DOI:10.1140/epjb/e2012-30282-0 Regular Article
P
HYSICAL
J
OURNAL
B
Controlling the first-spike latency response of a single neuron
via unreliable synaptic transmission
M. Uzuntarla1,a, M. Ozer2, and D.Q. Guo3,4
1 Bulent Ecevit University, Engineering Faculty, Department of Biomedical Engineering, 67100 Zonguldak, Turkey
2 Bulent Ecevit University, Engineering Faculty, Department of Electrical-Electronics Engineering, 67100 Zonguldak, Turkey 3 Key Laboratory for NeuroInformation of Ministry of Education, School of Life Science and Technology,
University of Electronic Science and Technology of China, Chengdu 610054, P.R. China
4 Computational Neuroscience Unit, Okinawa Institute of Science and Technology, 904-0411 Okinawa, Japan
Received 1st April 2012 / Received in final form 1st June 2012
Published online 13 August 2012 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2012 Abstract. Previous experimental and theoretical studies suggest that first-spike latency is an efficient infor-mation carrier and may contain more amounts of neural inforinfor-mation than those of other spikes. Therefore, the biophysical mechanisms underlying the first-spike response latency are of considerable interest. Here we present a systematical investigation on the response latency dynamics of a single Hodgkin-Huxley neu-ron subject to both a suprathreshold periodic forcing and background activity. In contrast to most earlier works, we consider a biophysically realistic noise model which allows us to relate the synaptic background activity to unreliable synapses and latency. Our results show that first-spike latency of a neuron can be regulated via unreliable synapses. An intermediate level of successful synaptic transmission probability significantly increases both the latency and its jitter, indicating that the unreliable synaptic transmis-sion constrains the signal detection ability of neurons. Furthermore, we demonstrate that the destructive influence of synaptic unreliability can be controlled by the input regime and by the excitatory coupling strength. Better tuning of these two factors could help the H-H neuron encode information more accurately in terms of the first-spike latency.
1 Introduction
Neurons communicate by means of stereotyped pulses, called action potentials or spikes, and one of the central problems in neuroscience is to understand the neural cod-ing mechanism. It is widely assumed that information may be encoded either in the mean firing rates of neurons (rate coding) [1,2] or in the timing of spikes (temporal cod-ing) [3,4]. Rate coding assigns importance of spike counts regarding to the intensity of the encoded stimulus features, whereas temporal coding does not assign any such impor-tance and considers the precise timing and coordination of spikes in time. Although whether the neurons use rate or temporal code is still controversial, codes based on spike times have attracted increasing attention over the past decades due to the growing evidence on the relation be-tween synchronization in neural networks and higher brain functions, such as memory, attention and cognition [5–10]. In addition, it has been argued that temporal coding al-lows information to be processed at higher speeds with less energy than does rate coding [3,11–13].
In the context of temporal coding, a potential coding strategy is that under strong temporal constraints neurons
a e-mail:muzuntarla@yahoo.com
may perform information processing with only one spike using first-spike latency as an information carrier [14,15]. First-spike latency coding has been studied with experi-mental protocols in different neuronal structures such as somatosensory [16,17], olfactory [18], auditory [19,20] and visual systems [21,22], and it has been shown that the first spike latencies carry a considerable amount of, or even more, information than those of other spikes. Besides these experimental works, many theoretical and computa-tional studies have been performed to investigate the in-fluence of different biophysical mechanisms on first-spike latency response dynamics of neurons [23–30]. Pankratova et al. [23,24] analyzed the impact of noise on the response latency of Fitzhugh-Nagumo and Hodgkin-Huxley (H-H) neuronal models driven by a suprathreshold periodic forc-ing, and obtained the dependence of the mean latency on the noise strength for different values of driving frequency. Their results showed that although the noise increases the first-spike latency and thus delay in signal detection, a proper choice of the driving frequency for the suprathresh-old periodic forcing could minimize this effect. They also obtained a resonance-like behavior for the mean latency at the frequency boundaries of suprathreshold spiking regime of neuron. When the suprathreshold periodic forcing fre-quency is near these boundaries, it is shown that the mean
latency first dramatically increases, then reaches a maxi-mum, and finally decreases as a function of the noise inten-sity. The authors attributed the noise-dependent increase of the latency to a phenomenon called “noise delayed de-cay” (NDD). In a recent study [28], we have reconsidered the subject with a more biophysically realistic model of a single neuron which allows us to relate cell membrane area to the NDD, and obtained similar results in a manner that mimics actual conditions with ion channel noise, as compared to a model with additive current noise as in [24]. Ozer and Graham [26] also investigated the NDD effect in an in-vivo setting of an active network where the intense of network activity was simply controlled by changing the cell membrane time constant. Their results indicate that the NDD effect on spike latency depends not only on noise, but also it is a function of the overall network activity: NDD is significant for small membrane time constants or high network activity, and decreases for large membrane time constants, or low network activity.
