; Wang et al., 2014]:

Iauti  gaut Vi (t  ) Vi (t ) ,

(4)

where gaut is autaptic intensity, Vsyn is synaptic reversal potential, and here we choose Vsyn = 2mV for excitatory autapses. Vi (t  ) is the action potential of neuron i at earlier time t  ,

 (in unit of ms) is autaptic delayed time. We assume all neurons have equal gaut and equal  . Other parameters are: k = 8,   0.25 . Synaptic current Iisyn is a coupling term. Here we use chemical synaptic current (Yu et al., 2015): Iisyn  

N



j 1( j i )

ij Cij j (Vi  Vsyn ) ,

 j   (Vj )(1 j )   j ,



 (Vj )  0 1  eV

j

Vshp

,

(4) (5) (6)

where Cij = 1 if neuron j couples to neuron i, and Cij = 0 otherwise. The reversal potential is chosen as Vsyn= 0. The synaptic recovery function  (Vj ) can be taken as the Heaviside function. Vshp= 5.0 determines the threshold above which the postsynaptic neuron is affected by the pre-synaptic one. Other parameters 0 and  are chosen as 0  2 and   1 . Synaptic coupling strength ij varies through STDP modification function F, which is defined as follows: ij (t  t )  ij (t )  ij ,

(7)

ij  ij F (t ) ,

(8)

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



  Ap exp  t  p F (t )     Am exp   t  m 

if t  0 if t  0 ,

(9)

where t  ti  t j , ti (or t j ) is marked as the spiking time of the ith (jth) neuron. The amount of synaptic modification is limited by Ap and Am, the adjusting rate of STDP. τp and τm determine the temporal window for synaptic refinement. Experimental investigations suggest that the temporal window for synaptic weakening is roughly the same as that for synaptic strengthening (Bi and Poo, 1998; Feldman and Brecht, 2005). Potentiation is consistently induced when the postsynaptic spike generates within a time window of 20 ms after presynaptic spike, and depression is induced conversely. STDP is usually viewed as dominant depression. Hence, the parameters are set to be τp = τm = 20 (Song et al., 2006) and Am/Ap = 1.005. Here Ap is chosen as an alterable parameter of STDP rule. All excitable synapses considered in this paper are initiated as ij   max 2  0.1 , where  max  0.2 is the coupling upper limit. To view the change of coupling strength, we average  ij over the whole population and time by:

1 T N N ij (t ) .  T  N 2T t 1 i 1 j 1

 ave  lim

(10)

As mentioned above, the effect of depression is more powerful than potentiation (i.e., the value of Am is larger than Ap), under which the decrement of synaptic strength is larger than the increment of it. Hence, the larger the value of Ap, the smaller the average coupling strength. The synchronization of the neuronal networks can be characterized by standard deviation

 defined as (Gong et al., 2007): 7

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    t   where 

with

 t  

1 N 1 N  2 Vi  t    Vi  t    N i 1  N i 1  , N 1

(11)

denotes the average over time and  the average over 20 different network

realizations for each set of parameter values. Standard deviation  describes the population-averaged variance of single-neuron activity and the time fluctuation of the average membrane potential, which measures the degree of the synchronization of the neuronal network. Larger  represents larger deviation between the neurons, and smaller  shows higher synchronization. Numerical integrations of Eqs. (1) – (9) are carried out using explicit Euler algorithm with time step of t  0.001ms. Periodic boundary conditions are used and the parameter values for all the neurons are identical except for noise terms i (t ) for each neuron. 3. Results and discussion 3.1 Effect of STDP on synchronization transitions induced by autaptic delay Throughout this paper, we set neuron number N=60 unless stated otherwise. Using Iauti described by eq. (3), we study synchronization transitions induced by chemical autaptic delay under STDP. First, we present synchronization transitions induced by autaptic delay when the adjusting rate Ap of STDP is fixed In Fig. 1, the spatiotemporal patterns of the membrane Fig. 1 potentials for different autaptic delays  are displayed. Other parameters are Ap  3106 ,

gaut  0.2 , and p = 0.1. As  increases, the neurons intermittently become synchronous and non-synchronous, exhibiting synchronization transitions. This phenomenon is quantified by standard deviation  in dependence on  in Fig. 2(a). It is shown that  passes through a

