Geometrical structures determined by the functional order in nervous nets

Abstract

The functional order of a collection of neural elements may be defined as the order induced through the total of covariances of signals carried by the members of the collection. Thus functional order differs from geometrical order (e.g. somatotopy) in that geometrical order is only available to external observers, whereas functional order is available to the system itself. It has been shown before that the covariances can be used to construct a partially ordered set that explicitely represents the functional order. It is demonstrated that certain constraints, if satisfied, make this set isomorphic with certain geometrical entities such as triangulations. For instance there may exist a set of hyperspheres in a n-dimensional space with overlap relations that are described with the same partially ordered set as that which describes the simultaneous/successive order of signals in a nerve. Thus it is logically possible that the optic nerve carries (functionally) two-dimensional signals, quite apart from anatomical considerations (e.g. the geometrically two-dimensional structure of the retina which exists only to external observers). The dimension of the modality defined by a collection of nervous elements can in principle be obtained from a cross-correlation analysis of multi-unit recordings without any resort to geometrical data such as somatotopic mappings.