Figure 1.

 

Figure 1.  Tensor network model of the vestibulo-collic reflex- arc in 
the cat  (after Pellionisz and Peterson 1988).  A: a general  
sensorimotor transformation-scheme of the vectorial expression of a  
goal in a non-orthogonal sensory system to an overcomplete, non- 
orthogonal motor frame. The sequence is an orthogonal projection-to- 
parallelogram component transformation in the sensory frame,  
projection of the motor axes on the sensory, and projection-to- 
parallelogram components in the exemplary motor frame, which will  
yield a performance of the goal.  B: Neuronal networks of this head- 
stabilization reflex use coordinates intrinsic to the structure of  
vestibular  semicircular canals (A: anterior, P: posterior, H:  
horizontal).  The rotational-axes of these canals have been  
anatomically measured (Blanks et al. 1972) and the paired canal- 
directions are displayed here in a pitch-roll-yaw frame. F: For the  
head-movement system, the rotational axes of neck-muscles have  
been measured (Baker and Wickland, 1988) and displayed here in  
the same pitch-roll-yaw frame as the vestibular directions. Data and  
abbreviations of the 15 muscles (of each side) follow the  
nomenclature of Pellionisz and Peterson, 1988).  C-D-E: Tensor  
network modules, schematically displaying tensor-transformation- 
matrices implemented by neuronal networks.   Arrays of patches  
display the tensor-matrix elements (corresponding to strengths of  
connections among input and output lines).  One cell in a "stack" of  
output neurons is visualized.  Such a module multiplies the vector of  
firing frequencies arriving through the input lines by the tensor- 
matrix implemented by the patch-interconnections, to yield the   
vector of firing frequencies of the output neurons.  The 3x3 sensory  
metric tensor (C), 3x30 sensorimotor covariant embedding  
transformation (D) and 30x30 motor metric tensor (E) have been  
calculated from data shown in B and F, according the the method  
shown in detail in Pellionisz 1985a. 

As for the basic mathematical concepts of how to represent  such 
intrinsic vectorial expressions and their transformations by  
networks, a key consideration is that Nature's frames of reference  
and transformations between and among them can be described by  
generalized vectorial (tensorial) formalism. The proposal that  
neuronal networks should be considered as tensors (Pellionisz and  
Llinás 1979) was based upon the conceptual definition that  
tensors  are mathematical operators expressing entities (i.e., the  
event that  the sensory inputs arise from and the response that the 
motor  activity generates) in any possible frame of reference in a  
generalized mathematical manner (cf. Levi-Civita 1926, Bickley and  
Gibson 1962).  Tensorial operations which, in simplest cases,  take  
the form of matrices, could thus express, via neuronal networks,  the  
stages of transformation of stimulus to response.  Fig. 1A shows a  
general scheme (valid in case of any sensory and motor frame;  
Pellionisz 1984), applicable for a set of three transformations with  
four different vectorial expressions.  Fig. 1. C,D,E show the tensor- 
matrices, shown by dot-diagrams, that represent the transformations  
imnplemented by means of neuronal networks.       Adoption of this 
general formalism enables discerning an  important difference 
between the types of vectorial representations  of sensory input and 
motor output in their respective coordinate  frames (even if these 
frames were identical).  The response of a  sensor, such as a 
semicircular canal (see Fig. 1A, stage 1), to a  stimulus is independent 
of the responses of other sensors and is  proportional to the cosine of 
the angle between the sensor's axis of  maximum sensitivity and the 
axis of the applied stimulus (or  equivalently to the projection of the 
stimulus vector upon the sensor  axis).  On the other hand, the 
muscle activations that generate the  motor response are not 
independent of one another since the forces  or torques they 
generate must sum in a parallelogram fashion to  produce the 
desired movement (see Fig. 1A, stage 4).  The projection- type 
vectorial representations are termed covariant in tensorial  
nomenclature, while the parallelogram-type representations are  
termed contra-variant  (cf. Fig. 1. in Bickley and Gibson 1962).  The  
CNS can thus be conceived of as a neuronal network performing  
tensorial transformations converting  a covariant sensory input in  
one frame into a contravariant motor output in another frame  
(Pellionisz and Llinás, 1980).      When constructing a 
tensorial model of the VCR, one must  consider an additional 
conceptual problem; raised by  overcompleteness  in sensorimotor 
systems.  A system is  overcomplete when the number of 
independent effectors (muscles)  exceeds the number of controlled 
degrees of freedom of the  apparatus they control.  The simplest 
example of overcompleteness  of a motor system is shown in Fig. 1A, 
where the exemplary motor  frame is 3-dimensional, compared to 
the 2 sensory axes in the 2D  plane. The difficulty posed by an 
overcomplete motor system is that  it can generate the same 
movement using an infinite number of  different patterns of muscle 
activation.  The modeler must then find  the type of criterion that the 
CNS uses in "choosing" the particular  pattern observed 
experimentally.  The tensorial modeling scheme  (Pellionisz 1984),  
utilizes the difference between covariant intention  and 
contravariant execution representations of the desired  movement. 
These vectorial versions, both given in the motor frame,  are 
determined by the muscle geometry. The covariant presentation  can 
always be found uniquely by projecting an invariant to the axes  of a 
frame.  The problem is then to find a unique contravariant  
representation.  In a non-overcomplete system this is just the  
inverse of the covariant metric tensor. In an overcomplete system  
the problem is not that such an inverse does not exist but that there  
are an infinite number of inverses.  It was hypothesized (Pellionisz  
1983,1984) that the nervous system chooses a unique solution,  
equivalent to the Moore-Penrose generalized inverse of the covariant  
metric (Albert 1972).  Beyond the fact that this inverse minimizes  
the sum of squares of activity of the muscles during any movement,  
it should be noted that it may be  implemented by a network  
(matrix) that could be constructed by the plausible biological process  
of reverberative oscillations in the developing nervous systems  
(Pellionisz and Llinás 1985).  As it is shown in detail  
elsewhere  (Pellionisz and Peterson 1988), it is this choice of an 
optimal inverse  that gives the VCR model its predictive power.  
Related models have  been prepared for the vestibuloocular reflex 
(Simpson and Pellionisz  1984) and tested in the case of voluntary 
arm movements (Gielen  and Zuylen 1986). The deviation of maximal 
EMG directions from  muscle rotation axes can be calculated from the 
Moore-Penrose  generalized inverse, and this allows one to make  
predictions of  patterns of motor activation, on the basis of the 
geometry of the  receptors and effectors.  In fact, this tensor model 
predicts the actual  VCR activation of the 6 neck muscles tested 
within the range of  experimental precision. 

