Random matrix theory of sparse neuronal networks with heterogeneous timescales

2025-12-14

AI Summary

How do sparse excitatory-inhibitory (E-I) neural networks spontaneously exhibit critical dynamics during working memory tasks, and what spectral properties of their non-Hermitian Jacobian matrices at fixed points underlie this phenomenon? Method: We construct, for the first time, a sparse non-Hermitian random matrix ensemble that faithfully incorporates synaptic timescale heterogeneity and gain variability of neuronal activation functions. Leveraging statistical field theory, supersymmetry techniques, and the Hermitized Green’s function formalism, we derive closed-form analytical expressions for the spectral edge. Contribution/Results: Our analysis quantitatively uncovers how sparsity, E/I balance, synaptic weight variance, and timescale distributions jointly regulate criticality at network fixed points and thereby enable robust working memory computation. This work provides the first analytically tractable theoretical framework for understanding dynamic criticality in biologically realistic neural circuits.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Social Networks and Social Media: Computational social science
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Modeling sparse non-Hermitian Jacobian matrices in trained neuronal networks
• Relating network statistical parameters to spectral edge and near-critical dynamics
• Explaining how inhibitory heterogeneity shapes robust working memory computation

Innovation

• Sparse non-Hermitian Jacobian matrices with heterogeneous timescales
• Random matrix ensemble modeling inhibitory-excitatory network motif
• Analytic spectral edge theory via supersymmetry-based field methods