Coupling-Induced Population Synchronization in An Excitatory Population of Subthreshold Izhikevich Neurons

Coupling-Induced Population Synchronization in An Excitatory Population of Subthreshold Izhikevich Neurons

Abstract

We consider an excitatory population of subthreshold Izhikevich neurons which exhibit noise-induced firings. By varying the coupling strength , we investigate population synchronization between the noise-induced firings which may be used for efficient cognitive processing such as sensory perception, multisensory binding, selective attention, and memory formation. As is increased, rich types of population synchronization (e.g., spike, burst, and fast spike synchronization) are found to occur. Transitions between population synchronization and incoherence are well described in terms of an order parameter . As a final step, the coupling induces oscillator death (quenching of noise-induced spikings) because each neuron is attracted to a noisy equilibrium state. The oscillator death leads to a transition from firing to non-firing states at the population level, which may be well described in terms of the time-averaged population spike rate . In addition to the statistical-mechanical analysis using and , each population and individual state are also characterized by using the techniques of nonlinear dynamics such as the raster plot of neural spikes, the time series of the membrane potential, and the phase portrait. We note that population synchronization of noise-induced firings may lead to emergence of synchronous brain rhythms in a noisy environment, associated with diverse cognitive functions.

Keywords:
Population Synchronization Cognitive Functions Subthreshold Izhikevich Neurons
pacs:
87.19.lm

1 Introduction

Recently, much attention has been paid to brain rhythms observed in scalp electroencephalogram (EEG) and local field potentials (LFP) with electrodes inserted into the brain (Buzski, 2006). These brain rhythms emerge via synchronization between individual firings in neural circuits. Population synchronization between neural firing activities may be used for efficient sensory and cognitive processing such as sensory perception, multisensory integration, selective attention, and working memory (Wang, 2010). Many recent works have been investigated in diverse views of population synchrony (Wang et al., 2008, 2010, 2011, 2012; Wang and Zheng, 2011; Zhang et al., 2010; Wang et al., 2003; Wang and Jiao, 2006; Qu et al., 2012; Wang and Zhang, 2011; Liu et al., 2010; Wang et al., 2009; Jiao and Wang, 2009). This kind of population synchronization is also correlated with pathological rhythms associated with neural diseases (Traub and Whittington, 2010). Here, we are interested in these synchronous brain rhythms. Population synchronization has been intensively investigated in neural circuits composed of spontaneously firing suprathreshold neurons exhibiting clock-like regular discharges (Wang, 2010, 2003). For this case, population synchronization may occur via cooperation of regular firings of suprathreshold self-firing neurons. In contrast to the suprathreshold case, the case of subthreshold neurons has received little attention. For an isolated single case, a subthreshold neuron cannot fire spontaneously; it can fire only with the help of noise. Here we are interested in population synchronization between complex noise-induced firings of subthreshold neurons which exhibit discharges like Geiger counters. Recently, noise-induced population synchronization was studied by varying the noise intensity observed in a population of subthreshold neurons, and thus collective coherence between noise-induced firings has been found to occur in an intermediate range of noise intensity (Wang et al., 2000; Lim and Kim, 2007, 2009a, 2011; Hong et al., 2011; Kim et al., 2012a, b).

