Figure 1.

 

Fig.1. Retino-ocular (optokinetic) sensorimotor reflex as a tensorial 
transformation, via neuronal networks, of sensory coordinates that 
are intrinsic to retinal ganglion cells and of motor coordinates that 
are intrinsic to extraocular muscle motoneurons.  A;  Sensory frame, 
intrinsic to mammalian retinal ganglion cells (data shown in table 
originate from Oyster et al. 1972). The four-axis frame (of maximal 
on-off type direction sensitivities of retinal ganglion cells) lies in the 
2D plane of the retina.  These exemplary axes are displayed in the 
extrinsic pitch, yaw, roll frame.  A  corresponding frame in the cat is 
yet to be  established and put into the context of this conceptual 
model.  Dor, ganglion-sensitivity axis along a mostly dorsal direction, 
lat; lateral, ven; ventral, med; medial direction.  B; Motor frame,  
intrinsic to the extraocular muscles in the cat (data shown in table 
originate from Ezure and Graf 1984). The six eye muscles (MR; 
medial rectus, LR; lateral rectus; IR: inferior rectus, SR; superior 
rectus, IO; inferior oblique, SO; superior oblique) rotate the eye in 
this six-dimensional frame that is intrinsic to the anatomy.  Central 
inset:  The scheme of the cat head shows a retinal (A) and an 
oculomotor (B) apparatus. Vestibular semicircular canals and several 
neck muscles are displayed here only to indicate that the hierarchy 
of gaze reflexes employs at least two sensory  and two motor 
systems (see Pellionisz 1986).  ROR; To act as an optokinetic (retino-
ocular) reflex (ROR), neuronal networks have to transform a shift of 
the visual image, passively measured in retinal coordinates, into an 
active eye movement that is physically executed in oculomotor 
coordinates.  Transformation of vectors within and among general 
coordinates can be described by tensors  

The coordinate systems intrinsic to a mammalian retino-ocular 
(optokinetic) reflex are shown in Fig.1. For the oculomotor apparatus, 
the rotational axes of the eye in case of individual extraocular muscle 
activation can be physically measured using the Helmholtz method 
(1896). The xyz and x'y'z' origin- and insertion-points of a muscle, 
together with the x¡,y¡,z¡ rotation-point of the eye, determine a plane 
whose normal will be the axis along which the eye turns (right-hand 
rule for the right eye, left-hand rule for the left eye is assumed; cf. 
Pellionisz 1985b).  This method has been uti-lized by Ezure and Graf 
(1984) to anatomically measure the oculo-motor coordinates shown 
in Fig.1.B.  It is evident that the oculomotor neurons must express 
eye movements using this frame that is intrinsic to the physical 
geometry of the arrangement of muscles.  The retinal sensory frame 
(Fig.1A), although it consists of a simple 4-axis arrangement, 
constitutes yet another kind of intrinsic frame.  As established by 
Oyster et al (1972) for the rabbit Ða comparable set of measurements 
for the cat remains a challenge for experimentalistsÐ the retinal 
ganglion cells carry directional information on the displacement 
(velocity) of retinal image; in effect constituting a similar "neuronal" 
intrinsic frame to the one displayed by cerebellar climbing fibers 
(Simpson et al., 1981). This class of frames intrinsic to CNS function 
(and not to the anatomical structure) are, of course, much more 
difficult to experimentally establish than skeleto-muscular intrinsic 
coordinates, nevertheless this type of experimental research will 
undoubtedly flourish in the future. Such theoretical requirements 
from data-gathering will enhance a mutual interdependence of 
modeling and experimentation. 

Figure 2.