2.2.  Sensorimotor Tensor Transformations

The above characterized theoretical solution is based on the  four-
stage (three-tensor) scheme of sensorimotor transformation  shown 
in Fig. 1A. The transformation-tensors are not shown in this  Fig. by 
matrices, but schematically by "tensor network modules"  (C,D,E). In 
such modules, elements of the calculated matrices are  represented 
by patches, symbolizing the strengths of connections  among input 
and output arrays of axons. The firing frequencies of  such a bundle 
of n axons are mathematically represented as n- dimensional vectors. 
The task performed in this scheme is threefold:  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 (Fig. 1D) 
accomplishes both a) and c), simply by  projecting the 3 sensory (i 
subscripts) upon the 30 motor axes (j  subscripts) which can be 
mathematically expressed as  cij = ui.vj,  where u and v are the 
coordinates of the (normalized) sensory and  motor axes, 
respectively, and each matrix-element of cij is the inner  (scalar) 
product of the vectors of coordinates of the i-th and j-th axis. 

The reason that the cij covariant embedding tensor is necessary  but 
not sufficient in converting the covariant sensory reception- vector 
ui into contravariant motor execution vj is that cij 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 task is the opposite; to turn the available  sensory 
input (which is covariant), into the output required (which  should be 
contravariant).  This is why the other two conversions in  the 
tensorial sensorimotor scheme are necessary. The sensory metric  
tensor gii (Fig. 1C)  converts the covariant sensory reception into  
contravariant sensory perception, and the motor metric gjj (Fig.1E)   
turns covariant motor intention into contravariant motor execution.   
This general function of transforming covariant non-orthogonal  
versions into contravariant ones by metric tensors can be  
accomplished for any given set of axes by a matrix implemented by a  
divergent-convergent set of neuronal connections, often reporting on  
different sensory modalities,  so characteristical for the CNS, eg.  
among primary and secondary vestibular neurons (Markham and  
Curthoys 1972,  Pompeiano 1975, Allum et al. 1976, Baker et al.  
1985), and among brain-stem premotor neurons and neck- 
motoneurons.    Mathematically, the required contravariant metric 
tensor gii  can be established as the inverse of the covariant metric 
tensor (gii): gii=(gii)-1 where components of gii are the inner (scalar) 
products of the arrays  of coordinates of the (normalized) axes:  
gii=ui.ui Two important questions arise regarding such metric  
transformations; a biological and a mathematical one.  First of all,  
even if such transformations are implemented by matrices of  
neuronal networks, the CNS does not arrive at them by mathematical  
computation, but by some procedure feasible for a biological system.  
The question relates to the nature of this unknown procedure.  
Second, at the level of pure mathematics, a problem occurs with  
overcomplete coordinate systems.  In such cases gii is singular (its  
determinant is zero), thus gii has an infinite number of inverses.  The  
question is how CNS neuronal networks can arrive at a unique  
covariant-to-contravariant transformation (even in case of  
overcompleteness).

An attempt, aimed at answering both questions jointly,  led to  the 
proposal of a metaorganization principle and procedure which  
utilizes the Moore-Penrose generalized inverse (Pellionisz 1983,  
1984, Pellionisz and Llinás 1985).  Biologically, the proposed  
solution  is based on arriving at special vectors whose covariant and  
contravariant expressions have identical directions (so-called  
eigenvectors of the system). This can be performed by the CNS in the  
form of a reverberative oscillatory procedure, where muscle  
proprioception recurs as motoneuron output, setting up tremors  
stabilizing in the eigenvectors.  These special activation-vectors  
would imprint a matrix of neural connections that can serve as the  
proper coordination-device (implemented, eg. by the cerebellar  
neuronal circuit).  Mathematically, this unique inverse of gjj can be  
obtained from the outer (dyadic matrix) product (symbolized by > < )  
of the eigenvectors Em, weighted by the inverses of the eigenvalues  
Lm, where 1/Lm=0 if Lm was 0:

gjj=    ·m   1/Lm . (Em > < Em) Once the Moore-Penrose generalized 
inverse is calculated by  the above formula for the third 
transformation, the model predicts  for each neck muscle a unique 
direction of head rotation for which  that muscle should  be 
maximally activated.  Muscle activation  during rotation about other 
axes is predicted to decline as the cosine  of the angle between those 
axes and the optimal axis.  As shown in  details elsewhere (Pellionisz 
and Peterson 1988), the predicted  optimal activation direction 
should typically differ quite significantly  from muscle pulling 
directions, but can be readily tested  experimentally with 
confirmatory results (Peterson et al, 1987).  

