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

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 Class 1 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.

FD and EM acknowledge support by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska Curie grant agreement No. 642563. AR acknowledges a project grant from the Spanish ministry of Economics and Competitiveness, Grants No. BFU2012-33413 and MTM2015-71509. AR has been partially funded by the CERCA progam of the Generalitat de Catalunya. EM acknowledges the projects grants from the Spanish ministry of Economics and Competitiveness, Grants No. PSI2016-75688- P and No. PCIN-2015- 127. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.