Functional Morphology and Neural Control of Neck Muscles in Mammals 1

Transcription

1 AMER. ZOOL., 29:13949 (1989) Functional Morphology and Neural Control of Neck Muscles in Mammals 1 BARRY W. PETERSON Department of Physiology, Northwestern University Medical School, Chicago, Illinois ANDRASJ. PELLIONISZ New York University Medical School, New York, New York JAMES F. BAKER AND EMILY A. KESHNER Northwestern University Medical School, Chicago, Illinois SYNOPSIS. Although the data described in this paper are from a traditional experimental animal, the cat, the problem addressed is applicable to many forms of motor control in a broad range of species. In general, we are trying to understand how the musculo-skeletal structure of a body component imposes constraints on how that component is controlled by the central nervous system. Our approach to this problem has been to develop a theoretical model of sensory-motor integration in complex systems where movement is produced by a large number of muscles and then to test whether that model can predict the patterns of neck muscle activity that underlie head movements. The resulting theoretical interpretation of physiological data sheds light on the organization of head movement control in the cat and also reveals a number of fundamental principles that must be considered in trying to characterize and understand any sensory-motor system. MORPHOLOGY OF THE NECK MUSCULO-SKELETAL SYSTEM The neuronal processes employed to control movement of musculo-skeletal systems such as the neck must necessarily depend on the physical structure of the system. A simple hinge joint moved by one pair of flexor and extensor muscles could be controlled by the simple equilibrium point scheme elaborated by Asatryan and Fel'dman (1965). The arrangement of the six extraocular muscles into three nearly orthogonal pairs has also made possible a very simple control scheme where each eye position is uniquely determined by the pattern of muscle activity required to maintain it (Robinson, 1982; Peterson el al., 1984). The head-neck system, however, represents a more general case where a large number of muscles acting about multiple joints control the rotation of an appendage. The first step in evaluating the complexity of head position control is to quan- 1 From the Symposium on Axial Movement Systems: Biomechanics and Neural Control presented at the Annual Meeting of the American Society of Zoologists, December 1986, at Nashville, Tennessee. 139 titate the types of skeletal movements that occur as the head is moved into a variety of orientations. Careful x-ray observations of amphibia, reptiles, birds and mammals (Vidal etal., 1986) have shown that, at rest, birds and mammals maintain the second through seventh cervical vertebrae in an earth-vertical alignment with the head inclined about the Ci-skull joint to place the horizontal semicircular canals within a few degrees of the horizontal plane. Movement from this rest position involves horizontal (yaw) rotation of the head primarily about Ci-C 2, extension-flexion (pitch) of the head about either C,-skull or C 7 -T,, or limited amounts of roll rotation about the head longitudinal axis that involve a lateral deflection of joints between C 2 and C 6. Thus movements within a normal range of motion are executed by a kinematic motor strategy that may involve 6 to 7 degrees of freedom (each representing rotation about one inter-vertebral joint axis). Passive range-of-motion studies by the same authors indicate that the neck joints can rotate about many more axes than those observed during head orienting behavior. The actual behavior, therefore, reflects the operation of some kind of a constraint in order to arrive at a set of movement pat-

2 140 B. W. PETERSON ET AL. SIDE VIEW OF CAT FRONT VIEW OF CAT TOP VIEW OF CAT \AS MUSCLE ORIGINS AND INSERTIONS VECTORS OF MUSCLE PULLING ACTION FIG. 1. Anatomical determination of pulling directions of neck muscles in the cat. Top row shows origins and insertions of 7 neck muscles as measured stereotaxically in a typical cat. Numbers on ordinates and abscissae are in mm. Cross shows the measured position of the C,-C 2 joint. Bottom row shows estimates of the torques produced by each muscle calculated in a two step procedure. First the cross product of vectors from the muscle's insertion to its origin and from the same insertion to the C,-Cj joint was computed. This vector was then normalized so that its length was 1.0 after which it was multiplied by the measured cross sectional area of the muscle to scale the torque according to muscle strength. The resulting torques are shown as vectors from which the muscle's pulling action can be appreciated by aligning the right thumb with the vector (thumb pointing away from cat) and noting the direction in which the right fingers curl around the axis indicated by the vector. terns that are a subset of the many movements possible in a structure involving as many as seven multi-directional joints. The next stage in understanding head movement control is to quantitate the actions of the individual neck muscles that move the head. Baker and Wickland (1987) have analyzed the action of fifteen muscles (on one side) which act on the head or first cervical vertebra in the cat. As shown in Figure 1, the origins and insertions of each muscle and the coordinates of the intervertebral joints were measured stereotaxically. The direction of the torque exerted by each muscle about a particular point could then be estimated by calculating the cross product of vectors extending from the muscle insertion to its origin and from the muscle insertion to the joint. Thus for a muscle like biventer cervicis with an insertion on the skull (S xyz ) and an origin on the vertebral column (V xyz ), the torque

