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Departments of Physiology and Applied Mathematics, University of Western Ontario, London, Ontario N6A 5C1, Canada; and Department of Neurology, University of Tübingen, Tübingen 72076, Germany
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ABSTRACT |
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 Abstract
 Introduction
Discussion
References
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Tweed, Douglas. Three-dimensional model of the human eye-head saccadic system. J. Neurophysiol. 77: 654-666, 1997. Current theories of eye-head gaze shifts deal only with one-dimensional motion, and do not touch on three-dimensional (3-D) issues such as curvature and Donders' laws. I show that recent 3-D data can be explained by a model based on ideas that are well established from one-dimensional studies, with just one new assumption: that the eye is driven toward a 3-D orientation in space that has been chosen so that Listing's law of the eye in head will hold when the eye-head movement is complete. As in previous, one-dimensional models, the eye and head are feedback-guided and the commands specifying desired eye position eye pass through a neural "saturation" so as to stay within the effective oculomotor range. The model correctly predicts the complex, 3-D trajectories of the head, eye in space, and eye in head in a variety of saccade tasks. And when it moves repeatedly to the same target, varying the contributions of eye and head, the model lands in different eye-in-space positions, but these positions differ only in their cyclotorsion about the line of sight, so they all point that line at the target
a behavior also seen in real eye-head saccades. Between movements the model obeys Listing's law of the eye in head and Donders' law of the head on torso, but during certain gaze shifts involving large torsional head movements, it shows marked, 8° deviations from Listing's law. These deviations are the most important untested predictions of the theory. Their experimental refutation would sink the model, whereas confirmation would strongly support its central claim that the eye moves toward a 3-D position in space chosen to obey Listing's law and, therefore, that a Listing operator exists upstream from the eye pulse generator.
In most large gaze shifts, eye and head work together to bring the line of sight swiftly onto its new target. Because of the complex, fast and accurate coordination involved, these movements Superior colliculus codes gaze shifts in space
Electrically stimulating the SC in a cat with its head free evokes an eye-head gaze shift (Guitton 1992 Eye is driven to a position in space
There is convincing evidence that both eye and head are driven to their desired positions by feedback in the form of efference copy and, likely, vestibular signals (Becker et al. 1981 VOR is shut off in one direction
During large saccades, the VOR is weakened or shut off, gradually turning back on as gaze error becomes small and becoming fully operational again during the final, VOR stage of the movement, when the gaze is on target but the head is still completing its motion (Guitton and Volle 1987 Desired eye position is saturated
Guitton and Volle (1987) Donders' and Listing's laws
Imagine a rotating object whose position is expressed using quaternion vectors. We shall say that the object follows Donders' law if its quaternion vector remains within a 2-D surface (Donders 1848 The map in the burst cell layer of the SC specifies the initial target direction relative to the eye, T0e. In the buildup cell layer of the SC, a comparator driven by feedback signals coding head velocity (from the semicircular canals) and eye position and velocity (from efference copy), computes dynamic target direction, Te, as it changes throughout the saccade. This signal, which is 2-D, must be converted into 3-D head and eye rotations. The operations involved in this conversion are shown in flow diagram form in Fig. 1; for simplicity, the computation of dynamic target direction Te is not shown in the flow diagram but takes place off the left-hand side of the figure.
Saturation
In the model, saturation of desired eye position operates three-dimensionally: horizontally, vertically, and torsionally, a generalization of Guitton and Volle's (1987) one-dimensional scheme which raises some new issues. In one dimension, saturation is a simple matter of "clipping" desired eye position signals larger than some limit, say 40°. In 3-D, saturation involves projecting overly eccentric desired eye position vectors into some central, 3-D subvolume of the oculomotor range, called the effective oculomotor range (EOMR). We know very little about the 3-D shape of the EOMR. Ellipsoids or cylinders seem the likeliest possibilities, so in the simulations here the EOMR is a compromise between these two shapes (see Fig. 2): it is a clipped ellipsoid, or pill-shape, 80° in diameter in the horizontal-vertical plane and 16° thick in the torsional dimension in the center, a value chosen because it is roughly the torsional eye position range seen during roll VOR (Misslisch et al. 1994a
Gaze point and facing direction
We start the model on a relatively simple task: predicting gaze paths. This is simpler than predicting eye and head rotations, because gaze paths are 2-D whereas rotations are 3-D. Figure 3 is a computer simulation showing the predicted paths traced by the gaze point as it moves across a large spherical viewing screen centered on the subject. The simulated subject makes 10 saccades: the first moves from center to 70 deg up and left; the next 8 saccades take the eye first counterclockwise (CCW) and then clockwise (CW) around a square Eye-in-space trajectories
Simulated oblique, eye-head saccades in all four quadrants are shown in Fig. 4, which plots eye-in-space orientations, depicted as quaternion vectors and viewed from behind so that the horizontal and vertical components of the motion are seen. A complex pattern of loops appears: centrifugal and centripetal saccades form CCW loops in the first and third quadrants, and CW loops in the other two quadrants. This pattern is echoed in the data shown at the bottom of the figure and indeed was present in all subjects tested by Tweed et al. (1995)
Isogaze curves
A central feature of the model is the way internal variability is manifested in its overt behavior and, in particular, in its final gaze direction and eye-in-space position. Recall that any one gaze direction can be achieved using infinitely many distinct eye-in-space positions, differing only in their orientations about the line of sight. If we plot all the eye-in-space quaternion vectors corresponding to one gaze direction, they form a curve in 3-D quaternion vector space, an isogaze curve. When the model makes a number of saccades between center and a fixed, eccentric target (e.g., the down-and-right target DR), varying the contributions of eye and head each time, then it lands in different eye-in-space positions, but all these positions lie on the same isogaze curve, so the gaze line nevertheless reaches the target every time.
