1. Introduction

Gaze control has recently been interpreted by tensor network theory of the CNS, which applies to multidimensional natural coordinate systems (Pellionisz & Llinas, 1979a,b, 1980, 1982a; Ostriker et al., 1985). The vestibulo-oculomotor reflex has also been described by a matrix analysis method, which reduces this system to arbitrary frames that are three-dimensional throughout (Schultheis & Robinson, 1981; Robinson, 1982; Ezure & Graf, 1984a,b) (Chapters 1 & 20). While a detailed tensorial computer model of the complex gaze system is offered elsewhere (Ostriker et al., 1985), two specific, yet important and timely tensorial aspects of the VOR and the adaptive gaze are pointed out in this paper, in order to better illuminate the differences and the similarities of these two approaches. First, it is shown mathematically how the problem of motor coordination is treated tensorially as a covariant-contravariant Eigenvector problem in overcomplete sensorimotor systems, through the generalized inverse of the covariant metric tensor (Pellionisz, 1983a, 1984a). Second, it is emphasized that there is a need for a hierarchical analysis of nested sensorimotor networks underlying gaze control, as dictated by the concept of Metaorganization, which explains the genesis and modification of neuronal networks that implement the functional geometries of the CNS (Pellionisz, 1983b, 1984c).

1.1. Multidimensional approaches to the central nervous system and to the vestibulo-ocular reflex

When Sherrington (1906) promulgated the classic concept of reflexes, he warned against its oversimplification. "The main secret of nervous coordination lies evidently in the compounding of reflexes. . . . But though the unit reaction in the integration is a reflex, not every reflex is a unit reaction, since some reflexes are compounded of simpler reflexes. A simple reflex is probably a purely abstract conception, because all parts of the nervous system are connected together and no part of it is probably ever capable of reaction without affecting and being affected by various other parts . . ." (Sherrington, 1906).

Indeed, the gaze system has been regarded since Flourens (1826) as an interconnected multivariable system where all vestibular canals affect the function of all eye muscles (cf. the classical analyses by Helmholtz, 1896, Weiland, 1898 or Lorente de No, 1932). However, this complex interpretation of the whole had to be supported by a convergent triad of experimental, formal and conceptual means capable of handling such complexities. Such an experimental approach, in the form of registering the contraction of all extraocular muscles, was pioneered by Szentagothai (1950). A corresponding formal treatment, in the form of conventional vector analysis, was attempted by Krewson (1950). An outstanding conceptual interpretation, in the form of the potent quarternion-theory, was introduced by Westheimer (1957). The complex approach to the gaze system, nevertheless, has only rarely been pursued in recent times (Nakayama, 1974), due perhaps to the lack of cohesion among these difficult experimental, formal and conceptual methods.

Instead, despite the warnings about oversimplification, studies of some `reflexes', most particularly that of the VOR, had gradually become reduced into a single scalar variable, the `gain' of the system (for a most recent example, see Miles, 1982). This trend towards simplification was based on the systems' analysis approach, borrowed from engineering, assuming that some simple reflexes can be regarded as single-variable control loops (cf., Robinson, 1968; Stark, 1968; Young, 1969).

It has been claimed that tensor network theory provides new conceptual and formal means to facilitate a unified approach which aims at the true structuro-functional complexity of the CNS (Pellionisz & Llinas, 1979a,b, 1980, 1982a). Since tensor network theory was introduced, there has been a dramatic return of the interest towards multidimensional analysis of CNS, especially those systems, such as the VOR, where such relationships can be quantitatively established (Berthoz et al, 1981a; Schultheis & Robinson, 1981; Robinson, 1982; Goldberg et al. 1982; Simpson, 1983; Ezure and Graf, 1984b) (Chapters 1 and 20).

1.2. Problems with a multidimensional approach reduced to three-dimensional matrix analysis

A multidimensional analysis of the CNS presents serious conceptual problems, for instance, how to interpret neuronal function that is expressed in natural coordinates and in hyperspaces with different dimensions. Secondly, it also poses the burden of quantitative elaboration of such transformations. Therefore, it was proposed that the tensorial approach, which can state the functional meaning of network transformations in a highly abstract, coordinate system-iree manner, should be coupled with computer modeling, which can numerically accomplish such quantitation (Pellionisz & Llinas, 1979b). A tensorial computer model of the gaze system is presently being developed (Ostriker et al., 1985). Some central conceptual problems that appear to be unresolved in the literature, however, can be focused upon in this paper from a purely theoretical viewpoint.

Any multidimensional approach appears more suitable to the VOR than reducing it into a single variable gain-control mechanism. Nevertheless, some problems of the three-dimensional matrix approach, which may lie in the setting of its goals, call for attention. The issue is not that the CNS requires three-dimensional, instead of onedimensional analysis because it relates to the physical three-dimensional space. CNS functional manifolds are not one or three dimensional, but at times very highly multidimensional functional hyperspaces spun over the firings of a large number of neurons. A humble approach is the one that lets the brain be investigated in its own grandeur instead of reducing it into a flat shadow. The reduced approach was designed to elevate the description from one to three dimensions, attainable by no more than matrix-analysis (cf., Chapter 20), and not to tensorially analyze the transformations of natural coordinates in various CNS functional hyperspaces with differing dimensions. As a basic simplification in the reduced approach, in order to contain vectorial expressions into three dimensions, artificial systems of coordinates had to be introduced. Some problems that arise from this simplification are as follows.