3 KINEMATICS OF NECK MOTOR CONTROL 141 a M 3 0 = (1.15, 0.5) M 60 =(1.0, 1.0) M go = (0.5, 1.15) =(0, 1.0) FIG. 2. Illustration of the properties of the two different kinds of representations of a sensory stimulus (or movement) in non-orthogonal coordinate frames, shown in an exemplary 2-dimensional system of coordinates. A. Contravariant, or parallelogram representation which corresponds to the physical combination of muscle actions. In this example the two muscles, a and /3, are required to produce movements in the 5 directions indicated by dots. Their activations for each direction are indicated by M x, where x is the direction in degrees relative to ar's axis. Parallelograms representing summation of muscle actions are constructed geometrically to determine how much activity is required in a and /3 to produce each movement. The first number in each parenthesis represents the parallelogram component for a, the second number, the component for 0. In this non-orthogonal frame where the angle between a and /3 is 120, a must be maximally activated for a movement not along the a direction itself, but for a movement along the one o'clock direction, representing a 30 degree deviation from ar's axis-direction. Similarly, j8 shows a maximum activation not for a movement along j8's direction, but along the 3 o'clock direction. M = motor. B. The covariant, or projection-type representation of the same five movements. Such sensory-type representation corresponds to responses (to stimuli in the 5 directions indicated) of sensors whose optimal response-directions were aligned with a and 0. Their responses for each direction are indicated by S x, where x is the direction in degrees relative to as axis. In this case the responses are determined by perpendicthat it produces about a particular neck joint (J XE ) is given by T = SV x SJ, where SV and SJ are vectors having S, V and J as their endpoints. The torque vector, T, will be perpendicular to the plane formed by vectors SV and SJ and thus motion with that torque vector as its axis will lie in that plane as expected. As illustrated in the figure, the torque-directions of the 15 neck muscles about the C,-C 2 joint are very diverse and do not seem to reduce into colinear pairs or groups as in the case of eye muscles. Measurement of neck muscle pulling directions indicates that the coordinate frame of the neck motor system is nonorthogonal: muscle actions do not produce movements along directions that lie at right angles to one another. The coordinate system is also overcomplete because the 30 muscles (15 pairs) are acting to control 6 to 7 degrees of freedom of joint rotation that underlie normal head movements. Overcompleteness means that the biomechanics of the neck do not dictate a single pattern of muscle activity corresponding to each head position, rather, they permit an infinite number of solutions. The situation is analogous to an algebra problem where there are 30 unknowns and only 6 equations, which has an infinite number of solutions. MODELS OF NEURAL CONTROL IN NON-ORTHOGONAL, OVERCOMPLETE SYSTEMS In order to form realistic hypotheses about the neuronal processes that control complex motor systems such as the neck, we need models of sensory-motor transformation in non-orthogonal, overcomplete systems. Figure 2A illustrates how our intuitions, which are honed on simple orthogonal arrangements, can lead us ular projection of each of the 5 points upon the two sensor axes, which are given in the parentheses. As shown, each sensor is maximally activated when the stimulus is exactly along its own axis. S = sensory. In both A and B, numbers in parentheses give the values of activation-components of a and 8 respectively, in response to a unit-movement in the directions indicated by the dots.

