{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T05:45:54Z","timestamp":1744263954975},"reference-count":32,"publisher":"MIT Press - Journals","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Neural Computation"],"published-print":{"date-parts":[[1998,1,1]]},"abstract":"<jats:p> 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\u2014all-saturated, all-but-one-saturated, all-but-twosaturated, and so on\u2014of 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 neurodynamics. For any given all-saturated configuration [Formula: see text] (all-but-one-saturated configuration [Formula: see text]) the article shows how to construct an exact parameter region R([Formula: see text])([Formula: see text]([Formula: see text])) such that if and only if the parameters fall within R([Formula: see text])([Formula: see text]([Formula: see text])), then [Formula: see text]([Formula: see text]) 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. <\/jats:p><jats:p> Based on a similar idea, the role of weight normalization realized by a saturated sigmoidal function in competitive learning is discussed. A necessary and sufficient condition is provided to distinguish two kinds of competitive learning: stable competitive learning with the weight vectors representing extremes of input space and being fixed-point attractors, and unstable competitive learning. <\/jats:p><jats:p> We apply our results to Linsker's model and (using extreme value theory in statistics) the Hopfield model and obtain some novel results on these two models. <\/jats:p>","DOI":"10.1162\/089976698300017944","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T11:57:52Z","timestamp":1027771072000},"page":"189-213","source":"Crossref","is-referenced-by-count":10,"title":["Fixed-Point Attractor Analysis for a Class of Neurodynamics"],"prefix":"10.1162","volume":"10","author":[{"given":"Jianfeng","family":"Feng","sequence":"first","affiliation":[{"name":"Biomathematics Laboratory, Babraham Institute, Cambridge CB2 4AT, U.K."}]},{"given":"David","family":"Brown","sequence":"additional","affiliation":[{"name":"Biomathematics Laboratory, Babraham Institute, Cambridge CB2 4AT, U.K."}]}],"member":"281","reference":[{"key":"p_2","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4149(93)90017-X"},{"key":"p_3","doi-asserted-by":"publisher","DOI":"10.2307\/1427581"},{"key":"p_4","first-page":"1","volume":"23","author":"Cottrell M.","year":"1986","journal-title":"Ann. 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