Random matrix theory of sparse neuronal networks with heterogeneous timescales

TL;DR

This paper develops a random matrix theory framework to analyze the spectral properties of Jacobian matrices in trained sparse neuronal networks with heterogeneous timescales, linking network parameters to dynamical stability and working memory performance.

Contribution

It introduces a novel random matrix ensemble capturing the spectra of Jacobians in trained networks with heterogeneous timescales, using supersymmetry methods for analytic spectral density description.

Findings

Spectral edge relates network parameters to stability near equilibria.

Heterogeneous timescales and sparsity influence the spectral properties.

Analytic theory predicts conditions for robust working memory.

Abstract

Training recurrent neuronal networks consisting of excitatory (E) and inhibitory (I) units with additive noise for working memory computation slows and diversifies inhibitory timescales, leading to improved task performance that is attributed to emergent marginally stable equilibria [PNAS 122 (2025) e2316745122]. Yet the link between trained network characteristics and their roles in shaping desirable dynamical landscapes remains unexplored. Here, we investigate the Jacobian matrices describing the dynamics near these equilibria and show that they are sparse, non-Hermitian rectangular-block matrices modified by heterogeneous synaptic decay timescales and activation-function gains. We specify a random matrix ensemble that faithfully captures the spectra of trained Jacobian matrices, arising from the inhibitory core - excitatory periphery network motif (pruned E weights, broadly distributed I weights) observed post-training. An analytic theory of this ensemble is developed using statistical field theory methods: a Hermitized resolvent representation of the spectral density processed with a supersymmetry-based treatment in the style of Fyodorov and Mirlin. In this manner, an analytic description of the spectral edge is obtained, relating statistical parameters of the Jacobians (sparsity, weight variances, E/I ratio, and the distributions of timescales and gains) to near-critical features of the equilibria essential for robust working memory computation.