Stability of a class of directionally sensitive asymmetric nonlinear neural networks

Abstract

Asymmetrically connected inhibitory shunting networks are believed to occur in many areas of the brain. One such area is the visual system; these networks are found useful in explaining many peripheral visual phenomena. This paper examines a class of asymmetrically connected networks for stability and uniqueness of outputs; both time-invariant and moving inputs are considered. Under different sets of constaints upon the network characteristics and operating region, it is possible to obtain: • • bounded-input-bounded-output stability of the solutions; • • stability and uniqueness of the output to any input for a network of finite extent, with time-invariant inputs giving time-invariant outputs; • • localization of edge effects on a finite network, and stability and uniqueness of the output to any input for a network of infinite extent, with time-invariant or traveling-wave inputs giving time-invariant or traveling-wave outputs respectively. The constraints required for these three sets of characteristics are successively more stringent. The most interesting constraints are those required for localization of edge effects on a finite network; they are not satisfied by a network in which the output from each node is obtained by sharply thresholding the node potential. However, they can be satisfied if the network operates in a relatively low signal region, below saturation of the outputs. An example of a network which does satisfy the constraints is given by certain field effect transistor (FET) implementations.