Eye-in-head trajectories
Simulated eye-in-head trajectories for centrifugal and centripetal saccades are plotted in Fig. 7. Eight saccades are shown in all, moving between center and the targets from Fig. 3, oblique targets at 70° eccentricity. Centrifugal movements are plotted in thin lines and centripetal in thick. These simulated saccades show the features seen in the human data of Glenn and Vilis (1992) The model is good at predicting the complex, curving paths of head-in-space, eye-in-head, and eye-in-space, even though it was not devised with these 3-D trajectories in mind but was built from a handful of "axioms," all with clear functional justification: 1) the VOR is switched off in the direction of eye-in-head error because its gaze-stabilizing function is counterproductive during saccades; 2) eye and head are driven by internal and vestibular feedback for speed and accuracy; 3) desired eye-in-head position is saturated to avoid driving the eye to its mechanical limits; 4) the eye moves toward its desired position in space, rather than its final position in the head, to bring the gaze line to the target as fast as possible; and 5) this desired eye position in space is chosen so that Listing's law of the eye in head will hold at the end of the head motion. As discussed above, the first four axioms are inherited from previous models and are well confirmed by 1- and 2-D eye-head data (Galiana and Guitton 1992 Trajectories, saturation and Donders' law
The looping paths in Figs. 3 and 4 occur, at least in the model, because the eye is bearing for a desired position in space, q*es, but is deflected systematically by the saturation function. As described earlier, saturation prevents the eye leaving the oculomotor range; i.e., if q*es is so far eccentric, relative to the head, that the current desired eye position in the head, q+eh, lies outside the effective oculomotor range, EOMR, then the brain computes a saturated desired eye position, qseh, which is the position within the EOMR where it will first be possible to foveate the target. This saturated position depends on how the head is going to move, because it is head movement that causes q+eh to move toward the EOMR. Because the head moves more horizontally than vertically in most subjects (Glenn and Vilis 1992 Site of the Listing operator
The central new claim of the model is that the eye is driven to a position that has been chosen to fit Listing's law, and therefore this position must be set by a Listing operator upstream from the comparator that computes eye-in-head motor error. Theoretically, this is not the only possible way to implement Listing's law. For example, if the oculomotor integrator is leaky in torsion, as suggested by Seidman et al. (1995) Further predictions and extensions
1) According to the model, head-only saccades, in which the eyes stay fixed in the head, should be impossible; i.e., when subjects attempt head-only saccades, their eyes should still shoot toward the target and then roll back during the VOR stage of the movement. The reason is that the eye is driven toward a target in space and therefore cannot be ordered to stay still in the head.
INTRODUCTION
Abstract
Introduction
Discussion
References
called eye-head saccadeshave attracted the attention of theorists (Galiana and Guitton 1992
; Galiana et al. 1992; Guitton et al. 1990; Laurutis and Robinson 1986; Tomlinson and Bahra 1990), but current models of eye-head control deal with motion in only one dimension and do not touch on the new issues that arise in three dimensions, such as curvature, noncommutativity and Donders' and Listing's laws.
we can look toward auditory, tactile, remembered or olfactory targetsbut there is evidence that auditory targets, at least, are converted to retinal coordinates within the brain and are stored, together with visual targets, in a single map in the superior colliculus, SC (Jay and Sparks 1984), a map of target direction relative to the eye (Robinson 1972; Schiller and Stryker 1972; Sparks and Mays 1980; Van Opstal et al. 1991). In this paper, therefore, I assume that the input to the saccadic system is always a2-D signal coding target direction in eye coordinates.
; Radau et al. 1994; von Helmholtz 1867) and Donders' law of the head on torso (Radau et al. 1994). I shall argue that recent data on eye-head gaze shifts can be explained, and fundamental new predictions can be made, by a model that introduces just one new assumption, and that is otherwise based on ideas that are well established from one-dimensional studies. The new assumption is that the eye is driven toward a 3-D orientation in space that has been chosen so that Listing's law of the eye in head will hold when the eye-head movement is complete. Before presenting this3-D model and its predictions, I review the experimental data and previous models on which it is built.
BACKGROUND
; Guitton and Volle 1987; Guitton et al. 1990; Roucoux et al. 1980). Repeated stimulation of a single site causes repeated gaze shifts of approximately the same size and direction, but the contributions of eye and head may vary, so that for one gaze shift, the eye may move a lot and the head only a little, for another the reverse. And similar observations have recently been made in monkeys (Robinson and Cowie 1993). These findings suggest that there is a map within the SC that specifies the desired movement of the gaze line or, in other words, gaze error. Regarding terminology, note that if gaze error is specified in eye coordinates, as we shall assume, then it is the same thing as target direction in eye coordinates, which is equivalent, in turn, to retinal error (if the lights are on and the target in within visual range); e.g., if retinal error is 20° right, then the target direction relative to the eye is also 20° right, as is the gaze shift, in eye coordinates, required to foveate the target.
, 1983). Another group of SC neurons, the buildup cells, may code the evolving, or "dynamic," gaze error throughout the movement (Munoz and Wurtz 1995a,b), but the existence of dynamic gaze error signals and their role in saccade generation are controversial. Following Galiana et al. (1992), the model in this paper drives the eye and head with a dynamic gaze error signal computed within the SC, but it is important to note that the experimental evidence for this is inconclusive, and that the crux of the modeldriving the eye to a 3-D orientation in space chosen so that the eye's final orientation in the head satisfies Listing's lawis independent of this question.
