Suppression of Chaos and Phase Locking in Two Coupled Nonidentical Neurons under Periodic Input

Dynamical behaviour of two coupled neurons with at least one of them being chaotic is presented. Bifurcation diagrams and Lyapunov exponents are calculated to diagnose the dynamical behaviour of the coupled neurons with the increasing coupling strength. It is found that when the coupling strength increases, a chaotic neuron can be controlled by the coupling between neurons. At the same time, phase locking is studied by the maxima of the differences of instantaneous phases and average frequencies between two coupled neurons, and the inherent connection of phase locking and the suppression of chaos is formulated. It is observed that the onset of phase locking is closely related to the suppression of chaos. Finally, a way for suppression of chaos in two coupled nonidentical neurons under periodic input is suggested.     

Chaotic phase synchronization in small-world networks of bursting neurons

We investigate the chaotic phase synchronization in a system of coupled bursting neurons in small-world networks. A transition to mutual phase synchronization takes place on the bursting time scale of coupled oscillators, while on the spiking time scale, they behave asynchronously. It is shown that phase synchronization is largely facilitated by a large fraction of shortcuts, but saturates when it exceeds a critical value. We also study the external chaotic phase synchronization of bursting oscillators in the small-world network by a periodic driving signal applied to a single neuron. It is demonstrated that there exists an optimal small-world topology, resulting in the largest peak value of frequency locking interval in the parameter plane, where bursting synchronization is maintained, even with the external driving. The width of this interval increases with the driving amplitude, but decrease rapidly with the network size. We infer that the externally applied driving parameters outside the frequency locking region can effectively suppress pathologically synchronized rhythms of bursting neurons in the brain.     

Phase synchronization and its transition in two coupled bursting neurons: theoretical and numerical analysis

It is crucially important to study different synchronous regimes in coupled neurons because different regimes may correspond to different cognitive and pathological states. In this paper, phase synchronization and its transitions are discussed by means of theoretical and numerical analyses. In two coupled modified Morris--Lecar neurons with a gap junction, we show that the occurrence of phase synchronization can be investigated from the dynamics of phase equation, and the analytical synchronization condition is derived. By defining the phase of spike and burst, the transitions from burst synchronization to spike synchronization and then toward nearly complete synchronization can be identified by bifurcation diagrams, the mean frequency difference and time series of neurons. The simulation results suggest that the synchronization of bursting activity is a multi-time-scale phenomenon and the phase synchronization deduced by the phase equation is actually spike synchronization.     

Noise-induced synchronization and coherence resonance of a Hodgkin-Huxley model of thermally sensitive neurons

We study nontrivial effects of noise on synchronization and coherence of a chaotic Hodgkin-Huxley model of thermally sensitive neurons. We demonstrate that identical neurons which are not coupled but subjected to a common fluctuating input (Gaussian noise) can achieve complete synchronization when the noise amplitude is larger than a threshold. For nonidentical neurons, noise can induce phase synchronization. Noise enhances synchronization of weakly coupled neurons. We also find that noise enhances the coherence of the spike trains. A saddle point embedded in the chaotic attractor is responsible for these nontrivial noise-induced effects. Relevance of our results to biological information processing is discussed.     

Structured information in sparse-code metric neural networks

Sparse-code networks have retrieval abilities which are strongly dependent on the firing threshold for the neurons. If the connections are spatially uniform, the macroscopic properties of the network can be measured by the overlap between neurons and learned patterns, and by the global activity. However, for nonuniform networks, for instance small-world networks, the neurons can retrieve fragments of patterns without performing global retrieval. Local overlaps are needed to describe the network. We characterize the structure type of the neural states using a parameter that is related to fluctuations of the local overlaps, with distinction between bump and block phases. Simulation of neural dynamics shows a competition between localized (bump), structured (block) and global retrieval. When the network topology randomness increases, the phase-diagram shows a transition from local to global retrieval. Furthermore, the local phase splits into a bump phase for low activity and a block phase for high activity. A theoretical approach solves the asymptotic limit of the model, and confirms the simulation results which predicts the change of stability from bumps to blocks when the storage ratio increases.     

