Power Laws in Empirical Eigenvalue Spectra

The critical brain hypothesis proposes that neural systems operate near a phase transition to optimize information processing. A key method for investigating this hypothesis is the phenomenological renormalization group (pRG), which looks for scale-invariant features across levels of coarse-graining. One such feature is the power-law scaling of eigenvalues of covariance matrices of coarse-grained variables. However, the estimation of this scaling exponent, $\mu$, often relies on linear regression over arbitrarily selected ranges of the plot of eigenvalues versus rank. This heuristic “eyeballing” introduces uncontrolled bias and complicates the interpretation of observed scaling relationships. In order to obtain a more robust estimation of $\mu$, we do not fit the standard eigenvalue-vs-rank relationship, but rather the density of eigenvalues, for which standard protocols exist for fitting power laws to empirical data distributions. We demonstrate this approach using a synthetic model that replicates the scaling signatures of neural data while providing control over the system’s exponents as well as neural data obtained from publicly available Neuropixels recordings. We also establish standards for the minimal data required to quantify power-law behavior in a pRG eigenvalue analysis. Our approach contributes a tool for understanding the fundamental limitations imposed by spatial and temporal constraints of experimental datasets, which is required to rigorously assess the neural criticality hypothesis.