Fig. 1.

Influence of perturbations in excitatory–inhibitory networks. (A) A reduced circuit model composed of two excitatory (E, red) and one inhibitory (I, blue) subpopulations. One E subpopulation (E1, Top) is perturbed and the influence of this perturbation on the activity of the other E subpopulation (E2, Bottom) is studied. (B) The direct effect of perturbation is highlighted via the monosynaptic connection from E1 to E2. (C) The effects of disynaptic interactions from E1 to E2 are highlighted with three possible motifs (E1→E2→E2, E1→E1→E2, and E1→I→E2). An example trisynaptic motif is shown on the Right, which has a disinhibitory effect overall. (D) The contributions of disynaptic motifs to the influence cancel each other out under a perfect balance of excitation and inhibition. This is obtained by $\alpha$ = 1 and $g=2$, with the latter compensating for the smaller size of inhibitory neurons compared to two E subpopulations. (E) Total influence of E1 perturbations on E2 activity in rate-base simulations (circles) for different values of $g$. Keeping $\alpha =1$ and increasing $g$ leads to divisive inhibition of the initial influence, but the influence remains positive (Left). Increasing $\alpha$, on the other hand, can lead to suppressive influence for higher values (Right). The prediction from the theory is plotted with solid lines in each case (J = 0.5). (F) Influence is evaluated in large-scale random networks of excitatory and inhibitory neurons. (G) The effect of perturbation of an E neuron (the influencer) on another E neuron (the influencee) in the network can be mediated by multiple pathways, including monosynaptic and higher-order motifs. The sign of the net influence at each branch is determined by considering the interaction of the signs of all synapses in the respective pathway. (H, Left) Distribution of influence between all pairs of excitatory neurons in the network, in the case of perfect balance ($\alpha =1$ and g = 1, NE = NI = 500, and J = 0.001). (H, Right) Average influence in the networks with different values of J. Given perfect balance, the average influence always matches with the average direct connection weight between E neurons, even for networks with unstable excitatory subnetworks (the border of instability is shown by the dashed line). (I) Average influence in networks with J = 0.002 and different values of g and $\alpha$. Similar behavior to E is observed, where increasing g leads to divisive inhibition and increasing $\alpha$ is needed to obtain suppressive influence. The prediction from the weight matrix of the network, and from the theory (SI Appendix, Methods and Eq. 80), match with the results of rate-based simulations.