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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 established eigenvector. The mossy fiber
input to the cerebellar cortex would transform, through the corticonuclear
network, into its eigenvector projection, since the network is
"intransparent" to (yields zero product with) vectors orthogonal to
the eigenvector. Thus the nucleofugal output will be exactly the required
filtered vector. With the use of such filtering via the already generated
eigendyads, the reverberation can proceed to find all the subsequent
eigenvectors in one continuous series of stabilizing oscillations, as shown by
a computer simulation." When all eigenvectors are found and the respective
eigendyads are imprinted into the corticonuclear circuitry, the proprioceptive
reverberation automatically becomes superfluous, since following the
normalization of the reverberated vector, each corticonuclear eigendyad filters out its
own eigenvector-projection.
The removal of
all eigenvector components leaves a zero resultant of the proprioceptive vector
in the nucleofugal output. On one hand this prediction of the metaorganization
principle is congruent with experimental data which indicate that the
propioception system is not essential for general motor performance of an
already established system.96 On the other hand the above proposal can in fact
be tested experimentally by determining whether proprioception is a vital
element of the genesis of neuronal networks subserving motor co-ordination, as
the metaorganization principle predicts.
2.3.2. Modification of the functional geometries
implemented by neuronal networks.
This section shows that the same procedure that is suitable
for generating the neuronal network (in the form of spectral representation by
eigendyads) is capable of carrying out modifications in order to adapt the
metric to possible changes in overall motor status.51 The errors in the
performance (produced by a network established with incorrectly calibrated
eigenvalues) can be used for its iterative perfection.
The procedure is based on the two-stage character of
the metaorganization: (1) the establishing, by a reverberative
"revolution", the eigenvectors of the physical apparatus and thus
imprinting the eigendyads (determining the principal axes) into a network and
then (2) calibrating by a gradual "evolution" the eigenvalues of the
functional geometry in order to match the exact values along the principal
directions to those of the physical system.
The modification procedure of calibrating the
eigenvalues, introduced in Ref. 78, is elaborated here by using a concise
summary diagram in Fig. 5B. This diagram is shown for two dimensions (since the
motor apparatus of Fig. 1 is confined to a plane) and demonstrates the
adjustment of only one eigenvalue. Nevertheless, since the eigenvectors are
mutually orthogonal, all eigenvalues (even if more than two) can be altered by
a single operation in a parallel manner and since it contains no restriction
for dimensionality the procedure is valid for any dimensional motor system.
Assume that a given eigenvector, E"" of
the physical system has been properly established, but it was imprinted into
an eigendyad with an incorrect eigenvalue-coefficient. In such a case, as shown
in the diagram of Fig. 5(B), the components of an intended vector i (that
covariantly represents the goal G) will result, through the erroneous metric,
in a contravariant execution vector e that results in a performance, P,
deviating from the goal, G. The physical state of affairs represented by P will
then be covariantly measured by the sensory mechanism yielding the covariant
components of the proprioception (performance) vector p. Note that the three
vectors i, e and p, together with an eigenvector E," (assumed to be stored
in the olive),'R contain all the information necessary for correcting the
erroneous metric that was imprinted earlier. Since the metric can be constructed,
in this spectral representation, as the sum of eigendyads with the
eigenvalue-coefi~cients, the difference between the existing and desired
eigenvalues can serve to correct the eigenvalues of the existing metric by
adding the dyad formed by the climbing fiber vector, c.
The modification is based on the fact that the inner
product, E", x e, represents the orthogonal projection component of the
output vector e of the networkmetric to the eigenvector, E,". Similarly,
E," x i, represents the projection to the eigenvector E", of the
input vector i. Thus the eigenvalue inherent in the existing erroneous
contravarant metric network is "~," = (E," x e,")/(E,"
x i). Likewise the eigenvalue that is implicit in the contravariant metric of
the physical effector mechanism is p~," = (E", x e)/(E", x p).
If the task is to correct the "d," in order to become,,~""
then the correction of the eigenvalue should be:
~m = n~m - ",lm = (Em x e)/(Em x P)
- (E", x e)/(E", x i) (16) This can be
accomplished, by adding a modification-matrix 0 g, that is the dyadic product
of the climbing fiber correction-vector, c to the existing network-matrix 77.
where g"k=c> <c
e = ~ [(Em x e)/(Em X P)
yz -(E," x e)/(E", x i) ~ Em. (17)
The process of calibrating the eigenvalues of the
metric (by the above modification algorithm) is illustrated below by a
numerical example that uses the cerebellar scheme (Figs 2-5).
Suppose that both eigenvectors shown in section
2.3.1 have been properly established but that the second eigenvalue was
erroneously set. Instead of using the correct ~,z = 0.209, assume that an
incorrect ~,z = 0.300 exists in the imprinted corticonuclear eigendyad. The
erroneous coefficient of the second eigendyad results in a matrix-component:
0.613 -0.120 -0.472 1/0.3 -0.120 0.023 0.092
-0.472 0.092 0.364 - I 2.043 -0.400 - 1.573I
-0.400 0.077 . 0.307 (18)
- 1.573 0.307 ~ 1.213
and thus, together with the (properly established)
first eigendyad, the erroneous metric is
2.155
-0.280 -1.458
g' = -0.280
0.204 0.430I . (19)
-1.458 0.430 1.331
-
Let an arbitrary intention vector be i = (- 100 100
100)T.
