Dynamic Mean Field Theories for Nonlinear Noise in Recurrent Neuronal Networks

Abstract: Strong, correlated noise in recurrent neural circuits often passes through nonlinear transfer functions, complicating dynamical mean-field analyses of complex phenomena such as transients and bifurcations. We introduce a method that replaces nonlinear functions of Ornstein-Uhlenbeck (OU) noise with a Gaussian-equivalent process matched in mean and covariance, and combine this with a lognormal moment closure for expansive nonlinearities to derive a closed dynamical mean-field theory for recurrent neuronal networks. The resulting theory captures order-one transients, fixed points, and noise-induced shifts of bifurcation structure, and outperforms standard linearization-based approximations in the strong-fluctuation regime. More broadly, the approach applies whenever dynamics depend smoothly on OU processes via nonlinear transformations, offering a tractable route to noise-dependent phase diagrams in computational neuroscience models.