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Fig. 2 few peaks as  increases, indicating the occurrence of synchronization transitions. The amplitudes of  peaks reflect the difference of highest and lowest synchronization and hence represent the degree of synchronization transitions. To view the change of average coupling strength, we average coupling strength  ij over time and neuron population. Average coupling strength  ave is plotted as a function of  in Fig. 2(b). It is shown that  ave undergoes a few peaks as  increases. This shows that, when autaptic delay increases, average coupling strength increases and decreases, and the neurons exhibit synchronization transitions. We now study the effect of STDP on synchronization transitions induced by autaptic delay. In Fig. 3,  is plotted as a function of  for different Ap values at gaut = 0.2, p = 0.1. As Ap Fig. 3 increases, the amplitudes of  peaks increase and become largest at around Ap  6 106 and then decrease again. The evolution of the amplitudes of  peaks can be clearly seen from the insert plot, in which average amplitudes of  peaks are plotted against Ap. The amplitudes of  peaks represent the difference between  maxima and minima which represent the lowest and highest synchronization, respectively. Larger amplitudes show stronger synchronization transitions. The result in Fig. 3 shows that the synchronization transitions become strongest at a certain Ap value. We have obtained results for other Ap values. In Fig. 4 (a), we display the contour plot of  Fig. 4 as functions of Ap and  . It is shown that  always increases as Ap increases, and it

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intermittently increases and decreases as  increases. Moreover, the difference between  maximum and minimum becomes biggest at around Ap  6 106 . This shows that the degree of synchronization of the neurons is weakened as Ap increases, and the neurons exhibit synchronization transitions as autaptic delay is varied. Moreover, the synchronization transitions become strongest at a certain Ap value. In Fig. 4(b), we plot the contour of  ave against Ap and  . It is shown that  ave always decreases as Ap increases, but it intermittently increases and decreases as  increases. From Figs. 4(a) and 4(b), it is seen that  and  ave peaks and valleys almost appear at the same

 values. This shows that the neurons tend to become more synchronized when average coupling strength is smaller. As stated, the synchronization of the neurons decreases with the increase of Ap. For too small or large Ap, the neurons are too highly or too lowly synchronized. In these two cases, it is difficult for autaptic delay to change the synchronization states of the neurons, and hence autaptic delay can induce weak synchronization transitions. For an intermediate Ap, the neurons are in a moderate synchronization, and autaptic delay can easily change the synchronization state and hence can induce strong synchronization transitions. Therefore, there is optimal Ap by which autaptic delay can induce strongest synchronization transitions. 3.2 Influence of STDP on the effect of network randomness and network size on synchronization transitions Network randomness and network size are two important topological parameters of the small-world neuronal network. In this section, we study how STDP affects the effects of network randomness and network size on synchronization transitions induced by autaptic

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delay. In Fig. 5,  is plotted against  for different p values for (a) Ap  5 107 ,. (b) Fig. 5

Ap  3106 , and (c) Ap  7 106 . For Ap  5 107 the amplitudes of  peaks decrease as p increases; for Ap  3106 the amplitudes of  peaks become largest at around p = 0.1 and are very low at p = 0.2; for Ap  7 106 the amplitudes of  peaks are still largest at around p = 0.1 but become low up to p = 0.7. This shows that the synchronization transitions become strongest at a certain value of network randomness, and when Ap increases, the value of network randomness for strongest synchronization transitions increases. In Fig. 6, we plot  as a function of  for different N values for (a) Ap  5 107 , Fig. 6 (b) Ap  3106 , and (c) Ap  7 106 . For Ap  5 107 , the amplitudes of  peaks decrease as N increases from N = 30 to N = 150; for Ap  3106 , the amplitudes of  peaks become largest at around N = 60 and are small for N = 150; for Ap  7 106 , the amplitudes of  peaks increase until N = 150. This shows that the synchronization transitions become strongest at a certain value of network size; and when Ap increases, the value of network size for strongest synchronization transitions increases. Using Iauti described by eq. (4), we have studied synchronization transitions induced by electrical autaptic delay and obtained similar results (not shown). The above phenomenon can be briefly explained. Previous studies have shown that the neurons in the Newman–Watts networks are more and more synchronized as random shortcuts increase. However, the synchronization of the neurons decreases as the adjusting rate Ap of STDP increases. As such, the number of random shortcuts for synchronization

