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Neuroscience Vol. 16, No. 2, pp. 245-273, 1985 0306-4522/85
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Printed in Great Britain Pergamon Press Ltd IBRO
METAORGANIZATION OF FUNCTIONAL GEOMETRIES
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A. PELLIONISZ and R. LLINÁS
Department of
Physiology and Biophysics, New York University Medical Center,
550 First Ave,
New York, NY 10016, U.S.A.
Abstract
Here
we present an elaboration and a quantitative example for a hypothetical
neuronal process, implementing what we refer to as the metaorganization
principle. This process allows the internalization of external (body)
geometries into the central nervous system (CNS) and a reciprocal and equally important
action of the CNS geometry on the external (body) geometry. The hypothesis is
based on the destination, within the CNS, between covariant sensory and
contravariant motor vectorial expressions of the extrinsic geometry. These
sensory and motor expressions, given in natural co-ordinate systems, are
transformed from one to the other by a neuronal network, which acts as a metric
tensor. The metric tensor determines the relationship of these two expressions
and thus comprises the functional geometry of the system.
The
emergence through metaorganization of networks that implement such metric
function is viewed as the result of interactions between the covariant motor
execution which generates a physical action on the external world (via the
musculoskeletal system) and the covariant sensory proprioception which measures
the effect of such motor output. In this transformation of contravariants to
covariants by the physical geometry of the motor system, a covariant metric
tensor is expressed implicitly. However, co-ordinated motor action requires its
dual tensor (the contravariant metric), which is assembled in the CNS based on
the metaorganization principle, i.e. the ability of CNS and external geometries
to mold one another. The two metric transformations acting on each other detect
error signals whenever the match of the physical and functional geometries is
imperfect. Such error signals are utilized by the metaorganization process to
improve the match between the two metrics, so that with use the internal
representation becomes increasingly homeometric with the geometry of the
external world.
The
proposed physical process by which the metaorganization principle is
implemented is based on oscillatory reverberation. If covariant proprioception
is used as a recurrent signal to the motor apparatus, as if it were a
contravariant motor expression, then reverberations at their steady state yield
the eigenvectors and eigenvalues of the system. The stored eigenvectors and
eigenvalues can serve, respectively, (1) as a means for the genesis of a metric
(in the form of its spectral representation) with the given eigenvectors and
(2) as a means of comparing the eigenvalues that are implicit in the external
body geometry and those of the internal metric. The difference between these
eigenvalues is then used to modify the metric so that it "evolves" to
perform a more accurate covariant-contravariant transformation.
The
metric can be represented by the dyadic outer products of its eigenvectors,
where each dyad is weighted by the corresponding eigenvalue. Such a spectral
representation yields in a uniform manner (a) the contravariant metric (in the
case of complete CNS hyperspaces) and (b) the Moore-Penrose generalized inverse
of the covariant metric tensor (in the case of non-Riemannian overcomplete CNS
hyperspaces).
The
metaorganization of metric networks, together with the sensorimotor covariant
embedding network, provides an explanation of the emergence of a whole
sensorimotor apparatus leading to the feasibility of constructing truly
brain-like robotic systems.
1. INTRODUCTION
1.1. Geometries and
brain function
The evolvement of the CNS through natural selection is the
fundamental means by which multicellular organisms develop optimal interactions
with the world. In terms of abstract geometry this can be expressed by stating
that the function of the brain is to match the system of relations among
objects in the external world, with a multidimensional inner functional
geometry, in a manner such that these geometries approach homeomorphism if not
isomorphism.5o.so While general considerations as such may be of significance
in brain theory, the nature of the interaction between the inner and outer
geometries must be defined in a concrete manner.
Sensorimotor operations are the appropriate paradigms of brain
function to consider first for such elaboration. Here, the relation between the
external world and its internal representation in the brain can be directly
observed and quantitatively treated as it is expressed by the precision of
goal-oriented movements.
Sensorimotor transformations, most particularly those involving
the cerebellum, have already been investigated from the point of view of how
they adapt to an alteration of geometries, either of the external physical
arrangement or its inner representation. As an example of the former,
modification of the vestibulo-ocular reflex, as in the case of vision inverted
by reversing prisms, has been amply studied.23 For the latter the compensation
for unilateral vestibular ablation has also been analysed in detail.58 The
issue in this paper is to define the mechanism by which the relationship
between inner brain geometry and the external world are matched in accordance
with the modified conditions. This point has been raised in preliminary
communications.75,82,85
246 [non-searchable facsimile page-view]
1.2. The essentials for
a geometrical approach to brain function
The general hypothesis of the geometrical interpretation of brain
function hinges on the assumption that the relation between the brain and the
external world is determined by the ability of the CNS to construct an internal
model of the external world using an interactive relationship between sensory
and motor expressions. This relation is evident, for instance, in the
orienting-response of an animal in a new environment. Indeed the process of
sensory detection involves a significant amount of motor activity. 32 Such
observations strongly indicate that the sensorimotor transformations are the
gauging tool by which the CNS relates to the external world. It has been
demonstrated in detail in human and animal experimentation that a convergence
of sensory and motor information is necessary to form internal models of novel
objects. 23,63
At the level of mathematical analysis, the basis for formalizing
the above in terms of a geometrical representation appears to be the dual
expression of extrinsic physical objects by intrinsic "CNS vectors".
These vectorial expressions, with respect to reference frames intrinsic to the
organism, are covariant for sensory analysis and contravariant for motor
synthesis. The geometrical relationship between these two vectorial components
is comprised in a neuronal network acting as a metric tensor. 81
1.2.1. Fundamental mathematical concepts inherent in covariant and
contravariant neuronal representations.
The covariant-contravariant distinction is of cardinal importance,
as their relationship determines the metric, which comprises the functional
geometry of the system. Indeed these two ways in which the CNS relates to the
external world are profoundly different. The primary expression derives from an
external object a multitude of covariant vector components, which constitutes a
sensory input to the CNS. Such a vector arises from the firing frequencies of a
set of sensory neurons which collectively represent an extrinsic physical
state. The secondary expression is the implementation of a physical reality
from a multitude of contravariant components; e.g. generating a displacement of
the arm through the activation of individual motor units. This dual
relationship between CNS expressions and physical invariants via the sensory
and motor systems is fundamental. The question is not whether such different
expressions exist but, rather, whether it is possible to construct a precisely
formulated general concept, which encompasses the functional essence of (a) the
above sensory and motor properties and (b) the transformations, which convert
one into the other. The most fundamental difference between these two
expressions appears to be their causality; i.e. that in the former the
components arise from the physical reality, while in the latter the physical
reality arises from the components.
Sensory reception is therefore an analysis (i.e. a differentiation)
while motor execution is a synthesis (i.e. an integration). The covariant
expression is based on a process of comparing the features of the external
physical reality (which is co-ordinate-system invariant) with a set of internal
physical states (which serve as the measuring standard within the CNS). Each of
these relationships yields a quantity, which is then used as an inner measure
of one aspect of the external reality. An example of this procedure in the CNS
is the inertial force generated by the endolymphatic mass in each semicircular
canal, each yielding one individual measure (a cosine component) of the
head-acceleration. Note that as defined in calculus, establishing the relation
of the rates of change is a differentiation. Given two invariants their
infinitesimal relationship yields the derivative. It is also well known that,
following the chain-rule of differentiation, such derivative components of x
change with the alteration of the frame a in a manner obeying the following
rule of covariant transformation: 14,94,101
[see equations in 246.gif]
The biological significance of these known mathematical properties
was expressed in the proposal that these projection-type covariant components correspond
to sensory processes.81
Motor actions, on the other hand, are integrative type operations.