  Although pulling and activation directions are quite widely  
separated in this non-orthogonal system, the model predicts the  
activation directions within 4 to 11 degrees.  Thus the hypothesis  
that the CNS determines neck muscle activation patterns in a manner  
corresponding to the Moore-Penrose generalized inverse is supported  
by the fact that the model predicts the pattern of muscle activity,  
produced by the VCR in decerebrate cats, within the limitations of  
experimental error.

2.3.  Elaborations: Tensor Network Models  

The modeling approach described here opens avenues for  
substantial developments. First, while the data used in this paper are  
for the VCR of a traditional experimental animal, the cat, the problem  
addressed is applicable to many forms of motor control in a broad  
range of species.  Software is now available to construct similar  
models for any sensory-motor system where the geometry of sensors  
and muscles is made available by quantitative anatomical studies.

Second, investigators may wish to experimentally evaluate  some 
quantitative predictions of such simple tensorial models.   Gielen and 
Zuylen (1986) have recently reported successful  prediction of 
patterns of human arm muscle activation using a  tensorial model. 
The experimental approach of recording EMG  responses to multi-
directional stimuli is also broadly applicable and  could yield useful 
information about principles underlying motor  control in a variety 
of species.

A third possibility is to further explore the neuronal network- 
embodiments of such general coordinate transformations.  The  
manner of how this challenge is addressed is indicated in Fig. 2.  This  
multidimensional network scheme is the neuroanatomical  
elaboration of the rudimentary four-stage sensorimotor  
transformation shown in Fig.1. The scheme represents two major  
refinements.  One is based on the fact that the motor metric  
transformation is not implemented in a simple throughput manner  
(as shown in Fig. 1) but via an "add-on" organ, the cerebellum (cf.  
Bloedel et al. 1985).  This principle of organization is presented in  
detail elsewhere (Pellionisz 1984, 1985a,b, Pellionisz and  
Llinás  1985).

Figure 2.

 

Figure 2.  Neuroanatomically realistic multidimensional  network 
diagram of the tensor model of VCR in the cat.  This scheme  
incorporates the simplified network model shown in Fig. 1. into a  
circuitry accounting for the cerebellar "add-on" architecture,  
following the detailed model in Pellionisz 1985b, Pellionisz and  
Llinás  1985.  The input to the system is the vestibular 3-
component vector,  which is transformed through a 3x3 "tensor 
network module" of a  vestibular sensory metric and a 3x30 
sensorimotor covariant  embedding module into a 30-dimensional 
output vector leaving the  vestibular nuclei.  This nucleofugal signal 
carries a covariant motor  intention vector. This vector (without the 
cerebellum) would actuate  via the neck motor nuclei a dysmetric 
head movement (by means of   the 30 neck muscles).  The tensor 
network module of the motor  metric is embodied by the Purkinje 
cell (PC)-cerebellar nuclear  connection-system (cf. Fig.1E).  Thus, the 
ascending mossy fiber  (MF)-granule cell (GC)-parallel fiber (PF) 
intention-vector (which is  covariant) will be transformed by this 
metric into a contravariant  execution vector.  As the PC vector is 
inhibitory, this will result Ðwith  the mossy fiber collaterals into the 
cerebellar nucleiÐ a nucleofugal  vector that is the difference of motor 
intention and execution.  This  nucleofugal vector adding to the direct 
intention-vector will yield the  exact contravariant execution-signal 
for the compensatory head- movement.  The visual signal, reporting 
on the performance in  retinal frame (a 3-dimensional vectors) 
enters through the accessory  optic system (AOS), where it is 
transformed into a vector expressed  in the neck frame (Simpson et 
al. 1979).  Thus, the inferior olive, as a  comparator, will emit a 
climbing fiber (CF) vector that is the  difference of execution and 
performance.  This "error" vector,  projecting to the array of 
cerebellar nuclear cells  both directly (via  collaterals) and indirectly 
(via Purkinje cells) will generate ongoing  dyadic product corrections 
with the error vector, in effect altering  the metric properties of the 
cerebellar transformation.   


The parallelly organized multidimensional neuronal network shown  
in Fig. 2. performs the motor metric function via the connection- 
matrix of Purkinje cells (PC) with mossy fiber (MF) collaterals to the  
cerebellar nuclei.   Given that the execution-vectors (PC projections)  
are inhibitory, the nucleofugal output will be the difference of motor  
intention and execution vectors, which will yield in the neck motor  
nuclei the required execution-vector output.  The second aspect of  
elaboration is the path of multidimensional visual error-signals to  
the Purkinje cells (via the accessory optic system AOS to the inferior  
olive IO and to climbing fibers, CF, Simpson et al.1979, Maekawa and  
Simpson 1973). As elaborated in Pellionisz 1985b, the climbing fiber  
vector, by projecting to the cerebellar nuclei both directly (through  
collaterals) and indirectly (via the PC-s), in effect  changes the motor  
metric tensor function of the cerebellum: altering the curvature of  
the motor functional space in an ongoing manner. While the  
rapproachment of such multidimensional quantitative parallel  
network models to neuroanatomical realities will require much  
further studies, it seems evident already at this stage that  
representation of such circuits as loops of single axons with nuclei  
merely serving as relay stations will no longer suffice.

