Solvable macroscopic models for populations of type I excitability neurons with Gaussian noise

Abstract

The dynamics of type I excitability neurons near the bifurcation point are governed by the model of quadratic integrate-and-fire neuron (QIF). The model of QIF subject to white Gaussian noise is abundant in statistical physics, since it not only reasonably reproduces the dynamics of some biological systems but also emerges in the problems of stochastic stability and in quantum mechanics of one-dimensional systems with quenched disorder. We derive the solution for the mean interspike interval of a QIF subject to additive Gaussian noise in terms of analytic functions, from the first passage time problem. On the basis of this solution, we construct analytical expressions for the firing rate and population-mean membrane potential in terms of the real and imaginary parts of a certain universal analytic function. With this function, we generalize theoretical results to heterogeneous populations with Lorentzian-distributed parameters. Implementation of theoretical results is demonstrated and examined for two problems: the interplay between chemical and electrical synapses, and the emergence of effective endogenous noise in sparse inhibitory networks. For the latter case, it is possible to cover the scenarios where the effective noise intensity depends on Lorentzian-distributed parameters, which is not feasible directly with the diffusion approximation.