Partial Synchronization in Pulse-Coupled Oscillator Networks I: Theory

We study $N$ identical integrate and fire model neurons coupled in an all to all network through $\alpha$-function pulses, weighted by a parameter $K$. Studies of the dynamics of this system often focus on the stability of the fully synchronous and the fully asynchronous splay states, that naturally depend on the sign of $K$, i.e. excitation vs inhibition. We find that for finite $N$ there is a rich set of other partially synchronized attractors, such as $(N-1,1)$ fixed states and partially synchronized splay states. Our framework exploits the neutrality of the dynamics for $K=0$ which allows us to implement a dimensional reduction strategy that replaces the discrete pulses with a continuous flow, with the sign of $K$ determining the flow direction. This framework naturally incorporates a hierarchy of partially synchronized subspaces in which the new states lie. For $N=2,\;3,\;4$, we completely describe the sequence of bifurcations and the stability of all fixed points and limit cycles.