Coexistence of stable stationary behavior and partial synchrony in an all-to-all coupled spiking neural network

We consider the stationary and the partially synchronous regimes in an all-to-all coupled neural network consisting of an infinite number of leaky integrate-and-fire neurons. Using analytical tools as well as simulation results, we show that two threshold values for the coupling strength may be distinguished. Below the lower threshold, no synchronization is possible; above the upper threshold, the stationary regime is unstable and partial synchrony prevails. In between there is a range of values for the coupling strength where both regimes may be observed. The assumption of an infinite number of neurons is crucial: simulations with a finite number of neurons indicate that above the lower threshold partial synchrony always prevails-but with a transient time that may be unbounded with increasing system size. For values of the coupling strength in a neighborhood of the lower threshold, the finite model repeatedly builds up toward synchronous behavior, followed by a sudden breakdown, after which the synchronization is slowly built up again. The "transient" time needed to build up synchronization again increases with increasing system size, and in the limit of an infinite number of neurons we retrieve stationary behavior. Similarly, within some range for the coupling strength in this neighborhood, a stable synchronous solution may exist for an infinite number of neurons.