In this paper, we investigate coupling-induced population synchronization which leads to emergence of synchronous brain rhythms by varying the coupling strength in an excitatory population of globally coupled subthreshold Izhikevich neurons, and thus rich types of population synchronization are found to emerge. As an element in our coupled neural system, we choose a simple Izhikevich neuron (Izhikevich, 2007, 2003, 2004; Izhikevich and Edelman, 2008) which is as biologically plausible as the Hodgkin-Huxley model (Hodgkin and Huxley, 1952), yet as computationally efficient as the integrate-and-fire model (Lapicque, 1907; Abbott, 1999). These Izhikevich neurons interact via excitatory AMPA synapses in our computational study. For small individual neurons fire spikings independently, and thus the population state is incoherent. However, when passing a lower threshold , population spike synchronization occurs because the coupling stimulates coherence between noise-induced spikings. As in globally-coupled chaotic systems, this kind of transition between population synchronization and incoherence may be well described in terms of an order parameter (Baek and Ott, 2004; Topaj et al., 2001; Sakaguchi, 2000; Manrubia et al., 2004); in our case, the time-averaged fluctuation of the population-averaged membrane potential plays the role of . As is further increased and passes another threshold , noise-induced burstings appear in individual membrane potentials, and population burst synchronization also emerges. In contrast to spiking activity, bursting activity alternates between a silent phase and an active phase of repetitive spikings (Izhikevich, 2007; Coombes and Bressloff, 2005; Izhikevich, 2000). This type of burstings are known to play the important roles in neural communication (Izhikevich, 2003, 2004; Izhikevich and Edelman, 2008). As continues to increase, the length of active phase in individual bursting potential increases, and eventually a transition from bursting to fast spiking occurs at a threshold . Consequently, breakup of population burst synchronization occurs and incoherent states appear because individual fast spikings keep no pace with each other. However, as is further increased, coupling stimulates population synchronization between fast spikings in a range of . For population states become incoherent and slow spikings appear in individual membrane potentials. As a final step, when passes a high threshold , coupling induces oscillator death (i.e., quenching of noise-induced slow spikings of individual neurons) because each neuron is attracted to a noisy equilibrium state. This stochastic oscillator death in the presence of noise (Lim and Kim, 2008, 2009b) is in contrast to the deterministic oscillator death occurring in the absence of noise (Suárez-Vargas et al., 2009). At the population level, a transition from firing to non-firing states results from stochastic oscillator death. We also characterize the firing-nonfiring transition in terms of the time-averaged population spike rate which plays a role similar to that of the order parameter for the incoherence-coherence transition. In addition to the statistical-mechanical analysis using and , these diverse population and individual states are well characterized by using the techniques of nonlinear dynamics such as the raster plot of spikes, the time series of the membrane potential, and the phase portrait.

2 Subthreshold Izhikevich Neuron Model

We consider an excitatory population of globally-coupled subthreshold neurons. As an element in our coupled neural system, we choose the simple Izhikevich neuron model which is not only biologically plausible, but also computationally efficient (Izhikevich, 2007, 2003, 2004; Izhikevich and Edelman, 2008). The population dynamics in this neural network is governed by the following set of ordinary differential equations:

(1)
(2)
(3)

with the auxiliary after-spike resetting:

(4)

where

(5)
(6)
(7)

We note that of Eq. (5) was obtained by fitting the spike initiation dynamics of cortical neurons so that the membrane potential has mV scale and the time has ms scale (Izhikevich, 2003, 2004; Izhikevich and Edelman, 2008). The state of the th neuron at a time is characterized by three dimensionless state variables: the membrane potential , the recovery variable representing the activation of the ionic current and the inactivation of the ionic current, and the synaptic gate variable denoting the fraction of open synaptic ion channels. After the spike reaches its apex (=30 mV), the membrane voltage and the recovery variable are reset according to Eq. (4). There are four dimensionless parameters, and representing the time scale of the recovery variable , the sensitivity of to the subthreshold fluctuations of , and the after-spike reset value of and , respectively. Tuning the four parameters, the Izhikevich neuron model may produce 20 of the most prominent neuro-computational features of cortical neurons (Izhikevich, 2003, 2004; Izhikevich and Edelman, 2008). Unlike Hodgkin-Huxley-type conductance-based models, the Izhikevich model matches neuronal dynamics instead of matching neuronal electrophysiology.

Each Izhikevich neuron is stimulated by the common DC current and an independent Gaussian white noise [see the 2nd and 3rd terms in Eq. (1)] satisfying and , where denotes the ensemble average. The noise is a parametric one which randomly perturbs the strength of the applied current , and its intensity is controlled by the parameter . The last term in Eq. (1) represents the coupling of the network. Each neuron is connected to all the other ones through global couplings via excitatory AMPA synapses. of Eq. (6) represents such synaptic current injected into the th neuron. Here the coupling strength is controlled by the parameter and is the synaptic reversal potential. We use mV for the excitatory synapse. The synaptic gate variable obeys the 1st order kinetics of Eq. (3) (Golomb and Rinzel, 1994; Wang and Buzski, 1996). Here, the normalized concentration of synaptic transmitters, activating the synapse, is assumed to be an instantaneous sigmoidal function of the membrane potential with a threshold in Eq. (7), where we set mV and mV. The transmitter release occurs only when the neuron emits a spike (i.e., its potential is larger than ). For the excitatory glutamate synapse (involving the AMPA receptors), the synaptic channel opening rate, corresponding to the inverse of the synaptic rise time , is , and the synaptic closing rate , which is the inverse of the synaptic decay time , is (Brgers and Kopell, 2003, 2005).

