  The activity of a sparse network of leaky integrate-and-fire neurons is
carefully revisited with reference to a regime of a bona-fide asynchronous
dynamics. The study is preceded by a finite-size scaling analysis, carried out
to identify a setup where collective synchronization is negligible. The
comparison between quenched and annealed networks reveals the emergence of
substantial differences when the coupling strength is increased, via a scenario
somehow reminiscent of a phase transition. For sufficiently strong synaptic
coupling, quenched networks exhibit a highly bursting neural activity, well
reproduced by a self-consistent approach, based on the assumption that the
input synaptic current is the superposition of independent renewal processes.
The distribution of interspike intervals turns out to be relatively
long-tailed; a crucial feature required for the self-sustainment of the
bursting activity in a regime where neurons operate on average (much) below
threshold. A semi-quantitative analogy with Ornstein-Uhlenbeck processes helps
validating this interpretation. Finally, an alternative explanation in terms of
Poisson processes is offered under the additional assumption of mutual
correlations among excitatory and inhibitory spikes.
