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.     

分布式电流刺激抑制心肌组织中螺旋波

利用两种恒定电信号刺激介质不同区域来产生分布式电流, 外界分布式电流内化为跨膜电流后在介质中形成稳定的梯度力可抑制螺旋波的传播. 当螺旋波被消除后系统进一步演化为均匀态且对应的膜片电位采样序列逐渐趋于一个稳定值. 在考虑噪声情况下, 该方法仍然能消除螺旋波, 对其采样序列分析也验证了方法的可靠性. 关键词： 螺旋波 分布式电流 激发介质     

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.     

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.     

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.     

Formation and instability of spiral wave induced by Gaussian coloured noise

In this paper, we studied the effect of Gaussian coloured noise on the formation and instability of spiral waves described by one class of modified FitzHugh--Nagumo equation. It was found that Gaussian coloured noise plays a constructive role in the formation, transition and instability of spiral wave. Too weak or too strong noise may act against the formation of spiral waves. At a certain noise level, spiral wave is maintained in a medium, in which spiral wave cannot be observed in the absence of the noise. It is difficult to make a stable spiral wave into unstable state by Gaussian coloured noise, unless the noise level is very high. The parameter regions of Gaussian coloured noise for spiral forming and spiral instability were given and discussed with numerical simulations.     

Delay-enhanced coherence of spiral waves in noisy Hodgkin-Huxley neuronal networks

We study the spatial dynamics of spiral waves in noisy Hodgkin-Huxley neuronal ensembles evoked by different information transmission delays and network topologies. In classical settings of coherence resonance the intensity of noise is fine-tuned so as to optimize the system's response. Here, we keep the noise intensity constant, and instead, vary the length of information transmission delay amongst coupled neurons. We show that there exists an intermediate transmission delay by which the spiral waves are optimally ordered, hence indicating the existence of delay-enhanced coherence of spatial dynamics in the examined system. Additionally, we examine the robustness of this phenomenon as the diffusive interaction topology changes towards the small-world type, and discover that shortcut links amongst distant neurons hinder the emergence of coherent spiral waves irrespective of transmission delay length. Presented results thus provide insights that could facilitate the understanding of information transmission delay on realistic neuronal networks.     

钠离子和钾离子通道噪声扰动对神经网络 时空模式的影响

研究了钠离子和钾离子通道噪声扰动对Hodgkin-Huxley神经网络放电时空模式的影响. 发现无论钠离子通道噪声还是钾离子通道噪声扰动, 当取定一组温度、噪声强度, 随着耦合强度的增大, 神经网络放电时空斑图总能演化出螺旋波, 而且存在形成螺旋波所需的临界耦合强度. 分析发现钠离子通道噪声有利于神经网络螺旋波的形成, 而钾离子通道噪声不利于螺旋波形成. 结果还表明较低的温度能够使神经网络对噪声更加敏感. 最后, 讨论了特定参数下螺旋波与靶波之间的转化现象.     

Multiple Spatial Coherence Resonances and Spatial Patterns in a Noise-Driven Heterogeneous Neuronal Network

Heterogeneity of the neurons and noise are inevitable in the real neuronal network. In this paper, Gaussian white noise induced spatial patterns including spiral waves and multiple spatial coherence resonances are studied in a network composed of Morris-Lecar neurons with heterogeneity characterized by parameter diversity. The relationship between the resonances and the transitions between ordered spiral waves and disordered spatial patterns are achieved. When parameter diversity is introduced, the maxima of multiple resonances increases first, and then decreases as diversity strength increases, which implies that the coherence degrees induced by noise are enhanced at an intermediate diversity strength. The synchronization degree of spatial patterns including ordered spiral waves and disordered patterns is identified to be a very low level. The results suggest that the nervous system can profit from both heterogeneity and noise, and the multiple spatial coherence resonances are achieved via the emergency of spiral waves instead of synchronization patterns.     

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.     

高斯白噪声对神经元二维映射模型动力学的影响

从神经元二维映射模型出发，用高斯白噪声模拟了神经元的噪声环境，进而研究了高斯白噪声对参数空间相图的影响.研究发现，噪声可以提高系统的可兴奋性.通过数值模拟研究了噪声引起的相干共振现象.结果表明，只有当系统参数选取在静息区域且接近连续点火和静息状态分界线时才可以得到相干共振现象. 关键词： 高斯白噪声 神经元二维映射模型 相图 相干共振     

The influence of long-range links on spiral waves and its application for control

The influence of long-range links on spiral waves in excitable medium has been investigated. Spatiotemporal dynamics in excitable small-world network transforms remarkably when we increase the long-range connection probability P. Spiral waves with few perturbations, broken spiral waves, pseudo spiral turbulence, synchronous oscillations, and homogeneous rest state are discovered under different network structures. Tip number is selected to detect non-equilibrium phase transition between different spatiotemporal patterns. The Kuramoto order parameter is used to identify these patterns and explain the emergence of the rest state. Finally, we use long-range links to control spiral wave and spiral turbulence successfully.     

