Ott-Antonsen attractiveness for parameter-dependent oscillatory networks

TL;DR

This paper extends the Ott-Antonsen ansatz to parameter-dependent oscillatory networks, proving convergence to the manifold under various conditions and broadening its applicability to realistic neural and network models.

Contribution

The authors generalized the OA ansatz to include oscillator-specific parameters, time-dependent and multi-dimensional parameters, and complex network topologies, with rigorous proofs of convergence.

Findings

Proved convergence of the OA manifold in parameter-dependent systems.

Validated the extended OA ansatz with numerical and theoretical analysis.

Applied the framework to networks of theta neurons and other realistic models.

Abstract

The Ott-Antonsen (OA) ansatz [Chaos 18, 037113 (2008), Chaos 19, 023117 (2009)] has been widely used to describe large systems of coupled phase oscillators. If the coupling is sinusoidal and if the phase dynamics does not depend on the specific oscillator, then the macroscopic behavior of the systems can be fully described by a low-dimensional dynamics. Does the corresponding manifold remain attractive when introducing an intrinsic dependence between an oscillator's phase and its dynamics by additional, oscillator specific parameters? To answer this we extended the OA ansatz and proved that parameter-dependent oscillatory systems converge to the OA manifold given certain conditions. Our proof confirms recent numerical findings that already hinted at this convergence. Furthermore we offer a thorough mathematical underpinning for networks of so-called theta neurons, where the OA ansatz has just been applied. In a final step we extend our proof by allowing for time-dependent and multi-dimensional parameters as well as for network topologies other than global coupling. This renders the OA ansatz an excellent starting point for the analysis of a broad class of realistic settings.