Critical phenomena in self-organizing feature maps: Ginzburg-Landau approach.

Abstract

Self-organizing feature maps (SOFM's) as generated by Kohonen's algorithm are prominent examples of the cross fertilization between theoretical 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, robotics, and for the data-driven coarse graining of state spaces of nonlinear dynamical systems. From the point of view of physics Kohonen's algorithm may be viewed as a stochastic dynamical equation of motion for a many particle system of high complexity which may be analyzed by methods of nonequilibrium statistical mechanics. We present analytical and numerical studies of symmetry-breaking phenomena in Kohonen's SOFM that 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 parameter close to the critical point where a topology preserving second-order phase transition takes place. By extensive computer simulations we do not only support our theoretical findings, but also discover a first order transition leading to a topology violating metastable state. Consequently, close to the critical point we observe a phase-coexistence regime.