The scale-invariant covariance spectrum of brain-wide activity
Quantitative analyses of the growing volume of large-scale neural data present both exciting challenges and opportunities, offering insight into the crucial roles of high-dimensional neural activity 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, the shape of this spectrum is invariant across scales: That is, the structure of correlations in a smaller and randomly sampled cell assembly is statistically similar to that of the entire brain. We propose that this non-trivial 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 functional space. Through a rigorous examination of the model with both numerical and analytical methods, we pinpoint three key factors responsible for the observed scale invariance in experiments: (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 invariance in zebrafish data, as well as two-photon and multi-area electrode recordings in mice. Furthermore, fitting the model to the experimental data uncovers the reorganization of a neural cluster in the functional space when the zebrafish is engaged in hunting behavior. Notably, the anatomical distribution of this cluster remains consistent between individuals. 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.