Metastable spiking networks in the replica-mean-field limit

TL;DR

This paper introduces an extension of the replica-mean-field framework to analyze metastable dynamics in finite-size neural networks with mixed excitation and inhibition, providing a tractable way to understand neural variability and phase transitions.

Contribution

It extends the RMF approach to include mixed excitation and inhibition in neural models, characterizing metastability through stationary firing rates and delayed differential equations.

Findings

Metastable finite-size networks have multistable RMF limits characterized by stationary firing rates.

Stationary rates are solutions to delayed differential equations under regularity conditions.

Rates define probabilistic pseudo-equilibria that match neural variability in finite networks.

Abstract

Characterizing metastable neural dynamics in finite-size spiking networks remains a daunting challenge. We propose to address this challenge in the recently introduced replica-mean-field (RMF) limit. In this limit, networks are made of infinitely many replicas of the finite network of interest, but with randomized interactions across replicas. Such randomization renders certain excitatory networks fully tractable at the cost of neglecting activity correlations, but with explicit dependence on the finite size of the neural constituents. However, metastable dynamics typically unfold in networks with mixed inhibition and excitation. Here, we extend the RMF computational framework to point-process-based neural network models with exponential stochastic intensities, allowing for mixed excitation and inhibition. Within this setting, we show that metastable finite-size networks admit multistable RMF limits, which are fully characterized by stationary firing rates. Technically, these stationary rates are determined as the solutions of a set of delayed differential equations under certain regularity conditions that any physical solutions shall satisfy. We solve this original problem by combining the resolvent formalism and singular-perturbation theory. Importantly, we find that these rates specify probabilistic pseudo-equilibria which accurately capture the neural variability observed in the original finite-size network. We also discuss the emergence of metastability as a stochastic bifurcation, which can be interpreted as a static phase transition in the RMF limits. In turn, we expect to leverage the static picture of RMF limits to infer purely dynamical features of metastable finite-size networks, such as the transition rates between pseudo-equilibria.