title: Convergence property of topographic mapping formation from cell layer to cell layer through correlation learning rule.
abstract: To elucidate the mechanism of topographic organization, we propose a simple topographic mapping formation model from one-dimensional cell layer to one-dimensional cell layer. In our model, each cell takes a binary state value and we consider several learning principles which are extensions of Hebb's rule. We pay special attention to a correlation learning rule where a synaptic weight value is increased if pre- and post-synaptic cells' state values are the same. First, we show that under a certain network size condition, a mapping is stable with respect to the correlation learning if and only if it is topographic. Second, we introduce a special class of weight matrices called band type and show that the set of band type weight matrices is strongly closed and such a weight matrix cannot yield a topographic mapping. Third, we show that any mapping, if it is defined by a non band type weight matrix, converges to a topographic mapping. The proof method is intrinsically of combinatorial nature in a framework of Markov process.
authors.affiliation: Department of Electronics and Informatics, Ryukoku University, Otsu, Shiga, Japan. sakamoto@fagaceae.elec.ryukoku.ac.jp
authors.forename: S
authors.initials: S
authors.lastname: Sakamoto
authors.forename: Y
authors.initials: Y
authors.lastname: Kobuchi
mesh_terms: D000465:Algorithms; D008390:Markov Chains; D008959:Models, Neurological; D009415:Nerve Net / Q000166:cytology; D013569:Synapses / Q000502:physiology