The Journal of Neurophysiology Vol. 80 No. 5 November 1998, pp. 2274-2294
Copyright ©1998 by the American Physiological Society

Human Oculomotor System Accounts for 3-D Eye Orientation in the Visual-Motor Transformation for Saccades

Eliana M. Klier¹ and J. Douglas Crawford^{1, 2}

Centre for Vision Research and ¹ Department of Biology and ² Department of Psychology, York University, Toronto, Ontario M3J 1P3, Canada

	ABSTRACT

Abstract Introduction Methods Results Discussion References

Klier, Eliana M. and J. Douglas Crawford. Human oculomotor system accounts for 3-D eye orientation in the visual-motor transformation for saccades. J. Neurophysiol. 80: 2274-2294, 1998. A recent theoretical investigation has demonstrated that three-dimensional (3-D) eye position dependencies in the geometry of retinal stimulation must be accounted for neurally (i.e., in a visuomotor reference frame transformation) if saccades are to be both accurate and obey Listing's law from all initial eye positions. Our goal was to determine whether the human saccade generator correctly implements this eye-to-head reference frame transformation (RFT), or if it approximates this function with a visuomotor look-up table (LT). Six head-fixed subjects participated in three experiments in complete darkness. We recorded 60° horizontal saccades between five parallel pairs of lights, over a vertical range of ±40° (experiment 1), and 30° radial saccades from a central target, with the head upright or tilted 45° clockwise/counterclockwise to induce torsional ocular counterroll, under both binocular and monocular viewing conditions (experiments 2 and 3). 3-D eye orientation and oculocentric target direction (i.e., retinal error) were computed from search coil signals in the right eye. Experiment 1: as predicted, retinal error was a nontrivial function of both target displacement in space and 3-D eye orientation (e.g., horizontally displaced targets could induce horizontal or oblique retinal errors, depending on eye position). These data were input to a 3-D visuomotor LT model, which implemented Listing's law, but predicted position-dependent errors in final gaze direction of up to 19.8°. Actual saccades obeyed Listing's law but did not show the predicted pattern of inaccuracies in final gaze direction, i.e., the slope of actual error, as a function of predicted error, was only 0.01 ± 0.14 (compared with 0 for RFT model and 1.0 for LT model), suggesting near-perfect compensation for eye position. Experiments 2 and 3: actual directional errors from initial torsional eye positions were only a fraction of those predicted by the LT model (e.g., 32% for clockwise and 33% for counterclockwise counterroll during binocular viewing). Furthermore, any residual errors were immediately reduced when visual feedback was provided during saccades. Thus, other than sporadic miscalibrations for torsion, saccades were accurate from all 3-D eye positions. We conclude that 1) the hypothesis of a visuomotor look-up table for saccades fails to account even for saccades made directly toward visual targets, but rather, 2) the oculomotor system takes 3-D eye orientation into account in a visuomotor reference frame transformation. This transformation is probably implemented physiologically between retinotopically organized saccade centers (in cortex and superior colliculus) and the brain stem burst generator.

	INTRODUCTION

Abstract Introduction Methods Results Discussion References

Visual signals must be processed sequentially through several internal stages to generate accurate saccades. For example, visible light is initially coded on several two-dimensional (2-D) retinotopic maps including the retina, primary visual cortex, and the superficial layers of the superior colliculus (Hubel and Wiesel 1979; Sparks 1989). At a later stage, reticular formation burst neurons produce phasic signals, in a 3-D head-fixed coordinate system, that provide the "eye velocity" signal (to the motor neurons) necessary to drive the eyes in a certain direction at a certain speed (Crawford and Vilis 1992; Henn et al. 1989; Luschei and Fuchs 1972). However, it is unclear how the intermediate structures convert 2-D, oculocentric, sensory vectors into the 3-D, headcentric, motor vectors needed to drive the burst generator. In other words, how is retinal error (RE; the retinal distance and direction of the target image from the fovea, or alternatively, desired gaze direction relative to the eye) converted into the motor error (ME) command that drives the burst neurons?

One possibility is that the brain maps RE signals directly onto equivalent ME signals in the neural equivalent to a visuomotor "look-up table" (LT). This idea originated with the foveation hypothesis of Schiller (1972). Here, horizontal and vertical components of 2-D RE are input to a look-up table that simply maps RE onto ME displacements directly, without any comparisons with current eye position. This hypothesis also featured prominently in the displacement-feedback tradition of models founded by Jürgens et al. (1981). This scheme is often associated with a direct mapping between the superficial sensory and deeper motor layers of the superior colliculus (Moschovakis et al. 1988) and has been cited as the classic example of a sensorimotor look-up table (e.g., Churchland and Sejnowski 1992).

The second hypothesis was initially proposed by David Robinson and colleagues (Robinson 1975; Zee et al. 1976). We call this the "reference frame transformation" (RFT) hypothesis because it involves a transformation of eye-centered representations into head-centered representations. To do this, the RFT model uses comparisons between visual input and an internal representation of current eye position. In the first such comparison, information about eye position, derived from the burst neurons' integrated velocity signal, is added onto incoming RE to derive a desired eye position command. This signal is then transformed, via a second subtractive comparison to eye position, into an instantaneous ME command.

To date, experimental evidence has been cited in support of both models. First, retinotopic maps (sufficient for the LT model) are prevalent in the brain in such visuomotor areas as the occipital lobe, the posterior parietal cortex, and the superior colliculus (reviewed in Moschovakis and Highstein 1994). However, information regarding target position relative to the head or body (required for the RFT model) has also been identified in several areas including the thalamus, frontal cortex, and posterior parietal cortex (Andersen et al. 1985; Schlag and Schlag-Rey 1987; Sparks 1989). Second, the RFT model is capable of accounting for the ability to saccade to remembered target locations after intervening saccades (Hallet and Lightstone 1976; Sparks 1989), whereas the original LT model failed to emulate multiple saccades. The latter has been corrected by the addition of a "vector subtraction" mechanism upstream of the visuomotor transformation (Goldberg and Bruce 1990; Moschovakis and Highstein 1994; Waitzman et al. 1988). However, the mechanism for remembering target locations independent of eye movements may differ from the visuomotor transformation for saccades made directly to visual targets (Crawford and Guitton 1997; Henriques et al. 1998), which will be the focus of our experiments. In this context (direct visuomotor execution), the sequential adding and subtracting of eye position in the 1-D RFT model seems redundant. Indeed, a trivial mapping between RE and ME displacement codes seems completely sufficient to determine saccade direction and amplitude in both 1-D and 2-D models (Waitzman et al. 1991).

Thus the practical difference between these two hypotheses seems ambiguous in abstract 1-D or 2-D models. However, a recent theoretical investigation has suggested that in real 3-D space, saccades cannot obey Listing's law and be accurate from all initial eye positions without an intermediate position-dependent reference frame transformation (Crawford and Guitton 1997). As pointed out by Crawford and Guitton (1997), RE, being eye-fixed, depends on the 3-D orientation of the eye as well as the configuration of the target in space (Fig. 1A). This would not be a problem if saccade axes were also eye-fixed, but Listing's law only allows such axes to rotate by half the angle of eye position (Helmholtz 1867; Tweed and Vilis 1990).