 
Fig. 2.  Tensorial scheme of the retino-ocular (optokinetic) reflex in 
the cat.  The theoretically required four stages (1-4) of a general 
tensorial sensorimotor transformation scheme are shown in the 
upper row, using simple exemplary coordinate systems (two-axis 
nonorthogonal sensory frame and three-axis nonorthogonal motor 
frame).  For readability, the frames shown are simplifications of 
those used in Fig.1. For calculating A,B and C, however, the actual 
retinal and oculomotor frames (Fig.1) were used; cf. also Fig.3 . It 
should be noted that if a general (tensorial) transformation solution 
is valid for one unrestricted  set of frames, it is valid for all.  
Diagrams (2-4) demonstrate that in nonorthogonal frames of 
reference the orthogonal projection type (covariant) and physically 
executable parallelogram component (contravariant) vectorial 
expressions are different.  In order for an optokinetic reflex  to work, 
an invariant (such as the position change of a target)  has to be both 
measured (by covariant components in the retinal frame, ui) and 
executed (by contravariant components expressed in the oculomotor 
frame; vm).  The proposed solution (Pellionisz 1984) implements this 
transfer by means of a three-tensor network transformation (A,B,C), 
employing two interim vectorial expression s; uj and vn

Figure 3.

 

Fig. 3. Tensors of the optokinetic reflex.    The three transformations 
necessary for a sensorimotor transfer are shown at five different 
levels of abstraction.  The quantitative matrix expressions are 
calculated from data shown in Fig.1. The sensory- to motor-frame 
conversion (middle column) is a 4 x 6  table of cosines among the 
four retinal and six oculomotor axes.  The sensory and motor 
contravariant metric tensors (first and third columns) are the Moore-
Penrose generalized inverses of the covariant metric tensors (which 
are, again, the tables of cosines among axes of the sensory and motor 
frames, respectively).  The patch-diagram representation of these 
matrices is used for illustrating throughout the figures the network 
implementations of these transformations. Filled and empty circles 
represent +/- components of the matrix, with the area of each patch  
proportional  to the numerical value of the matrix component.  For 
more detail, see Pellionisz 1984, 1985

With retinal sensory- and eye movement motor-frames 
quantitatively established, the reflex of tracking the retinal image-
displacement of a moving target by ocular rotation can be 
mathematically stated as a tensor transformation between two 
vectors that express the identical image-displacement in the retinal 
sensory frame and in the oculomotor system of coordinates.  Fig.2. 
encapsulates a general tensorial scheme (valid for any coordinate 
system) of how the primary measurement is transformed into final 
execution.  The scheme provides solutions for three necessities.  (1) 
The change of frame from sensory to motor, (2) A resolution of the 
mathematical problem that the number of motor axes is larger than 
the number of the sensory axes (this overcompleteness permits an 
infinite number of variations in motor expression; cf. Pellionisz 1984) 
(3) Change of the vectorial version used in sensory and motor 
frames. This latter problem (explained in more detail in Pellionisz 
and Llinás 1980, Pellionisz 1984, 1985b) concerns the 
fundamental fact that primary sensory measurements (most 
obviously the vestibular canal excitations) are expressed in 
orthogonal projections (sensory-type, so-called covariant) vectorial 
components, while the physical motor execution (resultant of muscle 
actions) has to be expressed in parallelogram-components (motor-
type, mathematically so-called contravariant vectors, since the sum 
of covariant components does not physically add up to generate the 
invariant; cf. Pellionisz and Llinás 1980).  These different 
versions are depicted in the upper row of Fig.3.  Both the sensory- 
and motor frames are displayed in this row in an exemplary manner, 
the motor frame being overcomplete compared to the sensory frame 
(three motor axes versus two sensory axes) Ð similarly to the 
overcompleteness of the ROR (making a transformation from 4 
retinal axes to 6 eye muscles). The proposed tensorial scheme uses 
three transformation matrices (A,B,C, shown by patch-diagrams, cf. 
Figs.3-4) that transform the initial covariant reception vector into a 
contravariant version expressed in the same frame, and to a 
covariant intention vector that already uses the motor frame, but 
yields projection-type, "naive", components that are unsuitable for 
direct execution and thus have to be turned into contravariants first 
(cf. Figs.3-5 in Pellionisz and Llinás 1980). 

Figure 4.

