Rank-one approximation and predicted low-dimensional dynamics for sparse excitatory-inhibitory networks.

(A) Left: Comparison of the predicted eigenvalue outlier λ0 = c(NEAE + NIAI) (black line) with finite-size simulations (red area shows mean±std for 30 realizations and reflects finite-size effects). The gray area represents the area covered by the eigenvalue bulk. Right: example spectrum of one realization of connectivity matrix with E/I ratio AE/AI = 0.33, and the radius rg of the eigenvalue bulk computed from the statistically equivalent Gaussian connectivity (see S4 Text). (B) 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 dash lines respectively indicate the means and covariances for each population obtained from simulations. For visualization purposes, the x− and y-axis are scaled unequally. (C) 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 the sparse connectivity J. (D) Bifurcation diagram for increasing the ratio AE/AI: analytical predictions of Eq (117) compared with simulations of the full network with locally generated sparse connectivity. Purple line: analytical prediction including the first-order perturbation term in the rank-one approximation; gray: projection of simulated activity onto the connectivity vector m computed by perturbation theory Eq (104) for 30 realizations of sparse connectivty J, the shaded area reflects finite-size effects. (E) Comparison between the predictions (solid lines) and simulations (shaded areas) for the population-averaged variances of Δxi. Shaded areas show mean±std and reflect finite-size effects. (F) Comparison between simulations and mappings obtained from the low-rank approximation for the activity of individual units in a given realization of the sparse connectivity J. For each unit i, a dot shows the deviation Δxi of the steady-state activity from the population average, against the corresponding value Δmi of the perturbed part of the connectivity vector m (Eq (16)). The low-rank theory maps Δxi = κΔmi. Orange, cyan and gray scatters show excitatory or inhibitory populations, each for two values of the ratio AE/AI. Lines represent y = κx, where κ is obtained from Eqs (19) and (20). Upper panels show the result in a realization with a high fixed point, bottom panels show the result in a realization with a low fixed point. The gray vertical dashed line in A left, D, E correspond to the critical point c at which the absolute value of the outlier is equal to the radius of the eigenvalue bulk. In A, D, E the departure of the centres of the numerical simulation results from the theoretical predictions reflect the systematic errors due to the approximation of the low-rank connectivity statistics as well as the latent dynamical variable κ. Network parameters: NE = 4NI = 800, c = 0.3, AE = 0.025. The transfer function ϕ has parameter θ = 1.5.