View larger version (15K): [in this window] [in a new window]   FIG. 1. Basic input-output geometry for saccades showing Listing's law and position-dependent retinal geometry in head coordinates. A: right side view of the right eye and head, with gaze elevated 30°. Eye position vectors fall within Listing's plane, which is viewed edge on. The head-fixed vertical axis for horizontal eye rotation also falls within Listing's plane. In contrast, the shortest-path axis of rotation () for a rightward saccade would be perpendicular to current gaze direction, i.e., eye-fixed. The actual axis of rotation (- - -) allowed by Listing's law is about halfway between the latter 2 axes (Tweed and Vilis 1990). B: behind view of same situation as in A, again, with gaze pointed 30° upward. The "horizontal" meridian of the retina is now tilted with respect to the head. Circle shows the points where light falling on this meridian would intersect with a sphere centered around the right eye (radius = gaze vector), as it would project onto Listing's plane. As targets are displaced further horizontally from current gaze in retinal coordinates, from 30° () to 60° () to 90° () retinal error (RE), the target displacement becomes more and more oblique in headcentric coordinates. Rightward rotation about the eye-fixed axis (shown in A) would cause gaze to sweep around this circle. Rightward rotation about the head-fixed axis would cause gaze to curve away from this circle (). Rightward rotation compatible with Listing's law would produce an intermediate trajectory (- - -). C: same situation in eye-fixed coordinates centered around current gaze. In these coordinates, the targets (, , ) are displaced horizontally, and the REs that would be satisfied by the gaze trajectories in B (- - -) curve obliquely, such that the deviation between these traces increases with eccentricity. (Larger and more complex patterns occur for tertiary and torsional eye positions). Thus a horizontal saccade will not satisfy horizontal RE at these eccentricities. For the saccade generator to acquire these targets, it must map a horizontal RE onto a nonhorizontal saccade. This imposes a position-dependent visuomotor reference frame problem in saccade generation that cannot be solved by any known eye muscle properties. See Crawford and Guitton (1997) for further details.

One possible solution is that the visuomotor transformation ignores the difference between RE and ME and approximates the above transformations with a fixed mapping between any one RE and any one ME (Hepp et al. 1993, 1997; Raphan 1997, 1998). This strategy would only produce minor errors in the peri-primary range (Crawford and Guitton 1997; Hepp et al. 1993, 1997). However, Crawford and Guitton (1997) demonstrated that any 3-D version of the LT model (Fig. 2A) would produce large directional inaccuracies for large saccades between eccentric targets and from initial torsional eye positions. Regardless, it has been suggested that the system would tolerate such errors in favor of simplifying the visuomotor transformation (Hepp et al. 1997). Indeed, Hepp et al. (1997) proposed that the function of Listing's law is to allow for the best approximation for a LT transformation to give reasonably accurate saccades while also providing a fixed torsional component for each gaze direction (i.e., Donder's law).

View larger version (18K): [in this window] [in a new window]   FIG. 2. Two 3-dimensional (3-D) models of the saccade generator tested in this paper. A: look-up table (LT) model. Components of RE are mapped directly onto components of Ei without any considerations or comparisons to the eyes' current position in the head (E). We called this mapping a "look-up table." B: reference frame transformation (RFT) model. 2-D RE, or desired gaze direction relative to the eye (Gdeye), is rotated multiplicatively () by an internal representation of current eye position (E) to produce a desired gaze direction relative to the head (Gdhead). This accomplishes the necessary transformation of data from an oculocentric to a craniotopic reference frame. This command (still 2-D) is then input to a Listing's law operator (LL), described by Tweed and Vilis (1990), to give a 3-D desired eye position command (Ed). Finally, subtracting E from Ed results in Ei. For more details see Crawford and Guitton (1997). C: both models share the same downstream saccade generator. Displacement feedback from a resettable integrator is subtracted from initial motor eror (ME; Ei) to compute instantaneous 3-D ME (E). A rate-of-position-change signal () is then derived to drive the burst neurons, whose velocity output travels both straight to the motoneurons (MN) that move the eyes, as well as through an integrator that produces an eye position signal (E) with which the eyes maintain their final position. K and R represent the elasticity and viscosity estimates, respectively, used by the brain stem to overcome those found in the plant (the eye and its surrounding tissues and musculature). We have modeled the plant either as having head-fixed muscle pulling directions, requiring an internal implementation () of the "half-angle" rule (defined in text), or as a "linear plant" that implements the half-angle rule of Listing's law itself (the latter was used exclusively in our simulations of the LT model) (Quaia and Optican 1998).

In contrast, Crawford and Guitton (1997) assumed that the saccade generator does not sacrifice either accuracy or Listing's law within the oculomotor range. To convert 2-D, oculocentric, RE vectors into 3-D, head-fixed, ME vectors, they formulated a model that, in outline, bears a striking resemblance to that of Robinson's model (Fig. 2B). In this model, incoming 2-D RE was first rotated by an internal measure of current 3-D eye position, providing a measure of desired gaze direction relative to the head. The next step involved a Listing's law operator that performed a 2-D to 3-D transformation, giving rise to a 3-D command encoding desired eye position in Listing's plane. Finally, current eye position was subtracted from desired eye position to produce a 3-D ME signal that drove a feedback loop, containing a resettable displacement integrator, and subsequent burst neurons. It was suggested that these position-dependent transformations may be implemented implicitly (van Opstal and Hepp 1995; Zipser and Andersen 1988), such that only the inputs (RE) and outputs (ME) might be explicitly observed in the brain. In contrast to the 3-D LT model, this model produced accurate and kinematically correct saccades from all initial 3-D eye positions (Crawford and Guitton 1997).

Surprisingly, no one has yet simultaneously evaluated Listing's law and saccade accuracy over a large enough range to distinguish between the 3-D LT and RFT models experimentally. Furthermore, a rigorous test between these hypotheses would require a geometrically correct computation of RE, which remarkably, has not yet been done [beyond local measures of "false torsion" at tertiary positions (Helmholtz 1867)]. Finally, these actual measures would have to be input to 3-D versions of the RFT and LT models to compare their predictions against actual saccade trajectories. Our goal was to combine these approaches to determine whether the oculomotor visuomotor transformation for saccades uses a look-up table to approximately satisfy RE, or if it makes the proper compensation for eye position.

	THEORETICAL PREDICTIONS

This section describes the simulations and predictions that motivated the specific paradigms used in this study. First, we examined the geometrically unavoidable, yet often ignored prediction that RE depends not only on target displacement in space, but also on eye orientation in Listing's plane. During saccades, 3-D eye position vectors (which describe eye orientation as an axis of rotation) lie in a 2-D plane known as Listing's plane (e.g., Tweed and Vilis 1990). One particular gaze direction, that which is orthogonal to the plane, is referred to as primary position. If this orthogonal direction is defined as the torsional axis, then Listing's plane becomes the plane of zero torsion (Westheimer 1957). It has been demonstrated that, to keep eye position vectors in Listing's plane, axes for horizontal saccades must tilt out of Listing's plane by half the angle of vertical gaze deviation from primary position. We will refer to this as the "half-angle" rule (Tweed and Vilis 1990).