On the other hand, neurons process information by exchanging spikes with each other via synaptic contacts. Since the neurons in vivo are embedded in a network of active cells, each neuron is subjected to a synaptic bom-bardment from a large number of incoming excitatory and inhibitory spike inputs resulting in a synaptic back-ground activity. Although most computational studies considered that neurons transmit spikes based on deter-ministic synaptic interaction model, many findings from experimental studies demonstrated that synapses trans-mit signals in an unreliable fashion due to the stochastic release of neurotransmitters [3,31–34]. Depending on the neuronal structure type, the transmission failure rate at some synapses can be higher than the fraction of successful transmissions. For instance, in the cortex, it is found that the probability of neurotransmitter release in response to a single spike can be as low as 0.1 or lower, meaning that as many as 90% out of all arriving presynaptic inputs fail to evoke a postsynaptic response [32]. In this letter, we investigate how the synaptic background activity modu-lated by the unreliable synaptic transmission influences the latency dynamics of a single H-H neuron driven by a suprathreshold periodic forcing. Using a detailed model-ing approach for the background activity will allow us to extend the results obtained in [23–30] and will also pro-vide us a better understanding on the effects of different biophysical components such as input regime, presynaptic input firing rates, unreliability of synapses and coupling strength.
2 Mathematical model and setup
The time evolution of a single H-H neuron in the pres-ence of synaptic background activity can be described as follows [35]: CmdVm dt =−GNam 3h(V m− ENa)− GKn4(Vm− EK) − GL(Vm− EL) + IApp(t) + Isyn(t) (1)
where Vm is the membrane potential, Cm = 1 μF cm−2 is the membrane capacity per unit area. GNa = 120 mS cm−2, GK = 36 mS cm−2 and GL= 0.3 mS cm−2 denote the maximal conductance for the sodium, potas-sium and leakage channels, respectively. ENa = 115 mV,
EK=−12 mV and EL = 10.6 mV are the related reversal
potentials. The parameters m and h stand for the activa-tion and inactivaactiva-tion gating variables for the sodium chan-nel, respectively, the potassium channel includes an acti-vation gating variable, n. These gating variables change dynamically over time and obey the following Langevin generalization [35]: dm dt = αm(Vm)(1− m) − β (Vm) m m, (2a) dh dt = αh(Vm)(1− h) − β (Vm) h h, (2b) dn dt = αn(Vm)(1− n) − β (Vm) n n, (2c)
where αx and βx (x = m, n, h) are experimentally de-termined voltage dependent rate functions for the gating variable x that can be found in [35]. In equation (1), IApp is an externally applied strong periodic forcing current; IApp(t) = A sin(2πf t), where A and f denote the ampli-tude and frequency of the sinusoidal forcing current, re-spectively. Finally, Isyn represents the total synaptic cur-rent introduced into the neuron.
