Comment on "Phase Reduction of Stochastic Limit Cycle Oscillators"

TL;DR

This comment clarifies that both the conventional and modified phase equations for noisy limit-cycle oscillators are valid depending on the relative timescales, emphasizing the conditions under which each approximation applies.

Contribution

It refutes the claim that the conventional phase equation is invalid, showing its validity under certain conditions related to amplitude relaxation and noise correlation times.

Findings

Both phase equations are valid depending on timescale ratios.

Conventional phase equation is suitable for fast amplitude relaxation.

The choice of phase equation depends on the relative size of amplitude relaxation and noise correlation times.

Abstract

This is a comment on a recent paper by Yoshimura and Arai [Phys. Rev. Lett. 101, 154101 (2008)] on phase reduction of noisy limit-cycle oscillators, in which the authors claimed that the conventional phase stochastic differential equation used in previous studies does not give a proper approximation and proposed a modified phase equation. Here we point out that both phase equations are valid depending on the situation. We argue that the relative size of amplitude relaxation time and noise correlation time determines which of the two equations is appropriate in the white-noise limit. The conventional phase equation is also a proper approximation to noisy limit cycles with sufficiently fast amplitude relaxation, which can be used as a starting point for further analysis.