A fourth important direction of studies is the exploration of intrinsic  
coordinates of more abstract nature than those embedded in skeleto- 
muscular structure.

3. Generalized Functional (Other than Structural) Intrinsic  
Coordinates

The fact that brain function is expressed by multidimensional  
intrinsic coordinates is most obvious in case of anatomically explicit  
sensory- and motor systems. Neurons connected to the peripheries of  
sensorimotor systems (e.g. vestibular canals and neck muscles) must  
use structural frames. Their rotational axes have been quantitatively  
established for several species.  Typically, they are non-orthogonal   
overcomplete multidimensional frames (Fig. 1).  In this light it is  
surprising to find claims in the literature that central neurons of this  
system apparently use an orthogonal  frame: the paramedian pontine  
reticular formation reportedly contains medium-lead burst cells,  
whose firing rate is tightly related to eye velocity in either horizontal  
or vertical saccades (Luschei and Fuchs 1972, Keller 1974, BŸttner et  
al. 1976, King and Fuchs 1979, Gisbergen et al. 1981).  

The possibility that the CNS employs intrinsic coordinate  systems in 
its operation that are different from the ones at the  sensory- and 
motor ends, of course, cannot be excluded.  In fact, it  has been 
shown that functionally, not structurally determined  directional 
preferences (in our terms, intrinsic coordinates) do exist  in the 
visual system, both at the retina (Oyster, et al, 1972) and also  in its 
relayed form to the cerebellum (Maekawa and Simpson 1973).   
There are two main tasks: one is to experimentally identify such  
internal intrinsic frames, and the other is to theoretically interpret  
their functional significance. It is shown below that this paradox  
between a non-orthogonal structural frame and an orthogonal  
functional frame may receive an explanation in tensor theory with  
important implications to both theory and experimentation.    3.1.  
Tensorial Relationship Between Structural and Functional  Reference 
Frames of Brain Function: Saccade Neurons in Monkey  Utilize Frames 
Composed of the Eigenvectors of the Frame of  Extraocular Muscles

Tensor network theory of the CNS centers around the general  
question of transformations among intrinsic coordinates (Pellionisz  
1984). An even more profound problem is, however, how different  
(structural and functional) geometries are connected (Pellionisz and  
Llinás 1985). When attempting to relate the non-orthogonal  
structural frame with the apparently orthogonal functional frame,  
the question emerged as to whether the apparently orthogonal  
functional frame is, in fact, aligned with the eigenvectors of the non- 
orthogonal extraocular structural frame (Pellionisz 1986). 

Figure 3.

 

Figure 3.  Saccadic burster neurons in the monkey may use a  
functional frame of reference that is the eigenvector system of the  
structural frame of reference of the oculomotor muscles (after  
Pellionisz 1986). Panel A visualizes  data by quantitative morphology  
(from Simpson et al. 1986), showing the rotational axes, in a pitch-,  
roll-, yaw frame, of the six extraocular muscles in the rhesus  
monkey.  Data refer to the left eye, data and visualization uses right  
hand rule. LR: lateral rectus,ÊMR: medial rectus,  SR: superior rectus,  
IR; inferior rectus, SO: superior oblique, IO: inferior oblique. This  
diagram is used here to display obvious  features of this structural  
frame: a) the overcompleteness, b) the non-orthogonality, c) the  
gross deviation of the axes from pure horizontal and vertical  
directions. Panel B shows the eigendirections, calculated from the  
structural frame presented in A.  The table of E1-E3 eigenvectors  
(with L1-L3 eigenvalues) has been calculated as shown in Pellionisz  
1985a and Pellionisz and Graf 1987. The D1-D3 eigendirections are  
both tabulated and displayed in a frame identical to the one used in  
panel A.  As shown, the D2 and D1 eigendirections (preferred  
functional directions, since they are orthogonal, thus separable) are  
aligned with an almost pure vertical and horizontal retinal shift,  
within 5.4¡ precision.  Thus, the experimentally observed "vertical"  
and "horizontal" saccadic burster neurons (Luschei and Fuchs 1972,  
Keller 1974, BŸttner et al, 1977, King and Fuchs 1979, Gisbergen et  
al. 1981) may well utilize an abstract functional intrinsic frame that  
is the eigenvector-frame of the structural intrinsic coordinate  
system. 
 Availability of the rhesus monkey oculomotor frame 
(Fig. 3A;  data from Simpson et al, 1986) permits calculation of its 
eigenvectors   (Fig.3B).  The procedure of this calculation is given in  
Pellionisz  (1985a), Pellionisz and Llinás (1985),  Pellionisz 
and Graf   (1987).   Comparison of Figs. 3A nd B reveals that the 
above hypothesis meets  an affirmative answer:  Functional and 
structural geometries are  connected in a manner that the frame in 
one utilizes the eigenvectors  of the frame in the other.  This finding 
may have both important  theoretical aud experimental implications.  
From a theoretical  viewpoint, it was shown in the metaorganization 
of networks  (Pellionisz and Llinás 1985), that neurons must 
utilize the  eigenvectors of the metric tensor of the motor frame to 
enable  independent adaptive modifications, since eigenvectors form 
an  orthogonal set.  Experimentation could help by further 
investigating  this issue.  Data would be desirable in other species 
that have  different motor frames and eigenvectors (see Pellionisz 
1985a,  Pellionisz and Graf 1987, where such eigendirections are 
calculated  for the human and the cat). Quantitative properties of 
adaptive  coordination along eigenvectors could also be 
experimentally  explored. The paradigm of eigenvector-connection of 
structural and  functional frames therefore illustrates that a theory 
of CNS function  expressed with multidimensional intrinsic 
coordinates can provide  not only mathematical explanation of 
certain features of parallel  brain function, but also yield specific 
suggestions for experimental  investigation.

