INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

This study deals with the coordinated movements of the eye and head during large gaze shifts, i.e., head-free gaze saccades. This topic has been studied with increasing frequency during the last decade, both for its intrinsic importance and as a possible general model for coordination and control. Initial studies focused on one-dimensional (1-D), i.e., horizontal, aspects of eye-head coordination (e.g., Bizzi et al. 1971; Dichgans et al. 1973; Fuller 1996; Guitton 1992; Guitton and Volle 1987; Laurutis and Robinson 1986 Tomlinson 1990) but recently have been extended to two dimensions (Freedman and Sparks 1997; Goossens and Van Opstal 1997). Moreover, the three-dimensional (3-D) kinematics of human gaze shifts, i.e., the trajectories of eye and head orientation in space, now have been described in some detail (Glenn and Vilis 1992; Misslisch et al. 1998; Radau et al. 1994; Tweed et al. 1995). The latter investigations largely focused on behavioral constraints that reduce the redundant degrees of freedom of the eye-in-space (Es), head-in-space (Hs), and eye-in-head (Eh) during visual fixation. However, it is unclear how the brain organizes 3-D Hs trajectories and coordinates 3-D Eh saccades with vestibuloocular reflex (VOR) slow phases to implement these final fixation constraints. Furthermore, it remains unclear whether these constraints exist to optimize anatomy- or task-related variables (Radau et al. 1994). Finally, an animal model is required to investigate the physiological mechanisms of these behaviors. The purpose of the current investigation was to develop such a model by describing the 3-D kinematic constraints on eye-head coordination during visual fixation in the monkey, to determine how the trajectories of the eye and head are controlled to achieve these constraints, and to determine if these constraints optimize anatomic or behavioral requirements.

If the primary purpose of the head-free saccade generator is to redirect gaze direction (i.e., the visual axis in space), then its secondary purpose is probably to determine the 3-D orientation of the eye about this axis. This orientation is important because it determines the geometric correspondence of lines in space with the eye and thus the pattern of retinal stimulation (Crawford and Guitton 1997a; Glenn and Vilis 1992; Haustein and Mittelstaedt 1990; von Helmholtz 1925; Klier and Crawford 1998). In general, Donders' law predicts that the eye will achieve the same orientation for each gaze direction, irrespective of the preceding trajectory (Donders 1848). When the head is mechanically fixed or at rest, a specific form of Donders' law called Listing's law is obeyed (Ferman et al. 1987a; von Helmholtz 1925; Tweed and Vilis 1990). Listing's law states that 3-D eye position vectors (defined in the following text) align within a plane such that rotations about a head-fixed torsional axis are minimized. The question thus arises, does Listing's law continue to hold for Es when the head is free to move?

It appears that when the human head is free to move, Donders' law of Es still is obeyed, at least approximately (Straumann et al. 1991). However, the orientations of Es do not form a Listing's plane but rather a twisted 2-D surface (Glenn and Vilis 1992; Radau et al. 1994). The observed twist is more consistent with the orientations produced by a Fick gimbal (e.g., like a telescope mount), where ideally, horizontal movements are achieved through rotations about a body or space-fixed vertical axis, but vertical rotations occur about an eye-fixed horizontal axis. This provides the observed form of Donders' law, where rotations about the third, eye-fixed torsional axis are minimized. This strategy has different perceptual consequences than Listing's law; for example, it tends to better preserve horizontal lines on the retina (Glenn and Vilis 1992). However, it was initially unclear whether this apparent constraint on Es results from a deliberate control strategy or whether it follows trivially from more fundamental mechanical or neural constraints on Hs and Eh.

