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

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