 


Fig. 4.  Tensor network module.  Diagram illustrates how a neuronal 
network can be conceived of (at different levels of abstraction) as  
implementing  (a) a matrix, (b) a tensor, (c) a functional geometry. 
The visual symbolism of a tensor network module is composed of (1) 
a bundle of input axons, whose firing frequencies constitute an 
ordered set of quantities, a vector;  (2)  a bundle of axons of output 
neurons which carry another vector  (the number of fibers in the 
input and output pathways need not be equal), (3) a set of 
interconnections among the axons of input fibers and the dendritic 
trees of the output neurons. Such compact arrangement of heavily 
interconnected neurons is similar to the structure of a nucleus along 
a neuronal pathway. A matrix:: Components of the interconnection 
matrix may be implemented, e.g., by  synaptic strengths, and/or of 
the number of synapses; the diagram shows such  effect by means of 
patches (cf. Fig.3).  A tensor: If for the shown case the input vector 
carries a covariant vector of retinal measurements (ui) and the 
numerical values of the transformation matrix components 
correspond to the contravariant metric tensor of the retinal space 
(gij), then the vector of output neuron firing frequencies will carry 
the product of the input vector with the transformation matrix, 
which is the contravariant retinal vector (uj). A geometry :  By 
comprising the metric tensor, a simple neuronal connectivity in effect 
establishes the functional geometry of the retinal space


A quantitative elaboration of the sequence of three multidimensional 
tensor transformations is shown in Fig.3 (the steps follow procedures 
given in Pellionisz 1984,85b).  Tensors are presented in abstract 
formalism (generalized for all coordinate systems), verbally, by 
numerical matrix-expressions, patch-diagrams, and also by putting 
forward the theoretical prediction for their site of implementation.  
Since tensor network theory claims that such tensor operations are 
incorporated by networks, it may be particularly important to show, 
in Fig. 4, how a "tensor network module" (which is a set of 
interconnections among input axons and a column of output neurons) 
may embody not just a particular matrix, but a general tensor 
transformation, or even a functional geometry.  It is easy to conceive 
that the i input fibers, each carrying a firing frequency, constitute a 
mul-tidimensional (here, four-dimensional) vector, whose 
components will be multiplied, to yield the scalar product, by the 
components of one row of the matrix of interconnection-strengths. 
The sum of these products (altogether a scalar product) will yield the 
firing frequency of one output cell (neurons are symbolized in Fig.4. 
and throughout the paper by exemplary dendritic trees).   The 
output vector of firing frequencies of all neurons will be the product 
of the input vector with the matrix of interconnections.  While the 
particular matrix of interconnections and the values and dimensions 
of input and output vectors may vary, a class of a given tensor 
network module (eg. representing the superior colliculus of different 
specimens or even species) may embody one and the same general 
tensor.  For instance, if the component values of the connection 
matrix are such that the network transforms an input vector with 
covariant components to a vector expressed in the same frame but 
with contravariant components (cf. transformation of ui to uj by A in 
Fig.2), then the network performs the operation of the metric tensor, 
which in effect comprises the geometry of the space spun over the 
axes of the coordinate system.  Speaking of generalizations, it must 
be emphatically stressed here that the notion of general coordinates 
is not limited to space coordinates only.  While Ð for the sake of 
simplicity of exposition Ð only space coordinates are used in this 
paper, the cerebellum is conceived (and elaborated, eg. in Pellionisz 
and Llinás 1982) as a metric tensor of the unified spacetime 
domain;  such that coordination and timing functions are inseparable.

With the visual symbolism of tensor network modules, the 
"evolution" of a multidimensional sensorimotor reflex can be put 
forward (Fig.5).  Part A presents the most essential element, the 
transformation tensor-matrix from a 4-axis retinal sensory frame to 
a 6-axis oculomotor frame.  The components of this matrix easily 
arise if each motor axis is projected, one by one, to all sensory axes 
(such a procedure is shown in Fig.5. of Pellionisz 1985b). 
Mathematically speaking, the components are the cosines among 
sensory and motor axes.  While this is an utterly simple 
transformation, it is inadequate to represent a sensorimotor reflex, in 
itself, for several reasons.  First, neuroanatomy demonstrates, that  
sensory detectors never connect in a single step to the motor 
apparatus (thus, a single brain-stem matrix lumps an entire 3-
neuron reflex-arc into a single matrix; Robinson 1982, versus 
Pellionisz 1985b). Second, the single table of cosines (a projection-
matrix) would mathematically do if the input sensory vector were 
contravariant and the output would have to be covariant.  However, 
in sensorimotor transformation-vectors of the opposite types occur: 
the sensory input is co-variant and the motor output has to be 
contravariant (cf. Fig.2).  Thus, the single transformation-matrix of A 
alone would, indeed, yield a motor vector but with wrong 
component-values; an approximation at best.  A further deficiency of 
such a primitive sensorimotor transfer shown in A is, that no 
decisions can be made on the external invariants (distances) without 
actually performing the movement (a life or death question when 
jumping a ditch).

