The spectrum of covariance matrices of randomly connected recurrent neuronal networks with linear dynamics

A key question in theoretical neuroscience is the relation between the connectivity structure and the 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 network dynamics’ Principal Component Analysis (PCA) and the associated effective dimensionality. We consider the spontaneous fluctuations around a steady state in a randomly connected recurrent network of stochastic neurons. An exact analytical expression for the covariance eigenvalue distribution in the large-network limit can be obtained using results from random matrices. The distribution has a finitely supported smooth bulk spectrum and exhibits an approximate power-law tail for coupling matrices near the critical edge. We generalize the results to include second-order connectivity motifs and discuss extensions to excitatory-inhibitory networks. The theoretical results are compared with those from finite-size networks and the effects of temporal and spatial sampling are studied. Preliminary application to whole-brain imaging data is presented. Using simple connectivity models, our work provides theoretical predictions for the covariance spectrum, a fundamental property of recurrent neuronal dynamics, that can be compared with experimental data.Author summary: Here we study the distribution of eigenvalues, or spectrum, of the neuron-to-neuron covariance matrix in recurrently connected neuronal networks. The covariance spectrum is an important global feature of neuron population dynamics that requires simultaneous recordings of neurons. The spectrum is essential to the widely used Principal Component Analysis (PCA) and generalizes the dimensionality measure of population dynamics. We use a simple model to emulate the complex connections between neurons, where all pairs of neurons interact linearly at a strength specified randomly and independently. We derive a closed-form expression of the covariance spectrum, revealing an interesting long tail of large eigenvalues following a power law as the connection strength increases. To incorporate connectivity features important to biological neural circuits, we generalize the result to networks with an additional low-rank connectivity component that could come from learning and networks consisting of sparsely connected excitatory and inhibitory neurons. To facilitate comparing the theoretical results to experimental data, we derive the precise modifications needed to account for the effect of limited time samples and having unobserved neurons. Preliminary applications to large-scale calcium imaging data suggest our model can well capture the high dimensional population activity of neurons.