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

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>=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 et al., 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.

💡 Research Summary

In this work the authors investigate a modified stochastic susceptible‑infected‑refractory‑susceptible (SIRS) model in which the infection rate is no longer a linear function of the number of infected neighbours but an exponential reinforcement, and diffusion of individuals is completely suppressed. The infection probability for a susceptible site with n infected neighbours is taken as λ exp(α n), where λ is a baseline transmission rate and α (>0) quantifies the strength of the non‑linearity. After infection a site enters a refractory (R) state for a fixed time τ_R before returning to the susceptible (S) state, thus the dynamics consist of discrete pulses rather than continuous phase coupling.

The model is simulated on three types of networks: (i) a complete graph (mean‑field limit), (ii) Erdős‑Rényi random graphs with finite average degree, and (iii) d‑dimensional hypercubic lattices with d = 1, 2, 3. For each topology the authors scan the two control parameters (λ, α) together with the refractory time τ_R, measuring the stationary density of infected sites ρ, the density of refractory sites σ, and their fluctuations. To detect possible hysteresis they perform bi‑directional sweeps of λ (increasing and decreasing) and record the resulting ρ(λ) curves.

Two distinct regimes emerge. When α is small (weak non‑linearity) the system behaves like the classic stochastic SIRS model: as λ crosses a critical value λ_c the infected density grows continuously from zero, indicating a second‑order, absorbing‑to‑active phase transition belonging to the directed‑percolation universality class. In contrast, for sufficiently large α (strong non‑linearity) the transition becomes discontinuous in dimensions d ≥ 2 and on random graphs. In this case ρ jumps abruptly at a threshold λ_c, and a pronounced hysteresis loop appears: the forward and backward sweeps of λ give different stationary states, evidencing bistability between an absorbing state (ρ ≈ 0) and an active state (finite ρ). The authors interpret this as an “explosive contagion” phenomenon: the exponential reinforcement makes the infection rate increase so sharply with the local infected fraction that a small cluster can either die out or, once a critical size is reached, rapidly engulf the whole system.

A mean‑field theory is derived by assuming each site experiences the average infected fraction ρ. The resulting rate equations are

dρ/dt = –ρ/τ_R + λ e^{α ρ} (1 – ρ – σ) ρ,
dσ/dt = ρ/τ_R – σ/τ_S,

where τ_S is the refractory‑to‑susceptible relaxation time (often taken equal to τ_R for simplicity). Fixed‑point analysis of these equations shows that for weak α there is a single stable fixed point at ρ = 0, while for strong α three fixed points can coexist: two stable (absorbing and active) and one unstable separating them. The bifurcation diagram reproduces the numerical phase diagram, correctly predicting the location of the continuous‑to‑discontinuous crossover and the existence of a hysteresis region.

An important comparative observation concerns synchronization. Wood et al. (Phys. Rev. Lett. 96, 145701 2006) reported that a similar exponential coupling in a population of phase‑coupled stochastic oscillators leads to a transition toward a partially synchronized state. In the present pulse‑coupled excitable model, no such synchronized phase is observed for any α, λ, or τ_R. The key difference lies in the interaction rule: phase oscillators adjust continuously their phase, allowing collective locking, whereas the SIRS‑type elements interact only through discrete infection pulses, which favor abrupt state changes rather than gradual phase alignment.

Nevertheless, for very long refractory periods and high non‑linearity the system can display “collective excitability.” In this regime stochastic fluctuations occasionally trigger large, transient bursts of infection that resemble unstable stochastic oscillations. These bursts are not sustained; after a brief active episode the system returns to the absorbing state. This behavior is reminiscent of excitable media in neuroscience (e.g., neuronal avalanches) or cardiac tissue where a long refractory period can support large, but ultimately self‑terminating, wave fronts.

In summary, the paper demonstrates that introducing an exponential reinforcement into a pulse‑coupled SIRS model generates a rich nonequilibrium phase diagram. Weak reinforcement yields the familiar continuous absorbing‑active transition, while strong reinforcement produces a discontinuous transition with hysteresis and bistability in dimensions two and higher. Mean‑field theory captures the essential features, and extensive simulations on complete graphs, random networks, and hypercubic lattices confirm the theoretical predictions. The absence of a synchronized phase distinguishes this model from previously studied phase‑coupled oscillator systems, highlighting the crucial role of interaction modality (pulse versus phase coupling). The findings broaden our understanding of explosive contagion phenomena, provide a tractable framework for studying collective excitability in excitable media, and suggest future directions such as exploring scale‑free networks, community structures, or external periodic driving to probe how topology and forcing influence the discontinuous transition and the emergence of transient stochastic oscillations.