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.     

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.     

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.     

Spatial coherence resonance induced by coloured noise and parameter diversity in a neuronal network

Spatial coherence resonance in a two-dimensional neuronal network induced by additive Gaussian coloured noise and parameter diversity is studied. We focus on the ability of additive Gaussian coloured noise and parameter diversity to extract a particular spatial frequency (wave number) of excitatory waves in the excitable medium of this network. We show that there exists an intermediate noise level of the coloured noise and a particular value of diversity, where a characteristic spatial frequency of the system comes forth. Hereby, it is verified that spatial coherence resonance occurs in the studied model. Furthermore, we show that the optimal noise intensity for spatial coherence resonance decays exponentially with respect to the noise correlation time. Some explanations of the observed nonlinear phenomena are also presented.     

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     

Additive Temporal Coloured Noise Induced Eckhaus Instability in Complex Ginzburg-Landau Equation System

The effect of additive coloured noises, which are correlated in time, on one-dimensional travelling waves in the complex Ginzburg-Landau equation is studied by numerical simulations. We found that a small coloured noise with temporal correlation could considerably influence the stability of one-dimensional wave trains. There exists an optimal temporal correlation of noise where travelling waves are the most vulnerable. To elucidate the phenomena, we statistically calculated the convective velocities Vg of the wave packets, and found that the coloured noise with an appropriate temporal correlation can decrease Vg, making the system convectively more unstable.     

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.     

Breakup of spiral wave under different boundary conditions

In this paper, we investigate the breakup of spiral wave under no-flux, periodic and Dirichlet boundary conditions respectively. When the parameter \varepsilon is close to a critical value for Doppler-induced wave breakup, the instability of the system caused by the boundary effect occurs in the last two cases, resulting in the breakup of spiral wave near the boundary. With our defined average order measure of spiral wave (AOMSW), we quantify the degree of order of the system when the boundary-induced breakup of spiral wave happens. By analysing the AOMSW and outer diameter R of the spiral tip orbit, it is easy to find that this boundary effect is correlated with large values of R, especially under the Dirichlet boundary condition. This correlation is nonlinear, so the AOMSW sometimes oscillates with the variation of \varepsilon.     

具有早期后除极化现象的可激发系统中螺旋波破碎方式研究

在Greenberg-Hasting元胞自动机模型中引入了正常元胞和老化元胞,并规定只有老化元胞存在早期后除极化现象且早期后除极化可以激发其他元胞.在正常元胞和老化元胞均匀分布的情况下,研究了早期后除极化对螺旋波演化行为的影响,重点探讨了早期后除极化导致的螺旋波破碎方式.数值模拟结果表明:早期后除极化在比率约为26.4%的少数情况下不对螺旋波产生影响,在其他情况下则会对螺旋波产生各种影响,包括使螺旋波漫游、漂移、波臂发生形变以及导致螺旋波破碎和消失等.观察到早期后除极化通过传导障碍消失和通过转变为反靶波消失,早期后除极化导致螺旋波破碎有8种方式,包括非对称破缺导致的破碎、对称破缺导致的破碎、同时激发双波导致的破碎、非对称激发导致的破碎、整体传导障碍导致的破碎、整体快速破碎等.分析发现这些螺旋波破碎现象都与早期后除极化产生回火波有关,得到螺旋波破碎的总比率通常约为13.8%,但是在适当选取老化元胞密度和早期后除极化的激发下,螺旋波破碎比率可达到32.4%,这些结果与心律失常致死的统计结果基本一致,本文对产生这些现象的物理机理做了简要分析.     

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.     

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     

Robustness and breakup of the spiral wave in a two-dimensional lattice network of neurons

The robustness and breakup of spiral wave in a two-dimensional lattice networks of neurons are investigated. The effect of small- world type connection is often simplified with local regular connection and the long-range connection with certain probability. The network effect on the development of spiral wave can be better described by local regular connection and changeable long-range connection probability than fixed long-range connection probability because the long-range probability could be changeable ...     

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.     

离散可激发介质中早期后去极化对螺旋波影响的数值研究

通过考虑某些不应态也可以被激发,在离散可激发介质Greenberg-Hasting模型中引入早期后去极化行为,研究了早期后去极化对螺旋波的影响.数值结果表明:在适当选择参数下,早期后去极化对螺旋波有很大影响,这些影响包括使螺旋波漫游、漂移和破碎,导致螺旋波波纹被扭曲和波臂粗细交替变化,以及导致螺旋波的周期在两个值之间交替变化,产生从稳定螺旋波到呼吸螺旋波和反螺旋波的相变等.当不应态的激发阈值很高时,早期后去极化对螺旋波没有影响.对发生上述现象作了简要的讨论. 关键词： 离散可激发介质 螺旋波 早期后去极化     

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.     

From local to global spatiotemporal chaos in a cardiac tissue model

Two kinds of chaos can occur in cardiac tissue, chaotic meander of a single intact spiral wave and chaotic spiral wave breakup. We studied these behaviors in a model of two-dimensional cardiac tissue based on the Luo-Rudy I action potential model. In the chaotic meander regime, chaos is spatially localized to the core of the spiral wave. When persistent spiral wave breakup occurs, there is a transition from local to global spatiotemporal chaos.     

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.     

螺旋波动力学性质的元胞自动机有向小世界网络研究

以Greenberg-Hastings激发介质元胞自动机模型为基础,研究了有向小世界网络中重新连接概率p对螺旋波动力学行为的影响.对于在规则网络下的稳定螺旋波,施加有向小世界网络后发现:当p值较小时,原本稳定的螺旋波仍保持其稳定性.随着p的增大,先后观察到螺旋波持续漫游、螺旋波断裂以及螺旋波消失等现象.通过监测系统的激发比率,发现以上现象的产生源于介质激发性随p的增大而降低.同时还发现元胞周期的变化也与p有关. 关键词： 元胞自动机 螺旋波 激发介质 有向小世界网络     

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.