Stationary covariance spectra of discrete-time non-normal random recurrent dynamics

Principal component analysis is widely used to characterize structure in the dynamics of recurrent neural networks. For stationary noise-driven dynamics, the distribution of variance among the principal components is determined by the spectrum of the stationary covariance matrix. While the spectral properties of this matrix are well-understood for linear networks with normal synaptic weight matrices, our understanding of the stationary covariance spectrum for random non-normal dynamics remains incomplete. In this note, we use a free-probability approach to formally derive a closed functional equation for the moment generating function of the limiting stationary covariance spectrum of discrete-time dynamics with random non-normal Gaussian weights. This characterization allows us to analyze the behavior of tail eigenvalues in the critical regime. In contrast, applying the same approach to the analogous continuous-time dynamics leads to an infinite hierarchy of Schwinger-Dyson equations, rather than a closed scalar equation. We conclude with some comments regarding the relevance of these results to comparisons of models of non-normal dynamics to neural data.