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The spectrum of covariance matrices of randomly connected recurrent neuronal networks with linear dynamics

Fig 5

Robustness of the covariance spectrum to low rank perturbations of the connectivity.

A. Eigenvalues of a Gaussian random connectivity J (Eq (3)), g = 0.4, N = 400. As N → ∞, the limiting distribution of eigenvalues is uniform in the circle with radius g ([34] red solid line). The black dashed line is the 0.995 quantile of the eigenvalue radius calculated from 1000 realizations. B. Same as A. but for the rank-1 perturbed J + xuv^T. , and x = 4.03. This example also corresponds to adding diverging motifs (Section 3.3 and Section C.1 in S1 Text). C. The histogram of covariance eigenvalues (Eq (2)) under the J in A. D. The bulk histogram of eigenvalues with J + xuv^T has little change and remains well described by the Gaussian connectivity theory (red line, Eq (5)). Besides the bulk, there are two outlier eigenvalues to the left and right (inset, arrows) E,F, Analytical predictions (solid and dashed lines) of the outlier locations given g and |x| when u, v are (asymptotically) orthogonal unit vectors that are independent of J (see other cases in Section C.3 in S1 Text). The y-axis is the outlier location subtracting the corresponding edge x_±, Eq (6), so it is zero for small |x| before the outlier emerges (dashed line). The dots are the mean of the smallest (for the left outlier) or largest (right outlier) eigenvalues averaged across 100 realizations of the random J, N = 4000. The errorbars are the standard error of the mean (SEM, many are smaller than the dots).

doi: https://doi.org/10.1371/journal.pcbi.1010327.g005