Large communities of biological oscillators show a prevalent tendency to self-organize in time. This/ncooperative phenomenon inspired Winfree to formulate a mathematical model that originated the theory of/nmacroscopic synchronization. Despite its fundamental importance, a complete mathematical analysis of the/nmodel proposed by Winfree/n—/nconsisting of a large population of all-to-all pulse-coupled oscillators/n—/nis/nstill missing. Here, we show that the dynamics of the Winfree model evolves into the so-called Ott-/nAntonsen manifold. This important property allows for an exact description of this high-dimensional/nsystem in terms of a few macroscopic variables, and also allows for the full investigation of its dynamics./nWe find that brief pulses are capable of synchronizing heterogeneous ensembles that fail to synchronize/nwith broad pulses, especially for certain phase-response curves. Finally, to further illustrate the potential of/nour results, we investigate the possibility of/n“/nchimera/n”/nstates in populations of identical pulse-coupled/noscillators. Chimeras are self-organized states in which the symmetry of a population is broken into a/nsynchronous and an asynchronous part. Here, we derive three ordinary differential equations describing/ntwo coupled populations and uncover a variety of chimera states, including a new class with chaotic/ndynamics.

We thank Juan M. López for a critical reading of the manuscript, Arkady Pikovsky for interesting discussions, and John Rinzel for pointing us to Ref. [7]. D. P. acknowl-edges support from Cantabria International Campus and the Ramón y Cajal program of MINECO (Spain). We acknowl-edge support from the Spanish research Projects No. FIS2009-12964-C05-05 and No. SAF2010-16085.