The spectrum of covariance matrices of randomly connected recurrent neuronal networks

A key question in theoretical neuroscience is the relation between connectivity structure and collective dynamics of a network of neurons. Here we study the connectivity-dynamics relation as reflected in the distribution of eigenvalues of the covariance matrix of the dynamic fluctuations of the neuronal activities, which is closely related to the network9s Principal Component Analysis (PCA) and the associated effective dimensionality. We consider the spontaneous fluctuations around a steady state in randomly connected recurrent network of spiking neurons. We derive an exact analytical expression for the covariance eigenvalue distribution in the large network limit. We show analytically that the distribution has a finitely supported smooth bulk spectrum, and exhibits an approximate power law tail for coupling matrices near the critical edge. Effects of adding connectivity motifs and extensions to EI networks are also discussed. Our results suggest that the covariance spectrum is a robust feature of population dynamics in recurrent neural circuits and provide a theoretical predictions for this spectrum in simple connectivity models that can be compared with experimental data.