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

Abstract

We study the effects of a probabilistic refractory period in the collective behavior ofcoupled discrete-time excitable cells (SIRS-like cellular automata). Using mean-fieldanalysis and simulations, we show that a synchronized phase with stable collectiveoscillations exists even with non-deterministic refractory periods. Moreover, furtherincreasing the coupling strength leads to a reentrant transition where the synchronizedphase loses stability. In an intermediate regime, we also observe bistability (andconsequently hysteresis) between a synchronized phase and an active but incoherent phasewithout oscillations. The onset of the oscillations appears in the mean-field equations asa Neimark–Sacker bifurcation, the nature of which (i.e. super- or subcritical) isdetermined by the first Lyapunov coefficient. This allows us to determine theborders of the oscillating and of the bistable regions. The mean-field predictionthus obtained agrees quantitatively with simulations of complete graphs and, forrandom graphs, qualitatively predicts the overall structure of the phase diagram.The latter can be obtained from simulations by defining an order parameterq suited for detecting collective oscillations of excitable elements. We briefly reviewother commonly used order parameters and show (via data collapse) thatq satisfies the expected finite-size scaling relations.