(1) Enforcing a three-dimensional description entails artificially reducing the six-muscle extraocular motor system into a three-dimensional one by pairing `agonist and antagonist' muscles, even though their axes of eye rotation, such as that of the superior and inferior recti in human, may lie as far apart as 36° (Robinson, 1975b). In addition to this morphological reality, physiological evidence is also available to show that the activation of so-called `agonist and antagonist' eye muscles is grossly asymmetrical (Barmack, 1976). Indeed, textbooks warn against the ". . . misleading custom of linking extraocular muscles together, usually into pairs . . . Limited views of this kind may have a mnemonic value, but they ignore the inescapable fact that in any ocular rotation all six muscles must change in length. . . . it is impossible to dogmatize as to whether every muscle is contracted precisely in step with the progressive inhibition of an antagonist" (Williams & Warwick, 1975; p. 1125). Beyond the mathematical error that `averaging of vectors' represents, under these conditions, by changing from intrinsic frames of reference to imaginary ones, the goal of understanding the CNS function in its own terms becomes elusive again.

(2) One of the most significant conceptual problems of sensorimotor integration is the way in which the CNS transforms an expression in a lower dimensional sensory frame into one in a higher dimensional motor frame. This problem underlying motor coordination has been raised, and a tensorial solution was suggested, by Pellionisz and Llinas (1979b, 1980, 1982a), and quantitatively elaborated by Pellionisz (1984a). On the other hand, reducing the system to threedimensionality throughout, eliminates rather than tackles the very problem of overcompleteness which most warrants a multidimensional approach.

(3) Not analyzing the system in its intrinsic (or natural) coordinates permits only a lumped description of the whole sensorimotor transformation in one single step, by one `brain stem matrix' of the VOR (Robinson, 1982). However, the VOR has long been known to be based on a `reflex arc' composed of at least three sequential steps (Lorente de No, 1933), from the primary vestibular neurons to secondary vestibular neurons to oculomotor neurons, even if the premotor neurons of the internuclear brain stem apparatus (cf., Baker & Berthoz 1977; Baker et al., 1981a) are ignored. Recently, one of the basic considerations of the reduced approach, requiring that secondary vestibular neurons receive no convergence from various semicircular canals (Robinson, 1982), was found to be conrary, in many instances, to experimental findings of projections to these neurons from up to all six canals (Baker et al., 1983). ..

To facilitate multidimensional VOR analysis without the need for these simplifying assumptions, a quantitative elaboration of the previously introduced tensorial scheme is presented below.

2. Motor coordination as a covariantcontravariant Eigenvector problem in overcomplete sensorimotor frames intrinsic to CNS

2.1. Extrinsic and intrinsic systems of coordinates for CNS function

Whenever the function of the CNS is related to physical objects of the world (external to the living organism), the complex multivariable spatial and temporal relationships can conveniently and concisely be given in systems of coordinates. The vestibulo-ocular reflex, which turns the eye in a manner that compensates for head movement, thereby allowing a stable field of vision, has been spatially characterized by authors too numerous, to mention. In all cases the morphological orientation of the ocular motor system and the vestibular sensory apparatus was established in the frames of reference selected by the observer (cf., Fig. 1.).

While, as a matter of course, any arbitrarily chosen system of coordinates can be applied to an extrinsic description of such physical geometries, for reasons of convenience the traditional orthogonal Cartesian XYZ system is utilized by all authors. On the other hand, however, the individual sensors (vestibular semicircular canals) and motor effectors (individual eye muscles) constitute frames of reference of another type. These are intrinsic to the CNS, since neuronal firings express,physical actions in these natural systems of coordinates. Accordingly, one has to accomplish two different tasks: first, to determine how the actual physical arrangements of natural systems of coordinates relate to the extrinsic, usually Cartesian, frames of the observer; and second, to explain how the CNS expresses its function in its own natural coordinates. For instance, how does the CNS implement a sensorimotor transformation from the three components of the covariant vector that the oblique set of semicircular canals sense, to the six components of the contravariant vector that activate the non-orthogonally arranged extraocular muscles?

Fig. 1. The two kinds of coordinate systems, used in the external description and in the inner workings of the vestibuloocular reflex (VOR). Extrinsic, yet biologically oriented mirror-symmetric XYZ Cartesian frames (for the lateral sides of the body) are used as medial, dorsal and anterior, respectively. As in biological organisms with lateral symmetry, righthand rule applies to the right side, and left to the left-side. The `standard' position for visual demonstration presents XYZ with equal axes, 120° apart. Semicircular canals represent the HAP intrinsic system of vestibular coordinates, marked as: H, horizontal; A, anterior; and P, posterior. Eye muscles and their corresponding eye-rotational axes are denoted by: LR, lateral rectus; MR, medial rectus; SR, superior rectus; IR, inferior rectus; SO, superior oblique; IO, inferior oblique. These abbreviations apply throughout the paper. The diagram of the eye muscle orientation is drawn with the utilization of the computer model by Ostriker et al. 1985, and therefore both the paired sensory and unpaired motor systems are shown in a quantitatively exact manner.

The first goal was reached a century ago by Volkmann (1869), who established the rotational axes corresponding to the individual contractions of the human eye muscles. Similar data have been recently provided also for other species (Simpson, 1983; Ezure and Graf, 1984a). Similarly, the excitatory sensitivity axes of each semicircular vestibular canal are also quantitatively known (Blanks et al., 1972, 1975b; Simpson, 1983, Ezure and Graf, 1984a). These data can be given in any extrinsic frame. Consequently, while the Cartesian system is used exclusively, the XYZ conventions vary. Although some yield advantages over the others, the variations may represent a source of confusion. In the approach presented here (see also Ostriker et al., 1985) two mirror-symmetric XYZ systems are used for the two lateral sides of the body (medial, dorsal and anterior directions, respectively), with a right-hand rule for the right side, and left-hand rule for the left side (Fig. 1.). The XYZ are chosen to correspond to pitch, yaw and roll, no matter which side is referenced. As a result of this natural extrinsic convention, the morphological data are identical numerically and in sign for both sides. For visual demonstration purposes, a `standard' frontal-lateral view of the head is used throughout the paper (pitched down 45°, and then yawed 45° to the left, so that the XYZ system appears with circular symmetical axes 120° apart).