两个非耦合Hindmarsh-Rose神经元同步的非线性特征研究

利用Hindmarsh-Rose(HR)神经元输出的膜电压作为刺激调整两个具有不同初始条件的非耦合HR神经元的电流输入，通过分析神经元放电峰峰间期(ISI)的分布揭示了两个神经元同步过程轨道演化的机理.在周期信号刺激下，两个具有相同参数原处于混沌状态的神经元可以 实现完全同步，且可以同步到不同于刺激信号频率的周期响应上；两个具有不同参数的神经 元可以实现相位同步，参数差别较小的两个神经元可以相位同步到与刺激信号不同频率的周 期响应上，参数差别较大的两个神经元只可能相位同步到与刺激信号相同频率的周期响应上 .混沌信号刺激两个神经元只可能同步到产生混沌信号神经元的放电模式上，可见混沌刺激 更有利于神经元信息编码与解码.分析两个被调整神经元系统的最大条件Lyapunov 指数(Lmc )与刺激强度k的关系表明当k达到某一阈值时两个系统的Lmc均为负值是两个系统实现同 步的必要条件.平均发放率相同的混沌刺激和周期刺激相比较混沌刺激更容易使两个神经元 实现同步，表明混沌刺激产生的效应更强，该结论与实验结果相符合. 关键词： 放电峰峰间期 同步 相位同步 条件Lyapunov 指数     

A neural network model composed of two-state and three-state neurons

A neural network model composed of two-state (1 and -1) and three-state (1, 0 and -1) neurons is proposed. The two-state neurons are connected with the three-state ones only and vice versa. We derive dynamic equations for the model under the assumption of non-symmetrical dilution of connections. A zero-noise phase diagram is obtained and a region in which two fixed point solutions can coexist is found. Basins of attraction for the solutions are also investigated. Received 26 October 1998 and Received in final form 12 February 1999     

Spiral Wave in Small-World Networks of Hodgkin--Huxley Neurons

The effect of small-world connection and noise on the formation and transitionof spiral wave in the networks of Hodgkin-Huxley neurons are investigated in detail. Some interesting results are found in our numerical studies. i) The quiescent neurons are activated to propagate electric signal to others by generating and developing spiral wave from spiral seed in small area. ii) A statistical factor is defined to describe the collective properties and phase transition induced by the topology of networks and noise. iii) Stable rotating spiral wave can be generated and keeps robust when the rewiring probability is below certain threshold, otherwise, spiral wave can not be developed from the spiral seed and spiral wave breakup occurs for a stable rotating spiral wave. iv) Gaussian white noise is introduced on the membrane of neuronsto study the noise-induced phase transition on spiral wave in small-world networks of neurons. It is confirmed that Gaussian white noise plays active role in supporting and developing spiral wave in the networks of neurons, and appearance of smaller factor of synchronization indicates high possibility to induce spiral wave.     

Spike-burst synchronization in an ensemble of electrically coupled discrete model neurons

In this paper, we study the dynamics of a system of two model neurons interacting via the electrical synapse. Each neuron is described by a two-dimensional discontinuous map. A chaotic relaxational-type attractor, which corresponds to the spiking-bursting chaotic oscillations of neurons is shown to exist in a four-dimensional phase space. It is found that the dynamical mechanism of formation of chaotic bursts is based on a new phenomenon of generation of transient chaotic oscillations. It is demonstrated that transition from the chaotic-burst generation to the state of relative rest occurs with a certain time delay. A new characteristic which estimates the degree of synchronization of the spiking-bursting oscillations is introduced. The dependence of the synchronization degree on the strength of coupling of the ensemble elements is studied.     

