Synchronization of spiral waves in a two-layer coupled inhomogeneous excitable system

We studied synchronization behaviours of spiral waves in a two-layer coupled inhomogeneous excitable system. It was found that phase synchronization can be observed under weak coupling strength. By increasing the coupling strength, the synchronization is broken down. With the further increase of the coupling strength, complete synchronization and phase synchronization occur again. We also found that the inhomogeneity in excitable systems is helpful to the synchronization.     

两层耦合可激发介质中螺旋波转变为平面波

采用Br-Eiswirth模型研究了两层耦合可激发介质中螺旋波的动力学,两层介质通过网络连接,即在每一层介质上,每一列选一个可激发单元作为中心点,在一层介质上同一列的可激发单元只与另一层介质上对应的中心点及其8个邻居有耦合.数值模拟结果表明:通过这种局部耦合,在适当小的耦合强度下两耦合螺旋波可实现同步,增大耦合强度会导致螺旋波漫游和漂移,造成螺旋波不同步,观察到螺旋波与静息态、低频平面波和不规则斑图共存现象.在适当强的耦合强度下,还观察到两螺旋波转变成同步的平面波消失现象.对产生这些现象的物理机理做了讨论.     

Cooperative behavior between oscillatory and excitable units: the peculiar role of positive coupling-frequency correlations

We study the collective dynamics of noise-driven excitable elements, so-called active rotators. Crucially here, the natural frequencies and the individual coupling strengths are drawn from some joint probability distribution. Combining a mean-field treatment with a Gaussian approximation allows us to find examples where the infinite-dimensional system is reduced to a few ordinary differential equations. Our focus lies in the cooperative behavior in a population consisting of two parts, where one is composed of excitable elements, while the other one contains only self-oscillatory units. Surprisingly, excitable behavior in the whole system sets in only if the excitable elements have a smaller coupling strength than the self-oscillating units. In this way positive local correlations between natural frequencies and couplings shape the global behavior of mixed populations of excitable and oscillatory elements.     

弱信号在Hodgkin-Huxley神经元单向耦合系统中的传输特性

研究了阈下信号在含噪声的Hodgkin-Huxley神经元单向耦合系统中的传输特性.结果表明,各单元中均存在随机共振现象,可见噪声有助于提高信号的检测和传输;另外,耦合实现了信号的传输,且随着耦合强度的增强信号的传输效率增加,在耦合强度达到某一程度时两神经元实现了有时延的一致放电;并且接收元的信噪比最优值处的噪声强度随着耦合强度的提高而减小,最终与驱动元的一致;另外在耦合强度过强时,接收元出现过耦合放电,但是最终会被不断增强的噪声抑制,此现象有助于解释神经元的自放电及神经系统的自调节.研究表明噪声和耦合在 关键词： Hodgkin-Huxley神经元模型 随机共振 噪声 单向耦合系统     

Effect of noise on coupled chaotic systems

The effect of noise in inducing order on various chaotically evolving systems is reviewed, with special emphasis on systems consisting of coupled chaotic elements. In many situations it is observed that the uncoupled elements when driven by identical noise, show synchronization phenomena where chaotic trajectories exponentially converge towards a single noisy trajectory, independent of the initial conditions. In a random neural network, with infinite range coupling, chaos is suppressed due to noise and the system evolves towards a fixed point. Spatiotemporal stochastic resonance phenomenon has been observed in a square array of coupled threshold devices where a temporal characteristic of the system resonates at a given noise strength. In a chaotically evolving coupled map lattice with the logistic map as local dynamics and driven by identical noise at each site, we report that the number ofstructures (a structure is a group of neighbouring lattice sites for values of the variable follow which the certain predefined pattern) follows a power-law decay with the length of the structure. An interesting phenomenon, which we callstochastic coherence, is also reported in which the abundance and lifetimes of these structures show characteristic peaks at some intermediate noise strength.     

Frequency enhancement in coupled noisy excitable elements: effects of network topology

Coupled excitable elements in the presence of noise can exhibit oscillatory behavior with non-trivial frequency dependence as the coupling strength of the system increases. The phenomenon of frequency enhancement (FE) occurs in some coupling regime, in which the elements can oscillate with a frequency higher than their uncoupled frequencies. In this paper, details of the FE are investigated by simulations of the FitzHugh-Nagumo model with different network topologies. It is found that the characteristics of FE, such as the maximal enhancement coupling, enhancement level etc, are functions of the network topology and spatial dimensions. The effect of excitability and the spatio-temporal patterns during FE are investigated to provide an intuitive picture for the enhancement mechanism. Interestingly, some of these characteristics of FE can be described by scaling laws; suggesting the existence of universality in the FE phenomenon. The relevance of these results to biological rhythms are also discussed.     

Universal occurrence of the phase-flip bifurcation in time-delay coupled systems

Recently, the phase-flip bifurcation has been described as a fundamental transition in time-delay coupled, phase-synchronized nonlinear dynamical systems. The bifurcation is characterized by a change of the synchronized dynamics from being in-phase to antiphase, or vice versa; the phase-difference between the oscillators undergoes a jump of pi as a function of the coupling strength or the time delay. This phase-flip is accompanied by discontinuous changes in the frequency of the synchronized oscillators, and in the largest negative Lyapunov exponent or its derivative. Here we illustrate the phenomenology of the bifurcation for several classes of nonlinear oscillators, in the regimes of both periodic and chaotic dynamics. We present extensive numerical simulations and compute the oscillation frequencies and the Lyapunov spectra as a function of the coupling strength. In particular, our simulations provide clear evidence of the phase-flip bifurcation in excitable laser and Fitzhugh-Nagumo neuronal models, and in diffusively coupled predator-prey models with either limit cycle or chaotic dynamics. Our analysis demonstrates marked jumps of the time-delayed and instantaneous fluxes between the two interacting oscillators across the bifurcation; this has strong implications for the performance of the system as well as for practical applications. We further construct an electronic circuit consisting of two coupled Chua oscillators and provide the first formal experimental demonstration of the bifurcation. In totality, our study demonstrates that the phase-flip phenomenon is of broad relevance and importance for a wide range of physical and natural systems.     

Experimental and numerical study on the basin stability of the coupled metronomes

The basin stability is an effective parameter to measure the stability of multistable system under perturbations. In this paper, we try to explore the effects of the coupling strength on the basin stability of the coupled metronomes. In two coupled non-identical metronomes, the coupling strength linearly decreases the basin stability of in-phase synchronization while increases that of the anti-phase synchronization. In three coupled metronomes, there are rich coexisting collectively dynamics as in-phase, anti-phase synchronization, quasi-period states and period 4 states. The coupling strength may still change the basin stability of these coexisting dynamics states. The results are observed in experimental systems and numerical models. Our findings are significant on understanding the multistable dynamics under noisy environment.     

Influence of Coupling Delay on Noise Induced Coherent Oscillations in Excitable Systems

Influence of small time-delays in coupling between noisy excitable systems on the coherence resonance and self-induced stochastic resonance is studied. Parameters of delayed coupled deterministic excitable units are chosen such that the system has only one attractor, namely the stationary state, for any value of the coupling and the time-lag. Addition of white noise induces qualitatively different types of coherent oscillations, and we analyzed the influence of coupling time-delay on the properties of these coherent oscillations. The main conclusion is that time-lag τ≥1, but still smaller than the refractory period, and sufficiently strong coupling drastically change signal to noise ratio in the quantitative and qualitative way. An interval of noise values implies quite large signal to noise ratio and different types of noise induced coherence are greatly enhanced. We also observed coincident spiking for small noise intensity and time-lag proportional to the inter-spike interval of the coherent spike trains. On the other hand, time-lags τ<1 and/or weak coupling induce negligible changes in the properties of the stochastic coherence.     

