Tensor Geometry:  a Language of Brains & Neurocomputers. 
Generalized Coordinates in Neuroscience & Robotics

A. Pellionisz

Department of Physiology and Biophysics New York University 
Medical Center 550 First Ave, New York City N.Y. 10016 USA

Abstract

Neurocomputers are implementations of mathematical paradigms 
performed by neuronal networks. Thus, it is essential for their 
construction that the mathematical language of brain function be 
made explicit. Based on the philosophy that the brain, as a product 
of natural evolution, is a geometrical object (not a machine that is a 
product of engineering), tensor geometry is used to describe 
multidimensional general (tensor) transformations of natural 
coordinates that are intrinsic to the organism. Such an approach uses 
a formalism that not only generalizes existing Cartesian vector- 
matrix paradigms, but can unite neuroscience with robotics: general 
frames include both Natural coordinate systems (found by 
quantitative computerized anatomy) and those simple artificial ones 
that are selected in engineering for convenience. Utilizations of the 
tensor approach center on natural and artificial sensorimotor 
operations, promoting a co-evolution of coordinated (and intelligent) 
robots with Nature's systems such as adaptive cerebellar 
compensatory reflexes. Such sensorimotor-based strategy enables 
also a cross-fertilization; eg. employing neurocomputers to 
implement a coordination-algorithm of cerebellar-networks to be 
used for functional neuromuscular stimulation of paraplegics. 

1. Introduction: Brains, Computers, Neurocomputers

The neurocomputer revolution of the nineteen-eighties had been in 
the offing for ages. Some of the deepest roots reach back to the times 
of Descartes, if not beyond, since man's desire has long been to 
understand himself (his brain) and then to quickly make use of what 
had been learned. Clearly, Descartes intended to first gain an 
understanding of brain function in the terms of the state of art of 
natural sciences (geometry of Euclidean spaces represented in 
Cartesian 3-dimensional orthogonal systems of coordinates), and then 
to conceptualize some kind of a physical implementation of brain 
function (hydraulic pressures, conducted along the nerves, operating 
on the muscles).

Three centuries ago, however, the three basic ingredients of 
neurocomputers (sufficient knowledge of the nervous system, a 
proper -mathematical understanding of brain function, and adequate 
technology to implement it) were obviously unavailable. While all 
these components had evolved tremendously, it is argued here that 
once the components are available, it is the philosophy of how to 
put them together which makes maybe the largest difference (8). 
First of all, he basic stance, a direction from an understanding to  
implementation was turned upside down in our modern age, 
dominated not so much by an understanding but by machines and  
technology Ð and lately their eiptome, the computer. The philisophical 
twist of putting technology first was triggered by man-made 
wonders at the turn of this century, when the original idea of 
creating brain-like machines was altered to thinking about the brain 
in terms of the most advanced machinery. Thus, the brain was 
thought of as a telegraph, telephone switchboard, etc., Ð and 
increasingly as a computer. 

Since the time of Babbage (2) computation was perceived not only as 
a primary brain function, but also as something that can be 
mechanized. Pressing calculation-needs of modern societies (eg. 
census; cf. Hollerith, 1890, projectile ballistics; ENIAC 1943) pushed 
computation, as a most significant brain function, further to the 
forefront. There is neither space nor need to elaborate here how 
mathematics of information theory and logical calculus (68,75) led to 
a virtual identification of computer- and brain function Ð as if the 
brain was for computation only. Such predominance of technology in 
thinking about brain function received further impetus with 
Cybernetics(77) when control, exercised by feedback circuits, was 
identified with CNS function Ð as if the brain was for control only. The 
extent of the infiltration of control engineering into brain research 
(both in terms of concepts, as well as measurement-techniques and 
mathematical analysis) is exemplified by the fact that eg. oculomotor 
or cerebellar research can barely recover today from the devastation 
over two decades by the underlying "brain theory" that the CNS is 
basically an amplifier whose gain and phase are adjusted and thus 
their measurement and plotting will explain the function.

Since von Neumann was a key architect in constructing computers 
(which is therefore often called "von Neumann machine"), it is most 
illuminating to recall how, nearing the final moments of his creative 
life, he looked back on computers that he helped create as compared 
with the brain that scientists originally intended to mimic (75) . The 
difference, he found, was as enormous as it was profound, most of all 
because the mathematical language of computers was well 
established, while that of the brain was unknown. Beyond the basic 
philosophical sommersault of identifying the brain with a calculating 
engine or a control machine, and trying to reach an understanding of 
Nature based on the principles of man-made systems, there were 
other major reasons why computers could not be developed to be 
brain-like. It is trivial to conclude that neither today's knowledge on 
CNS nor the required technology were available to von Neumann. In 
1957 he had to miss that incredible body of knowledge that was 
amassed since, especially in the "big boom of science" in the sixties- 
seventies in the USA.

