Dynamics of stochastic integrate-and-fire networks

Gabriel Ocker
Department of Mathematics and Statistics
Boston University

We construct the joint density functional for membrane potentials and spike trains in networks of integrate-and-fire neurons with stochastic spike generation. Our construction identifies exact stochastic and approximate deterministic mean field theories for the activity. The deterministic mean field theory is a simple neural activity equation with a new nonlinearity: a rate-dependent leak, which approximates the spike-driven resets of neurons’ membrane potentials. We study the impact of this nonlinearity on the networks’ dynamics, uncovering bistability between quiescent and active states in homogenous and excitatory-inhibitory networks. We then show that the rate-dependent leak intrinsically stabilizes the dynamics of even excitatory-only networks, so that inhibition is not necessary to stabilize activity. A paradoxical reduction of inhibitory firing rates after stimulation does, however, occur in wide regions of parameter space. Finally, we discuss exact and perturbative descriptions of fluctuations in these networks.​