Figure 1.

Fig. 1. Fields of research, potentially benefitting from a conceptually and formally homogeneous brain theory, such as Tensor Network Theory of the Central Nervous System

	In its philosophy, tensor network theory is based on the conviction (see in detail in 
Churchland, 1986), that brain theory becomes more a part of natural sciences if  abandons the 
dogma of emulating the most advanced technology.  Instead, it had better build its own theoretical 
structure on carefully laid axioms and utilize, of course, the most powerful mathematical approach  
available in the natural sciences to represent universal invariants (cf. Pellionisz 1986b).  The 
specific concept and formalism in tensor network theory is that brain function is implemented by 
neuronal network transformations that represent physical objects by dual, sensory and motor type 
multidimensional general vectors  (mathematically, these are covariant and contravariant tensors, cf. 
Bickley & Gibson, 1962, Pellionisz & Llinás 1980). Based on such reductionalist predilection, 
tensor theory approaches the brain-mind structurofunctional entity from the viewpoint of 
multidimensional functional geometry, using it to build  a geometrical representation  theory.  Thus, 
it is not by chance that the core of its mathematical apparatus is the one used in the unification of 
physical spacetime (theory of generalized reference-frame-aspecific vector-matrix operations, i.e.  
tensor transformations, as used e.g. in. relativity; cf. Levi-Civita 1926 and Einstein  1916).  
Therefore, tensor network theory  is philosophically much more akin to the modern 
multidimensional superstring-theory of the universe (cf. Green 1986), by elevating brain theory into 
the realm of abstract natural sciences, than eg. to the overreductionalist brain-model offered by the 
quantitative descriptive apparatus of electronic gain-controlled amplifiers which, in effect, makes 
brain theory a chapter in control-engineering.

TENSORIAL APPROACH TO BRAIN THEORY:  REPRESENTATION OF INVARIANTS BY 
GEOMETRICAL TRANSFORMATIONS OF INTRINSIC COORDINATES

 	Mathematically, tensor network theory is based on the fact that the structure of the physical 
geometry of the organisms determines those natural coordinate systems that are intrinsic to the  
expression of their function.  Therefore, adoption of the concept and formalism of coordinate-
system-aspecific generalized vectors and matrices (tensors) enables and liberates  one to deal with 
any  frame of reference, in fact "letting the brain speak in its own terms" (cf. Simpson & Graf, 
1985).  A characteristic example of coordinate systems that are specified by the physical geometry 
of the body is shown in Fig. 2.  concering a head-stabilizing neuronal apparatus, the so-called 
vestibulo-collic reflex (see in detail in Pellionisz & Peterson, 1987).

Figure 2.

Fig. 2. An example of implementing CNS function by transformations of general vectors among multidimensional, non-orthogonal coordinate systems. The frames of reference frames intrinsic to the vestibular-neck muscle sensorimotor reflex. A: The head-stabilization sensorimotor reflex includes the vestibular semicircular canal sensory apparatus, and neck-muscle motor apparatus. B: The vestibular apparatus is characterized by directions in the three-space belonging to A:anterior, P:posterior, H: horizontal semicircular canals (data from Blanks et al., 1972). C: The 30 major neck-muscles are characterized by rotational-directions (data from Baker et al., 1985). D: Motoneurons which generate rotations, expressed in the neck-mucle frame, have to receive signals transformed from the vestibular sensory frame in order to properly stabilize the head.

---

	 Passively occurring head-movements are measured by the vestibular semicircular canal 
apparatus, and are compensated for by expressing the same movement (with opposite sign) by 
means of coordinated contractions of neck-muscles. It is physically obvious, that the head contains 
built-in natural frames of reference for expressing its movements.  As anatomically measured by 
Blanks, Curthoys & Markham (1972, see Fig. 2 B) the three vestibular canals  form an arrangement 
whose characteristical axes constitute a coordinate system that  resembles  the well-known Cartesian 
(3-axis, orthogonal) frame of reference.  On theÊother hand,  as anatomically established by Baker, 
Goldberg & Peterson (1985), the head-rotational-axes belonging to  the pulling of neck muscles 
comprise a 30-axis arrangement (see Fig. 2C).  Indeed, this neck-motor frame  is one of the clearest 
examples of a highly non-orthogonal system of coordinates that is vastly overcomplete (since it uses 
ten times as many axes as the minimum required for expressing 3-dimensional rotations of a body 
around a center). Thus, this scheme demonstrates the possibility and importance of describing CNS 
function by means of transformations within and among  general coordinate systems.  While in case 
of a few highly specialized systems (eg. the vestibular canals) it is tempting to fall back on the use 
of Cartesian vectorial expressions, in most sensory and motor systems (let alone higher CNS 
functions) the frames intrinsic to the neuronal expressions just simply cannot be taken for granted.

	 Thus, when the CNS represents eg. head movements both in sensory and motor manner,  the 
question is not if  the brain implements transformations of head-rotation expressed in vestibular 
frame into head-rotation expressed in the neck-motor frame, but how  the CNS does it by its 
neuronal networks.  Further, the question is what group of  neuroscientists  is up to the challenge of 
making use, for their own purposes,  those potent and general concepts and formalisms that are 
made available for quantitatively dealing with such general coordinate system transformations, both 
in sensorimotor neuronal operations and elsewhere in the CNS. 

