Collective phase description of globally coupled excitable elements

TL;DR

This paper develops a theoretical framework for describing the collective phase dynamics of globally coupled noisy excitable elements, extending phase reduction methods to infinite-dimensional systems like nonlinear Fokker-Planck equations.

Contribution

It introduces a novel collective phase description for macroscopic rhythms in noisy excitable systems, linking bifurcation types to phase sensitivity functions.

Findings

Type-I collective phase sensitivity near saddle-node bifurcation

Type-II sensitivity near Hopf bifurcation

Extension of phase reduction to infinite-dimensional systems

Abstract

We develop a theory of collective phase description for globally coupled noisy excitable elements exhibiting macroscopic oscillations. Collective phase equations describing macroscopic rhythms of the system are derived from Langevin-type equations of globally coupled active rotators via a nonlinear Fokker-Planck equation. The theory is an extension of the conventional phase reduction method for ordinary limit cycles to limit-cycle solutions in infinite-dimensional dynamical systems, such as the time-periodic solutions to nonlinear Fokker-Planck equations representing macroscopic rhythms. We demonstrate that the type of the collective phase sensitivity function near the onset of collective oscillations crucially depends on the type of the bifurcation, namely, it is type-I for the saddle-node bifurcation and type-II for the Hopf bifurcation.