Eigenvalues and dominant eigenvectors for locally-defined Gaussian connectivity with <i>homogeneous</i> reciprocal correlations.

(A) Eigenvalue spectra of excitatory-inhibitory connectivity matrices J, with homogeneous reciprocal correlations η. Different colours from top to bottom correspond to networks with different values of η. The dots in the elliptical bulk show 600 eigenvalues for one realization of the random connectivity. Outlying eigenvalues are shown for 30 realizations of the random connectivity, the dispersion reflects finite-size effects. The red arrow on the top points to the eigenvalue λ0 of the mean connectivity . Coloured circles are the eigenvalues predicted using determinant lemma and truncated series expansion (Eqs (7) and (8), truncating at k = 2), coloured triangles are the eigenvalues predicted using determinant lemma without finite truncation (see S6 Text, Eqs. (153)-(157)). (B) Comparison of the eigenvalues from the finite-size simulation with the predictions of the determinant lemma as the reciprocal correlation η is increased. The coloured solid lines show the roots of the third-order polynomial in Eq (8) (truncated series expansion). The light purple area indicates the empirical distribution of the dominant outlier for 30 realizations, reflecting finite-size effects, while the black dashed line is the unperturbed eigenvalue λ0. The grey areas represent the areas covered by the eigenvalue bulk. (C) Scatter plot showing for each neuron i its entry ni on the left eigenvector against its entry mi on the right eigenvector. Red and blue colours represent respectively excitatory and inhibitory neurons. The white dots and the dashed lines respectively indicate the means and covariances for each population. (D) Comparison between eigenvector entries obtained from direct eigen-decomposition of J with projections obtained using perturbation theory (Eqs (9) and (10)) in a given realization of Z. (E) Comparison between simulations (coloured areas, finite-size effects) and theoretical predictions (coloured lines, Eq (97)) for the population covariance of the entries on the left and right connectivity eigenvectors to different populations. (F) Comparison of the overall covariance σnm (Eq (72)) with the deviation Δλ of the dominant outlying eigenvalue from the unperturbed value λ0. Empirical covariance (gradient blue area reflects finite-size effects, where the colour depth represents η) compared with the theoretical prediction (black line) obtained using Eqs (97) and (92). The x-axis uses the theoretical prediction of the deviation of the eigenvalue λ from λ0. Other network parameters: JE = 2.0, JI = 1.2, NE = 4NI = 1200 and homogeneous variance parameters gpq = g = 0.3. (G-I) Same as (A, B, F) for an inhibition dominates connectivity matrix where JI = 2.0, JE = 1.2, with homogeneous reciprocity η and variance parameters g = 0.3. (J-M) Same as (A-C) and (E) for excitatory-and-inhibitory connectivity matrices with homogeneous reciprocal correlations η but cell-type-dependent variance parameters gEE : gEI : gIE : gII = 1.0 : 0.5 : 0.2 : 0.8 and gEE = 0.3. In (B, E, F, H, I, K, M), the departure of the centres of the numerical simulation results from the theoretical predictions reflect the systematic errors due to the first-order perturbation approximation of eigenvalues and eigenvectors.