Thus, through the error-laden g', this covariant vector will be
transformed into a contravariant execution vector e = (-389.3 91.7 321.9)T.
This execution vector will result in the physical output of P instead of G.
Through the covariant metric, implicit in the physical geometry as expressed in
section 2.1.4, the invariant P will be measured by the performance vector as
1.000 0.906 0.799 - 389.3 p; = 0.906 1.000 0.978
91.7
0.799 0.978 1.000 321.9
- 49.0
53.8 . (20) 100.5
Since EZ = (-0.783 0.153 0.603)T, the inner products
required for establishing the climbing fiber vector c are:
Ez x e = 512.9 (21) EZ x i = 153.9 (22)
EZ x p = 107.2 (23)
The ratio, representing the eigenvalue inherent in
the physical geometry, therefore, is
512.9/107.2 = 4.785 (24)
and the ratio, representing the eigenvalue
erroneously implemented in the network is
513.0/ 153.9 = 3.333. (25)
From the above, the required correction is ~ = 4.785
- 3.333 = 1.452, and thus the modificationdyad, to be imprinted into the
corticonuclear network is
0.613 -0.120 -0.472 1.452 -0.120 0.023 0.092
-0.472 0.092 0.364 0.890 -0.174 -0.685
-0.174 0.033 0.134. (26) -0.685 0.134 0.529
This modification-dyad added to the erroneous second
dyad of (18) results in the proper second dyad as shown in section 2.1.4.
The dyadic product of the climbing fiber correction
vector, c, can be impressed on the corticonuclear network as a whole, via CFs
that project both to the PCs and the CB nuclear cells. The emerging corticonuclear
matrix will then act as an appropriate metric producing a zero error in the
next performance. In this trial-and-error process the internal geometry becomes
increasingly homeometric with the external one. Thus through the CF system the
physical geometry is matched with its proper internal representation.
In the example it was assumed that the first eigendyad
was imprinted with the correct eigenvalue. However, since the eigenvectors are
mutually orthogonal the modification procedure is independent along each
eigenvector and thus the modification of all eigenvalues may be implemented
simultaneously in a single reverberation.
The means of implementation of the proposed
metric-modification at a single neuron level (either at the PCs or at NCs or at
their conjunction) has not yet been conclusively established. Such a task is
all the more difficult, since one component of the correctionmatrix may either
be positive, negative or zero.'8 Therefore the required perturbation at a
singleneuron level should be expected as a bimodal effect, including at times
an indetectable zero action.~°v'~3' Moreover, the correction in any matrix
component is a function of all vector elements. Thus if only a single dimension
is controlled by the experimental paradigm, as in conventional analyses, the
prediction of a single component of the matrix may prove to be a very complex
matter. Finally any alteration is expected to be much more pronounced at the
site of the dyadic convergence (at NC), as opposed to the site of the intensive
search, the CB cortex. These factors, plus a lack of a conceptual framework
accounting for what is defined in Ref. 78 as the "CB functional triad;
co-ordination, timing and adaptation", may explain the meager experimental
results, despite dedicated efforts through one and a half decades, in an
attempt to conclusively demonstrate an adaptive feature of the CB at the PC
level.34,46,37
Some additional comments are warranted regarding
the operations of the inferior olive. It is assumed that the inferior olive
expresses the difference between the eigenvalues in the external physical
geometry and those in the functional geometry implemented by the corticonuclear
cerebellar network. The equation yielding the climbing fiber vector c
[introduced in Ref. 78 and elaborated iri (17)], is one of several possible
implementations for the task. The advantage of the proposal above is that it
measures the eigenvector of the physical geometry and the eigenvalue inherent
in the already generated network. The disadvantage of this solution is that it
requires nontrivial vectoral calculations in the olive, although it is known
that the olive does receive of all the ascending and descending signals
necessary for such a "comparator function".'~65
It is therefore noteworthy, that a simplified operation
could also be utilized by the olive, one not based on measuring the eigenvalue
of the cerebellar neuronal network. Rather, it could utilize the stored
eigenvalue that can be imprinted into the olive at the same time as the
eigendyads are imprinted into the corticonuclear network. Relying on the stored
eigenvalue, its comparison with the error signal can be used to determine the
required modification of the existing eigenvalue. The error d = i - p may arise
as a difference between intention and proprioception or between intention and
performance, as detected through the total sensorimotor loop.
Thus, the new eigenvalue should be:
NewLambda[sub]m = oldLambda[sub]m X (Em X d)/(Em X
p) (27)
This formula, that interprets the climbing fiber
vector as based on the error vector, corresponds well to the experimental
evidence showing that climbing fibers express functional errors in intrinsic
frames of reference.61.90 This requires a "computation" in the olive
that is simpler than the full formula given in equation (17). The disadvantage
of the simple computation is that it may accumulate errors, since instead of
measuring the actual eigenvalue of the network the process relies on a stored,
and possibly imperfect, eigenvalue by which the network was generated.