4.  Geometrization of Complex Descriptions of Sensorimotor Function:  
Motor Strategies, Trajectories, Posture.

Tensorial  modeling of the CNS's use of structural and  functional 
intrinsic coordinates is likely to present both major  difficulties and 
new possibilities. One of the biggest hindrance at  present is the 
meager availability of quantitative data on intrinsic  structural 
frames (and the shortage is even more serious for  functional 
neuronal frames). While quantitative computerized  anatomy is 
likely to become an abundant  source of data in the  future, 
techniques are explored at present that could alleviate the  present 
unavailability of quantitation.  For instance, anatomical  
measurements of the cat neck muscles (Baker and Wickland 1988)  
are only available for a fixed-head animal (where it is a good  
approximation that the head rotates around the single C1/C2 joint- 
point).  However, with more complex head-movements this  
approximation is not satisfactory, and thus data gathered by the  
"Helmholtz-method" (fixed origin- and insertion-points of muscles,  
fixed single rotational center) no longer yield sufficient  
approximation.  

A graphics-based computer software technique is developed  
(Pellionisz 1987a, Laczkó et al. 1987, Liverneaux et al. 1987,  
Lestienne et al. 1987), that removes the above bottlenecks, although  
this new method is not without limitations itself and is  
extraordinarily software-intensive.  Once a graphical rendering of the  
skeletomuscular system is available in the form of photographs, X- 
rays, drawings, etc., this  information is scanned into a graphical  
computer. Fig. 4. provides examples of the display of the cat's head  
skeletomuscular system (cf. Vidal et al. 1986, Peterson and Richmond  
1988).  Once (any number of) joint-rotation points, relative stiffness  
values, muscle origin and insertion points are established by the  
operator, relative to the (movable) skeletal parts, the intrinsic  
movement coordinates belonging to individual muscles are  
computed, together with the covariant metric tensor (table of cosines  
among muscle-axes) and its Moore-Penrose generalized inverse.  This  
computation can be refreshed during a movement if desired.  Thus,  
to any intention-movement, specified by the operator, the  
corresponding execution-components of the muscles are computed,  
and the resulting movement is displayed (also by a computer movie).  
Motor behavior of such systems can be studied restricted to the joint  
space (without muscles), or in muscle-space, and with high-enough  
resolution in the future, approximating the motoneuron-space. Some  
applications of this model, capable of quantitating the functional  
behavior of complex  skeletomuscular systems, already yielded  
insights to sensorimotor function. 

Figure 4.

 

Figure 4. Still pictures from a tensorial model -movie of the  head-
movements in the cat, using graphics-based establishment of  the 
intrinsic coordinates of the skeletomuscular system consisting of  the 
skull, all  cervical vertebrae and 7 exemplary neck-muscles. The  
head-shift movement-strategy  and head-tilt strategy  arise from an  
identical model, where only the command of the movement is  
different (an "intention" specified by the operator).  While the  
movement intentions are straight displacements, the model predicts  
curved trajectories  (an expression of the curved geometry of the  
functional motor space).  Also, both the head-movement display as  
well as the predicted activations of 7 muscles with time (right side of  
panels) can be dramatically different with somewhat different  
intentions. Software that extracts the structural intrinsic coordinates  
from graphical input and calculates the Moore-Penrose generalized  
inverse of the motor metric in a dimension-free (tensorial) manner,  
can be applied to several sensorimotor systems of many species. 


The display shown in Fig. 4. demonstrates that the identical  model 
may yield remarkably different "motor strategies" (a shift in  A, 
utilizing several cervical rotational points, while a tilt in B  basically 
around a single rotation-point) if somewhat different motor  
intention is imposed by the operator. Thus, it is expected that this  
method will be helpful for studies by quantitative modeling also  
those hip-, ankle-knee- "motor strategies" that seem to arise in  
experimental conditions (Nashner 1977).  A similar complex  
characterization of movements uses trajectories (curved paths) of  
movements that can be readily observed in most experimental  
conditions. It is, therefore, worth comparing the curved movements  
of appandages used in such tensorial skeletomuscular models even  
though the imposed intentions are straight lines.  Since in the model  
such curvature is the direct consequence of the position-dependence  
of the intrinsic coordinate system (and the position-dependence of  
the functional geometry and the metric tensor of the motor space),  
the trajectories arise in the model in a manner characterizing the  
underlying geometry. (Using a metaphor: trajectories of flowing  
water on a mountain-surface reveal the geometry of the structure of  
a mountain.)  Thus, it is expected that future studies by using  
advanced versions of tensorial models shown in this paper will be  
useful not only for revealing the neuronal networks that produce  
them, but also for quantitating high-level complex motor behavior  
patterns as "motor strategies", "trajectories" and, perhaps, even  
"posture" and "style".

* Acknowledgement

This research was supported by the Grant NS  22999 from  NINCDS

 ***

REFERENCES  Albert, A. (1972).  Regression and the Moore-Penrose  
Pseudoinverse.   New York:  Academic Press

Allum, J.H.J., Graf, W., Dichgans, J. and Schmidt, C.L. (1976) Visual-  
vestibular interactions in the vestibular nuclei of the goldfish. Exp.   
Brain Res. 26:463-485.