Here we consider the case of regular-spiking cortical excitatory neurons for , , , and . Depending on the system parameters, the Izhikevich neurons may exhibit either type-I or type-II excitability (Izhikevich, 2000, 2003, 2004; Izhikevich and Edelman, 2008); for the case of type-I (type-II) neurons, the firing frequency begins to increase from zero (non-zero finite value) when passes a threshold (Hodgkin, 1948; Rinzel and Ermentrout, 1948). For our case, a deterministic Izhikevich neuron (for ) exhibits a jump from a resting state (denoted by solid line) to a spiking state (denoted by solid circles) via a subcritical Hopf bifurcation for by absorbing an unstable limit cycle born via a fold limit cycle bifurcation for , as shown in Fig. 1(a). Hence, the Izhikevich neuron shows the type-II excitability because it begins to fire with a non-zero frequency that is relatively insensitive to the change in . Throughout this paper, we consider a subthreshold case of . An isolated subthreshold Izhikevich neuron cannot fire spontaneously without noise. Figures 1(b) and 1(c) show a time series of the membrane potential of a subthreshold neuron and the interspike interval histogram for . Complex noise-induced subthreshold oscillations and spikings with irregular interspike intervals appear. Population synchronization is investigated in an excitatory population of these subthreshold Izhikevich neurons coupled via AMPA synapses. Hereafter, we fix the value of the noise intensity as . Numerical integration of the governing equations (1)-(3) is done using the Heun method (San Miguel and Toral, 2000) (with the time step ms). For each realization of the stochastic process in Eqs. (1)-(3), we choose a random initial point for the th neuron with uniform probability in the range of , , and .

Figure 1: (a) Bifurcation diagram in the single type-II regular-spiking Izhikevich neuron for . Solid line denotes a stable equilibrium point. Maximum and minimum values of for the spiking state are represented by solid circles. (b) Time series of the membrane potential and (c) the interspike interval histogram in the single subthreshold Izhikevich neuron for and .

3 Coupling-Induced Population Synchronization

By varying the coupling strength , we investigate population synchronization, via which synchronous brain rhythms emerge, by using diverse techniques of statistical mechanics and nonlinear dynamics. Emergence of population synchronization may be described by the population-averaged membrane potential (corresponding to the global potential) and the global recovery variable ,

(8)

Figure 2(a) shows rich phase portraits of the representative coherent and incoherent population states in the phase plane. Population synchronization appears on noisy limit cycles for and , while incoherent states occur on noisy equilibrium points for , and . Particularly, for population burst synchronization emerges on a noisy hedgehoglike limit cycle; spines and body correspond to active and silent phases of the bursting activity, respectively. A schematic phase diagram of these population states on the axis is shown in Fig. 2(b). Transitions between incoherent and coherent states may be well described in terms of the order parameter. For our case, the mean square deviation of the global potential (i.e., time-averaged fluctuations of ),

(9)

plays the role of an order parameter, where the overbar represents the time averaging. Here, we discard the first time steps of a stochastic trajectory as transients during ms, and then we numerically compute by following the stochastic trajectory for ms when and . For the coherent (incoherent) state, the order parameter approaches a nonzero (zero) limit value in the thermodynamic limit of . Figure 2(c) shows a plot of the order parameter versus the coupling strength. For , the order parameter tends to zero as , and hence incoherent states exist. As passes the lower threshold , a coherent transition to spike synchronization occurs because the coupling stimulates coherence between noise-induced spikings. Thus, spike synchronization appears for . However, when passing another threshold , individual neurons exhibit noise-induced burstings and population burst synchronization occurs. As is further increased and passes a threshold ), a transition from bursting to fast spiking occurs in individual potentials and the burst synchronization breaks up because individual fast spikes keep no pace with each other. Thus, for the order parameter tends to zero as , and incoherent states appear. However, with further increase in , coupling-induced fast spike synchronization occurs in a range of . For incoherent states reappear as shown in Fig. 2(c), and individual neurons exhibit slow spikings. As a final step, when passing a high threshold , coupling induces stochastic oscillator death (i.e., cessation of noise-induced slow spikings) because each neuron is attracted to a noisy equilibrium state. This stochastic oscillator death leads to a transition from firing to non-firing state at the population level. In this way, three kinds of population synchronization (i.e., spike, burst, and fast spike synchronization) emerge in the gray regions of Figs. 2(b) and 2(c).