Section B of Fig.5. shows that with the introduction of a sensory 
preprocessing, incorporating the sensory geometry, two of the above 
problems are eliminated.  First, network A of Fig.2-3, serving as the 
metric tensor of the retinal frame, would make available the 
contravariant counterpart of the covariant retinal sensory vector.  As 
shown in Fig.4, with both representations available, their inner 
product yields a measure of the invariant distance itself.  Thus, no 
actual motor performance ("jumping of the ditch") would be 
necessary to decide if it is appropriate to attempt a movement; such 
decisions are possible entirely within the sensory domain (for 
further elaboration of inferring invariants from dual representation, 
see Pellionisz and Llinás 1982, Pellionisz 1987). Second, such 
a metric tensor (mathematically, the matrix of Moore-Penrose 
generalized inverse of the table of cosines among sensory axes; cf. 
Albert 1972, Pellionisz 1985b) will yield the proper contravariant  
sensory vector transformed from the input.  However, the third 
deficiency still remains: schemes A and B in Fig.5. yield the motor 
output in improper (covariant) vectorial coordinates.

Section C of Fig.5. shows that these two transformation-tensors are 
predicted to be embodied by the neuronal networks of the optic 
tectum and/or pretectum, in accordance with the findings that its 
input is sensory, its output is motor, and that it converts retinotopic 
measurements to vectorial version that yields distances (Sparks and 
Pollack 1977).  Since the brainstem core of ROR in Section C consists 
of the same transformations as in section B (the oculomotor tensor-
module is only a "transmitter", a unit-matrix with diagonal elements 
of 1 and off-diagonals of 0), in the absence of the cerebellum the 
oculomotor response would be executed with covariant (dysmetric) 
components.  Again, specific quantitative experimental tests of the 
theoretical prediction of dysmetric eye movements (which are easily 
modeled) would be most valuable.  

Section D of Fig.5. complements the brain-stem core of ROR with an 
"add-on" cerebellar circuit Ð corresponding to the fact that the 
cerebellum is a late development in evolution (Llinás 1969) 
which "only" improves upon existing sensorimotor capabilities by 
imposing a coordinative role.  The fact that the cerebellum never 
initiates movements, and that motor performance is possible in its 
absence (although in an uncoordinated manner) metaphorically 
likens the cerebellum to an "executive secretary". Such an agent is 
informed about outbound "naive" (intentional) commands, and its 
role is to bend them into appropriate executive orders before they 
reach the plant. To be able to do this, the agent has to possess a 
realistic internal representation of the system of relations existing 
out there on the executors.  The coordinative effect (knowing the 
difference of intention and execution) is to be added to the directly 
downgoing intentions so that orders reaching the plant are the pure 
executive commands.  Mathematically, this requires that the Purkinje 
cell Ð cerebellar nuclear cell connection matrix constitutes the metric 
tensor of the motor frame (mathematically, the Moore-Penrose 
generalized inverse of the table of cosines among motor axes; cf. 
Pellionisz 1984, 85b).  With this, the mossy fibers would carry the i 
intention vector directly to the nuclei. The granule cell Ð parallel fiber 
Ð Purkinje cell multidimensional pathway carries to the 
corticonuclear metric connectivity-set (which transforms it to 
execution vector, e=g.i).  Given that Purkinje cells are inhibitory (Ito 
et  al. 1970), the nucleofugal vector is i-e, which turns the motor 
intention vector in the oculomotor nuclei to pure execution vector i-
(i-e)=e.