4 142 B. W. PETERSON ET AL. astray even when dealing with a relatively simple, complete (2 muscles-2 degrees of freedom) system that is non-orthogonal. Intuition suggests that each muscle should be maximally activated when movement occurs along the direction in which it exerts its maximum pulling action. Figure 2A shows that in fact muscle a must be maximally activated when movement is at a 30 degree angle with its pulling direction. Similarly muscle /3 is maximally activated when movement is in the 90 degree direction, or 30 degrees from its maximum pulling direction. In 1979 Pellionisz and Llinas (1979, 1980) introduced their tensor network theory designed to represent the stages of sensory-motor transformation in non-orthogonal systems containing an arbitrarily large number of dimensions. The key element of the theory is the recognition that an external action such as a sensory stimulus or the movement that it produces can be represented in two different ways in a nonorthogonal system. Figure 2 A illustrates the parallelogram form of representation that corresponds to the way forces generated by muscles combine to generate a net force. In tensorial terminology this representation is referred to as contravariant (cf. Bickley and Gibson, 1962). In turn, Figure 2B illustrates the situation where points spaced at 30 intervals around a unit circle are represented by their perpendicular projections upon each of the coordinate axes. This type of representation corresponds to one used in a sensory system such as the vestibular semicircular canals, where each sensor responds independently to a stimulus in proportion to the projection of that stimulus upon the sensor's direction of maximal sensitivity. This representation is referred to as covariant in tensorial nomenclature. The important point for biologists seeking to understand sensorimotor transformation in a real system, where sensors and muscles form non-orthogonal coordinate systems, is that the two representations are not identical and thus neural transformations must involve conversion from covariant to contravariant representations. The tensorial approach can be used to model the transformation occurring in various neuronal networks. To illustrate its usefulness in generating hypotheses concerning neck motor programming, we will describe here a model of the vestibulocollic reflex (VCR), which is the reflex that converts head rotation, sensed by the semicircular canals of the vestibular labyrinth, into patterns of neck muscle activity that tend to move the head in the direction opposite to the sensed rotation thus stabilizing the head in space. The model (cf. Pellionisz and Peterson, 1985, 1987) treats the VCR as a three-stage transformation of a covariant vestibular sensory input to a contravariant neck motor output (Fig. 3). For simplicity in this initial model, the neck is treated as a single, 3-degree-of-freedom "joint" about which head rotations occur. Each stage of the transformation is implemented by a matrix that expresses a general tensorial operation in the particular coordinate frames of the VCR. Stage 1, the vestibular metric tensor, converts covariant signals from the three semicircular canals into contravariant form. It is implemented by a 3 x 3 matrix which is the inverse of the matrix composed of the cosines of the angles between the vestibular axes. These directions, representing the maximum sensitivity of the three semicircular canal pairs are available from quantitative morphological studies by Blanks et al. (1972) and Curthoys et al. (1977). In stage 2, sensorimotor embedding, the contravariant sensory signals are projected upon neck coordinate axes defined as the pulling directions of the 30 neck muscles (cf. Fig. 1). The 3 x 30 matrix involved is simply a table of cosines of the angles between the sensitivity axis of each of the three semicircular canal pairs and each of the 30 muscle pulling axes. Its output is a covariant representation of the VCR motor command. Stage 3, the neck metric tensor, implemented by a 30 x 30 matrix, converts this covariant motor command into the contravariant form that is required to generate a pattern of muscle activity that will physically sum in parallelogram fashion to produce the desired movement. It is at stage 3 that the problem caused by overcompleteness of the neck motor sys-

5 143 KINEMATICS OF NECK MOTOR CONTROL roll ooo I Dilcn Vestibular sensory frame 0* o O»«o * OO OO oo o O* 0 «O«..... o*o*>*«..oo- O OO i vrf:. * a JOO OO o-. Neck motor frame FIG. 3. Illustration of the tensor network model of the VCR in the cat (cf. Pellionisz and Peterson, 1985). The sensorimotor transformation occurs in three stages: gpr, sensory metric tensor; cip, sensorimotor embedding tensor; and gic, neck motor metric tensor. These general tensorial operations are expressed in the particular frames of reference of the VCR by matrices, which are here represented by patch diagrams where positive and negative components are shown by filled and open circles whose surface area is proportional to the size of the component. The CNS can implement these matrices by neuronal networks as indicated in the upper part of the diagram. There the strengths of connectivities from H, A, P canals to the 30 neck muscles are represented by line shadings (black for excitation, grey for inhibition) and by line thickness proportional to the value of the interconnecting matrix element. tem arises. As at stage 1, the matrix we require is the inverse of a matrix containing the cosines of the angles between all axes in the coordinate frame. In the case of stage 1, where the number of axes (3) is the same as the three dimensions of the external Cartesian space in which head rotations occur, the inverse can be calculated conventionally. In the case of stage 3, where the number of muscle axes is greater than 3, the 30 x 30 matrix of cosines has no unique inverse: an infinite number of solutions exist. In considering this problem, which arises for most motor systems. Pellionisz (1984) hypothesized that the CNS might choose a matrix corresponding to the Moore-Penrose generalized inverse (Albert, 1972) of the 30 x 30 array of cosines. This solution has a number of attractive features. First, it represents the equivalent of a least squares fit where unwanted muscle co-contractions are minimized thus generating efficient motor patterns. Second, it preserves a set of special directions (the eigenvectors) for which covariant and contravariant representations have the same direction in space. Third, it is possible to conceive of a process

