Connectivity Structure and Dynamics of Nonlinear Recurrent Neural Networks.

Studies of the dynamics of nonlinear recurrent neural networks often assume independent and identically distributed couplings, but large-scale connectomics data indicate that biological neural circuits exhibit markedly different connectivity properties. These include rapidly decaying singular-value spectra and structured singular-vector overlaps. Here, we develop a theory to analyze how these forms of structure shape high-dimensional collective activity in nonlinear recurrent neural networks. We first introduce the random-mode model, a random-matrix ensemble related to the singular-value decomposition that enables control over the spectrum and right-left mode overlaps. Then, using a novel path-integral calculation, we derive analytical expressions that reveal how connectivity structure affects features of collective dynamics: the dimension of activity, which quantifies the number of high-variance collective-activity fluctuations, and the temporal correlations that characterize the timescales of these fluctuations. We show that connectivity structure can be invisible in single-neuron activities while dramatically shaping collective activity. Furthermore, despite the nonlinear, high-dimensional nature of these networks, the dimension of activity depends on just two connectivity parameters-the variance of the couplings and the effective rank of the coupling matrix, which quantifies the number of dominant rank-one connectivity components. We contrast the effects of single-neuron heterogeneity and low dimensional connectivity, making predictions about how z-scoring data affects the dimension of activity. Finally, we demonstrate the presence of structured overlaps between left and right modes in the