How random connectivity shapes the fluctuating dynamics of finite-size neural populations

TL;DR

This paper develops a stochastic mean-field model for finite-size neural populations with random, non-full connectivity, revealing how disorder influences neural variability and stabilizes asynchronous states.

Contribution

It introduces a novel mesoscopic model that accounts for connectivity disorder and finite-size effects, improving upon previous theories that assumed homogeneous connectivity.

Findings

Model accurately predicts fluctuations and nonlinearities in neural activity.

Disorder in connectivity can stabilize asynchronous neural states.

Connectivity probability significantly affects population firing rate variability.

Abstract

Mesoscopic models of finite-size neuronal populations are crucial to understand the dynamics of neural networks in the brain, especially their fluctuations and response to stimuli. However, current theories to derive such models are based on homogeneous all-to-all (full) connectivity. This assumption neglects the variance in the connectivity of biologically realistic networks with connection probabilities $p<1$ (non-full connectivity). To gain insight into the different fluctuation mechanisms underlying neural variability at the population level, we derive and analyze a stochastic mean-field model for finite-size networks of Poisson neurons with random connectivity (including non-full connectivity), external noise and disordered mean inputs. We treat the quenched disorder of the connectivity by an annealed approximation enabling a doubly stochastic description of synaptic inputs for finite network size. A further reduction leads to a low-dimensional closed system of coupled Langevin equations for the mean and variance of the membrane potentials as well as a variable capturing finite-size fluctuations. Compared to microscopic simulations, the mesoscopic model describes the fluctuations and nonlinearities well and outperforms previous theories that neglected the variance in the connectivity. The joint effect of connectivity disorder and finite network size can be analytically understood by a softening of the effective nonlinearity and the multiplicative character of spiking noise. The mesoscopic theory shows that quenched disorder can stabilize the asynchronous state, and it correctly predicts large quantitative and non-trivial qualitative effects of connection probability on the variance of the population firing rate and its dependence on stimulus strength. In conclusion, our theory elucidates how disordered connectivity shapes nonlinear dynamics and fluctuations of neural populations.