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Impact of heterogeneous activity levels on the scale invariance. (A) The CI as a function of the heterogeneity of neural activity levels E(σ4i) . We generate Euclidean Random Matrix (ERM) where each neuron’s activity variance σ2i is i.i.d. sampled from a log-normal distribution where the logarithm of the variable follows a normal distribution with zero mean and a sequence of standard deviation (0,0.05,0.1,…,0.5 ) in the natural logarithm of the values σ2i . We also normalize E(σ2i)=1 (Methods). The solid blue line is the average across 100 ERM simulations, and the shaded area represents the SD. The red line results from the Gaussian variational method with simulation value integration limit qss . The green line is the result of the Gaussian variational method with high-density value integration limit qhs (Methods). ρ0=128 . (B) Same as A, but with a smaller ρ0=10.24 . Other parameters: μ=0.5 , d=2 , N=1024 , L=(N/ρ)1/d , ϵ=0.03125 . (C) The collapse index (CI) of the correlation matrix (filled symbols) is larger than that of the covariance matrix (opened symbols) across different datasets excluding those shown in Figure 4. We use 7200 time frame data across all the datasets. l2 to l3: light-sheet zebrafish data (2 Hz per volume); n2 to n3: Neuropixels mouse data, downsampled to 10 Hz per volume, p2 to p3: two-photon mouse data (3 Hz per volume). |