🤖 AI Summary
How do structural features of connectivity—such as rapidly decaying singular value spectra and structured singular vector overlaps—regulate high-dimensional collective dynamics in nonlinear recurrent neural networks?
Method: We introduce a random modal model that unifies the effects of input/output mode overlap, single-neuron heterogeneity, and low-rank connectivity structure. Employing path-integral saddle-point analysis, two-node cavity methods, and random matrix theory, we derive analytical expressions for both the dimensionality of neural activity and its temporal correlation spectrum.
Contribution/Results: We establish, for the first time, a quantitative relationship between the effective rank of the coupling matrix and the intrinsic dimensionality of neural activity. Crucially, we find that low-dimensional connectivity manifests exclusively at the population level—individual neuron statistics remain indistinguishable from high-dimensional controls. Our framework yields testable dynamical predictions for connectomes reconstructed via electron microscopy (EM), bridging structural anatomy and functional dynamics.
📝 Abstract
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.