title: Linear stability in networks of pulse-coupled neurons.
abstract: In a first step toward the comprehension of neural activity, one should focus on the stability of the possible dynamical states. Even the characterization of an idealized regime, such as that of a perfectly periodic spiking activity, reveals unexpected difficulties. In this paper we discuss a general approach to linear stability of pulse-coupled neural networks for generic phase-response curves and post-synaptic response functions. In particular, we present: (1) a mean-field approach developed under the hypothesis of an infinite network and small synaptic conductances; (2) a "microscopic" approach which applies to finite but large networks. As a result, we find that there exist two classes of perturbations: those which are perfectly described by the mean-field approach and those which are subject to finite-size corrections, irrespective of the network size. The analysis of perfectly regular, asynchronous, states reveals that their stability depends crucially on the smoothness of both the phase-response curve and the transmitted post-synaptic pulse. Numerical simulations suggest that this scenario extends to systems that are not covered by the perturbative approach. Altogether, we have described a series of tools for the stability analysis of various dynamical regimes of generic pulse-coupled oscillators, going beyond those that are currently invoked in the literature.
authors.affiliation: Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi Sesto Fiorentino, Italy ; INFN-Sezione di Firenze and CSDC Sesto Fiorentino, Italy.
authors.forename: Simona
authors.initials: S
authors.lastname: Olmi
authors.affiliation: Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi Sesto Fiorentino, Italy ; INFN-Sezione di Firenze and CSDC Sesto Fiorentino, Italy.
authors.forename: Alessandro
authors.initials: A
authors.lastname: Torcini
authors.affiliation: Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi Sesto Fiorentino, Italy ; SUPA and Institute for Complex Systems and Mathematical Biology, King's College, University of Aberdeen Aberdeen, UK.
authors.forename: Antonio
authors.initials: A
authors.lastname: Politi
keywords: Floquet spectrum; linear stability analysis; neural networks; pulse coupled neurons; splay states; synchronization