; Guitton and Volle 1987; Guitton et al. 1990; Laurutis and Robinson 1986; Pélisson et al. 1989). Further, it is known that the eye is driven toward its desired position in space, not its desired final position in the head. For example, if the visual target is 65° right in space-fixed coordinates, then at the end of the movement, the head will be ~50° right and the eye 15° right in the head. But during the saccade, the eye does not simply move to its final position in the head, 15° right, and wait there for the head to finish its motion. Instead, it shoots well past 15° until the gaze line hits the target, and then, under the influence of the vestibuloocular reflex (VOR), it rotates back to 15° right as the head completes its motion (Guitton and Volle 1987; Laurutis and Robinson 1986; Tweed et al. 1995). This way, the gaze line reaches the target earlier, before the head finishes its motion.
; Laurutis and Robinson 1986; Roucoux et al. 1981; Tomlinson and Bahra 1986; Tweed et al. 1995). As Laurutis and Robinson pointed out, it is sensible to switch the VOR off during such saccades, because its function is to hold the gaze stable in space and that function is not desirable during a saccade. Optimally, the brain should shut off the VOR in the direction of the saccade but leave it on in other directions, because only head movements in the direction of the saccade should affect the motion of the eye in space. This design specification is at least roughly implemented in the actual VOR, which switches off in the direction of the saccade (Guitton and Volle 1987; Laurutis and Robinson 1986; Tomlinson and Bahra 1986), but remains on in the opposite (Pélisson and Prablanc 1986; Pélisson et al. 1988) and orthogonal directions (Tomlinson and Bahra 1986).
; Henn et al. 1992; Misslisch et al. 1994a). This means that, if the eye is to obey Listing's law when the eye-head gaze shift is over, it must not be in Listing's plane when the gaze line hits the target, because then the VOR would just drag it back out again. Instead, the eye saccadic system must place the eye in a position, outside Listing's plane, that has been chosen carefully so that the subsequent action of the VOR brings it into the plane. Further, this must happen despite large variations in the size and duration of the VOR stage (Fuller 1992; Tweed et al. 1995). As stated above, the model achieves this by driving the eye toward a desired final position in space that has been chosen such that eye position in the head will fit Listing's law when the head reaches its desired position.
showed that the command driving the eye saccade passes through a neural "saturation box" before reaching the saccadic pulse generator, so that a desired eye position command of 90° right, for example, would emerge as ~40° right. Saturation prevents the eye from running at full speed to the end of its leash, like the dog in the Bugs Bunny cartoon, a maneuver that might overstretch the muscles or damage the globe. Further, saturation prevents errors that might arise if the muscles were given commands they could not execute. But note that saturation, if it is to serve these purposes, must act on signals coding desired eye position relative to the head, not desired gaze or eye movement. For example, if the desired gaze movement is 60° to the right, is saturation required? Yes, if the eye is, say, 30° right, because then the desired eye-in-head position is 30° + 60 deg = 90° right, which is uncomfortably eccentric. No, if the eye is 30° left, because then the desired eye-in-head position is only 60° 
30° = 30° right. This shows that the saturation operator cannot act on retinal error or on commands coding only the desired movement of the eye but requires signals coding absolute eye position in the head. In the model, the input to the saturation function is a desired eye-in-head position signal, computed within or downstream from the SC.
). If this surface is flat,i.e., if it is a planethen the object also obeys Listing's law (von Helmholtz 1867; Westheimer 1957). It is known that eye-in-head positions obey Listing' law, with an inaccuracy of only a degree or so, during and between head-fixed saccades and between eye-head saccades (Ferman et al. 1987; Radau et al. 1994; Tweed and Vilis 1990; von Helmholtz 1867). Head positions obey Donders' law during spontaneous eye-head gaze shifts (Tweed and Vilis 1992), and recent studies have shown that the law applies much better to head-on-torso motion than to head-in-space (Misslisch et al. 1994b; Radau et al. 1994). Donders' law of the head fails during repeated gaze shifts between two targets (Tweed and Vilis 1992), and of course this law can be repealed voluntarily, i.e., you have the power to cock your head into almost any 3-D orientation you wish. This shows that Donders' law of the head is a default rule that can be altered or overridden.
MODEL
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FIG. 1.
Three-dimensional model of human eye-head saccadic system. Te is an internal estimate of current target direction relative to the eye, a 2-dimensional (2-D) variable that is updated using feedback signals coding eye and head motion; this updating is not shown in flow diagram, but is defined by Eq. A1.As described in Eqs. A2 and A3, Te is converted into Ts, target direction in space, a 2-D signal that passes through Donders operator (Eqs. A4 and A5) to yield a desired 3-dimensional (3-D) head position, q*h, which obeys Donders' law unless that law is voluntarily overridden. Inside head pulse generator, Ph, q*h is compared with a feedback signal coding actual head position, qh, to yield head velocity command 
h (Eqs. A6 and A7) which, together with any perturbations coming from outside, determines rotational head velocity 
hsh. At same time, desired gaze direction and desired head position pass through Listing operator to yield a 3-D desired eye-in-head position, q*eh, obeying Listing's law (Eq. A8). This signal interacts with desired head position to yield desired eye position in space q*es (Eq. A9a), which then combines with actual head position qh to yield current desired eye-in-head position, q+eh (Eq. A9b). This passes through saturation box, Sat, emerging as saturated current desired eye-in-head position, qseh (see Fig. 2 and Eqs. A9c and A10), which then travels to eye pulse generator, Pe, where it is compared with a feedback signal coding actual eye position qeh to yield a saccadic eye velocity signal which sums with vestibuloocular reflex (VOR) to generate eye-in-head velocity eh (Eqs. A11 and A12). VOR is switched off in direction of eye-in-head motor error, as described in Eq. A13.
