In: Proc. of the Ninth Annual Conf. of IEEE/Engineering in Medicine
and Biology Society, Boston, 13-16 Nov. 1987
Tensor Geometry: Mathematical Brain Theory for Neurocomputers
and Neurobots. A Parallel Algorithm for Functional Neuromuscular
Stimulation
A. J. Pellionisz
Department of Physiology and Biophysics, New York University
Medical Center New York, N.Y. 10016, USA
Abstract
Conceptually and formally homogeneous quantitative brain theory
serves as a mathematical basis for our understanding of CNS
function and as a medium for blueprints of constructing brain-like
machines. Tensor Network Theory [9-10,19-22] is based on the
axiom that neuronal networks trans-form generalized vectorial
(tensorial) expressions of external invariants in intrinsic coordinates.
How neural networks transform such expressions within and among
representation-spaces (with not necessarily Euclidean or Riemannian
geometries) can be described by tensor geometry, which thus
becomes a unifying language of theoretical neuroscience, and
neurobotics & neurocomputing. An example of coordinated motor
control by neural networks provides with a parallel algorithm for
functional neuromuscular stimulation.
Introduction
Brain Theory, in a primary sense, enables us to arrive at
structuro-functional principles of the operation of distinct
subsystems of the CNS. The geometrical approach of Tensor Network
Theory [reviews: 10,15,17] conceives brain function as a geometrical
representation comprised by the complex structural interconnections
of a neuronal network. Thus, neuronal nets implement functional
geometries that match those of the external world [22]. Cerebellar
and gaze systems and the role they play in adaptive, coordinated
sensorimotor operations could already be functionally interpreted
[11,12,14,18,20, 21,23]. Further, however, theory also leads from
knowledge to its utilization. Mathematical neuronal network theories
already started to yield new means, brain-like machines, both for
production and control in the fields of neurobotics (robots equipped
with brain-like controllers [9,13,16]) and neuro-computers (new
bread of non von Neumann-type "computers" implementing brain-
like functions; [14,16,17]).
Fig.2. Tensor network model of transformations of intrinsic
coordinates: a model of the three-step vestibulo-collic head-
stabilization sensorimotor reflex [23]. Transformation from
coordinates intrinsic to a sensory system to an intrinsic motor frame
(where the latter may be of higher dimensions) can be accomplished
by a three-step tensorial scheme [20]. The vestibular sensory metric
tensor, vestibulo-collic sensorimotor tensor and contravariant neck
motor metric tensor transformations can be expressed verbally or by
abstract, reference frame-aspecific tensor-symbolism or by matrix-
(patch-) and network-diagrams. Here, these three matrices are
shown, for the particular vestibular [3] and neck-motor frames [2] of
the cat by patch-diagrams only. A quantitative visualization of
corresponding neuronal networks that can accomplish such transfer
is also presented. Network diagrams display the massively parallel
architecture of the CNS, where convergences & divergences are the
rule and separated point-to-point connections rarely, if ever,
characterize the structure. Such existing neuronal networks that
transform infromation from sensory frame to motor, can also be
implemented by various man-made means, eg. parallely organized
software and/or hardware. Neuroscience provides blueprints for
neurocomputer algorithms.
Fig.1. Schematic illustration of (joint-angle) coordinates intrinsic to
the structure of a motor apparatus (A) and the required metric
tensor transformation from a unique set of projection-type
(covariant, B) to overcomplete parallelogram-type (contravariant, C)
vectorial expression.
Tensor Model: Neuronal Networks Transform Intrinsic Coordinates
Tensorial approach necessitates three major classes of investments.
Beyond the central task of development of the tensorial concepts,
formalisms, software & parallel hardware (neurocomputer) systems,
a knowledge of the neuronal net-works underlying CNS performance
is required: abundant in neuroanatomy. Third, quantitative
experimental establishment of the intrinsic coordinate systems is
inevitable to move from illustrative joint-coordinates (Fig.1) to
anatomically correct biomechanical models of various musculo-
skeletal systems that the CNS operates with. An exemple is given in
Fig.2. [details in 23] showing neuronal networks that transform
general intrinsic coordinates (established by quantitative
computerized anatomy) from vestibular sensory- [3] to neck-motor
frame [2].
