Self-Organizing Maps
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We propose a simple topographic mapping formation model from a cell layer to a cell layer. Our model is a discrete one in that the state value of input and output cells takes 0 or 1 and input and output layers are represented by undirected graphs. A binary input pattern can be given to the network consisting of input and output cell layers. Such an input pattern can be represented by a subset of input cells. That is, a state value of an input cell takes 1 if a cell belongs to the subset, otherwise, a state value of an input cell is 0. Such a definition of an input pattern does not necessarily assume a short-range excitatory mechanism in an input layer. Thus, a topographic mapping described in this model is a map, which preserves the input pattern relation. By using the concept of input pattern separability, we showed an existence condition of certain learning rules, which are correlational. We have paid special attention to such correlational type learning rules, and have shown under the rules that topographic mappings are the only stable ones. As to the non-correlational learning rules, we also investigate the stability of generated mappings.