Synchronization patterns in inhibitory firing rate models
Population models of neuronal activity have become an standard tool of analysis in computational neuroscience. Rather than focus on the microscopic dynamics of neurons, these models describe the collective properties of large numbers of neurons, typically in terms of the mean firing rate of a neuronal ensemble. In general, such population models, often called firing rate equations (or Wilson-Cowan equations), are obtained using heuristic mean-field arguments. Despite their heuristic nature, Heuristic-Firing Rate Equations (H-FRE) often show remarkable qualitative agreement with the dynamics in equivalent networks of spiking neurons. However, this agreement breaks down once a significant fraction of the neurons in the population fires spikes synchronously. As a case in point, I will focus on partially synchronized states in networks of heterogeneous inhibitory neurons. There networks are able to generate robust macroscopic oscillations, while the corresponding H-FRE for such an inhibitory population does not generate such oscillations without an explicit time delay. I will show that this discrepancy is due to a voltage-dependent spike-synchronization mechanism inherent in networks of spiking neurons. Specifically, I will present a recent theory to derive the exact FRE for a population of Quadratic Integrate and Fire (QIF) neurons that 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 QIF-FRE formally reduce to the standard Wilson-Cowan equations, as long as external inputs are also slow. However, even in the limit of slow synapses, synchronous spiking can be elicited by inputs which fluctuate on a time-scale of the membrane time-constant of the neurons. The QIF-FRE therefore represent an extension of the standard Wilson-Cowan equations in which spike synchrony is also correctly described.