Crawford and Guitton (1997) pointed out that even for eye positions in Listing's plane, an error arises if one assumes that a target displaced horizontally, from an initial fixation point in space, produces horizontal RE. For example, they showed that horizontal REs correspond to nonhorizontal lines in space, depending on eye position (Fig. 1B), and conversely, that when the eye is oriented at any vertical or torsional position, targets displaced horizontally in space stimulate nonhorizontal (i.e., oblique) REs (Fig. 1C). (Contrary to common belief, this effect even occurs at secondary eye positions, but it becomes even more complex at the tertiary positions described below.) The challenge for the oculomotor system is then to generate horizontal saccades from these oblique RE signals, or to deal with the consequences of inaccurate target foveation (Crawford and Guitton 1997).

To test this, we first simulated a saccade paradigm, shown in Fig. 3A, similar to the one described in Crawford and Guitton (1997) [for model equations, see the appendix in Crawford and Guitton (1997)]. Data simulations are illustrated in Listing's coordinates (thus the origin corresponds to primary gaze direction). Five leftward fixation lights () and their horizontally paired target lights () were separated by 60° in space, symmetrically about the ordinate, and ranging in elevation from +40° to 40°, at 20° intervals. Subjects foveated one of the five leftward fixation lights until its corresponding rightward target light was briefly flashed, at which time they were required to make a saccade and foveate the target as accurately as possible.

View larger version (25K): [in this window] [in a new window]   FIG. 3. Accuracy of the 3-D RFT and LT models (illustrated in Fig. 1) for simulated horizontal saccades. A: 5 initial fixation lights () and their paired target lights () are separated by 60° symmetrically about the ordinate, at 5 different vertical elevations (0°, ±20°, and ±40°). Horizontal and vertical components of gaze are plotted in Listing's coordinates (thus the origin corresponds to primary gaze position). Simulated lines (- - -) of lights that would stimulate the vertical and horizontal meridians of the eye at each initial eye position () are also shown. B: REs () caused by each target light while the eye foveated the initial light were computed for each of the 5 light pairs. C: LT model () produces systematic, position-dependent errors in final saccade direction. This error is only absent along the abscissa, but then increases with increased eccentricity from primary position (directional error of 6.1° for targets at ±20° and 13.9° for targets at ±40°). D: both the RFT models, with the standard plant () and the linear plant (), consistently predict accurate saccade endpoints that coincide with the targets' locations.

Normally, this task would be assumed to evoke exclusively horizontal REs, but this is not correct. For example, Fig. 3A also shows the simulated lines (- - -) of lights that would stimulate the vertical and horizontal meridians of the eye at each initial eye position (). The rightward horizontal retinal lines (- - -) follow a characteristic curving pattern, first curving centrifugally (related to false torsion), and then curving more strongly in the centripetal direction. By corollary, lines that are straight in these head coordinates should curve in retinal coordinates. Figure 3B shows these simulated REs, calculated when the targets () in Fig. 3A were converted into oculocentric coordinates. This was done by rotating the five rightward target directions by the inverse of initial 3-D eye position at the five corresponding leftward lights. It is apparent that, except across primary position, the resultant REs () were oblique, in a position-dependent pattern, where the degree of "fanning out" was proportional to the target's initial vertical position. Thus we predict that targets displaced horizontally in space do not simply cause horizontal RE. The true value of RE will depend on both the initial position of the eye-in-head and the relative locations of the targets in space.

How then, would our two alternative models (Fig. 2) handle this pattern of inputs? Figure 3C depicts simulated horizontal saccades, at the same five vertical elevations (0°, ±20°, and ±40°) depicted in Fig. 3A, for the LT model. The eye began each movement positioned 30° to the left, and simulated RE was computed as described above. Although not shown here, the LT model correctly upheld Listing's law in these circumstances (Crawford and Guitton 1997). However, it produced gaze shifts () that were only accurate along the abscissa (i.e., across primary position). Otherwise, it predicted position-dependent inaccuracies that increased with increased displacement from primary position. Essentially, this occurred because Listing's law only allowed the eye to rotate about the axis orthogonal to the RE vector across primary position, but then caused these two axes to diverge by half the angle of upward or downward vertical gaze. Notice that, not surprisingly, the erroneous pattern of saccade trajectories closely resembled the pattern of REs calculated in Fig. 3B. This is because the LT model must output ME commands directly from RE input. Errors in final gaze direction () of 6.1 and 13.9° were predicted at the ±20 and ±40° elevations, respectively (again, assuming that primary position fell within the middle of the range).

Figure 3D illustrates the predicted outcomes of the RFT model, which takes eye position into account. The two trajectories shown represent the outcomes of two versions of the ocular plant (the eye globe and its surrounding musculature and tissues). With the "standard plant," eye muscle activation relative to the head is independent of eye position and thus requires an internal implementation of the half-angle rule (Crawford 1994), whereas with the "linear plant," the eye muscles tilt by half the angle of eye eccentricity, in line with the pulley hypothesis (Demer et al. 1995; Miller 1989; Miller et al. 1993; Quaia and Optican 1998; Raphan 1998). Both models predicted the same endpoints, but the linear plant () predicted straight gaze trajectories independent of the eye's initial eccentricity, whereas the standard plant () predicted trajectories that curve as a function of initial eye position. In either case (i.e., independent of plant characteristics), by taking eye position into account, the models made no appreciable errors and led to accurate final foveation () of each of the five targets.

In addition to testing our two models with eye positions in Listing's plane, we also tested whether torsional eye positions out of Listing's plane are compensated for in a similar fashion. A well-known method by which to induce such ocular torsion in humans is by rotating the head about the occipital-nasal axis. This, in turn, induces an ocular counterroll in a direction opposite to head rotation, and thus the eyes assume a torsional component out of Listing's plane (Crawford and Vilis 1991; Haslwanter et al. 1992). This torsion produces a misalignment between the retina and head that is problematic both for perception (Wade and Curthoys 1997) and saccade generation (Crawford and Guitton 1997). For example, Fig. 4 illustrates saccades made from a central fixation point to eight targets, displaced by 30° from the center, in the four cardinal and four diagonal directions. The simulation in Fig. 4A was programmed such that the head was upright and the eyes had no initial torsional component. This is indicated on the left side by the head caricature, and is evident on the right side where corresponding 3-D eye positions remained in Listing's plane (i.e., the plane of zero torsion). This led to simulations, as shown by the eye-in-head trajectories on the left side, in which both models correctly foveated all eight targets. Figure 4B included a counterclockwise (CCW) eye torsion of 10°. The RFT () model took this deviation into account and thus produced accurate eye movements, whereas the LT (···) model, which directly maps RE onto ME, output consistently inaccurate gaze trajectories in a clockwise (CW) pattern of errors (i.e., in the direction opposite to that of the eye torsion) for each of the eight targets. Figure 4C was almost identical to B, except that the 10° eye torsion was now present in the CW direction. Again, similar errors were made in a direction opposite to that of eye torsion (i.e., CCW) for all eight targets (or, as a rule of thumb, the trajectories are tilted incorrectly in the same direction as the head).