Figure 2: Population coherent and incoherent states in globally-coupled excitatory subthreshold Izhikevich neurons for and . (a) Phase portraits of the population states in the plane for . (b) Schematic diagram of populations states on the axis (c) Transition between coherence and incoherence: plots of versus for , , and . Spike, burst, and fast spike synchronizations occur in the gray regions in (b) and (c).

We present population synchronization clearly in terms of the raster plots of spikes and the time series of the global potential . The first spike synchronization appears in a range of . An example for is shown in Fig. 3(a). Stripes (composed of spikes), indicating population synchronization, appear regularly with the mean time interval ms) in the raster plot, and shows a small-amplitude negative-potential population rhythm with frequency Hz. The second burst synchronization occurs in a range of . Figure 3(b1) shows bursting synchronization for . Clear burst bands, composed of stripes of spikes, appear successively at nearly regular time intervals ms) in the raster plot, and the corresponding global potential exhibits a large-amplitude bursting rhythm with Hz. In contrast to spiking rhythm in Fig. 3(a), much more hyperpolarization occurs in the bursting rhythm. For a clear view, magnifications of a single burst band and are given in Fig. 3(b2). For this kind of burstings, burst synchronization refers to a temporal relationship between the active phase onset or offset times of bursting neurons, while spike synchronization characterizes a temporal relationship between spikes fired by different bursting neurons in their respective active phases (Rubin, 2007). In addition to burst synchronization, spike synchronization also occurs in each burst band, as shown in Fig. 3(b2); as we go from the onset to the offset times, wider stripes appear in the burst band. Hence, this kind of burst synchronization occurs on a hedgehoglike limit cycle [see Fig. 2(a)], and exhibits bursting activity like individual potentials. Finally, the third fast spike synchronization emerges in a range of . An example for is shown in Fig. 3(c). In contrast to Fig. 3(a), stripes appear successively at short time intervals ms) in the raster plot, and shows a small-amplitude positive-potential fast rhythm with Hz.

Figure 3: Three types of population synchronization in globally-coupled excitatory subthreshold Izhikevich neurons for and . Raster plots of spikes and time series of the global potential for (a) spike synchronization (), (b1) and (b2) burst synchronization (), and (c) fast spike synchronization ().
Figure 4: Various transitions in individual membrane potentials in globally-coupled excitatory subthreshold Izhikevich neurons for and . Transition from spiking to bursting in individual potentials: time series of the membrane potential and the recovery variable of the 1st neuron for (a1) 0.6, (a2) 0.75, (a3) 0.8. Plots of fraction (number of spikes per burst) versus n for (b1) 0.75 and (b2) 0.8. Average number of spikes per burst versus is shown in (c). Transition from bursting to fast spiking in individual potentials: time series of the membrane potential for (d1) 6.06, (d2) 6.07, and (d3) 6.09. Inverse of average burst length, , versus is shown in (e). Transition from fast spiking to oscillator death in individual potentials: time series of the membrane potential for (d1) 15, (d2) 18, and (d3) 20. Mean firing frequency versus is shown in (g).