We assume that the neuron receives a large number of excitatory and inhibitory inputs from totally N = 5000 presynaptic neurons, where the ratio of excitatory to inhibitory neurons is taken as NE: Ni= 4 : 1 preserv-ing the similar ratio found in the mammalian cortex [36]. Each presynaptic neuron is considered as an independent Poisson spike train generator with the same input fir-ing rate v0. In this framework, the total synaptic current is modeled with an impulsive current approximation as in [37–41]: Isyn(t) = Cmgexc N e k=1 l hlkδt− tlk − K Ni m=1 n hnmδ (t− tnm) (3)
where gexc represents the coupling strength for the exci-tatory synapses and K is the relative strength between inhibitory and excitatory synapses. Hence, the coupling strength for the inhibitory synapses becomes Kgexc. Un-less otherwise noted, we set gexc= 0.1. tlk is the discharge time of the lth spike at the excitatory presynaptic neuron
k, and hlk is the synaptic transmission reliability
parame-ter of this spike which is used to mimic whether the spike transmission is successful or not. Similarly, tnm is the dis-charge time of the nth spike at the inhibitory presynaptic neuron m, and hnmis the synaptic transmission reliability parameter. The reliability of spike transmission is modeled based on the stochastic Bernoulli on-off process by assum-ing hlk = 1 or hnm= 1 with probability ps, and hlk = 0 or
Fig. 1. (Color online) Schematic illusturation of the considered system. hnm= 0 with probability 1− ps, where psis defined as the
successful transmission probability of spikes [40–44]. A schematic illustration of the considered system is shown in Figure 1where the first appearance time of the spike in response to suprathreshold driving is defined as the first-spike latency. Here, spike timing is determined by the upward crossing of Vm past a detection threshold of 20 mV. In order to quantify the first-spike latency dynam-ics of the H-H neuron, we obtain the mean latency (ML) of an ensemble of first spikes by averaging their latencies over r = 5000 realizations as follows:
M L =t = 1 r r i=1 ti (4)
where ti is the appearance time of the first spike for ith realization. We also consider the standard deviations of the latencies, or temporal jitter, as follows:
σL=
t2 − t2 (5)
where t2 represents the mean squared latency. The nu-merical integration of equations (1)–(3) is performed using the standard fourth order Runge-Kutta algorithm with a step size of 10 μs.
3 Results
In what follows, we will systematically analyze how the synaptic background activity with unreliable synaptic transmission affects the first-spike timing of a H-H neu-ron driven by a suprathreshold periodic forcing. Follow-ing the studies in [24,26], the model neuron studied here subjected to a sinusoidal forcing current of magnitude of A = 4 μA/cm2 operating in a suprathreshold spiking regime for a frequency range of f ∈ [16 : 149] Hz. Since the NDD effect on first-spike latency only emerges near these frequency boundaries [24,26,28], we studied the la-tency dynamics of the neuron in the presence of synaptic background activity at a frequency of f = 20 Hz, which
is just above lower threshold boundary and warrants the suprathreshold spiking activity over a large range inten-sity of synaptic background noise.
First, we consider the balanced state of excitatory and inhibitory synaptic inputs, and investigate how the la-tency statistics of the neuron might be influenced by the unreliable synaptic transmission in this special case. To this end, we set the scale factor K = 4 and computed the mean latencies and their standard deviations, henceforth called jitter, as a function of the successful transmission probability ps for various values of input firing rate, v0. Obtained results are presented in Figure2. It is seen that both the ML and jitter display a non-monotonic behav-ior with respect to psfor all values of v0. Below a certain value, ML is robust to the increase in ps, close to the de-terministic one, and the jitter gets smaller values meaning that the neuron fires the first spike with a high tempo-ral precision. As psincreases from small values, both the ML and jitter start to increase and reach their maximums, indicating the occurrence of delay in external signal detec-tion and very low temporal spiking precision, respectively. With a further increase in ps, ML and jitter decrease to lower values than the deterministic one. Additionally, it is seen that v0 plays a significant role in latency statistics of the model neuron. Although the maximum latency value is insensitive to the v0, increased values of the input fir-ing rate decreases the reliable transmission probability for which the response delay is evident.