The Olivary-Climbing Fiber System: Ongoing Correction of the 
Cerebellar Metric Tensor to Make it Position-Dependent (Change the 
Curvature of the Motor Hyperspace)

The tensor network diagram of Fig.5 predicts, therefore, an 
optokinetic reflex which may function in the total absence of the 
cerebellum (with quantitatively predictable ataxic performance), and 
also predicts that once the cerebellum is fully developed, the olivaryÐ
climbing fiber system is not part of the essential cerebellar network.  
The network-diagram of Fig.5. provides therefore a quantitative 
framework for data about an experimentally accessible and 
technically duplicatible entire multidimensional coordinated 
sensorimotor apparatus, within which the specific function of the 
olivary-climbing fiber system is proposed below (see Fig.6).  It is 
believed that such explanations might bring closer the day when 
cerebellar modelists first explain (and perhaps even check by 
implementation) cerebellar coordination of multidimensional 
sensorimotor systems  (that actually works in the absence of 
climbing fibers) before attributing a single dimensional gain-control 
role, attained by a Purkinje cell Ð climbing fiber Ð parallel fiber 
heterosynaptic junction, or postulating an associative memory-role 
that will not coordinate a movement.

The proposed role of the olivaryÐclimbing fiber system  has two 
facets (the elements were shown in Pellionisz and Llinás 
1985). One is played in the genesis of the cerebellar metric tensor 
network  (see the Metaorganization principle and procedure in the 
above paper), the other is exerted in the ongoing modification of the 
metric tensor function (Llinás and Pellionisz 1985). It is 
obvious from  Fig.6, these functions are inherently multidimensional; 
corresponding to findings that olivary neurons fire in assemblies of 
electrotonically coupled neurons (Sotelo et al. 1974, Llinás 
1974). Mathematically speaking, the model interprets information 
carried by bundles of climbing fibers (vectors) to cerebellar zones of 
Purkinje cells (cf. Voogd and Bigarre 1980), rather than interpreting 
a single climbing fiber as communicating a scalar-value ("gain") to a 
single Purkinje cell. 

Figure 5.

 


Fig. 5. Tensor network model of the stages of ÒevolutionÓ of the 
optokinetic reflex (ROR), including the  essential cerebellar network.  
A; The absolute necessity for a sensorimotor performance is a 
conversion from a sensory coordinate system to a motor frame. This 
is accomplished by a tensor network which incorporates the matrix 
of the cosines among sensory and motor axes (cf. Fig. 3B).  While such 
a network is easily constructed by the CNS,  it yields an 
approximative, projection-type (covariant intention) motor output.  
Also, the eye displacement, necessary to compensate a shift of the 
retinal image,  can be judged only by actually performing the 
movement.  B; A tensor-network module, implementing the metric 
tensor of the sensory space (cf. Fig. 3A), complements the system in 
such a manner that with the availability of both the covariant and 
contravariant retinal expressions decisions can be made on the 
movement amplitudes within the sensory domain.  When a 
movement is made, it is still with intentional components.  C; It is 
predicted that the two operations described above are implemented 
by networks of the colliculus superior.  If the brain-stem core of the 
ROR is not complemented by a cerebellar circuit (corresponding to 
the case of cerebellar ablation: see its removal along the dotted line), 
intentional motor commands are transmitted through the oculomotor 
nuclei, which network consists of straight connections of input and 
output fibers, mathematically, a unit matrix,  to result in an ataxic 
(naive) motor performance.  D; The cerebellar tensor network 
provides the metric of the motor system (cf. Fig. 3C).  Mossy fibers 
(MF), carrying a motor intention vector, are transformed through the 
granule cell (GC)- parallel fiber (PF)- Purkinje cell (PC) pathway to 
the cerebellar nuclei. If the corticonuclear system of projections of 
Purkinje cells forms the matrix shown in Fig. 3C, then it will 
transform the  mossy fiber covariant input to contravariant motor 
output, and with a negative sign since Purkinje cells are inhibitory.  
The mossy fiber collaterals to the cerebellar nuclei provide a direct 
(excitatory) intention vector.  Therefore, the vector leaving the 
cerebellar nuclei (CN) will carry the i-e intention-execution 
(coordination) vector.  This negative vector, projecting to the 
oculomotor nuclei, will result in the i-(i-e)=e contravariant execution 
vector