In a motor process such as displacement of the eye by the co-contraction of the
extraocular muscles, the physical summation of the components is of the essence
and not the differential-type sensory relationship of one physical invariant to
another. Infinitesimally, the summation is implemented by the parallelogram
components that obey the rule of covariant transformation. 14,101
Reference frames are invoked by both the sensory and motor
processes. These are instruments, intrinsic to the structure of the organism,
through which extrinsic points of physical spaces are related to arithmetical
manifolds within the CNS.
Note that so far the only requirement in the interpretation of
sensory covariants and motor contravariants is that the derivatives of the
manifold should exist, i.e. that the manifold which arises from the use of
reference frames be smooth. This is the necessary and sufficient prerequisite
for the existence of covariant sensory and contravariant motor expressions.
Transformation of co- and contravariants through the metric is a
different problem from the simple distinction between these two types of
representation. The central question is whether a concise expression can be
given for the transformation between these co- and contravariant vectoral
forms. As is well-known, such expressions can be obtained since any geometry
can be most concisely characterized by its metric tensor (also called
fundamental tensor,) 19,46 which expresses the relationship between covariants
and contravariants. Indeed, a study of any geometry may well start with the
definition of its metric. It has been stressed, 84 however, that the geometry
of the CNS hyperspaces cannot be a priori characterized by a known metric; e.g.
by an Euclidean metric (as in Cartesian tensor analysis that is used in
engineering)98 or by a Riemannian metric (as in the tensor analysis of
four-dimensional manifolds used in relativity theory).19 The features of the
metric in a mathematical CNS hyperspace cannot be taken for granted; in fact
the metric is the unknown in brain research. Indeed if a CNS hyperspace is
amorphous, a metric in the strict sense may not even exist; sensory covariants
and motor contravariants may be unrelated (which would manifest itself in a
dysmetric motor action). The existence of any metric in the CNS poses the
question to the experimenter "how to find it" and to the theorist, "how
such metric is developed in complex organisms".
1.2.2. Fundamental differences in biological interpretation of the
contravariant motor activation and covariant proprioception, commensurate with
tensions.
The metaorganization principle will be elaborated in this paper by
means of a quantitative illustration. The model diagram shown in Fig. 1 has
been used in a preliminary form in preceding publications on tensor network
theory. 75,76 First it will be shown, by means of a specific quantitative
example, that the active forces that are exerted by the muscles upon
innervation are proportional to the contravariant physical components of the
impressed load G. In contrast, the passive forces that are measurable by
proprioceptive peripheral sensory system, can be determined proportionally to
covariant orthogonal projection components of G.
The motor execution mechanism, shown schematically in Fig. 1(A),
assumes that the three joint arm moves in the two-dimensional plane of the
paper, meaning that the motor system is overcomplete. A flexor and an extensor
muscle at each of the three joints are depicted in Fig. 1(A): the shoulder,
elbow and wrist. The pectoralis muscle (1) acts as a flexor, and the deltoideus
(2) acts as an extensor of the upper arm at the shoulder joint. Thus the
difference between the forces exerted by them would be the force acting at the
index finger in direction alpha (Fig. 1B). Likewise, the triceps muscle (3)
extends while the brachialis (4) flexes the lower arm and thus the resultant
force is along direction beta. In turn the extensor carpi ulnaris (5) generates
a force along direction gamma, working against the force of the flexor carpi
radialis (6).
The alpha, beta and gamma directions shown in Fig. 1(B) determine
a local non-orthogonal frame of reference. The general considerations in 1.2.1.
are demonstrated in detail, in Figs 1(C) and (D), namely in such non-orthogonal
frames of reference two vectorial expressions are possible, each with different
characteristics. If a load force (G) is attached to the index finger (Fig. 1
C), and it is assumed that the arm holds this load in a stationary position,
both the active and passive forces are expressed in the (position dependent)
local system of co-ordinates as introduced in Fig. 1(B). However, the active motor
actions rotate the arm-segments by means of the forces exerted by the muscles
(Fig. 1C), whereas peripheral proprioceptive organs detect different, passive
forces (Fig. 1 D).
The contravariant character of the forces exerted by the muscles
follows directly from the fact that in a steady-state the active forces must
balance; the muscle components must yield a resultant force that is equal but
opposite in direction to the load (G, see Fig. IC). These physical components
of the load (G) are, by definition, the contravariant components that add
according to the parallelogram rule. The forces of the muscles have always been
regarded to be the physical components that balance the load (G). Tensor theory
simply re-phrases this by stating that the vectorial expression of an object in
any system of co-ordinates by means of physical components is of the
contravariant type. It has been discussed previously, 8,76 and is obvious from
Fig. 1(C), that such a contravariant expression of a given load (G) in an
overcomplete frame of reference is not unique. Indeed, an infinite number of
configurations of the components along a, R and y can result in the same load
(G). In fact it is this mathematical indeterminacy that raises the question of
how does the CNS arrive, in a co-ordinated act, at a particular choice from an
infinite number of possible solutions. The most obvious demonstration of such
overcompleteness can be shown by remembering that any constant can be added to
the forces exerted by both an agonist (flexor) and antagonist (extensor)
muscle-pair, without changing the resultant, since the two forces act along a
common co-ordinate axis and thus the additions cancel one another. While such
addition to both components of a pair of muscles is mathematically redundant, its
physiological significance is obvious. The same stationary position of the arm
can be held with the muscles contracting minimally, or in an overexerted stiff
manner; consequently, the holding position may be delicate or robust. The
understanding that additions of cancelling force-pairs will not modify position
makes it possible to consider in the rest of the paper only the differences of
forces in reciprocally innervated muscle pairs acting along common axes. Such
"pairing", however, is only for simplifying convenience; the scheme
to be presented in this paper also applies to the separate treatment of muscles
acting along different axes.
Fig.1.
Schematic representation of the physical geometry of an exemplary motor
executor apparatus, which is to be matched by a functional geometry
(implemented by the cerebellum), their match enabling the co-ordinated control
of the multiarticulate limb.
(A) and (B) show the frame of reference of
limb displacements, that is intrinsic to the motor apparatus. (C)-(E)
demonstrate the two different kinds of co-ordinates (contravariant and
covariant) that express the physical invariant of a displacement by active
muscle components and passive proprioceptive components, respectively-the two
vectorial expressions implying the covariant metric inherent in the physical
geometry of the motor apparatus. (A) Individual muscles determine a local
curvilinear frame of reference for displacements of the limb by six major
muscles. 1, pectoralis; 2, deltoideus; 3, triceps; 4, brachialis; 5, extensor
carpi ulnaris; 6, flexor carpi radialis. (B) Simplified, non-orthogonal
rectilinear frame of reference of limb-movements. The alpha, beta and gamma
local displacement-directions belong to individual incrementation of the
joint-angles. (C) Physical (contravariant, parallelogram-type) vectorial
expression of an invariant G in the intrinsic frame of reference. (D)
Proprioceptive (covariant, projection-type) expression of the same G; with
unique components even in an overcomplete frame. (E) Contravariant components
physically execute motor acts, covariant components are in turn detectable by
proprioception. Thus a contravariant metric tensor is implied in the physical
geometry of a motor apparatus.