6 144 B. W. PETERSON ET AL STIMULUS ORIENTATION STIMULUS ORIENTATION 0' STIMULUS ORIENTATION YAW PITCH ROLL 1.B8. * ] NORMALIZE ( *.78S, ) RESPONSE VECTOR TOP VIEW OF CAT FRONT VIEW OF CAT LEFT SIDE VIEW OF CAT FIG. 4. Experimental determination of rotations that maximally excite individual neck muscles. Records at top show EMG response of left bi venter cervicis muscle to rotations in 15 different planes. As indicated by the figurines, the first 5 rotations were from a series of rotations about an earth-horizontal axis that included pitch and roll rotations. Responses in 8 of these planes are graphed below the data (0 on abscissa corresponds to pitch, 90 to roll). The response size is represented as a gain, which is the ratio of amplitude of EMG response to stimulus amplitude in degrees. The middle 5 rotations and plot are for rotations about the earthvertical axis with the cat tilted left or right ear down. The right 5 rotations and plot are for rotations about the earth-vertical axis with the cat tilted nose up or down. In these two series 0 corresponds to yaw; results of these two series are averaged to obtain the muscle's response to yaw. Each plot contains a sinusoid fitted to the data by least squares. The bottom of the figure shows how responses for yaw, pitch, and roll are read off the fitted curve and combined to make a vector that is then normalized to a length of 1.0. This vector, which is shown in three views, represents the axis of the rotation that would maximally excite the muscle. (Modified from Baker et al., 1985.) by which the CNS could determine and imprint the Moore-Penrose generalized inverse in the form of a neuranal network. As described by Pellionisz and Llinas (1985) this process involves setting up tremor in which the covariant sensory input is sent directly to the motor nuclei. Oscillations will stabilize along the directions of the

7 KINEMATICS OF NECK MOTOR CONTROL 145 TOP VIEW OF CAT FRONT VIEW OF CAT LEFT SIDE VIEW OF CAT FIG. 5. Maximal response axes for 5 biventer cervicis muscles from 4 decerebrate cats. At top, responses of 3 right and 2 left biventer muscles are represented by an optimal response vector, derived as shown in Figure 4. At bottom, vectors of left muscles have been reflected to make them equivalent to right muscles so that consistency of responses can be appreciated. (From Baker et ai, 1985.) eigenvectors, which can then be sensed and used to imprint the Moore-Penrose generalized inverse in a cross-connected neuronal network. Once the Moore-Penrose generalized inverse has been chosen for stage 3, the model predicts for each neck muscle a unique direction of head rotation for which that muscle should be maximally activated. Muscle activation during rotation about other axes is predicted to decline as the cosine of the angle between those axes and the optimal axis. As in the case illustrated in Figure 2A, the predicted optimal activation direction should typically differ quite significantly from muscle pulling directions. EXPERIMENTAL TESTING OF THE MODEL The model can readily be tested experimentally. Baker et al. (1985) performed an initial series of tests in which decerebrate cats were rotated in numerous (24) different planes while recording electromyographic (EMG) activity of 7 neck muscles. Figure 4 illustrates a typical set of data and shows how such data were used to determine a direction for each muscle that represents the axis of the rotation to which that muscle responded maximally. The direction of rotation can be visualized from the vector shown at the bottom of the figure by placing one's right thumb along the vector and noting the direction in which the fingers curl around the vector axis. In this example the biventer cervicis muscle was maximally excited by rotating the animal downward and to the (animal's) right. Figure 5 shows that optimal activation vectors are consistent across animals. This fact also suggests that the CNS does choose, by some identifiable constraint and not by a random selection, a single solution from among the infinite number of possible motor patterns that could be used in gen-