The underlying effect of the unreliable synaptic trans-mission on the latency dynamics of the neuron can be in-terpreted as follows. There are two components of synap-tic background noise. One is the mean, or bias, current introduced to the neuron which increases or decreases the average membrane potential. The other is the standard de-viation, or noise strength, which determines the membrane potential fluctuations. In previous studies [23,24,26,27], noise induced delayed response in a H-H neuron was inves-tigated via varying the strength of the additive Gaussian white noise, while keeping the mean value to zero. When the scale factor K = 4 in the model studied here, the H-H neuron receives balanced excitatory and inhibitory
10-4 10-3 10-2 10-1 100 0 5 10 15 20 p s M e an Lat ency ( m s) 0 20 40 60 5 10 15 20 psv0 (Hz) Mea n Latenc y ( m s ) v0=8 Hz v0=16 Hz v0=32 Hz v0=64 Hz p sv0≈ 3.5 Hz (a) 10-4 10-3 10-2 10-1 100 0 5 10 15 20 ps J it te r (ms ) 0 20 40 60 0 10 20 psv0 (Hz) Ji tt e r ( m s ) v0=8 Hz v0=16 Hz v0=32 Hz v0=64 Hz (b)
Fig. 2. (Color online) The dependence of latency statistics on the reliability of spike transmission, ps: (a) mean latency, (b) jitter (K = 4).
inputs that result in a fluctuating synaptic current with zero mean value. Since we observed the same response de-lay phenomenon by varying the reliable spike transmission probability, it seems that the strength of synaptic current fluctuations might be controlled via ps. Actually, its im-pact on fluctuations arises from determining the effective mean spike arrival rate per synapse by tuning the input firing rate v0. To illustrate this effect, we rescaled the ML and jitter as a function of psv0 as shown in the insets of Figures 2a and 2b, respectively. Interestingly, it is seen that all data points almost collapse into a single curve and there exists a critical value of psv0 (≈3.5 Hz) for the occurrence of maximal delayed response and jitter. This
10-4 10-3 10-2 10-1 100 0 10 20 30 40 50 60 ps M e an Lat ency ( m s) 10 -4 10 -3 10 -2 10 -1 10 0 0 5 10 15 20 25 ps M ean Lat enc y ( m s ) K=2 K=3 K=4 K=5 K=6 (a) 10-4 10-3 10-2 10-1 100 0 5 10 15 20 25 30 35 40 45 p s Ji tt e r ( m s) K=2 K=3 K=4 K=5 K=6 (b)
Fig. 3. (Color online) The dependence of latency statistics on the relative coupling strength between inhibitory and excita-tory synapses, K: (a) mean latency, (b) jitter (v0= 32 Hz).
is because the NDD effect for a H-H neuron occurs in the presence of appropriate level of noise, and the synaptic background activity with only at the critical value of psv0 is able to generate proper stochastic fluctuations to induce such a delay mechanism.
Next, we investigated the dependence of the ML and jitter on the scale factor K. To that effect, we again ob-tained the latency statistics as a function of the successful transmission probability ps for several values of the scale factor K, with the fixed input firing rate v0 = 32 Hz. Figure3 features the obtained results. It can be observed that delayed response property of the model neuron can be modulated by the relative coupling strength between
inhibitory and excitatory synapses. In case of K < 4, where the studied neuron operates in the high input regime [41,45–47], we still observe the delayed response for a particular range of ps values, but with a lower peak for both the ML and jitter comparing with the balanced state (K = 4). This means that delay in signal detec-tion could be reduced or minimized for the high input regime. As the spike transmission reliability increases, ML decreases to a lower value than the deterministic one, and the jitter saturates around a non-zero small value, indicat-ing a significant distribution of first-spike latencies. Also, decreasing the scale factor K results in shrinkage of the successful transmission probability psregion in which the response delay still occurs (see the insert in Fig.3a). More-over, one can notice that further decrease of K, implying a very high input regime (see K = 2 in Fig. 3), might cause the complete vanishing of delay in signal detection. On the other hand, for the case of the low input regime (K > 4) [41,45–47], the resonance-like behavior of the latency statistics disappears. Although ML and jitter ex-hibit a similar tendency with the two other regimes up to a certain ps, the expected decrease in both could not be observed for the resonance-like behavior aspsincreases. After reaching some higher maximum values than the bal-anced state, a further increase in ps provide both the ML and jitter to saturate around the peak amplitude values for which K is not too large, i.e. K = 5. Also, for a very low input regime (see K = 6 in Fig.3), it is seen that both of them increase with the increasing values of successful transmission probability.