Synchronization and Asynchronization in Two Coupled Excitable Systems

The synchronization and asynchronization of two coupled excitable systems are investigated. The two systems with different initial configurations, which are separately a single spiral wave (or a travel wave) and the rest state, can be developed to the synchronizing state with the same spiral wave (or travel wave) in each system, when the coupling is very strong. Decreasing the coupling intensity, two rest states or two different configurations appear in the two systems. The qualitative analysis and interpretation are given.     

Self-organized transition to coherent activity in disordered media

Synchronized oscillations are of critical functional importance in many biological systems. We show that such oscillations can arise without centralized coordination in a disordered system of electrically coupled excitable and passive cells. Increasing the coupling strength results in waves that lead to coherent periodic activity, exhibiting cluster, local and global synchronization under different conditions. Our results may explain the self-organized transition in a pregnant uterus from transient, localized activity initially to system-wide coherent excitations just before delivery.     

基于稀疏贝叶斯学习的复杂网络拓扑估计

提出了一种噪声环境下复杂网络拓扑估计方法, 仅利用含噪时间序列估计未知结构混沌系统的动力学方程和参数, 以及由混沌系统组成的复杂网络的拓扑结构、节点动力学方程、所有参数、 节点间耦合方向和耦合强度.通过采用动力学方程的统一形式, 将动力系统方程结构和参数估计看成线性回归问题的系数估计, 该估计问题利用贝叶斯压缩传感的信号重建算法求解, 含噪信号的模型重建使用相关向量机方法,即通过稀疏贝叶斯学习求解稀疏欠定线性方程得到上面提到的可估计对象.以单个Lorenz系统及由200个 Lorenz系统组成的无标度网络为例说明方法的有效性. 仿真结果表明,提出的方法对噪声有很强的鲁棒性,收敛速度快,稳态误差极小, 克服了最小二乘估计方法收敛速度慢、 稳态误差大以及压缩传感估计方法对噪声鲁棒性不强的缺点.     

Projective synchronization of two coupled excitable spiral waves

Interaction of two identical excitable spiral waves in a bilayer system is studied. We find that the two spiral waves can be completely synchronized if the coupling strength is sufficiently large. Prior to the complete synchronization, we find a new type of weak synchronization between the two coupled systems, i.e., the spiral wave of the driven system has the same geometric shape as the spiral wave of the driving system but with a much lower amplitude. This general behavior, called projective synchronization of two spiral waves, is similar to projective synchronization of two coupled nonlinear oscillators, which has been extensively studied before. The underlying mechanism is uncovered by the study of pulse collision in one-dimensional systems.     

Transition to complete synchronization via near-synchronization in two coupled chaotic neurons

The synchronization transition in two coupled chaotic Morris-Lecar （ML） neurons with gap junction is studied with the coupling strength increasing. The conditional Lyapunov exponents, along with the synchronization errors are calculated to diagnose synchronization of two coupled chaotic ML neurons. As a result, it is shown that the increase in the coupling strength leads to incoherence, then induces a transition process consisting of three different synchronization states in succession, namely, burst synchronization, near-synchronization and embedded burst synchronization, and achieves complete synchronization of two coupled neurons finally. These sequential transitions to synchronization reveal a new transition route from incoherence to complete synchronization in coupled systems with multi-time scales.     

Chaotic patterns in a coupled oscillator-excitator biochemical cell system

In this paper we examine dynamical modes resulting from diffusion-like interaction of two model biochemical cells. Kinetics in each of the cells is given by the ICC model of calcium ions in the cytosol. Constraints for one of the cells are set so that it is excitable. One of the constraints in the other cell - a fraction of activated cell surface receptors-is varied so that the dynamics in the cell is either excitable or oscillatory or a stable focus. The cells are interacting via mass transfer and dynamics of the coupled system are studied as two parameters are varied-the fraction of activated receptors and the coupling strength. We find that (i) the excitator-excitator interaction does not lead to oscillatory patterns, (ii) the oscillator-excitator interaction leads to alternating phase-locked periodic and quasiperiodic regimes, well known from oscillator-oscillator interactions; torus breaking bifurcation generates chaos when the coupling strength is in an intermediate range, (iii) the focus-excitator interaction generates compound oscillations arranged as period adding sequences alternating with chaotic windows; the transition to chaos is accompanied by period doublings and folding of branches of periodic orbits and is associated with a Shilnikov homoclinic orbit. The nature of spontaneous self-organized oscillations in the focus-excitator range is discussed. (c) 1999 American Institute of Physics.     