8 146 B. W. PETERSON ET AL. " FIG. 6. Comparison of pulling directions (Top row), maximal activation directions (Middle row) and model predictions of maximal activation directions (Bottom row) for six neck muscles. Pulling directions correspond to vectors shown in Figure 1. Activation directions and model predictions are shown in mirror image (i.e., multiplied by ) to facilitate comparison with pulling directions. erating the VCR. If such a choice were not made, we would expect that each muscle would exhibit different optimal activation directions in each animal reflecting the many possible ways of programming head movements. Figure 6 shows the pulling and activation directions for the 6 muscles tested together with model predictions of activation directions for the same muscles. Although pulling and activation directions are quite widely separated in this non-orthogonal system, the model predicts the activation directions within 4 to 11 degrees. Thus the hypothesis that the CNS determines neck muscle activation patterns in a manner corresponding to the Moore-Penrose generalized inverse is supported by the fact that the model predicts the pattern of muscle activity, produced by the VCR in decerebrate cats, within the limitations of experimental error. The success of the model in predicting motor patterns underlying the VCR encouraged us to examine whether the same principles would also predict muscle activity patterns during more complex neck motor behaviors. The behavior we chose was voluntary tracking movements of the head. We trained alert cats to follow a water tube that rotated about their head in 24 different planes (Keshner et al., 1986). The head rotations were slow (0.25 Hz) sinusoids with an amplitude of about 30 degrees and were produced by sinusoidal modulation of neck EMG activity as shown in Figure 7. The data obtained could thus be processed in a way analogous to VCR data to determine for each muscle the axis of voluntary rotation that was accompanied by maximum modulation of that muscle's activity. The same alert cats were also exposed to whole body rotation, thereby eliciting a VCR, so that direct comparisons between tracking and VCR activation could be made in the same animal. As illustrated in Figure 8, we found significant differences between patterns of muscle activation utilized during the VCR and tracking movements. We also observed that while each individual animal always used a particular pattern of muscle activation when making tracking head movements, the patterns varied much more between animals than did patterns underlying the VCR. The data thus suggest that the model shown in Figure 3 cannot alone account for muscle activation patterns produced during voluntary tracking movements of the head. These movements, indeed, involve more than just the VCR. The components include: 1) voluntary sensory-motor pathways in which visual and somatosensory (vibrissal) cues are converted into intended head movements, and 2) feedback pathways in which vestibular and neck proprioceptive signals elicited by those head movements generate reflexes that modulate the movement in a closed loop fashion. Further experimentation is required in which each subsystem involved is examined individually to determine the departures from the muscle activation pattern which is predicted by the VCR reflex model alone. Possible theoretical reasons for such departures include selection of a subset of muscles to execute one compo-

9 KINEMATICS OF NECK MOTOR CONTROL 147 LEFT RECTUS MAJOR YAWED PITCH TRACKING VCR 30 deg TIME 25 FIG. 7. Activation of left rectus major muscle in an alert cat during sinusoidal whole body rotation (right traces) and sinusoidal head tracking with body fixed (left traces). On the far right of the figure, the solid lines with curved arrows indicate the axis of rotation for that set of body or head rotation records. This series of rotations includes pitch (0 ) and roll (90 ). EMG responses are sinusoidally modulated in both tasks but the null plane where the response grows small and reverses direction clearly differs for muscle activation during body rotation (VCR) and tracking. nent of the motor behavior, or the use of a different generalized inverse in stage 3 for one (or more) of the systems involved in complex tracking behavior (Peterson and Pellionisz, 1986). CONCLUSIONS The modeling approach described here is applicable to a broad range of neural control problems. Software is now available to construct similar models for any sensory-motor system where the geometry of sensors and muscles is made available by quantitative anatomical studies. Thus many investigators may wish to determine if the behavior they are studying can be predicted by a simple tensorial model. Gielen and Zuylen (1986) have recently reported

