Transition times in self-organizing maps

We study the creation of topological maps. It is well known that topological defects, like kinks in one-dimensional maps or twists ('butterflies') in two-dimensional maps, can be (metastable) fixed points of the learning process. We are interested in transition times from these disordered configurations to the perfectly ordered configurations, i.e., the average time it takes to remove a kink or to unfold a twist. For this study we consider a self-organizing learning rule which is equivalent to the Kohonen learning rule, except for the determination of the 'winning' unit. The advantage of this particular learning rule is that it can be derived from an error potential. The existence of an error potential facilitates a global description of the learning process. Mappings in one and two dimensions are used as examples. For small lateral-interaction strength, topological defects correspond to local minima of the error potential, whereas global minima are perfectly ordered configurations. Theoretical results on the transition times from the local to the global minima of the error potential are compared with computer simulations of the learning rule.