The competition between excitatory and inhibitory inputs determines the dynamics of synaptic background activity. Therefore, analyzing the structure of the total synaptic current will help us get a better understand-ing on how the response latency is modulated in differ-ent input regimes with unreliable synaptic transmission. Figure4shows the average and standard deviation of the total synaptic current as a function of ps for above con-sidered values of K. For the balanced case (K = 4), since the excitatory and inhibitory inputs cancel each other out, average synaptic current has zero mean value and chang-ing ps just influences fluctuating part of the total synap-tic current which gives rise to occurrence of the delay in signal detection as discussed before. When the scale fac-tor K < 4, changing psnot only influences the fluctuating part but also the average part of the total synaptic current, i.e. increasing psamplifies positively both the average and standard deviations of total synaptic current. Notably, positive amplification in the average part starts after a cer-tain spike transmission probability value, ps= 0.02, where the ML and jitter reach almost their maximums. Since a positive average synaptic current depolarizes the mem-brane potential from the resting state, the external peri-odic forcing current becomes more suprathreshold, thus ML and jitter decrease in the above determined ps re-gion comparing with the balanced state (Fig. 3). After this optimal region for the delayed response, average part of the synaptic current increase substantially that guar-antees the occurrence of the first spike in the first-quarter
10-4 10-3 10-2 10-1 100 -8 -6 -4 -2 0 2 4 6 8 p s <I sy n > K=2 K=3 K=4 K=5 K=6 (a) 10-4 10-3 10-2 10-1 100 0 0.5 1 1.5 2 2.5 3 3.5 4 p s σ Is yn K=2 K=3 K=4 K=5 K=6 (b)
Fig. 4. Variation of the total synaptic current statistics as a function of reliable synaptic transmission probability for differ-ent input regimes: (a) average synaptic currdiffer-ent, (b) standard deviation (gexc= 0.1, v0 = 32 Hz).
phase of the periodic stimulus despite the high level of current fluctuations. Therefore, ML decreases to a value lower than the deterministic one, and the jitter is close to zero for larger values of ps. On the other hand, in the case of the low input regime (K > 4), increasing psnegatively and positively amplifies the average part and fluctuating part of the total synaptic current, respectively. Since the external periodic stimulus with f = 20 Hz is just above firing current threshold of the neuron, a negative aver-age synaptic current weakens the suprathreshold effect of the external stimulus due to the decrease in average membrane potential from the resting state. Thus, the ML and jitter get larger values than the balanced state within the optimal ps region. Moreover, a further increase in ps
provides the average synaptic current to become more neg-ative and even the external periodic current to switch from suprathreshold to subthreshold for the case of a very low input regime (K = 6). Therefore, the first spike timing is mainly determined by the fluctuations of synaptic current, and accordingly, ML and jitter get higher values than the ones in the balanced state.