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increases as Ap increases. Therefore, the value of network randomness for strongest synchronization transitions increases as Ap increases. This reflects the competition between the effects of increasing Ap and increasing network randomness on the synchronization of the neuronal network. Previously, we have studied multiple coherence resonance induced by synaptic delay in neuronal networks (Gong et al., 2011; Wu et al., 2013) and synchronization transitions induced by noise (Wang et al., 2015a) and autaptic delay (Wu et al., 2015b; Wang et al., 2015b) in neuronal networks with fixed or periodic coupling. In the present paper, we study how STDP influences the synchronization transitions induced by autaptic delay in adaptive neuronal networks. The present study shows that the synchronization transitions induced by autaptic delay strongly depend on the adjusting rate Ap of STDP, and STDP can significantly enhance the synchronization transitions when the adjusting rate of STDP is appropriate. By comparison, we can see that the model and results of the present work are completely different from those of our previous studies. 4. Conclusion In summary, we have studied the effect of STDP on synchronization transitions induced by autapses in adaptive NW HH neuronal networks. It is found that the synchronization transitions vary with the adjusting rate Ap of STDP and become strongest at a certain Ap value. As network randomness or network size increases, the value of Ap for strongest synchronization transitions increases and the related synchronization transitions are enhanced. As network randomness or network size increases, the synchronization transitions become strongest at a certain network randomness or network size; as Ap increases, the value of

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network randomness or network size for strongest synchronization transitions increases and the related synchronization transitions are enhanced. These results show that STDP with appropriate adjusting rate can significantly enhance the synchronization transitions induced by autaptic delay in the adaptive neuronal networks. These findings could find potential implication of STDP and autapses for the information transmission in neural systems.

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Figure captions: Fig.1 Spatiotemporal patterns of the membrane potentials for different autaptic delays  . Other parameters are: Ap  3106 , gaut  0.2 , and p = 0.1. As  increases, the neurons intermittently become synchronous and non-synchronous, exhibiting synchronization transitions. Fig.2 Dependence of  (a) and  ave (b) on  at Ap  3106 , gaut  0.2 , and p = 0.1. As  increases,  and  ave pass through a few peaks, which quantifies the presence of synchronization transitions induced by autaptic delay. Fig.3 Variation of  as a function of  for different Ap values at gaut  0.2 and

p = 0.1.

As Ap increases, the amplitudes of  peaks become largest at around Ap  6 106 , which shows the synchronization transitions become strongest at a certain Ap value. Fig.4 Contour plot of (a)  and (b)  ave as functions of Ap and  at gaut  0.2 and p = 0.1. At around Ap  6 106 , the difference of maximal and minimal  values with varying  is biggest, indicating the presence of strongest synchronization transitions induced by autaptic delay. Fig.5 Plot of  as a function of  for different p values at gaut  0.2 for (a) Ap  5 107 , (b) Ap  3106 , and (c) Ap  7 106 . For Ap  3106 and Ap  7 106 , the amplitudes of

 peaks are largest at around p = 0.1, indicating the existence of strongest synchronization transitions at a certain network randomness. Fig.6 Dependence of  on  for different N values for (a) Ap  5 107 , (b) Ap  3106 , and (c) Ap  7 106 . Other parameters are gaut  0.2 and p = 0.1. As N increases, the amplitudes of  peaks decrease for Ap  5 107 , become largest at around N=60

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for Ap  3106 , and increase for Ap  7 106 .

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