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  • Open Access

Connectivity Structure and Dynamics of Nonlinear Recurrent Neural Networks

David G. Clark1,2,*,‡, Owen Marschall1, Alexander van Meegen3,§, and Ashok Litwin-Kumar1,2,†

  • *Contact author: dgclark@fas.harvard.edu
  • †Contact author: a.litwin-kumar@columbia.edu
  • ‡Present address: Kempner Institute for the Study of Natural and Artificial Intelligence, Harvard University, Cambridge, Massachusetts 02138, USA.
  • §Present address: School of Life Sciences and School of Computer and Communication Sciences, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.
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Phys. Rev. X 15, 041019 – Published 3 November, 2025

DOI: https://doi.org/10.1103/2jt7-c8cq

Abstract

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 Drosophila connectome, incorporate them into the theory, and show how they further shape collective dynamics.

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Popular Summary

Understanding how patterns of connections in the brain shape neural activity is a major challenge in neuroscience. Brain networks can be mathematically broken down into input and output modes, and recent studies show that actual neural circuits are far from random, dominated by a small number of strong components. At the same time, new technologies allow us to record the activity of thousands of neurons at once, giving us a chance to directly link connectivity to collective neural dynamics. In this study, we show that the structure of connectivity can predict key features of network activity, including the number of independent activity patterns the network can generate (the activity dimension).

To achieve this, we develop a mathematical framework that models networks as collections of input and output modes with adjustable strengths. Using this approach, we derive how the connectivity structure determines the activity dimension and other collective properties. Analysis of our theory suggests that large-scale data can reveal relationships between network connectivity and the activity of thousands of neurons simultaneously.

Our results show that even when observing individual neurons, the underlying connectivity may appear hidden, but population-level recordings reveal its influence. In particular, just two measures—the typical connection strength and the number of dominant components—largely determine the activity dimension in uncorrelated modes. The framework also captures additional constraints when input-output modes are correlated, as seen in the fruit fly connectome.

This work provides a powerful tool to link detailed structural maps of neural circuits with their emergent activity patterns, offering a foundation for future studies on how connectivity shapes computation in the brain.

Article Text

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Phys. Rev. X 15, 041019– Published 3 November, 2025
  • Received 6 September 2024
  • Accepted 29 September 2025
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