NEURONAL NETWORKS: GENERAL THEORY OF THE STRUCTURE AND FUNCTION OF 
REALISTIC NEURONAL CIRCUITS

	While tensor network theory has been formulated to provide a conceptually and 
mathematically homogeneous abstract  brain theory,  it aims at never loosing touch with the concrete 
neuroanatomical and physiological reality. By providing neuroscience with a network theory , it  
directly addresses those organizational  properties which are inherent in and intrinsic to the physical 
organization of the brain.   It has long been customary to mathematically represent the massively 
parallel neuronal networks of the CNS by matrices, which become here specific concrete 
implementations of the general reference-frame-aspecific tensor operators  (cf. Pellionisz 1986e).  
Availability of a general network theory may be significant, since it is a widely held view that 
neuroscience must have a powerful enough concept and formalism that can be accepted as an 
abstract understanding of the function of specific quantitative neuronal networks. 

Figure 3.

Fig. 3. Tensor network model of the three-step vestibulo-collic head-stabilization sensorimotor reflex (cf. Pellionisz & Peterson, 1987). Transformation form sensory coordinates to a motor frame (where the latter may be of higher dimensions) can be accomplished by a three-step tensorial scheme. The vestibular sensory metric tensor, vestibulo-collic sensorimotor tensor and contravariant neck motor tensor transformations can be expressed verbally, by abstract reference frame aspecific tensor-symbolism (see in text) or by matrix- (patch-) and network-diagrams. Here, the three matrices are shown, for the particular vestibular and oculomotor frames of the cat (cf. Fig. 2) by patch-diagrams only, and by a quantitative visualization of the corresponding neuronal networks that can accomplish such transfer.ÊÊNetwork diagrams illustrate the massively parallel architecture of the CNS, where convergences and divergences are the rule, and separated point-to- point connections rarely, if ever, characterize the structure.

---

	A particular elaboration of the above general features is shown in  Fig. 3. (cf. Pellionisz & 
Peterson, 1987).  This scheme illustrates one of the main difficulties posed by a general 
coordinate-system-transformation; i.e. the frames may be overcomplete.   For example, the neck-
motor system can produce the same movement using an infinite number of different patterns of 
muscle activation.  The solution proposed by Pellionisz (1984) utilizes the difference between 
covariant and contravariant representations of the desired movement, both expressed in the motor 
frame as determined by the muscle geometry.  The covariant presentation can be uniquely 
established by projecting the movement vector upon each of the muscle axes.  The problem is to 
find its unique inverse, the contravariant representation.  In an overcomplete system the problem is 
not that this does not exist but that there are an infinite number of inverses.  It has been proposed 
(Pellionisz 1983,1984) that the CNS chooses a unique solution the Moore-Penrose generalized 
inverse of the covariant metric (Albert, 1972), which may be implemented by a network that could 
plausibly be constructed by developing nervous systems (Pellionisz & Llinás, 1985).   Related 
models have been prepared for the vestibulo-ocular reflex (Simpson & Pellionisz, 1984), for the 
vestibulo-collic reflex (Peterson et al. 1985)  and for arm movements  (Gielen & Zuylen, 1986). 

	The solution in Fig. 3. is based on the three-step scheme of sensorimotor tensor 
transformation.  The task is to change a) the sensory frame into motor, b) the measured, covariant 
type vector to an executable contravariant version, c) to increase dimensions from three to thirty.  
The central, covariant embedding tensor accomplishes both a) and c), simply by projecting the 3 
sensory (i subscripts) upon the 30 motor axes (p subscripts), mathematically expressed as 
    						cip = si.mp,                      		           (1)
where s  and m  are the coordinates of the (normalized) sensory and motor axes, and each matrix-
element of cip is the inner (scalar) product of the vectors of coordinates of the i-th and p-th axis. 

	The reason that the cip covariant embedding tensor is necessary but not sufficient is that cip 
is a projective  tensor.  It turns a physical-type (contravariant) input vector into an output that is 
provided in its projection-components (covariants).  However, our case is the opposite; the 
available sensory input is covariant, while the output required is contravariant.  This is why the 
other two conversions are necessary; the vestibular sensory metric tensor gpr that converts 
covariant sensory reception into contravariant sensory perception, and the contravariant neck-motor 
metric gie  (the large 30x30 matrix in Fig. 3) that turns covariant motor intention into contravariant 
motor execution.  This general function of transforming covariant non-orthogonal versions into 
contravariant ones by a metric tensor can be accomplished for any given set of axes by a matrix of 
divergent-convergent neuronal connections among primary and secondary vestibular neurons and 
among brain-stem premotor neurons and neck-motoneurons (Baker et al. 1984).  The required 
contravariant metric tensor  gpr is the inverse of the covariant metric tensor  gpr:
						gpr=(gpr)-1					(2)
where  gpr  is the inner (scalar) product of the vectors of coordinates of the (normalized) axes  si : 
						gpr=si.si					(3)
	
	The question of how CNS neuronal networks can arrive at a unique covariant-to-
contravariant transformation led to the proposal of a metaorganization principle and procedure 
which utilizes the Moore-Penrose generalized inverse (Pellionisz 1983, 1984, Pellionisz and Llinás 
1985).  This solution is based on arriving at the eigenvectors of the system (those special vectors 
whose covariant and contravariant expressions have identical directions) by a reverberative 
oscillatory procedure (muscle proprioception recurring as motoneuron output, setting up stabilizing 
tremors).  The eigenvectors would imprint a matrix of neural connections that can serve as the 
proper coordination-device (eg. cerebellar neuronal circuit).  The unique inverse of  gie  can be 
obtained from the outer (dyadic matrix) product (symbolized by > < ) of the eigenvectors  Em,  
weighted by the inverses of the eigenvalues  lm  (the inverse is 0 if   lm=0, cf. Albert, 1972):

					gie=  S 1/lm . (Em > < Em).			(4)

	The tensor network model of the vestibulo-collic reflex emerges from the quantitative data 
of Fig.2. in the form shown in Fig. 3.  Each of the 3 matrices in the model is represented by a 
patch-diagram in which the size and sign of each matrix element are indicated by filled (positive) 
and open (negative) circles.  Four columns represent canal inputs (H,A,P, at the left side of the 
network-diagram), motor nerve outputs (LC.., at right side) and 2 intermediate neural stages.

