Tensor network theory of the metaorganization of functional geometries in the central nervous system

Abstract Here we present an elaboration and a quantitative example for a hypothetical neuronal process, implementing what we refer to as the metaorganization principle. This process allows the internalization of external (body) geometries into the central nervous System (CNS) and a reciprocal and equally important action of the CNS geometry on the external (body) geometry. The hypothesis is based on the distinction, within the CNS, between covariant sensory and contravariant motor vectorial expressions of the extrinsic geometry. These sensory and motor expressions, given in natural co-ordinate Systems, are transformed from one to the other by a neuronal network which acts as a metric tensor. The metric tensor determines the relationship of these two expressions and thus comprises the functional geometry of the System. The emergence through metaorganization of networks that implement such metric function is viewed as the result of interactions between the covariant motor execution which generates a physical action on the external world (via the musculoskeletal System) and the covariant sensory proprioception which measures the effect of such motor output. In this transformation of contravariants to covariants by the physical geometry of the motor System, a covariant metric tensor is expressed implicitly. However, co-ordinated motor action requires its dual tensor (the contravariant metric) which is assembled in the CNS based on the metaorganization principle, i.e. the ability of CNS and external geometries to mold one another. The two metric transformations acting on each other detect error signais whenever the match of the physical and functional geometries is imperfect. Such error signals are utilized by the metaorganization process to improve the match between the two metrics, so that with use the internal representation becomes increasingly homeometric with the geometry of the external world. The proposed physical process by which the metaorganization principle is implemented is based on oscillatory reverberation. If covariant proprioception is used as a recurrent signal to the motor apparatus, as if it were a contravariant motor expression, then reverberations at their steady-state yield the eigenvectors and eigenvalues of the System. The stored eigenvectors and eigenvalues can serve, respectively, (1) as a means for the genesis of a metric (in the form of its spectral representation) with the given eigenvectors and (2) as a means of comparing the eigenvalues that are implicit in the external body geometry and those of the internal metric. The difference between these eigenvalues is then used to modify the metric so that it “evolves” to perform a more accurate covariant-contravariant transformation. The metric can be represented by the dyadic outer products of its eigenvectors where each dyad is weighted by the corresponding eigenvalue. Such a spectral representation yields in a uniform manner (a) the contravariant metric (in the case of complete CNS hyperspaces) and (b) the Moore-Penrose generalized inverse of the covariant metric tensor (in the case of non-Riemannian overcomplete CNS hyperspaces). The metaorganization of metric networks, together with the sensorimotor covariant embedding network, provides an explanation of the emergence of a whole sensorimotor apparatus leading to the feasibility of constructing truly brain-like robotic Systems.