Mean-field approximation of two coupled populations of excitable units

Abstract

The analysis on stability and bifurcations in the macroscopic dynamics exhibited by the system of two coupled large populations composed 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 interensemble 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 interpopulation couplings.

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

“…Such an assumption has also been made, e.g., for coupled FitzHugh-Nagumo oscillators [34,35], integrate-andfire neurons [36], a general class of master equations [37] and delayed-coupled systems [38][39][40]. Within the Gaussian approximation the system's dimension can be reduced to four coupled first-order differential equations, which allows a thorough bifurcation analysis.…”

Activation process in excitable systems with multiple noise sources: One and two interacting units

“…To do so, we derive a deterministic model based on Gaussian approximation of the original system (1). According to Gaussian approximation, all the cumulants above the second order are assumed to vanish [36][37][38]. One is ultimately left with a set of five equations describing the dynamics of the first moments m x (t) = E[x(t)] and m y = E[y(t)], the variances s…”

Persistence and failure of mean-field approximations adapted to a class of systems of delay-coupled excitable units

“…Recall that both types of Hopf bifurcation can further be cast as direct or inverse [28], which refers to whether the fixed point unfolds on the unstable or the stable side, respectively. The final expression for the critical time delay in dependence of c and D reads [25,26] where the (+) (−) sign reflects the direct (inverse) character of bifurcation, j = 0,1,2, . .…”

“…Nevertheless, such a requirement is too strong, in a sense that the validity of the MF model is then satisfied trivially. Moreover, examples have already been found where the MF system can accurately predict the properties of the oscillatory state, including the oscillation frequency [25,26].…”