Geometric Analysis of Synchronization in Neuronal Networks with Global Inhibition and Coupling Delays

TL;DR

This paper analyzes how coupling delays influence synchronization in neuronal networks with excitatory and inhibitory populations, providing analytical conditions and validating findings through simulations, with implications for sleep rhythm models.

Contribution

It offers a geometric singular perturbation analysis revealing how delays promote synchronization in coupled neuronal networks, extending understanding of neural rhythm generation.

Findings

Coupling delays enhance network synchronization.

Analytical conditions for stable synchronization states.

Validation through numerical simulations and application to sleep rhythms.

Abstract

We study synaptically coupled neuronal networks to identify the role of coupling delays in network's synchronized behaviors. We consider a network of excitable, relaxation oscillator neurons where two distinct populations, one excitatory and one inhibitory, are coupled and interact with each other. The excitatory population is uncoupled, while the inhibitory population is tightly coupled. A geometric singular perturbation analysis yields existence and stability conditions for synchronization states under different firing patterns between the two populations, along with formulas for the periods of such synchronous solutions. Our results demonstrate that the presence of coupling delays in the network promotes synchronization. Numerical simulations are conducted to supplement and validate analytical results. We show the results carry over to a model for spindle sleep rhythms in thalamocortical networks, one of the biological systems which motivated our study. The analysis helps to explain how coupling delays in either excitatory or inhibitory synapses contribute to producing synchronized rhythms.