The construction of a simultaneous functional order in nervous systems. II. Computing geometrical structures

Abstract

The functional order of a collection of nervous elements is available to the system itself, as opposed to the anatomical geometrical order which exists only for external observers. It has been shown before (Part I) that covariances or coincidences in the signal activity of a neural net can be used in the construction of a simultaneous functional order in which a modality is represented as a concatenation of districts with a lattice structure. In this paper we will show how the resulting functional order in a nervous net can be related to the geometry of the underlying detector array. In particular, we will present an algorithm to construct an abstract geometrical complex from this functional order. The algebraic structure of this complex reflects the topological and geometrical structure of the underlying detector array. We will show how the activated subcomplexes of a complex can be related to segments of the detector array that are activated by the projection of a stimulus pattern. The homology of an abstract complex (and therefore of all of its subcomplexes) can be obtained from simple combinatorial operations on its coincidence scheme. Thus, both the geometry of a detector array and the topology of projections of stimulus patterns may have an objective existence for the neural system itself.