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

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’ (Montbrió and Pazó 2011 Phys. Rev. Lett. 106 254101). Using this approximation, 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. As expected, the results obtained using the full model are fully consistent with those obtained using the averaging approximation. However, we find that 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.