A final, but most important comment is that the
process of calibrating the eigenvalues by iteration can perfect the metric if
the eigenvectors have been precisely established and only the eigenvalues are
incorrect. However, if the eigenvectors themselves are improper, the above
modification-process may continue indefinitely without ever converging to the
proper metric. If the intrinsic system of co-ordinates is experimentally
altered and the old eigenvectors are entirely improper, the system needs to
regress to a revolution (a drastic re-assessment of the principal directions
and values) by re-doing the entire oscillatory reverberative process. This
prediction of the theory corresponds to the findings that after major
disturbance to the cerebellar coordination-apparatus (e.g. vestibular nerve
ablation),58 one of the earliest and most dramatic phases of the compensation
process is marked by violent shaking and oscillatory behavior, observable in
the animal and at the olivary level.58
3.
GENESIS AND MODIFICATION OF THE THREE-STEP NETWORK OF THE TENSORIAL
SENSORIMOTOR SCHEME
3.1. The
three-stage tensorial scheme of sensorimotor systems
In section 2, the mathematical principle and the
physical process of the metaorganization was elaborated. It was applied to the
metric-type motor network that organizes the functional geometry of a neuronal
system through the generalized inverse of the covariant metric of a motor
apparatus. With this background the three basic transformation matrices used in
the tensorial sensorimotor scheme may be developed in principle as well as in
physical reality. The building-blocks of the system are (a) a matrix which
serves as a contravariant sensory metric, (b) a matrix which expresses the
covariant embedding involved in sensorimotor transformation and (c) a matrix
which serves as a motor metric. The entire procedure will be demonstrated in
the model scheme shown in Fig. 6.
The function of these transformations, once the
matrices are available, has been quantitatively demonstrated
elsewhere.78. The system shown in Fig.
6 has been simplified and the diagram serves only as a model by showing how the
above three matrices may be generated.
3.2
Metaorganization of motor geometry: cerebellar metric-type networks
The first step must be to generate, via the
metaorganization process, the neuronal networks comprising the functional
geometry of the executor mechanism. In the case shown in Fig. 6(A), this means
generating the cerebellar network g"'`. The motor geometry must be
established first since the metaorganization process applies as soon as a motor
system and the proprioceptive system are available and connected to the
external physical reality. At this stage the process does not require the
sensorimotor transformation matrix or the sensory metric. This is in contrast
to the generation of the sensory metric which, as will be shown in section 3.4,
presumes the availability of the motor metric. Indeed, as pointed out
recently49 such order in the developing of the particular networks is
consistent with the classical morphological studies.88 In particular, in the
case of the cerebellum, the first part to be developed is the cerebellar nuclei
and then the Purkinje cell nuclear pathways, followed by the development of the
input to Purkinje cells. After this the connectivity is refined.44~88
Accordingly, the metaorganization of motor geometry, as the first stage of
development, may start as soon as the muscles and proprioception peripheral
organs begin to function. This agrees with the observations of embryonic
twitches 8-30 and the fact that in some species cerebellar neuronal networks
are reasonably well developed at the neonatal stage.2
The development of the motor-metric-type cerebellar
network is shown by Fig. 6(A), according to the steps described in section 2.
There are two additional points to be considered.
The calibration of the proprioceptive vector components
can be accomplished by using the ratio of the spike-frequency bursts from the
motoneurons which innervate each muscle (shown schematically in Fig. 6B) and
the registered response returning via the proprioceptive reverberation. This is
possible since the "base vector" represented by a single-motor
impulse has only one non-zero component and thus it is therefore both co- and
contravariant. For the same reason the calibration is independent of whether
the calibrator signal is transmitted through a developed cerebellar metric or
entirely bypasses this circuit without undergoing a cerebellar covariant-contravariant
transformation.
A second point follows from the fact that tensor
analysis deals with general co-ordinates. Thus, while in order to keep the
complexity of the presentation minimal, only two-dimensional spatial
co-ordinates are shown here, the principles apply to any multidimensional
system such as one with space- or torquetime etc. co-ordinates.
Thus the generation and modification of metrics
involved not only setting or altering the connections, i.e. the
electro-responsiveness of the components in the metric network, such as
required for altering the spatial metric, but also involves a modification of
the dynamic temporal characteristics of neurons. One manner in which this can
be accomplished refers to changing the zero-first-second order time-derivative
properties of single neurons,53,72,73 in effect changing the characteristic
oscillation frequencies of the neurons or of small assemblies. 48. If findings
on ongoing modulation of the electroresponsiveness at the PC level 10,17,37 are
made unambiguous, they could be interpreted as means for such subtle modulation
of the Taylor coefficients of the Purkinje cells in the temporal lookahead-module
but not the means of the setting of the eigenvalue-coefficients of the
corticonuclear eigendyads.
Fig.
6. Schematic illustration of the metaorganization of functional CNS
geometries.
(A) Interaction of a
physical geometry inherent in the motor apparatus with a functional geometry of
the cerebellar neuronal network. Reverberation of contravariant execution to covariant
motor proprioception yields the eigenvectors of the covariant metric inherent
in the motor geometry g,~. Thus its inverse (g"~) or
Moore-Penrose-generalized inverse (g"~)+ may be obtained in spectral
representation. (B) Covariant embedding of the sensory geometry into the motor
hyperspace. The jth unit-vector of the motor apparatus (0 0 1) generate unitary
displacement along the jth motor axis. Its projection to the kth sensory axis
(cos ~~k, cf. numerical example in the text) will provide the c~",
matrix-component of the sensorimotor transformation matrix (C) Metaorganization
of the three basic sensorimotor matrices, and the generation of a
hierarchically connected metageometry that represents the whole sensorimotor
system. Once the cerebellar g"k and sensorimotor c~k matrices are
available, metaorganization may be used to organize a sensory metric tensor,
replacing the set of direct connections (Kronecker-delta) from the sensory
receptors (s;) to sensory perceptors (s~). For every s~ the entire sensorimotor
circuit yields the corresponding s;; implying the covariant sensory metric.