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.     

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.     

Delay-induced synchronization transitions in small-world neuronal networks with hybrid electrical and chemical synapses

We study the dependence of synchronization transitions in small-world networks of bursting neurons with hybrid electrical–chemical synapses on the information transmission delay, the probability of electrical synapses, and the rewiring probability. It is shown that, irrespective of the probability of electrical synapses, the information transmission delay can always induce synchronization transitions in small-world neuronal networks, i.e., regions of synchronization and nonsynchronization appear intermittently as the delay increases. In particular, all these transitions to burst synchronization occur approximately at integer multiples of the bursting period of individual neurons. In addition, for larger probability of electrical synapses, the intermittent synchronization transition is more profound, due to the stronger synchronization ability of electrical synapses compared with chemical ones. More importantly, chemical and electrical synapses can perform complementary roles in the synchronization of hybrid small-world neuronal networks: the larger the electrical synapse strength is, the smaller the chemical synapse strength needed to achieve burst synchronization. Furthermore, the small-world topology has a significant effect on the synchronization transition in hybrid neuronal networks. It is found that increasing the rewiring probability can always enhance the synchronization of neuronal activity. The results obtained are instructive for understanding the synchronous behavior of neural systems.     

Exploring the role of inhibitory coupling in duplex networks

The electrical coupling of myocytes and fibroblasts can play a role in inhibiting electrical impluse propagation in cardiac muscle. To understand the function of fibroblast–myocyte coupling in the aging heart, the spiral-wave dynamics in the duplex networks with inhibitory coupling is numerically investigated by the Br–Eiswirth model. The numerical results show that the inhibitory coupling can change the wave amplitude, excited phase duration and excitability of the system. When the related parameters are properly chosen, the inhibitory coupling can induce local abnormal oscillation in the system and the Eckhaus instability of the spiral wave. For the dense inhibitory network, the maximal decrement(maximal increment) in the excited phase duration can reach 24.3%(13.4%), whereas the maximal decrement in wave amplitude approaches 28.1%. Upon increasing the inhibitory coupling strength, the system excitability is reduced and even completely suppressed when the interval between grid points in the inhibitory network is small enough. Moreover, the inhibitory coupling can lead to richer phase transition scenarios of the system, such as the transition from a stable spiral wave to turbulence and the transition from a meandering spiral wave to a planar wave. In addition, the self-sustaining planar wave, the unique meandering of spiral wave and inward spiral wave are observed. The physical mechanisms behind the phenomena are analyzed.     

耦合小世界神经网络的随机共振

噪声广泛存在于生物神经系统中,对系统功能具有重要作用.采用神经元二维映射模型构建一个复杂神经网络,由多个小世界子网络构成,研究了Gaussian白噪声诱导的随机共振现象.研究发现,只有合适的噪声强度才能使神经网络对输入刺激信号的频率响应达到峰值.另外,网络结构对系统随机共振特性有重要影响.在固定的耦合强度下,存在一个最优的局部小世界子网络结构,使得整个系统的频率响应最佳.     

Development and transition of spiral wave in the coupled Hindmarsh--Rose neurons in two-dimensional space

The dynamics and the transition of spiral waves in the coupled Hindmarsh--Rose (H--R) neurons in two-dimensional space are investigated in the paper. It is found that the spiral wave can be induced and developed in the coupled HR neurons in two-dimensional space, with appropriate initial values and a parameter region given. However, the spiral wave could encounter instability when the intensity of the external current reaches a threshold value of 1.945. The transition of spiral wave is found to be affected by coupling intensity D and bifurcation parameter r. The spiral wave becomes sparse as the coupling intensity increases, while the spiral wave is eliminated and the whole neuronal system becomes homogeneous as the bifurcation parameter increases to a certain threshold value. Then the coupling action of the four sub-adjacent neurons, which is described by coupling coefficient D’, is also considered, and it is found that the spiral wave begins to breakup due to the introduced coupling action from the sub-adjacent neurons (or sites) and together with the coupling action of the nearest-neighbour neurons, which is described by the coupling intensity D.     

Effects of noise on periodic orbits of the logistic map

Noise can induce an inverse period-doubling transition and chaos. The effects of noise on each periodic orbit of three different period sequences are investigated for the logistic map. It is found that the dynamical behavior of each orbit, induced by an uncorrelated Gaussian white noise, is different in the mergence transition. For an orbit of the period-six sequence, the maximum of the probability density in the presence of noise is greater than that in the absence of noise. It is also found that, under the same intensity of noise, the effects of uncorrelated Gaussian white noise and exponentially correlated colored (Gaussian) noise on the period-four sequence are different.