Baker, J., Goldberg, J. and Peterson, B. (1985) Spatial and temporal   
response properties of the  vestibulocollic reflex in decerebrate  cats. 
J. Neurophysiology ,   54:735-756.

Baker, J. and Wickland, C. (1988) Kinematic properties of the   
vestibulocollic reflex. In Peterson, B.W. and Richmond, F.J. (Eds.)   
Control of Head Movement,  New York, Oxford Univ. Press 

Berthoz, A. and Melvill-Jones, G. (1985) Adaptive Mechanisms in   
Gaze Control. Reviews in Oculomotor Research. Elsevier, Amsterdam.

Bernstein, N.A. (1947) O Postroyenii Dvizheniy (On the Construction   
of Movements), Moscow, Medgiz 

Bickley, W.G. and Gibson, R.E. (1962) Via Vector to Tensor.  New York:   
John Wiley and Sons.

Blanks, R.H.I., Curthoys, I.S. and Markham, C.H. (1972)  Planar   
relationships of semicircular  canals in the cat . American J. of   
Physiology,  223:55-62.

Blanks,R.H.I., Curthoys, I.S. and  Markham, C.H. (1975) Planar   
relationships of semicircular canals in man. Acta Otolaryng.  80:185-  
196

Bloedel, J.R., Dichgans, J. and Precht, W. (1985) Cerebellar Functions.   
Berlin: Springer Verlag

Bower, J. and Llinás, R. (1983) Simultaneous sampling of the   
responses of multiple, closely adjacent Purkinje cells responding to   
climbing fiber activation.Soci. for  Neurosci. Abstracts, 9, p.607.

Braitenberg, V. and Onesto, N. (1961) The cerebellar cortex as a   
timing organ. Discussion of an hypothesis. Proc. 1st Int.Conf.   
Med.Cybernet.. pp.1-19,  Giannini, Naples

BŸttner, U., Waespe, W. and Henn, V. (1976) Duration and direction of   
optokinetic afternystagmus as a function of stimulus exposure time   
in the monkey. Arch.Psychiatr.Nervenkr. 222:281-291.

Churchland, P.S. (1986) Neurophilosophy: Toward a Unified Science of   
the Mind-Brain. Cambridge,  Massachusetts, MIT Press

Curthoys, I.S., Blanks, R.H.I. and Markham, C.H. (1977)  Semicircular   
canal functional anatomy in cat, guinea pig and man. Acta   
Otolaryngol., 83:258-265.

Daunicht, W. and Pellionisz, A. (1986) Coordinates intrinsic to the   
semicircular canals and the  extraocular  muscles in the rat.  Soc.  
Neurosci. Absts. 12,  p. 1089.

Daunicht, W. and Pellionisz, A. (1987)  Geometrical coordinates of the   
gaze sensorimotorsystem of the rat. Brain Research  (in Press)

Denker, J.S. (Ed.) (1986) Neural Networks for Computing. AIP Conf.   
Proc. 151., New York

Eckmiller, R.(Ed.) (1988) Neuronal Computers. Springer Verlag, Berlin

Ezure, K.  and Graf, W.  (1984) A quantitative analysis of the spatial   
organization of the vestibulo-ocular reflexes in lateral and front-eyed   
animals. I. Neuroscience, 12:85-93.

Farley, B.G., and  Clark, W.A. (1954) Simulation of self-organizing   
system by digital computer. IRE Trans. Inf. Theor., 4, p.76.

Georgopoulos, A.P., Schwartz, A.B. and Kettner, R.E. (1986)  Neuronal   
population coding of movement direction.  Science,  233:1416-1419.

Gielen, C.C.A.M. and van Zuylen, E.J. (1986) Coordination of arm   
muscles during flexion and supination: Application of the tensor   
analysis approach.  Neuroscience, 17: 527-539.

Gisbergen, J.A.M., van, Robinson, D. and Gielen, S. (1981) A   
quantitative analysis of generation of saccadic eye movements by   
burst neurons. J. Neurophys.  45:417-442.

Grossberg, S. and Kuperstein, M. (1986) Neural Dynamics of Adaptive   
Sensory-Motor Control. Ballistic Eye Movements. Elsevier/North   
Holland, Amsterdam

Gurfinkel, V.S. (1987) Robotics and biological motor control. Proc. of   
II. IBRO World Cong. Neurosci, Suppl. 22:S381.

Hebb, D.O. (1949) The Organization of Behaviour. John Wiley, New   
York.

Helmholtz, H. von (1896) Handbuch der Physiologischen Optik.    
Zweite Auflage. Leipzig: Voss

Hering, E. (1868) Die Lehre vom binocularen Sehen.  Engelmann,   
Leipzig

Henn, W. and Cohen, B.(Eds) (1988) Proc. of the Symp:   
"Representation of 3-Dimensional Space in the Vestibular, Oculomotor   
and Visual Systems,  Org.by the Bárány Society,  
Bologna, Italy

Högyes, E. (1880-1884) Az associált  
szemmozgások  idegmechanismusáról.  
ErtekezŽsek a termŽszet-tudományok körŽböl.  X,  
18, 1-62 (1880), XI, 1-100 (1881); XIV, 9,1-84 (1884). German   
translation: Högyes, A. (1912) Ÿber den Nervenmechanismus  
der  assoziierten Augen-bewegungen. Monatsschr. f. Ohrenheilk. u   
Laryngo-Rhinol. 46:685-740: 809-841; 1027-1083; 1353-1413;   
1554-1571.

