Covariance Spectrum in Nonlinear Recurrent Neural Networks: A Near-Critical Effective Connection Strength Throughout Chaos

Abstract

Advances in simultaneous recordings of large numbers of neurons have driven significant interest in the structure of neural population activity, such as dimensionality, and in how it arises mechanistically from network connectivity. The covariance eigenvalue distribution, or spectrum, which can be derived analytically in random recurrent networks, was proposed as a robust description of population geometry beyond dimensionality. The theoretical spectrum has since matched data across brain areas and species, serving as a computational biomarker linking neural population activity to network connectivity. However, this empirical success highlights a gap in theory, as the model used to derive the spectrum was minimal with linear neurons. Here we close this gap by studying the covariance spectrum in networks with nonlinear neurons and across broader dynamical regimes, including chaos. Surprisingly, the spectrum is precisely described by equations analogous to the linear theory, substituted with an effective recurrent connection strength, which reflects both the connection weights and the dynamic gain of neurons. Across regimes, this effective parameter gives a unified account of the spectrum and dimensionality changes, and stays near its critical value throughout the chaotic regime, suggesting a mechanism for brain criticality without fine-tuning and a new interpretation for previously observed near-critical spectra. Our key step is proposing a covariance matrix ansatz under nonlinear dynamics, which explicitly links neuronal dynamics, recurrent connectivity, and all pairwise correlations in the network. The ansatz is supported by extensive numerical and theoretical validations, including extensions to other network models. These results advance the theory of nonlinear population dynamics and strength the covariance spectrum analysis a practical tool for biological circuits.