AUTHOR=Olmi Simona , Politi Antonio , Torcini Alessandro 

TITLE=Linear stability in networks of pulse-coupled neurons

JOURNAL=Frontiers in Computational Neuroscience

VOLUME=Volume 8 - 2014

YEAR=2014

URL=https://www.frontiersin.org/journals/computational-neuroscience/articles/10.3389/fncom.2014.00008

DOI=10.3389/fncom.2014.00008

ISSN=1662-5188

ABSTRACT=In a first step towards the comprehension of neural activity, one should focus on<br/>the stability of the possible dynamical states. Even the characterization of idealized regimes, <br/>such as that of a perfectly periodic spiking activity, reveals unexpected difficulties. <br/>In this paper we discuss a general approach to linear stability of pulse-coupled neural <br/>networks for generic phase-response curves and post-synaptic response functions. <br/>In particular, we present: (i) a mean-field approach developed under the hypothesis <br/>of an infinite network and small synaptic conductances; (ii) a ``microscopic" approach <br/>which applies to finite but large networks. As a result, we find that there exist two classes of perturbations: those which<br/>are perfectly described by the mean-field approach and those which are subject to finite-size<br/>corrections, irrespective of the network size. The analysis of perfectly regular, asynchronous, states reveals that their stability<br/>depends crucially on the smoothness of both the phase-response curve and the transmitted <br/>post-synaptic pulse. Numerical simulations suggest that this scenario extends to<br/>systems that are not covered by the perturbative approach. <br/>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.