Republished: Dynamics of Stochastic Integrate-and-Fire Networks

Abstract

The neural dynamics generating sensory, motor, and cognitive functions are commonly understood through field theories for neural population activity. Classic neural field theories are derived from highly simplified models of individual neurons, while biological neurons are highly complex cells. Integrate-and-fire models retain a key nonlinear feature of neuronal activity: Action potentials return the membrane potential to a nearly fixed reset value. This nonlinear reset of the membrane voltage after a spike is absent from classic neural field theories. Here, we develop a statistical field theory for networks of integrate-and-fire neurons with stochastic spike emission. This reveals a new mean-field theory for the activity in these networks, fluctuation corrections to the mean-field dynamics, and a mapping to a self-consistent renewal process. We use these to study the impact of the spike-driven reset of the membrane voltage on population activity. The spike reset gives rise to a multiplicative, rate-dependent leak term in the mean-field membrane voltage dynamics. This leads to bistability between quiescent and active states in the mean-field theory of homogeneous and excitatory-inhibitory pulse-coupled networks. We uncover two types of fluctuation correction to the mean-field theory, due to the nonlinear mapping from membrane voltage to spike emission and the nonlinear reset. These can have competing effects, promoting and suppressing activity, respectively. We then examine the roles of spike resets and recurrent inhibition in stabilizing network activity. We calculate the phase diagram for inhibitory stabilization and find that an inhibition-stabilized regime occurs in wide regions of parameter space, consistent with experimental reports of inhibitory stabilization in diverse brain regions. Fluctuations narrow the region of inhibitory stabilization, consistent with their role in suppressing activity through spike resets.