Estimating Phase from Observed Trajectories Using the Temporal 1-Form

Simon Wilshin, Matthew D Kvalheim, Clayton Scott, Shai Revzen

Neural Computation, vol. 37, no. 12, pp. 2158-2204, 2025

Abstract

Oscillators are ubiquitous in nature and are usually associated with the existence of an asymptotic phase that governs the long-term dynamics of the oscillator. We show that the asymptotic phase can be estimated using a carefully chosen series expansion that directly computes the phase response curve (PRC) and provides an algorithm for estimating the coefficients of this series. Unlike previously available data-driven phase estimation methods, our algorithm can use observations that are much shorter than a cycle; has proven convergence rate bounds as a function of the properties of measurement noise and system noise; will recover phase within any forward invariant region for which sufficient data are available; recovers the PRCs that govern weak oscillator coupling; and recovers isochron curvature and recovers nonlinear features of isochron geometry. Our method may find application wherever models of oscillator dynamics need to be constructed from measured or simulated time-series.

BibTeX

@article{wilshin2022estimating,
  author = {Wilshin, Simon and Kvalheim, Matthew D and Scott, Clayton and Revzen, Shai},
  title = {Estimating Phase from Observed Trajectories Using the Temporal 1-Form},
  journal = {Neural Computation},
  year = {2025},
  volume = {37},
  number = {12},
  pages = {2158-2204},
  doi = {10.1162/neco.a.32},
  cvtopic = {ddmodel},
}