Thus the eigenvectors can be established by reverberation and the inverse (or
generalized inverse) metric tensor g'~ may be generated, corresponding to the
collicular neuronal network. (D) Metaorganization of dual hyperspaces is
possible, if both the complementer and the dual metric tensor is generated, one
by using the generalized inverses of the eigenvalues as coefficients of the
eigendyads, the other by using the found eigenvalues themselves. As shown in
(C) such a process may be used to generate, together with the collicular
contravariant sensory metric tensor (g'~ a cortical duplicate of the covariant
sensory metric (gy). These dual geometries may then be reverberated to mold one
another and the duplicate geometry of the sensorimotor system (its internal
model) may be used for initiation of motor acts without external sensory input
[big arrows in part (C) symbolize such interactions].
3.3. Development of the
sensorimotor network, implementing the covariant embedding transformation
Although the sensory system shown in Figs 6(B) and (C) is
diagrammatic (being composed of two non-orthogonal axes at 150 and 270°) the
basic features relevant to tensor theory are well represented; (a) the sensory
frame covariantly measures the action generated by the muscles and (b) it
utilizes a non-orthogonal system that is different from that for the motor
execution both in its direction and number of axes. In the analysis of sensory
mechanisms
such as the six semicircular canals of the vestibular apparatus,'8~g° both the
anatomical realism and the ensuing quantitative complexity is significantly
greater. In both the realistic and simplified cases, however, attention must be
focused on the transformation matrix that changes the sensory vector into the
motor vector (Fig. 6B). The function of the sensorimotor covariant embedding
matrix is discussed in detail in Ref. 78. In the case of the vestibulo-ocular reflex
model (see Fig. 5 in Ref. 79), the process of establishing such a covariant
embedding matrix was qualitatively elaborated to show that each unit-vector of
the motor system should be covariantly measured along the sensory axes,
yielding a matrix of the cosines among the sensory and motor axes.
Mathematically, this is a trivial operation both in
the simple model presented in this paper and in the more complex
vestibulo-ocular refiex.'9~g9 The matrix elements are the cosine-projections of
each motor unit-vector onto the sensory axes. Thus, in the case shown, the
motor axes with 185, 160 and 148° have to be projected onto each of the sensory
axes with 270 and 150° angles:
C~k = COS ((~7~k)
cos (185° - 270°) cos (185° - 150°) = cos (160° -
270°) cos (160° - 150°) cos (148° - 270°) cos (148° - 150°)
0.087 0.819
-0.342 0.985. (28) -0.530 0.999
In order to establish these components of the
sensorimotor transformation matrix in the CNS, it is necessary to assume that
each "base vector" of the motor system is generated by a spike burst
of a premotor (Pyramidal-type) neuron, such as are schematically represented by
Fig. 1 in Refs 75, 76 and 78 and by Fig. 6(B) in this paper. These cells are
designated here as "premotor neurons" because they use the motor
frame of reference and are capable of generating a movement. However, since
they will be connected to the sensory mechanism by the covariant-embedding
matrix, they express the displacement in a covariant manner. Therefore a
direct execution of such signals without a cerebellar-type
covariant-contravariant transformation would result in a dysmetric movement.
The firing of each large Pyramidal-type premotor neuron can be evoked by a
strong burst of activity in a small cortical cell (e.g. layer IV), while Golgi-type
inhibitory neurons ensure that other large Pyramidal cells are silenced during
this operation (cf. Fig. 1 in Ref. 76). Thus during this special sensorimotor
"imprinting"stage only one vertical column of a Pyramidal cell may
produce an excitation at one time, producing a "base vector" signal
that descends to the motor mechanism and generates a movement.
It is noteworthy that these "base
vectors", as in the calibration process proposed in section 3.2, are both
co- and contravariants. Thus in establishing the sensorimotor matrix, the
cerebellum is again "transparent" as a covariantcontravariant
transformer; it is not necessary for the cerebellum to partake in this
procedure.
Second, the elementary movements will be measured
covariantly by the sensory system. Since the sensorimotor transformation is a
pre-requisite to the establishing of the sensory metric, it follows that the
sensorimotor transformation must precede the development of the sensory metric.
However, direct connections between the input and output elements of the
sensory system are necessary and thus it is assumed that in an initial state
the matrix of the sensory metric is a Kronecker-delta, i.e. a set of such
direct connections. This permits the covariantly measured sensory components of
the motor base vector to be transferred directly to the Pyramidal cells. This
covariant vector will produce a synaptic activation that yields the exact
coefficients to be imprinted into the premotor neuron and thus result in the
required components of the sensorimotor matrix (cf. Fig. 6B).
