Phase reduction and synchronization of a network of coupled dynamical elements exhibiting collective oscillations

TL;DR

This paper develops a general phase reduction method for networks of coupled dynamical elements with collective oscillations, enabling simplified analysis of synchronization and response to perturbations.

Contribution

It introduces a novel phase reduction framework applicable to heterogeneous networks, deriving adjoint equations for phase sensitivity functions.

Findings

Derived coupled adjoint equations for phase sensitivity functions.

Reduced complex network dynamics to a one-dimensional phase equation.

Analyzed mutual synchronization in FitzHugh-Nagumo networks.

Abstract

A general phase reduction method for a network of coupled dynamical elements exhibiting collective oscillations, which is applicable to arbitrary networks of heterogeneous dynamical elements, is developed. A set of coupled adjoint equations for phase sensitivity functions, which characterize phase response of the collective oscillation to small perturbations applied to individual elements, is derived. Using the phase sensitivity functions, collective oscillation of the network under weak perturbation can be described approximately by a one-dimensional phase equation. As an example, mutual synchronization between a pair of collectively oscillating networks of excitable and oscillatory FitzHugh-Nagumo elements with random coupling is studied.