The relationship between the geometric properties of the neural activity space and the size of neural assemblies.

A. Illustration of how dimensionality of neural activity (DPR) changes with the number of recorded neurons. B. The eigenvalues of the neural covariance matrix dictate the geometrical configuration of the neural activity space with being the distribution width along a principal axis. C. Examples of two neural populations with identical dimensionality (DPR = 25/11 ≈ 2.27) but different spatial configurations, as revealed by the eigenvalue spectrum (green: {λi} = {7, 7, 1}, blue: {λi} = {9, 3, 3}).

Whole-brain calcium imaging of zebrafish neural activity and the phenomenon of its scale-invariant covariance eigenspectrum.

A. Rapid light-field Ca2+ imaging system for whole brain neural activity in larval zebrafish. B. Inferred firing rate activity from the brain-wide calcium imaging. The ROIs are sorted by their weights in the first principal component (23). C. Procedure of calculating the covariance spectrum on the full and sampled neural activity matrices. D. Dimensionality (circles, average across 8 samplings (dots)), as a function of the sampling fraction. The curve is the predicted dimensionality using Eq. (5). E. Iteratively sampled covariance matrices. Neurons are sorted in each matrix to maximize values near the diagonal. F. The covariance spectra, i.e., eigenvalue vs. rank/N, for randomly sampled neurons of different sizes (colors). The gray dots represent the sorted variances Cii of all neurons. G to I. Same as F but from three models of covariance (see details in Methods): (G) a Wishart random matrix calculated from a random activity matrix of the same size as the experimental data; (H) replacing the eigenvectors by a random orthogonal set; (I) covariance generated from a randomly connected recurrent network.

ERM model of covariance and its eigenspectrum.

A. Schematic of the Euclidean Random Matrix (ERM) model, which reorganizes neurons (circles) from the anatomical space to the functional space (here d =2 is a two-dimensional box). The correlation between a pair of neurons decreases with their distance in the functional space according to a kernel function . This correlation is then scaled by neurons’ variance (circle size) to obtain the covariance Cij. B. An example ERM correlation matrix (i.e., when ). C. Spectrum (same as Fig. 2F) for the ERM correlation matrix in B. D. Visualizing the distribution of the same ERM eigenvalues in C by plotting the probability density function (pdf).

Three factors contributing to scale invariance.

A. Impact of µ and d (see text) on the scale invariance of ERM spectrum (same plots as Fig. 3C) with. The degree of scale invariance is quantified by the collapse index (CI), which essentially measures the area between different spectrum curves (upper right inset). For comparison, we fix the same coordinate range across panels hence some plots are cropped. B. Top: sampled correlation matrix spectrum in an example animal (fish 1). Bottom: Same as top but for the covariance matrix that incorporates heterogeneous (gray dots). C. The CI of the correlation matrix (filled squares) is found to be larger than that for the covariance matrix (opened squares) across different datasets: f1 to f6: six light-field zebrafish data (10 Hz per volume, this paper); fl: light-sheet zebrafish data (2 Hz per volume, (33)); mn: mouse Neuropixels data (downsampled to 10 Hz per volume); mp: mouse two-photon data, (3 Hz per volume, (23)).

The relationship between the functional and anatomical space and theoretical predictions.

A. Three sampling methods (A1) and RCCA (see text). When RCCA ≈ 0 (A2), the anatomical sampling (ASap) resembles the random sampling (RSap), and while when RCCA ≈ 1 (A4), ASap is similar to the functional sampling (FSap). B. Distribution of neurons in the functional space inferred by MDS. Each neuron is color-coded by its projection along the first canonical direction in the anatomical space (see text). Data based on fish 6, same for C to E. C. Similar to B. but plotting neurons in the anatomical space with color based on their projection along in the functional space (see text). D. Dimensionality (DPR) across sampling methods: average DPR under RSap (circles), average and individual brain region DPR under ASap (squares and dots), and DPR under FSap for the most correlated neuron cluster (triangles; Methods). Dashed and solid lines are theoretical predictions for DPR under RSap and FSap, respectively (Methods). E. The CI of correlation matrices under three sampling methods in 6 animals (colors). **p<0.01; ***p<0.001; one-sided paired t tests: RSap vs. ASap, p = 0.0010; RSap vs. FSap, p = 0.0004; ASap vs. FSap, p = 0.0014.

Table of notations.

Resources for additional experimental datasets

Modifications of the shape of near used in Fig. S7, Fig. S8 and Fig. S9.

Flat: when . Tangent: when follows a tangent line of the exact power law and have a same first-order derivative when . c is a constant. Tent: when follows a straight line while the slope is not the same as the tangent case. Parabola: when follows a quadratic function (ax2 +1 and have same first-order derivative). t pdf: mimic the smoothing treatment like the t distribution. All the constant parameters are set such that f (0) = 1.