Analysis of the stabilized supralinear network

TL;DR

This paper investigates a neural network model with supralinear input-output functions, showing how feedback inhibition stabilizes the network and causes a transition from supralinear to sublinear response summation, explaining cortical nonlinearities.

Contribution

It introduces a dynamic stabilization mechanism in supralinear networks and analyzes conditions for response transitions and supersaturation in a simplified 2D model.

Findings

Dynamic stabilization prevents instability in strong inputs.

Transition from supralinear to sublinear summation explained.

Conditions for supersaturation derived in the 2D case.

Abstract

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.