Practical Synchronization of Winfree Oscillators in a Random Environment

We present a practical synchronization estimate of Winfree oscillators in a random environment. In a large coupling regime, the deterministic Winfree model exhibits the oscillator death, emerging with a convergence of the phase ensemble. The additive noise, however, is expected to destroy the stability of an equilibrium. In this paper, we estimate the running maximum of the phase processes, and conclude that the escaping probability from a small interval is in the order of $T{N}^{-1}\exp (-\kappa /\parallel \Sigma {\parallel }^2)$ over a time interval [0, T], where $\kappa$, N and $\parallel \Sigma \parallel$ denote the coupling strength, number of oscillators and noise strength, respectively. This result explains the robustness of the practical synchronization, which indicates that the finite-time emergent behavior from finite oscillators is close to the synchronization phenomena when $\kappa$ is large enough. Our approach produces explicit bounds on probabilities, relying on comparisons with the Ornstein–Uhlenbeck processes. It is hence optimal in the sense that the linearized model gives the same order.