Figure 4.

Fig. 4. "Tensor modules " in a network model of the vestibular-neck motor head- stabilization reflex, involving the cerebellum. In all, the 3-dimensional vestibular signals, expressed by covariant components, are transformed by the vestibulo-cerebellar neuronal network into 30- dimensional neck motor signals, expressed by contravariant vectorial components. This rendering of the tensor-matrices utilizes tensor-modules (see eg. the module marked "cerebellar nuclei"), where the n -dimensional input vector is shown by a strip of n incoming axons, theÊn -dimensional output vector is by a strip of n outgoing axons. The dendritic trees of output neurons are shown only by a representative single cell, and the synaptic connectivities among input and output neurons are shown by an n x n matrix, illustrated by patch-diagram. The basic 3-step transformation is implemented in the vestibular- and cerebellar nuclei. The cerebellum serves as an add-on circuit, which turns the covariant motor intention into a contravariant motor execution. The accessory optic system and olivary system serve the role of reporting on the misperformance of the head-compensation reflex thus the climbing fibers generate an ongoing modification of the cerebellar metric tensor (Simpson et al.,1979). AOS:Accessory Optic System, CF:climbing fibers, CN:cerebellar nucleofugal path, GC: granule cells, ME:motor execution signals, MF:mossy fibers, PC: Purkinje cells, PF: parallel fibers.

---

	Another rendering of the tensor network model of the vestibulocollic reflex is shown in Fig. 
4.  This presentation is basically a neuromorphological elaboration of the tensorial scheme of Fig.3.   
First, the transformation-matrices are not represented here by visually difficult-to-comprehend 
complex sets of interconnections (used in the top part of Fig. 3.), but by so-called "tensor modules" 
(cf.  part of Fig. 4, marked "cerebellar nuclei").  In such a module, the input vector arriving by the 
incoming axons is transformed by the synaptic interconnection-set  into an output vector (the 
connections are shown by patches, cf. bottom part of Fig.3 ).  A single dendritic tree of the output 
neurons is drawn to symbolize cells which implement the transformation.ÊÊA second difference in 
Fig. 4. in comparison with Fig. 3. is the detailed elaboration of the cerebellar embodiment of the 
motor metric tensor.  In Fig. 3., the third and last transformation is shown by a one-step network.  
This conversion, however, does not occur in a simple "throughput-type" network, but is performed 
by the cerebellar "add-on-type" network (cf. Pellionisz 1984).

	The "add-on" structurofunctional architecture of the cerebellum (probably a result of its 
character as "an evolutionary afterthought") enables direct sensorimotor operations even without  
any cerebellar contribution (motor performance is retained in case of cerebellar ablation but it 
becomes "dysmetric"; cf.   Bloedel 1985). The "add-on"  architecture is in accord with the 
hypothesis (Pellionisz, 1985b) that the function of the cerebellum is to turn motor intentions  (which 
are covariant  vectors, specifying the independent features of the goal, but whose components do 
not add up to properly make the performance) into motor executions   (which are contravariant  
vectors, whose components actually perform the goal).  Metaphorically, this cerebellar architecture 
is similar to a "secretarial antechamber" that intercepts and transforms the intentional commands  
emanating from the boss' main office.  For his "good intentions" could  directly operate the system 
even in the absence of the "add-on" side-loop of secretarial executive transformation. but such 
absence of executives typically results in "ataxic" performance of the system.  This metaphor also 
indicates that a "secretarial"Êknowledge of what executive commands should be attached to 
intentional goal-specifications requires an internal model of relations existing in the external system; 
a representation of the external geometry.

	In the network model of Fig. 4. which conforms with the structure of well-known cerebellar 
nets, the geometrical model of covariant-contravariant relationships is comprised in the "cerebellar 
nuclei"  tensor module. The additional circuits of the model (the accessory optic system and 
climbing fiber system arising from the inferior olive, cf. Simpson, Soodak & Hess, 1979)Êserve as 
aÊcorollary that monitors the misperformance of the cerebellar transformation, yielding ongoing 
adaptive modifications of the metric tensor (Llinás & Pellionisz 1985, cf. also Pellionisz & Llinás 
1979, Pellionisz 1986a).

SENSORIMOTOR SYSTEMS:  PROVING GROUND OF BRAIN THEORY

	Tensorial brain theory was first applied to sensorimotor systems.  First of all,  any brain 
theory should prove its adequacy on simple primary CNS operations before considered relevant for 
complex and high-level CNS tasks. It would be premature to forge theories of associations, pattern 
recognition, let alone tackling such almost purely philosophical problems as the neuronal 
embodiment of consciousness or "free will", if a particular approach in brain theory could not even 
explain the function of eg. simple 3-neuron structures as the vestibulo-ocular reflex arc (Pellionisz 
& Graf 1987).  Secondly, since sensorimotor operations are measurable and describable by 
physical means, it is possible to put forward, in case of sensorimotor transfer, not merely an 
abstract brain theory  but also its quantitative elaboration.  This should, in turn, result in  direct 
experimental comparison of experimental measurements with theoretical predictions (Peterson et al., 
1985, Gielen & Zuylen 1985).

