Dislocation Coupling-Induced Transition of Synchronization in Two-Layer Neuronal Networks

The mutual coupling between neurons in a realistic neuronal system is much complex, and a two-layer neuronal network is designed to investigate the transition of electric activities of neurons. The Hindmarsh-Rose neuron model is used to describe the local dynamics of each neuron, and neurons in the two-layer networks are coupled in dislocated type. The coupling intensity between two-layer networks, and the coupling ratio (Pro), which defines the percentage involved in the coupling in each layer, are changed to observe the synchronization transition of collective behaviors in the two-layer networks. It is found that the two-layer networks of neurons becomes synchronized with increasing the coupling intensity and coupling ratio (Pro) beyond certain thresholds. An ordered wave in the first layer is useful to wake up the rest state in the second layer, or suppress the spatiotemporal state in the second layer under coupling by generating target wave or spiral waves. And the scheme of dislocation coupling can be used to suppress spatiotemporal chaos and excite quiescent neurons.     

Instability and Death of Spiral Wave in a Two-Dimensional Array of Hindmarsh-Rose Neurons

Spiral wave could be observed in the excitable media, the neurons are often excitable within appropriate parameters. The appearance and formation of spiral wave in the cardiac tissue is linked to monomorphic ventricular tachycardia that can denervate into polymorphic tachycardia and ventricular fibrillation. The neuronal system often consists of a large number of neurons with complex connections. In this paper, we theoretically study the transition from spiral wave to spiral turbulence and homogeneous state （death of spiral wave） in two-dimensional array of the Hindmarsh-Rose neuron with completely nearest-neighbor connections. In our numerical studies, a stable rotating spiral wave is developed and selected as the initial state, then the bifurcation parameters are changed to different values to observe the transition from spiral wave to homogeneous state, breakup of spiral wave and weak change of spiral wave, respectively. A statistical factor of synchronization is defined with the mean field theory to analyze the transition from spiral wave to other spatial states, and the snapshots of the membrane potentials of all neurons and time series of mean membrane potentials of all neurons are also plotted to discuss the change of spiral wave. It is found that the sharp changing points in the curve for factor of synchronization vs. bifurcation parameter indicate sudden transition from spiral wave to other states. And the results are independent of the number of neurons we used.     

Collapse of Synchronization in a Memristive Network

For an oscillating circuit or coupled circuits,damage in electric devices such as inductor,resistance,memristor even capacitor can cause breakdown or collapse of the circuits. These damage could be associated with external attack or aging in electric devices,and then the bifurcation parameters could be deformed from normal values. Resonators or signal generators are often synchronized to produce powerful signal series and this problem could be investigated by using synchronization in network. Complete synchronization could be induced by linear coupling in a two-dimensional network of identical oscillators when the coupling intensity is beyond certain threshold. The collective behavior and synchronization state are much dependent on the bifurcation parameters. Any slight fluctuation in parameter and breakdown in bifurcation parameter can cause transition of synchronization even collapse of synchronization in the network. In this paper,a two-dimensional network composed of the resonators coupled with memristors under nearestneighbor connection is designed,and the network can reach complete synchronization by carefully selecting coupling intensity. The network keeps synchronization after certain transient period,then a bifurcation parameter in a resonator is switched from the previous value and the adjacent resonators(oscillators) are affected in random. It is found that the synchronization area could be invaded greatly in a diffusive way. The damage area size is much dependent on the selection of diffusive period of damage and deformation degree in the parameter. Indeed,the synchronization area could keep intact at largest size under intermediate deformation degree and coupling intensity.     

Collapse of Synchronization in a Memristive Network

For an oscillating circuit or coupled circuits, damage in electric devices such as inductor, resistance, memristor even capacitor can cause breakdown or collapse of the circuits. These damage could be associated with external attack or aging in electric devices, and then the bifurcation parameters could be deformed from normal values. Resonators or signal generators are often synchronized to produce powerful signal series and this problem could be investigated by using synchronization in network. Complete synchronization could be induced by linear coupling in a two-dimensional network of identical oscillators when the coupling intensity is beyond certain threshold. The collective behavior and synchronization state are much dependent on the bifurcation parameters. Any slight fluctuation in parameter and breakdown in bifurcation parameter can cause transition of synchronization even collapse of synchronization in the network. In this paper, a two-dimensional network composed of the resonators coupled with memristors under nearest- neighbor connection is designed, and the network can reach complete synchronization by carefully selecting coupling intensity. The network keeps synchronization after certain transient period, then a bifurcation parameter in a resonator is switched from the previous value and the adjacent resonators (oscillators) are affected in random. It is found that the synchronization area could be invaded greatly in a diffusive way. The damage area size is much dependent on the selection of diffusive period of damage and deformation degree in the parameter. Indeed, the synchronization area could keep intact at largest size under intermediate deformation degree and coupling intensity.     