Like Es, Hs and Eh each pose a degrees-of-freedom problem for their respective control systems because these segments are mechanically (and behaviorally in some circumstances) capable of rotating in the redundant torsional dimension (Collewijn et al. 1985; Crawford and Vilis 1991, 1995; Hess and Angelaki 1997; Misslisch et al. 1994a). When the orientation vectors of Eh are quantified during a range of human head-free gaze fixations, they appear to form (or at least are indistinguishable from) a plane similar to the head-fixed Listing's plane (Radau et al. 1994). The human head also appears to follow Donders' law during head-free visual fixations (Straumann et al. 1991; Tweed and Vilis 1992), but in this case it is more consistent with the Fick gimbal strategy described earlier for Es (Glenn and Vilis 1992; Melis 1996; Theeuwen et al. 1993). This means that all three possible measures (Es, Hs, and Eh) obey Donders' law at least in approximation, but geometrically only two of these rules need to be enforced and the third will simply fall out: but which one is this? Radau et al. (1994) attempted to answer this by quantifying the variance of torsion in each of these segments and found that of the three, static Eh torsion was controlled most tightly and Es torsion was controlled least tightly. They concluded that the brain was actually controlling Hs and Eh orientation and that, at least in terms of control, the Fick-like nature of Es orientation was the trivial fall-out of these more fundamental rules. Our first goal was to similarly evaluate the head-free orientation ranges of Es, Hs, and Eh during visual gaze fixations in monkeys.

Knowing the orientations of the eye and head during gaze fixations does not tell us how these segments rotate to achieve these positions. Because of the noncommutativity of rotations, Donders' law can never be achieved by consistently rotating a given segment about the shortest path axis orthogonal to current and desired pointing direction. With Listing's law, there is a kinematic rule that selects a single ideal axis of rotation for any one saccade that will preserve eye position in Listing's plane throughout the trajectory (Tweed and Vilis 1990). However, during head-free gaze shifts, this is problematic for both the eye and the head.

First, the eye cannot use a simple kinematic strategy to maintain Listing's law during head free gaze saccades because of the VOR. Each gaze shift typically is accompanied by an initial Eh saccade that is followed by a VOR slow phase, which stabilizes desired eye orientation in space once the visual target is acquired to prevent overshoot while the head continues to move (Bizzi et al. 1971; reviewed in Guitton 1992). The VOR is known to not obey Donders' law but rather rotates the eye about the same or nearly the same axis as the head (Crawford and Vilis 1991; Haslwanter et al. 1995; Hess and Angelaki 1997; Misslisch et al. 1994a), resulting in accumulation of Eh torsion (Smith and Crawford 1998). Recent gaze saccade experiments have suggested that the 2-D components of Eh saccades anticipate subsequent VOR slow phases to bring the eye toward a final desired position in the head (Crawford and Guitton 1997b; Tweed et al. 1995). This concurs with the earlier observation that during passive head rotations, saccade-like quick phases drive the eye out of Listing's plane in an anticipatory fashion such that vestibular driven slow phases end up bringing the eye back toward Listing's plane (Collewijn et al. 1985; Crawford and Vilis 1991; Crawford et al. 1989).

Here we consider whether Eh saccades during head-free gaze shifts would similarly drive eye position out of Listing's plane in an anticipatory fashion such that VOR slow phases would end up bringing the eye back into Listing's plane at fixation. This would require very precise control over all the torsional components of the saccade burst generator (Crawford et al. 1997; Henn et al. 1989) during goal-directed gaze shifts. If so, this would be difficult to reconcile with the view that Listing's law emerges from a simple projection of visual vectors onto a saccade command aligned in Listing's plane (Raphan 1997, 1998). Furthermore, this would have important general implications for understanding the organization of eye-head coordination downstream from the common gaze error command (Freedman et al. 1996; Munoz et al. 1992; Robinson and Zee 1981).