The geometrical arrangement of the sensory and motor systems shown in Fig. l. are based on data in Fig. 2 (Volkmann, 1869; Blanks et al., 1975a). Based on an implicit argument that the vestibular apparatus measures the same head acceleration vector, and the misalignment of canal orientations on the two sides is within close limits, it became a practice to average the covariant (cosine) components provided by corresponding semicircular canals of the vestibular systems of the two sides, and thus arrive at a unified vestibular coordinate system as shown in Figs. 1 and 2. (cf. , Robinson 1982; Ezure and Graf 1984a,b). This is inevitable to the reduced approach, in order to arrive at a three-variable apparatus. While for the proposed tensorial treatment such averaging is not a requirement at all, this assumption of a cambined vestibular system will be maintained temporarily because (a) compared to pairing muscles this pairing yields only a relatively minor error but, more importantly, (b) it facilitates comparison of the approaches, and (c) it shows how the CNS may transform a threedimensional sensory input into an overcomplete, six-dimensional output. Nevertheless, it is necessary to point out that `averaging' of vectors is only permissible when the components are of equal magnitude. In contrast to the extraocular apparatus, this assumption is a reasonable one in the case of the vestibular system since, given the maximal deviation in humans of 24° of the canals of the two sides, the maximal deviation of the activation of paired or non-paired canals is 7%. From calculations presented later in this paper it will be evident (see Fig. 3F, IR and SR) that in the case of the extraocular muscles the difference in magnitude between the activation of a so-called `agonist-antagonist pair' can be as high as 2670%, and thus pairing of the extraocular muscles is impermissible

Fig. 2. The sensory and motor systems of coordinates of the VOR, intrinsic to CNS function, as defined by the extrinsic vestibular matrix V and extrinsic eye muscle matrix E. The directions in three-dimensional XYZ space of the unit (normalized) rotational axes, belonging to individual eye muscle contractions, are shown in the left. The excitatory activationaxes of the combined semicircular canals of the two vestibuli are shown on the right. To facilitate visual perception of the three-dimensional directions of the axes, their orthogonal projection to the XZ plane is also indicated. The tables of extrinsic eye muscle matrix E and extrinsic vestibular matrix V represent the data base used throughout the paper (after Volkmann, 1869 and Blanks et al., 1975a,b respectively).

The sensory and motor systems of coo.rdinates, intrinsic to the CNS, are shown in Fig. 2. From the diagram, the differences in the direction and the number of axes of the input and output frames are evident, although the visual perception of the three-dimensional physical arrangement of the rotational axes does require some practice. From the orientation of vestibular directions it is clear that, in human subjects, a 24° down-pitched position would produce the largest excitation of the horizontal canals in case of a horizontal yaw, a position which is used to evoke a `mostly' horizontal eye rotation (cf., numerical example later in the paper). It is of interest, that this position for maximal horizontal activation is not the position for minimal anterior and posterior activation. That position can be calculated from the cross product of A and P canals, yielding a 12° downpitch. The numerical results for this (and for any) head-rotation can be calculated using the scheme of Fig. 7, connected to the extrinsic frame (if required) through the extrinsic matrices V and E shown in Fig. 2.

2.2. Tensorial scheme of the VOR

The second task, relevant to CNS function, is to reveal how sequential parallel networks of the `three neuron reflex arc' may transform a head rotation, given by a covariant reception vector, into a contravariant motor execution vector. The tensor approach, which can deal with intrinsic coordinates, suggests a solution to the problem of an increase in dimensionality and shows the interim expressions that are required after each network-transformation. The solution is based on the idea introduced by Pellionisz and Llinas (1979a), shown in the form of a conceptual scheme (Pellionisz & Llinas 1980; Fig. 4), and elaborated into a qualitative neuronal network implementation (Pellionisz & Llinas 1982a, Fig. 8). The notion that in overcomplete expressions the counterpart of the covariant metric tensor can be obtained by its generalized inverse was introduced by Pellionisz (1983a); a detailed numerical model showing the computed generalized inverse was later given by Pellionisz (1984a,b).

The basic tensorial scheme is shown in Fig. 3. According to the tensor concept, the physical object (invariant) that is extrinsic to the CNS, is expressed in various intrinsic neuronal frames of reference in co- and contravariant forms. In the case of the VOR, the rotation of the head around a physical axis, e.g., as given in Fig. 3, can be represented as a unit-vector with a physical orientation which is, of course, expressible in an extrinsic, e.g., Cartesian frame as (0.000,-0.924, 0.374) in this example 4. This input was chosen for illustration as it is the so-called `purely' horizontal stimulation. The head-rotation around this axis is the physical invariant, which is represented throughout the VOR in different vectorial expressions, and which finally emerges at the output as an identical physical entity, the movement of the eye. In reality, the compensatory eye movement is opposite to the head movement. However, since it is not presently known at which point of the neuronal `reflex arc' the sign of the transformation reverses, the VOR is shown throughout this paper as yielding an eye movement identical to the head movement.

Fig. 3. Tensorial scheme of the VOR. A physical entity, a coordinate-system invariant head rotation is vectorially expressed in extrinsic, orthogonal Cartesian frames (A,G), and in intrinsic non-orthogonal vestibular and extraocular muscle-frames, both covariantly and contravariantly in either (B,C,D,F). A, an arbitrarily selected head rotation, corresponding to maximal excitation of the horizontal canals is expressed in an extrinsic, Cartesian XYZ frame. The extrinsic vestibular matrix V (Fig. 2) transforms these XYZ extrinsic coordinates into HAP intrinsic covariant components, as shown in B (for the vestibular HAP, see Fig. 2V). The BCDF three-step sequence is implemented by neuronal networks peforming a sensory metric g^pr, sensorimotor transfer c sub ip, and motor metric transformations g^ie. The last intrinsic neuronal expression is the contravariant motor execution vector me, which generates a physical rotation by activating the eye muscles. The extrinsic eye-muscle matrix E (Fig. 2) can be used to calculate the Cartesian components of this rotation, which emerges as the physical resultant of infinitesimal rotations produced by eye muscle contractions (G). The numerical expressions are provided by the calculations shown in the rest of the Figures. All calculations throughout the paper are given in infinitesimal components so that their summation yields identical result in any permutation.

The first expression of the external physical invariant in natural coordinates occurs in the vestibular semicircular canals, a physically obvious covariant expression in the non-orthogonal HAP vestibular frame (see Fig. 2, cf., Pellionisz and Llinas 1980, Robinson 1982). Such an expression of the invariant was called covariant sensory reception (Pellionisz & Llinas 1982a), sr in short, the covariant nature of the vector shown by the subscript. At the other end of the VOR, the compensatory eye movement emerges as the physical resultant, contravariant vectorial expression of the contractions of six extraocular eye muscles (cf., the physical addition of six components in Fig. 3F, yielding the same rotation as in the A input). In Fig. 3E, the motor expression vector is symbolized by me, the superscript denoting the contravariant character of the vector. The vectorial components are shown numerically, as calculated later in the paper. The question of how the CNS arrives from this covariant three-component vector to the contravariant six-vector by a transformation, is first of all a general biological problem, that of coordination. However, it can also be looked at as a morphological problem of how `reflex arcs' implement a sequence of network transformations, and it can also be phrased as a mathematical problem of how the tensorial expressions can be obtained in an overcomplete space, i.e., in more than three-dimensional spaces for physical movements. All three aspects of this central issue have long been addressed, in the form of (a) analyzing coordination in the CNS and cerebellar reflexes (Sherrington, 1906; Flourens 1826), (b) emphasizing the at least `three-neuron reflex arc' (Lorente de N6, 1933; Szentagothai, 1950), and (c) pointing out the overcompleteness through explicit statements such as "it is possible to abduct the eye to the same degree in innumerable ways" (Weiland, 1898). The tensorial scheme in Fig. 3. provides an answer to these classical questions by explaining the sequence of parallel transformations in overcomplete natural frames, and yields a blueprint that not only provides the known function but, as shown elsewhere (Pellionisz, 1984c), can also be generated and modified to perfection by physical CNS procedures.

The two interim vectors are the contravariant expression of the invariant in the sensory frame (sensory perception sp; see Fig. 3C), and the covariant expression in the motor frame (motor intention, m;, Fig. 3D). The three transformations necessary to obtain these vectorial versions of the invariant are a contravariant sensory metric g^pr, to transform a covariant expression into a contravariant one in the same sensory space, a motor metric-type transformation g^ie which makes a similar conversion in the motor space, and a sensorimotor transformation c;P, a covariant embedding that transforms a contravariant sensory expression into a covariant motor version. The theoretical need of these specific interim versions was pointed out in the original papers by Pellionisz and Llinas (1980, 1982a). The distributed structure of the `three neuron reflex arc' provides a morphological basis for the existence of interim expressions.

2.2.1. Vestibular sensory metric tensors, and their characterization by Eigenvectors

The calculation of the sensory metric is straightforward (cf., Fig. 3. in Pellionisz & Llinas, 1980). Accordingly, the matrix of the covariant sensory metric has already been published (Robinson, 1982). A re-calculation of g sub rp, is given by Fig. 4A (note the different conventions, in our case left-hand rule for left-side). For a triplet of paired canal-axes, which is not an overcomplete system (cf., Fig. 4B), the contravariant vestibular metric tensor can be obtained by simple inversion of the covariant vestibular metric tensor. Robinson (1982) did not calculate it since he argued that "the metric tensor . . . need not be specifically recognized" (cf., Chapter 20).

In reality, however, it is not the published covariant sensory metric that is relevant to the function of CNS neuronal networks. The vestibulum measures these covariants, which can be calculated from the extrinsic components of the rotation via the covariant metric, for reasons of simple physics. Thus, the CNS needs to construct, through neuronal networks, the other type of metric tensor, the contravariant metric in order to obtain the counterpart of the physically measured covariant vectorial version. Then, with both coand contravariant expressions available, internal sensory judgements on the external invariants (e.g., in this case, directions) can be made (cf., Pellionisz & Llinas, 1982a).

Recognition of the sensory metric tensor yiclds immediate experimental paradigms that have hitherto escaped attention. It is basic mathematical knowledge, also noted in neuroscience (e.g., by the pioneering work of Anderson et al., 1977), that matrices can most profoundly be characterized by their Eigenvectors. Paraphrased, these are special input vectors to a matrix for which the output will be changed only in `magnitude' but not in `direction' (see, e.g., Petrofrezzo, 1966). Such Eigenvectors of the sensory metric can readily be calculated by any of the several commonly used methods, and are tabulated in Fig. 4C. Since, for Eigenvector input, the output differs only by the Eigenvalue coefficient, it follows that the Eigenvectors of the covariant and contravariant metric are the same, with reciprocal Eigenvalues. Those physical directions in the XYZ three-dimensional space that yield these vestibular Eigenvectors, are calculated from the data presented in the extrinsic vestibular matrix shown in Fig. 2V, and are shown in Fig. 4 (note, that the vectors are not normalized, in order to provide a measure of the Eigenvalues). From the diagram of Fig. 4D it is evident that there is a position, attainable in the human by pitching the head up from the stereotaxic plane by an angle of 21°=arctan(0.396/1.036); where pitch, yaw and roll around the X, Y, Z axes, respectively, will stimulate the vestibulum by an Eigenvector

Fig. 4. Covariant (A) and contravariant (B) vestibular metric tensors, and their Eigenvectors E with Eigenvalues L (C). The physical directions in the XYZ three-dimensional space, corresponding to the normalized Eigenvectors (C), are shown in D. These D Eigendirections are shown not normalized. E, experimental subject in a pitched-up position aligns the vestibular Eigendirections with the Earth-referenced yaw, pitch and roll directions. Note that such position is in contrast to the usual pitched-down orientation that is used for maximal stimulation of the horizontal canals.