It must be emphasized that in the non-orthogonal system of
co-ordinates shown in Fig. 1, in addition to the contravariant active physical
muscle forces, a difierent covariant-type vectorial expression of the load (G)
is also possible (Fig. 1D). The covariants are the passive force-components,
measured as the orthogonal projections of the load-vector (G) onto the
co-ordinate axes. The nature and the functional role of the passive forces,
detectable by peripheral proprioceptive systems such as the tendon organs, have
not been conclusively defined in motor physiology, despite thorough analysis.
59,97 The prevalent ideas are that this system monitors the load on the motor
apparatus or provides an overload-preventing warning signal. The role of
proprioception is therefore particularly intriguing in oculomotor actions,
where the load of the system is constant. 29
In contrast, it has been suggested in tensor theory of the CNS
that proprioception serves to supply the components of the dual complementer
motor vector, the covariant counterpart of the contravariant motor action.75 It
is well known that peripheral receptor organs are capable of measuring passive
stretch. Such a passive force may significantly differ from the active force
exerted by the given muscle. For instance, while the active force generated by
a muscle is proportional to its own motoneuron-activation, a stretch in the
tendon of the given muscle arises from an interaction of the activity of many
muscles as well as from external load factors. Since the tendons utilize the
same as the alpha, beta and gamma local system of co-ordinates as the motor
actuators, and they express the physical object of the load in a sensory
manner, they yield the covariant components of the load. This proposal is
substantiated below with the help of Fig. 1(E).
Muscles exert force on the skeletal system (even if the action
arises from a variety of active and passive factors) through their tendons. Thus,
if a load (G) is balanced in a stationary position, the passive force (F),
proportionally detectable at the elbow-joint denoted by beta, would be the
difference of the tendonforces of the extensor and flexor carpi. This passive
force F, measurable by a strain-gauge in the tendon at beta, must balance the
torque (G g), exerted by the load, by (Ff ) (cf. Fig. lE). From the similarity
of the GF and fg triangles, it follows that F = G.cos (omega), which is the
definition of the covariant, orthogonal projection-type component. Note that
the set of covariant proprioception components of G, measured as the tensions
in the tendons, is unique, in contrast to the non-uniqueness of the
contravariant components.
The scheme in Fig. 1. presents a motor effector mechanism in which
the dual sets of co- and contravariant components are available. This poses the
following challenge to the CNS: given a contravariant motor execution vector
the proprioceptive system must provide the corresponding covariant vector. Such
contravariant-to-covariant relation is implied in the physical geometry of the
effector mechanism. Thus when an internal functional geometry is developed by
the CNS the physical geometry must be matched by this homeometric internal
representation. It is suggested that the matching of the physical geometry with
its functional counterpart is furnished by the cerebellum. Below, a concise
account is given of how the cerebellar circuit may perform as a co-ordinator
once the circuit is available (point 1.2.3) and how such explanation leads to
the question of how such networks may emerge (point 1.3).
1.2.3. Utilization of dual vectorial representations to explain
CNSfunctions: the tensor model of cerebellum.
Tensor network theory of the CNS evolved as a mathematical
formulation, with the use of the above basic terms of covariants and
contravariants, of the geometrical concept of brain function, especially that
of the cerebellum.74-78,80-83. The tensor model can concisely explain in the
above terms the function of the cerebellar circuit, once that circuit is
available through the development of a sensorimotor system.
A general tensorial interpretation of the CNS is based on the
notion that the intrinsic natural frames of reference, in which neurons
attribute ordered sets of activity-values (co- and contravariant vectors) to
physical invariants of the external world, invoke multidimensional arithmetic
manifolds. The functional geometry of such a CNS hyperspace is comprised by its
metric tensor, which can be implemented by a matrix-perhaps the most natural
abstract representation of a neuronal network. Sensorimotor systems could
therefore be functionally explained in a three-stage scheme.81 This consists of
(1) a sensory metric (an internal representation of the geometry of the
external world) which could be the optic tectum (see the scheme in Ref. 74),
(2) a cerebellar motor metric network that endows the executor mechanism with a
functional spacetime geometry (see the scheme in Ref. 83), and (3) a
sensorimotor transformation that relates the two CNS geometries to one another;
e.g. by embedding one space, such as the sensory, into another, such as the
motor (see the scheme in Ref. 76).
A tensorial interpretation of a particular sensorimotor system
yields a functional scheme (e.g. Fig. 1, in Ref. 76). Such network schemes
provide a mathematical interpretation of sensory processes, as yielding
covariant vectorial expressions, and motor processes, as executing invariants
with contravariant components. Moreover, tensorial schemes can formally
describe the nature of sensorimotor integration as transforming a covariant
vector, assigned to an invariant expressed in the sensory frame, into a
contravariant expression in the motor frame. In turn, the problem of
co-ordination (the uniqueness of a motor expression in an overcomplete executor
mechanism) can be resolved as a covariant embedding followed by a cerebellar
motor transformation from covariant intention to contravariant execution, even
in the case when the covariant metric is singular. 74-76
A tensorial interpretation of the cerebellum, which is suggested
to be the crucial final step of the sensorimotor system, is shown in Fig. 2.
The cerebellum is featured as an "add-on" unit; 75-77 the scheme in
Fig. 2 demonstrates that a direct spinal cord pathway could carry a motor
intention-vector to represent the motor output directly. Such approximative
sensorimotor transduction may have been an early evolutionary
"solution", where the directly obtainable but incorrect motor vectors
were used for motor execution. According to the "add-on" scheme,
cerebellectomy results in the direct execution of covariant motor intention,
through the down-going spinal pathway shown schematically in Fig. 2. This
feature of the model corresponds to classical knowledge 33 that ablation of the
cerebellum does not break the sensorimotor transduction (and thus the
cerebellum cannot be considered as the organ implementing this function); the
ablation does result, however, in a "dysmetric" motor activity
decomposed both in space and time.
The detailed operation of the essential cerebellar circuit is
described as follows. The covariant components of the motor intention-vector,
belonging to a co-ordinate-system-invariant displacement (inv), are shown in
the upper left circle of Fig. 2.
Fig. 2.
Functional scheme of the essential cerebellar network: co-ordination by acting
as a metric tensor, transforming covariant intention into contravariant
execution.
Sensorimotor transformation, by means of
covariant-embedding procedure, yields independently and uniquely established
projection-type intention components (even in case of overcompleteness) of an
invariant (inv; upper left circle). This ik vector, if it directly descends
(through the brain stem nuclei, bn) on the motor apparatus, would physically
add to an incorrect execution (int; e.g. in acerebellar dysmetria). The role of
the essential cerebellar network is to transform, by the "add-on"
circuitry in the cerebellar cortex and nuclei, this intention to contravariant
execution e" (exc; lower left circle). The metric-transformation is
accomplished by the cerebellar corticonuclear network g"k, by which the
intention (i), carried by mossy fibers (mf) to parallel fibers (pf) and to
Purkinje cells (PC) is connected to cerebellar nuclear cells (cn). The
inhibitory execution vector (e), together with the mossy fiber collaterals to
the nuclei, forms a corticofugal output (i - e). This signal gives rise in the
brain stem nuclei (bn) to the e = i - (i - e) execution-vector output. Note
that the
olivary-climbing fiber system is not
required for the essential cerebellar coordinative function.
These components are the orthogonal projections from the invariant
to each motor axis. Taking the motor apparatus shown in Fig. 1 as a symbolic
example these axes are 0, 25 and 37° with respect to each other. Thus, a 5°
physical invariant displacement,with an arbitrary magnitude of 100, will yield
a covariant vector ik = (100 94 85)T where superscript T denoted the transpose
of the row-vector into column-vector. While these components do represent the
displacement, their physical summation would yield a different displacement
(int) with a different amplitude and direction (Fig. 2, upper left).
In contrast, the metric-type transformation through the cerebellar
neuronal network yields contravariants that physically add to exactly yield the
required invariant exc (Fig. 2, lower left inset). The three-segment limb with
the a, ~i and y system of co-ordinates therefore requires a contravariant
metric-type transformer. This can be expressed numerically as g"k, shown
by a quantitative 3 x 3 matrix in Fig. 2. The system of connectivities, which
implements such a matrix is the network between Purkinje and cerebellar nuclear
cells (Fig. 2).
This scheme conforms with the known cerebellar
anatomy.36,48,52,69,70 Any ik covariant motor intention vector that enters the
cerebellar cortex by mossy fiber activity will generate, through g"k, the
contravariant execution vector e" via cerebellar Purkinje cells (Pc) in
the cerebellar nuclei (cn). This vector e" = (80 25 -4)T impinges on the
cerebellar nuclear cells in an inhibitory form.38 Together with the excitatory
mossy fiber collaterals into the nuclei, the cerebellar nucleofugal output will
be ik - e" = (20 69 89)T. In the brain stem nuclei (bn) this output will
transform the intention vector into the required execution vector. This vector
leaves the spinal relay nuclei as e=i-(i-e)=(85 25 -4)T.
Note that the network essential to this covariantcontravariant
transformation contains only mossy fibers, granule cells, Purkinje cells,
cerebellar nuclear neurons and brain stem nuclear neurons.
Strictly speaking therefore, neither the climbing fiber system
(and the inferior olive) nor the proprioceptive sensory mechanism is required
for the explanation of the coordinative function of the cerebellum (the
covariant-contravariant transformation) implemented by the "essential
cerebellar network" (Fig. 2).76,77 As elaborated elsewhere,85 however, the
climbing fiber system is essential in answering the question "how might
such a network emerge?"
1.3 Problem: the
genesis and modification of neuronal networks serving as
covariant-contravariant metric-type transformers
The above exposition (point 1.2.3) of how the CNS could function
by tensor-transformations assumes that the required matrices are in place,
implemented by neuronal networks. A more profound question, however may be 82
"how are neuronal networks organized such that they can embody and
functionally support the necessary geometrical transformations?" While it
is crucial to emphasize (as discussed in detail in Ref. 51) that the
development of motor coordination is not the function of the cerebellum (since
the function is co-ordination), after identifying the function that cerebellar
neuronal networks perform, answering the underlying developmental question may
be very revealing from the point of view of emergence of neuronal networks in
the CNS in general.
This major theoretical problem of network organization can be
readily illustrated by the cerebellum. Indeed the geometry of the motor
apparatus is physically explicit and it is known that the matching functional
geometry (implemented by the cerebellar neuronal network) develops from
specific genetic and epigenetic arrangements which are expressed in the
embriogenesis of the cerebellum.64,69,92
Once the basic co-ordination-function is performed by the emerged
cerebellar networks, it is also known that a misalignment between the geometry
of the execution system and a pre-existing functional motor geometry can result
in a functional error. If this mismatch is minor, it may be absorbed by the
overcompleteness of the functional transformaton.83 For misalignments of the
geometries that follow a certain trend (e.g. when motor co-ordination has to
keep up with the increase in body size during growth) the cerebellar system
must respond with a degree of adaptability, just as in every subsystem of the
CNS.S~ It is also known that major discrepancies may trigger realignment of the
internal and external geometries-an important function that allows the organism
to resume an optimal interaction with the surrounding world (cf. the reversal
of the vestibulo-ocular reflex by prisms and the ensuing compensation).23 As
suggested in the three-step scheme of sensorimotor transformation,76,83 the
neuronal networks implementing a sensory metric tensor and a motor metric are
the means of incorporation of such functional geometries. Therefore the
adaptability of the whole sensorimotor operation raises the question how a
limited degree of adaptibility of each of these metrics may contribute to a
maintained match of the external physical and internal functional geometries.
As for the cerebellum, the assumed position-dependent motor metric-function
implies, for example, that the neuronal network must undergo constant phasic
updating by the climbing fiber system in order to perform the required
non-Euclidean (non-constant) metric function.
While the question of emergence of neuronal networks, acting as
metric tensors, can be conveniently approached at the level of motor
co-ordination, this question is more profound than a limited study of
sensorimotor operations implies. Indeed one of the fundamental challenges in
neuroscience is that of providing a formal account of the ability of
geometries, intrinsic and extrinsic to CNS, to organize one another so that a
set of optimally interactive geometries can evolve.
2. METAORGANIZATION OF
CNS GEOMETRIES
2.1. The principle of
metaorganization
In search of the principles by which neuronal networks could be
organized, one needs to rely on the power of both the mathematical formalism
and of the biological insights derived from direct experimental acquaintance
with the problem.
2.1.1. Optimal mathematical characteristics of the procedure of
establishing metric networks: iterative algorithm for eigenvector-expansion.
A general principle in all biological systems appears to be that
their structure is parallelly organized and distributed and their function
develops by iterative procedural means. An example is the acquisition of
cerebellar temporal lookahead by the implementation of a type of Taylor series
expansion.80 Thus, when addressing the questions of the distributed
implementations of neuronal metrics, one may consider the manner in which
matrices, acting as metric tensors, can be established by iterative
reverberative procedures in a form of matrix-expansion.
Another lead is provided by the fact that the primary entities in
CNS function are the covariant sensory and contravariant motor expressions, and
not the metrics that may or may not connect them. The co- and contravariant
inter-relations evoked by external physical reality can manifest themselves in
case of a lack of an explicit realization or even in case of a total absence of
an ordinary metric: both sensory and motor processes are possible without an
intermediate co-ordinated transfer. Any metric expression is therefore
secondary, as it derives from a process by which given co- or contravariants
may be converted into one another.
As for a matrix acting as a metric, such matrix is symmetrical and
determined by those co- and contravariant vectors which constitute eigenvectors.
The metric is indeed fully characterized by these special
input-output vectors that are identical in their normalized form.91
The above two mathematical considerations provided the impetus for
pointing to the steady-state covariant-contravariant reverberation as the key
for a formal geometrical characterization of the function and emergence of
neuronal networks. An eventual identity of covariant sensory information and
contravariant motor output (where the input can directly determine the output
in the form of an eigenvector without the necessity of an interconnected metric
transformation) provides the basis of metaorganization.
2.1.2. Optimal characteristics' for the biological procedure for
establishing metric networks: tensorial interpretation of tremor.
The formative aspects of oscillatory behavior are most conspicuous
during embryogenesis.49 It is known, for example, that embryos evolve through
characteristic tremor and oscillatory twitching.8,30 Reverberative resonance is therefore a dominant characteristic
which may reveal fundamental properties of the movement effector and the
functional properties of related neuronal networks. Oscillations have also been
analysed in detail by numerous workers both from biological and mathematical
points of view (see Refs 4, 5, 12, 22, 24, 28, 60, 68 and 100).
The most significant feature of tremor may be expressed formally
by assuming that the proprioception system provides a covariant measure of the
contravariant motor execution. Indeed it has been suggested'5 that
musculoskeletal systems endowed with proprioceptive feedback, where the frame
of reference both for motor execution and sensory reception is common, can base
their function on the following fact. The contravariant motor action and the
covariant proprioception, belonging to the same physical invariant, represent
together the covariant metric tensor. That is, for any given contravariant
execution vector the proprioception, yields the covariant counterpart. Thus for
a motor system the contravariant motor action and its covariant proprioception
together define the covariant metric tensor inherent in the physical geometry
of the motor apparatus. However, in motor co-ordination the contravariant
metric is required to implement the proper transformation of covariant motor
intention vectors into their contravariant executable form.
The above considerations lead to the metaorganization principle,
summarized in the following.
(1)
The eigenvectors of the covariant metric of a motor system can be
established by reverberations, resulting from the return of covariant
proprioception to motor effectors as if they were contravariants.
(2)
This oscillation will reach a steady-state of
covariant-contravariant eigenvectors.
(3)
These eigenvectors and eigenvalues (or the generalized inverses of
the latter) can be used to generate either a duplicate or a complementer of the
covariant metric tensor.
(4)
The resulting metric-type networks (e.g. that of the cerebellum)
can be used as an internal function representation of an external geometry
(e.g. used for motor co-ordination).
It is important to emphasize that co-ordination established by
metaorganization may not be necessary for some stereotyped movements
(especially those in lower vertebrates). Operations such as basic locomotory or
grooming actions do not represent the class of co-ordinated goal-oriented
movements that are the focus of this study. Rather they seem to be
"preprogrammed" activities based on fixed-pattern generators at
spinal cord level, for which a higher cortical involvement of the intention, or
a cerebellar co-ordination of the execution, is not necessary.
2.1.3. Reverberation of proprioceptive covariants as
contravariants, in order to establish the eigenvectors implicit in the physical
geometry of the motor apparatus.
The interpretation (in section 1.2) of the proprioception signals
and motor signals as covariants versus contravariants provides a concrete example
of the physical implementation of the metaorganization of CNS. The proposed
procedure will be shown to yield a co-ordinated control of an overcomplete
muscular system. The implementation is based on the reverberation of the
covariant proprioception-afferent as if it were a contravariant-executor
motoneuron efferent. This proprioception-execution reverberation will set up an
oscillation of the motor apparatus that reaches a steady-state when the input
and output signals are identical, i.e. when they both constitute an eigenvector
of the system.
The procedure shown in Fig. 3, utilizes the motor system of Fig. 1
and the cerebellar neuronal network in Fig. 2 with an arbitrary vector
descending the motor executor system. It is emphasized that this initial vector
could be arbitrary and may arise from internal "noise" of the
circuits. Let the motoneurons innervating the pectoralis muscle produce a burst
of spikes of unitary strength which results in a movement of the hand along the
a direction. This arbitrary motor signal can be described as a contravariant
vector
e = (1 0 0)[to the power of] T, where superscript T denotes the
transpose of the row-vector into column-vector.
According to section 1.2.2, the peripheral sensors, which measure the
tensions in the tendons, are capable of yielding the covariant sensory
components of the generated invariant. These components arise geometrically by
establishing the orthogonal projection components of the unitary motor vector e
(along direction a, to the /3 and y axes). These covariant components can be
calculated from the contravariant vector by multiplying it with the covariant
metric tensor of the motor system, which is simply the table of cosines among
the axes." The local system of coordinates for the movements of the index
finger is shown in Fig. 1, by axes at 185, 160 and 148° angles, in the
two-dimensional physical plane, where the angles are measured from the
customary right-horizontal 0° direction. Thus the covariant metric of the physical motor apparatus is: g[sub]nk = cos
(phi[sub]nk)
cos(185-185) cos(185-160) cos(185.148) =
cos(160-185) cos(160-160) cos(160-148)
cos(148-185) cos(148-160) cos(148-148)
1.000 0.906 0.799
3 = 0.906 1.000 0.978 ( ) 0.799 0.978 1.000
Given that such a covariant metric of the motor
execution is represented by a matrix where the components are the cosines of
the angles among the axes, two properties of the matrix are given: (1) the
matrix is symmetrical, since ¢"k = ~k" and (2) its components are
real values since cos(~"k) is a real number for any angle. Symmetrical
real-valued matrices constitute a special subclass of Hermitian matricess' and
thus are characterized by having a set of orthogonal eigenvectors with
eigenvalues that are real numbers. ~, Thus an actual physical procedure to
generate such metric matrices is possible; these are implemented by
reverberation, as shown in the remaining part of this section.
The numerical example, shown in Fig. 3, starts with
the initial arbitrary contravariant motor execution vector oe = (1 0 0)T. The
corresponding first covariant expression, detected by proprioception, will
therefore be ,p = (1.000 0.960 0.799)T. One can establish these values either
geometrically, by taking the orthogonal projection to the other axes of the nonzero
component of oe, or by computing its components through the covariant metric
according to p = g x e. It is assumed that p is normalized before it is
reverberated as if it were contravariant (normalized vectors are denoted by
barred symbols). This vector will descend on the motor system as ,e = (0.638
0.578 0.509)T. This contravariant output will then produce an arm position that
will be measured covariantly in the second reverberation by the proprioceptors,
which detect tension in the tendons (Zp). Application of the covariant metric
reveals that zp = (1.568 1.654 1.584)r. If this proprioceptive vector is
reverberated for the second time in normalized form as p, it will be executed
as Ze = (0.565 0.596 0.571)T. This vector will then be covariantly measured as
3p = (1.561 1.666 1.605)T and reverberated in the third cycle in a normalized
form as 3e = (0.559, 0.597 0.575)r. As verified through the application of this
vector to the covariant metric, the third sensory proprioception will yield 3p
= (1.560 1.666 1.606)T. Repeating the cycle, the fourth contravariant return
will be 4e = (0.559 0.597 0.575)r. Note that the above reverberation
stabilizes with a vector that is identical in its proprioceptive and execution
forms. The reverberated signals are identical after the third cycle: 3e = 3P =
aP = (0.559 0.597 0.575)T.
The example of reverberation shown in Fig. 3
demonstrates that after a rapid convergence the oscillation of the system
reaches a steady-state of the eigenvectors. Without such normalization, during
the reverberation each component of the nth covariant sensory vector would be ~
= 2.791-times greater than the corresponding n th contravariant motor vector
component:
"p=~ x"e=2.791 x"e
= 2.791 x (0.559 0.597 0.575)T. (4)
The factor ~ is known as the eigenvalue.a' It can be
measured by the same operation as the normalization since ~ is the necessary
degree of change of the magnitude (normalization) of the vector before its
reverberation.
Such a covariant-contravariant pair, given above in
normalized form as 3e = Qp = E, = (0.559 0.597 0.575)T, where the covariant and
contravariant forms differ by only a constant coefficient for each component,
is called an eigenvector of the covariant metric in the given frame of
reference and the ~, = 2.791 constant is the first eigenvalue belonging to the
first eigenvector E,.
The above iterative mathematical method of finding
an eigenvalue and the belonging eigenvector is widely used in computer science
in the case of large symmetrical real-value matrices as in Hermitian matrices
(cf. 7.27 in Ref. 11). Although the eigenvector-decomposition of matrices was
not applied to the metric tensor and the co- and contravariant tensorial
aspects had not been recognized, the neurobiological significance of such
decomposition has been greatly exploited (see, for example, Ref. 5).
The utilization of the eigenvectors found by reverberation
for the genesis of a metric-type network is illustrated in Fig. 4. The most
crucial step of the metaorganization-process is the relaying of the found
eigenvector to a cortico-nuclear array of neurons, both directly (e.g. via
climbing fiber collaterals to the cerebellar nuclei) and indirectly (e.g. via
climbing fibers to the Purkinje cells which in turn project to the nuclei).
Such convergence of the same (climbing fiber) vector may imprint an array of
neurons by the dyadic (outer) product of the vector with itself. Such a dyad D,
= E, > < E, (symbol > < denotes the outer product of vectors) can
be seen both in the connectivity diagram and also numerically (Fig. 4). The
dyadic product of an eigenvector with itself will be called an
"eigendyad". As shown, D, will serve as the first approximation of
g"k, denoted by g"k.
Once the first eigenvector and corresponding eigenvalue
is established, the remaining eigenvectors of the system can be found by
reverberating a vector whose direction is orthogonal to that of the previously
found eigenvectors. Reverberation can therefore proceed by filtering out from
p, before every reverberation, the already found eigenvector-component pF=(E,
> < E,) x p, (see Fig. 4 in Ref. 11):
P - P[sub]F = P - SUM
(Em > < Em) x p. (5) m
Fig.
3. Oscillatory reverberation of motor execution-proprioception,
establishing the eigenvectors of the motor apparatus.
(A) Subcortical
reverberatory circuits. Ascending spino-cerebellar pathways carry covariant
proprioception (p) of any execution (e), via mossy fiber (mf) collaterals into
the cerebellar nuclei (cn). The reverberation-loop closes on the brain stem
nuclei (bn), with descending motoneuron pathways, carrying execution components
that will be physically assembled as contravariants (e). The covariant metric
g"k, inherent in the physical geometry of the motor apparatus, will
provide for any contravariant executor vector its proprioceptive covariant
counterpart. (B) A quantitative example for the stabilization of the execution
(e)-proprioception (p) reverberation in the eigenvector (E,). Barred vectorial
symbols denote normalized vectors. Starting with an arbitrary ee = (1 0 0)
execution, already after the second reverberation the execution and
proprioception vectors are identical, meaning that an eigenvector is
established.
Without this filtering the reverberation would again
converge to the same eigenvector; while the filtering forces the reverberation
to be confined to the direction orthogonal to the previously found eigenvectors.
The above iterative computation technique of the eigenvectors and eigenvalues
is possible since in real-valued symmetrical matrices the eigenvectors are
mutually orthogonal and the eigenvalues are real (7.27 in Ref. 11 and theorem
4-4 in Ref. 87).
Similarly it can be calculated that the above mathematical
but physically implementable method of filtered reverberations leads to the
second normalized eigenvector and the corresponding eigenvalue:
E[sub]2 = ( - 0.783 0.153
0.603)[power]T; lambda[sub]2 =
0.209.
The end of the reverberative iterative search, indicating
that all the eigenvectors and eigenvalues have been found, can be determined as
follows. In a physical process it can be monitored when the reverberating
vector, filtered for all previously found eigenvectors, becomes zero. At that
point no more eigenvectors can be found and thus the search stops. In a
mathematical process, when the covariant metric can be made explicitly
available by numerical calculation, the end-point of the reverberative search
can also be determined by comparing the sum of found eigenvalues to the trace
(tr) of the matrix of the covariant metric, which is defined as the sum of the
diagonal elements.87 Since
tr(g[sub]nk) = tr(g[power]nk) = SUM lambda[sub]m;
(6)
in the above case
tr(g[sub]nk) = tr(g[power nk]) = 1 + 1 + 1 = 2.791 +
0.209. (7)
In our example this means that only two non-zero
eigenvalues can be found and therefore the search can end.
The existence of only two non-zero eigenvectors in a
three-matrix reflects the fact that the three-axis frame of reference is
overcomplete compared to the two-dimensional space. Since the eigenvectors of a
symmetrical real-valued matrix are mutually orthogonal, when the motor
apparatus shown in Figs 1-3 is confined into a two-dimensional plane, only a
second orthogonal vector can be found in addition to the first direction
determined by the eigenvector.
2.1.4. Spectral representation of the covariant metric
tensor and its proper inverse (or Moore-Penrose generalized inverse) as
expressed by their eigendyads; the outer products of eigenvectors weighed by
the corresponding eigenvalues.
The covariant metric tensor was established in (3)
by calculation as a matrix composed of the cosines among co-ordinate axes. In
the physical motor mechanism this metric is only implicitly available in the
sense that for every particular contravariant motor execution vector the physical
efFector system provides its covariantly measured proprioceptive vectorial
counterpart. However, with the use of the eigenvectors and eigenvalues found by
the above physical oscillation the metric tensor can be made explicit either in
its co- or contravariant form. Accordingly a neuronal network can be
constructed that implements the matrix which establishes functional
geometries, e.g. the transformation from covariant motor intention into
contravariant motor execution which was proposed as the basis for motor
co-ordination.81
The method of constructing the metric is based on
the spectral representation of the covariant metric'6 (cf. p. 132, theorem 7.3
in Ref. 11, or theorem 8.8 in Ref. 91 ):
g[sub]nk = SUM[m] lambda[sub]m X (E[sub]m > <
E[sub]m) (8)
where Em is the m-th normalized eigenvector and Em
> < Em is the outer (dyadic) matrix product of the m-th eigenvector.
Fig.
4. Genesis of the cerebellar corticonuclear metric via metaorganization, by
means of imprinting the dyads of eigenvectors found by reverberation.
The reverberatory scheme
shown in Fig. 3 is supplemented by the olivary system (IO), which compares the
ascending proprioception (p) and descending execution-vector (e). (1) Having
detected an identity, an eigenvector (E) is found, that is implemented in the
olive. (2) The eigenvector is utilized to generate the corticonuclear network,
by being transmitted via climbing fiber vector (c = E) to both the Purkinje
cells and the cerebellar nuclear cells. (a) The dyad of the eigenvector
(eigendyad) D, is shown numerically, yielding the first approximation of the
metric g"k; in efFect determining the principal axes of the cerebellar
tensor ellipsoid (cf. Fig. SC). (b) The filtering of the reverberation. After
having found the first eigenvector, the already established D, dyad serves as a
filter that removes the pr=(E, > < E,). p components from the
proprioception vector p, forcing the reverberation-vector p - pr to be
orthogonal to the already established eigenvector.
In the given example the above formula yields the
spectral representation of g[sub]nk as follows:
0.559 0.559 0.597 0.575 g"k = 2.791 0.597
0.575
-0.783 - 0.783 0.153 0.603
9 + 0.209 0.153 ( ) 0.603
0.312 0.334 0.321 = 2.791 0.334 0.356 0.343 0.321
0.344 0.331
0.613 -0.120 -0.472 +0.209 -0.120 0.023 0.092 -0.472
0.092 0.364
0.999 0.907 0.800
= 0.907 0.999 0.976 . (10) 0.800 0.976 1.000
The resulting covariant metric is, with practical
precision, identical to the one computed directly from the cosines. The
computation was actually performed for 7 decimal digits; however, as shown
here, results come within ±0.002 precision even if the calculation is rounded
to the biologically relevant 3 digits.
The two most important aspects of metaorganization
are that the above iterative procedure which yields the spectral representation
of the covariant metric (a) is established by a physically executable
oscillation which is set up simply by a recurrent reverberation and (b) will
yield not just the covariant metric tensor itself, but also either its proper
inverse (if it exists) or its Moore-Penrose generalized inverse (in case of
overcompleteness). The former applies if the space is complete (e.g. it is
Riemannian) and thus the inverse of the covariant metric tensor exists. The
latter applies if the covariant metric is singular and thus the space is
non-Riemannian. The unified expression of the proper, or generalized metric,
is:
(g"k)+ = ~'lm X (~m > < Em) (I1) m
where ~ m is the generalized inverse of the m th
eigenvalue (3.6.2. in Ref. 1 ).
1/nmif ~.m~0 ,t + = ~
m
0 if ~m = 0. (12)
For further details of the mathematics of generalized
inverses and the Moore-Penrose pseudoinverse see Refs 1 and 9. For its
introduction into tensor network theory of the CNS see Refs 74-78. For
non-tensorial neurobiological applications of the generalized inverse see Ref.
41 and for robotics see Ref. 40. It is emphasized that in the metaorganization
algorithm (a) the spectral decomposition is applied not to any matrix, but
specifically to the covariant metric tensor, and (b) the eigenvector, established via an oscillation, is used to
generate a generalized inverse of the covariant metric, expressed not in
Cartesian but in non-orthogonal co-ordinates.
In the given numerical example, the above formula of
Moore-Penrose-generalized inverse of the covariant metric yields:
0.559 0.559 0.597 0.575 (g"k)+ = 1/2.791 0.597
0.575
-0.783 -0.783 0.153 0.603
+ 1/0.209 0.153 (13) 0.603
and, if the dyads of eigenvectors
("eigendyads"), weighed by the eigenvalue are explicitly calculated,
is equal to
0.112 0.120 0.115 (g"k)+ = 0.120 0.127 0.123 +
0.115 0.123 0.118 2.933 -0.574 -2.259 + -0.574 0.110
0.440 =
-2.259 0.440 1.742
3.045 -0.454 -2.114
-0.454 0.237 0.563. (14) -2.144 0.563 1.860
The verbal expression of the above is the following.
The Moore-Penrose-generalized inverse of a matrix, that conserves the
eigenvectors of the original matrix, is constructed as the sum of dyadic outer
product of each eigenvector with itself; i.e. of the dyads weighted by the
(generalized) inverse of the corresponding eigenvalue.
Note that the Moore-Penrose-generalized inverse of
the covariant metric tensor has already been numerically calculated by applying
the metaorganization principle and algorithm as proposed earlier.74-78. The numerical example of Fig. 2, using the
metric-type neuronal network, has also been shown with the
Moore-Penrose-generalized inverse components of the contravariant metric.
Nevertheless, an exposition of the reverberative procedure has not hitherto
been offered.
The metaorganization principle is elaborated here as
an oscillatory procedure in accordance with the classic notions on recurrent
reverberating circuits; the theory of closed "self reexciting"
chains of neurons,b° the control-theoretical emphasis on "feedback and
oscillation" in Chap IV of Ref. 100, the emphasis on reverberation in Ref.
12 and the recent analysis of the central role of motor oscillations at the
neuronal level in motor development.49
2.2
Conceptual interpretation of the principle of metaorganization of neuronal
networks
A basic interpretation
of the principle of metaorganization relies on the fact that the orthogonal
spectral decomposition of the matrix of the covariant metric is conceptually
equivalent to viewing the motor-transformation through the metric not as a
wholly integrated operation, but as composed of transformations through
separable eigendyads. As shown each eigendyad is the outer product of a
normalized eigenvector with itself, the eigenvalue serving as a coefficient. It
can easily be verified that an eigendyad transforms an eigenvector into itself,
which will only be stretched or shortened by the eigenvalue coefficient. Since
the eigenvectors are mutually orthogonal, each eigendyad operates only on that
vector component which lies in its own direction; it is
"intransparent" (producing zero output) to components that are
diagonal to it. In the metric transformation of contravariants into covariants
the magnification coefficient is the eigenvalue of the covariant metric. This
explanation that the reverse contravariant metric-type transformation (via the
generalized inverse of the covariant metric) must be performed through the same
set of eigendyads (serving as the eigenvector-transformers), while the
coefficient of each dyad must be the generalized inverse of the
eigenvalue.77,78
It must be emphasized, however, that this decomposition
into independent "channels" of amplification (for biological
correlates of such channels, see Ref. 47) is only possible along the mutually
orthogonal eigenvectors of the system. Therefore, a customary interpretation of
a horizontal eye movement, for example, as a separable direction from vertical
and torsional eye movements may be improper. The eigenvectors of the oculomotorcovariant
metric (a) have not even been established at the time of such customary
interpretation and (b) when they have recently been calculated, they turned out
to be greatly different from the horizontal direction (by about 45°, cf. Ref.
77). Thus some of the most immediate experimental paradigms, derived from the
proposed metaorganization principle are (a) to establish experimentally the
eigenvectors in biological systems and (b) to determine if the amplification
of co- to contravariant vector-components can be independently altered along
the mutually orthogonal eigenvectors and interdependently along all other
directions, as suggested here by this theory.
A more abstract conceptual interpretation of the
metaorganization principle is possible by a graphic depiction of the function
of the cerebellum, as a geometrical distortion which is implemented by the
covariant metric and its generalized inverse. Such is possible in the form of a
tensor-ellipsoid (see Fig. 5C, after Ref. 78). The covariant-contravariant
transformation (and vice versa) is visualized in Fig. 5(C) as a geometrical
distortion of an ellipsoid of the cerebellar input intention vectors ik into a
circle of execution vectors e". Such transformation is determined by the
principal direction-axes of the ellipsoid (given by the eigenvectors) and by
the magnitude-distortion (where the lengths of the principal axes along each
eigenvector correspond to the eigenvalue).
This geometrical definition of the primary cerebellar
function leads one to the secondary question of the development of the function
through the emergence of the neuronal network that implements this
transformation.5' While the geometrical symbolism in Fig. 5(C) provides a
concise interpretation of the function itself in the first place, secondarily
it also suggests that its development may be determined by the double
procedure'g~'9 of (1) establishing and storing of the eigenvectors of the
tensor-ellipsoid and (2) trimming the eigenvalues (i.e. adjusting the principal
axes to their proper lengths). The steps by which these tasks are accomplished
are reviewed next.
2.3
Elaboration of the metaorganization principle, explaining the genesis and
modification of cerebellar metric-type neuronal networks
In section 1.2.3. it was summarized and quantitatively
demonstrated how a covariant~ontravariant transformer-matrix can serve as a
cerebellar motor coordinator. In section 2.3.1 below, a concrete numerical
example is given for how the metaorganization principle can be implemented by
a process in the CNS to generate such cerebellar networks.
2.3.1. The genesis of functional geometries as implemented
by neuronal networks.
The summary diagram in Fig. 5 (after Refs 75-78)
illustrates a general scheme of the function, genesis and modification of the
cerebellar networks. By including Fig. 2, it shows that the function of the
essential cerebellar network can be interpreted as performing a covariantcontravariant
transformation. However, Fig. 5 also indicates that this network converts the
motor intention-signals into motor signals, taking in account not only space
co-ordinates as shown in Fig. 2, but also spacetime co-ordinates as shown in Ref.
83. The "stacks" of Purkinje cells, which serve as "temporal
lookahead-modules",80-83 each model requiring about two hundred cells, are
illustrated by a schematic triad of Purkinje neurons (Fig. 5D).
The illustration shown
in Fig. 5, also encompasses the circuitry necessary for the establishment of
the eigenvectors by reverberation and the network approximation of the metric
by its eigendyads (see Figs 3 and 4). The covariant proprioception vectors, p,
enter to the cerebellum via mossy fibers that give collaterals to the
cerebellar nuclei. Because the Purkinje cell-cerebellar nuclear cell synaptic
connectivity is established late in embryogenesis,92 we propose that
reverberation specifies this connectivity in an epigenetic manner. Thus,
initially for any proprioceptive input, the mossy fiber input to the cerebellar
cortex will yield a zero vector through the Purkinje cells at the cerebellar
nuclei before the metaorganization process is implemented. As a result the
nucleofugal output will carry the same information as the mossy fiber input
itself. This output is then introduced in the motor system as if it were a
contravariant effector vector, with only a signal-reversal at the brain stem
nuclei, thus leading to stabilizing oscillations.
Fig. 5. Co-ordination by the cerebellar networks,
and their genesis and modification by metaorganization.
(A) Metaorganization-algorithm
for the genesis of network-matrix that approximates the metric g[power]nk by the
dyads of its eigenvectors; D[sub]m, via climbing fiber vector, carrying the
eigenvector; c = E[sub]m,. Such algorithm sets up a tensor-ellipsoid with
principal axes of the eigenvectors (cf. Fig. 5C).
(B) Metaorganization-algorithm for the
modification of the network matrix, in order to correct the eigenvalues of the
tensor-ellipsoid. Geometrical inset-diagram illustrates, by means of a
simplified two-dimensional frame, that a goal (G), given by
intention-components (i[sub 1-2) is improperly executed if the existing
eigenvalue is incorrect. Thus, the execution-components (e[power 1-2]) add to
an erroneous performance-point (P), that is covariantly relayed back by
perception (p [sub 1,2]). Graph demonstrates, that by the projections of points
G and P to the established eigenvectors (Em) the difference of the existing and
the desirable eigenvalues can be measured. Thus, a correction-vector (c) is
established by the olive, so that climbing fibers imprint a dyad of correction
(4 g"k).
(C) Geometrical
representation of the function of the essential cerebellar network as a
covariant intention to contravariant execution transformer. A circle of
execution vectors (exc), when expressed in the form of intentions (int), would
be distorted into an ellipse. This tensor ellipsoid is determined by its
eigenvectors (E,~2) and eigenvalues (R). Thus the function of the cerebellum is
symbolized as a geometrical "mirror-like" transformation of
distortion-prone intentions into proper execution.
(D) Composite diagram
of circuits necessary for the essential function, plus its genesis and
modification. Proprioceptive signals are shown in green, intention in blue,
execution in red and correction in yellow. The essential function is
implemented by the blue-to-red spacetime metric circuit (g"k), the genesis
is implemented by the green-to-red reverberation and imprinting (via the yellow
circuit) of the cortico-nuclear network. The olivary system (IO; yellow)
subserves modification by relying on the green proprioceptive, blue intention
and red execution signals. Temporal "lookahead-module" of stacks of
Purkinje cells symbolize that the metric transformation is not restricted to
the space domain, but applies to a
unified spacetime manifold.
The imprinting of the eigendyads into the cerebellar
corticonuclear circuitry is illustrated in Figs 4 and 5(A). In the scheme shown
in Fig. 5 small additional circuits are necessary for normalizing the amplitude
of the reverberation and for identity detection to monitor the eigenvalue
stabilization when e = p. Both operations can easily be accomplished by taking
their inner product. This can be accomplished by introducing an interneuron
(see below). Indeed, in order to normalize the amplitude, the inner product is
initially obtained by multiplying p with itself. With this factor the local
inhibitory interneuron can reduce the magnitude of the vector. This operation
corresponds to the amplitude-stabilization by Golgi cells proposed earlier.86
Here the effect of the Golgi inhibition is normalization, with the firing rate
of the Golgi cell being the measue of the eigenvalue. In order to monitor the
degree of eigenvector stabilization, for the inner product of p and e the
interneuron is organized such that it will only reach a unitary firing rate
when p and e are eigenvectors. Such a simple interneuron circuit can determine
whether the convergence of the inner product of the two normalized vectors is
close enough to (1.00 in absolute value) to indicate that an eigenvector has
been found. At that point both the eigenvalue, automatically provided by the
normalizer, and the corresponding eigenvector (taken either from the ascending
or descending pathway) are available for constructing the eigendyads as shown
in section 2.1.4. Such imprinting requires (a) a convergence of the identical
eigenvector on both the row and column elements of a matrix in order to
establish their product and (b) that ionic mechanisms are capable of triggering
chemical changes at the postsynaptic elements which may modify intrinsic
electro-responsiveness in a manner proportional to this product. The mossy
fiber-parallel fiber-Purkinje cell system and the climbing fiber-Purkinje cell
system could in principle be capable of generating such modification, since
both these pathways carry the eigenvectors at such steady-state of the
oscillation. Still, several considerations support the argument that such an
adaptive modification may occur through the corticonuclear synaptic network.
First the inferior olive signal is received directly from olivocerebellar
collaterals arriving at the nuclei 16 and indirectly via the climbing fiber
activation of Purkinje cells, 38 so that a convergence capable of evoking the
required integration 54 can occur. Second, modification of the cerebellar
nuclei is consistent with the finding that vestibular adaptation is retained
after ablation of most of the cerebellar cortex.15-58 Third, as will be seen in
section 2.3.2, in pathological conditions where a modification of the
corticonuclear circuitry after its initial genesis may be required, the direct
and indirect climbing fiber vectors facilitate an adaptive change at the site
of their convergence, presumably in the cerebellar nuclei.
During the process of imprinting the eigendyads into
the corticonuclear cytoarchitecture, as proposed recently, 78 the inferior
olive would store the eigenvectors and eigenvalues found by reverberation.
Such storage will be required for the conformation of the genesis; the ongoing
modification of the network. The suggestion that the olive has storage
properties is particularly apt in view of the intrinsic capacity of olivary and
related neurons for rebound oscillation.49 Long-term ionic conductance-change
mechanisms, consistent with the proposed storage, have been demonstrated
experimentally.55-57 According to this view, following the storage of the m-th
eigenvector g3 the olive would signal a climbing fiber vectorial correction c =
Em, which would modify the corticonuclear integrative properties by an
additive
G[sub]nk' = g[sub]nk + delta g = g[sub]nk + (c >
< c)
= g[sub]nk + (Em > < Em) (15)
thereby imprinting the actual eigendyad. Using the
first eigenvector-eigenvalue, it can be easily verified that such
corticonuclear convergence of climbing fiber vectors will yield the first
eigendyad shown in 2.1.4. The result of this procedure will be an approximation
of the metric by its eigendyads:
g[sub]nk = SUMMA D[sub]m
(cf. Fig. 5A), a matrix which has the correct eigendirections
(principal axes of the tensor-ellipsoid), while the eigenvalues may be
uncalibrated (incorrect).
A remarkable feature of the scheme proposed in Figs 4 and 5 is that in the reverberative search for the subsequent second and third eigenvectors, the filtering-out of the already imprinted eigenvectors can be automatically provided by the eigendyads which have already been generated. For example, after having imprinted the first eigendyad, the reverberative search for the second eigenvector requires the p - p[sub]F = p - (E[sub]i x p) motor output; the vector which contains only that component of the reverberated vector which is orthogonal to the es
