  We study the global dynamics of integrate and fire neural networks composed
of an arbitrary number of identical neurons interacting by inhibition and
excitation. We prove that if the interactions are strong enough, then the
support of the stable asymptotic dynamics consists of limit cycles. We also
find sufficient conditions for the synchronization of networks containing
excitatory neurons. The proofs are based on the analysis of the equivalent
dynamics of a piecewise continuous Poincar\'e map associated to the system. We
show that for strong interactions the Poincar\'e map is piecewise contractive.
Using this contraction property, we prove that there exist a countable number
of limit cycles attracting all the orbits dropping into the stable subset of
the phase space. This result applies not only to the Poincar\'e map under
study, but also to a wide class of general n-dimensional piecewise contractive
maps.