	Third, as will be shown below, any theoretical understanding, gained from the knowledge of 
sensorimotor systems,  lends itself to direct utilization not only in the immediate fields relating to 
motor systems, such as kinesiology and  rehabilitation medicine but also in the biologically-related 
fields of  robotics  (neurobotics,  cf. Pellionisz 1983) and of the technology of brain-like computers 
(neuronal computers;  cf. Eckmiller, 1987).  An eventual co-evolution of brain theory  with some of 
the most important technological challenges  of our time may prove to be of substantial benefit both 
for constructing intelligent robots and brain-like computers, assuming that brain theory is not an 
epigon of the technology but can put forward its own leading theoretical foundation.

DATA-BANKS FOR QUANTITATIVE ANATOMY: COMPUTERIZED MAPS OF BODY-
COORDINATES

	Loooking beyond the basic challenge of creating and formulating a geometrical approach in 
brain theory, the first requirement towards its quantitative elaboration is the availability of 
morphological data that specify the coordinate systems intrinsic to the physical geometry of living 
organisms. Quantitative morphological specification of sensorimotor systems begun about a century 
ago by Helmholtz' (1896) measurements of the extraocular musculature.  This field has experienced 
a rapid growth in the past decade (Blanks et al, 1972, Ezure & Graf 1984, Simpson et al, 1986, 
Daunicht & Pellionisz 1986c). The data-sets should ideally be obtained in a manner that they are 
compatible with one another as well as with the theoretical requirements.  Also, it would be useful  
to make them widely and conveniently available for the research community.  Therefore, it is 
expected that the field will soon support the establishment of data-banks to gather, hold and 
disseminate the findings of the rapidly emerging discipline of quantitative computerized anatomy.  
The data-banks could serve as nodes of a computer network. With the widespread availability of 
today's economical graphic work-stations, linked by telephone network-connections, quantitative 
sensorimotor research and its applications  will no doubt experience a quantuum jump of efficiency.  
As outlined below, such modernized approach to reveal the structure of living organisms is 
expected to have a major impact on a range of fields of research and applications.

REHABILITATION MEDICINE AND KINESIOLOGY: FUNCTIONAL MUSCLE STIMULATION 
AND EMG-MOTOR UNIT INTERPRETATION

	The main contribution of the tensorial approach to the sensorimotor field may be that of 
providing with a quantitative theory (eg. by offering a general solution for overcomplete 
musculature, cf. Simpson & Pellionisz 1984, Peterson et al. 1985, Gielen & Zuylen 1986).  Never-
theless, potential use of its elements, eg. the Moore-Penrose generalized inverse, can be illustrated 
even in such simple structures of anatomy as the skeleton  of the neck-motor apparatus (Fig. 5A).

Figure 5.

Fig. 5. Coordination of an overcomplete skeleto-motor system. Predictions are calculated by the Moore-Penrose generalized inverse of the covariant metric tensor of the coordinates intrinsic to the 8 cervical joint -7 neck muscle motor apparatus. (Programming by: A.Pellionisz & J.Laczk—, anatomy by: F.Richmond, J. Baker, P. Vidal & W.Graf, tensor model by A. Pellionisz & B.Peterson). A: Tensor model of constraints of movements inherent in the overcomplete skeletal apparatus composed of 8 cervical joints. The coordinate-axes of the displacement of the head are determined by the joint rotation-pointss. A movement intention (see arrow) is decomposed into covariant intention components and transformed into contravariants by the Moore-Penrose inverse. B: Pulling of each of the 7 representative muscles determine a displacement of the head (in case of multiarticular muscles, the knowledge of the relative stiffness of joints is required). C: Movement intention, similar to that in A, will produce a head-shift, with almost all movement at C1/C2 and C7. Predicted muscle-activations correspond to EMG signals, thus could be a basis for functional stimulation and EMG interpretation. D: Model, identical to that in C, producing a markedly different movement-pattern for a "look-up" motor intention. Movement is almost exclusively C1/C2 rotation, without neck-tilt ("EMG"-pattern is also different). "Motor strategies" may arise from a single model.

---

	The most rudimentary physical geometry underlying motor performance is the skeletal 
structure.  Once the position and the measurements of the vertebral column of the cat's neck is 
established (cf. Vidal, Graf & Berthoz, 1986), it is possible to use the quantitative tensor model to 
predict the nature of the constraints that the Moore-Penrose generalized inverse would impose on 
head-movements executed with the use of this overcomplete joint-structure.ÊThe intrinsic system of 
coordinates for the 2-dimensional displacement of the head (specified by displacement of  the cat's 
eye) can be calculated from the x,y coordinates of the vertebral rotation-joints.ÊÊAs shown in Fig. 
5A, given a motor intention-vector, the Moore-Penrose generalized inverse of the covariant metric 
tensor of the intrinsic frame will determine a characteristical movement-pattern of the head, in which 
the cervical column remains a rigid body and almost all of the movement is generated by rotation 
around C1/C2 and C7  joints (cf. Vidal et al., 1986).