Metaorganization of Tensor Transformations in Co-ordinated
Robotic- and Living Sensorimotor Systems Once theory is
elaborated [9-10,19-22] and experimentally tested
[5,23], tensor geometry can serve as a common language describing
coordinated control both in natural (gaze, loco-motory, haptic) and
man-made (robotic) sensorimotor systems. Tensors are
mathematical operators connecting general coordinates, where one
must distinguish between sensory- and motor-type vectorial
representations (covariant and contravariant tensors, [20] ). Thus,
the features [10] and the emergence, via the so-called
Metaorganization-principle [23], can be specified for the metric
tensor that comprises a multidimensional functional geometry that
transform the dual representations from one to another : 1) Sensory
metric tensor (gpr), transforming a covariant reception vector (sr) to
contravariant perception (sp). Lower and upper indeces denote co-
and contravariants: sp=gpr.sf where gpr = |gpr|-1= |cos(½pr)|-1
andÊ|cos(½pr)| is the table of cosines of angles among sensory unit-
vectors. 2) Sensorimotor covariant embedding tensor (cip),
transforming the sensory vector (sp) into covariant motor intention
vector (mi). Covariant embedding is unique, including the case of
overcompleteness, but results in a non-executable covariant
expression [20]: mi = cip . sp where cip= ui. wp and ui , wp are
the i-th sensory- and p-th motor unit-vector. 3) Motor metric tensor
[20,10] that converts intention mi to executable contravariants; me =
gei .mi (where gei is computed as gpr was for sensory axes in 1).
In case of overcompleteness of either or both the sensory and motor
coordinate systems, tensor network theory suggests that the CNS
uses the Moore-Penrose generalized inverse of the unique covariant
metric tensor [10]: gj,k= Sm {1/lm+ . |Em > < Em|}, where Emand lm+
are the m-th Eigenvector of gj,k and its generalized Eigenvalue
Fig.3.Tensor model of the coordination of a 4-joint 7-muscle
overcomplete skeletomuscular apparatus using a Moore-Penrose
inverse.Predicted muscle activation (right) is a basis for functional
neuromuscular stimulation. The progress of neurocomputation
from Nature's networks towards a new breed of brain-like hardware
(neurochips or even biochips), can be illustrated by Fig.3 (and its
video), which projects the use of tensorial modeling for functional
neuro-muscular stimulation of paraplegics Ð a research field in need
of massively parallel control algorithm [6,7,8]. For the neural
paradigm of cerebellar-type motor coordination [10,12], several
steps have already been accomplished. The first was to
experimentally reveal the networks. Second, to arrive at mathe-
matical algorithms, eg. the metaorganization process, that these
networks implement [23]. Third, to build the software (using von-
Neumann computers) for applications [13-17]. Fig.3. shows the
application of the tensorial "netware" utilized to gene-rate a
coordinated movement of thee overcomplete system of 4-joint, 7-
muscle cat hindlimb-model. There are three further stages (steps
4,5,6) in the realization of the metaorganization algorithm. The
concept of "virtual instrument" (realized by graphic-based software
on von-Neumann type computers) is a
relatively small step to make (towards level 4, the stage of R&D)
from the present exploratory state of art. Stage 5, future stand-alone
traditional microprocessor implementations of such
"neurocomputation" are likely to yield marketable products to help
the vast community of handicapped paraplegics. Ultimately, the
understanding of how cerebellar neuronal networks bring about
coordinated motor control will be matched by utilization of the
principles by means of neurochips that will implement the
metaorganization-algorithm by massively parallel, neurocomputer
manner Ñ just like the living brain.
Acknowledgement: Research supported by NS 13742 & 22999 from
NINCDS
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