Chaotic phase synchronization in a modular neuronal network of small-world subnetworks

We investigate the onset of chaotic phase synchronization of bursting oscillators in a modular neuronal network of small-world subnetworks. A transition to mutual phase synchronization takes place on the bursting time scale of coupled oscillators, while on the spiking time scale, they behave asynchronously. It is shown that this bursting synchronization transition can be induced not only by the variations of inter- and intra-coupling strengths but also by changing the probability of random links between different subnetworks. We also analyze the effect of external chaotic phase synchronization of bursting behavior in this clustered network by an external time-periodic signal applied to a single neuron. Simulation results demonstrate a frequency locking tongue in the driving parameter plane, where bursting synchronization is maintained, even with the external driving. The width of this synchronization region increases with the signal amplitude and the number of driven neurons but decreases rapidly with the network size. Considering that the synchronization of bursting neurons is thought to play a key role in some pathological conditions, the presented results could have important implications for the role of externally applied driving signal in controlling bursting activity in neuronal ensembles.     

Interaction function of coupled bursting neurons

The interaction functions of electrically coupled Hindmarsh–Rose(HR) neurons for different firing patterns are investigated in this paper.By applying the phase reduction technique,the phase response curve(PRC) of the spiking neuron and burst phase response curve(BPRC) of the bursting neuron are derived.Then the interaction function of two coupled neurons can be calculated numerically according to the PRC(or BPRC) and the voltage time course of the neurons.Results show that the BPRC is more and more complicated with the increase of the spike number within a burst,and the curve of the interaction function oscillates more and more frequently with it.However,two certain things are unchanged:Φ = 0,which corresponds to the in-phase synchronization state,is always the stable equilibrium,while the anti-phase synchronization state with Φ = 0.5 is an unstable equilibrium.     

Phase synchronization in two coupled chaotic neurons

Chaotically-spiking dynamics of Hindmarsh–Rose neurons are discussed based on a flexible definition of phase for chaotic flow. The phase synchronization of two coupled chaotic neurons is in fact the spike synchronization. As a multiple time-scale model, the coupled HR neurons have quite different behaviors from the Rössler oscillators only having single time-scale mechanism. Using such a multiple time-scale model, the phase function can detect synchronization dynamics that cannot be distinguished by cross-correlation. Moreover, simulation results show that the Lyapunov exponents cannot be used as a definite criterion for the occurrence of chaotic phase synchronization for such a system. Evaluation of the phase function shows its utility in analyzing nonlinear neural systems.     

Multi-time-scale synchronization and information processing in bursting neuron networks

We analyze the effect of synchronization in networks of chemically coupled multi-time-scale (spiking-bursting) neurons on the process of information transmission within the network. Although, synchronization occurs first in the slow time-scale (burst) and then in the fast time-scale (spike), we show that information can be transmitted with low probability of errors in both time scales when the bursts become synchronized. Furthermore, we show that for networks of non-identical multi-time-scales neurons, complete synchronization is no longer possible, but instead burst phase synchronization. Our analysis shows that clusters of burst phase synchronized neurons are very likely to appear in a network for parameters far smaller than the ones for which the onset of burst phase synchronization in the whole network takes place.     

计算相位响应曲线的方波扰动直接算法

通过相位响应曲线可对具有极限环周期运动的动力系统的性质有更为深入的理解.神经元是一个典型的动力系统,因此相位响应曲线提供了一种研究神经元重复周期放电行为的新思路.本文提出一种求解相位响应曲线的方法,即方波扰动的直接算法,通过Hodgkin-Huxley,Fitz Hugh-Nagumo,Morris-Lecar和Hindmarsh-Rose神经元模型验证该算法可计算周期峰放电、周期簇放电的相位响应曲线.该算法克服了其他算法在运用过程中的局限性.利用该算法计算结果表明:周期峰放电的相位响应曲线类型是由其分岔类型所决定;在Morris-Lecar模型中发现一种开始于Hopf分岔终止于鞍点同宿轨道分岔的阈上周期振荡,其相位响应曲线属于第二类型.通过大量的相位响应曲线的计算发现相位响应的相对大小及正负性仅取决于扰动所施加的时间,而且周期簇放电的相位响应曲线比周期峰放电的相位响应曲线更为复杂.     

