  We study a rate-model neural network composed of excitatory and inhibitory
neurons in which neuronal input-output functions are power laws with a power
greater than 1, as observed in primary visual cortex. This supralinear
input-output function leads to supralinear summation of network responses to
multiple inputs for weak inputs. We show that for stronger inputs, which would
drive the excitatory subnetwork to instability, the network will dynamically
stabilize provided feedback inhibition is sufficiently strong. For a wide range
of network and stimulus parameters, this dynamic stabilization yields a
transition from supralinear to sublinear summation of network responses to
multiple inputs. We compare this to the dynamic stabilization in the "balanced
network", which yields only linear behavior. We more exhaustively analyze the
2-dimensional case of 1 excitatory and 1 inhibitory population. We show that in
this case dynamic stabilization will occur whenever the determinant of the
weight matrix is positive and the inhibitory time constant is sufficiently
small, and analyze the conditions for "supersaturation", or decrease of firing
rates with increasing stimulus contrast (which represents increasing input
firing rates). In work to be presented elsewhere, we have found that this
transition from supralinear to sublinear summation can explain a wide variety
of nonlinearities in cerebral cortical processing.