; Tweed and Vilis 1992; Tweed et al. 1995). For the simulations in Figs. 3-5 and 7, the operator is set so that head movement contributes ~80% of horizontal gaze shifts and ~50% of vertical gaze shifts, because these values are typical for normal subjects (Glenn and Vilis 1992). Figure 6 shows what happens when the Donders operator is altered, changing the contribution of the head (for mathematical details of this adjustment, see the APPENDIX). Figure 8 gives an example where the Donders operator is overridden.
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FIG. 3.
Model predictions of gaze point and facing direction trajectories for saccades to targets at 70° eccentricity on the 45° oblique meridians. Gaze point traces are plotted as thin lines, facing direction traces are thick.
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FIG. 7.
Predicted paths of eye in head during oblique saccades to and from 70° eccentric targets defined in Fig. 3. Four centrifugal movements are drawn as thin lines, 4 centripetal as thick lines. Trajectories overshoot desired final position of the eye in head, q*eh. Initial motion matches neither direction of visual target (white arrow) nor that of q*eh (black arrow).
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FIG. 6.
Internal variability in model leads to different eye-in-space positions, but does not degrade gaze accuracy. Predicted final eye-in-space positions for 9 eye-head saccades to each of 4 targets, 90° eccentric along oblique meridians, up and left (UL), up and right (UR), down and left (DL), and down and right (DR). Nine saccades to each target differ in contributions made by head and eye. Different contributions yield different final positions of eye in space, but eye-in-space vector continues to lie on the same isogaze curve, which is curve containing all eye-in-space quaternion vectors that aim gaze line at same target. If it seems confusing that isogaze curves are not parallel with abscissa, recall that different positions along an isogaze curve differ in their angle of rotation about line of sight, and in quaternion coordinates such rotations do not correspond to translations parallel with abscissa.
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FIG. 8.
Large, transient deviations from Listing's law are predicted for eye-head saccades involving head torsion. Here, a simulated subject begins with head tilted 30° CCW and ends with head 30° CW. Eye-in-head position is same at beginning and end of gaze shift 10° up and lying in Listing's planebut en route, it loops 8° CW out of the plane.
h, that drives the neck muscles until the head movement is complete. Note that this velocity command, h, is quaternion velocity, i.e., it is the rate of change of the head position quaternion; it is not the angular velocity vector of the head.
), to yield a saccadic eye velocity command which, together with the VOR, determines eye-in-head velocity eh. The equations for this flow diagram are given in the APPENDIX.
).
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FIG. 2.
Three-dimensional saturation. In model, when current desired eye-in-head position, q+eh, lies outside effective oculomotor range, EOMR, eye is instead aimed at a saturated desired position, qseh, which is calculated as shown here. First, in plane containing the horizontal and vertical components of quaternion vectors, q+eh is projected along straight line joining it to q*eh, desired final position of the eye in head (bottom). Because head motion normally causes q+eh to move along a more or less straight path in quaternion vector space, this straight projection line actually is predicted path of q+eh. Its intersection with boundary of EOMR defines horizontal and vertical components of saturated desired eye-in-head position, qseh. If torsional component of projected point lies outside EOMR, it is projected along dotted line parallel with torsional, or ql, axis until boundary of EOMR is reached, thereby defining torsional component of qseh (top). Strictly, qseh should be projected along an isogaze line (see Fig. 6) rather than dotted line, but projection shown here is a reasonable approximation and yields a simpler formula (Eq. A10).
suggested that the best strategy would be to project in such a way that qseh is the point where q+eh (which is continuously changing because of the head's motion) is predicted to enter the EOMR. This strategy would get the eye-in-space to its final position at the earliest possible moment and with the least wasted eye-in-head motion. But it turns out that, in some cases when the head is prevented from completing its motion, this strategy could cause the gaze line to fall short of the target.
suggested driving the eye to the position within the EOMR where the target will first be foveable with the correct eye-in-space torsion, the present model drives the eye to the position in the EOMR where the target will first be foveable, period. I should add, however, that for normal, unperturbed gaze shifts, there is little difference between the saccades produced by the two saturation schemes and, in particular, the patterns in Figs. 3-8 below are insensitive to the details of the saturation.
SIMULATIONS
where CW and CCW are defined from the subject's viewpointand the tenth saccade returns to center. Thicker lines depict the paths of the head's facing direction, which can be thought of as a spot cast by a laser attached to the subject's nose, pointing straight ahead when the head is in its reference position. These simulated paths match at least five key characteristics of the 3-D data collected by Glenn and Vilis (1992) and Tweed et al. (1995). 1) The box traced by the facing direction is wider than itis tall, meaning that the head contributes more to horizontal than to vertical gaze shifts. This feature was explicitly built into the model (for details see Eq. A5) to mimic Glenn and Vilis's (1992) data, so its confirmation doesn't provide any new support for the model, but the remaining four points are genuine predictions, confirmed by the data in Tweed et al. (1995). 2) Horizontal paths of both the gaze point and facing direction are roughly straight, but vertical paths bow out like the sides of a barrel. 3) The subtle figure-eights in the horizontal gaze point paths in Fig. 3 also are seen in all human subjects. 4) Oblique paths of the facing direction are straight. 5) Oblique paths of the gaze point are not straight, and their curvature changes with the direction of the gaze shift, so that centrifugal and centripetal saccades into this quadrant trace out a CCW loop. The mechanisms behind these curving paths, and behind the similar patterns we shall see in plots of 3-D eye-in-space position, are dealt with in the DISCUSSION.
