An analytical approach to the Mean--Return-Time~Phase of isotropic stochastic oscillators

TL;DR

This paper derives analytical expressions for the mean-return-time phase of isotropic stochastic oscillators, enabling explicit evaluation and analysis of phase behavior under strong noise, and introduces lines of constant return time variance.

Contribution

It provides the first analytical formulas for the MRT phase of isotropic stochastic oscillators, facilitating explicit analysis and comparison with existing numerical methods.

Findings

Explicit formulas for MRT phase in isotropic stochastic oscillators.

Analysis of MRT phase behavior in strong noise limits.

Introduction of lines of constant return time variance.

Abstract

One notion of phase for stochastic oscillators is based on the mean return-time (MRT): a set of points represents a certain phase if the mean time to return from any point in this set to this set after one rotation is equal to the mean rotation period of the oscillator (irrespective of the starting point). For this so far only algorithmically defined phase, we derive here analytical expressions for the important class of isotropic stochastic oscillators. This allows us to evaluate cases from the literature explicitly and to study the behavior of the MRT phase in the limits of strong noise. We also use the same formalism to show that lines of constant return time variance (instead of constant mean return time) can be defined, and that they in general differ from the MRT-isochrons.