	While it is important to study the fundamental physical constraints of motor performance 
imposed by the skeleton, movements are controlled by the CNS not in a rather low-dimensional, 
nonetheless overcomplete, joint-coordinate space,  but in very high dimensional neuronal  
coordinate space.  Between these two extremes of dimensionality is the muscle-space, spun over the 
coordinates intrinsic to the pulling of the individual muscles. Fig. 5B-D show a preliminary study 
of the motor control of a musculoskeletal system from the muscle space .  As seen, even though the 
apparatus is overcomplete  since two dimensional dis-placements of the eye-center are determined 
by 8 joints and 7 muscles, the tensorial approach can predict, with the use of the Moore-Penrose 
generalized inverse, a unique execution of movement.  It is a characteristical feature of the model 
that the pattern of movements differ greatly on the motor intention (specified by the displacement of 
the eye-center).  For example, Fig. 5C shows a head-movement similar to one in Fig. 5A. 
However, if the intention is to look up (Fig. 5D) the identical  model will display a movement-
pattern where practically all rotation occurs at C1/C2 and the neck will not tilt.  This model 
suggests, therefore, that the "different strategies" occurring in motor performance (Nashner, 1977) 
may be an epiphenomenon of one underlying model and might not invoke a set of different 
mechanisms to choose from. The model in Fig. 5B-D goes beyond the skeletal model only in the 
sense that  in case of multiarcticular muscles the calculation of the intrinsic coordinate system 
(establishing the axes that belong to the pull of individual muscles) also neccessitates an assumption 
of the relative stiffness of the joints.

	The study shown in Fig. 5. can predict an activation-pattern of an overcomplete number of 
muscles in case of a coordinated movement. This illustrates the potential use of the tensorial 
approach in the fields of prothestics (Mann, 1981), and functional neuromuscular stimulation 
(FNS, cf. Kralj & Brobelnik, 1977, Mauritz, 1986, Gruner 1986).  In these applications a central 
problem is to arrive at a biologically realistic algorithm  which can generate the unique set of an 
overly large number of muscle activation components that are necessary to make an intended 
movement.  The tensorial analysis could also prove to be useful for the interpretation of large 
numbers of EMG and motor unit measurements (cf. Loeb & Richmond,1986), where the problem 
is, again, on what theoretical grounds to conceptually unify and interpret the multiple sets of 
quantitative experimental data.  The problem of interpreting multi-unit EMG and motor unit signals 
also relates to the question discussed in the last section of this paper.

NEUROBOTICS: UNIFIED GEOMETRICAL THEORY OF INTELLIGENT ORGANISMS

	Treating motor control problems in terms of multidimensional geometry (with the use of 
general coordinates) may have an importance in a wider context.   Namely, it could lead to a 
generally applied formalism that yields the means for the unification of fields that are as closely 
related to sensorimotor research as kinesiology, sports medicine and ergonomy (both in civilian and 
other applications) and also with those that presently seem to be beyond the realm of the biological 
sciences.  As discussed elsewhere (Pellionisz 1983, 1985c, Loeb 1983)  by adopting, both  in 
robotics  and neuroscience,  a common language, eg. the formalism of generalized vectorial 
expressions (not just those expressed in Cartesian 3-dimensional, orthogonal frames)  these fields 
could be united by their common language. Finally, in the widest context, the question of how the 
CNS may exert communication, control and command operations on a most complex (living) 
organism, in terms of multidimensional geometry and by means of massively parallel computation, 
is not without the interest of c3 theorists (cf. Ingber in this Volume).

COMPUTING BY NEURONAL NETWORKS: THE NATURE OF COMPUTATION  AND  THE 
STRUCTURE OF THE NETWORKS

	Presently, there is a rapidly growing interest in computing by neuronal networks (cf. This 
Volume, also Eckmiller 1987).  Thus, the question may arise how the tensorial approach relates to 
this unfolding trend. First of all, while other approaches aim at interpreting the function of 
imaginary neuronal networks that lack any specific structure (characterized by a set of  "everything-
to-everything" interconnections), the tensor approach deals with existing, not arbitrary, neuronal 
networks  (such as vestibulo-ocular, vestibulo-collicular and cerebellar networks).  Further, this 
approach provides  formal means of handling both their structure (cf. the "tensor module" above) 
and their function, in terms of transformation of general vectorial expressions.  Perhaps the most 
important difference is, however, that the tensor formalism defines the intrinsic mathematical nature 
of computation : stating that the calculations performed by networks are transformations of 
generalized vectors that are expressed in intrinsic coordinates (Pellionisz, 1986d).  Thus, in case of 
the cerebellum, for example, it is possible to state the general function of specific cerebellar circuits  
(eg. in different species), ie. that all individually different cerebellar circuits implement a general 
covariant-contravariant metric tensor transformation.  As a matter of course, it can be reasonably 
expected that by adopting the axiom of general coordinates, a large part of the research done in the 
field of associative memories and intelligence will gain new dimensions in the not-so-far future.

