Coherence of oscillations in the weak-noise limit

TL;DR

This paper analytically derives the coherence quality factor for oscillations in noisy environments using a weak noise approximation, extending the understanding of oscillation stability in both one- and two-dimensional systems.

Contribution

It provides an analytical expression for the coherence quality factor in a continuum Fokker-Planck model and adapts this to two-dimensional limit cycles, including chemical and oscillator models.

Findings

Derived an explicit formula for the coherence quality factor in weak noise limit.

Proved an upper bound on coherence analogous to discrete Markov models.

Validated results with numerical comparisons to Hamilton-Jacobi based methods.

Abstract

In a noisy environment, oscillations loose their coherence which can be characterized by a quality factor. We determine this quality factor for oscillations arising from a driven Fokker-Planck dynamics along a periodic one-dimensional potential analytically in the weak noise limit. With this expression, we can prove for this continuum model the analog of an upper bound that has been conjectured for the coherence of oscillations in discrete Markov network models. We show that our approach can also be adapted to motion along a noisy two-dimensional limit cycle. Specifically, we apply our scheme to the noisy Stuart-Landau oscillator and the thermodynamically consistent Brusselator as a simple model for a chemical clock. Our approach thus complements the fairly sophisticated extant general framework based on techniques from Hamilton-Jacobi theory with which we compare our results numerically.