Note that the two procedures of proprioceptive
calibration and sensorimotor imprinting can be combined into a single process,
since both rely on the "straddling" of the motor system by the
unitary firings of individual actuators. However, while the motor-metric
network may be organized during embryogenesis (since it only requires motor
effectors and proprioceptive reverberation), the development of certain sensory
mechanisms (for instance, vision) must commence postnatally.
3.4.
Metaorganization of the sensory geometry: tectal metric-type networks
The final stage of the emergence of sensorimotor
networks is the development of a sensory metric (Figs 6C and D). This requires
not only the availability, but the active participation of the developed
sensorimotor embedding transformation and the cerebellar motor metric
networks. The development of a sensory metric by means of a motor metric as
proposed here is consistent with (a) the presence of a separate, explicit CNS
sensory metric that expresses invariants both covariantly and contravariantly
in the same frame of reference, the inner product enabling geometrical
judgements on the invarianta3 and (b) the fact that sensory functions such as
vision cannot emerge without the active participation of motor mechanisms, such
as eye movements.32
The process is based on a generalization of the
metaorganization principle. Metaorganization could develop a network that
implements a secondary geometry when the primary geometry was inherent in the
motor executor mechanism (see section 2). The physical geometry of the
musculoskeletal system provides a proprioceptive covariant vector which is a
counterpart for every contravariant motor action.
The
generalization of the metaorganization principle is based on the recognition of
the fact that any system, not only a physical apparatus, that is capable of
providing the dual counterpart to any particular input vector can serve as the
primary geometry and thus be duplicated or complemented by the process of
metaorganization.
Note that the sensory metric network on one hand and
the rest of the sensorimotor scheme on the other constitute two halves of a
circle, which are joined through an external invariant. After completion of the
development of the sensorimotor embedding and the motor metric networks,
however, the undeveloped "sensory metric network" is still only a
set of input and output neurons which have the presumed direct system of
connections (a Kronecker delta) from the sensory receptors to the sensory
perceptors. Looking at the input and output neurons of this "sensory
metric network" reveals, however, that while the input neurons may be
improperly connected to the sensory output neurons within the sensory metric
network: the output neurons are properly connected to the sensory input neurons
through the periphery via the sensorimotor metric networks: the external motor
machinery and the sensory apparatus. That is, while one half-circle is still
unorganized, the other is perfectly functional. Thus any arbitrary vector, s',
over the set of sensory perceptor neurons (even one generated by internal
"noise" of the system) can be transformed through the sensorimotor
embedding network, c~k, into motor intention vector ik, then through the
cerebellar motor metric into motor execution vector, e". The physical
invariant, emerging from the contravariant motor execution will then be
covariantly measured by the sensory mechanism to yield a sensory reception
vector, s;. Therefore, as shown schematically in Fig. 6(D), to any
contravariant s~ the total pathway that includes the external physical motor
and sensory mechanisms will yield the appropriate covariant counterpart, s;.
This is the necessary and sufficient condition for the applicability of the
metaorganization process, which then can serve to modify the geometry of the
contravariant sensory metric from the Kronecker-delta to the actual functional
representation of the remaining circuit.
Given simplified sensory and motor frames, the
reverberation process of s; (as if it were si) through the entire motor and
sensory systems, will yield the normalized eigenvectors E, = (2~~~z 2~~~Z)T and
EZ = (2~ ~~z -2~~~2)T with the corresponding eigenvalues of ~., = 0.5 and ~,2 =
1.5. Therefore, the contravariant sensory metric can be generated in its
eigendyadexpansion:
g'~ = 1/0.5I0.500 0.500I + 1/1.5I 0.500 -0.500I
0.500 0.500 -0.500 0.500
- I 1.333 0.666I. (29)~ 0.666 1.333
This sensory metric, shown in Fig. 6 as g`~ (elaborated
in Fig. 1 in Ref. 76), can be implemented (using a scaling factor of 4/3) as a
simple neuronal network with twice as many direct connections between sensory
receptors and sensory perceptor neurons as the number of cross connections.
Since the metaorganization procedure applied here is
conceptually identical to that described in section 2, only two general
comments will be made here, in order to illustrate how the proposal of
establishing the sensory metric fits into the hierarchy of the top-down and
bottom-up approaches used in brain theory. At the top level, the process of
finding the eigenvectors of the sensory metric by reverberations, would look
like a rhythmic oscillatory exercise of the motor mechanism, accompanied by an
intense introspective use of the sensory apparatus-similar to the behavior
observed during embryogenesis.48 At the bottom level of the neuronal circuits
subserving sensory information processing, the primary covariant-contravariant
sensory transformation may take place in the neuronal network such as the optic
tectum (cf. elaboration in Ref. 74).
The suggestion that such a network could be
generated by generalized metaorganization raises a novel functional
interpretation of the actual neuronal circuits that are known to be involved in
sensory preprocessing. The process of metaorganization requires that the
eigenvectors, found by reverberation, be stored. The implementation of this
function can be accomplished by neuronal networks of small nuclei such as the
inferior olive rather than cortical networks where the spectral expansion of
the metric by its eigendyads is implemented. However, in order to serve as the
"imprinter" and continuous "corrector" of the cortical
network, as explained by the metaorganization process, such a nucleus must be
intimately connected to a cortex. A possibility exists that the nucleus
isthmi, known to be endowed by the above-described properties, 25 could play a
similar role in generating the tectal circuits.