Finally, it is not yet clear how the trajectories of head motion subserve the Fick gimbal strategy observed during fixation. In particular, it is not known if Donders' law of the head is maintained during movements or if it is only obeyed between movements. It has been observed that during repetitive back-and-forth head movements, humans transiently violate the Fick gimbal strategy in favor of a "minimum rotation" strategy, where the head simply rotates about the axis orthogonal to the plane of its initial and final facing directions (Tweed and Vilis 1992). However, it is not yet clear what happens during normal, nonrepetitive movements. Presumably the head-control system might achieve purely vertical or horizontal rotations about constant axes very similar to those seen in the telescope-like physical implementation of a Fick gimbal. In other words, it might maintain Donders' law throughout the course of each movement by rotating the head about a constant, space-fixed vertical axis during horizontal movements and a constant head-fixed horizontal axis during vertical movements. If so, then the head would behave both statically and dynamically like Fick gimbals.

However, by corollary, because these horizontal and vertical axes of Fick coordinates are fixed thus in different segments, in some situations mechanically rigged Fick gimbals must rotate the controlled object about nonconstant axes, i.e., axes that do not have a constant orientation in space. In particular, during large oblique head movements, the head-fixed horizontal axis would have to rotate along with the head around the space-fixed vertical axis during the movement to maintain head orientation at zero torsion in Fick coordinates. In other words, a mechanically rigged Fick gimbal with torsion fixed at zero rotates about nonconstant, torsionally looping axes during oblique movements. Alternatively, without this dynamic constraint, any final Fick orientation can be reached from any initial Fick orientation by a constant axis of rotation that transiently violates the Fick form of Donders' law during the movement. It is currently unknown if the head-control system optimizes head axes in this fashion during oblique head movements or maintains Donders' law during movements according to the previous alternative.

The latter question is important, because it may reveal whether the Fick gimbal strategy of the head is mechanical or neural in origin, and if neural, whether it is implemented in a feed-forward fashion while planning the movement or within an internal feedback loop during the movement. Moreover, if this strategy is neural in origin, then it might show some capacity for plasticity or task-dependence (Tweed 1997), which could in turn reveal the normal functional significance of the Fick gimbal strategy of the head. For example, even if the Fick strategy is neural in origin, it still may function to optimize either anatomic (Radau et al. 1994; Theeuwen et al. 1993), perceptual (Glenn and Vilis 1992), or task-related requirements. We set about to test among these options both in normal gaze shifts and also in a pin-hole goggle paradigm where several task constraints (that might normally encourage or enforce the Fick strategy) were removed (Crawford and Guitton 1997b). Some of these experiments have been described briefly in abstract form (Crawford and Guitton 1995; Guitton and Crawford 1994; Klier et al. 1998).

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Surgery and equipment

Three monkeys (Macaca fascicularis), henceforth designated animals M1, M2, and M3, underwent aseptic surgery under general anesthesia (isoflurane, 0.8-1.2%) during which they were fitted with an acrylic skull-cap. Two 5-mm-diam scleral search coils also were implanted in the right eye of each animal for 3-D recordings, using a method described previously (Crawford and Vilis 1991; Tweed et al. 1990). Animals were given analgesic medication and prophylactic antibiotic treatment during the postsurgical period. Experiments commenced after 2 wk of postoperative care. All animal care and experimental procedures were approved by the McGill and/or York Animal Care Committees and were in accordance with the Canadian Council on Animal Care policy on the use of laboratory animals.

Before experimentation, animals were placed in a Crist Instruments primate chair, with the top plate modified to reduce encumbrances to head movement. Specifically, for the first two animals (M1 and M2), the top plate was integrated to fit flanges on custom-made plastic monkey collars, such that the collar slid in, leaving a 1-cm-thick, 7.5-cm-diam ring of rigid plastic around the neck (Crawford and Guitton 1997b). A recessed hollow also was built into the front of this plate so that it did not interfere with the animal's chin on looking down. Other than gaze directions >50° downward (which we did not explore), rotational head movements and the visual range were left unobstructed by this apparatus. However, the rigid structure of the materials surrounding the neck restricted translational movements of the head and forced the animal to sit upright without slouching. To test whether these mechanical constraints might interact or interfere with the neural constraints on head orientation, a different chair was used with the third animal (M3). In the latter chair, the rigid structures surrounding the neck were replaced altogether by a reenforced cloth that was buckled snugly at the animals back during experiments, similar to the system used by Freedman and Sparks (1997). This arrangement allowed animal M3 to obtain its accustomed slouching posture and make unobstructed head translations over a reasonably wide range.