10 148 B. W. PETERSON ET AL. LEFT RECTUS MAJOR YAWED PITCH ORIENTATION Fie. 8. Plots of amplitude (gain) and phase of activation of same muscle in the same animal shown in Figure 7 graphed as a function of the orientation of the rotation axis for voluntary tracking (+) and whole body rotation (O). A difference in the null plane (80 for body rotation, 120 for tracking) as well as the maximal plane is evident. For tracking responses EMG was in phase with position of the water tube, for reflex responses with velocity of body rotation. successful prediction of patterns of human arm muscle activation using a tensorial model. The experimental approach of recording EMG responses to multi-directional stimuli is also broadly applicable and could yield useful information about principles underlying motor control in a variety of species. Our experience with the feline neck motor control system suggests that the model incorporating the Moore- Penrose generalized inverse can predict experimental data obtained from multidirectional testing of relatively simple sensorimotor systems such as the vestibulocollic reflex. More complex models appear to be required to account for neural programming observed in more complex systems such as the voluntary head tracking system. REFERENCES Albert.A Regression and the Moore-Penrose pseudomverse. Academic Press, New York. Asatryan, D. G. and A. G. Fel'dman Functional tuning of the nervous system with control of movement or maintenance of posture. I. Mechanographic analysis of the work of the joint on execution of a postural task. Biophysics 11: Baker, J., J. Goldberg, and B. W. Peterson Spatial and temporal response properties of the vestibulocollic reflex in decerebrate cats. J. Neurophysiol. 54: Baker, J. and C. Wickland Kinematic properties of the vestibulocollic reflex. In B. W. Peterson and F. J. Richmond (eds.), Control of head movement. Oxford Univ. Press, New York. (In press) Bickley, W. G. and R. E. Gibson Via vector to tensor. John Wiley & Sons, New York. Blanks, R., I. Curthoys, and C. Markham Planar relationships of semicircular canals in the cat. Amer. J. Physiol. 223: Curthoys, I. S., R. H. I. Blanks, and C. H. Markham Semicircular canal functional anatomy in cat, guinea pig and man. Acta Otolaryngol. 83: Gielen.C.C. A. M. ande.j. van Zuylen Coordination of arm muscles during flexion and supination: Application of the tensor analysis approach. Neuroscience 17: Keshner, E. A., J. Baker,J. Banovetz, B. W. Peterson, and C. Wickland Neck muscles demonstrate preferential activation during voluntary and reflex movements in the cat. Soc. Neurosci. Abstr. 12:684. Pellionisz, A Coordination: A vector-matrix description of transformations of overcomplete CNS coordinates and a tensorial solution using the Moore-Penrose generalized inverse. J. Theoret. Biol. 110: Pellionisz, A. and R. Llinas Brain modeling by tensor network theory and computer simulation. The cerebellum: Distributed processor for predictive coordination. Neuroscience 4: Pellionisz, A. and R. Llinas Tensorial approach to the geometry of brain function. Cerebellar coordination via a metric tensor. Neuroscience 5: Pellionisz, A. and R. Llinas Tensor network theory of the Metaorganization of functional geometries in the CNS. Neuroscience 16: Pellionisz, A. and B. W. Peterson Tensor models of primary sensorimotor systems, such as the vestibulo-collic reflex (VCR) and of the metaorganization of hierarchically connected networks. Soc. Neurosci. Abstr. 11:83. Pellionisz, A. and B. W. Peterson A tensorial model of neck motor activation. In B. W. Peterson and F. J. Richmond (eds.), Control of head

11 KINEMATICS OF NECK MOTOR CONTROL 149 movement. Oxford Univ. Press, New York. (In press) Peterson, B., J. Baker, J. Goldberg, and C. Wickland Kinematic organization of the cat vestibuloocular reflex (VOR). Soc. Neurosci. Abstr. 10:162. Peterson, B., J. Baker, C. Wickland, and A. Pellionisz Relation between pulling directions of neck muscles and their activation by the vestibulocollic reflex: Tests of a tensorial model. Soc. Neurosci. Abstr. 11:83. Peterson, B. W. and A. J. Pellionisz A tensorial model of the kinematics of head movement in the cat. Soc. Neurosci. Abstr. 12:684. Robinson, D. A The use of matrices in analyzing the three-dimensional behavior of the vestibulo-ocular reflex. Biol. Cybern. 46: Vidal, P. P., W. Graf, and A. Berthoz The orientation of the cervical vertebral column in unrestrained awake animals. I. Resting position. Exp. Brain Res. 61: