Tensor
Network Theory of the Central Nervous System

*Andr**á**s
J. Pellionisz*

Volume
II

Edited
by

George
Adelman

Foreword
by Francis O. Schmitt

BIRKHAUSER
Boston - Basel - Stuttgart

I
196 Tensor Network Theory of the Central Nervous System

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Tensor
analysis is a mathematical discipline used to characterize physical quantities
by generalized vectorial relations expressible in any particular system of
coordinates. Tensor network theory of the central nervous system (CNS) is based
on the abstract geometric concept that the brain internalizes relations
existing in the external world by multidimensional vectorial relations,
implemented within the CNS by neuronal networks. The theory describes in an
abstract general manner, as well as in a particular quantitative fashion, how
the brain may construct an internal representation of external relations of
invariants by multidimensional vectors: the internal vectorial expressions are
assigned to the external physical entities by the use of non-orthogonal
intrinsic coordinate systems within the CNS.

The
coordinate-system-free tensorial notation (better called coordinate-system-*general*
representation) has already been successfully utilized in engineering and in
the theory of relativity. In engineering, it was used to describe physical
quantities such as distances, directions, forces, tensions (hence *tensor*), in a manner such that
the choice of the particular system of coordinates used for a given vectorial
representation is irrelevant. Since the realm of engineering is largely
confined to classical mechanics, it uses tensors that are generally Cartesian
[three-dimensional expressions in orthogonal frames], representing points in
Euclidean space.

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In
theory of relativity, which concerns relativistic mechanics, the geometry of
the vector space is Riemannian. Tensor network theory of the central nervous
system necessitates a further generalization of tensor theory beyond Euclidean
and Riemannian spaces since functional multidimensional spaces of the CNS are
not necessarily limited to such well-known geometries.

One
of the most fundamental mathematical requirements that are necessary in tensor
network theory of brain function is the explicit distinction of covariant and
contravariant versions of a vector, since the intrinsic natural systems of
coordinates of the CNS are composed of axes that are usually not orthogonal to
one another, and these forms are identical only in orthogonal frames of
reference. These different vectorial versions correspond to the independently
established but non-executional covariant vector components, yielding a sensory
intention-type vector, and to the physically executable but interdependent
contravariant vector components, yielding a motor execution vector. The
relationship between covariant and contravariant vectorial versions is
characterized by the metric tensor (see Fig. 1).

*Figure 1. [legend] The fundamental distinction in
tensor network theory of the CNS of covariant and contravariant vectorial
expressions and their relationship throught the metric tensor.. From Pellionisz
and Llin**á**s ( 1980).*

The
covariant intention to contravariant execution transformation (via the
contravariant metric tensor or, in case of singularity, the Moore-Penrose
generalized inverse of the covariant metric tensor) can be described in
abstract tensorial notation, as well as by a particular network transformation
in any given frame of reference. Such a quantitative neuronal network model is
shown in Figure 2. transforming vestibular sensory

F*igure 2**. [legend] Tensor network model of
the vestibulocollic reflex, embodying a covariant intention to contravariant
motor execution transformation via the cerebellar neuronal network. From
Pellionisz (1985).*

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coordinates
into neck-muscle rnotor intention components and then through the cerebellum
into motor execution vector components.

A
physical entity, such as a head movement, is first expressed in a sensory
system of coordinates, intrinsic to the organism, such as the vestibular
semicircular canals. Then, a sensorimotor tensor transformation, called
covariant embedding, yields projection-type covariant motor intention
components, expressed in the neck-muscle frame of reference. The motor metric
tensor-type transformation, converting covariant motor intention to
contravariant motor execution by the Moore-Penrose generalized inverse of the
covariant metric tensor, is performed by the cerebellar neuronal network, acting
as a space-time metric tensor.

Beyond
interpreting the general function of neuronal networks such as the cerebellum,
tensor theory provides an explanation for the genesis and modification of
neuronal networks. This hypothetical process, called metaorganization, predicts
how physical and functional geometries may organize one another by means of
reverberative oscillations of covariant proprioception and contravariant
execution vectors. The process, implemented by physical tremors, can identify
those special vectors, so-called *eigenvectors,* whose normalized
covariant and contravariant versions are identical. If these fundamental
functional vectors are imprinted into nuclear regions of the CNS, such as the
inferior olive, they can govern the adaptive organization (genesis and
modification) of higher order hierarchical structures such as the cerebellar
corticonuclear neuronal network.

Experimental
research has produced evidence that the central nervous system distributes
activations of forelimb muscles in humans close to that predicted by the
tensorial approach.

See
also Cerebellum, Network Physiology; Integration, Neural; Motor Control

Further
reading

Gielen
CCAM, Zuylen EJ (1985): Coordination of flexion and supination of the forearm:
Application of the tensor analysis approach. Neuroscience 17:527-539

Lánczos
C (1970): Space Through the Ages Academic Press (Lond)

Pellionisz
A (1984): Coordination: A vector-matrix description of transformations of
overcomplete CNS coordinates and a tensorial solution using the Moore-Penrose
generalized inverse. J Theor Biol I 10:353-375

Pellionisz
A (1985): Tensor network theory of the central nervous system and sensorimotor
modeling. In: Brain Theonˇ, Palm G, Aertsen A, eds. Berlin Heidelberg New York:
Springer-Verlag, p. 121-145

Pellionisz
A, Llinás R ( 1980): Tensorial approach to the geometry of brain function.
Cerebellar coordination via a metric tensor. Neuroscience 5:1761-1770

Pellionisz
A, Llinás R (I985): Tensor network theory of the Metaorganization of functional
geometries in the CNS. Neuroscience 16:245-273

Simmonds
JA ( 1982): A Brief on Tensor Analysis. Springer-Verlag, New York