View larger version (19K): [in this window] [in a new window]   FIG. 4. Simulations of the 3-D RFT () and LT (···) models for radial saccades of 30°. A: with the head upright (0° ocular torsion). Left column: behind view of gaze positions shows that both models predict accurate foveation of all 8 targets. Right column: side view shows that 3-D eye positions remain in Listing's plane (i.e., along the ordinate). B: with the head rotated about the line of sight 45° clockwise [CW; 10° counterclockwise (CCW) ocular counterroll]. Left column: behind view shows that the RFT model remains accurate, but the LT model misses the targets in the direction of head rotation (CW). Right column: side view shows how the eye counterrolls in a direction opposite to that of head rotation, and thus eye positions lie out of Listing's plane by 10° in the CCW direction. C: with the head rotated about the line of sight 45° CCW (10° CW ocular counterroll). Left column: behind view shows again that that the RFT model accurately reaches each target, while the LT model errs in the direction of head rotation (CCW). Right column: side view shows how the eye counterrolls in the CW direction and eye positions remain out of Listing's plane throughout the simulation.

These latter predictions can be explained intuitively as follows. When the eye is rotated 10° CCW, what was once the top of the eye has now been twisted 10° CCW, so that the uppermost target (in space coordinates) now causes a RE that, relative to the eye, is up and to the right. This, by definition of the LT model, causes a ME indicating "move the eyes up and to the right," and such a movement misses the final target. This error occurs consistently for all eight targets, but note that the directional errors are only approximately half the angle of the 10° ocular counterroll. This is because the simulated linear plant rotates the axes of eye rotations by half the angle of eye position.

For this reason it is necessary to point out that the plants used to simulate the LT predictions in RESULTS assume that the pulling directions of the muscles, in the horizontal and vertical directions, tilt 50% with current eye position (Quaia and Optican 1998). This model also assumes a similar dependence of axes on torsional position, which is supported by mechanical simulations of orbital "pulleys" (Miller et al. 1997). Note that without such a mechanical position dependence, the errors predicted by the LT model below would essentially double, and Listing's law would be violated. [Conversely, a 100% position dependency would provide accurate saccades but would also result in gross violations of Listing's law (Crawford and Guitton 1997)].

	METHODS

Abstract Introduction Methods Results Discussion References

Subjects

A total of seven human subjects (4 male; 3 female), ranging in age between 23 and 33, participated in our study. Six participated in the first experiment and continued on to perform the second. Five of those also completed the third experiment, but one was unable to participate and was replaced with a seventh subject. None of the participants had any known neuromuscular deficits, and only one required corrective lenses during the third experiment. Subjects signed informed consent forms before inclusion, and the study was preapproved by the York University Human Participants Review Subcommittee.

Apparatus

Each subject was seated in an earth-fixed chair fitted with a personalized bite bar for head stabilization. The subject's right eye, when seated in the chair, was located 1 m off the ground and 1.1 m away from a flat, 2.14-m² tangent screen holding 19, 3-mm light-emitting diodes (LEDs; each 0.17° in visual diameter and luminance of 2.0 mcd). In addition, the subject's right eye was in the exact center of three mutually perpendicular magnetic fields (90, 125, and 250 kHz), generated by Helmholtz coils 2 m diam. Movement of the right eye was recorded using Skalar 3-D scleral search coils, while head orientations were measured using a homemade, 3-D coil taped securely in place on the center of the forehead (~5 cm from the center of the fields). In calibration tests, measured quaternions were accurate to 0.58% (magnitude)/0.9° (direction) with coils at the center of the fields, and 2% (magnitude)/ 2.05° (direction) with coils at ±10 cm from center. Data from the search coils were monitored on-line, on an oscilloscope in an adjacent room, and simultaneously sampled at 200 Hz. These signals were collected onto a PC for analysis along with feedback signals from the LEDs.

The magnetic field signals were precalibrated by rotating a gimbal-mounted coil 360° in the horizontal, vertical and torsional directions, by the method described in Tweed et al. (1990). At the end of each experimental session, we instructed each subject to freely rotate their eyes and head simultaneously in large horizontal (yaw), vertical (pitch), and torsional (roll) semicircles. The gains and biases of the corresponding coil signals, recorded during the latter procedure, were then further adjusted off-line so that the 3-D coil "vectors" described spheres centered about the origin (Tweed et al. 1990). This was done to eliminate any in vivo distortions of the coil signals. Torsional calibrations of the eye and head coils were also double checked at the end of the second and third experiments to ensure that computations of eye-in-head torsion during counterroll were correct.

Procedure

All experiments were performed in complete darkness. At the beginning of each paradigm, subjects were required to fixate the central target light for 5 s to obtain a reference position and check for coil slipping. At either the beginning or end of each experimental session, subjects were asked to perform pseudorandom self-generated saccades for 100 s (still in complete darkness). We made certain that subjects covered their entire visual field by viewing on-line measurements of their eye movements and encouraging them verbally to explore their full range. This allowed for the measurement and visualization of gaze, 3-D eye positions, and especially Listing's plane, over the entire oculomotor range. In addition, we performed the following evaluations of saccade accuracy.

EXPERIMENT 1. In the first experiment, subjects were required to make horizontal saccades between five parallel pairs of lights, each pair arranged symmetrically across the midline such that the rightward, target light was displaced 60° horizontally (angle of gaze projected onto the horizontal plane), in space coordinates, from the leftward, initial light (similar to the simulation in Fig. 3A). One pair of lights was situated at the subjects' eye level (i.e., 1 m above the ground), and subsequent pairs were placed at both 20 and 40° (angle of gaze projected onto the saggital plane) above and below the center pair (Fig. 3A). Subjects were instructed to stare at the leftward member of each pair (light duration varied randomly from 1,000 to 2,000 ms) until it disappeared and the rightward light (visible only monocularly to the right eye) was briefly flashed (150 ms). The random timing of the initial lights was chosen to eliminate any anticipatory effects, while the timing of the target lights corresponded to typical saccade latencies and thus avoided the possibility of subjects using visual feedback during the experiment. Subjects then made a saccade rightward, to the target light. Horizontal eye traces for five consecutive trials, plotted as a function of time, are shown in Fig. 5 for one subject. Note that the saccades were initiated slightly after (78.70 ± 10.44 ms, mean ± SE averaged across all saccades and then across all subjects) the target light was extinguished, so that there was no visual feedback, but not long enough after to evoke memory effects (Gnadt et al. 1991; White et al. 1994). Thus these were visually triggered saccades based solely on initial RE. This paradigm was designed to emulate the theoretical test shown in Fig. 3A, where saccade trajectories were programmed based on initial RE and eye position. This sequence was repeated 20 times for each of the 5 pairs of lights.

View larger version (25K): [in this window] [in a new window]   FIG. 5. Five consecutive saccade gaze trajectories for the right eye () are plotted against time. Subjects foveated the initial light (30° to the left) until its paired target light (30° to the right) was briefly flashed. Subjects made a saccade to the target light only after it had been extinguished.

At the end of this experiment, a visual calibration task was performed in which subjects were instructed to foveate the illuminated targets as accurately as possible. This was done five times for each of the LEDs described above. In this case, the target LEDs were illuminated for 2 s, allowing ample time for visually guided corrective saccades. This was used as a measure of the subjects' "desired" gaze direction for each light, and these values were later used as reference positions to determine the endpoint errors of the saccades. This measure of desired gaze direction (as opposed to our geometric measures) was used because 1) it is conceivable that there could be subjective variations in target foveation and 2) this would automatically cancel out minute errors in eye-coil signals so that they would not be misconstrued as inaccuracies. Note that this paradigm also allowed us to evaluate saccade accuracy in the presence of visual feedback, as a further control.

EXPERIMENT 2. Eight binocularly viewed target LEDs were arranged in a radial pattern, in the four cardinal and four diagonal directions, at an eccentric distance of 30° from the center light. With the head upright, subjects stared at the center light (duration varied randomly between 1,000 and 2,000 ms) until one of the peripheral lights flashed (150 ms), after which they made a saccade toward it. In this experiment, the light sequence began with the uppermost target (i.e., at 12:00), and saccades to all eight target lights were repeated five times, in a clockwise sequence. This was followed by a desired gaze calibration task similar to the one described above.

We then repeated the test and calibration paradigms with the head tilted torsionally to induce ocular counterroll. First, the head was rotated 45° CW, along with the bite bar apparatus, to induce CCW ocular counterroll. A 45° perturbation of the head in this manner has been found to induce ~5-10° of ocular torsion in the opposite direction (Crawford and Vilis 1991; Haslwanter et al. 1992). Next, the head was rotated upright again, and the calibration procedure was repeated. This was done to check the torsional stability of the eye coil and to minimize any cross-training effects. Finally, we rotated the head 45° CCW, and the procedure was repeated. As described below, the head-fixed coil was used to measure precise head orientation and to compute 3-D eye position relative to the head.

EXPERIMENT 3. After performing the second experiment, we concluded that, conceivably, binocular visual inputs could be used to infer ocular torsion indirectly (Howard and Zacher 1991), and that there may have been order effects in the radial saccade task. Furthermore, we wanted to compute the geometrically correct RE for the right eye as the unique measure of visual input, as we had done in experiment 1. Therefore we repeated experiment 2, but with the left, nonrecorded eye patched, and with a randomized order of target lights.

Data analysis

QUANTIFICATION OF COIL SIGNALS. The coil signals recorded while subjects fixated the central target were used as the initial reference positions for eye positions in space coordinates. Coil signals were first used to compute quaternions (Tweed et al. 1990), to visualize Listing's plane, and to perform the mathematical transformations described below. Quaternions were then used to compute unit vectors aligned with gaze direction (Tweed et al. 1990). (Our 2-D figures show the vertical and horizontal components of these gaze vectors as they project onto the plane of the tangent screen or Listing's plane.) In addition, quaternions were transformed into linear angular measures of 3-D eye position (Crawford and Guitton 1997) for statistical analysis. In this way, any final eye orientation could be described as a rotation vector from an initial reference eye position (this can be visualized with the right-hand rule). The torsional thickness (quantified as the standard deviation) of this data was computed using the algorithm described in Tweed et al. (1990). Finally, we were also able to compute angular velocities from the quaternions when required.

COORDINATE TRANSFORMATIONS AND COMPUTING RETINAL ERROR. Three different coordinate systems were used in the computation and subsequent analysis of the data. The raw eye coil data were in an earth-fixed orthogonal coordinate system defined by the magnetic fields that we called "space" coordinates. Eye position quaternions were subsequently rotated into (eye-in-) head coordinates by dividing them by the head position quaternion (Glen and Vilis 1992). This was particularly important during the head tilts in experiments 2 and 3, but was also useful to account for minute tilts of head posture against the bite bar. To put the data into Listing's coordinates, primary eye position was computed and used as the reference position, while the coordinates were rotated to align with Listing's plane (Tweed et al. 1990). Finally, 2-D target directions and 3-D eye positions in Listing's coordinates were used to compute target directions in eye coordinates (i.e., T_eye), as follows.

With the data in Listing's coordinates, we first obtained the subjects' final eye positions at the target lights. These data were selected visually from the most stable traces of horizontal, vertical, and torsional eye positions, for each of five trials per target light in the calibration task, and then averaged. These points were then converted into gaze directions to produce a measure of each target's direction relative to the head (T_head). They were thus considered the ideal desired target directions in Listing's coordinates.

For each saccade, we also found the eye's initial position quaternion (q) at the initial fixation light. These points were chosen automatically by a computer algorithm restricted to certain selection criteria (described below), and their inverses (q¹) were computed (Tweed and Vilis 1987). T_head was then rotated into eye coordinates by the following formula (Crawford and Guitton 1997) This final unit vector (relative to the eye) was graphed in retinal coordinates, where the origin of the coordinate system represents a unit vector emanating from the fovea through the center of rotation of the eye. We defined the "horizontal" and "vertical" meridians of the eye as the arc intersections of the retina with the vertical and horizontal planes in Listing's coordinates, with the eye at primary position. Thus this particular target direction, in eye coordinates, specifies the unique point of retinal stimulation relative to the fovea (i.e., RE) (Crawford and Guitton 1997).

QUANTIFICATION OF PREDICTED AND ACTUAL SACCADE ERRORS. Actual saccades were selected according to the following criteria. Only the initial saccades were analyzed (corrective saccades, if any, were not). Occasional saccades that began before the target light was extinguished were rejected. Actual saccade starting points were selected as the points before the interval where their velocities reached 100°/s, and their endpoints selected at the points where their velocities decreased to 20°/s in the preceding interval. Trials that did not adhere to our set criteria were not quantified.

Before analyzing any of our data, we simulated the outcomes of the LT model using the experimental paradigms of this paper and formulas found in Crawford and Guitton (1997). We quantified the predicted errors of the LT model by inputting subjects' actual initial 3-D eye position data and computed RE into a simulation algorithm, and then allowing the computer to generate the predicted outcomes in Listing's coordinates. These results were then compared, along with the subjects' actual saccade endpoints, to the desired gaze directions obtained in the calibration trials to judge their relative accuracy.

