Stability of the splay state in pulse--coupled networks

TL;DR

This paper analyzes the stability of splay states in pulse-coupled neural networks, revealing how pulse-width and inter-spike interval ratios influence stability and identifying multiple stability components.

Contribution

It introduces a reduced map approach for large networks and characterizes the Floquet spectrum, highlighting conditions for stability and the impact of inhibitory coupling.

Findings

Stability depends on pulse-width to inter-spike interval ratio.

Floquet spectrum includes mean-field, finite-frequency, and isolated eigenvalues.

Splay states can be stable even with inhibitory coupling.

Abstract

The stability of the dynamical states characterized by a uniform firing rate ({\it splay states}) is analyzed in a network of globally coupled leaky integrate-and-fire neurons. This is done by reducing the set of differential equations to a map that is investigated in the limit of large network size. We show that the stability of the splay state depends crucially on the ratio between the pulse--width and the inter-spike interval. More precisely, the spectrum of Floquet exponents turns out to consist of three components: (i) one that coincides with the predictions of the mean-field analysis [Abbott-van Vreesvijk, 1993]; (ii) a component measuring the instability of "finite-frequency" modes; (iii) a number of "isolated" eigenvalues that are connected to the characteristics of the single pulse and may give rise to strong instabilities (the Floquet exponent being proportional to the network size). Finally, as a side result, we find that the splay state can be stable even for inhibitory coupling.