Thus, if one were to test the vestibular apparatus for its special overall performance, this simple tilt up into an `Eigenposition' would lend itself easily to an experimental analysis. This is contrary to the not trivially justifiable experimental habit of tilting the head down by 24°, in order to align one coordinate axis with the rotation. It is noteworthy that in natural tests for the ultimate vestibular performance, such as in free fall or in an iceskating piroutte, subjects appear to adopt an easily measurable pitched-up head position.

Since all vectorial relationships in this paper can be given in a reference frame-independent tensorial notation (see Fig. 3), different data bases, similar to the one shown for the human in Fig. 2, will be readily applied to model the VOR of various species. Within the scope of this paper, the calculated results on the cat (cf., data from Blanks et al., 1972 and Ezure and Graf, 1984a) or on the rabbit (cf., data from Simpson, 1983 and Ezure and Graf, 1984a) are given only for this `Eigenposition', see Fig. 4E. This yields 26° for the cat, and 24° for the rabbit.

Fig. 5. An explanation of the method of covariant embedding in general (A), and in the specific case of sensorimotor vestibulo-extraocular transformation (B). Such transformation may be performed from a contravariant vector to a covariant one. Note, that this direction is opposite to the sensory covariant to motor contravariant sequence. The covariant embedding procedure makes no restrictions regarding the dimensionality of either frame; thus an increase in dimensionality is also possible. The example in A illustrates a five-to-four dimensional transformation, but it applies equally to an n to k, where n and k may be any integer. Further explanation is in the text.

2.2.2. Sensorimotor covariant embedding from the vestibular to the oculomotor system of coordinates

In any sensorimotor system, by definition, somewhere and at least once, a neuronal network has to be found that executes a transformation through which the external physical invariant is expressed in a sensory frame at the input side, and in a motor frame at the output side. For an understanding of the nature of sensorimotor integration, it is inevitable that an exact explanation be provided of how such a transformation may occur in the CNS. The covariant embedding procedure suggests such a transfer, implementable by neuronal networks (see Pellionisz & Llinas, 1980, 1982a). However, it assumes the availability of a corttravariant sensory expression (to be provided by the contravariant sensory metric tensor) and yields a vectorial expression of the invariant in the required motor space, but in a covariant form. A covariant, when physically executed, yields an improper invariant. Therefore, the covariant sensorimotor embedding procedure necessitates an additional, contravariant motor metric tensor (Pellionisz & Llinas 1980, 1982a).

Covariant embedding, as demonstrated in Fig. 5., is based on the principle of independence in establishing covariant components along any number of axes. Let a physical invariant be represented by an i-dimensional contravariant vector, v' (with physical components), and let us calculate the covariant components of the same invariant along the j axes of the ul covariant vector (let i=5, and j=4, although there will be no restrictions to either i or j). Given a physical component of unitvector of v' (along an axis), it can be measured along each axes of u~. These cos(alpha) components will yield the i-th column in the cl; matrix of the covariant embedding. It is emphasized that either of the v or u vectors can be of any dimensions, and the covariant embedding can still be implemented with unique components. Therefore, a basis for both the sensorimotor transfer and the change of dimensionality (e.g., from a lower to a higher) is conceptually established. Quantitatively speaking, a table of cosines of the angles among sensory and motor axes can be calculated by the inner product of the unit-vectors shown in Fig. 2, yielding c;p in Fig. SB. The question of how such a sensorimotor covariant embedding network can be generated by the CNS was answered by Pellionisz (1984c).

In summary, starting with any physical invariant, such as head rotation, the vestibular canals will sense its covariant components. Through the contravariant sensory metric, the contravariant sensory expression is made available. This contravariant vectorial version of the physical invariant is amenable to the covariant sensorimotor embedding procedure, which yields a vectorial expression in the motor frame, but in projectiontype covariant components. Therefore, to complete the VOR transformations, a last conversion, a covariant to contravariant motor transformation must be made available, as elaborated below.

2.2.3. Covariant- to-contravariant motor transformation

Given a motor frame of reference, such as the six rotational axes corresponding to the contraction of individual eye muscles, the existence of both co- and contravariant-type expressions of an invariant can easily be verified (see Fig. 6A,B). Given any six physical components, such as activations of extraocular muscles by motoneurons, the physical entity of an eye movement will emerge. Thus, to any contravariant vectorial expression there exists an invariant (Fig. 6A). Likewise, starting from this invariant, the covariant components can uniquely be established by projections from the invariant onto each axis separately; thus a covariant expression of the invariant also exists. As shown in Fig. 6A,B, these components may greatly differ in magnitude.

Fig. 6. Moore-Penrose generalized inverse acting as a contravariant metric in overcomplete CNS manifolds. A, any given sextuplet of contravariant (physical) components generates a physical resultant. B, the arising physical entity can be measured, using orthogonal projections to the axes, yielding a unique set of covariant components. The numerical data in A and B show an Eigenvector, where each covariant component in B is 2.077 times larger than the contravariant in A. To establish the reverse relationship, from a given set of covariants arriving at a unic,ue contravariant (one out of the infinite nutnber of possible expressions), it is required that the Eigenvectors of one expression be transformed to the Eigenvectors of the other. The unique, real-valued and symmetrical covariant metric tensor is given in C, and its Eigenvectors, Eigenvalues and Eigendirections are given in E and D, respectively. F, the Moore-Penrose generalized inverse of the covariant metric tensor, acting as a contravariant metric, transforms covariants into contravariants with the same Eigenvectors and reciprocal Eigenvalues compared to the covariant metric tensor

Thus, since there exists both a covariant and a contravariant expression for an invariant, even if the frame of reference is overcomplete, the question that remains is; how can their relationship be established? The covariant metric tensor can be expressed in a unique manner (since its elements are the cosines of the angles among the axes, cf., Fig. 6C). For the same reason, it is a real-valued, symmetrical matrix. The difficulty arises, however, if one attempts to calculate the contravariant metric by inverting the covariant metric, as could be done if the manifold was Riemannian, and thus Riemannian tensor analysis was applicable. In the multidimensional spaces that govern an overcomplete system of coordinates, however, the covariant metric is singular. Therefore, although there must exist a mathematical device that describes a multitude of possible transformations from a covariant set into a set of contravariant components, a unique solution cannot be identified by inverting this singular covariant metric. This mathematical feature reflects the physical condition, that in an overcomplete frame, the solution is not unique for the covariant-tocontravariant transformation. Given an invariant, the covariants assigned to it are unique, while the invariant itself can physically be assembled in an infinite number of different combinations of contravariant components. While th.is problem was recognized for eye movements as early as 1898 by Weiland, it has not hitherto been resolved.

It has often been stated that, in tensor theory of the CNS, the most profound questions relate to revealing the geometrical properties of the multidimensional functional manifolds, since it cannot be taken for granted that such spaces are Euclidean, or even Riemannian (cf., Fig. 1. in Pellionisz and Llinas 1982b). While, for didactic purposes two-dimensional diagrams (in the Euclidean space of the paper) have been extensively used to facilitate the understanding of the newly introduced concepts, it has been explicity stated and shown that the metric tensor (which is position independent in Euclidean spaces), is position dependent (non-Euclidean) in the overcomplete CNS hyperspaces (cf., Fig. 5. in Pellionisz & Llinas, 1980).

2.3. Covariant-to-contravariant transformation in special mathematical spaces (overcomplete rrueltidimensional CNS hyperspaces): the MoorePenrose generalized inverse of the covariant metric tensor acting as a contravariant metric

There is no a priori reason to assume a Reimannian, let alone Euclidean, geometry for the overcomplete CNS functional hyperspaces. Indeed, since they have never been closely investigated,

they may well turn out to be non-Riemannian in their functional geometry. The gravity of this point is that a rigorous enforcement or a tacit acceptance of axioms that disregard the facts may result in dogmatism or naivete even in scientific research. Thus, the chain of argument in neuroscience, just as in physics or any other natural science, must differ from the one in pure mathematics where there is a respectable place for any solid structure, even one based on imaginary axioms. Accordingly, there the axioms of the geometry of a space are defined first, and the consequences are then explored which follow from the axioms. In neuroscience, the experimentally obtained facts are preeminent. The challenge is to arrive at the basic mathematical features of the hyperspaces that lie beneath the rich complexities of findings and to ide.ntify and/or construct the appropriate abstract structure.

Pursuant to this argument, it is shown below that overcomplete CNS hyperspaces, indeed, appear to be non-Riemannian. This is so at least in the sense that Riemannian tensor analysis must be further generalized following the mathematical groundwork by Moore-Penrose (cf., Albert, 1972; Ben-Israel & Greville, 1980).

The approach presented here is based on the Eigenvectors of the overcomplete covariantcontravariant system. It can be established, both graphically and mathematically, that a particular contravariant physical set of vectorial ,:omponents may yield an invariant that is uniquely decomposed into a covariant set of projection components, that differ only by a constant coefficient from their contravariant counterparts. Such a numerical example is provided in Fig. 6AB, where each covariant component is 2.077-times larger than its contravariant counterpart. The Eigenvectors and Eigenvalues of a matrix can be calculated by several numerical methods (cf., Petrofrezzo, 1966). The results are presented in tabulated form in Fig. 6E; the directions in physical three-dimensional space that correspond to such Eigenvectors are shown in Fig. 6D. It is noteworthy, but cannot be elaborated here, that such `Eigendirections' carry physiological significance that are amenable to experimental exploration, similarly to the Eigendirections of the vestibulum. For our present purposes, given that the contravariant-to-covariant relation is established by the covariant metric tensor, transforming an Eigenvector into an Eigenvector magnified by an Eigenvalue, the contravariant metric-type transformation is expected to change the same covariant Eigenvector into a contravariant one, reduced by the same Eigenvalue. This conceptual argument leads to the Moore-Penrose generalized inverse of the covariant metric, which yields the required matrix. Indeed, the reader can verify that both g^ie (Fig 6F) and g;~ (Fig. 6C) share the same Eigenvectors, tabulated in Fig. 6E.

Fig. 7. Tensorial solution of an overcomplete sensorimotor transformation and its quantitative (matrix and network) implementation. Combined results of Figs. 4-6, provide a complete set of three intrinsic transformations that convert a three-dimensional covariant vestibular vector s, into a six-dimensional overcomplete contravariant extraocular vector m^e. The intrinsic transformations can be connected (if desired) to any extrinsic frame, e.g., by the extrinsic vestibular- and extraocular matrices of V and E (Fig. 2). The shown numerical example of a particular vestibular input vector s sub r is selected since it corresponds to the one in Fig. 3, the case of maximal horizontal stimulation. The scheme can be used for calculation of any eye-muscle activation, emerging from a given vestibular input. In the example shown, note that the activation of 'agonist-antagonist' muscles may yield a grossly asymmetrical contraction of, e.g., the SR and IR muscles, which are activated in a 26.7:1.0 ratio. In addition, both muscles are inactivated rather than acting in a push-pull manner. The tensorial scheme requires two interim vectorial expressions of s^p' and m sub i, and thus calls for three separate transformation matrices of vestibular metric tensor g^pr, sensorimotor embedding c sub ip, and oculomotor metric g^ie. It is suggested that the interim expression s^p is implemented in the secondary vestibular neurons which receive projections from all canals, and m sub i is in the premotor neurons of the internuclear brain stem mechanism

