Collective oscillations of excitable elements: order parameters, bistability and the role of stochasticity

TL;DR

This paper investigates how probabilistic refractory periods influence collective oscillations in coupled excitable elements, revealing stable synchronized phases, reentrant transitions, and bistability through mean-field analysis and simulations.

Contribution

It introduces a comprehensive analysis of stochastic refractory periods' effects on collective dynamics, including phase transitions and bistability, with quantitative agreement between theory and simulations.

Findings

Stable synchronized oscillations exist despite stochastic refractory periods.

Increasing coupling strength causes a reentrant transition out of synchronization.

Bistability and hysteresis occur between synchronized and incoherent phases.

Abstract

We study the effects of a probabilistic refractory period in the collective behavior of coupled discrete-time excitable cells (SIRS-like cellular automata). Using mean-field analysis and simulations, we show that a synchronized phase with stable collective oscillations exists even with non-deterministic refractory periods. Moreover, further increasing the coupling strength leads to a reentrant transition, where the synchronized phase loses stability. In an intermediate regime, we also observe bistability (and consequently hysteresis) between a synchronized phase and an active but incoherent phase without oscillations. The onset of the oscillations appears in the mean-field equations as a Neimark-Sacker bifurcation, the nature of which (i.e. super- or subcritical) is determined by the first Lyapunov coefficient. This allows us to determine the borders of the oscillating and of the bistable regions. The mean-field prediction thus obtained agrees quantitatively with simulations of complete graphs and, for random graphs, qualitatively predicts the overall structure of the phase diagram. The latter can be obtained from simulations by defining an order parameter q suited for detecting collective oscillations of excitable elements. We briefly review other commonly used order parameters and show (via data collapse) that q satisfies the expected finite size scaling relations.