CRITICAL PHENOMENA IN SELF-ORGANIZING FEATURE MAPS - GINZBURG-LANDAU APPROACH

Self-organizing feature maps (SOFM's) as generated by Kohonen's algori thm are prominent examples of the cross fertilization between theoreti cal physics and neurobiology. SOFM's serve as high-fidelity models for the internal representation of the external world in the cortex. This is exploited for applications in the fields of data analysis, robotic s, and for the data-driven coarse graining of state spaces of nonlinea r dynamical systems. From the point of view of physics Kohonen's algor ithm may be viewed as a stochastic dynamical equation of motion for a many particle system of high complexity which may be analyzed by metho ds of nonequilibrium statistical mechanics. We present analytical and numerical studies of symmetry-breaking phenomena in Kohonen's SOFM tha t occur due to a topological mismatch between the input space and the neuron setup. We give a microscopic derivation for the time dependent Ginzburg-Landau equations describing the behavior of the order paramet er close to the critical point where a topology preserving second-orde r phase transition takes place. By extensive computer simulations we d o not only support our theoretical findings, but also discover a first order transition leading to a topology violating metastable state. Co nsequently, close to the critical point we observe a phase-coexistence regime.