VISUAL FEEDBACK. It has been assumed for many years that during saccades our vision is suppressed, and thus we cannot make use of any visual stimuli we may encounter midflight (Carpenter 1977). Some studies have suggested that stimuli presented during a saccade may be used to guide subsequent eye movements (Hallet and Lightstone 1976). However, the processing time required for vision is generally thought to be too lengthy to influence the current movement (Carpenter 1977). Our experiments afforded several opportunities where relatively inaccurate saccades (particularly in the monocular, torsional, radial task) were made in the absence of visual feedback (experimental trials), but then also tested with visual feedback (calibration trials). While analyzing these data, we observed a trend in which saccades made with visual feedback appeared to be more accurate than those without. We therefore included these data in our RESULTS, as described quantitatively below.

	RESULTS

Abstract Introduction Methods Results Discussion References

Listing's law

The theoretical arguments of Crawford and Guitton (1997) assume that Listing's law is obeyed, within reasonable limits, even for large, eccentric saccades. Similarly, the models that we tested take initial 3-D eye position into account, but then assume that the half-angle rule for Listing's law holds. In contrast, some models have made the contrary assumption that Listing's law only holds for small saccades in peri-primary range (e.g., Schnabolk and Raphan 1994). This is relevant to saccade accuracy because rotation of the eye about a head-fixed torsional axis will contribute to gaze direction at peripheral targets [~sin (e_{gaze eccentricity}) × torsional angle]. Therefore, before testing between the RFT and LT models, we first confirmed the adherence of the large saccades used in our study to Listing's law.

The full 3-D range of eye-in-head positions in the random saccade task is depicted in Fig. 6A. Subjects were asked to make saccades throughout their oculomotor range in complete darkness. Eye position vectors () during fixation (i.e., with velocities of <1°/s) are plotted, for one subject. Horizontal and vertical components of eye positions are shown from a behind perspective (indicated by the head caricature). Note that these are actually the tips of vectors emanating from the origin. The direction and magnitude of each position vector gives the axis and magnitude of the eye's relative rotation from primary position. This can be visualized by using the right-hand rule. For example, a downward pointing vector (direction of thumb) represents a rightward position (fingers curl to the right). These positions are plotted relative to the computed primary position, which was not generally at the center of the eye position range. The subjects' typically obtained a wide range of vertical and horizontal eye positions in this task. For example, the subject shown here spanned 80° vertically and 90° horizontally.

View larger version (28K): [in this window] [in a new window]   FIG. 6. Large saccades obey Listing's law. Quaternions derived from random saccades made throughout the oculomotor range, for one subject, are plotted in Listing's coordinates from behind (A) and the side (B). Coordinate axes are defined according to the right-hand rule. Only those points with velocities of <1°/s are shown to emphasis their compliance with Listing's law. From the side view, eye positions are clearly restricted to a flat plane of approximate thickness ±4° (torsional SD). Eye positions collected during one cycle of the calibration task are plotted in Listing's coordinates for the same subject. The entire range of eye movements covered in this experiment are plotted from behind (C), and from the side (D), eye positions both during and at the end of each saccade are clearly seen to lie in Listing's plane. E: the half-angle rule for saccade axes. Velocity trajectories for 5 consecutive saccades are plotted along with their respective gaze position in Listing's coordinates, for one subject, for the 5 light elevations (1-5) used in our experiment. Eye velocity vectors tilt torsionally out of Listing's plane by half the angle of eye eccentricity from primary gaze position (). Thus the angle between gaze and velocity becomes more acute as the eye increases its eccentricity from primary position.

Figure 6B shows the same data, but now viewed from a perspective to the right side of the head. The abscissa corresponds to the head-fixed torsional axis and the ordinate to the vertical axis, where rotation about the torsional axis causes the eye to move CW/CCW and rotation about the vertical axis causes horizontal eye displacements. From this view, the subject's eye position vectors appear flattened into a plane centered at 0° torsion. As further quantified below, this confirmed the classic observation that eye position vectors are confined to a plane (i.e., Listing's plane) during head-fixed saccades.

Next, we examined the large saccades between our visual targets to see how well they conformed to Listing's law. To illustrate this, we have shown 3-D eye positions recorded while the same subject made saccades between all 10 targets in the calibration task (Fig. 6C). The rightward saccades (in the direction indicated by the arrows and labeled 1-5) correspond to the particular movements we will study. As indicated in this behind view, this forced the subjects to use a very large distribution of the complete horizontal/vertical range, even going beyond the randomly selected range in some instances. However, the side view (Fig. 6D) suggests that these eye positions conformed to Listing's plane, both during and particularly in between the large horizontal saccades, over the range that we studied.

These observations are quantified for all subjects in Table 1. The columns indicate the thickness of Listing's plane (i.e., degrees torsional SD) for 1) saccades made in the absence of visual feedback at each light elevation (40° up, 20° up, 0°, 20° down, and 40° down), 2) during calibrations with visual feedback, and 3) for fixations between random saccades in the dark. Values for the first five columns were calculated relative to a plane fit to the calibration data in column 6. On average, the standard deviations for the random saccade paradigm (3.45 ± 0.55°) were relatively high compared with previously reported repetitive saccades in the central range (Tweed and Vilis 1990), presumably due to the larger excursions in our range, the randomness of the saccade directions, and the complete absence of visual stimuli. In comparison, the torsional ranges during fixations with visual feedback during the calibration task were considerable lower (2.40 ± 0.45°). Most importantly, the torsional range for the five sets of experimental saccades, to be quantified for accuracy below, were minute compared with their ~60° horizontal excursions. Averages for each of the five elevations (40° up, 20° up, 0°, 20° down, and 40° down) were 2.31 ± 0.47°, 1.58 ± 0.17°, 2.99 ± 0.93°, 2.43 ± 0.33°, and 2.70 ± 0.52° respectively. This confirmed the assumption that these large saccades still obeyed Listing' law with only small random deviations. Since, for example, 2° of headcentric torsion at an eccentricity of 45° would rotate gaze direction by only 1.4°, such minute torsional deviations would not significantly alter the predictions cited below.

Crawford and Guitton (1997) argued that Listing's law poses a problem for saccade accuracy because it precludes the use of eye-fixed axes for saccades in favor of the half-angle rule. This strategy is illustrated for the saccades in our study in Fig. 6E. Five graphs, each one depicting five horizontal saccade velocity traces as well as five gaze trajectories, from the side, at each of the five elevations labeled 1-5 (40° up, 20° up, 0°, 20° down, and 40° down, in space coordinates) are shown. Notice how the angle between gaze and velocity becomes more acute as gaze moves downward from primary gaze position (). This occurs because in each case, the velocity traces tilt by approximately half the amount of gaze eccentricity from primary position. Because this effect is kinematically equivalent to the results shown in Table 1, we will henceforth focus on eye positions and gaze accuracy. However, Fig. 6E graphically demonstrates the key observation that Listing's law precludes eye-fixed RE from being mapped trivially onto an eye-fixed rotation (Crawford and Guitton 1997). It also demonstrates that, because the deviations between the eye-fixed and actual axes grow relative to primary position, the predicted errors should also be measured relative to Listing's primary position.

