The Winfree model with non-infinitesimal phase-response curve: Ott–Antonsen theory

Abstract

A novel generalization of the Winfree model of globally coupled phase oscillators, representing phase reduction under finite coupling, is studied analytically. We consider interactions through a non-infinitesimal (or finite) phase-response curve (PRC), in contrast to the infinitesimal PRC of the original model. For a family of non-infinitesimal PRCs, the global dynamics is captured by one complex-valued ordinary differential equation resorting to the Ott–Antonsen ansatz. The phase diagrams are thereupon obtained for four illustrative cases of non-infinitesimal PRC. Bistability between collective synchronization and full desynchronization is observed in all cases. In 1967, Winfree proposed a model for the spontaneous synchronization of large ensembles of biological oscillators. The Winfree model played a seminal role in the field of collective synchrony, inspiring the Kuramoto model as well as promoting recent advances in theoretical neuroscience. In spite of the simplifying assumptions of the Winfree model, uniform all-to-all weak coupling, analytical solutions have been found only recently using the Ott–Antonsen ansatz. Weak coupling is implicit in the use of phase oscillators as the units of the model. Moreover, their interactions are modeled by the so-called infinitesimal phase-response curve (iPRC), which is only valid in the limit of vanishing coupling. In this paper, we extend the Winfree model considering a non-infinitesimal (also called finite) PRC such that the phase shift of one oscillator is not proportional to the magnitude of the input. For a family of non-infinitesimal PRCs, and a Lorentzian distribution of natural frequencies, the global dynamics is captured by one complex-valued ordinary differential equation by means of the Ott–Antonsen ansatz. We obtain phase diagrams for four instructive cases.