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

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.

Synchronization and phase redistribution in self-replicating populations of coupled oscillators and excitable elements

“…The last transition, from state 3 back to the quiescent state 0, is also probabilistic and occurs with probability p γ , which is the same for all excitable elements [24]. A thorough analysis of this model [24] reveals that, in the thermodynamic limit, the population transitions from an absorbing state to an active state at a critical value of σ. More interestingly, for certain choices of p γ , it can undergo synchronous oscillations within a range of coupling strengths σ 1 < σ < σ 3 , and there exists a region of bistability between synchronous oscillations and an asynchronous fixed point for σ 2 < σ < σ 3 , with σ 1 < σ 2 < σ 3 .…”

“…where δ ij is the Kronecker delta. Because the choice of ǫ i has broken the rotational symmetry of the model, the solution P i = 1/3 is not, in general, a steady state solution to Equation 24. In what follows, we first consider the case χ = 1/3, which is amenable to analytical treatment, and then go on to numerically explore the general case 0 ≤ χ ≤ 1.…”

“…The results of Section II raise the question of whether population growth might have similar effects on populations of coupled excitable elements. To explore this question in a simple context, we will model an excitable system as a discrete m-state system comprised of a quiescence state (0), an excited state (1), and a finite set of refractory states (2, 3, ..., m − 1) using the model introduced in [24]. For each state i, the discretized phase is θ = 2πi/m.…”

“…To avoid limits due to computational memory, we reduce the total population size by a factor of 5 (while preserving the state distribution) when the number of units reaches 5 × 10 4 . As in [24], we take the probability of exciting a quiescent state to be…”

“…In the case where all states can divide, the most salient effect of population growth is to decrease the range of σ over which oscillations are stable (Figure 9). For example, when p γ = 0.94, [24] showed that the model undergoes a synchronizing transition as σ is increased from zero. Further increase of σ leads to discontinuous re-entrant transition that includes a region of bistability between synchronous and asynchronous states (black curves, main panel).…”