Self-consistent moment dynamics for networks of spiking neurons

A novel approach to moment closure problem is used to derive low dimensional laws for the dynamics of the moments of the membrane potential distribution in a population of spiking neurons. Using spectral expansion of the density equation we derive the recursive and nonlinear relation between the moments, such as the mean potential, and the population firing rates. The self-consistent dynamics found relies on the dominant eigenvalues of the evolution operator, tightly related to the moments of the single-neuron inter-spike interval distribution. Contrary to previous attempts our system can be applied both in noise- and drift-dominated regime, and both for weakly and strongly coupled population. We demonstrate the applicability of the theory for the case of a network of leaky integrate-and-fire neurons deriving closed analytical expressions. Truncating the mode decomposition to the first few more relevant moments, results to effectively describe the population dynamics both out-of-equilibrium and in response to strongly-varying inputs.