. But although the qualitative looping pattern is the same, data and model do not agree perfectly on the detailed paths taken by the eye. In particular, centripetal saccades usually are curved more in the data than in the model. Reasons for this discrepancy, as well as for the correct prediction of systematic looping, are covered in the DISCUSSION.
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FIG. 4.
Loops in eye-in-space trajectories. Predicted (top) and actual (bottom) trajectories of eye-in-space during oblique saccades between center and 4 eccentric targets shown in Fig. 3, with horizontal and vertical components of paths shown. Thin lines are centrifugal traces, thick lines centripetal. Data plot reproduced from Tweed et al. 1995 .
.
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FIG. 5.
Looping in torsional dimension. Predicted (top) and actual (bottom) trajectories of eye-in-space during horizontal saccades between eccentric targets in Fig. 3. Saccades are at different elevations, between targets down and left (DL) and down and right (DR) and between up and left (UL) and up and right (UR). Torsional and horizontal components of paths are shown. Thin lines are leftward saccades, thick rightward. Data plot reproduced from Tweed et al. 1995 .
.
and Tweed et al. (1995), e.g., the eye overshoots its desired final position in the head, q*eh, and then glides back. How far and in what direction the eye overshoots depends on how much the head contributes to the saccade; in this simulation, as in most human subjects, the head contributes more to horizontal than to vertical gaze motion, and so the overshoot is largely horizontal, as seen by Tweed et al. (1995).
.
). We return to this simulation, which is the central untested prediction of the model, in the DISCUSSION.
DISCUSSION
Abstract
Introduction
Discussion
References
; Guitton and Volle 1987; Guitton et al. 1990; Laurutis and Robinson 1986; Pélisson and Prablanc 1986; Pélisson et al. 1988; Tomlinson and Bahra 1986). The fifth will stand or fall based on its key prediction that the eye-in-head should leave Listing's plane during eye-head saccades (see Site of the Listing operator, below). In what follows, I discuss some issues arising from the simulations in Figs. 3-8 and describe some further predictions of the model.
; Tweed et al. 1994), qseh lies on a more vertical meridian than the unsaturated target, q+eh (see Fig. 2). In other words, the eye initially moves more vertically than the direction of the target because it predicts that the head will do much of the horizontal work. It may seem like a lot of work to predict where the target will enter foveation range, but Eq. A10 shows that a rather simple algorithm does the job.
), so any model would need personalized parameter settings to mimic individuals' patterns. Another problem is that in the model, curvature is mostly due to the saturation function, whereas in reality, additional factors are at work. For instance real subjects show systematic curves even during head-fixed saccades (Bains et al. 1992), where the current desired eye position is always inside the EOMR, and so saturation should play no role, or at best a small role if we imagine a "soft" saturation. But even if saturation is not the whole explanation for curved saccade paths, Figs. 3 and 4 suggest that it does play an important role, because the 3-D saturation function in Fig. 2, designed on purely functional grounds, predicts the qualitative pattern that is shared by all subjects: CCW loops in the first and third quadrants, CW loops in the second and fourth. This same saturation mechanism explains why the eye-in-head paths in Fig. 6 start out on a path intermediate between the direction of the visual target and the final position of the eye in head:the saturated target, qseh, lies on a path between q+ehand q*eh.
), an arrangement which produces the kind of twisted, nonplanar distribution of static positions seen in Fig. 5. This arrangement is also responsible for the barrel-shaped gaze paths in Fig. 3: with Fick gimbals, horizontal paths are flat, following lines of latitude, whereas vertical paths bulge out, tracing lines of longitude.
).
, eye positions gradually will leak into Listing's plane, thereby restoring Listing's law whenever it is broken. However, it is clear that humans and monkeys do not rely on this mechanism for Listing's law because when supine subjects are rotated in roll, they make torsional quick phases to return to Listing's plane (Crawford and Vilis 1991; Seidman et al. 1995); i.e., they do not wait for a leaky integrator or any other slow mechanism to restore Listing's law, rather they make saccades aimed at Listing's plane. Clearly, then, there is a saccadic mechanism for Listing's law. This mechanism appears in the model as the Listing operator in Fig. 1, but that operator goes one step further: during eye-head saccades, it aims the eye, not at a position fitting Listing's law, but at a position that will fit Listing's law when the entire eye-head movement is complete. This predictive feature can explain Crawford and Vilis's (1991) observation that monkeys, when they are rotated so that their VOR slow phases carry them out of Listing's plane, make quick phases that carry them well past the plane.
observed the predicted effects in human eye-head saccades, but the observed and predicted departures from Listing's plane were small for the saccade tasks they recorded. A more dramatic test will be to have a subject perform the gaze shift that is simulated in Fig. 8. During this movement, the head rotates purely CW, so in the middle of the saccade, the desired eye position in space is rotated far CW relative to the head. As a result, the eye is driven to the CW torsional boundary of the EOMR, a full 8 deg out of Listing's plane. A complicating factor, neglected for simplicity in the simulation, is that Listing's plane moves in the head depending on head orientation (Haslwanter et al. 1992), so a 60° CW head rotation, as in this simulation, shifts Listing's plane ~5° CCW. Therefore the actual prediction is a large CW motion, still reaching the forward boundary of the EOMR, but then looping back to ~5° behind the starting position. If experimental data bear out this prediction, they will provide direct support for the fifth axiom of the model, which implies a Listing operator upstream from the eye pulse generator.