The function of olivary-climbing fiber system, as an ongoing modifier 
of the cerebellar metric, is necessitated by the firm mathematical 
fact that the fixed matrices of Fig. 5, without climbing fiber 
modification of the electrore-sponsive properties of Purkinje cells, 
would only yield a perfect mathematical result if the coordinate 
systems intrinsic to the retinal ganglion cells and to the eye rotations 
by extraocular muscles would be fixed (would not depend on the 
position of the eye).  This is, however, only approximately true (Figs. 
4,8,9 in Ostriker et al, 1985 provide quantitative measures of how eg. 
the frame intrinsic to the oculomotor apparatus changes with the 
position of the eye; the position-dependency is not dramatic, but 
certainly perceptible).  Therefore, if the cerebellar metric tensor is 
perfectly calibrated to yield exact contravariants from covariant 
components in the primary eye position,  the "wired-in" matrix 
connections of the cerebellum only perform an approximative metric 
transformation when the eye is in a secondary position. The model 
therefore predicts a quantifiable error-shift of the retinal image 
during optokinetic tracking in climbing-fiber-deprived preparation 
(another prediction for quantitative experimental tests).  As for the 
function of olive, it has been suggested (Oscarsson 1969, Armstrong 
1974) that this system, being connected both to the downgoing 
executor pathways as well as ascending pathways reporting on the 
motor performance, the olive may serve as a "comparator". It is also 
experimentally known that such retinal error-shifts are reported to 
cerebellar Purkinje cells by means of climbing fiber-evoked complex 
spikes expressed in an intrinsic coordinate system  (Simpson and 
Alley 1974, Simpson et al. 1981, Simpson and Graf 1985) .  This 
experimental knowledge is represented in the scheme of Fig.6. by the 
postulates that the olivary network computes the error-vector of the 
performance, expressed in oculomotor coordinates (by comparing, in 
this case, the oculomotor execution vector with the retinally detected 
displacement of the image of the target).  Such a comparison, 
however, of one vector expressed in retinal frame with another, 
expressed in motor frame, would not be possible if they were not 
converted into a common frame. Thus, conclusion of the 
experimentation that the function of the accessory optic system 
(AOS) is to convert retinal coordinates to another frame that appears 
to be an oculomotor coordinate system (Simpson et al. 1979, 1981) is 
a strong basis for this model. Since the analysis of this question in 
depth is presently a focus of active research, in this paper the retino-
ocular conversion is only tentatively represented, using matrix B 
from Fig.2.

The  above-elaborated ongoing modification of the cerebellar metric 
accords well with the observation that a "phasic" grouped firing of 
climbing fibers occurs whenever errors or obstacles are encountered 
in motor coordination function (Llinás and Volkind 1974).  It 
is also worth-while to point out, that the anatomy of cerebellar 
pathways, specifically the direct projection of climbing fiber 
collaterals to the nuclei together with the indirect projection to the 
same array via Purkinje cells of the cerebellar cortex, enable the 
olivary system to construct an external (matrix) product of the 
climbing fiber errorvector on the array of cerebellar nuclear cells 
(see more detail  in Pellionisz and Llinás 1985, Fig.4).  Thus, 
when the optokinetic reflex operates in an offprimary position, 
climbing fiber assemblies (reporting on the error of the metric), 
functionally update the cerebellar metric tensor by inducing an 
ongoing modification of the array of nuclear cells. 

Mathematically, the above function is equivalent to having in the 
multidimensional motor space a position-dependent metric tensor; in 
effect predicting sensorimotor operations to take place in a curved 
functional space (where the curvature of the operational region is 
adjusted by the climbing fiber system).  Electrophysiologically, such a 
a dynamic ongoing alteration is predicted to be much more distinctly 
detectable on the array of cerebellar nuclear cells  (where a double Ð
direct and indirectÐ projection of climbing fibers occurs) than on 
Purkinje cells, which only transmit this climbing fiber action to the 
nuclei, although eg. from modeling studies it is clear that the deep 
depolarization may undoubtedly exert some residual influence 
(Pellionisz and Llinás 1977).  Moreover, given that the 
quantitatively predictable ongoing modification is an inherently 
multidimensional function, occurring on an array of neurons (and 
corresponding to the positive and negative components of the error-
matrix), the alteration is bimodal,  positive, negative (or zero) on 
different particular neurons. Therefore, it is not entirely surprising 
that single cell  electro-physiological studies looking into 
modifiability (when interpreted within an entirely different, single 
dimensional theoretical framework) are apparently inconclusive 
(having found both a "depression" detected by Ito et al. 1982 versus 
an "enhancement" reported by Bloedel et al. 1983).  It is rather likely 
that such bimodal ongoing modifications evoked by the olivary Ð 
climbing fiber system will be revealed over an array of neurons, 
using multi-unit recording techniques from which data the effect of 
the multi-unit signals on the intrinsic functional geometry can be 
calculated; cf. Pellionisz 1988a. If an experimental paradigm is 
combined with a quantitative model of those coordinates that are 
intrinsic to coordinated (and erroneous) motor performance, 
theoretical predictions of this multidimensional climbing-fiber 
induced adjustment of the metric will become testable.