SINGLE CELL ELECTROPHYSIOLOGY: EXPLORATION OF INTRINSIC COORDINATES

	Lowering our sights from the distant vistas to present-day possibilites and necessities, a 
practical and immediate question is how the inherently multidimensional  theories may relate to data-
procurement by classical and widely available single-cell  recordings.  Since it is not the actually 
utilized technique that determines the fundamental merits of a scientific project but the potency of the 
underlying scientific hypothesis, it is therefore  proposed here that by adopting a multidimensional 
concept   even single-cell recordings may quickly gain new significance.  An example of this may 
be the exploration of coordinates intrinsic to neuronal function in the CNS.  In case of sensorimotor 
systems,  sensory detectors  (eg. primary vestibular neurons) must, by definition, use the frame 
intrinsic to the structure of sensory mechanism (the vestibular canals).  On the other hand, motor 
effectors (eg. oculo-motor or neck-motor neurons) must utilize the frame intrinsic to the 
musculature.  Thus, when detecting direction-sensitive firings of neurons in the middle of a 
sensorimotor apparatus (eg. brain-stem saccadic bursters, or neurons of the motor cortex; cf. 
Georgeopoulos, Schwartz & Kettner 1986), an immediate question is whether these neurons use 
the sensory or the motor frame or something other.  In fact, based on available data (Simpson et al., 
1986) it has already been proposed  that  these cells may use a coordinate system that is neither the 
sensory nor the motor frame, but the eigenvector-frame  of the extraocular mucle apparatus 
(Pellionisz, 1986c). Since eigenvector-frames have been calculated for several species (Pellionisz 
1985a, Pellionisz & Graf 1987, Pellionisz 1986c, Daunicht and Pellionisz 1986), quantitative 
predictions  are already available to be tested in a comparative manner, since predictions of the 
eigendirections are different in various species.  These theoretical predictions could be verified or 
rejected by means of experimental investigations using only classical single-unit recordings.

MULTI-UNIT PHYSIOLOGY: CORRELATION COEFFICIENTS AS METRIC TENSOR: EXPLORING 
THE GEOMETRY OF FUNCTIONAL CNS HYPERSPACES DEFINED OVER MULTI-UNIT SIGNALS

	Although, for technical reasons, classical electophysiological methods have been developed 
for single units it has been evident to most workers that, given the axiom that the CNS is a 
massively parallel system, sooner or later experimental methods needed to be invented to access a 
multitude of neurons simultaneously (see review in  Llinás, 1974).  Such, so-called multi-unit 
recording techniques have, indeed, been pioneered through the past decades (cf. Freeman, 1975, 
Gerstein et al. 1983, Reitboeck 1983, Bower & Llinás 1983). Partly because establishing, master-
ing and honing such  techniques is an exceedingly demanding endeavor, attention has only recently  
been focussed on the further, and equally excruciating  question of how to theoretically interpret the 
vast arrays of data  made available by such parallel methods.  At first, the mere visualization of such 
parallel recordings is satisfactory (Bower & Llinás, 1983), since it represents the long-awaited 
fullfillment of the dream by Sherrington (1906), who envisioned the massively parallel brain 
function in the form of the dynamic flickerings of myriads of neurons as an  "enchanted loom".

	The classical quantitative  analysis of multi-unit data  is the cross-correlation technique (cf. 
review in Gerstein et al., 1983).  This method concludes in establishing n x n  tables of cross-
correlogram coefficients among  n  signals.   One of the many advantages of this stochastical 
approach is the availability of software for this conventional quantitative computer analysis. The 
most important shortcoming inherent in correlograms is, however,  that they have hitherto been the 
end-product of the analysis.  The interpretation and evaluation of  the n x n  tables of cross-
correlograms (in case of n data source) is, however,  a source of frustration for the neuroscience 
community (cf. Kruger 1982).

	Another, more recent  fundamental concept of interpreting multi-unit recordings is the massive 
data-compression of n  recordings along time into the movement in time of a single point in a 
functional n -space . This extremely powerful concept, which was pioneered by Aertsen, Gernstein 
& Johannesma (1986), is depicted in Fig. 6.

Figure 6.

Fig. 6. The functional geometry inherent in CNS hyperspaces defined over multi-unit signals is not a matter of convenient assumption of an Euclidean metric. On the contrary; establishment of the metric tensor of the unknown geometry is the goal of multi-unit experimentation. A: multi-unit recording symbolized by n =3 signal sources. B: "Point in the n-space" concept (Aartsen et al. 1983) of interpreting the recorded activities (eg. firings of neurons). C: Convenient, but unsupported assumption of an Euclidean "flat" geometry in the n-space permits calculation of geometrical features, but the working hypothesis of the Kronecker delta serves as the metric tensor is untenable. D: The concept of a proposed approach in multi-unit recordings: The functional geometry of the n-space is unknown, it is to be established by determining its metric tensor.

---

	 In such an approach, the individual activities in the multi-unit recording represent at every 
time-point an ordered set of quantitites; a mathematical vector.  The coordinates are, then, taken as 
representing a point in the n -space.  Although this concept would open the way to comprehensive 
geometrical interpretation of multi-unit recordings, such as calculation of distances, directions, 
trajectories, center of mass, gravitational clustering and similar geometrical features, such 
calculations are possible only in the case if the geometry of the n-dimensional hyperspace is known.   
As it has often been pointed out (see eg. the note added in proof in Pellionisz & Llinás 1985, # 2), 
however, a central problem  of brain theory is that there is absoulutely no assurance that the CNS 
functional hyperspaces are limited to either simple Euclidean or even to Riemannian geometry.  