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ACKNOWLEDGEMENTS |
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I thank S. Au, M. Fetter, T. Haslwanter, H. Misslisch, and T. Vilis for valuable comments.
This study was supported by the Medical Research Council of Canada Grant MT-12847.
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APPENDIX: MODEL EQUATIONS |
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Mathematical representations
For computational convenience, I express the model equations using quaternions, vectors, and matrices. There is no evidence that the brain uses these representations, but then there is no evidence that it uses any other standard notation either. Probably the brain uses some exotic representations, as yet unknown to mathematicians. In this paper, however, I am concerned primarily not with how the computations underlying gaze shifts are implemented neurally, but with expressing clearly what I think the computations are, so I shall stick to standard, quaternion-vector-matrix notation. [For readers unfamiliar with the relevant 3-D geometry, the necessary background is given in Brand (1948), or any kinematics text with a section on quaternions or Clifford algebra.] Formulas in this paper use arithmetic operations, such as multiplication and square root finding, that are somewhat unusual in oculomotor models but this is no cause for concern. It is clear that individual neurons are versatile information-processing devices (see e.g., Poggio and Torre 1978), and even if they were not, work on artificial neural networks has shown that assemblies of very simple elements can perform arbitrarily complex mathematical operations. And finally, I have no doubt that the model described here can be simplified considerably, without significant loss of performance, by using approximations to its equations. But approximations can blur the essential structure of the model, and therefore it is better to describe the theory in exact form first. If the framework is sound, simplification can come later.
Gaze comparator
A comparator in the SC computes dynamic target direction relative to the eye, Te (= retinal error = gaze error, or better: desired gaze shift in space expressed in eye coordinates). This, I am assuming, is the command coded by the buildup cells. In Eq. A1 below, Te is represented as a unit vector pointing in the direction of the saccade target. By a theorem of kinematics (see any text, e.g., Goldstein 1980), the time rate of change of Te, called 
e, equals the vector cross product of Te and esc, the angular velocity of the eye relative to space, expressed in an eye-fixed coordinate system. This cross product formula is shown in the first line of Eq. A1, but because ese is not one of the variables shown in the flow diagram in Fig. 1, the subsequent lines in Eq. A1 derive an expression for e based on variables that are available
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(A1) |
eh is its time rate of change, and q
1eh is its inverse. hsh is the angular velocity vector of the head relative to space, expressed in head-fixed coordinates; in other words, hsh is the head velocity estimate delivered by the vestibular system. (Strictly, qeh should be written qehh, meaning the quaternion of eye position relative to the head in head-fixed coordinates, but because qeh never is expressed in any other coordinates in this paper, I have omitted the final h for simplicity.) The symbol × indicates the vector cross product, and the juxtaposition hshqeh is the quaternion product of the vector and the quaternion, in which the vector is treated as a quaternion with a scalar component of 0. Once e has been obtained, it is integrated within the SC to obtain the updated estimate of Te.
Desired head position and the Donders operator Desired head position is computed from Te by the formulas
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(A2) |
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(A3) |
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(A4) |
; Tweed et al. 1995), Donders scales the horizontal and vertical components of desired head position differently, and then it scales the torsional component to fit Donders' law of the head. For clarity, Donders is best defined in three steps
(A5a)
(A5b)
Here i, j, and k are the unit vectors pointing along the three axes of the coordinate system, i pointing forward, j left, and k up; x2 and x3 are the coordinates of the vector x along the j and k axes, and · indicates the dot product of two vectors. Equation A5a quaternion-multiplies
(A5c)
Ts with i and then takes the quaternion square root. (If p is the quaternion square root of q, then the quaternion product pp = q; in terms of rotations, p has the same axis and direction as q but only half the amplitude.) This operation yields a quaternion x representing the shortest rotation taking the vector i to the vector Ts. In other words, x is the head orientation that would point the nose at the visual target while obeying a head version of Listing's law, with Listing's plane of the head orthogonal to the vector i. But because the head doesn't normally rotate far enough to point the nose at the target, Eq. A5b scales the rotation by the factors 
V and H. In all simulation plots except Fig. 6, H is near 0.9 and V is near 0.3, reflecting the fact that the head contributes more to horizontal than to vertical gaze shifts; (in Fig. 6, H is varied between 1 and 1.4 and V between 0 and 0.4 to simulate noise or alteration in the Donders operator). And because the head doesn't obey Listing's law, its position quaternion needs a torsional component, Tx2x3 i, where T is set at 0.15 (approximately equal to VH/2) to yield a quadratic Donders surface for the head that matches those seen in the data (Glenn and Vilis 1992) (T = 0 would yield a planar surface, i.e., Listing's law). Finally, Eq. A5c converts the vector y into a unit quaternion representing desired head position in space.
T, V, and V, and indeed the whole computation in Eqs. A5a-c, are merely defaults. It is clear that we can control voluntarily how much motion the head contributes in all three dimensions; i.e., the Donders operator can be voluntarily altered or overridden. Further, a more realistic model, particularly one involving torso movements, would require a more sophisticated way of choosing the final head position than just scaling by three factors T, V, and H as in Eq. A5b, but this simple scheme seems satisfactory for torso-fixed saccades.
