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

Abstract

We consider the approximations behind the typical mean-field model derived for a class of systems made up of type II excitable units influenced by noise and coupling delays. The formulation of the two approximations, referred to as the Gaussian and the quasi-independence approximation, as well as the fashion in which their validity is verified, are adapted to reflect the essential properties of the underlying system. It is demonstrated that the failure of the mean-field model associated with the breakdown of the quasi-independence approximation can be predicted by the noise-induced bistability in the dynamics of the mean-field system. As for the Gaussian approximation, its violation is related to the increase of noise intensity, but the actual condition for failure can be cast in qualitative, rather than quantitative terms. We also discuss how the fulfillment of the mean-field approximations affects the statistics of the first return times for the local and global variables, further exploring the link between the fulfillment of the quasi-independence approximation and certain forms of synchronization between the individual units.

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.…”

Mean-field dynamics of a random neural network with noise

“…Our approach is essentially based on the well-known Quasi-independence and Gaussian approximations [35], and leads to the secondorder mean-field model of the macroscopic dynamics. In other words, we use the moment approach with the Gaussian closure hypothesis.…”

Mean field dynamics of networks of delay-coupled noisy excitable units

“…The fact that the system comprises only the equations for the first and the second order moments is consistent with the Gaussian approximation, which is required as a closure hypothesis due to nonlinearity of the original system (1). In order to ensure that the model is analytically tractable, one may further introduce "adiabatic approximation", by which the variances and the covariances are replaced by their stationary values [3,4]. This is physically justified as the corresponding relaxation times are typically small.…”

“…This is physically justified as the corresponding relaxation times are typically small. The MF model then contains just two equations for the means [3,4]:…”

“…It is further possible to directly compare the features of excitable dynamics (such as threshold-like behavior) for a single FHN unit and the assembly. Another important point is that the bifurcation analysis of the MF model can be carried out analytically [3]. In particular, system (2) is found to undergo a sequence of direct and inverse Hopf bifurcations, see the τ(D) bifurcation diagram in Fig.…”