2.4. A tensorial blueprint of transformation networks accomplishing an overcomplete VOR function

With the covariant-to-contravariant motor transformation determined, the chain of VOR transformations is now complete, as shown in Fig. 7. The sensorimotor transformation, which utilizes non-orthogonal intrinsic frames of reference, converts a three-dimensional sensory input vector into a unique overcomplete six-dimensional motor output vector. The required interim expressions correspond well to the morphological fact that the VOR pathways do not constitute a single lumped `brain stem matrix' (Robinson, 1982), but form distinct nuclei with specific structural and functional properties (cf., Lorente de No, 1933; Robinson, 1982; Baker et al., 1981a). While XYZ is not part of the VOR per se, for practical purposes it is useful to convert me, which is expressed in intrinsic coordinates, into XYZ extrinsic coordinates by using the extrinsic eye muscle matrix as shown in Fig. 2E.

Relating the model-explanation shown in Fig. 7. to the exact details of reality calls for substantial further investments at all levels. Nevertheless, a correspondence can be well established on physical grounds for oculomotor neurons performing the contravariant execution function, and primary vestibular neurons acting as covariant receptors. While teleological reasoning postulates no convergence from primary vestibular neurons of various semicircular canals onto single secondary vestibular neurons (Robinson, 1982), evidence was and is available to prove this to be an incorrect assumption. Indeed, labyrinthine convergence on vestibular neurons has been demonstrated both by natural and electrical stimulation (Curthoys & Markham, 1971; Markham & Curthoys, 1972) and it has recently been shown that secondary vestibular neurons may receive input from up to all six vestibular canals (Baker et al., 1983). These findings reasonably correlate to the proposed vestibular metric, which contains ,nonzero off-diagonal elements. Another point of resonance with ongoing morphological research relates to the premotor neurons, which implement a covariant motor expression in the tensorial scheme. The interstitial, reticular and internuclear premotor-type neurons of the brain stem gaze control apparatus, extensively discussed by Baker and Berthoz (1977), are presently being given thorough attention, furthering the establishment of a correlation of the experimental facts with their model representation. Nevertheless, the significance of the proposed scheme lies in the fact that such increase in dimensionality from sensory to motor apparatus, as explained in Fig. 7, is a ubiquitous feature of the CNS. Indeed, in the gaze control apparatus there exist subsystems, most particularly the neck muscle mechanism, that dramatically deviate from the quasiorthogonal, non-overcomplete, threedimensional arrangements that can be adequately described by the reduced matrix analysis. A quantitative example is provided, in a demonstrative diagram (after Pellionisz, 1984b, see Chapter 15), of how the cerebellum may play a covariant-tocontravariant metric-type transformation for neck muscles based on the generalized inverse approach introduced by Pellionisz (1983a, 1984a,b).

The ability to calculate the transformations that are necessary for a sensorimotor integration in an overcomplete system may urge one to seek immediate conclusions regarding the actual morphological implementation of the calculated results. Thus, a word of caution seems appropriate; do not attempt the task unprepared. For instance, the pairing of the vestibular canals is not necessary when one utilizes the generalized inverse approach elaborated here. Thus, in the follow-up of this study, it would be important to treat the overcomplete six-canal vestibular system in a non-paired manner, in order to verify whether the error involved in a reduced approximate solution is, indeed, negligible.

A second reason for carefully thinking through a tensorial analysis of the system before arriving at its practical consequence is much more profound, as discussed below.

3. The need for a hierarchical analysis of nested sensorimotor systems in a study of the genesis and modification of the gaze control

The above tensorial calculation of an overcomplete sensorimotor transformation can yield the quantitative data for the three basic neuronal connectivities that explain how the VOR may implement its function, once these matrices are available. The scheme certainly did not show, however, how the necessary tensor transformation matrices can emerge through the functioning of the CNS. It was shown by Pellionisz (1984c) that the above transformation networks can, indeed, be generated and modified to perfection by the CNS if it follows certain physical procedures in a definite sequence. Such a scheme, where the interim and intrinsic expressions within the VOR are revealed, provides an opportunity to study not just the overall adaptive features of such a system, but actually to identify the sites and means of such genesis and modication. However, while the presented numerical calculation is applicable to explain the function of the VOR, the actual physical implementation of the genesis and modification of such sensorimotor geometries (Metageometries) is based on an assumption that the system is a true primary sensorimotor apparatus; but this assumption does not apply to the VOR.

3.1. Why the VOR is not a primary sensorimotor system: tensorial organization of metageometries through Eigenvectors of the covariants and contravariants belonging to an invariant

The vestibular apparatus is a sensory mechanism that, by means of measuring acceleration, keeps track of the movements of the head. The extraocular muscle system is a motor mechanism that, by means of generating an ideally equal but opposite eye movement, compensates for head movements and thus maintains the direction of the gaze in a moving head. That these sensory and motor systems work with one another, and that they are capable of exhibiting a most intriguing adaptive feature has also been thoroughly studied (see Chapters 2 & 3) (Gonshor and Melvill Jones, 1973, 1976b). Moreover, this modificationphenomenon is not restricted to one dimension only (cf., Berthoz et al., 1981a). Thus, VOR is often regarded as a sensorimotor system which is perhaps the best model for studying `learning' by the CNS. However, it is asserted here that the VOR is not, in the deepest sense, a sensorimotor system, and thus an inadequate model of the genesis and modification of the adaptive sensorimotor function.

The term `sensorimotor integration', while often utilized, is rarely defined. Tensor network theory, however, provides some means for a desired definition: sensory function was defined as a covariant vectorial expression by the CNS, of an external physical invariant, while motor function is its conlravariant vectorial expression. The transformation from one to the other is based on a metric, comprising a functional geometry (Pellionisz & Llinas 1980, 1982a,b).