Detecting anomalous phase synchronization from time series

Modeling approaches are presented for detecting an anomalous route to phase synchronization from time series of two interacting nonlinear oscillators. The anomalous transition is characterized by an enlargement of the mean frequency difference between the oscillators with an initial increase in the coupling strength. Although such a structure is common in a large class of coupled nonisochronous oscillators, prediction of the anomalous transition is nontrivial for experimental systems, whose dynamical properties are unknown. Two approaches are examined; one is a phase equational modeling of coupled limit cycle oscillators and the other is a nonlinear predictive modeling of coupled chaotic oscillators. Application to prototypical models such as two interacting predator-prey systems in both limit cycle and chaotic regimes demonstrates the capability of detecting the anomalous structure from only a few sets of time series. Experimental data from two coupled Chua circuits shows its applicability to real experimental system.     

Frequency—Locking in Coupled Chaotic Systems

A novel approach is presented for measuring the phase synchronization(frequency-Locking)of coupled N nonidentical oscillators,which can characterize frequency-locking for chaotic systems without well-defined phase by measuring the mean frequency.Numerical simulations confirm the existence of frequency-locking.The relations between the mean frequency and the coupling strength and the frequency mismatch are given.For the coupled hyperchaotic systems.the frequency-locking can be better characterized by more than one mean frequency curves.     

Chemical turbulence and standing waves in a surface reaction model: The influence of global coupling and wave instabilities

Among heterogeneously catalyzed chemical reactions, the CO oxidation on the Pt(110) surface under vacuum conditions offers probably the greatest wealth of spontaneous formation of spatial patterns. Spirals, fronts, and solitary pulses were detected at low surface temperatures (T<500 K), in line with the standard phenomenology of bistable, excitable, and oscillatory reaction-diffusion systems. At high temperatures (T greater, similar 540 K), more surprising features like chemical turbulence and standing waves appeared in the experiments. Herein, we study a realistic reaction-diffusion model of this system, with respect to the latter phenomena. In particular, we deal both with the influence of global coupling through the gas phase on the oscillatory reaction and the possibility of wave instabilities under excitable conditions. Gas-phase coupling is shown to either synchronize the oscillations or to yield turbulence and standing structures. The latter findings are closely related to clustering in networks of coupled oscillators and indicate a dominance of the global gas-phase coupling over local coupling via surface diffusion. In the excitable regime wave instabilities in one and two dimensions have been discovered. In one dimension, pulses become unstable due to a vanishing of the refractory zone. In two dimensions, turbulence can also emerge due to spiral breakup, which results from a violation of the dispersion relation.     

Synchronization of a large number of continuous one-dimensional stochastic elements with time-delayed mean-field coupling

We study synchronization as a means of control of collective behavior of an ensemble of coupled stochastic units in which oscillations are induced merely by external noise. For a large number of one-dimensional continuous stochastic elements coupled non-homogeneously through the mean field with delay we developed an approach to find a boundary of synchronization domain and the frequency of the mean-field oscillations on it. Namely, the exact location of the synchronization threshold is shown to be a solution of the boundary value problem (BVP) which was derived from the linearized Fokker-Planck equation. Here the synchronization threshold is found by solving this BVP numerically. Approximate analytics is obtained by expanding the solution of the linearized Fokker-Planck equation into a series of eigenfunctions of the stationary Fokker-Planck operator. Bistable systems with a polynomial and piece-wise linear potential are considered as examples. Multistability and hysteresis in the mean-field behavior are observed in the stochastic network at finite noise intensities. In the limit of small noise intensities the critical coupling strength is shown to remain finite, provided that the delay in the coupling function is not infinitely small. Delay in the coupling term can be used as a control parameter that manipulates the location of the synchronization threshold.     

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