Connectivity structure and dynamics of nonlinear recurrent neural networks

We develop a theory to analyze how structure in connectivity shapes the high-dimensional, internally generated activity of nonlinear recurrent neural networks. Using two complementary methods -- a path-integral calculation of fluctuations around the saddle point, and a recently introduced two-site cavity approach -- we derive analytic expressions that characterize important features of collective activity, including its dimensionality and temporal correlations. To model structure in the coupling matrices of real neural circuits, such as synaptic connectomes obtained through electron microscopy, we introduce the random-mode model, which parameterizes a coupling matrix using random input and output modes and a specified spectrum. This model enables systematic study of the effects of low-dimensional structure in connectivity on neural activity. These effects manifest in features of collective activity, that we calculate, and can be undetectable when analyzing only single-neuron activities. We derive a relation between the effective rank of the coupling matrix and the dimension of activity. By extending the random-mode model, we compare the effects of single-neuron heterogeneity and low-dimensional connectivity. We also investigate the impact of structured overlaps between input and output modes, a feature of biological coupling matrices. Our theory provides tools to relate neural-network architecture and collective dynamics in artificial and biological systems.