Mean-field models of neural populations with Gaussian noise and non-Cauchy heterogeneities

We derive mean-field equations for a large population of globally coupled quadratic integrate-and-fire neurons subject to weak Gaussian noise and heterogeneous time-independent non-Cauchy distributed currents. We employ a circular cumulant approach that has previously been used only in the context of Cauchy heterogeneity, to the best of our knowledge. We extend this approach to rational distribution functions, which, unlike the Cauchy function, can have many poles in the complex plane. Population dynamics are analyzed for two families of rational functions that are approximations of the uniform and normal distributions. It is found that Gaussian noise and all considered types of heterogeneities have qualitatively the same effect on the population dynamics. This differs from the case of Cauchy noise, which has been shown to have the same effect on population dynamics only in the case of Cauchy heterogeneity, while Cauchy noise and non-Cauchy heterogeneity can have qualitatively different effects. The mean-field equations for both families of distribution functions are validated through a comparison of their solutions with the "exact" solutions of the Fokker-Planck equation, as well as with the results of modeling the stochastic microscopic dynamics of finite-size populations.