Pc. Bressloff et S. Coombes, A dynamical theory of spike train transitions in networks of integrate-and-fire oscillators, SIAM J A MA, 60(3), 2000, pp. 820-841
A dynamical theory of spike train transitions in networks of pulse-coupled
integrate- and-fire (IF) neural oscillators is presented. We begin by deriv
ing 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 collec
tive period. We then investigate the stability of phase-locked solutions by
constructing a linearized map of the firing times and analyzing its spectr
um. 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 ins
tabilities can induce transitions to nonphase locked states characterized b
y periodic or quasi-periodic variations of the interspike intervals on attr
acting invariant circles. The resulting spatio-temporal pattern of network
activity is compatible with the behavior of a corresponding firing rate (an
alog) model in the limit of slow synaptic interactions.