Stability of splay states for pulse-coupled neuronal networks: finite size versus finite pulse-width effects

The dynamics of collective states observed in globally coupledneuronal networks is still an open problem. In particular,although it is claimed that the periodic firing state("splay state") is stable only for excitatory coupling [1],counterexamples have been found for inhibitory couplingas well [2]. Moreover, the stability of the splay states hasbeen analyzed only in the mean field limit [1,3,4]. Ouraim is to investigate simultaneously, for a pulse-couplednetwork of leaky integrate-and fire neurons, the effect ofthe number N of neurons as well as of the pulse-width ofthe post-synaptic potentials. Finite-N networks can bestudied by suitably modifying the map-like formalism[5,6] usually adopted to implement numerically themodel. As a result, we find that the stability of the splaystate depends crucially on a parameter that is proportionalto the width of the delivered pulses rescaled to the averageinterspike interval. More precisely, we show that the Floquetspectrum of eigenvalues is made of two components,one of which coincides with that one predicted by themean-field analysis [1]. Depending on the value of the relevantparameter, the second component may be responsiblefor the occurrence of instabilities which in turn suggestthe failure of the continuum limit approximation. Finally,for sufficiently small pulse-width we observe that thesplay state can be stable even for inhibitory coupling.