Excitable elements controlled by noise and network structure

TL;DR

This paper investigates how noise and network structure influence the collective dynamics of stochastic excitable elements, revealing that heterogeneity in connections can induce bifurcations and alter oscillation thresholds.

Contribution

The study introduces a mean-field and Gaussian approximation approach to analyze the impact of network heterogeneity and noise on excitable network dynamics, highlighting new bifurcation phenomena.

Findings

Heterogeneity causes bifurcations in excitable regimes.

Critical noise for saddle-node bifurcation decreases with connectivity variance.

Onset of global oscillations depends non-monotonically on heterogeneity.

Abstract

We study collective dynamics of complex networks of stochastic excitable elements, active rotators. In the thermodynamic limit of infinite number of elements, we apply a mean-field theory for the network and then use a Gaussian approximation to obtain a closed set of deterministic differential equations. These equations govern the order parameters of the network. We find that a uniform decrease in the number of connections per element in a homogeneous network merely shifts the bifurcation thresholds without producing qualitative changes in the network dynamics. In contrast, heterogeneity in the number of connections leads to bifurcations in the excitable regime. In particular we show that a critical value of noise intensity for the saddle-node bifurcation decreases with growing connectivity variance. The corresponding critical values for the onset of global oscillations (Hopf bifurcation) show a non-monotone dependency on the structural heterogeneity, displaying a minimum at moderate connectivity variances.