It is not trivial to assert, however, that without a theoretical 
foundation that can identify the mathematical language that the 
brain is using, no profound scientific understanding of brain function 
is possible Ð and thus an implementation of a deficient understanding 
is per definitionem, tentative at best. Yet reading the last page in 
von Neumann's last book, it is clear enough that he was desperately 
looking for the discovery of the mathematical language of brain 
function, knowing fully well that "the outward forms of our  
mathematics are not absolutely relevant from the point of view of 
evaluating what the mathematical or logical language truly used by 
the CNS is" (75).

2. Brain Theory: Identification of the Mathematics Inherent in Brain 
Function

If the last page of Neumann's book ended with the open question of 
the mathematics of brain function, the first page of our book on 
Neurocomputers must start with a proper identification of the 
mathematical language underlying CNS operations. This is all the 
more im-portant, since accordinag to a definition, introduced here, 
neurocomputers are implementations of mathematical paradigms 
performed by neuronal networks. The primary task for brain theory 
remains, therefore, to identify the mathematics inherent in neuronal 
network operations. It is not irrelevant, however, to mention that an 
explicit acknowledgement of the problem of brain theory as a 
problem of modern mathematics is not universal even today. Even 
eminent natural scientists may still hold a position that brain theory 
could be a non-mathematical discipline (12). Others, however, do 
think that mathematics is to offer those frameworks that are 
necessary for homogeneous abstract understanding, also in 
neuroscience: "what we require now are approaches that can unite 
basic neurobiology and behavioural sciences into a single operational 
framework" (60), or "what is conspicuously lacking is a broad 
framework of ideas within which to interpret all these diffferent 
approaches...It is not that most neurobiologists do not have some 
general concept of what is going on. The trouble is that the concept is 
not precisely formulated" (9). This latter point is well illustrated by 
classic notions that properly perceived the brain as a massively 
parallel processor, which was well expressed eg. by Sherrington's 
metaphor (69) of "the brain as an enchanted loom" (of the 
flickerings of myriads of neurons in parallel). It has already been 
referred to that the mathematical formalism of logical calculus 
(Boolean Algebra, as expressed eg. by (6, 33)), was basically the 
language of serially organized computers, inadequate either for the 
frequency-code found in most places of the CNS (5) or for the 
massively parallel organization of the brain. Lacking a suitable 
mathematics, parallel expressions were carried empirically to 
"patterns" by computer simulations (14). Likewise, in the absence of 
a proper formalism, for example, Piaget's general notion of "schemas" 
(58) is still utilized without a mathematical elaboration. 

Hitherto the most adequate mathematics to express massive 
parallelism is the vectorial formalism pioneered in brain research by 
Wiener (77). This formalism could successfully express, in 
quantitative terms, classical and most intuitive, but largely 
qualitative concepts. For instance, although it was recognized rather 
early that motor control can be best understood in terms of collective 
co-action of many muscles, "synergies" (3), it is only very recently 
that "gestallt"-type categories such as coordination of multi-joint 
limbs or posture can be best described geometrically by vectors 
(53,39,62,56,4,35,27,28,29,31,19). Similarly, the clasical thinking of 
neurons organized in "assemblies" (20), is expressed today by 
mathematical terms of vectors (36). Perhaps the most visible 
application of vector formalism is, however, the mathematization of 
the classical association-concept (71,63) which boils down to the 
realization that the external product of two (normalized) vectors 
(A,B) presented together results in a matrix (M), which heretofore 
will yield (when normalized) vector B automatically upon the 
presentation of vector A alone. This "association scheme" (originally 
called Steinbuch's lernmatrix,71) has been modified by a good 
number of workers (71,63,34,1,25,7,17,23,36,37) mostly in order to 
enable the matrix-elements of increasingly sophisticated learning- 
rules (Hebbian-, delta-rule, back-propagation rule etc: 
20,23,64,65,67,74), to produce what is popular today as the class of 
"modified Hopfield models".  Three basic facts are important to 
remember, however: a) the vectors used are mathematical points in 
the ordinary vectorspace with Euclidean geometry (no metric other 
than the Kronecker-delta), b) the association-scheme works 
impeccably only for a single vector-pair (gets confused with 
increasing number of associated vectors), c) the mathematical 
algorithm elaborates a single concept (association), which could be 
extended to the "traveling salesman paradigm" (23), a problem that 
is nowhere shown to be resolved by existing neuronal networks.  
Although the assumption of "brain vectors" being points in a space 
with Euclidean geometry is entirely arbitrary, and thus can not be 
accepted at face value as the language of the brain, its ubiquitous 
usage is understandable as the formalism of vector-matrix 
transformations has been the closest to fit the bill of the 
mathematical language of "parallel distributed processing" (64, 65)  
performed by the brain, which was recognized even by Neumann as 
a logically shallow (few number of transformations), but massively 
parallel processor. 