Figure 6.

 


Fig. 6. The olivary-climbing fiber system updates the cerebellar 
motor metric tensor in an ongoing  manner to  correct  for position-
dependence  (changing the curvature of the motor hyperspace).  A 
fixed matrix, acting as a metric tensor, provides a perfect covariant-
contravariant transformation only if the intrinsic coordinates do not 
change with the position of the eye.  Since such a change is a fact, in 
secondary positions of the eye the metric will be only approximative; 
thus, an error will occur in the optokinetic reflex.  This error vector 
can be produced  by the neuronal network of the olive acting as a 
ÒcomparatorÓ  of the motor execution signal e and the change that 
actually occurred in the position of the retinal image.  Comparison  is 
possible only if the accessory optic system (AOS) transforms the 
retinal vector into one expressed in oculomotor coordinates. The olive  
thus receives the oculomotor error vector.  Its output is a climbing 
fiber correction vector that is the error-vector expressed in 
eigenvector components  (for this operation, the olive must store the 
eigenvectors of the oculomotor system).  The climbing fiber vector 
reaches the cerebellar nuclei in two ways: directly, via climbing fiber 
collaterals to the nuclei, and indirectly via Purkinje cells that also 
project to the nuclear neuronal array. Thus, the external (matrix) 
product of the climbing fiber vector with itself can momentarily arise 
over the array of nuclear cells. This complementary matrix 
functionally updates the wired-in cerebellar metric tensor, such that 
the cerebellum can act as a metric not in a position-independent 
ÒflatÓ but in a position-dependent ÒcurvedÓ motor hyperspace

Some more subtle aspects of the predicted function of olivaryÐ
climbing fiber system can also be mentioned here, although a more 
detailed discussion (illustrated by a quantitative example) is offered 
elsewhere (Pellionisz 1984a, Pellionisz and Llinás 1985).  
Namely, in order for the climbing fiber vector to mathematically 
produce the error-correction matrix, the olive has to send a climbing 
fiber vector  expressed in the eigenvector-coordinates of the motor 
system  (see mathematical elaboration in equation 17 in Pellionisz 
and Llinás 1985).  This issue is connected to the other facet of 
the predicted function of the olive (not in the focus of this paper); its 
role in the genesis of the cerebellar motor metric tensor.  As 
proposed by the Metaorganization principle and procedure (Pellionisz 
1984, Pellionisz and Llinás 1985), such a process is based on 
reverberative oscillations of proprioceptive and motor executive 
commands, which tremor stabilizes in the eigenvectors of the motor 
plant (see Fig. 3 in Pellionisz and Llinás 1985).  These 
eigenvectors need to be (1) sent via climbing fiber vectors to imprint 
the cerebellar corticonuclear network to serve as the Moore-Penrose 
generalized inverse, and (2) stored in the olive, such that it can 
decompose an oculomotor error-vector into eigenvector-components.  
A likely neuronal mechanism of storing eigenvectors in the olive may 
be the experimentally revealed electrotonic coupling of olivary 
neurons (Sotelo et al. 1974, Llinás and Volkind 1973, 
Llinás and Yarom 1981). The issue of expressing internal CNS 
vectors not in structural intrinsic frames, but such functional 
derivative frames as the one com-posed of eg. eigenvectors  of the 
oculomotor frame has also been raised in connection with saccadic 
burster neurons (in the monkey; Pellionisz 1988b).

