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.
This is a pre-print. The definitive version: BRESSLOFF, P.C. and COOMBES, S., 2000.A dynamical theory of spike train transitions in networks of integrate-and-fire oscillators. SIAM Journal on Applied Mathematics, 60(3), pp.820-841.