Thus, if the head turn and the eye turn were identical physical objects (in perfect maintenance of gaze they are equal, albeit opposite, movements), the vestibular and the ocular parts of the `reflex' would be the covariant and corresponding contravariant vectorial expressions of the same physical invariant. In a first approximation, the VOR appears to represent both a sensory and a motor vectorial expression of a physical invariant, and thus can be taken for a sensorimotor system. However, a more careful analysis shows this to be an oversimplification.

The vestibulo-ocular reflex is not, in itself, a system that closes on the same invariant. It is not a physical necessity that the sensory vestibular reception and extraocular motor action are assigned to the same invariant; one and the same rotation. Indeed, the vestibular sensory system is incapable of providing any measure of the invariant eye turn that the extraocular muscle system generates. The adaptive feature of the VOR itself provides a demonstration that achieving such an identity is the goal of the VOR and not the basis of its operation. "

3.2. Primary sensorimotor systems of gaze

The above declassification of VOR from a primary sensorimotor system, and from a model suitable in itself for understanding adaptive mechanisms, does not mean that primary sensorimotor systems do not exist in gaze control. In fact, there is more than one such apparatus, but the VOR as a self-contained system is not one of them.

3.2.1 Vestibulo-collicular reflex (VCR) as a primary sensorimotor system expressing one invariant, the head-movement

3.2.2 Retino-ocular reflex (ROR) as a primary sensorimotor system expressing ofae invariant, the eye movement

When analyzing gaze control, the most conspicuous physical entity is the movement of the head. It is evident that there exists both a covariant sensory and a contravariant motor mechanism that are directly tied to one and the same object, the head movement. It is physically guaranteed that, in rest, any head movement that is generated by neck muscle action is the same one that is directly observable by the vestibular apparatus. In this sense, both the contravariant generation of the invariant and the covariant vectorial measurement of the same is available in the vestibulo-collicular sensorimotor reflex (cf., the scheme in Pellionisz (1984b) and in Chapter 15).

It is noteworthy that gaze-stabilization in birds occurs predominantly by means of this primary sensorimotor system of head stabilization via neck muscles, in contrast to eye stabilization in primates. Other than the evolutionary argument that primary systems must appear earlier in phylogenesis than secondary hierarchical ones, the explanation for in-flight gaze control by headstabilization (as opposed to visual stabilization) may be that the vestibular apparatus is very fast in detecting head movements, and thus the neck muscle apparatus can provide a quick response. This is in contrast to the slow retinal apparatus, which would provide the necessary control signals at a time when flying conditions could already be significantly altered. The preference of vestibular control over visual is intuitively obvious to anyone participating in rapid sport activities, such as skiing or skating.

The tensorial scheme presented in this paper could, of course, be quantitatively elaborated for a vestibulo-collicular sensorimotor system (it has already been outlined in Chapter 15), and it may be shown later how such a primary system can be generated and modified by physical CNS processes. However, it is unfortunate that presently no quantitatively precise data base is available for the motor frame of neck (and the limb-trunk) muscle systems, although for the eye comparable data have been available for more than a century (Volkmann, 1869).

There appears to be a second sensorimotor system that expresses both co- and contravariantly the same physical invariant. An eye movement, generated by the extraocular muscle system, could directly be measured by the corresponding retinal displacement of the visual image. In this sense, producing a physical object (eye movement) and detecting the same movement (slip of the retinal image) can serve as a basis on which a sensorimotor system can be generated. Thus, the retino-ocular reflex system is also a self-contained apparatus, that could not only be modeled as shown in this paper, but also physically generated by the CNS (ROR is identical to OKN).

Developing such a system is possible in a stationary head with only the eye moving, and checking, by means of retinal detection of the slip of the visual image on the retina, that the compensation is correct. A major disadvantage of this system over head-stabilization lies in the longlatency visual system. However it also yields some significant advantages over the other. The motor mechanism moves only the eye, in contrast to the much greater mass of the head, and uses only six muscles, in contrast to the more numerous neck muscles. Forthcoming tensorial analysis will quantitate what is already evident at an intuitive level; that this latter system, while slower than the first, yields greater precision.

To elaborate a tensorial scheme for this primary sensorimotor system, the retinal frame of reference must be established. While the persistence through the years of experimentally revcaling the natural coordinate systems intrinsic to the neural mechanism of visual-slip detection has already yielded a knowledge of the coordinates for olivo-cerebellar correction (Simpson et al., 1981a), the question of whether the sensory metric utilizes such frames is yet to be answered (cf., tensorial interpretation of tectum by Pellionisz in 1983a).

3.3. Hierarchical nesting of the vestibulocollicular head-stabilization and the retino-ocular eye-stabilization primary systems into the secondary gaze-stabilization system of VOR

The physical process of genesis of Metageometries requires three well-ordered consecutive procedures for each of the two primary sensorimotor systems described above. Once these six procedures are accomplished and the two independent sensorimotor mechanisms perform their function, a relatively simple seventh process, a linkage from one to the other can be established. Thus, a hierarchical hybrid of the VOR will emerge, which has the disadvantage of not being an independently developed sensorimotor system. However, given that its precursory systems are in operation, the hierarchical hybrid has the advantage of combining the speed of one with the precision of the other.

In summary, a quantitative elaboration of overcomplete transformations such as VOR is already possible without resorting to gross oversimplifications of the tensorial approach. Nevertheless, for a thorough analysis, especially of the genesis and modification of the holistic gaze system (cf., Llinas 1974; Llinas et al., 1976) much preparedness seems to be necessary, both in developing the necessary formal techniques, such as the quantitative means of computer models (cf., Ostriker et al., 1985) and in carefully thinking through the rather formidable conceptual implications.

Acknowledgements

This research was supported by USPHS Grant NS-13742 from NINCDS.

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