两种耦合布居的可激发基元的平均场近似
Mean field approximation of two coupled populations of excitable units
摘要：
通过研究由$N$随机可激发单元组成的两个耦合大粒子群系统的近似系统，分析了该系统宏观动力学的稳定性和分岔.在精确系统中，系综内的单元通过时滞线性耦合进行通信，而系综间项涉及由适当的全局变量介导的非线性时滞相互作用。其目的是证明，分叉影响的稳定性的原系统的稳态，由一组4N随机延迟微分方程的微观动力学，可以准确地再现由一个流只包含四个确定性延迟微分方程描述的平均场为基础的变量的演变。特别是，所考虑的问题包括确定的参数域的定态是稳定的，发病和时间延迟诱导抑制的集体模式的情况下，以及参数域承认平衡和振荡状态之间的双稳态。我们展示了如何分析易处理的分叉发生在近似模型中可以用来识别的特征机制，其中的稳态是不稳定的不同的系统配置下，如那些对称或不对称的种群间耦合。
英文摘要：
The analysis on stability and bifurcations in the macroscopic dynamics exhibited by the system of two coupled large populations comprised of $N$ stochastic excitable units each is performed by studying an approximate system, obtained by replacing each population with the corresponding mean-field model. In the exact system, one has the units within an ensemble communicating via the time-delayed linear couplings, whereas the inter-ensemble terms involve the nonlinear time-delayed interaction mediated by the appropriate global variables. The aim is to demonstrate that the bifurcations affecting the stability of the stationary state of the original system, governed by a set of 4N stochastic delay-differential equations for the microscopic dynamics, can accurately be reproduced by a flow containing just four deterministic delay-differential equations which describe the evolution of the mean-field based variables. In particular, the considered issues include determining the parameter domains where the stationary state is stable, the scenarios for the onset and the time-delay induced suppression of the collective mode, as well as the parameter domains admitting bistability between the equilibrium and the oscillatory state. We show how analytically tractable bifurcations occurring in the approximate model can be used to identify the characteristic mechanisms by which the stationary state is destabilized under different system configurations, like those with symmetrical or asymmetrical inter-population couplings.
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