The scale-invariant covariance spectrum of brain-wide activity

The structure of high-dimensional neural activity plays a pivotal role in various sensory and behavioral processes. Here, we analyze whole-brain calcium activity in larval zebrafish, captured by fast light-field volumetric imaging during hunting and spontaneous behavior. We find that brain-wide activity is distributed across many principal component dimensions described by the covariance spectrum. Intriguingly, this spectrum shows an invariance to spatial subsampling: That is, the distribution of the eigenvalues of a smaller and randomly sampled cell assembly is statistically similar to that of the entire brain. We propose that this property can be understood using a Euclidean random matrix model (ERM), where pairwise correlation between neurons can be mapped onto a distance function between two points in a low-dimensional functional space. We numerically and analytically calculate the eigenspectrum in our model and identify three key factors that lead to the experimentally observed scale invariance: (i) the slow decay of the distance-correlation function, (ii) the higher dimension of the functional space, and (iii) the heterogeneity of neural activity. Our theory can quantitatively recapitulate the scale-invariant spectrum in zebrafish data, as well as two-photon and multi-area electrode recordings in mice. Furthermore, fitting the model to the experimental data uncovers a reorganization of neurons in the functional space when the zebrafish is engaged in hunting behavior. Our results therefore provide new insights and interpretations of brain-wide neural activity and offer clues about circuit mechanisms for coordinating global neural activity patterns. ### Competing Interest Statement The authors have declared no competing interest.