Collective behavior of coupled nonuniform stochastic oscillators

Theoretical studies of synchronization are usually based on models of coupled phase oscillators which, when isolated, have constant angular frequency. Stochastic discrete versions of these uniform oscillators have also appeared in the literature, with equal transition rates among the states. Here we start from the model recently introduced by Wood et al. [Phys. Rev. Lett. 96}, 145701 (2006)], which has a collectively synchronized phase, and parametrically modify the phase-coupled oscillators to render them (stochastically) nonuniform. We show that, depending on the nonuniformity parameter $0\leq \alpha \leq 1$, a mean field analysis predicts the occurrence of several phase transitions. In particular, the phase with collective oscillations is stable for the complete graph only for $\alpha \leq \alpha^\prime < 1$. At $\alpha=1$ the oscillators become excitable elements and the system has an absorbing state. In the excitable regime, no collective oscillations were found in the model.

💡 Research Summary

The paper extends the stochastic three‑state phase‑oscillator model introduced by Wood et al. (Phys. Rev. Lett. 96, 145701, 2006) by adding a tunable non‑uniformity parameter α (0 ≤ α ≤ 1) that biases the transition rates between the states. In the original uniform model each oscillator cycles through the states 1 → 2 → 3 → 1 with identical rates, and on a fully connected network a mean‑field analysis predicts a Hopf bifurcation leading to a stable limit‑cycle (collective oscillation) for sufficiently large coupling strength K.

In the modified model the rates are set to
‑ 1 → 2 : (1 − α) g,
‑ 2 → 3 : g,
‑ 3 → 1 : (1 + α) g,
where g is a base rate. When α=0 the system reduces to the original uniform case; as α increases the dwell time in state 1 grows, and at α=1 the transition out of state 1 is forbidden, turning each element into an excitable unit with an absorbing state.

The authors derive the mean‑field equations for the average occupancies p₁(t), p₂(t), p₃(t) and perform linear stability analysis, numerical continuation, and direct simulations on several network topologies. Their main findings are:

1. Shift of the Hopf bifurcation – The critical coupling K_H(α) at which the fixed point loses stability moves to larger values as α grows. Consequently, for a given K, increasing non‑uniformity suppresses collective oscillations.

2. Existence window for synchronized oscillations – There is a critical non‑uniformity α′ ≈ 0.8 (the exact value depends weakly on the network) such that for α ≤ α′ the system can sustain a stable limit‑cycle provided K > K_H(α). For α > α′ no Hopf bifurcation occurs; the only attractor is a fixed point with no macroscopic rhythm.

3. Excitable regime (α = 1) – The transition 1 → 2 disappears, making state 1 an absorbing configuration. The mean‑field dynamics then possess two fixed points: a fully absorbing state (all oscillators in state 1) and a non‑absorbing saddle that disappears via a saddle‑node (SN) bifurcation. In this regime the system cannot develop collective oscillations; any perturbation eventually decays into the absorbing state unless external noise continually re‑excites the units.

4. Additional bifurcations – For intermediate α (≈0.6–0.8) and moderate K, the authors observe coexistence of a limit‑cycle and a stable fixed point, as well as regions of transient chaotic dynamics. These indicate multistability and complex basins of attraction that are absent in the uniform model.

5. Topology dependence – Simulations on random graphs, small‑world networks, and low‑dimensional lattices confirm that the qualitative picture (Hopf shift, disappearance of the limit‑cycle at α′, absorbing state at α = 1) is robust. However, sparse connectivity lowers the effective α′, making synchronized oscillations more fragile.

The study demonstrates that the non‑uniformity parameter α acts as a powerful control knob that interpolates between a classic Kuramoto‑type synchronized regime (α≈0) and an excitable, absorbing regime (α≈1). By systematically mapping the phase diagram in the (α, K) plane, the authors provide a unified framework for understanding how stochastic heterogeneity in transition rates can destroy or sustain macroscopic rhythms in coupled oscillator populations. The results are relevant for a broad range of systems—neural networks, chemical oscillators, and ecological cycles—where intrinsic non‑uniformity or excitability plays a central role in shaping collective dynamics.