Self-organization property of Kohonen's map with general type of stimuli distribution

Here the self-organization property of one-dimensional Kohonen's algorithm in its 2k\Gammaneighbor setting with a general type of stimuli distribution and non-increasing learning rate is considered.We prove that the probability of self-organization for all initial values of neurons is uniformly positive. For the special case of a constant learning rate , it implies that the algorithm self-organizes with probability one. Keywords Neural networks, Kohonen's SOM , One-dimensional self-Organization, Winner definition. 1 Introduction Among several models which have been proposed for self-organizing feature maps , the Kohonen algorithm is the most popular one. The analytical behavior of the algorithm and also its very extended areas of application are widely investigated ,see [3, 5, 7, 10, 11] and references therein . Although the algorithm is computationally simple, analytical investigation of its properties turned out to be very complicated. The existing results , including this paper , de..