Eigenvalue spectra of large correlated random matrices

TL;DR

This paper derives self-consistent equations for eigenvalue distributions of large correlated asymmetric random matrices, with implications for neural network stability and dynamics, confirmed by numerical simulations.

Contribution

It introduces a diagrammatic method to analyze eigenvalue spectra of correlated random matrices, extending previous models to include correlations and variance differences.

Findings

Correlations significantly affect network stability.

Analytical results match numerical simulations.

Eigenvalue spectra are influenced by matrix element correlations.

Abstract

Using the diagrammatic method, we derive a set of self-consistent equations that describe eigenvalue distributions of large correlated asymmetric random matrices. The matrix elements can have different variances and be correlated with each other. The analytical results are confirmed by numerical simulations. The results have implications for the dynamics of neural and other biological networks where plasticity induces correlations in the connection strengths within the network. We find that the presence of correlations can have a major impact on network stability.