Robustness,Death of Spiral Wave in the Network of Neurons under Partial Ion Channel Block

The development of spiral wave in a two-dimensional square array due to partial ion channel block (Potas- sium, Sodium) is investigated, the dynamics of the node is described by Hodgkin-Huxley neuron and these neurons are coupled with nearest neighbor connection. The parameter ratio x Na (and xK ), which defines the ratio of working ion channel number of sodium (potassium) to the total ion channel number of sodium (and potassium), is used to measure the shift conductance induced by channel block. The distribution of statistical variable R in the two-parameter phase space (parameter ratio vs. poisoning area) is extensively calculated to mark the parameter region for transition of spiral wave induced by partial ion channel block, the area with smaller factors of synchronization R is associated the parameter region that spiral wave keeps alive and robust to the channel poisoning. Spiral wave keeps alive when the poisoned area (potassium or sodium) and degree of intoxication are small, distinct transition (death, several spiral waves coexist or multi-arm spiral wave emergence) occurs under moderate ratio x Na (and xK ) when the size of blocked area exceeds certain thresholds. Breakup of spiral wave occurs and multi-arm of spiral waves are observed when the channel noise is considered.     

热敏神经元网络中螺旋波死亡和破裂的数值模拟

实验发现大脑皮层内出现螺旋波且螺旋波对神经元电信号传递有积极作用.利用细胞网络方法从对大脑皮层观察到的螺旋波进行数值模拟.以包含温度因子的热敏神经元模型在二维空间构造规则网络,研究了神经元膜片温度参数对神经元网络中螺旋波演化影响;定义了一类统计同步因子来刻画温度因子引起螺旋波相变(破裂和死亡)的临界条件.发现在规则网络下,当温度超过一定值后螺旋波会死亡和消失而导致整个网络达到均匀同步;在考虑了弱通道噪声情况下,螺旋波温度超越一定临界值则引起螺旋波的破裂.进一步分析了暂时性发烧昏迷的可能机制在于神经系统某些功能区螺旋波传播电信号的中断.     

Breakup of Spiral Waves in Coupled Hindmarsh--Rose Neurons

Breakup of spiral wave in the Hindmarsh-Rose neurons with nearest-neighbour couplings is reported. Appropriate initial values and parameter regions are selected to develop a stable spiral wave and then the Gaussian coloured noise with different intensities and correlation times is imposed on all neurons to study the breakup of spiral wave, respectively. Based on the mean field theory, the statistical factor of synchronization is defined to analyse the evolution of spiral wave. It is found that the stable rotating spiral wave encounters breakup with increasing intensity of Gaussian coloured noise or decreasing correlation time to certain threshold.     

梯度耦合下神经元网络中靶波和螺旋波的诱发研究

神经系统内数量众多的神经元电活动的群体行为呈现一定的节律性和自组织性. 当网络局部区域存在异质性或者受到持续周期性刺激, 则在网络内诱发靶波, 且这些靶波如'节拍器'可调制介质中行波的诱发和传播. 基于Hindmarsh-Rose 神经元模型构造了最近邻连接下的二维神经元网络, 研究在非均匀耦合下神经元网络内有序波的诱发问题. 在研究中, 选定网络中心区域的耦合强度最大, 从中心向边界的神经元之间的耦合强度则按照阶梯式下降. 研究结果表明, 在恰当的耦合梯度下, 神经元网络内诱发的靶波或螺旋波可以占据整个网络, 并有效调制神经元网络的群体电活动, 使得整个网络呈现有序性. 特别地, 当初始值为随机值时, 梯度耦合也可以诱发稳定的有序态. 这种梯度耦合对网络群体行为调制的研究结果有助于理解神经元网络的自组织行为.     

