Mean field approximation of two coupled populations of excitable units

TL;DR

This paper demonstrates that a mean-field approximation can accurately reproduce the bifurcation behavior and stability properties of two coupled populations of stochastic excitable units, simplifying complex delay-differential equations.

Contribution

The study introduces a mean-field model that reduces the original high-dimensional stochastic delay system to four deterministic delay equations, enabling analytical bifurcation analysis.

Findings

Mean-field model accurately predicts bifurcations and stability domains.

Identifies parameter regions for stationary state stability and oscillation suppression.

Reveals mechanisms of destabilization under various coupling configurations.

Abstract

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.