# Synchronization scenarios in the Winfree model of coupled oscillators

## TL;DR

This paper derives an exact low-dimensional model for the Winfree oscillator system, revealing two distinct synchronization scenarios linked by a bifurcation, and compares these with Kuramoto-Sakaguchi predictions.

## Contribution

It provides an exact low-dimensional representation of the Winfree model and identifies two different synchronization scenarios based on pulse shape and PRC offset.

## Key findings

- Two distinct synchronization scenarios identified
- A bifurcation links the scenarios via a Bogdanov-Takens point
- Discrepancy with Kuramoto-Sakaguchi prediction behaves as an odd function of PRC offset

## Abstract

The emergence of collective synchronization was reproduced long ago by Winfree in a classical model consisting of an ensemble of pulse-coupled phase oscillators. By means of the Ott-Antonsen ansatz, we derive an exact low-dimensional representation which is exhaustively investigated for a variety of pulse types and phase response curves (PRCs). Two structurally different synchronization scenarios are found, which are linked via the mutation of a Bogdanov-Takens point. From our results, we infer a general rule of thumb relating pulse shape and PRC offset with each scenario. Finally, we compare the exact synchronization threshold with the prediction of the averaging approximation given by the Kuramoto-Sakaguchi model. At the leading order, the discrepancy appears to behave as an odd function of the PRC offset.