Coding of amplitude-modulated signals in the cochlear nucleus of a grass frog

To study the mechanisms that govern the coding of temporal features of complex sound signals, responses of single neurons located in the dorsal nucleus of the medulla oblongata (the cochlear nucleus) of a curarized grass frog (Rana temporaria) to pure tone bursts and amplitude modulated tone bursts with a modulation frequency of 20 Hz and modulation depths of 10 and 80% were recorded. The carrier frequency was equal to the characteristic frequency of a neuron, the average signal level was 20–30 dB above the threshold, and the signal duration was equal to ten full modulation periods. Of the 133 neurons studied, 129 neurons responded to 80% modulated tone bursts by discharges that were phase-locked with the envelope waveform. At this modulation depth, the best phase locking was observed for neurons with the phasic type of response to tone bursts. For tonic neurons with low characteristic frequencies, along with the reproduction of the modulation, phase locking with the carrier frequency of the signal was observed. At 10% amplitude modulation, phasic neurons usually responded to only the onset of a tone burst. Almost all tonic units showed a tendency to reproduce the envelope, although the efficiency of the reproduction was low, and for half of these neurons, it was below the reliability limit. Some neurons exhibited a more efficient reproduction of the weak modulation. For almost half of the neurons, a reliable improvement was observed in the phase locking of the response during the tone burst presentation (from the first to the tenth modulation period). The cooperative histogram of a set of neurons responding to 10% modulated tone bursts within narrow ranges of frequencies and intensities retains the information on the dynamics of the envelope variation. The data are compared with the results obtained from the study of the responses to similar signals in the acoustic midbrain center of the same object and also with the psychophysical effect of a differential sensitivity increase in the process of adaptation.     

Chinchilla auditory-nerve responses to low-frequency tones

Single unit activity was recorded in the auditory nerves of chinchillas. Period histograms were constructed for responses to tones with frequencies 30-1000 Hz. For low-frequency tones at near-threshold levels, peak period histogram phases for low- and medium-best-frequency (BF) neurons (less than or equal to kHz) ranged from synchronous with condensation at the eardrum to 90 degrees leading it. At near-threshold (but high absolute) levels, high-BF (greater than or equal to 8 kHz) neurons responded in phase with rarefaction. At even higher levels, period histograms for responses of high-BF neurons tended to become bimodal, with one of the modes lagging rarefaction by 90 degrees. Using cochlear microphonics as an indicator of basilar membrane (BM) displacement, at threshold levels, response phase of low- and medium-BF neurons fall within a range between displacement and velocity of the BM toward scala vestibuli. High-BF neurons respond, at threshold (but high) intensities, in phase with BM displacement toward scala tympani. The rates of growth of frequency sensitivity in responses of low-BF (+ 18 dB/oct) and high-BF (+ 12 dB/oct) neurons are consistent with preferred response phases corresponding to BM SV velocity and ST displacement, respectively. At supra-threshold levels high-BF neurons may fire preferentially to both scala tympani displacement and scala vestibuli velocity. These results support the notion that, for high-intensity, low-frequency stimuli, OHC hyperpolarization can induce excitation of the dendrites innervating IHCs.     

Diverse population-bursting modes of adapting spiking neurons

We study the dynamics of a noisy network of spiking neurons with spike-frequency adaptation (SFA), using a mean-field approach, in terms of a two-dimensional Fokker-Planck equation for the membrane potential of the neurons and the calcium concentration gating SFA. The long time scales of SFA allow us to use an adiabatic approximation and to describe the network as an effective nonlinear two-dimensional system. The phase diagram is computed for varying levels of SFA and synaptic coupling. Two different population-bursting regimes emerge, depending on the level of SFA in networks with noisy emission rate, due to the finite number of neurons.     

Synchronization regimes in a map-based model neural network

The dynamical activity of a neural network model composed of electrically connected map-based neurons is investigated. After detailing the behavior of the isolated neuron for a wide parameter range, collective network states are depicted using the activity, spatial correlation and time phase distribution as measures. A detailed discussion on the stability of global and partial synchronization states is presented.