Computing retinal error

This section describes the procedure used to compute a geometrically correct measure of RE and tests the prediction that horizontally displaced targets may not elicit horizontal RE. Figure 7A shows gaze directions from a behind view while subjects stared repeatedly at the 10 target lights, plotted in space coordinates. These data were recorded during calibration trials in which visual feedback allowed subjects' to correctly foveate the desired targets, and therefore these points were taken to represent the desired gaze directions. In these coordinates, the five pairs of lights were indeed displaced horizontally with respect to each other. For reference, Fig. 7B shows 3-D eye position vectors from a side view as subjects made saccades between the same targets. This shows that eye positions fall into a planar range that does not align perfectly with arbitrary space coordinates.

View larger version (21K): [in this window] [in a new window]   FIG. 7. Calculating the Geye (or Teye), produced by each of the 5 target lights while the eye fixates on the paired initial lights, as a geometrically correct measure of RE (Crawford and Guitton 1997). The horizontal saccade calibration task depicted in space coordinates. A: behind view of gaze vectors during 5 repetitions of foveating each light for 1.5 s. Note that the scale of this vector projection system does not correspond exactly to the azimuth/elevation angles (described in METHODS) used to place the targets. Thus, at tertiary positions, 40° elevation appears to be less eccentric. B: side view of 3-D eye position vectors during the same task. The same data are replotted in Listing's coordinates. C: behind view of gaze vectors in Listing's coordinates, where the origin of the coordinate axes corresponds to primary position. Clusters of target direction data are slightly more spread out in head coordinates due to slight shifts in head position against the bite bar. This is accounted for in the space-to-head transformations. D: side view depicts Listing's plane. E: REs, for each of the 5 light pairs (1-5), computed by rotating a vector representing the target direction around the inverse of a vector representing eye position at the initial light. The origin of this oculocentric coordinate system corresponds to the fovea.

These same points were then replotted in headcentric, Listing's coordinates by recomputing gaze directions relative to primary position (Fig. 7, C and D) using a method described previously (Tweed et al. 1990). Figure 7D illustrates how this resulted in an improved alignment of the eye position vectors, in Listing's plane, with the coordinates (this improved alignment was often more dramatic than shown for this particular subject). However, other than a slight shift relative to the newly computed primary position, the overall pattern of target directions (Fig. 7C) remained unchanged.

Finally, we rotated the target direction vectors (at the rightward member of each horizontal pair) by the inverse of 3-D eye position at foveation (at the leftward member of each pair), both in Listing's coordinates, to obtain the "rightward" REs, in eye coordinates, of the rightward targets (see METHODS). As Fig. 7E shows, varying the eye's initial 3-D eye orientation, even within Listing's plane, changed the RE produced by a purely horizontally (in space or head coordinates) displaced target light. The further the subjects' eyes were displaced from primary position, the greater the vertical and (to a lesser extent) the horizontal components of RE deviated from the displacement of the target in space coordinates. This confirmed the predictions of Crawford and Guitton (1997) and shows the importance of taking 3-D eye orientation into account when computing RE. Similar procedures were used to compute RE in all of the examples below.

Large horizontal saccades

The previous figure gives rise to certain testable predictions. If a model of saccade generation changes RE into ME directly, like the LT model proposes, the oblique REs in Fig. 7E should lead to oblique movements of the eye. This would result in a poor oculomotor response, because oblique eye movements could not correctly foveate horizontally displaced targets. To rigorously quantify these predictions, we input real RE and 3-D eye position data from initial light foveation for each individual saccade into our LT model, as described in METHODS. The predicted results, for one subject, are illustrated in Listing's coordinates in Fig. 8A. The subject's initial gaze positions ( to the left), the associated final gaze positions as predicted by our LT model algorithm ( to the right), and average desired gaze points (; from the calibration data) at each elevation are shown.

View larger version (32K): [in this window] [in a new window]   FIG. 8. A: predicted directional accuracy of the LT model vs. actual saccades, for 1 subject.  on the left, the eye's initial gaze positions;  on the right, final gaze positions predicted by the LT model of saccade generation;  to the right, average final calibration positions. B: actual gaze trajectories for the same subject as in A for 5 consecutive saccades at each light elevation. This subject appears to foveate each target accurately, both vertically and horizontally. C: similar data for another subject who consistently undershot the targets. D: data for a 3rd subject who consistently overshot the targets.

The endpoints predicted by the LT model (Fig. 8A) miss their targets in a systematic position-dependent pattern consistent with the outwardly "fanning" pattern of RE shown in Fig. 7E (which used data from the same subject). This is not surprising because, by definition, the LT model maps RE directly onto ME. In this way, the specified direction shown on the retina would be reflected by a similar trajectory of the eye relative to the head. Such a plan would lead to errors that increase systematically with increased eye deviations from primary position. In contrast, the RFT model always predicted accurate foveation of each target (), because it accounts for initial eye orientation (Crawford and Guitton 1997).

After computing the predicted errors of the LT transformation for each individual saccade in each subject, we compared these results to our actual recordings of saccade trajectories. Figure 8B depicts five consecutive saccade gaze trajectories, at each elevation, for the same subject used to generate the predictions in Fig. 8A. Note that these saccades were made in complete darkness and without any visual feedback. Figure 8B illustrates the path of the eyes from the initial, leftward light toward the final, rightward target (). However, in contrast to Fig. 8A, the actual saccade endpoints in Fig. 8B were relatively accurate (as quantified below), and even the slight errors that they did show did not systematically follow the position-dependent pattern of errors predicted by the LT model. Indeed, this subject was relatively accurate both in saccade direction and magnitude.

Figure 8, C and D, illustrates saccade trajectories for two more subjects, in which the first consistently undershot the targets, while the second consistently overshot them. Because all the subjects that undershot the final targets had no difficulty in acquiring them in the visually guided calibration task, we concluded that these final gaze positions were not limited mechanically. All three subjects displayed somewhat curved gaze trajectories that increased their curvature with increased eccentricity from primary position, similar to the "standard plant" simulations of the RFT model (Fig. 3D). However, again, even with the variance found in the horizontal component of final eye positions, these subjects did not show the pattern of directional errors predicted by the LT model (Fig. 8A).

Figure 9, A and B, quantifies the observed directional and magnitude errors as a function of initial eye position in Listing's coordinates. Figure 9A shows horizontal under/overshooting, whereas Fig. 9B shows vertical upward/downward error, measured for each subject, relative to their own calibration data. It is evident from Fig. 9A that considerable horizontal endpoint variability existed between subjects at each elevation. The variation in final horizontal position was 4.86° (SD averaged across subjects). One subject completely overshot all five targets, three consistently undershot, whereas two showed variable under/overshooting, depending on the position of the lights. In general, more overshooting was observed at the center target, whereas more undershooting occurred at the peripheral targets. As mentioned previously, this does not seem to be due to the fact that subjects were physically unable to attain the targets, because during the calibration trials, all five lights were easily foveated. In contrast to the horizontal errors, the range of vertical errors (average SD = 2.00), shown in Fig. 9B, were significantly less variable, t(4) = 5.13 (P < 0.01). In other words, saccade directions were more accurate than saccade magnitudes.