With increasing , change in firing patterns of individual neurons and the corresponding population states are discussed. For small individual neurons exhibit noise-induced spikings. Figure 4(a1) shows the time series of the membrane potential and the recovery variable of the 1st neuron for . Here, the slow variable provides a negative feedback to the fast variable . Spiking pushes outside the spiking area. Then, slowly decays into the quiescent area [see Fig. 4(a1)], which results in termination of spiking. This quiescent pushes outside the quiescent area; then, revisits the spiking area, which leads to spiking of . Through repetition of this process, spikings appear successively in , as shown in Fig. 4(a1). Population synchronization between these individual spikings appear for . However, as J passes a threshold , the coherent synaptic input into the first neuron becomes so strong that a tendency that a spike in cannot push outside the spiking area occurs. As an example, see the case of J = 0.75 in Fig. 4(a2). For this case, both spikings (singlets) and burstings (doublets consisting of two spikes) appear, as shown in Fig. 4(a2); 69 percentage of firings are singlets, while 31 percentages of firings are doublets [see Fig. 4(b1)]. For , after the 2nd spike in , at first decreases a little (with nearly zero slope) and then increases abruptly up to a peak value of , which is larger than that of for . Thus, after the 2nd spike, remains inside the spiking area; hence, a third spike, constituting a doublet, appears in . After this 3rd spike, is pushed away from the spiking area and slowly decays into the quiescent area, which results in the termination of repetitive spikings. In this way, doublets appear in for J = 0.75. As J is further increased, the coherent synaptic input becomes stronger, so the number of spikes in a burst increases [e.g., see the doublets and triplets for in Fig. 4(a3)]; 88.7 percentage of firings are doublets, while 11.3 percentages of firings are triplets [see Fig. 4(b2)]. Figure 4(c) shows the average number of spikes per burst versus , and becomes larger than unity (i.e., burstings appear) for . Population synchronization between these burstings occurs for . With increase in , longer burst lengths (i.e., lengths of the active phase for the bursting activity) appear as shown in Figs. 4(d1) and 4(d2), and eventually the average burst length, , diverges to the infinity (i.e., its inverse, , decreases to zero) as goes to [see Fig. 4(e)]. Then, for individual neurons exhibit fast spikings as shown in Fig. 4(d) for . Since these fast spikes keep no pace with each other, incoherent states appear as shown in Fig. 2(c). However, as is further increased, the coupling induces fast spike synchronization in a range of . Then, slow spikings with longer spiking phases appear, as shown in Figs. 4(f1) and 4(f2). Figure 4(g) shows the mean firing frequency (i.e., the inverse of the average interspike interval) versus the coupling strength. As approaches a threshold , goes to zero. Consequently, for stochastic oscillator death (i.e., quenching of noise-induced slow spikings) occurs [e.g., see Fig. 4(f3) for ].

Figure 5: Transition between firing and non-firing states in globally-coupled excitatory subthreshold Izhikevich neurons for and . Plots of the average population spike rate versus the coupling strength for , , and .

The stochastic oscillator death of individual neurons leads to a transition from firing to non-firing states at the population level. This firing-nonfiring transition may be well described in terms of the average population spike rate which is a time average of the instantaneous population spike rate. To get a smooth instantaneous population spike rate , each spike in the raster plot is convoluted with a Gaussian kernel function (Shimazaki and Shinomoto, 2010):

(10)

where is the neuron index, is the th spike of the th neuron, is the total number of spikes for the th neuron, is the total number of neurons, and the Gaussian kernel of band width (=1 ms) is given by

(11)

Here, we discard the first time steps of a stochastic trajectory as transients during ms, and then we numerically compute by following the stochastic trajectory for ms when and . For the firing (non-firing) state, the average population spike rate approaches a non-zero (zero) limit value in the thermodynamic limit of . Figure 5 shows a plot of the average population spike rate versus the coupling strength. For , tends to zero as goes to the infinity, and hence non-firing states appear due to the stochastic oscillator death of individual neurons.

4 Summary

We have studied coupling-induced population synchronization which may be used for efficient cognitive processing by changing the coupling strength in an excitatory population of subthreshold Izhikevich neurons. As is increased, rich population states have appeared in the following order: incoherent state spike synchronization burst synchronization incoherent state fast spike synchronization incoherent state non-firing state. Particularly, three types of population synchronization (i.e., spike, burst, and fast spike synchronization) have been found to occur. Transitions between population synchronization and incoherence have been well described in terms of a thermodynamic order parameter. These various transitions between population states have occurred due to emergence of the following diverse individual states: spiking bursting fast spiking slow spiking oscillator death. Each population synchronization and individual state were well characterized by using the techniques of nonlinear dynamics such as the raster plot of spikes, the time series of membrane potentials, and the phase portrait. As a final step, stochastic oscillator death (cessation of individual noise-induced slow spikings) occurred because each individual neuron is attracted to a noisy equilibrium state. This stochastic oscillator death leads to a transition from firing to non-firing states at the population level. The firing-nonfiring transition has also been characterized in terms of the average population spike rate. Since the Izhikevich model we employed for our study is a canonical model (Izhikevich, 2007, 2003, 2004; Izhikevich and Edelman, 2008), we expect that our results are still valid in other neuronal models. Finally, we note that population synchronization of noise-induced firings may lead to emergence of synchronous brain rhythms in a noisy environment which contribute to cognitive functions such as sensory perception, multisensory integration, selective attention, and working memory.

