Asymptotic phase for stochastic oscillators.

Oscillations and noise are ubiquitous in physical and biological systems. When oscillations arise from a deterministic limit cycle, entrainment and synchronization may be analyzed in terms of the asymptotic phase function. In the presence of noise, the asymptotic phase is no longer well defined. We introduce a new definition of asymptotic phase in terms of the slowest decaying modes of the Kolmogorov backward operator. Our stochastic asymptotic phase is well defined for noisy oscillators, even when the oscillations are noise dependent. It reduces to the classical asymptotic phase in the limit of vanishing noise. The phase can be obtained either by solving an eigenvalue problem, or by empirical observation of an oscillating density's approach to its steady state.