Stability of splay states in globally coupled rotators

TL;DR

This paper investigates the stability of splay states in globally coupled rotators, revealing how the Floquet exponents scale with system size and identifying conditions for stability and neutrality.

Contribution

It provides a detailed analysis of the spectral stability of splay states, including explicit results for leaky-integrate-and-fire neurons and continuous velocity fields.

Findings

Floquet exponents scale as 1/N^2 for short-wavelength modes

Sign of stability depends on the average derivative of the velocity field

In continuous velocity fields, exponents scale as 1/N^4 and neutral directions exist

Abstract

The stability of dynamical states characterized by a uniform firing rate ({\it splay states}) is analyzed in a network of $N$ globally pulse-coupled rotators (neurons) subject to a generic velocity field. In particular, we analyse short-wavelength modes that were known to be marginally stable in the infinite $N$ limit and show that the corresponding Floquet exponent scale as $1/N^2$. Moreover, we find that the sign, and thereby the stability, of this spectral component is determined by the sign of the average derivative of the velocity field. For leaky-integrate-and-fire neurons, an analytic expression for the whole spectrum is obtained. In the intermediate case of continuous velocity fields, the Floquet exponents scale faster than $1/N^2$ (namely, as $1/N^4$) and we even find strictly neutral directions in a wider class than the sinusoidal velocity fields considered by Watanabe and Strogatz in {\it Physica D 74 (1994) 197-253}.