		When postulating aÊfunctional hyperspace over activities of multi-units, the problems with 
arbitrarily assumed geometries become painfully obvious. The first question is whether the space is 
spun over discrete or continuous  variables of coordinate components.  While most workers operate 
with the tacit assumption that neuronal activities represent "continuous" variables (eg. frequencies), 
moreover, that the manifold is derivable, "smooth" (Aertsen, Gerstein & Johannesma, 1986), even 
this working hypothesis is not universal.   Assumptions of a discrete space, spun over 0,+1 (or -1, 
+1) binary values of neuronal activity-variables can still be found, possibly because of the remnants 
of the "Computers=Brains" McCulloch-Pitts school (where neurons were considered as flip-flop 
binary units, just like computer-elements).  Postulation of such discontinuous, thus non-derivable 
(non-smooth) manifold is particularly questionable in case of interpretations of multi-unit recordings 
from the cerebellum (Carman, Rasnow & Bower 1986).  Operations of this organ, throughout 
evolution, centered around vestibulo-cerebellar transformations. The vestibulo-cerebellar 
appparatus, however, is well-known  to employ a frequency-coding (see eg. Bloedel 1985), 
resulting in a reasonably smooth and continuous functional space.  A further questionable 
assumption is the postulation of a highly specific structure of CNS multidimensional functional 
manifold (eg. invoking a geometry with Hamming-distances; Carman et al. 1986), since there is 
absolutely no guarantee that such geometry,  indeed, is manifested by CNS function.  In most 
approaches, in fact, the simplest and most parsimonious assumptions are introduced, such that the 
functional multidimensional hyperspace is continuous and "smooth", moreover, that  it is endowed 
with a position-independent "flat" Euclidean geometry  (cf. Aertsen et al. 1986).  While most 
workers are keenly aware of the provisionary nature of such initial postulates (which only serve 
technical convenience), one cannot overemphasize the stopgap nature ofÊthis compromise, lest 
some followers might be led to the mistaken belief that  the geometry of CNS functional spaces is 
truly known.	

	In contrast, as depicted in Fig. 6., the nature of the geometry of CNS functional hyperspaces 
is not a matter of convenient assumption, but represents the very challenge that neuroscience must, 
at some time, squarely face and properly meet.  In fact, neuronal functional manifolds may well be 
endowed with complex geometries that are characterized by a metric tensor which is position- 
dependent (the space being curved),Êthe axes could be non-orthogonal, non-rectilinear (curvilinear) 
or even only locally linear.  Further, the distinct possibility exists that some CNS hyperspaces (eg. 
cognitive neocortical spaces in infants) may not have, at an early developmental stage, an explicite 
structured geometry at all.  It is possible, that  "learning", defined here as the structuring of the 
geometry of the functional space, may start with amorphic, "chaotic" spaces with no metric tensor at 
all.  While the above arguments are tacitly accepted in general, it often presents an irresistable 
temptation that the assumption of a Euclidean metric, even if it is false, permits swift calculations of 
distances, directions, trajectories etc. in the CNS manifolds.  In contrast, an acknowledgement that 
the geometry is, indeed, unknown would keep such activities on an uncertain hold until methods for 
establishing the unknown metric were made available.

Figure 7.

Fig. 7. Concept of the proposal, that the table of cross-correlation coefficients (r) of the activity of n -signal sources approaches the table of covariant metric tensor components (g). Both the coupling of covariant (projection-type) vectorial components (r) , and the angle between the axes (g=cos(j)) expresses the same measure: "how close are a and b ?".

---

	In an attempt to break through the above impass and to contribute to further fruitution of the 
seeds inherent in the above-mentioned existing techniques, an approach is proposed below 
(announced in Soc. Neurosci. Convention, 1986), which in effect could synthesize the  "table-of-
cross-correlograms" stochastical interpretation with the "point-in-the-n -space" geometrical  
analysis.  Such unification may open a new way to reveal features of the metric tensor of the 
geometry of the functional  n -dimensional hyperspace.  The proposal hinges on the consideration 
that if the points in the n -space are viewed as expressed in a general, non-orthogonal frame (which, 
however,  we may not know) then the n x n correlation coefficient table contains statistical measures 
of the coupling between the separate coordinates (eg. a, b components) belonging to the points.  If 
the (unknown) axes were perfectly aligned, the a and b values would be identical (coupled by 
coefficient 1), whereas in case of an orthogonal set of axes the a  and b  values would be 
independent (the coupling would be 0).  Thus, the correlation-coefficients, by expressing the degree 
of how close are a  and b  ,  are directly  related to the angle between the coordinate-axes, therefore 
correlograms may help us establish the relation of the unknown axes. 

Figure 8.

Fig. 8. Quantitative elaboration of the proposal. Comparison of the covariant metric tensor and the cross-correlation-coefficient r, calculated for four randomly selected points in a two-axis frame (the angle of axes incremented by 5¡). A:Comparison of r and g reveals a similarity of these measures, even if only four data-points are considered.ÊC-B:Two-axis frame with four data points. Covariant vector components are closely coupled (close to 1) if the cosine of the angle of axes is near 1, whereas the coupling is loose (close to 0) if the cosine is nearing 0. Formulae at bottom show the conventional method of calculating correlogram-coefficients, and the covariant metric tensor.

---

	The above conceptual intuition has been mathematically explored in the study shown in Fig. 
8.  For demonstration purposes, a two-axis frame of reference was investigated, with a varying j 
angle between them (see Fig. 8. B and C).  For four randomly selected points, the covariant 
(projection-type) a and b  components were established.ÊÊVisual comparison of Fig. 8 B and C 
shows, that the a  and b  components are very close to one another if the j angle is small (Fig. 8 C), 
while the a and b components are rather different (eg. for points 2 and 4) if the  j angle is close to 
perpendicular (Fig.8 B).ÊThis visual impression is borne out by mathematical analysis, where the 
cross-correlogram coefficient  (r ) and the covariant metric  (g) are calculated by the conventional 
formulae below (where	the covariant metric yields the cosine of the angle of axes):    (5,6)

                                 
	Plotting r and g=cos(j) in Fig. 8 A. reveals that even in case of only four (different) 
randomly selected points in each two-axis frame (where  j was changed by 5 degree increments), 
the r and g=cos( j)  values are, indeed,  close enough to warrant further studies.

Figure 9.

