A dynamical theory of spike train transitions in networks of integrate-and-fire oscillators

A dynamical theory of spike train transitions in networks of pulse-coupled integrateand- fire (IF) neural oscillators is presented. We begin by deriving conditions for 1:1 frequency locking in a network with non-instantaneous synaptic interactions. This leads to a set of phase equations determining the relative firing times of the oscillators and the self-consistent collective period. We then investigate the stability of phase-locked solutions by constructing a linearized map of the firing times and analyzing its spectrum. We establish that previous results concerning the stability properties of IF oscillator networks are incomplete since they only take into account the effects of weak coupling instabilities. We show how strong coupling instabilities can induce transitions to non-phase locked states characterized by periodic or quasiperiodic variations of the inter-spike intervals on attracting invariant circles. The resulting spatio-temporal pattern of network activity is compatible with the behavior of a corresponding firing rate (analog) model in the limit of slow synaptic interactions.