Footnotes

  1. email: woochanglim@dnue.ac.kr
  2. email: woochanglim@dnue.ac.kr

References

  1. Abbott LF (1999) Lapicque’s Introduction of the Integrate-and-Fire Model (1907). Brain Research Bulletin 50:303-304.
  2. Baek SJ, Ott E (2004) Onset of synchronization in systems of globally coupled chaotic maps. Physical Review E 69:066210.
  3. Brgers C, Kopell N (2003) Synchronization in networks of excitatory and inhibitory neurons with sparse, random connectivity. Neural Computation 15:509-538.
  4. Brgers C, Kopell N. (2005) Effects of noisy drive on rhythms in networks of excitatory and inhibitory neurons. Neural Computation 17:557-608.
  5. Buzski G (2006) Rhythms of the Brain. Oxford University Press, New York.
  6. Coombes S, Bressloff PC (2005) Bursting: The Genesis of Rhythm in the Nervous System. World Scientific, Singapore.
  7. Golomb D, Rinzel J (1994) Clustering in globally coupled inhibitory neurons. Physica D 72:259-282.
  8. Hodgkin AL (1948) The local electric changes associated with repetitive action in a non-medullated axon. The Journal of Physiology 107:165-181.
  9. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology 117:500-544.
  10. Hong DG, Kim SY, Lim W (2011) Effect of sparse random connectivity on the stochastic spiking coherence of inhibitory subthreshold neurons. Journal of the Korean Physical Society 59:2840-2846.
  11. Izhikevich EM (2000) Neural excitability, spiking and bursting. International Journal of Bifurcation and Chaos 10:1171-1266.
  12. Izhikevich EM (2003) Simple Model of Spiking Neurons. IEEE Transactions on Neural Networks 14:1569-1572.
  13. Izhikevich EM (2004) Which model to use for cortical spiking neurons? IEEE Transactions on Neural Networks 15:1063-1070.
  14. Izhikevich EM (2007) Dynamical Systems in Neuroscience. MIT Press, Cambridge.
  15. Izhikevich EM, Edelman GM (2008) Large-Scale Model of Mammalian Thalamocortical Systems. Proceedings of the National Academy of Sciences 105:3593-3598.
  16. Jiao X, Wang R (2009) Synchronous firing patterns of neuronal population with excitatory and inhibitory Connections. International Journal of Nonlinear Mechanics 45:647-651.
  17. Kim SY, Hong DG, Kim J, Lim W (2012a) Inhibitory coherence in a heterogeneous population of subthreshold and suprathreshold type-I neurons. Journal of Physics A: Mathematical and General 45:155102.
  18. Kim SY, Kim Y, Hong DG, Kim J, Lim W (2012b) Stochastic bursting synchronization in a population of subthreshold Izhikevich neurons. Journal of the Korean Physical Society 60:1441-1447.
  19. Lapicque L (1907) Recherches quantitatives sur l’excitation e´lectrique des nerfs traite´e comme une polarization. Journal de Physiologie et Pathologie General 9:620-635.
  20. Lim W, Kim SY (2007) Characterization of stochastic spiking coherence in coupled neurons. Journal of the Korean Physical Society 51:1427-1431.
  21. Lim W, Kim SY (2008) Stochastic oscillator death in globally coupled neural systems. Journal of the Korean Physical Society 52:1913-1917.
  22. Lim W, Kim SY (2009a) Stochastic spiking coherence in coupled subthreshold Morris-Lecar neurons. International Journal of Modern Physics B 23:703-710.
  23. Lim W, Kim SY (2009b) Coupling-induced spiking coherence in coupled subthreshold neurons. International Journal of Modern Physics B 23:2149-2157.
  24. Lim W, Kim SY (2011) Statistical-mechanical measure of stochastic spiking coherence in a population of inhibitory subthreshold neuron. Journal of Computational Neuroscience 31:667-677.
  25. Liu Y, Wang R, Zhang Z, Jiao X (2010) Analysis on Stability of Neural Network in the Presence of Inhibitory Neurons. Cognitive Neurodynamics 4:61-68.