4.
DISCUSSION
It seems quite clear that an analysis of basic
sensorimotor transformations requires the use of general hypotheses regarding
how the brain might implement functional geometries. The principle of
metaorganization is capable of embodying such a concept since it is general
enough to encompass many features of related geometries, yet can be elaborated
(as in this paper) in specific network models.
A basic assumption of the metaorganization process
is that the set of relations among the elements and among elements of another
system may exhibit certain common basic features, since one may be embedded
into the other. Identity of the two geometries, however, is not required; in
the analyzed case the points of one space represent physical locations in a
Euclidean space, while the internal functional motor hyperspace is of
higher dimensionality and non-Elucidean (not even Riemannian).
A most practical feature of the metaorganization
process is that it enables an explicit study of how an existing and
well-defined primary geometry (such as the physical geometry of the motor
apparatus), organizes a much less explicit and often ill-defined functional
geometry which is implemented by a neuronal network.
The emergence of highly organized structurofunctional
features of neuronal assemblies is often labeled as "adaptation",
"self organization", or "learning" in the CNS. While the
principle of metaorganization and its algorithm is closely related to these notions
(which have no precise and generally accepted definition), it differs from them
in several fundamental respects. First, the geometrical redefinition of the
emergence of networks in metaorganization is based on an identification of two
entities: (a) one that governs the process of organization and (b) another that
is being formed. For example, the first geometry is defined as that arising
from the physical structure of the effector system, while the latter is defined
as an abstract geometry over a multi-dimensional manifold. The well-defined
nature of these two entities, which mold one another, tempts one to compare
them with the notion of "self organization", which has generated much
interest without, however, defining the entity that is responsible for the
organizing the "self". Second, since the metaorganization principle
is elaborate using formal geometrical analysis, tensor network theory of the
CNS and the process of metaorganization can be demonstrated quantitatively by
using specific neuronal networks.
4.1.
Metaorganization of CNS hyperspaces: a geometrical re-definition of the
notions of "adaptation", "self-organization" and
"learning"
Adaptation, self organization and learning have been
conceptualized and elaborated using many different approaches.
Viewing the CNS as an adaptive control system
26,27,34,35 represents two aspects of the CNS. "Control" insures that
a system conforms to the internal order defined by the neuronal networks.
"Adaptation", in turn, enables the CNS to conform with the external
conditions. One limitation of an approach that separates, rather than unifies,
control and adaptation may be that it assumes the concepts and formalisms of
control-system theory. Borrowing from engineering, neuroscientists almost
invariably chose its most limited form, the feedback-gain control of a single
variable. That description did not lead to a network theory of existing
neuronal circuits. Formally, the unselective borrowing from control
engineering, even in the form of modern multivariable control theory
11,21,45,68 may lead to major distortions.
For example, since engineers express vectors in
convenient Cartesian orthogonal coordinates for man-made systems, some neuroscientists
may be led to believe that a vectorial notation that does not distinguish
covariant and contravariant expressions may still be adequate to describe
biological vector transformations. Admitting to the possibility that nature
may have selected other than Cartesian co-ordinate systems and that most
natural frames of reference are demonstrably non-orthogonal (cf.
vestibulo-collic reflex) 67 leads to the inevitability of using a conceptual
and mathematical apparatus that can express physical invariants in general
non-orthogonal coordinate systems (such as tensor analysis, or other
mathematical apparatus as listed in Ref. 75).
In order to compare the metaorganization principle
with the hypothesis that the CNS is a self organizing system, it is necessary
to briefly assess the development of the latter concept. The view of the CNS as
a self organizing system originates from automata theory.99 Basic
considerations on the functional organization of the brain immediately
elevated self organization to one of the most intriguing chapters of
Cybernetics. 100. Postulating a
synaptic mechanism which may underlie the organization of behavior 31 provided
a link between the abstract theory of the emergence of neuronal systems and
experimental neuroscience. However, research at that time did not provide
either a formal definition or a rigorous elaboration of the general notion.
A second major increase in interest in self
organization occurred in the 1960s, 12,30,102, when self organization was tied
to concepts of "learning", "optimalization",
"adaptation" and "approaching a steady-state in relating to
external systems".6 It was explicitly stated, 6, however, that a precise
definition was still lacking at that time.
Finally, self organization has attracted intensive
theoretical interest again in the last decade.3,18,26,42,62. Although the
mathematical sophistication inherent in these new models is unsurpassed, a
generally accepted definition of self organization, both in a philosophical or
mathematical sense, remains elusive. Therefore the term is used in neurobiology
in a largely intuitive sense.95 It may be applied to a specific phenomenon such
as the emergence of temporally stable neuronal patterns, or to such abstract
phenomena as human learning. Philosophically, however, the usage of the term
"self" to an organizing principle or a process appears contradictory,
since the organization of the "self" must surely be separable from
the steps which generated it. Indeed, the question may not be, how does the CNS
organize itself, but rather, how is the "self" organized in the CNS
by the rest of the body and by the external world.
A geometrical redefinition may become helpful in
alleviating some of the problems. For example, by providing separate definition
for the two interacting geometries that is the geometry of physical features of
motor systems and the functional geometry of neuronal networks that co-ordinate
their actions, the causative relationship between these two becomes
explicit and quantifiable.