3. Geometrical Representation Theory of Brain Function: Generalized 
Tensor Formalism for Coordinates Intrinsic to Nature

 Tensor Network Theory (TNT) of the Central Nervous System has 
been introduced (52-55) based on a philosophy, concept and 
formalism that were perceived as improvements on the state of art 
in Brain Theory. It is felt that it may provide with an adequate and 
general mathematical language of not only brain function but also of 
neurocomputers that implement it (38,41,45,46,47). 

Philosophically, TNT it is based on the monist-reductionalist view, 
that structuro-functional properties of the homogeneous brain-mind 
system are to be explained not in terms of technology as a machine, 
but by the most advanced, powerful and general mathematical 
abstraction suitable to explain it as part of Nature, and a product of 
Natural evolution. In turn, one is confident that once the proper 
mathematical description of the algorithms is provided, the required 
technology will be made available to implement it. Special 
implications of this philosophy could be mentioned here (for full 
analysis, see 8). For instance, in explaining a man-made mechanism 
the underlying mechanics can be taken for granted, since 
(extraordinary aplications nonwithstanding) the one used is classical  
(Newtonian) mechanics. In explaining Nature's complex phenomena Ð 
for instance, space-time representation in the brain (73, 76,54)Ð 
however, the underlying mechanics is not our choice and one must 
be open to the fact that a non-separable space-time continuum is 
represented in a Minkowski manifold.

Conceptually, TNT proposes that brain function is not merely for 
computation or for control, but for a function which includes both of 
these features: the brain is for (geometrical) representation of the 
external world (55). The general notion that the brain is a 
geometrical object (not a machine) is translated into the detailed 
explanation that neuronal networks comprise (not necessarily 
Euclidean) functional geometries that interact with one another in 
both a sensorimotor and a cognitive manner; 38,44,55) and also with 
the physical geometry of the external world (55). Since TNT is 
inherently a neuronal network theory, it could be considered part of 
connectionalism, although the network connections only subserve the 
functional role of transformations. Thus, to classify TNT it is more 
proper to use the term (by R. Llinás) of transformism. 

Mathematically, the axioms of TNS are rather subtle, yet critical: TNT 
is based on the fact that the CNS expresses its function, via its 
internal neuronal networks, in generalized coordinate systems (that 
are typically multidimensional, non-orthogonal, overcomplete 
frames), which are intrinsic to Nature's organisms Ð as opposed to 
their typical external description in extrinsic Cartesian coordinate 
systems. The existence of such intrinsic frames is not only physically 
obvious in vestibular, oculomotor, neck-motor, limb-musculoskeletal 
systems, but can also be measured by quantiative computerized 
anatomical methods (27,28,29,31,50,70,62,53,10,22,43,15,57). Given 
the existence of such intrinsic coordinates, the axiom of generalized 
coordinates appears inevitable if identification of the internal 
mathematical language, actually used by neuronal networks, is 
intended, especially since the mathematical fundamentals of 
transforming such covariant- contravariant and mixed tensorial 
expressions in non-orthogonal general frames are well established 
(30). 

A quantitative example of generalized coordinate systems intrinsic in 
a basic sensorimotor compensatory reflex, the vestibulo-collic head- 
stabilization apparatus is given in Fig. 1. (from 56).
Fig.1. Figure 1.
 


Fig.1. Tensor network model of transformations of intrinsic 
coordinates: a model of the three-step vestibulo-collic head- 
stabilization sensorimotor reflex (56). 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 (53). 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.

