The Winfree model with heterogeneous phase-response curves: Analytical results

TL;DR

This paper extends the Winfree model to include heterogeneity in phase-response curves and natural frequencies, deriving an approximate Kuramoto-like model and analyzing stability and bifurcations using the Ott-Antonsen ansatz.

Contribution

It introduces a novel extension of the Winfree model incorporating heterogeneous PRCs and natural frequencies, with analytical results on stability and bifurcation behavior.

Findings

Incoherent state stability depends on PRC heterogeneity level.

Derived an approximate Kuramoto-like model with distributed shear.

Analyzed full model stability using Ott-Antonsen ansatz.

Abstract

We study an extension of the Winfree model of coupled phase oscillators in which both natural frequencies and phase-response curves (PRCs) are heterogeneous. In the first part of the paper we resort to averaging and derive an approximate model, in which the oscillators are coupled through their phase differences. Remarkably, this simplified model is the 'Kuramoto model with distributed shear' (2011 Phys. Rev. Lett. 106 254101). We find that above a critical level of PRC heterogeneity the incoherent state is always stable. In the second part of the paper we perform the analysis of the full model for Lorentzian heterogeneities, resorting to the Ott-Antonsen ansatz. The critical level of PRC heterogeneity obtained within the averaging approximation has a different manifestation in the full model depending on the sign of the center of the distribution of PRCs.