Fig. 9. Convergence of the correlation - coefficient r to the covariant metric tensor g if the size of the statistical sample increases (from n=4 to n=10, n=100 and n=1000). A sample-size in the range of 100 is deemed sufficient for biological precision of 3-5%.

---

	It has to be emphasized, that the cross-correlogram method is an inherently statistical 
stochastic measure, while the geometrical measure of the closeness of the axes (the cosines of the 
angles between them) yields a single deterministic value of j.  Since in case of stochastical analysis 
the size of the statistical sample is crucial, therefore the calculation of r=g has been implemented in 
Fig.9. for a varying, much larger number than 4 randomly selected points, in order to ascertain the 
convergence of r=g with the increase of the number of points.  Comparison of the precision of r=g 
in case of 4, 10, 100 or 1000 points clearly shows that for a customary 3-5% biological precision 
the statistical sample need not be larger than about 100 measurement-points.  Given the fact that in 
multi-unit recordings firing of units can usually be obtained during protracted time (with literally 
thousands of unitary activities either in extracellular spiking, motor unit or EMG activities), the 
required number of sampling should pose no unsurmountable difficulty.

	The proposal for the convergence of  the correlation coeffient table to the table of covariant 
metric tensor components appears to be a useful beginning.  The road, however, is long towards 
synthesizing the classical statistical correlogram-analysis of multi-unit recordings with a recent, 
multidimensional geometrical interpretation.  However,  in the proposed approach the geometry (the 
metric tensor) of the multidimensional functional hyperspace is not taken for granted, but the very 
purpose of the analysis is to establish the unknown metric tensor.  Thus,  one can foresee that with 
enough time and investment new types of functional geometries of the CNS will be revealed, such 
that we have very little knowledge (or even imagination) about at the present time.

	 In order to provide a glimpse of the future possibilities, an arbitrary example is shown in Fig. 
10, to illuminate how one would go about conceptually and formally treating large n x n tables of 
cross-correlogram components.

Figure 10.

Fig. 10. Demonstration of a table of cross-correlogram coefficients considered equal to the covariant metric tensor, and calculating its dual contravariant metric tensor. The calculation uses the Moore-Penrose formula, yielding a proper inverse if the space is complete, and a generalized inverse if the space is overcomplete. With measurement of correlograms (left), and calculation of the dual metric (right), both metric tensors are available, thus the geometry of the functional space is determined. This enables one to calculate geometrical features of eigenvectors, distances, angles, geodesic trajectories, etc. (In this arbitrary demonstration the cross-correlograms were not taken from multi-unit measurements, but originated from the set of neck-motor axes shown in Fig.2.).

	Suppose that a 30 x 30 table of cross correlogram coefficients were experimentally established 
in a 30-electrode-recording.  It is visible in the left-side  plotting in Fig. 10, that the off-diagonals of 
the cross-correlogram component table are non-zeroes.  This means that the activities of the 
measured units are not independent of one another, but they are coupled. This could be the result of 
the activities arising from a coordinate system with non-orthogonal (non-independent) axes (in fact, 
in the given arbitrary example the coupled "recordings" originated from the neck-muscle axes, 
shown in Fig. 2).  The two questions that an investigator may ask are as follows:  a) is it possible to 
reconstruct the set of coordinate axes which yielded the given table?  b)  without the knowledge of 
the axes, is it possible to understand the functional geometry inherent in such recordings?  

	Positive answer to question a), in many cases, is not altogether that impossible.  If the firings 
arise eg. from motoneurons, which are connected to a set of muscles (as in the case of this 
exemplary demonstration shown in the left plotting in Fig. 10.), then measurement of the physical 
geometry of the muscular arrangement could directly reveal these axes.  However, in most cases, 
eg. when dealing with units deep inside a neuronal (eg. non-sensorimotor) apparatus, it might 
prove to be beyond our means to ever establish the underlying frames. Such may be the case, eg. 
when dealing with olfactory space (cf. Freeman, 1975), where the individual "axes" of detecting 
odors may never be physically established in the manner of how the muscle rotational axes or the 
vestibular canals can be visualized.  Nevertheless, if according to this proposal the table of cross-
correlogram coefficients is taken as the covariant metric tensor, its dual tensor, (the contravariant 
metric) can be established either by its regular inverse, or the Moore-Penrose generalized inverse.  

		
	Answering question b), a knowledge of both metric tensors (also called fundamental tensors, 
cf. Einstein 1916)  comprises a knowledge of the geometry of the n- space from which practically 
every geometrical property of the space can be calculated without knowing the axes themselves.  
One example, well worth mentioning here, is the calculation from the metric tensor of the functional 
eigenvectors, which lead to direct and feasible experimental hypotheses.  Mathematical spaces in the 
CNS, eg. the one in which the contravariant metric tensor is the Moore-Penrose generalized inverse 
of the covariant metric tensor, do not even have names at this point of time.  Yet, mathematical 
specification of the dual metric tensors would already enable one to properly calculate  trajectories 
(eg. geodesic lines), directions, distances, angles, center of gravity, gravitational clustering and 
many other geometrical features that we have just started to contemplate.

	The above survey of the vistas from Tensor Network Theory, although exhausting, is by no 
means exhaustive.  It is expected that since any  formally and conceptually coherent brain theory 
might exert a conglomerating and unifying effect on a wide range of disciplines, the significance of 
brain theory, that we have barely begun to appreciate, will only increase in the future.
*
 ACKNOWLEDGEMENTS

This work was supported by grants NS 22999 and NS  13742   from NINCDS
**

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END OF PAPER