Thus far, we have studied how the background activ-ity influences the response latency of a single neuron for a fixed excitatory coupling strength, equaling gexc = 0.1. To investigate the impact of excitatory coupling strength in distinct input regimes, we calculate the first spike la-tency statistics versus the spike transmission probability for different values of gexc and K. The corresponding re-sults are shown in Figure 5. It can be observed that the influence of gexc on the latency statistics largely depends on the input regime. In the high input regime (Fig. 5a), increasing gexc amplifies both the ML and jitter (insert in Fig. 5a), also shifts the ps region to lower values, in-dicating that more unreliable synapses are needed for the occurrence of the maximum delayed response. However, decreasing gexc leads to a reduction in both ML and jit-ter, or even complete loss of delayed response property of the model neuron. Moreover, gexc influences the latency oppositely for large values of ps than the resonance re-gion where the ML and jitter decrease with the increasing values of gexc. In the high input regime, increasing gexc is equivalent to increasing psto a certain degree, both of which positively amplify the average and fluctuating parts of the total synaptic current. However, since the successful transmission probability psis a factor mainly used to con-trol the spike arriving rate, according to the previous the-oretical results based on the mean-filed analysis [37,41,48], it has an equivalent effect on the average part of the to-tal synaptic current but a smaller effect on its fluctuating part, compared to those of gexc. For a small (but not too small) value of gexc, the above reason leads to the total synaptic current with a large average value and a small fluctuating at a relatively higher level of synaptic reliabil-ity, which in principle guarantees the H-H neuron to fire its first spike with a small ML and jitter (see Fig.5a). On the other hand, in the balanced state (Fig. 5b), we ob-serve that increasing the value of gexc does not influence the peak values of the mean latency and jitter, and only shifts the corresponding optimal ps value to left. This is no of surprise, since under this condition the average part of the total synaptic current is zero, and therefore, chang-ing the value of excitatory synaptic strength only affects the fluctuating part of the total synaptic current. Finally, for the case of low input regime (Fig.5c), a negative aver-age synaptic current is introduced to the H-H neuron. In this case, increasing the value of gexctends to negatively and positively amplify the average and fluctuating parts of the total synaptic current, respectively. According to our above discussions, these two factors can greatly reduce the resonance-like behaviors for both the ML and jitter, and even leads to such behaviors completely disappear for the case of a sufficiently low input regime. The results pre-sented in Figure5clearly demonstrate that the excitatory
10-4 10-3 10-2 10-1 100 0 2 4 6 8 10 12 14 16 18 20 22 ps M e an Lat ency ( m s) 10-4 100 0 10 20 Ji tt e r ( m s ) ps g exc=0.05 g exc=0.1 g exc=0.2 (a) 10-4 10-3 10-2 10-1 100 0 2 4 6 8 10 12 14 16 18 20 22 ps M e an Lat e ncy ( m s) 10-4 100 0 10 20 J itt e r ( m s ) ps gexc=0.05 gexc=0.1 gexc=0.2 (b) 10-4 10-3 10-2 10-1 100 100 1000 ps M e an Lat e ncy ( m s ) 10-4 100 100 102 Ji tt e r ( m s) ps gexc=0.05 gexc=0.1 gexc=0.2 (c)
Fig. 5. (Color online) The dependence of the latency statistics on coupling strength as a function of ps for different input regimes: (a) K = 2, (b) K = 4, (c) K = 6 (v0= 32 Hz).
coupling strength gexc plays rather complex roles in the first-spike latency dynamics of the H-H neuron, especially when the excitatory and inhibitory synaptic currents are in the unbalanced state (K= 4).
4 Summary
Neurons typically receive a large number of incoming excitatory and inhibitory inputs from their presynaptic neurons. This continuous barrage of synaptic inputs re-sults in highly variable neuronal responses and is there-fore believed to be an important source of neuronal noise. Although the information processing in the brain is highly reliable, physiological experiments have demon-strated that the synaptic communication via chemical synapses is unreliable [3,31–34]. From a theoretical point of view, such unreliable property of synaptic transmis-sion has a great effect on the background synaptic activ-ity, which will further affect the coding performance of neurons [49–52]. However, most relevant modeling studies only consider the deterministic synaptic interaction model and, to our knowledge, the effects of unreliable synaptic transmission on the coding performance of neurons still remain unclear.
In this work, we have systemically investigated the re-sponse latency and jitter of a single H-H neuron driven by a suprathreshold periodic forcing in the presence of synaptic background activity, which is caused by excita-tory and inhibiexcita-tory random spikes from its presynaptic neurons through unreliable synapses. We carried out this research because the first-spike latency is thought to be an efficient information carrier and may contain a much more considerable amount of neural information than those of other spikes [14–22]. Our numerical results shown above clearly demonstrated that the response latency can be reg-ulated by both the successful transmission probability and input firing rate. Under certain conditions, we observed that both the spike latency and jitter have distinct max-imal values as a function of psv0, indicating that an in-termediate level of effective mean spike arrival rate per synapse may significantly delay the suprathreshold signal detection. On the other hand, we also examined the in-fluence of synaptic input regime and excitatory coupling strength on first-spike latency response. Our simulation re-sults uncovered that better choosing the synaptic regime as well as the excitatory coupling strength can help the considered HH neuron encode neural information more ac-curately in terms of the first-spike latency.