4.2.
Oscillations and tremor
One aim of tensor network theory is to serve as a
help with a general interpretation of experimental data relating to brain
function. It is therefore significant to determine if some basic experimental
observations may be formally related to general principles of CNS organization.
Indeed, neuroscientists have long wondered if phenomenology as basic as
biological oscillation and resonance may provide a key to the mechanism which
determines the organization of the neuronal networks of the CNS. This
possibility is supported by the argument that oscillations and resonance are
observed during development at several levels of the neuraxis. Such processes
may thus be the means of development of electrical properties of the neuronal
elements that constitute the neuronal nets.49 In fact, neuronal oscillation and
resonance determine much of the developing limbs and thus must provide, by
recurrent afferent activity, crucial information about the dynamics of
body-reference frames during early neurogenesis. Moreover, it is quite
possible that such an internal searching mechanism may be operant in the adult
form and may become quite explicit in pathological conditions. For example, it
has long been known that in patients with Parkinson's disease, mechanical
oscillatory stimulation of a finger may induce tremor which irradiates upwards
along the limb,39 in a manner similar to the Jacksonian "march" of
motor seizures following localized lesion of the motor cortex. This
"tremor-march" phenomenon indicates that the several segmental
levels of the CNS which control limb movement are coupled to each other such
that they may phase-lock and resonate when tremor occurs in only one segment.
Such dissipative functional structures are of the essence when considering that
the entire limb may be used as a single element or as a set of separate
compartments.43 The importance of the interaction of sensory feedback and motor
output becomes clearer when considering that this interaction begins to occur
very early in development even prior to the generation of co-ordinated
movement and could thus serve as an epigenetic organization influence in
determining the selective stabilization of neuronal networks.13
4.3.
Generalization of the metaorganization principle: tensorial interpretation of
the hierarchy of dual (complementer and duplicate) geometries in the CNS
The principle of metaorganization was elaborated in
section 2 for a motor system where the primary geometry was directly
identifiable. Active muscular forces are thought to represent contravariants,
while passive tendon-forces represent covariant expressions of motor action.
This primary physical contravariant-covariant transformation was complemented
by establishing a secondary functional geometry which implemented a
covariant--contravariant operator. This was accomplished by finding the
eigenvectors and eigenvalues of the existing motor mechanism. Several
generalizations'5 of the metaorganization principle lead to its application to
systems that are beyond such primary geometries as those inherent in motor
mechanisms.
The first generalization of this principle was made
possible by the realization that the primary geometry does not have to be
represented by a single physical metric transformation, but could be an entire
sensorimotor chain which becomes a completed circle through its interaction
with an external physical invariant. This was the case when constructing the
sensory metric (Fig. 6D), where the entire sensorimotor chain provided for any
contravariant sensory perception vector, s~, the corresponding covariant
perception-vector, s;. This was attained through (a) a sensorimotor neuronal
network, (b) a motor metric operator and (c) a physical invariant generated by
the motor system where this invariant was physically measured by the sensory apparatus.
Beyond the above, the metaorganization principle may
be applied to any metric tensor (e.g. one that is manifested only in an
abstract functional geometry). For instance, once the network of the sensory
metric is developed as a geometry which is secondary to the primary geometry
inherent in the external world, the sensory metric itself may then be used as a
primary geometry for metaorganizing its own secondary geometry in a
hierarchically coupled neuronal network. For instance, the optic tectum itself
may generate a secondary space in the cerebral cortex (that is
"empty", i.e. "amorphous" in its pristine state), by
serving as the organizer of the higher-order mold in this cortex of the tectal primary
functional geometry.
The third generalization follows from the simple
fact that the metaorganization process builds the secondary geometry from the
eigenvectors and eigenvalues of the primary geometry. In generating the
eigendyads, metaorganization could utilize either the eigenvalues of the
initial system themselves of the initial system or their generalized inverses
(or both). Metaorganization can thus result either in a secondary geometry
which is a duplicate or a complement of the primary geometry, or both. This generalization
leads to a study of hierarchically connected hyperspaces where the primary
spaces (comprising sensorimotor metrics) are directly connected to physical
geometries external to the CNS but where the secondary hyperspaces are built
upon these sensorimotor metrics. Each geometry may in turn breed higher order
duplicate or complement geometries in connected hyperspaces. As pointed out,
75 the intelligence implemented by such a hierarchy of metageometries may
depend on (a) the number of levels of connected hyperspaces or (b) the
precision with which one metageometry is molded by another.
4.3.1. Biological generalization of
metaorganization: nucleocortical structures as metagenerators.
While the theoretical implications of the above
proposals may be far-reaching, the immediate task is to systematically close
the gap between experimental data and theory. This must be accomplished at the
system, network, cellular and subcellular levels. Techniques of modeling
neurons and neuronal networks 71,73 may be used to determine if network and
system-level proposals are compatible with our knowledge of the physiology and
biophysics of single elements. Initial checks of general theories as described
above should actually be suggested by the proponents of such theories.
