  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.
