  Large communities of biological oscillators show a prevalent tendency to
self-organize in time. This cooperative phenomenon inspired Winfree to
formulate a mathematical model that originated the theory of macroscopic
synchronization. Despite its fundamental importance, a complete mathematical
analysis of the model proposed by Winfree ---consisting of a large population
of all-to-all pulse-coupled oscillators--- is still missing. Here we show that
the dynamics of the Winfree model evolves into the so-called Ott-Antonsen
manifold. This important property allows for an exact description of this
high-dimensional system in terms of a few macroscopic variables, and the full
investigation of its dynamics. We find that brief pulses are capable of
synchronizing heterogeneous ensembles which fail to synchronize with broad
pulses, specially for certain phase response curves. Finally, to further
illustrate the potential of our results, we investigate the possibility of
`chimera' states in populations of identical pulse-coupled oscillators.
Chimeras are self-organized states in which the symmetry of a population is
broken into a synchronous and an asynchronous part. Here we derive three
ordinary differential equations describing two coupled populations, and uncover
a variety of chimera states, including a new class with chaotic dynamics.