Head pulse generator
The head is driven by a velocity command h representing the rate of change of head position
![<IT><A><AC>q</AC><AC>˙</AC></A><SUB>h</SUB>= P<SUB>h</SUB></IT>[<IT>V<SUB>q</SUB></IT>(<IT>q</IT><SUP></SUP>*<SUB><IT>h</IT></SUB><IT>q</IT><SUP>−1</SUP><SUB><IT>h</IT></SUB>), <IT>q</IT><SUB><IT>h</IT></SUB>]](/content/vol77/issue2/fulltext/654/img013.gif)
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(A6) |
for Vq (q*hq1h), then the head pulse generator Ph can be defined by
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(A7) |
; van Gisbergen et al. 1981), but the division by 1 + 20|| used here is nearly equivalent and computationally simpler. Also, a more realistic model would have the head driven by a signal coding some mixture of jerk, acceleration, velocity, and position, rather than simply velocity, and would pass this signal through a higher-order plant, but such complications would not alter the basic kinematics under study here. Note that the output of the pulse generator is the quaternion velocity of the head, hi.e., the time derivative of the quaternion of head position, qhand not the angular velocity of the head [which would be given by hss = 2hq1h = 100 /(1 + 20||)].
Desired eye position and the Listing operator Desired final eye position in the head, q*eh, is computed by the formula
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(A8) |
1hTsq*h, representing the direction that the saccade target will have relative to the head when the head is in its desired position q*h, and yields as output a quaternion q*eh representing desired eye-in-head position, which fits Listing's law. To create this output, Listing quaternion multiplies q*1hTsq*h by gp, which is the primary gaze direction of the eye expressed in head-fixed coordinates, and takes the quaternion square root, to yield a desired eye-in-head orientation, q*eh, that obeys Listing's law and that points the gaze line in the desired direction. In the simulations, gp is set equal to i, the forward pointing coordinate axis although in reality it moves slightly depending on head position (Haslwanter et al. 1992).
Eye saturation and pulse generation For clarity, the computation yielding saturated desired eye position in the head can be broken down into three steps
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(A9a) |
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(A9b) |
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(A9c) |
![<IT>x</IT> = (−<IT>b</IT> + <RAD><RCD><IT>b</IT><SUP>2</SUP> − 4<IT>ac</IT></RCD></RAD>)/2<IT>a</IT>; <IT>q</IT><SUP><IT>s</IT></SUP><SUB><IT>eh</IT></SUB> = <IT>q</IT><SUP><IT>s</IT></SUP><SUB><IT>eh</IT></SUB> + <IT>x</IT>[<IT>q</IT><SUP><IT>s</IT></SUP><SUB><IT>eh</IT></SUB> − <IT>V</IT><SUB><IT>q</IT></SUB>(<IT>q</IT><SUP></SUP>*<SUB><IT>eh</IT></SUB>)]](/content/vol77/issue2/fulltext/654/img023.gif)
What is happening here is that, if the squared horizontal-vertical eccentricity
(A10)
of current desired eye position q+eh is larger than the squared radius of the EOMR [where radius = sin(40°/2) 
0.12], then the code within the curly braces projects q+eh into the EOMR in the horizontal-vertical plane by solving a quadratic equation whose coefficients are a, b, and c. As this projected point may still lie outside the EOMR in the torsional dimension, the next two lines after the curly braces compute the maximum allowable torsion, given the horizontal and vertical coordinates of the projected point, and, if necessary, clip the torsional component of qseh at that value. The 0.15 in the third last line is a parameter affecting the shape of the EOMR; it is set slightly larger than radius to give the EOMR a pill-like, rather than a purely ellipsoidal, shape (see Fig. 2). Similarly, the 0.25 in the same line is equal to 0.2 × 0.15/radius, and reflects the fact that the torsional "thickness" of the EOMR at its thickest point is only 20% of its horizontal or vertical diameter, i.e., the allowable torsion lies between ±0.2(40°) = ±8 deg. The last line in Eq. A10 transforms the vector qseh into a unit quaternion, which is the output of the saturation operator.
eh
Here the eye pulse generator Pe is defined similarly to Ph in Eq. A7; i.e., writing
![<IT><A><AC>q</AC><AC>˙</AC></A><SUB>eh</SUB>= P<SUB>e</SUB></IT>[<IT>V<SUB>q</SUB></IT>(<IT>q</IT><SUP><IT>s</IT></SUP><SUB><IT>eh</IT></SUB><IT>q</IT><SUP>−1</SUP><SUB><IT>eh</IT></SUB>), <IT>q</IT><SUB><IT>eh</IT></SUB>] + <IT><A><AC>q</AC><AC>˙</AC></A></IT><SUB><IT>VOR</IT></SUB>](/content/vol77/issue2/fulltext/654/img028.gif)
(A11)
for Vq(qsehq1eh), we have
And
(A12)
VOR, the last term in Eq. A11, is the eye velocity command from the VOR, which is described in the next section. Equation A12 was used to model the pulse generator because it yields eye-only saccades with fixed axes and a realistic amplitude-duration relation.