Cerebellar Theory: the Challenge of Verification

The above tensor network model of the cerebellarÐolivary system, 
presented in a multidimensional quantitative framework of a 
sensorimotor mechanism may indicate to the reader a need to 
proceed from  an overly simplistic basic model of cerebellar function 
to a much more complex scientific account.  If advancing towards 
increasing complexity (eg. from single- to multi-dimensionality) 
leaves one with an uneasy feeling, one may wish to recall that if 
scientific theories are based on charmingly simplistic assumptions 
(eg. that planets rotate around us) then models (of eg. planetary 
trajectories) are hopelessly complicated, and experimental 
verification of such pontification may result in wasted or frustrated 
science. Based on a much more complex axiom (that we observe 
planetary trajectories centering around an object that we also rotate 
around), surprisingly elegant explanations may emerge, leading to 
perhaps even simpler but certainly more progressive verification.

Verification of theory in neuroscience will be twofold in the future. 
In addition to experimental scrutiny (eg. of predictions presented in 
this paper) the very recent emergence of neurocomputer-related 
applications started to exert a new influence on neuronal (cerebellar) 
modeling and brain theory. Neurocomputing and neurobotics 
industries will become testing workshops for brain theories 
(Pellionisz 1983, 88c, Eckmiller and Malsburg 1988). As a result, 
neuronal modeling and brain theory will no longer be an ivory tower 
exercise. No longer will prestige or political clout supremely arbitrate 
which theory is more advanced Ð natural selection through survival 
of actual tests will guarantee evolution. Theories that the Earth is 
flat, or that the cerebellum serves as an associative memory become 
untenable when means are available to verify that one reaches an 
Eastern location by travelling all around Westward, or when it 
becomes evident that one cannot program a robotic arm for 
coordinated movement based on the fiction that the cerebellum is an 
associative memory.

Cerebellum: Homework for Brain Theory

In a larger sense, one might mention that the basic paradox of 
analysis visa-vis  synthesis is not at all unique for cerebellar 
research. It arises in this domain since sensorimotor research in 
general, and cerebellar coordination research in particular, constitute 
a leading edge of system-neuroscience.  However, it is a general law 
that natural science progresses from the initial stage of gathering 
experimental (phenomenological) knowledge towards the goal of 
attaining a theoretical (conceptual) understanding.  It has amply 
been demonstrated, eg. in physics, how the richness of 
interdisciplinary phenomenology yields in time to the elegance of 
disciplined theory. Neuroscience, not being an exception among 
natural sciences, is presently scrambling to generate its own 
theoretical foundation, although, metaphorically speaking, its body 
can still be likened to "a welldressed gentleman with no shoes". 
There is no doubt, however, that our time is  marked by the 
emergence of brain theory (Churchland, 1985). The challenge is 
immense, and heretofore only partially met. A classical modest 
approach is based on the philosophy of making brain theory a 
chapter in control system engineering by conceiving brain function 
as an amplificationgain controller (Robinson 1968, Marr 1969, Ito 
1970).  Lately, modern brain theories, promulgated by physicists, 
excel in abstract simplification but connect rather loosely to the 
biological knowledge-base (eg. Nestor model by Cooper, 1974, or 
Spin-glass theory by Hopfield, 1982). Others, put forward by 
biologists, are strongly based on experimental data but are devoid of 
abstract mathematics (Group Selection theory by Edelman, 1979, 
Attentional "Searchlight" hypothesis by Crick,  1984). It is believed 
(see eg. review in Pellionisz, 1983) that mature neuroscience will 
demand and insist on theories which are not borrowed from other 
disciplines (either from engineering or physics) but are of 
neuroscience, for neuroscience and by neuroscience. Given the facts 
that cerebellar research has more than twenty years of leading 
history with such theoretical consolidation, and that it provides with 
a much scrutinized testing ground to check if lofty brain theories 
pass such simple tests as explaining basic CNS functions such as that 
of a threeneuron arc or of the cerebellum, one may wish, and indeed 
expect, that  cerebellar research will be among the first` to benefit if 
breakthroughs are truly achieved in brain theory.

Acknowledgement:  This research was supported by the grant from 
NINCDS 22999

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