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The emergence of collective synchronization was reproduced long ago byWinfree in a classical model consisting of an ensemble of pulse-coupled phaseoscillators. By means of the Ott-Antonsen ansatz, we derive an exactlow-dimensional representation which is exhaustively investigated for a varietyof pulse types and phase response curves (PRCs). Two structurally differentsynchronization scenarios are found, which are linked via the mutation of aBogdanov-Takens point. From our results, we infer a general rule of thumbrelating pulse shape and PRC offset with each scenario. Finally, we compare theexact synchronization threshold with the prediction of the averagingapproximation given by the Kuramoto-Sakaguchi model. At the leading order, thediscrepancy appears to behave as an odd function of the PRC offset.

Synchronization scenarios in the Winfree model of coupled oscillators