Topographic Organization of Receptive Fields in Recursive Self-Organizing Map

Recently, there has been an outburst of interest in extending topo-graphic maps of vectorial data to more general data structures, such as sequences or trees. However, at present, there is no general consen-sus as to how best to process sequences using topographic maps and this topic remains a very active focus of current neurocomputational research. The representational capabilities and internal representations of the models are not well understood. We rigorously analyze a gen-eralization of the Self-Organizing Map (SOM) for processing sequential 1 data, Recursive SOM (RecSOM [29]), as a non-autonomous dynamical system consisting of a set of ¯xed input maps. We argue that contrac-tive ¯xed input maps are likely to produce Markovian organizations of receptive ¯elds on the RecSOM map. We derive bounds on parameter ¯ (weighting the importance of importing past information when pro-cessing sequences) under which contractiveness of the ¯xed input maps is guaranteed. Some generalizations of SOM contain a dynamic module responsible for processing temporal contexts as an integral part of the model. We show that Markovian topographic maps of sequential data can be produced using a simple ¯xed (non-adaptable) dynamic module externally feeding a standard topographic model designed to process static vectorial data of ¯xed dimensionality (e.g. SOM). We elaborate upon the importance of non-Markovian organizations in topographic maps of sequential data.