Synchronizing pulse-coupled oscillators by constraining the phase response curve

Abstract: We consider networks of weakly pulse-coupled identical oscillators. In an effort to resolve a long-standing problem, we develop an analytic condition on the infinitesimal phase response curve (iPRC) for synchronized dynamic behaviour, extending the well-known result by Mirollo and Strogatz. Oscillators cluster towards synchronization through recurrent absorptions in the case of fully connected networks. We also point out that the same analytic condition guarantees absorption for general networks, and how the condition is extended for non-homogeneous coupling. For a network of neural oscillators of the quadratic-integrate-and-fire type (QIF) we reinterpret our synchronization result into explicit conditions on the QIF-model parameters.