Stationary States in Infinite Networks of Spiking Oscillators with Noise

We model networks of identical, all-to-all, pulse-coupled phase oscillators with white noise, in the limit of infinite network size and Dirac pulses, using a Fokker-Planck equation for the phase probability density. We give analytical, constructive existence and uniqueness results for stationary states (i.e., time-independent densities) and derive and study a one-dimensional eigenvalue equation for their linear stability. Our results are supplemented by numerical methods, which are applied to two classes of oscillator response functions. We find that the stationary network activity depends for some response functions monotonically and for others nonmonotonically on the coupling and noise strength. In all cases we find that a sufficiently strong noise locally stabilizes the stationary state, and simulations suggest this stability to be global. For most response functions the stationary state undergoes a supercritical Hopf bifurcation as noise is decreased, and a locally stable limit cycle emerges in its proximity. On that limit cycle, the network splits into groups of approximately synchronized oscillators, while the network's (mean) activity oscillates at frequencies often much higher than the intrinsic oscillator frequency.