VOR shutoff As noted earlier, we know from 1- and 2-D studies that the VOR is weakened or shut off in the direction of the saccade. In 3-D, new questions arise, e.g., should the VOR be shut off in the direction of gaze error or head-in-space motor error or eye-in-head motor error? There are no experimental data on the precise direction of VOR shutoff, and, in any case, the model in this paper operates very well even if this direction is very imprecisely controlled. But it is an interesting theoretical point that if the VOR were switched off in the direction of current eye-in-head motor error, then it could do a surprising amount to steer the gaze line to the target, even in the absence of other signals guiding the eye. In other words, this sort of precisely directed VOR could provide a valuable backup if other parts of the saccadic system were malfunctioning. The following equations achieve precise VOR shutoff of this type.
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(A13) |
Code
The following 2 m-files, esdot.m and eyespace.m, implement the model in Matlab. In these files, quaternions are expressed as 4-by-4 matrices of real numbers, with the result that all the quaternion algebra is done using Matlab's built-in matrix operators, so no extra m- or mex-files are needed. It is also possible to represent quaternions in Matlab as 2-by-2 matrices of complex numbers, but then the code runs slightly slower and certain operations involving quaternion vectors are more complicated.
function xdot = esdot(t,x)
% Convert input vector x into 4-by-4 matrices (quaternions) global i j k o % Basis quaternions, defined in eyespace.m (below) te = x(1)*i + x(2)*j + x(3)*k; % Target direction in eye coordinates = retinal error = retinotopic gaze error
qh = x(4)*i + x(5)*j + x(6)*k + x(7)*o; % Head position
qeh = x(8)*i + x(9)*j + x(10)*k + x(11)*o; % Eye-in-head position
% Desired head position in space and head pulse generator
qes = qh * qeh; ts = qes * te * qes
; % Eye-in-space position; Target direction in space
q = real((ts * i)
.5); % "real" removes imaginary parts brought in by "" xqh = .15*q(2,4)*q(3,4)*i + .3*q(2,4)*j + .9*q(3,4)*k; m = xqh(:,4) * xqh(:,4); % Donders
if m < = 1, xqh = xqh + sqrt(1 m)*o; else, xqh = xqh/sqrt(m); end % Desired head position
q = xqh * qh; q = q q(4,4)*o; dqh = q * qh * 40/(1 + 20 * norm(q(:,4))); % Head motor error, Head velocity
% Desired eye position in space fxqeh = real((xqh * ts * xqh *i) .5); % Desired final eye-in-head position (obeys Listing's law)
xqes = xqh * fxqeh; % Desired eye-in-space position
% VOR
cxqeh = qh * xqes; q = cxqeh * qeh; % Current desired eye-in-head position; Eye-in-head error
if q(4,4) < .985, m = 1; else, m = (1 q(4,4))/.015; end; v = q(1:3,4);
vor = m * v * v/(v * v) eye(3); whsh = 2 * (qh * dqh); % VOR matrix; Head angular velocity
v = vor * whsh(1:3,4); weh = v(1)*i + v(2)*j + v(3)*k; dqVor = .5 * weh * qeh; % VOR eye velocity command
% Saturation
s = cxqeh(1:3,4); f = fxqeh(1:3,4);
alpha = s(2) * s(2) + s(3) * s(3); % Squared eccentricity of current desired eye position in hor-vert plane
radius = sin(40 * pi/360); % Radius of EOMR in hor-vert plane
if alpha > radius * radius, beta = s(2) * f(2) + s(3) * f(3); gamma = f(2) * f(2) + f(3) * f(3);
a = alpha 2 * beta + gamma; b = 2 * (alpha beta); c = alpha radius * radius; % Polynomial coefficients
root = (b + sqrt(b * b 4 * a * c))/(2 * a); s = s + root * (s f); % Project s into EOMR in hor-vert plane
alpha = radius * radius;
end
maxTorsion = sqrt(1.25 alpha/(radius * radius)) * sin(4 * pi/180);
if abs(s(1)) > maxTorsion, s(1) = sign(s(1)) * maxTorsion; end % Torsional component of s
s(4) = sqrt(1 s(1) * s(1) s(2) * s(2) s(3) * s(3)); % Scalar component of s
sxqeh = s(1)*i + s(2)*j + s(3)*k + s(4)*o; % Convert saturated desired eye in head position, s, to a quaternion
% Eye pulse generator
q = sxqeh * qeh; q = q q(4,4)*o; dqeh = q * qeh * 80/(1 + 20 * norm(q(:,4))) + dqVor; % Gaze comparator
dte = te * (qeh * (whsh * qeh + 2 * dqeh)); % Convert to derivative vector xdot for output
xdot(1:3) = dte(1:3,4); xdot(4:7) = dqh(:,4); xdot(8:11) = dqeh(:,4);
function eyespace global i j k o % Basis quaternions expressed as 4-b-4 matrices i = [0 0 0 1; 0 0 1 0; 0 1 0 0; 1 0 0 0]; j = [0 0 1 0; 0 0 0 1; 1 0 0 0; 0 1 0 0];
k = [0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0]; o = eye(4);
a = sin(70 * pi/180) * sqrt(.5); b = sqrt(1 2 * a * a);
x0 = [b a a 0 0 0 1 0 0 0 1]; % Initial retinal error, head position and eye-in-head position
[t, x] = ode23(`esdot', 0, .5, x0);
plot(x(:,9), x(:,10), `c'), m = sin(60 * pi/360); axis([m m m m]), axis(`square'), hold on, plot(x(:,5), x(:,6), `r')
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FOOTNOTES |
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Address for reprint requests: Dept. of Physiology, University of Western Ontario, London, Ontario, N6A 5C1, Canada.
Received 18 December 1995; accepted in final form 3 October 1996.
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REFERENCES |
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Abstract
Introduction
Discussion
References
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