Figure 1.

 

Fig. 1.  Neural Network of the Vestibulo-Ocular Reflex of the squirrel 
monkey.  A: Coordinate system in which axes of intrinsic frames 
are expressed. B: Coordinates intrinsic to eye muscles. C: Curved 
surfaces of the otolith organs, accomodating 50,000 hair cells.  D-E: 
Vestibular coordinate system of the Anterior, Posterior and 
Horizontal semicircular canals. F: Oculomotor coordinate axis 
directions, 

G: 30 axis neck-motor coordinate system. H:Simplified set of 
connections from three vestibular canals to six oculomotor muscles.
I-J-K Tensor transformation matrices (shown by filled and empty 
circles representing positive and negative matrix-components) 

A practical example of this problem can be observed when a surgeon 
operates using sensory feedback supplied by television (e.g. 
micromanipulation in endoscopic surgery).  The human nervous 
system is sandwiched by this between an active sensory 
representation, involving a moving camera, and operation in 
atelepresented environment. If telepresence and teleoperation  are  
left uncoordinated, as in general,  their conflict calls for,  at the least,  
an arduous learning process (measurable by many months)-  and  
outright impossible for some surgeons. 

While "virtual environment" techniques for active sensory 
representation and "robot control" by remote computers 
(teleoperation) are advanced fields in the information sciences,  their 
integration poses the task of matching the geometry of active 
sensory representation to the geometry of active motor 
representation. 

II. Linear Vestibulo-Ocular Reflex as a Neural Network Prototype for 
Matching Sensory and Motor Geometries

The role of the cerebellum in solving,  in an adaptive fashion, such 
problems of a conflict of sensory and motor geometries can be 
demonstrated by a gaze-stabilization neural network, the so-called 
(linear) vestibulo-ocular reflex [13, 28]. To stabilize gaze when 
fixating to a visual target while the head is in motion, the sensory 
apparatus measures head-movement in the 3 axis coordinate 
systems of vestibular semicircular canals (as well in the 50,000-axis 
coordinate system of the otolith organ (see Fig.1). Sensory 
information, expressed in covariant vector-components [20],  then 
has to be transformed into compensatory motor information, 
expressed as a contravariant vector in the six-dimensional 
oculomotor system (as well as in a thirty-dimensional neck-motor 
coordinate system).   It is both physically evident that the sensory 
coordinate systems are vastly different from that of the motor 
apparatus, and it has also been elaborated, for a number gaze 
reflexes of a number of species, that neural networks performing 
generalized vector (tensor) transformations can resolve the conflict 
of sensory and motor geometries [15],[19],[4],[24].

Figure 2.

 

Fig. 2. Non-Euclidean fractal and metrical neural geometries revealed 
by the cerebellum. Fractal geometry is expressed by  the
cerebellar Purkinje cells in Guinea-pig (A) and in the computer 
model (B). Complexity of the dendritic tree is comprised by
fractal template shown in (C). Metrical neural geometry of firings of 
cerebellar Purkinje cells can be revealed by the mathematical
technique, cf. [18], of  correlation analysis (F) followed by calculation 
of Moore-Penrose inverse of covariant metric (G) 


As shown in Fig. 1. tensorial neural nets transform a)  sensory 
coordinates to motor coordinates (where not only the direction of 
axes may be different but also the number of dimensions can change, 
even increase), b) changing from mathematically covariant (sensory 
intentional) components to mathematically contravariant (motor 
execution) components. As elaborated earlier,  it is the cerebellar 
neural network that enables adaptability to such sensorimotor 
reflexes,  in effect by altering the cerebellum as the metric tensor of 
the motor geometry [21].

III. Discerning Non-Euclidean Neural Geometries from the Cerebellum

A geometrical concept of neural network research [20],[6],[2], 
therefore has at least three major thrusts of direction; (1) discerning 
the exact nature of neural geometry, (2) implementation of neural 
geometries by artificial (electronic) means, (3) application of 
implemented neural geometries, as in the case of the cerebellum, for 
sensorimotor tasks such as to geometrically match  telepresence to 
teleoperation.  Fig.2. outlines the complexity of the task of discerning 
neural geometry. First, metrical  geometries (such as that of overall 
sensorimotor operation) can be measured by multielectrode-means 
(for theoretical foundation see [18], for methods and results see 
[22],[25]) and Figs. 3-4). It must be emphasized, that neural 
geometry, even at the coarsest level of granularity,  as in overall 
sensorimotor performance of movement-directions and amplitudes, 
is non-Euclidean (the functional geometry is curved since its metric 
is position-dependent). Its experimental measurement calls for the 
mathematical procedure of correlation-analysis followed by the 
calculation of Moore-Penrose generalized inverse of covariant metric 
tensor [18]. Experimentally, this requires multielectrode-techniques 
[27] as well as a dedicated parallel processor, in effect an electronic 
neurocomputer facing the biological neurocomputer. (Features of 
such a transputer-based, Macintosh-IIfx hosted neurocomputer are 
shown in Fig.3.).  

It is also evident that from the same cerebellar structure, at a finer 
level of granularity, a fractal neural geometry  can be revealed (Fig. 
2B,C). This implies that geometrical primitives of neural networks 
may well be grossly non-Euclidean fractals, not

Figure 3.

 

Fig. 3. Features of the Transputer-architecture of the neurocomputer 
developed for analysis and implementation of neural geomety [25]. 


only in a structural sense as shown by dendritic trees of neurons, but 
also functionally. For instance, contrary to classical experi-ments [8], 
the geometrical primitives of biological vision (pattern recognition, 
especially recognition of textures [10]) are likely to be fractals. This 
latter geometrical approach already yields dramatic bandwith-
reduction techniques for robotic vision and virtual environment-
techniques [3], and may well revolutionize experimentation on 
neurobiological vision.

Fig. 4 presents a display of the Transputer-based neuro-computer, to 
be used for discerning from the firing-patterns of cerebellar Purkinje 
neurons the position-dependent metric tensors of the functional 
geometry of musculo-skeletal system of an eye-head-neck apparatus 
of the squirrel monkey [25].

IV. Artificial Cerebellum ACE for Aerospace Applications of Adaptive 
Sensorimotor Control

The cerebellum provides a "secretariat-type" interface between one 
type of geometry, governing the world of intentions specified by 
sensory type representation, and another governing the world in 
which precise execution commands are given in terms of another 
functional geometry, specified by the constraints of the motor 
effectors. Thus, implementation of such a generic "interface" between 
sensory and motor geometries will find its usefulness in a variety of 
particular cases of adaptive sensorimotor control. Most importantly, 
it is expected to be crucial to the economic feasibility of the human 
settlement of space [1], which will require new control technologies 
in transportation [29] and an improvement in telerobotics techniques 
so as to permit the cost-effective exploitation of non-terrestrial 
materials and planetary exploration and monitoring. The Artificial 
Cerebellum ACE project is a framework within which NASA, together 
with other agencies such as NSF and NIH-NIMH could effectively 
target such goals of telepresence and teleoperation. 
While presently the Artificial Cerebellum ACE Neural Network is 
implemented only on the Macintosh IIfx experimental platform 
(using Transputer parallel processor), the suitability of other 
dedicated processors, such as the Draper INCA (with JPL neurochip), 
and INTEL's Ni1000 processor are also explored. 
As for general types of application of the Artificial Cerebellum ACE, 
three major areas appear as the most important (see Fig. 5). In 
conformance with the original goal for which nature developed 
biological neurocomputers, flight control emerges as one of the most 
potent ultimate application. In nature the fast and precise 
coordination required by forward-swept wings is resolved by an 
existing neurocomputer that features both an architecture with 

Figure 4.

 

Fig. 4. (See following page!) Representative display  from the 
Macintosh IIfx-hosted, Ttransputer-based neurocomputer,  
developed by AJP and DLT on NASA DDF-T4967, to discern neural 
geometry from vestibular and cerebellar neurons of the squirrel 
monkey. Panel A displays the eye-head-neck skeletomuscular 
system of the squirrel monkey as it tracks a moving target. The 
neural network scheme shows the vestibular nucleus (B) and the 
cerebellum (C) from which Purkinje cell activities are recorded 
(panel D). Bottom panels show the computation of the metric tensors 
of the geometry of the curved functional space.  Panel E shows the 
coordinate axes of the sensory system and its covariant metric tensor 
(matrix elements shown by proportional circles). Contravariant 
sensory metric tensor is shown in F. Axes and covariant motor metric 
is shown in panel G, contravariant metric as calculated from the axis 
is shown in H. Panels I and J display the metric tensors calculated 
from neural firings. Note that tensor I closely corresponds to G, while 
tensor J is virtually identical with the one in H. Also note, that all of 
the metric tensors discerned have non-zero off-diagonals! 

Figure 5.

 

Fig. 5. Three major targeted fields of application of the Artificial 
Cerebellum (ACE) project for (1) Flight control  (coordination
and control of forward-swept winged craft), (2) Propulsion control of 
vectored thrusters, (3) Telerobot control in telepresence.
These applications are based on the close parallel of existing 
neuronal network solutions in biology and technological utility. 


built-in error tolerance as well as adaptive, self-organizing 
"software". While tackling such flight-control tasks with electronic 
neurocomputers is still its very early planning stages, the ultimate 
advantages of parallel, error-tolerant architecture destine 
neurocomputing for such role. Propulsion control of vectored 
thrusters is another potential application, where close parallel 
between non-orthogonally arranged and movable effectors (such as 
eye-muscles) are controlled by a biological neural net, and such 
natural solutions lend themselves for use in control of similar 
vectored thrusters.

One of the most obvious parallel between biological and artificial 
neural net sensorimotor control exists between telerobot control in 
telepresence and the actual neural control of skeletomuscular 
apparatus (e.g. that of eye-head-neck system) under visual and 
vestibular coordination.

Government support (especially by NIH) has long resulted in vast 
knowledge and has started to yield mathematical understanding of 
biological neural networks. Likewise, civilian neuroengineering 
programs (e.g. by NSF) have also been in place for some time to 
provide the technology needed for implementation of artificial neural 
nets. Specific application-oriented NASA programs were therefore 
required that delineate separate civilian efforts. This was the 
rationale for aiming at an "Artificial  Cerebellum, ACE-
Neurocontroller", as a framework of an interdisciplinary and 
interagency program by NASA-NSF-NIH-NIMH.   

Acknowledgment 

This research was supported by NASA-DDF-T4967 to AJP and DLT, 
and NASA-NRC-A8967730 to AJP

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