时变反应扩散系统中螺旋波和湍流的控制

研究了一类参数时变的反应扩散系统中螺旋波和湍流对外电场的响应问题.在数值模拟中,以一类改进的Fitzhugh-Nagumo模型为研究对象(在恰当参数值下可分别描述激发介质和振荡介质),考虑随机和不确定因素(如内外噪声、气压、温度梯度分布和介质形变等)所引起的系统参数涨落对斑图演化的影响,在模拟中选取的参数涨落范围确保系统可以观测到稳定旋转的螺旋波、漫游的螺旋波和湍流,经历一定的暂态过程后,对介质施加极化电场,研究螺旋波和湍流在外电场中的演化.数值计算结果表明：在系统参数发生涨落和外电场强度比较小情况下,主 关键词： 螺旋波 湍流 时变系统 Fitzhugh-Nagumo模型     

非连续反馈控制激发介质中的螺旋波

采用非连续反馈方法来控制Fitz-Hugh-Nagumo方程描述的激发介质中的螺旋波. 在控制过程中,对于系统各个格点快变量的幅值进行观测并和设定的阈值进行比较,当采样格点的快变量的值大于这个阈值时,则对系统进行直接小幅度的负反馈. 研究发现：在对系统所有格点快变量幅值观测时选择比较小的阈值则更容易将系统的螺旋波消除掉并使系统达到稳定均匀态. 在比较大的阈值下,系统的螺旋波则变得稀疏,也可以导致螺旋波的破裂. 在任意选择单个格点的快变量观测下,比较小的反馈强度仍然可以消除螺旋波,系统也达到稳定均匀态. 当 关键词： 螺旋波 Fitz-Hugh-Nagumo方程 反馈控制     

Hyperchaos--chaos--Hyperchaos Transition in a Class of On--Off Intermittent Systems Driven by a Family of Generalized Lorenz Systems

Blowout bifurcation in nonlinear systems occurs when a chaotic attractor lying in some symmetric subspace becomes transversely unstable. A class of five-dimensional continuous autonomous systems is considered, in which a two-dimensional subsystem is driven by a family of generalized Lorenz systems. The systems have some common dynamical characters. As the coupling parameter changes, blowout bifurcations occur in these systems and brings on change of the systems＇ dynamics. After the bifurcation the phenomenon of on-off intermittency appears. It is observed that the systems undergo a symmetric hyperchaos-chaos-hyperchaos transition via or after blowout bifurcations. An example of the systems is given, in which the drive system is the Chen system. We investigate the dynamical behaviour before and after the blowout bifurcation in the systems and make an analysis of the transition process. It is shown that in such coupled chaotic continuous systems, blowout bifurcation leads to a transition from chaos to hyperchaos for the whole systems, which provides a route to hyperchaos.     

Spatial patterns in a network composed of neurons with different excitabilities induced by autapse

It has been identified that autapse can modulate dynamics of single neurons and spatial patterns of neuronal networks. In the present paper, based on the results that autapse can induce type II excitability changed to type I excitability, spatial pattern transitions are simulated in a two-dimensional neuronal network composed of excitatory coupled neurons with autapse which can induce excitability transition. Different spatial patterns including random-like pattern, irregular wave, regular wave, and nearly synchronous behavior are simulated with increasing the percentage (σ) of neurons with type I excitability. When noise is introduced, spiral waves are induced. By calculating signal-to-noise ratio from the spatial structure function and the mean firing probability of neurons, regular waves and spiral waves exhibit optimal spatial correlation, implying the occurrence of spatial coherence resonance phenomenon. The changes of mean firing probability of neurons show that different firing frequency between type I excitability and type II excitability may be an important factor to modulate the spatial patterns. The results are helpful to understand the spatial patterns including spiral waves observed in the biological experiment on the rat cortex perfused with drugs which can induce single neurons changed from type II excitability to type I excitability and block the inhibitory couplings between neurons. The excitability transition, absence of inhibitory coupling, noise as well as the autapse are important factors to modulate the spatial patterns including spiral waves.     

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.     

白噪声诱发Morris-Lecar模型构成的Ⅱ型兴奋网络产生多次空间相干共振

为研究噪声在网络中的作用及对时空行为的影响, 通过电耦合、近邻连接的Morris-Lecar模型构建了同质可兴奋细胞网络. 单元振子的确定性行为表现为Ⅱ型兴奋性的静息. 在高斯白噪声的作用下, 网络会在较大的噪声强度范围产生螺旋波, 以及在某些较小的噪声强度范围产生杂乱的空间结构. 随着噪声强度的增加, 螺旋波的结构会在简单和复杂之间转换, 或与杂乱的空间结构交替出现. 通过空间结构函数及其信噪比的计算, 发现简单螺旋波的信噪比较大, 复杂螺旋波以及杂乱的时空结构的信噪比较小. 信噪比随着噪声强度的增加会出现多次极大值, 说明白噪声可以在可兴奋细胞网络中诱导多次空间相干共振. 研究结果提示现实的可兴奋系统能有多次机会选择不同强度的噪声加以合理利用.     

