A partial differential equation for the mean first-return-time phase of a stochastic oscillator

Phase reduction of limit cycle dynamics provides a low-dimensional representation of high-dimensional oscillator dynamics. For a deterministic dynamical system with a stable limit cycle, the change to a phase variable is well established. In contrast, for stochastic limit cycle systems, a phase reduction can be defined in several nonequivalent ways (e.g. Schwabedal and Pikovsky Phys. Rev. Lett. 110, 205102 (2013), Lindner and Thomas Phys. Rev. Lett. 113, 254101 (2014)]. Schwabedal and Pikovsky introduced a phase for stochastic oscillators based on a foliation of the basin of attraction, with the property that the mean transit time around the cycle from each leaf to itself is uniform and developed a numerical procedure to estimate the corresponding isochrons. For robustly oscillating planar systems driven by white Gaussian noise, we establish a partial differential equation with a mixture of reflecting and jump boundary conditions that governs this phase function. We solve this equation numerically for several examples of noisy oscillators. In addition, we obtain an explicit expression for the isochron function, for the rotationally symmetric case, and compare this analytical result with oscillators that have been studied numerically in the literature.