Generalized paradoxical effects in excitatory/inhibitory networks

An inhibition-stabilized network (ISN) is a network of excitatory and inhibitory cells at a stable fixed point of firing rates for a given input, for which the excitatory subnetwork would be unstable if inhibitory rates were frozen at their fixed point values. It has been shown that in a low-dimensional model (one unit per neuronal subtype) of an ISN with a single excitatory and single inhibitory cell type, the inhibitory unit shows a "paradoxical" response, lowering (raising) its steady-state firing rate in response to addition to it of excitatory (inhibitory) input. This has been generalized to an ISN with multiple inhibitory cell types: if input is given only to inhibitory cells, the steady-state inhibition received by excitatory cells changes paradoxically, that is, it decreases (increases) if the steady-state excitatory firing rates decrease (increase). We generalize these analyses of paradoxical effects to low-dimensional networks with multiple cell types of both excitatory and inhibitory neurons. The analysis depends on the connectivity matrix of the network linearized about a given fixed point, and its eigenvectors or "modes". We show the following: (1) A given cell type shows a paradoxical change in steady-state rate in response to input it receives, if and only if the network with that cell type omitted has an odd number of unstable modes. Excitatory neurons can show paradoxical responses when there are two or more inhibitory subtypes. (2) More generally, if the cell types are divided into two nonoverlapping subsets A and B, then subset B has an odd (even) number of modes that show paradoxical response if and only if subset A has an odd (even) number of unstable modes. (3) The net steady-state inhibition received by any unstable mode of the excitatory subnetwork will change paradoxically, i.e. in the same direction as the change in amplitude of that mode. In particular, this means that a sufficient condition to determine that a network is an ISN is if, in response to an input only to inhibitory cells, the firing rates of and inhibition received by all excitatory cell types all change in the same direction. This in turn will be true if all E cells and all inhibitory cell types that connect to E cells change their firing rates in the same direction.