The Self-Organizing Field

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 property 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 selforganization. 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. The 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.