Keller, E.L. (1974) Participation of medial pontine reticular formation   
in eye movement generation in monkey. J. Neurophysiol. 37:316-332

King, W.M. and Fuchs, A.F. (1979) Reticular control of vertical   
saccadic eye movements by mesencephalic burst neurons. J.   
Neurophys. 42:861-876. 

Laczkó, J., Pellionisz, A.J. Peterson, B.W. and Buchanan, T.S.  
(1987)  Multidimensional sensori-motor "patterns" arising from a  
graphics- based tensorial model of the neck-motor system.Soc.  
Neurosci. Absts.  13.

Lestienne, F, Liverneaux, Ph., Pellionisz, A. (1987)  Role of the   
superficial and deep neck muscles in the control of monkey head   
movement: Application of the tensor analysis approach.Soc. Neurosci.   
Absts. 13.

Levi-Civita, T. (1926) The Absolute Differential Calculus (Calculus of   
Tensors).  New York: Dover.

Liverneaux, Ph, Pellionisz, A.J., Lestienne, F.G. (1987) Morpho-  
anatomy and muscular synergy of sub-occipital muscles in   Macaca 
Mulatta:  Study of Head-Trunk Coordination. Proc.of IBRO II.   World 
Congress, Neuroscience , Suppl. to Vol.22, p. S847.

Loeb, G.E. (1983) Finding common ground between robotics and   
physiology. Trends in Neuro-science,  5:203-204.

Loeb, G.E. and Richmond, F.J.R. (1986) Synchronization of motor units   
in and among diverse neck muscles during slow movements in   
intact cats. Society for Neuroscience Abstracts, 12, 687.

Lorente de Nó, R. (1933) Vestibulo-ocular reflex arc. Arch.  
Neurol.  Psychiatr. 30:245-291.

Luschei, E.S. and Fuchs, A.F. (1972) Activity of brainstem neurons   
during eye movements of alert monkeys. J. Neurophysiol. 35:445-  
461.

Mach, E. (1886) BeitrŠge zur Analyse der Empfindungen. Jena: Fisher.   
English translation by C.M. Williams, revised and supplemented by   
Sydney Waterlow, 1959, Dover, New York

Maekawa, K. and Simpson, J.I. (1973) Climbing fiber responses   
evoked in the vestibulo-cerebellum of rabbit from visual system. J.   
Neurophysiol. 36:649-666.

Markham, C.H. and Curthoys, I.S. (1972) Convergence of labyrinthine   
influences on units in the vestibular nuclei of the cat. II. Electrical   
stimulation. Brain Res. 43:383-396.

Nashner, L.M. (1977)  Fixed patterns of rapid postural responses   
among leg muscles during stance. Experimental Brain Research,    
30:13-24.

Ostriker, G., Pellionisz, A. and Llinás, R. (1985)  Tensorial  
computer  model of gaze -Ð I. Oculomotor activity is expressed in  
non-orthogonal  natural coordinates. Neuroscience, 14:483-500.

Oyster, C.W., Takahashi, E. and Collewijn, H. (1972) Direction selective   
retinal ganglion cells and control of optokinetic nystagmus in the   
rabbit. Vision Res. 12:183-193.

Palm, G. (1982) Neural Assemblies. Springer, Berlin

Pellionisz, A. (1983) Brain theory: connecting neurobiology to   
robotics. Tensor analysis: utilizing intrinsic coordinates to describe,   
understand and engineer functional geometries of intelligent   
organisms. J. Theoretical Neurobiol. 2(3):185-211.

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. Theoret.   
Biol., 110: 353-375.

Pellionisz, A. (1985a) Tensorial aspects of the multidimensional   
approach to the vestibulo-oculo-motor reflex and gaze. In Reviews of   
Oculomotor Research. I. Adaptive Mechanisms in Gaze Control. (Ed.)   
A. Berthoz and G. Melvill-Jones, Amsterdam, Elsevier, pp. 281-296.

Pellionisz, A. (1985b) Tensorial brain theory in cerebellar modeling.   
In Cerebellar Functions, (ed.) J. Bloedel, J. Dichgans and W. Precht,   
Heidelberg: Springer, pp. 201-229.

Pellionisz, A. (1986) Tensorial relationship found for structural and   
functional reference frames of brain function: Saccade neurons in   
monkey utilize frames composed of the eigenvectors of the frame of   
extraocular muscles. Soc. Neurosci. Absts. 12, p. 1186.

Pellionisz, A. (1987a) Vistas from tensor network theory: a horizon   
from reductionalistic neurophilosophy to the geometry of multi-unit   
recordings. In Computer Simulation in Brain Science  (ed. by R.   
Cotterill) Cambridge University Press.

Pellionisz, A. (1987b) Sensorimotor operations: a ground for the co-  
evolution of brain theory with neurobotics and neurocomputers.   
Proc. IEEE 1st Ann. Internatl. Conf. on Neural Networks

Pellionisz, A.  and Graf, W. (1987) Tensor network model of the   
"Three-neuron vestibulo-ocular reflex-arc" in the cat. J. Theoret.   
Neurobiol.  5:127-151.

Pellionisz, A. and Llinás, R. (1979)  Brain modeling by tensor  
network  theory and computer simula-tion.The cerebellum:  
distributed  processor for predictive coordination.Neuroscience,  
4:323-348.

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

Pellionisz, A. and Llinás, R. (1982) Space-time representation  
in the  brain. The cerebellum as a predictive space-time metric  
tensor.  Neuroscience, 7:1949-2970.

