FIXED-POINT ATTRACTOR ANALYSIS FOR A CLASS OF NEURODYNAMICS

Nearly all models in neural networks start from the assumption that th e input-output characteristic is a sigmoidal function. On parameter sp ace, we present a systematic and feasible method for analyzing the who le spectrum of attractors-all-saturated, all-but-one-saturated, all-bu t-two-saturated, and so on-of a neurodynamical system with a saturated sigmoidal function as its input-output characteristic. We present an argument that claims, under a mild condition, that only all-saturated or all-but-one-saturated attractors are observable for the neurodynami cs. For any given all-saturated configuration <(xi)under bar> (all-but -one-saturated configuration <(eta)under bar>) the article shows how t o construct an exact parameter region RO) ((R) over bar(<(eta)under ba r>)) such that if and only if the parameters fall within R(<(xi)under bar>) ((R) over bar<(eta)under bar>), then <(xi)under bar> (<(eta)unde r bar>) is an attractor (a fixed point) of the dynamics. The parameter region for an all-saturated fixed-point attractor is independent of t he specific choice of a saturated sigmoidal function, whereas for an a ll-but-one-saturated fixed point, it is sensitive to the input-output characteristic. Based on a similar idea, the role of weight normalizat ion realized by a saturated sigmoidal function in competitive learning is discussed. A necessary and sufficient condition is provided to dis tinguish two kinds of competitive learning: stable competitive learnin g with the weight vectors representing extremes of input space and bei ng fixed-point attractors, and unstable competitive learning. We apply our results to Linsker's model and (using extreme value theory in sta tistics) the Hopfield model and obtain some novel results on these two models.