Summary
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.
Footnotes
We would like to take this opportunity to clarify several key points about our work. 1. Subsampling the covariance matrix does not necessarily lead to the collapse of the eigenspectra, as demonstrated by new figures and other models. 2. Power-law-distributed eigenvalues alone are not sufficient for spectral collapse under subsampling. 3. Our choice of a power-law correlation function is for mathematical convenience, and we have shown that collapse can occur for nonpower-law slow decaying functions. 4. The power law is an approximation for part of the spectrum, and the main message of our work emphasizes the slow decay of spatial correlation and heterogeneity of neural activity.