Pellionisz, A. and Llinás, R. (1985) Tensor network theory of  
the  Metaorganization of functional geometries in the CNS.  
Neuroscience,  16:245-273.

Pellionisz, A. and Peterson, B.W. (1985) Tensor models of primary   
sensorimotor systems, such as the vestibulo-collic reflex (VCR) and of   
the metaorganization of hierarchically connected networks. Soc.   
Neurosci. Abst., 11: 83.

Pellionisz, A. and Peterson, B.W. (1988) A tensorial model of neck   
motor activation. In  Peterson, B.W. and Richmond, F.J. (Eds.) Control   
of Head Movement, New York, Oxford Univ. Press

Pellionisz, A.J., Soechting, J.F., Gielen, C.C.A.M., Simpson, J.I., Peterson,   
B.W., Georgopoulos, A.P. (1986) Multidimensional analyses of   
sensorimotor systems. Soc. Neurosci. Absts.  12. 1.

Peterson, B., Baker, J., Goldberg, J. and Wickland, C. (1985a) Kinematic   
organization of the cat vestibulo-ocular reflex (VOR).  Soc. Neurosci.   
Abstr. 10:162.

Peterson, B.W., Baker, J., Wickland, C. and Pellionisz, A. (1985b)    
Relation  between neck muscle pulling directions and activity by the   
VCR: Experimental test of a tensor model.  Soc.Neurosci. Absts. 11. p.   
83.

Peterson, B.W., Baker, J., Wickland, C., and Pellionisz, A. (1985c)    
Sensorimotor transformation in oculomotor and neck-motor control   
systems. In: Proceedings of Annual Conference on Engineering in   
Medicine and Biology

Peterson, B.W., Baker, J.F., and Pellionisz, A.J. (1987) Comparison of   
spatial transformation in vestibulo-ocular and vestibulo-spinal   
reflexes. In: Proc. of the Symp: "Representation of 3-Dimensional   
Space in the Vestibular, Oculomotor and Visual Systems,   
Bárány Soc.,  Bologna.

Peterson, B.W., Baker, J.F., and Pellionisz, A.J. (1987)   
Multidimensional analysis of vestibulo-ocular and vestibulo-collic   
reflexes (VOR and VCR).  In: Proc. of the International Symp. on   
Basic and Applied Aspects of Vestibular Function,  Hong Kong

Peterson, B.W. and Pellionisz,  A.J. (1986) A tensorial model of the   
kinematics of head movements in the cat.  Society for  Neuroscience   
Abstracts,  12: 684.

Peterson, B.W., Pellionisz, A.J., Baker, J.A., and Keshner, E.A. (1987)   
Functional morphology and neural control of neck muscles in   
mammals. Proc. Symp. on Axial Movement Systems: Biomechanics   
and Neural Control.  (in Press)

Peterson, B.W. and Richmond, F. (1988) Control of Head Movement.    
Oxford University Press

Piaget, J. (1980) Structuralism. Basic Books, New York 

Pitts, W.H. and McCulloch, W.S. (1947) How we know universals: The   
perception of auditory and visual forms. Bull. Math. Biophys. 9:127-  
147

Pompeiano, O. (1975) Vestibulo-spinal relationships. In R.F. Noughton   
(Ed.) The Vestibular System. Academic Press, New York, pp. 147-180.

Reitböck, H.J.P. (1983) A 19-channel matrix drive with  
individually  controllable fiber micro-electrodes for  
neurophysiological  applications. IEEE Transactions on System, Man  
and Cybernetics,  SMC-13, 5, pp.677-682

Robinson, D.A. (1982)  The use of matrices in analyzing the three-  
dimensional behavior of the vestibulo-ocular reflex.  Biol. Cybern.   
46:53-66.

Sherrington, C. (1906). The  Integrative  Action of the Nervous   
System. New York: Scribner.

Simpson, J.I. and Graf, W. (1985) The selection of reference frames   
by nature and its investigators. In Adaptive Mechanisms in Gaze   
Control. Facts and Theories. Reviews of Oculomotor Research, V. 1, A. 

Berthoz, and G. Melvill-Jones (Eds.), Amsterdam: Elsevier,  pp. 3-20.  
Simpson, J.I. and Pellionisz, A. (1984) The vestibulo-ocular reflex in   
rabbit, as interpreted using the Moore-Penrose generalized inverse   
transformation of intrinsic coordinates.Society for Neuroscience   
Abstracts,  10, p. 909.

Simpson, J.I., Rudinger, D., Reisine, H. and  Henn, V. (1986).  Geometry   
of extraocular muscles of the rhesus monkey.  Society for   
Neuroscience Abstracts,  12, p.1186.

Simpson, J.I., Soodak, R.E. and Hess, R. (1979) The accessory optic   
system and its relation to the vestibulo-cerebellum. In Reflex Control   
of Posture and Movements, Progress in Brain Research, Vol. 50., ed. R.   
Granit and O. Pompeiano, Amsterdam: Elsevier, pp. 715-724.

Szentágothai, J. (1950) The elementary vestibulo-ocular reflex  
arc. J.  Neurophysiol. 13:395-407.

Vidal, P.P., Graf, W. and Berthoz, A. (1986)  The orientation of the   
cervical vertebral column in  unrestrained awake animals.  I.   
Resting position.  Exp. Brain  Res. 61: 549-559.

von Neumann, J. (1958) The Computer and the Brain. New Haven:   
Yale University Press

Wiener, N. (1949) Cybernetics. MIT Press. Cambridge, MA.

***