# Firing rate equations require a spike synchrony mechanism to correctly describe fast oscillations in inhibitory networks

## TL;DR

This paper demonstrates that standard firing rate models lack the ability to produce gamma oscillations in inhibitory networks without explicit delays, but incorporating a spike-synchrony mechanism derived from neuron dynamics can accurately capture these oscillations.

## Contribution

The authors develop an extended mean-field model that includes a spike-synchrony mechanism, improving the description of gamma oscillations in inhibitory neural networks.

## Key findings

- Standard Wilson-Cowan equations do not produce gamma oscillations without delays.
- Inclusion of a spike-synchrony mechanism allows the model to generate gamma oscillations.
- The extended model accurately captures synchronous states even with slow synaptic kinetics.

## Abstract

Recurrently coupled networks of inhibitory neurons robustly generate oscillations in the gamma band. Nonetheless, the corresponding Wilson-Cowan type firing rate equation for such an inhibitory population does not generate such oscillations without an explicit time delay. We show that this discrepancy is due to a voltage-dependent spike-synchronization mechanism inherent in networks of spiking neurons which is not captured by standard firing rate equations. Here we investigate an exact low-dimensional description for a network of heterogeneous canonical type-I inhibitory neurons which includes the sub-threshold dynamics crucial for generating synchronous states. In the limit of slow synaptic kinetics the spike-synchrony mechanism is suppressed and the standard Wilson-Cowan equations are formally recovered as long as external inputs are also slow. However, even in this limit synchronous spiking can be elicited by inputs which fluctuate on a time-scale of the membrane time-constant of the neurons. Our meanfield equations therefore represent an extension of the standard Wilson-Cowan equations in which spike synchrony is also correctly described.