The results presented in this work are partly con-sistent with the findings given in several previous stud-ies [23,24,26]. However, it should be noted that these previous studies are mainly based on the ideal additive Gaussian white noise. Although considering the neuronal noise as the Gaussian white noise is in favor of both the modeling analysis and fast simulation, it may be biolog-ically questionable. By considering a biophysbiolog-ically realis-tic noise model as in this work, we are able to relate the synaptic background activity to the latency in a manner
that more closely mimics actual conditions with unreli-able synapses. The results presented here may also have some biological implications. After a long time of evolu-tion, it is reasonable to believe that biological neurons may achieve high level of first-spike latency coding per-formance, through suitable modulating the dynamics of synaptic transmission. Therefore, we hope that these re-sults can improve our understanding of the roles of un-reliable synaptic transmission in the latency dynamics of biological neurons and also contribute to the nontrivial ef-fects of noise on neuronal information processing [53–55]. Further studies on this topic include at least the follow-ing two aspects: (1) since neuronal networks have struc-tures, which tend to introduce a certain level of corre-lations to the system, it is interesting to investigate the role played by correlations among different synaptic in-puts in the first-spike latency dynamics of the H-H neu-ron [56,57]. (2) Real biological synapses have both the fast and slow inherent learning mechanisms, i.e., synaptic plasticity [58,59]. Thus, it is also of importance to further study the combination effects of synaptic unreliability and plasticity on the first-spike latency dynamics of the H-H neuron.
References
1. E.D. Adrian, J. Physiol. 61, 49 (1926)
2. A.P. Georgepoulos, A.B. Schwartz, R.E. Keftner, Science 233, 1416 (1986)
3. M. Abeles, Corticonics: Neural Circuitry of the Cerebral
Cortex (Cambridge University Press, New York, 1991)
4. M. Abeles, H. Bergman, E. Magalit, A. Vaadia, J. Neurophysiol. 70, 1629 (1993)
5. G. Buzs´aki, Z. Horvath, R. Urioste, J. Hetke, K. Wise, Science 256, 1025 (1992)
6. K.N. Dudkin, V.K. Kruchinin, I.V. Chueva., Neurosci. Behav. Physiol. 27, 303 (1997)
7. P. Fries, J.H. Reynolds, A.E. Rorie, R. Desimone, Science 291, 1560 (2001)
8. G. Buzs´aki, Rhytms of the Brain (Oxford University Press, New York, 2006)
9. T. Womelsdorf, P. Fries, Curr. Opin. Neurobiol. 17, 154 (2007)
10. U. Rutishauser, I.B. Ross, A.N. Mamelak, E.M. Schuman, Nature 464, 903 (2010)
11. S.J. Thorpe, D. Fize, C. Marlot, Nature 381, 520 (1996) 12. N. Masuda, K. Aihara, Neural Comput. 15, 103 (2003) 13. Y. Hirata, Y. Katori, H. Shimokawa, H. Suzuki, T.A.
Blenkinsop, E.J. Lang, K. Aihara, J. Neurosci. Methods 172, 312 (2008)
14. S.J. Thorpe, in Parallel Processing in Neural Systems
and Computers, edited by R. Eckmiller, G. Hartman,
G. Hauske (Elsevier, Amsterdam, 1990), pp. 91–94 15. R. VanRullen, R. Guyonneau, S.J. Thorpe, Trends
Neurosci. 28, 1 (2005)
16. S. Panzeri, R.S. Petersen, S.R. Schultz, M. Lebedev, M.E. Diamond, Neuron 29, 769 (2001)
17. R.S. Petersen, S. Panzeri, M.E. Diamond, Biosystems 67, 187 (2002)
18. S. Junek, E. Kludt, F. Wolf, D. Schild, Neuron 67, 872 (2010)
19. S. Furukawa, J.C. Middlebrooks, J. Neurophysiol. 87, 1749 (2002)
20. P. Heil, Curr. Opin. Neurobiol. 14, 461 (2004)
21. T.J. Gawne, T.W. Kjaer, B.J. Richmond, J. Neurophysiol. 76, 1356 (1996)
22. D.S. Reich, F. Mechler, J.D. Victor, J. Neurophysiol. 5, 1039 (2001)
23. E.V. Pankratova, A.V. Polovinkin, B. Spagnolo, Phys. Lett. A 344, 43 (2005)
24. E.V. Pankratova, A.V. Polovinkin, E. Mosekilde, Eur. Phys. J. B 45, 391 (2005)
25. H.C. Tuckwell, F.Y.M. Wan, Physica A 351, 427 (2005) 26. M. Ozer, L.J. Graham, Eur. Phys. J. B 61, 499 (2008) 27. M. Ozer, M. Uzuntarla, Phys. Lett. A 372, 4603 (2008) 28. M. Ozer, M. Uzuntarla, M. Perc, L.J. Graham, J.
