  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.
