Dynamic Mean Field Theories for Nonlinear Noise in Recurrent Neuronal Networks

TL;DR

This work addresses the challenge of nonlinear noise in recurrent neural networks by introducing the Gaussian Equivalent Method (GEM), which replaces nonlinear functions of Ornstein–Uhlenbeck noise with Gaussian equivalents matched in mean and covariance, and pairing this with a lognormal moment closure to produce a closed dynamic mean-field theory for firing rates. The resulting DMFT captures order-one transients, fixed points, and noise-induced shifts of bifurcation structure, outperforming standard linearization in strongly fluctuating regimes. It provides a tractable framework for noise-dependent phase diagrams in computational neuroscience and generalizes to dynamics that depend smoothly on OU processes. The approach offers a principled route to analyze how nonlinear noise reshapes neural dynamics and bifurcations, with potential extensions to spatially extended systems and colored noise.

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.

Figures (9)

• Figure 1: Realizations of the E/I network, and general schematic of the system. The parameters here are $\vec{W} = \{0.05, -0.75, 4, -3.5 \}$, $\mu_E= 5$, $\mu_I=9$, $\tau_E=6$, $\tau_I=20$, $\tau_N=1$, $\sigma_E=\sigma_I=1.0$, and correlation $\rho=0.5$
• Figure 1: Left, a visualization of moment closure schemes for systems with expansive nonlinearities. Distributions (stage 1) are summed into inputs (stage 2) to an expansive nonlinearity, and pass through that nonlinearity into outputs (stage 3), which do or do not resemble the distributions in stage 1. Normal distributions sent through expansive nonlinearities develop tails; lognormal distributions had tails to begin with. Right: Root mean squared fractional error in all third moments, under normal vs lognormal moment closure.
• Figure 1: The location of the Hopf bifurcation as a function of inputs. $\vec{W}=\{2, -2.75, 4, -3.5\}$, $\mu_E=2$, $\mu_I=1.52$, $\tau_E=6$, $\tau_I=20$, $\tau_N=5$, $\rho=0.5$. Original bifurcation figures were generated in XPPAUTO AUTO, and regenerated nicely in PyPlot.
• Figure 1: The mean firing rate (left) and variance (right) of the logistic transfer function system. We compare here the underlying simulated system, the mean field theory generated by linearization around a fixed point, and the mean field theory associated with the Gaussian equivalent method, up to third order in the Taylor expansion of the transfer function. The parameters are $W=1, \tau=1, \tau_N=0.5, \sigma=1.75, \mu=-0.9$, variance is 0.032 and mean is 0.42
• Figure 1: The loss ratio (Kullback-Leibler divergence divided by the entropy of the unknown distribution) for a variety of sample sizes -- a measurement of the information lost by approximation relative to the total amount of information contained in the original distribution. We pulled a sample of given size from a standard normal distribution, and raised it to the $M$. The mean and variance of the resulting set were used to generate a normal distribution. Samples drawn from this were compared to the underlying distribution. Sample size, and thus histogram bin size, has minimal impact on the result.