Theoretical Biol. 261, 83 (2009)
29. Z. Pawlas, L.B. Klebanov, V. Benes, M. Prokesova, J. Popelar, P. Lansky, Neural Comput. 22, 1675 (2010) 30. G. Wainrib, M. Thieullen, K. Pakdaman, Biol.
Cybern. 103, 43 (2010)
31. M. Raastad, J.F. Storm, P. Andersen, Eur. J. Neurosci. 4, 113 (1992)
32. C. Allen, C.F. Stevens, Proc. Natl. Acad. Sci. USA 91, 10380 (1994)
33. B. Katz, Nerve, Muscle and Synapse (McGrawHill, New York, 1996)
34. T. Branco, K. Staras, Nature Rev. Neurosci. 10, 373 (2009)
35. A.L. Hodgkin, A.F. Huxley, J. Physiol. 117, 500 (1952) 36. V. Braitenberg, A. Schuz, Anatomy of the Cortex:
Statistics and Geometry (Springer-Verlag, Berlin, 1991)
37. N. Brunel, J. Comput. Neurosci. 8, 183 (2000) 38. W. Kinzel, J. Comput. Neurosci. 24, 105 (2007)
39. A. Torcini, S. Luccioli, T. Kreuz, Neurocomputing 70, 1943 (2007)
40. J. Friedrich, W. Kinzel, J. Comput. Neurosci. 27, 65 (2009)
41. D.Q. Guo, C.G. Li, J. Theoretical Biol. 308, 105 (2012) 42. C.G. Li, Q.X. Zheng, Phys. Biol. 7, 036010 (2010) 43. D.Q. Guo, C.G. Li, J. Comput. Neurosci. 30, 567 (2011) 44. D.Q. Guo, C.G. Li, Cogn. Neurodyn. 6, 75 (2012) 45. A. Destexhe, M. Rudolph, J.M. Fellous, T.J. Sejnowski,
Neuroscience 107, 13 (2001)
46. S. Luccioli, T. Kreuz, A. Torcini, Phys. Rev. E 73, 041902 (2006)
47. M. Ozer, L.J. Graham, O. Erkaymaz, M. Uzuntarla, Neuroreport 18, 1371 (2007)
48. N. Brunel, V. Hakim, Neural Comput. 11, 1621 (1999) 49. W.B. Levy, R.A. Baxter, J. Neurosci. 22, 4746 (2002) 50. D.W. Sullivan, W.B. Levy, Neurocomputing 52, 397
(2003)
51. M.S. Goldman, Neural Comput. 16, 1137 (2004)
52. J. Kestler, W. Kinzel, J. Phys. A: Math. Gen. 39, 461 (2006)
53. P. H¨anggi, Chem. Phys. Chem. 3, 285 (2002)
54. L. Gammaitoni, P. H¨anggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998)
55. M. Perc, Eur. J. Phys. 27, 451 (2006)
56. T. Kreuz, S. Luccioli, A. Torcini, Phys. Rev. Lett. 97, 238101 (2006)
57. X. Sun, M. Perc, Q. Lu, J. Kurths, Chaos 20, 033116 (2010)
58. S. Song, K. Miller, L.F. Abbott, Nat. Neurosci. 3, 919 (2000)
59. M. Tsodyks, K. Pawelzik, H. Markram, Neural Comput. 10, 821 (1998)