View larger version (22K): [in this window] [in a new window]   FIG. 9. A and B: quantification of horizontal (i.e., magnitude) and vertical (i.e., directional) errors of final gaze position at all 5 light elevations for all subjects. A: horizontal overshoots and undershoots made by the 6 subjects (, , , , , ) to all light elevations analyzed. B: vertical upward and downward directional errors for each light elevation for all 6 subjects as listed above. C and D: quantitative plots of actual vs. predicted (by the LT model) final vertical directional errors. The RFT model predicts a slope of 0, whereas with the LT model, a slope of 1 (- - -) is expected. C: data plotted for 1 subject (refer to Fig. 3B; 5 of 20 saccades quantified here) for all 5 light elevations: 40° up (), 20° up (×), 0° (), 20° down (), and 40° down () (in space coordinates). , regression line for this subject. D: regression lines for all 6 subjects are plotted. Average slope was 0.01 ± 0.14.

The preceding observations suggest that, contrary to the predictions of the LT model, the saccade generator compensated for eye orientation effects when deriving saccade commands from RE. To rigorously quantify the degree to which these subjects compensated for eye position, we plotted actual versus predicted (by the LT model) final vertical gaze errors for each subject. The RFT model predicted a slope of 0 (indicating complete eye position compensation) because no actual errors were anticipated, whereas the LT model predicted a slope of 1.0 (indicating no eye position compensation). Figure 9C shows final actual versus predicted errors for one subject, the slope fit to this data (), and the slope predicted by the LT model (- - -). The data are shifted leftward on this graph (downward in real life) because primary position was relatively high in the range of this subject. The predicted errors grew with increased displacement from primary position to a maximum of 19.90 ± 1.02° at the lowest, most eccentric lights. In contrast, the actual errors remained small, such that the slope of best fit was only 0.04. Finally, similar results were found when we plotted regression lines to the corresponding data for the all six subjects (Fig. 9D). Their average slope was 0.01 ± 0.14 (mean ± SD between subjects), indicating near perfect compensation for 3-D eye orientation.

Radial saccades

The binocular and monocular radial saccade paradigms were designed to test the simulated predictions in Fig. 4. There, saccades from a central target were made to each of eight radially displaced targets with the head in three different orientations (upright, 45° CW and 45° CCW). Recall that in the latter two conditions, where the eyes counterrolled torsionally out of Listing's plane, the simulated saccades missed their targets unless 3-D eye orientations were taken into account. We will now show the actual performance of real subjects in an identical task.

Figure 10 shows five consecutive gaze trajectories to each target (on the left side) and 3-D eye positions corresponding to the same saccades (on the right side) for one typical subject, using the same conventions as in Fig. 4. With the head upright, eye positions gathered around 0° on the abscissa (i.e., eye positions lie in Listing's plane; Fig. 10A, right). Gaze trajectories were relatively straight for purely horizontal and vertical saccades, whereas the oblique eye movements were curved in a systematic manner (Fig. 10A, left) as previously described (Smit et al. 1990; Smit and van Gisbergen 1990). More importantly, note that with the head upright, final gaze positions appeared to be quite accurate in direction (quantified below), particularly in their final direction relative to the targets (an exception being the purely rightward saccades in this specific example).

View larger version (30K): [in this window] [in a new window]   FIG. 10. Actual saccade trajectories () to targets () and 3-D eye position data () are shown for 1 subject. Five gaze trajectories, made by recording continuously from each radial paradigm, are plotted along with eye position quaternions from 1 cycle of each paradigm. A: with the head upright (0° ocular torsion). Left column: behind view shows gaze trajectories are straight along the axes and curve slightly in the diagonal directions. Their endpoints appear to be accurate. Right column: side view indicates that average eye positions lie at 0.42 ± 0.45° CW torsion. Essentially, these eye positions are in Listing's plane. B: with the head rotated 45° CW. Left column: behind view saccade trajectories are somewhat less accurate and more variable. Again straighter saccades are seen along the cardinal directions relative to the head, whereas curved paths are followed for the oblique directions. Right column: side view quaternions indicate that the eye rotates in the CCW direction, on average, by 12.55 ± 0.49° and remains there while the subject saccades to every target from center. C: with the head rotated 45° CCW. Left column: behind view indicates similar results as in B. Right column: side view shows that the eye rotates in the CW direction, on average, by 11.50 ± 1.76°, and again, remains there throughout the trial.

Rotation of the head about the torsional axis caused a compensatory torsional counterroll of the eyes. On turning the head 45° CW, the torsional component of eye position deviated in the CCW direction (Fig. 10B, right). In this case, average eye-in-head CCW counterroll was found to be 8.71 ± 2.61° (mean ± SD, across subjects) during binocular viewing and 7.32 ± 5.56° during monocular viewing. Conversely, with the head rotated 45° CCW, the subjects' right eyes rotated in the CW direction (Fig. 10C, right). The average CW counterroll was 10.26 ± 2.23° during binocular viewing, and 7.52 ± 2.41° during monocular viewing. The eye positions seen in the latter two conditions were clearly shifted away from the normal Listing's plane, and, from these new eye positions, saccades resulted in eye position trajectories that remained out of the normal Listing's plane (Fig. 10, B and C, right).

To keep the location of the target lights consistent, irrespective of head orientation (indicated by the caricatures in Fig. 10), gaze was plotted in space coordinates in Fig. 10, A-C. Taking this and the head tilts into account, gaze trajectories continued to be relatively straight for horizontal and vertical saccades made relative to the head, while saccades made obliquely relative to the head were curved as mentioned previously (Fig. 10, B and C, left). The trajectories of these oblique saccades were initially too horizontal (re: head), but then changed course about two-thirds of their way to the target, despite the lack of any visual feedback. This could be accounted for by the relative strengths of the horizontal recti muscles, and probably does not reflect the visuomotor transformation.

Our main data analysis therefore focused on the endpoint accuracies of the initial saccades. In general, more directional errors were seen in final gaze direction with the head tilted, as compared with saccades made with the head upright (quantified below). Examples include the down-left and down-right (re: space) saccades in Fig. 10B, and the down-right and up-left (re: space) saccades in Fig. 10C. However, these errors did not qualitatively seem to follow the pattern predicted by the LT model (Fig. 4), i.e., the trajectories were not consistently tilted in the direction of head rotation (by the rule illustrated in Fig. 4) in either of the two head tilted orientations. In any one condition, we found great variability in the direction of observed errors both between and within target directions (upward and rightward saccades in Fig. 10B, and upward and leftward saccades in Fig. 10C). Thus a more rigorous quantification was necessary.

Figure 11 summarizes the endpoint accuracies of the subjects in the six tasks. Figure 11, A, C, and E, shows binocular data, Fig. 11, B, D, and