The simplest problem in the collective dynamics of neural networks: Is synchrony stable?

TL;DR

This paper investigates the stability of synchrony in spiking neural networks, revealing complex multi-operator dynamics and providing methods to analyze stability, with findings that inhibitory interactions generally promote synchrony.

Contribution

It introduces novel methods to analyze the stability of synchrony in complex neural networks, accounting for multiple operators and network topology effects.

Findings

Inhibitory interactions stabilize synchrony regardless of network parameters.

Eigenvalue bounds offer insights but do not fully determine stability.

Topologically strongly connected networks exhibit stable synchrony under inhibition.

Abstract

For spiking neural networks we consider the stability problem of global synchrony, arguably the simplest non-trivial collective dynamics in such networks. We find that even this simplest dynamical problem -- local stability of synchrony -- is non-trivial to solve and requires novel methods for its solution. The dynamics in the vicinity of the synchronous state is determined by a multitude of linear operators, in contrast to a single stability matrix in conventional linear stability theory. This unusual property qualitatively depends on network topology and may be neglected for globally coupled homogeneous networks. For generic networks, however, the number of operators increases exponentially with the size of the network. We present methods to treat this multi-operator problem exactly. First, based on the Gershgorin and Perron-Frobenius theorems, we derive bounds on the eigenvalues that provide important information about the synchronization process but are not sufficient to establish the asymptotic stability or instability of the synchronous state. We then present a complete analysis of asymptotic stability for topologically strongly connected networks using simple graph-theoretical considerations. For inhibitory interactions between dissipative (leaky) oscillatory neurons the synchronous state is stable, independent of the parameters and the network connectivity. These results indicate that pulse-like interactions play a profound role in network dynamical systems, and in particular in the dynamics of biological synchronization, unless the coupling is homogeneous and all-to-all.