The activity of a sparse network of leaky integrate-and-fire neurons is carefully revisited with reference to a regime of 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 semiquantitative analogy with Ornstein-Uhlenbeck processes helps validate 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.
A.T. received financial support by the Excellence Initiative I-Site Paris Seine (Grant No. ANR-16-IDEX-008), by the Labex MME-DII (Grant No ANR-11-LBX-0023-01) (together with A.P. and E.U.), and by the ANR Project ERMUNDY (Grant No ANR-18-CE37-0014), all part of the French program Investissements d’Avenir.