The three major classes of investments necessary for the tensorial 
approach follow from the above. Beyond the central task of 
development of the theory & its implementation; its concepts, 
formalisms, software and parallel hardware (neurocomputer) 
systems, a knowledge-base of the neuronal networks underlying CNS 
performance is required; this is available from neuroanatomy. Third, 
experimental establishment of a quantitative basis of the intrinsic 
coordinate systems is necessary; made available by the emerging 
field of computerized anatomy. This type of research originated with 
Helmholtz (22) , but has gained impetus recently (10,15,27- 
29,31,49,50,56,57,62,70). A quantitation is necessary in order to 
move eg. from illustrative joint-coordinates (Fig.2A) to anatomically 
correct biomechanical models of various musculo-skeletal systems 
that the CNS operates with. An exemple is given in Fig.1 (56), 
showing neuronal networks that transform general intrinsic 
coordinates from vestibular sensory- to neck-motor frame, and in 
Figs. 2D,3, which show preliminary data on coordinates intrinsic to 
various skeletomuscular systems.
Fig.2. Figure 2.
 

Fig.2. 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. Such intrinsic coordinates can be revealed for 
different musculoskeletal systems as eg. for human forelimb, shown 
schematically in D.

CNS operation can be represented by tensor transfomations, with 
these intrinsic coordinates at hand, since neuronal networks 
implement those within and among these general frames of 
reference. Tensors are mathematical operators connecting general co- 
ordinates, where one must distinguish between measurement-type 
orthogonal-projection vectorial representations, physically 
executable parallelogram-type vectors, and mixed expressions 
(covariant, contravariant and mixed tensors), 30,53). The features 
(53) and the emergence of the metric tensor (via the so-called 
Metaorganization-principle (55)), that comprises a multidimensional 
functional geometry transforming the dual representations from one 
to another, can also be expressed tensorially. The metric tensor 
operation was identified as a basic functional characteristics of 
sensorimotor networks, as elaborated for the cerebellum (53-55, 39- 
40). These mathematically exact operations are implemented by  
neuronal networks as shown in the upper part of Fig.1.

Metaorganization-algorithm. How vectorial expressions within and 
among various general frames are transformed by the CNS is 
summarized by a 3-step tensorial scheme to transfer covariant 
sensory coordinates to contravariant components expressed in a 
different motor frame (39-53) : 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 (39,53), but results in a non-executable covariant 
expression: mi = cip . sp where cip= ui. wp and ui , wp are the i- 
th sensory unit-vector and p-th motor unit-vector. 3) Motor metric 
tensor (39,53-55) 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 over-completeness 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 : gj,k= Sm {1/lm+ . 
|Em > < Em|}, where Em and lm+ are the m-th Eigenvector of gj,k and 
its Eigenvalue (the latter replaced by 1 if it was 0) 

The above metaorganization-algorithm (55), built on a neuroscience 
knowledge-base and explaining the structure of existing networks 
(cf. Fig.1), has a number of features that offer themselves to 
applications. Sincethe algorithm uses general coordinates, there are 
no restrictions for the coordinate systems; sensory and motor frames 
can be different from one another, also in the number of their 
dimensions. The algorithm resolves over-completeness. Beyond, this 
algorithm describes how the structure of neuronal networks can 
become organized to impose the required coordinative functional 
geometry under the direct government of the physical geometry of 
the sensorimotor apparatus (hence meta- organization).

 4. Application of Tensor Network Theory: Connecting Neuroscience 
with Industries of Neurocomputers and Neurobots

 Once theory is elaborated (38-57) and experimentally verified 
(15,56), tensor geometry can serve as a general common language of 
both brain- and neurocomputer-function, used in neuroscience and 
robotics. Progress towards this goal is slow, since engineers can 
choose any frame (and opt for Cartesian ones for convenience) thus 
only neuroscientists promote general frames. However, starting to 
build a library in the most general language may well pay off: the 
same software may coordinate both overcomplete robots and  
muscle-systems (Figs.3-5).
Fig.3. Figure 3.
 

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. 

Sensorimotor systems represent the obvious overlap and proving 
gorund. The advantage of concentrating on sensorimotor operations 
(11,43,45) is, that progress can be made on proven and directly 
verifiable grounds. Theories aside, this field presents actual 
neuronal networks of the brain. For instance, sensorimotor reflexes 
(such as involved in head- and eye stabilization) are rather well 
known (50,56), and one third of the mass of the brain, the 
cerebellum (that is responsible for motor coordination) is the part of 
the CNS whose neuronal networks have been the most throroughly 
investigated (5,40). Most importantly for utilizations, however, 
algorithms of physically implemented brain function can be directly 
applied in robotics. As in natural evolution, neural paradigms will be 
selected starting at most rudimentary motor control tasks and 
proceed to equip man-made instruments with brain-like sensory 
systems. Vision, pattern recognition and intelligent decision-making 
(38,44) are further stages of a co-evoultion of brain theory and 
neurobotics, starting with coordinated control both in natural (gaze, 
locomotory, haptic) and man-made (robotic) sensorimotor systems.
Fig.4. Figure 4.
 


