Eigenvalues and dominant eigenvectors for locally-defined Gaussian connectivity with independent synaptic weights.

(A) Eigenvalue spectra of excitatory-inhibitory connectivity matrices J with elements generated from Gaussian distributions with identical variances g2/N over neurons. The coloured dots in the circular bulk shows 600 eigenvalues for one realization of the random connectivity for each value of g. Different colours correspond to different values of g. Dashed envelopes indicate the theoretical predictions for the radius rg of the circular bulk computed according to Eqs. (147), (148). Outlying eigenvalues are shown for 30 realizations of the random connectivity, and for different g their location on the y-axis is shifted to help visualization, the dispersion reflects finite-size effects. The red arrow points to the eigenvalue λ0 of the mean connectivity matrix . (B) Statistics of outlying eigenvalues over realizations of random connectivity. Empirical distribution (the red area shows mean ± standard deviation and reflects finite-size effects), compared with the theoretical predictions for the mean (black dashed line) and standard deviation (gray dashed line) obtained using Eq (80). (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 obtained from simulations. (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 (full lines) and theory (dashed lines) for the variances , of eigenvector entries corresponding to different populations (Eq (90)). To help visualization, the curves for the excitatory population are thicker than the curves for the inhibitory population. (F-J) Identical quantities for connectivity matrices in which the variance parameters are heterogeneous: gEE : gEI : gIE : gII = 1.0 : 0.5 : 0.2 : 0.8, gEE increases from 0 to 1. Other network parameters NE = 4NI = 1200 and JE = 2.0, JI = 0.6 in all simulations.