Convergence property of topographic mapping formation from cell layer to cell layer through correlation learning rule

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.