标题： 两耦合兴奋单元群体的平均场近似 显示英文标题

标题： Mean field approximation of two coupled populations of excitable units

摘要： 对由两个相互耦合的大群体组成的系统的宏观动力学中所表现出的稳定性和分支问题进行分析，这些群体各自包含 $N$ 个随机可激发单元。通过研究一个近似系统来完成这一分析，该近似系统通过对每个群体用相应的平均场模型代替而获得。 在精确系统中，集合内的单元通过带时滞的线性耦合进行通信，而集合间的项涉及由适当的全局变量介导的非线性带时滞交互。目的是证明影响由一组 4N 个描述微观动力学的随机时滞微分方程控制的原始系统稳态稳定性分支，可以被仅含四个确定性时滞微分方程的流准确再现，这些方程描述了基于平均场变量的演化。 具体而言，所考虑的问题包括确定稳态稳定的参数域、集体模式出现和时滞诱导抑制的场景，以及允许平衡状态与振荡状态之间双稳态的参数域。 我们展示如何利用近似模型中解析可处理的分支来识别不同系统配置下（如对称或不对称种群间耦合）稳态失稳的特征机制。 显示英文摘要

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.