The self-organizing field [Kohonen maps]

Many of the properties of the well-known Kohonen map algorithm are not easily derivable from its discrete formulation. For instance, the "projection" implemented by the map from a high dimensional input space to a lower dimensional map space must be properly regarded as a projection from a smooth manifold to a lattice and, in this framework, some of its properties are not easily identified. This paper describes the self-organizing field: a continuous embedding of a smooth manifold (the map) into another (the input manifold) that implements a topological map by self-organization. The adaptation of the self-organizing field is governed by a set of differential equations analogous to the difference equations that determine weights updates in the Kohonen map. This paper derives several properties of the self-organizing field, and shows that the emergence of certain structures on the brain-like the columnar organization in the primary visual cortex-arise naturally in the new model.