The inverse excitation-induced abnormal spiral wave

The effects of a local constant forcing on spiral waves in two-dimensional excitable media described by Bär model are investigated. A constant external forcing is imposed on the core of spiral wave, leading to parameter variability of a medium. It is found that the forcing can significantly alter the shape and rotation period of spiral wave when the values of related parameters are properly chosen. The change of wave structure is attributed to the transition from normal excitation to inverse excitation in the forced medium. An abnormal spiral wave with a very thick spiral arm has been observed. The physical mechanism underlying these phenomena is theoretically analyzed.     

Travelling Waves in a Chain¶of Coupled Nonlinear Oscillators

In a chain of nonlinear oscillators, linearly coupled to their nearest neighbors, all travelling waves of small amplitude are found as solutions of finite dimensional reversible dynamical systems. The coupling constant and the inverse wave speed form the parameter space. The groundstate consists of a one-parameter family of periodic waves. It is realized in a certain parameter region containing all cases of light coupling. Beyond the border of this region the complexity of wave-forms increases via a succession of bifurcations. In this paper we give an appropriate formulation of this problem, prove the basic facts about the reduction to finite dimensions, show the existence of the ground states and discuss the first bifurcation by determining a normal form for the reduced system. Finally we show the existence of nanopterons, which are localized waves with a noncancelling periodic tail at infinity whose amplitude is exponentially small in the bifurcation parameter. Received: 10 September 1999 / Accepted: 15 December 1999     

Hindmarsh-Rose神经元阵列自发产生螺旋波的研究

采用Hindmarsh-Rose(HR)神经元模型,研究了二维神经元阵列系统从一个具有随机相位分布的初态演化最终是否能自发产生螺旋波的问题.数值结果表明:系统是否出现螺旋波与单个HR神经元的状态、系统的初态和耦合强度有关,其中单个HR神经元的振荡状态起主要作用.当单个HR神经元处于一周期振荡态时,在一定的耦合强度范围内系统都会自发出现多个螺旋波和螺旋波对,出现螺旋波与系统初态无关,只要适当选择耦合强度,在系统中可以出现单个螺旋波.当耦合强度超过某一阈值后,继续增加耦合强度,系统会呈现三种不同的动力学行为,分别与三类初态有关.系统从第一类初态演化将偶尔出现单个螺旋波,系统从第二类和第三类初态演化将分别出现间歇性全局同步振荡和振荡死亡.当单个神经元处于二周期态时,只有当系统神经元的初相位比较均匀分布时,系统才能自发出现螺旋波,而且出现螺旋波的耦合强度范围大为减少.当神经元处于更高的周期态时,系统一般不容易自发出现螺旋波.这些结果有助于人们了解大脑皮层自发产生螺旋波的机制.     

Chaotic spatial bifurcation by complex coupling

A spatial bifurcation (a transition from stationary to oscillatory regime) in a chain of unidirectionally coupled phase systems is studied. It is shown that complication of coupling terms can make this bifurcation spatially chaotic in contrast to the previously observed "regular" and "predictable" type. It is demonstrated that the found type of spatial bifurcation corresponds to a smooth (predictable) manifold in the parameter space, while its spatial location gets actually unpredictable being governed by regularities of chaotic behavior. We infer that complex collective dynamics may arise in networks with plain architecture and simple dynamics of individual elements if nontrivial coupling is realized.     

Spiral Waves and Multiple Spatial Coherence Resonances Induced by Colored Noise in Neuronal Network

Gaussian colored noise induced spatial patterns and spatial coherence resonances in a square lattice neuronal network composed of Morris-Lecar neurons are studied.Each neuron is at resting state near a saddle-node bifurcation on invariant circle,coupled to its nearest neighbors by electronic coupling.Spiral waves with different structures and disordered spatial structures can be alternately induced within a large range of noise intensity.By calculating spatial structure function and signal-to-noise ratio(SNR),it is found that SNR values are higher when the spiral structures are simple and are lower when the spatial patterns are complex or disordered,respectively.SNR manifest multiple local maximal peaks,indicating that the colored noise can induce multiple spatial coherence resonances.The maximal SNR values decrease as the correlation time of the noise increases.These results not only provide an example of multiple resonances,but also show that Gaussian colored noise play constructive roles in neuronal network.     

间接延迟耦合可激发介质中螺旋波的演化

采用Bär 模型研究了通过被动介质间接延迟耦合的两层可激发介质中螺旋波的相互作用. 数值模拟结果表明: 延迟耦合可以促进两个螺旋波的同步, 也可导致从螺旋波到集体振荡、各种靶波、时空混沌态或静息态的转变; 在这个耦合系统中还观察到周期 2和周期3螺旋波以及螺旋波漫游和漂移现象; 对产生这些现象的物理机制做了讨论. 关键词： 螺旋波 被动介质 时间延迟耦合 同步