Discontinuous nonequilibrium phase transitions in a nonlinearly pulse-coupled excitable lattice model

Abstract

We study a modified version of the stochastic susceptible-infected-refractory-susceptible (SIRS) model by employing a nonlinear (exponential) reinforcement in the contagion rate and no diffusion. We run simulations for complete and random graphs as well as d-dimensional hypercubic lattices (for d=3,2,1). For weak nonlinearity, a continuous nonequilibrium phase transition between an absorbing and an active phase is obtained, such as in the usual stochastic SIRS model [Joo and Lebowitz, Phys. Rev. E 70, 036114 (2004)]. However, for strong nonlinearity, the nonequilibrium transition between the two phases can be discontinuous for d>or=2, which is confirmed by well-characterized hysteresis cycles and bistability. Analytical mean-field results correctly predict the overall structure of the phase diagram. Furthermore, contrary to what was observed in a model of phase-coupled stochastic oscillators with a similar nonlinearity in the coupling [Wood, Phys. Rev. Lett. 96, 145701 (2006)], we did not find a transition to a stable (partially) synchronized state in our nonlinearly pulse-coupled excitable elements. For long enough refractory times and high enough nonlinearity, however, the system can exhibit collective excitability and unstable stochastic oscillations.

Collective behavior of coupled nonuniform stochastic oscillators

“…In addition to the previously reported phase transitions for α = 0 [12,13,34], we have obtained for α = 0 several bifurcations in the mean-field equations of the model, including saddle-node, infinite period, Hopf and homoclinic. Collective excitability [29] has been shown to occur in some parameter regions, as confirmed by simulations of complete graphs. Simulations have also confirmed the overall predictions of the mean-field analysis, although the stability of some stable limit cycles and fixed points has failed to resist the effects of finite-size fluctuations.…”

“…Here we make another attempt, now transforming the model by Wood et al so that their oscillating units become increasingly nonuniform. Differently from our previous attempt [29], units here remain phase-coupled, even in the limit where the nonuniformity transforms the oscillators into excitable units. Only in this limit does the system have an absorbing state, and the question is whether sustained macroscopic oscillations survive this microscopic parametric transformation.…”

“…The intention was to mimic the exponential rates of Wood et al which, in a system of phase-coupled stochastic oscillators, did generate stable collective oscillations [12,13]. In our nonlinearly pulse-coupled system of excitable units, however, this modification actually failed to generate oscillations, but rather turned the phase transition to an active state discontinuous [29].…”

“…Note that the presence of the saddle in Figs. 2(c)-(e) sets the conditions for collective excitability [29]. If the system is initially in the stable node, perturbations that throw the system to a different point still within its basin of attraction can lead to qualitatively different relaxation processes.…”

“…We previously attempted to answer this question by modifying the infection rate of the SIRS model, which was rendered exponential (instead of linear, as in the standard case) in the density of infected neighbors [29]. The intention was to mimic the exponential rates of Wood et al which, in a system of phase-coupled stochastic oscillators, did generate stable collective oscillations [12,13].…”