Asymptotic Phase for Stochastic Oscillators

TL;DR

This paper introduces a new definition of asymptotic phase for noisy oscillators using the slowest decaying modes of the Kolmogorov backward operator, applicable even with significant noise.

Contribution

It proposes a novel stochastic asymptotic phase concept that extends classical phase definitions to noisy systems, solvable via eigenvalue problems or empirical methods.

Findings

Defines stochastic asymptotic phase for noisy oscillators

Reduces to classical phase in the absence of noise

Provides methods to compute phase through eigenvalue problems or empirical observation

Abstract

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.