Phase-Amplitude Reduction and Optimal Phase Locking of Collectively Oscillating Networks

TL;DR

This paper introduces a phase-amplitude reduction framework for analyzing and controlling collective oscillations in networked dynamical systems, enabling better understanding and optimization of phase locking and synchronization.

Contribution

It extends phase reduction methods by incorporating amplitude variables, allowing analysis of deviations and control of collective oscillations in complex networks.

Findings

Effective for networks of FitzHugh-Nagumo elements

Enables derivation of optimal waveforms for phase locking

Facilitates feedback control for synchronization

Abstract

We present a phase-amplitude reduction framework for analyzing collective oscillations in networked dynamical systems. The framework, which builds on the phase reduction method, takes into account not only the collective dynamics on the limit cycle but also deviations from it by introducing amplitude variables and using them with the phase variable. The framework allows us to study how networks react to applied inputs or coupling, including their synchronization and phase-locking, while capturing the deviations of the network states from the unperturbed dynamics. Numerical simulations are used to demonstrate the effectiveness of the framework for networks composed of FitzHugh-Nagumo elements. The resulting phase-amplitude equation can be used in deriving optimal periodic waveforms or introducing feedback control for achieving fast phase locking while stabilizing the collective oscillations.