Abstract
The Winfree model, a well-known mathematical model for describing collective synchronization in living systems, such as flashing fireflies, has been under-utilized because of its daunting technical complexity. Now scientists have found a way to dramatically reduce it to a technically tractable form and demonstrate the power of the reduction with findings of new ``chimera'' states in populations of pulse-coupled oscillators.
Highlights
In 1967, Winfree proposed the first mathematical model for the macroscopic synchronization observed in large populations of biological oscillators [1]
We find that brief pulses are capable of synchronizing heterogeneous ensembles that fail to synchronize with broad pulses, especially for certain phase-response curves
The Winfree model describes a population of heterogeneous limit cycle oscillators, which interact via pulselike signals
Summary
In 1967, Winfree proposed the first mathematical model for the macroscopic synchronization observed in large populations of biological oscillators [1]. In 2008, Ott and Antonsen made a very important finding [14]: Kuramoto-like models have solutions in a reduced invariant manifold. This result drastically simplifies the task of investigating the collective dynamics of such systems. Despite their importance and generality, Kuramoto-like models—in which interactions are expressed by phase differences—are approximations of more-realistic models, such as the Winfree model, in the weak-coupling limit. The evolution of the Winfree model in the OA manifold opens the possibility of investigating phenomena that, far, have been addressed analytically using “Kuramoto oscillators.” As an example, we uncover the existence of a variety of the so-called chimera states [22] in populations of “Winfree oscillators.”
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