Based on the scheme shown in Fig. 2 and the coordination-algorith 
shown above, Figs. 3. and 4-5. (with the accompanying video) 
illustrate the application of the same (tensorial, i.e. dimension-free) 
"netware", utilized to generate a co-ordinated movement of 
overcomplate systems. A three-jointed robotic arm moving in a two- 
dimensional plane is shown in Fig. 3, and a 7-muscle, 5-joint skeleto- 
muscular system of the human head, and 3-joint, 7-muscle system of 
human leg is shown in Figs. 4-5. At the present preliminary stage of 
development,  these projects illuminate the potential inherent in 
the tensorial approach (49). It is obvious, that in terms of the 
anatomical precision, biophysical realism of the modeled muscles, 
usage of generalized coordinates incorporating dynamic space-time 
representation, as well as the sophistication of the software system 
substantial progress is to be made. Nevertheless, such projects 
indicate that tensorial brain theory, the mathematical language of 
general intrinsic coordinates, and paradigms learned from real 
neuronal networks will lead to sensorimotor neurocomputer 
applications that are in many ways closer to biology and robotics 
than the usual implementations of association-schemes.
Fig.5. Figure 5.
 


The need to progress of neurocomputation from Nature's networks 
towards a new breed of brain-like hardware (neurochips or even 
biochips), is illustrated by Fig.5 (and its video), which projects the 
use of tensorial modeling for Functional Neuromuscular Stimulation 
of paraplegics Ð a research field in need of massively parallel control 
algorithm (18,26,32,49). For the neural paradigm of cerebellar-type 
motor coordination (39-40), several steps have already been 
accomplished. The first was to experimentally reveal the networks.  
Second, to arrive at mathematical algorithms, eg. the 
metaorganization process, that these networks implement (55). 
Third, to build the software (using von-Neumann computers) for 
applications (41,27-29,31,49). Fig.3. shows the utilization of the 
tensorial "netware" utilized to generate a coordinated movement of 
the overcomplete system of a preliminary 3-joint, 7-muscle 
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 neurocomputers and neurochips that will 
implement the meta-organization-algorithm in a massively parallel, 
brain-like manner. Conclusions

 In many ways, to predict the future of neurocomputers is similar to 
how the progress of von Neumann-type computers was foreseen in 
the nineteen-forties. Underestimatation is an understatement. Safest 
is to predict the technological progress. While presently for 
"neurocomputers" it is easiest to thrive at the levels of software 
implementation and custom parallel-boards of von Neumann- 
machines (72, 45, 21), future development will be undoubtedly 
directed towards special hardware; either electronic- (16,66) : 
parallel architecture and/or VLSI "neurochip", or optical realization 
(59). In the network-algorithms used, progress already points far 
beyond the existing associative-, visual processing, and adaptive 
coordination algorithms (13,17,24,67,74), but in this direction no safe 
prediction can be made. This is illustrated by mentioning two 
presently esoteric possibilities. Presently, "reading the mind" by 
multielectrode-arrays (61) is not only technically difficult, but the 
interpretation of such signals as points in an n-space is theoretically 
underdeveloped since the underlying (non-Riemannian) CNS 
geometry is yet unexplored (49) . Future neurocomputers, capable of 
implementing (better said, forming) a higher-order geometry, could 
serve the role of interpreting parallel recordings. From then on, it is 
only a further step to directly match natural and artificial 
"neurocomputers". While an interface will require a virtual 
technological wonder, other presently futuristic projections will be 
much easier on technology (but harder on psychology and sociology). 
If "metaorganization" of geometries, in terms of coupled neuronal 
networks, is possible in natural systems by means of an n-th order 
hierarchy of representations, (55), the learned principle could be 
utilized to create, in neurocomputers, an n+1-th order, and later an 
n+m order, more intelligent, geometrical representation. Thus, since 
neurocomputers are electronic and not biochemical organisms, not 
only their speed of operation will surpass that of the brain (by 
about a few orders of magnitude), but also their speed of evolution; a 
few decades against millions of years. Acknowledgement: This 
research was supported by NS 13742 and 22999 from NINCDS

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