MULTIDIMENSIONAL GEOMETRIES INTRINSIC TO COGNITIVE CNS HYPERSPACES, AND METAORGANIZATION BY A SENSORIMOTOR APPARATUS. A.J. Pellionisz, Department of Physiology & Biophysics, New York University Medical Center, New York, NY 10016
Tensor Network Theory of the CNS is built on the axiom that the CNS expresses physical invariants of the external world in coordinates intrinsic to the organism. In such general frames dual geometrical representations are possible by assigning to invariants both sensory (covariant) and motor (contravariant) vectors. Neither, however, fully characterizes the invariant; one passively measuring it, without yielding a knowledge how to generate it, the other can produce it but not necessarily with appropriate metrical properties. Understanding the link between the two representations is all the more important, since in a sensorimotor system they are connected via a neuronal network, whose role will then be revealed.
Indeed, the general question how the sensory is related to the motor realm, is a central problem in experimentol neuroscience (which yields anatomical and physiological knowledge of the transfer), in brain theory (which mathematizes its understanding) and in the neurocomputer & neurobotics fields (which utilizes this understanding by physically implemented means). In neuroscience it is known that for a representation of the external world (for its cognitive model) such as obtained by vision, employing solely a sensory apparatus is not sufficient (the retina goes blind if our eye cannot move around). An interaction of a sensory and a motor expressions is necessary for a geometrical model of invariants in the CNS.
Neuronal networks that are required for moving the eye according to the retinal image (either by saccadic, or smooth pursuit manner) are known at least partially. Also, it is evident that the retina, the superior colliculus, pontine saccadic bursters (etc; even the visual cortex) and finally the eye muscles all utilize various vectorial coordinate systems intrinsic to these structuro-functional units. In terms of tensor network theory, the connection between the sensory and motor domain the metric tensor that transforms covariant descriptive vectorial representatic into contravariant constructive vectorial expressions. Mathematically, if this, between dual representations is established, all geometrical feafures of a multidimensional CNS hyperspace ore comprised in such metric tensor. This is worked out both in experimental research (of how the structure and function of arrays neurons in existing networks implement metric properties of body geometries and in theoretical neurophysics (revealing mathematical features of CNS hyperspaces) are expected to focus on physical and functional brain geometries.
For the visuo-oculomotor system, both the physical geometrical properties the muscles, and the matching functional geometry that is necessary to coordinate the action of the overcomplete muscles are well explored. Tensorial models of how sensory and motor metrics can evolve in an interaction with the physical geometry of a respective sensory or motor apparatus have been elaborated for the vestibulo-collic, retino-collic, retino-ocular and vestibul ocular neuronal arc. It was also postulated by the Metaorganization-principle however, that: once a dual sensorimotor representation emerges, a hierarchic representation of the function of the total circuit can be obtained, which comprises an internal cognitive geometrical model of the interrelation of sensory and motor expressions; called (in this case) vision. A model of how neuronal network of the tectum and visual cortex may implement such a cognitive space, invoke by the primary retino-ocular arc, will be presented. - Support: NS22999 & N51374