Santini : the Self - Organizing Field 3

|Many of the properties of the well-known Koho-nen 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 identiied. 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 eld is governed by a set of diierential equations analogous to the diierence 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.