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 integrate- and-fire (IF) neural oscillators is presented. We begin by deriving conditions for 1:1 frequency-locking in a network with noninstantaneous 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 nonphase locked states characterized by periodic or quasi-periodic variations of the interspike 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.