First, the relation of nucleocortical CNS systems
can be studied to determine whether the nuclear structures serve to embody the
functional eigenvectors and eigenvalues of primary geometries or whether the
cerebral cortical structures embody eigendyadic expansions implementing the
secondary functional geometry as determined by the eigenvectors and
eigenvalues. A particularly relevant possibility is that the thalamoneocortical
and other similarly arranged neuronal networks could in fact be dynamically
organized as proposed here for the olivocerebellar neuronal apparatus.
Second, because of its possible use in system-level
research the limitations of the metaorganization principle also need to be
pointed out by the proponents. As shown in section 3, the three neuronal
networks of a sensorimotor mechanism could be established if the system was
organized in a closed loop with regards to one physical invariant (in the
example in section 3 the invariant was a physical displacement). This is,
however, not the case for systems such as the vestibulo-ocular reflex, 79.
There, although the motor metric can be organized by the metaorganization
process, the sensorimotor transformation network and the last step, the sensory
metric, cannot be similarly generated. Indeed while the oculomotor metric could
in principle be generated by metaorganization,'s since the extraocular muscle
activity and its derived proprioception do represent the same invariant eye
movement, 66 the CNS cannot use metaorganization to generate the full vestibuloocular
reflex network, because eye movements cannot be detected by the vestibular
apparatus. However, systems such as the vestibulo-ocular reflex can emerge from
a hierarchy of primary sensorimotor systems, such as the vestibulo-collic
reflex and retinoextraocular reflx.'9 Finally metaorganization can be used, as
will be shown elsewhere, to generate the three sensorimotor networks in each
primary system and a seventh network to tie the two closed reverberative
primary mechanisms into a hierarchical sensorimotor architecture of the
vestibulo-ocular reflex. Thus while the metaorganization principle and
algorithm is applicable to quantitatively feature the development and function
of some specific neuronal networks such as the cerebellum, its ultimate use at
other levels of CNS function remains largely unexplored at present.
***
Acknowledgements-This work was supported by grant NS
13742 from NINCDS. We thank Dr J. I. Simpson for his valuable discussions and
reading of the manuscript. The authors also express their gratitude for all
their colleagues (far too numerous to list here) who offered constructive
criticism to tensor network theory.
Note added in proof.
After this work was
accepted for publication, a paper appeared offering some remarks on the uses of
tensor network theory [Arbib M. and Amari S. I. (1985) J. theor. Biol. 112,
123-155]. While generally supportive and appreciative of the tensor approach
and advocating its further use, the paper also raised five specific points of
concern. These can be readily dealt with here as they are off the mark
regarding the basic principles of tensor network theory and probably originate
from a misunderstanding of our approach.
(1)"Modern
mathematics has developed many techniques for coordinate-free analysis of
structure. PL [Pellionisz and Llinas] write as if the use of tensor analysis
were the only such technique". This remark is incorrect. The tensor
approach was explicitly compared to other mathematical techniques, see chapter
in Ref. 75 entitled "Comparison of tensor approach with other geometric
theories: Representation, modeling, mapping, differential geometry, lie
algebra".
(2) "PL speak of a CNS hyperspace F but
never prove that F is a Riemannian manifold." This remark is based on a
fundamental misunderstanding of
tensor network theory. Indeed, engineering typically uses Cartesian tensors of
Euclidean space; relativity uses
tensors in Riemannian space. What we face, however, is not a duty to conform
with previous scientific approaches but a challenge to create methods that
conform with CNS function and thus we have proposed tensor analysis of brain
hyperspaces. The important point to be understood here is that CNS hyperspaces
need not be confined either to Euclidean or Riemannian geometry. The remark is
also incomplete. The non-Euclidean character of the geometry has been
demonstrated by a quantitative computer model; see section entitled
"Position-dependent metric" of Fig. 5 in Ref. 81.
(3)In tensor network
theory "every array of numbers must be the co-ordinates of either a
covariant or contravariant vector . . . ." This is not so; for instance,
the cerebellar nucleofugal vector [(i-e) in Fig. 2. of the above
"Metaorganization" paper, or in Ref. 76] is neither co- nor
contravariant. More to the point, we wrote in Ref. 84: the ` . . . task is to
establish whether the way of assigning the components to the invariant is a
covariant or contravariant procedure". Once mathematical vectors have been
assigned by a co- or contravariant method to an invariant, such arrays of
numbers can, of course, be manipulated, e.g. by subtraction, so that the
primary character of the original arrays is no longer evident.
(4)Tensor network theory
"makes no use of tensor theory beyond the metaphorical use of the terms
`covariant', `contravariant' and `metric tensor' . . . . ' This remark is
incorrect. We have used multidimensional tensorial analyses of spacetime
manifolds, tensor transformations by covariant embedding, quantitative
exposition of the curved character of CNS hyperspaces, covariant-contravariant
transformations by the Moore-Penrose-generalized inverse, contraction of
tensors and network elaborations. Readers acquainted with some of our other
twenty or so papers published on this subject (not cited by these authors) will
be aware of these uses and developments.
(5)"The tensor theory is grounded in no data on cerebellar anatomy, physiology or function." This
is clearly an oversight by our colleagues.
Naturally tensor network theory requires further
elaboration, such as offered in the above "Metaorganization" paper.
The fact remains, however, that the introduction of non-orthogonal natural co-ordinates
to describe sensorimotor transformations does seem to result in advancements in
interpreting the anatomy, physiology and function of the CNS.
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(Accepted
3 May 1985)
