Viewing a class of neurodynamics on parameter space

Abstract

Nearly all models in neural networks start from the assumption that the input-output characteristic is a sigmoidal function. On parameter space we present a systematic and feasible method for analyzing the whole spectrum of attractors-all saturated, all-but-one saturated, all-but-two saturated, etc. — of a neurodynamical system with a saturated sigmoidal function as its input-output characteristic. We present an argument which claims, under a mild condition, that only all saturated or all-but-one saturated attractors are observable for the neurodynamics. For any given all saturated configuration ξ (all-but-one saturated configuration η) the paper shows how to construct an exact parameter region R(ξ) (¯R(η)) such that if and only if the parameters fall within R(ξ) (¯R(η)), then ξ (η) is an attractor (a fixed point) of the dynamics. The parameter region for an all saturated fixed point attractor is independent of the specific choice of a saturated sigmoidal function, whereas for an all-but-one saturated fixed point it is sensitive to the input-output characteristic.