[Paper Review] Characterizing variability in nonlinear recurrent neuronal networks

Guillaume Hennequin, Máté Lengyel|arXiv (Cornell University)|Oct 10, 2016

Neural dynamics and brain function15 references3 citations

TL;DR

This paper presents a semi-analytical framework to characterize variability in nonlinear, recurrent neuronal networks by deriving deterministic equations for the mean firing rates and noise covariance matrix under the assumption that membrane potentials are jointly Gaussian. The method accurately predicts Fano factors, pairwise correlations, and cross-correlograms in both weakly and strongly connected networks—even under strong amplification and non-Gaussian membrane potential distributions—without requiring small fluctuations.

ABSTRACT

In this note, we develop semi-analytical techniques to obtain the full correlational structure of a stochastic network of nonlinear neurons described by rate variables. Under the assumption that pairs of membrane potentials are jointly Gaussian -- which they tend to be in large networks -- we obtain deterministic equations for the temporal evolution of the mean firing rates and the noise covariance matrix that can be solved straightforwardly given the network connectivity. We also obtain spike count statistics such as Fano factors and pairwise correlations, assuming doubly-stochastic action potential firing. Importantly, our theory does not require fluctuations to be small, and works for several biologically motivated, convex single-neuron nonlinearities.

Motivation & Objective

• To develop a tractable theoretical framework for characterizing across-trial variability in nonlinear, recurrent neuronal networks with rate-based dynamics.
• To address the challenge of modeling correlated variability in networks with strong recurrent connectivity and non-normal membrane potential distributions.
• To extend existing mean-field approaches by incorporating full second-order statistics of firing rates and spike counts under nonlinear, non-Gaussian dynamics.
• To validate the theory in biologically relevant regimes, including inhibition-stabilized networks with strong noise amplification.
• To enable accurate prediction of Fano factors, pairwise correlations, and cross-correlograms without requiring small noise approximations.

Proposed method

• Derives stochastic differential equations for membrane potential dynamics with nonlinear rate transformations, assuming continuous-time, multiplicative noise.
• Applies assumed-density filtering to derive deterministic equations for the first and second moments of membrane potentials and firing rates.
• Uses the Gaussian assumption for joint membrane potential distributions to close the moment hierarchy, enabling analytical tractability.
• Introduces a doubly-stochastic Poisson model for spike emission to compute spike count statistics such as Fano factors and pairwise correlations.
• Solves the resulting system of equations numerically to predict mean firing rates, variances, and covariances across different network architectures.
• Validates predictions against long-time simulations of stochastic network dynamics under varying connectivity and noise regimes.

Experimental results

Research questions

• RQ1 Can a semi-analytical theory accurately predict the full correlational structure of firing rates in nonlinear, recurrent neuronal networks without assuming small fluctuations?
• RQ2 How well does the Gaussian assumption for membrane potential pairs hold in strongly connected, inhibition-stabilized networks with non-equilibrium dynamics?
• RQ3 To what extent can the theory predict Fano factors and pairwise spike count correlations in physiologically realistic, nonlinear networks?
• RQ4 Does the framework remain accurate when membrane potential distributions are skewed due to strong noise amplification or non-normal dynamics?
• RQ5 Can the theory capture complex temporal structures in membrane potential cross-correlograms in non-equilibrium networks?

Key findings

• The theory accurately predicts mean firing rates and noise covariance matrices in both weakly connected and strongly connected, inhibition-stabilized random networks.
• Despite non-Gaussian membrane potential distributions with significant skewness in the strongly connected regime, the theory yields reasonably accurate estimates of second-order moments.
• The model successfully captures the structure of membrane potential cross-correlograms, including non-symmetric, non-equilibrium dynamics in strongly connected networks.
• Fano factors and pairwise spike count correlations are accurately predicted across arbitrary time windows using the doubly stochastic spike emission model.
• The framework remains valid in regimes of large fluctuations and strong noise amplification, where traditional linearized or small-noise approximations fail.
• Numerical validation shows excellent agreement between theoretical predictions and simulation results for all key statistics, including in biologically relevant parameter regimes.