  26. Manrubia SC, Mikhailov AS, Zanette DH (2004) Emergence of Dynamical Order. World Scientific, Singapore.
  27. Qu JY, Wang RB, Du Y (2012) Synchronization study in ring-like and grid-like neuronal networks. Cognitive Neurodyanmics 6:21-31.
  28. Rinzel J, Ermentrout B (1998) Analysis of neural excitability and oscillations. In: Koch C, Segev I (eds) Methods in Neuronal Modeling: From Synapses to Networks. MIT Press, Cambridge, pp. 251-292.
  29. Rubin JE (2007) Burst synchronization. Scholarpedia 2:1666.
  30. Sakaguchi H (2000) Phase transition in globally coupled Rössler oscillators. Physical Review E 61:7212-7214.
  31. San Miguel M, Toral R (2000) Stochastic effects in physical systems. In: Martinez J, Tiemann R, Tirapegui E (eds) Instabilities and Nonequilibrium Structures VI. Kluwer Academic Publisher, Dordrecht, pp. 35-127.
  32. Shimazaki H, Shinomoto S (2010) Kernel bandwidth optimization in spike rate estimation. Journal of Computational Neuroscience 29:171-182.
  33. Suárez-Vargas JJ, González JA, Stefanovska A, McClintock PV (2009) Diverse routes to oscillation death in a coupled-oscillator system. Europhysics Letters 85:38008; see references therein.
  34. Topaj D, Kye WH, Pikovsky A (2001) Transition to coherence in populations of coupled chaotic oscillators: A linear response approach. Physical Review Letters 87:074101.
  35. Traub RD, Whittington MA (2010) Cortical Oscillations in Health and Diseases. Oxford University Press, New York.
  36. Wang H, Wang Q, Lu Q (2011) Bursting oscillations, bifurcation and synchronization in neuronal systems. Chaos Solitons & Fractals 44:667-675.
  37. Wang Q, Lu Q, Chen G (2008). Synchronization transition induced by synaptic delay in coupled fast-spiking neurons. International Journal of Bifurcation and Chaos 18:1189-1198.
  38. Wang Q, Perc M, Duan Z, Chen G (2010) Impact of delays and rewiring on the dynamics of small-world neuronal networks with two types of coupling. Physica A 389:3299-3306.
  39. Wang Q, Sanjuán MAF, Chen G (2012) Transition of phase locking modes in a minimal neuronal network. Neurocomputing 81:60-66.
  40. Wang QY, Zheng YH (2011) Effects of information transmission delay and channel blocking on synchronization in scale-free Hodgkin-Huxley neuronal networks. Acta Mechanica Sinica 27:1052-1058.
  41. Wang R, Jiao X (2006) Stochastic model and neural coding of large-scale neuronal population with variable coupling strength. Neurocomputing 69:778-785.
  42. Wang R, Zhang Z, Duan YB (2003) Nonlinear stochastic models of neurons activities. Neurocomputing 51:401-441.
  43. Wang R, Zhang Z (2011) Phase Synchronization Motion and Neural Coding in Dynamic Transmission of Neural Information. IEEE Transactions on Neural Networks 22:1097-1106.
  44. Wang R, Zhang Z, Tee CK (2009) Neurodynamics analysis on transmission of brain information. Applied Mathematics and Mechanics 30:1415-1428.
  45. Wang XJ, Buzski G (1996) Gamma oscillations by synaptic inhibition in a hippocampal interneuronal network. The Journal of Neuroscience 16:6402-6413.
  46. Wang XJ (2003) Neural Oscillation. In: Nadel L (ed) Encyclopedia of Cognitive Science. MacMillan, London, pp. 272-280.
  47. Wang XJ (2010) Neurophysiological and computational principles of cortical rhythms in cognition. Physiological Reviews 90:1195-1268.
  48. Wang Y, Chik DTW, Wang ZD (2000) Coherence resonance and noise-induced synchronization in globally coupled Hodgkin-Huxley neurons. Physical Review E 61:740-746.
  49. Zhang X, Wang R, Zhang Z (2010) Dynamic phase synchronization characteristics of variable high-order coupled neuronal oscillator population. Neurocomputing 73:2665-2670.
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