Stability of Topographic Mappings between Generalized Cell Layers

Shouji SAKAMOTO  Youichi KOBUCHI  

Publication
IEICE TRANSACTIONS on Information and Systems   Vol.E85-D   No.7   pp.1145-1152
Publication Date: 2002/07/01
Online ISSN: 
DOI: 
Print ISSN: 0916-8532
Type of Manuscript: PAPER
Category: Biocybernetics, Neurocomputing
Keyword: 
topographic mapping,  stability,  Hebbian learning,  unsupervised learning,  correlational learning,  

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Summary: 
To elucidate the mechanism of topographic organization, we propose a simple topographic mapping formation model from generalized cell layer to generalized cell layer. Here generalized cell layer means that we consider arbitrary cell neighborhood relations. In our previous work we investigated a topographic mapping formation model between one dimensional cell layers. In this paper we extend the cell layer structure to any dimension. In our model, each cell takes a binary state value and we consider a class of learning principles which are extensions of Hebb's rule and Anti-Hebb's rule. We pay special attention to correlation type learning rules where a synaptic weight value is increased if pre and post synaptic cell states have the same value. We first show that a mapping is stable with respect to the correlational learning if and only if it is semi-embedding. 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 can not yield a topographic mapping. Third, we show by computer simulations that a mapping, if it is defined by a non band type weight matrix, converges to a topographic mapping under the correlational learning rules.