During experiments, the primate chair was placed such that the head was at the center of three mutually orthogonal magnetic fields (1.5 m diam, giving uniform quaternion measurements within ±15 cm of center). Signals were low-pass filtered at 125 Hz, 12 dB/octave and digitized at 100 Hz from the eye coils and from two orthogonal 1-cm "head" coils embedded in plastic. For head-free experiments, the head coils and wire leads for the eye coils were secured with nylon screws to the skull-cap. These preparatory procedures were performed with the head fixed via a bolt screwed to a frontally placed cylindrical implant, which could be detached using a quick release mechanism, and the head-restraint assembly swung backward on the primate chair for relatively unencumbered head-free recordings. Because chest orientation was not measured, head orientation relative to the chest could not be precisely computed. However, given that the chest was nearly stationary during experiments, our measures of Es and Hs were roughly equivalent to eye and head relative to the chest. Some translational movement was observed qualitatively during experiments with M3 but was not quantified. Therefore this study could not directly address the stability of rotational axes in translational space (Melis 1996) or the contributions of head translations to gaze control but rather focused on measurements of behavioral constraints on 3-D orientation.

Visual tasks and training

All training was performed with the head unrestrained. Animals initially were trained to sit upright and face forward in the primate chair. Animals were next trained to fixate "primate treats" held at a 80 to 100 cm distance throughout a large range of the visual field (Crawford and Guitton 1997b). These targets were held at the end of forceps and moved very rapidly from location to location to avoid pursuit movements. If the animal faithfully followed the visual target, it (the treat) was fed manually to the animal. This procedure was continued until the animal would follow the target with discrete gaze saccades for ~3 min before acquiring the reward. Other than this requirement, no time constraints were added, i.e., animals were always allowed make self-paced gaze shifts between targets using an eye-head coordination strategy of their own choosing. This paradigm, where the visual target was the reward, was designed to emulate a biologically natural pattern of motivated gaze shifts. It resembled the food/barrier paradigm used extensively to investigate head-free gaze shifts in the cat (e.g., Guitton et al. 1990) except that a rich amount of visual motion information was available during repositioning of the target.

During control experiments, four standard paradigms were used. First, the radial task: animals oriented toward the target with gaze saccades toward and away from center, following a roughly even distribution of directions. A reproducible center point was maintained by fixing a pair of nylon strings orthogonally across the front of the Helmholtz coils forming a cross-hair directly ahead of the right eye in magnetic field coordinates. Among other things, this task was used to determine a reference point with a behaviorally natural combination of eye and head positions (see following text). Second, the repetitive task: using visual aids that could be repositioned at the front of the Helmholtz coils as a guide, targets were repositioned back and forth to induce repetitive horizontal/vertical gaze shifts at various vertical/horizontal levels. Third, the hour-glass paradigm (illustrated graphically in RESULTS): similar to the last, but inducing horizontal, vertical, and oblique movements in a reproducible but nonrepetitive fashion. Finally, the random task: the target was repositioned throughout the visual range in a nonrepetitive 200-s sequence that covered the range in as random an order as possible. Targets were held as eccentrically as possible without inducing whole body rotations (as determined through trial and error): approximately ±60-80°. These measurements were repeated on subsequent days for a total of 10 control experiments in animal M1 and 19 in M2. Nine further random-task experiments were obtained in animal M3, but this last animal could not be induced to comply with the task requirements of the more demanding paradigms.

When initial training was completed and control data were gathered, animals also were fitted with a pair of opaque plastic goggles that attached (via wing nuts) to screws implanted in the skull-cap. These goggles were shaped to the contour of the monkeys' faces with additional soft flanges at the edge so as to completely occlude vision in both eyes. A single round aperture then was added such that the right eye was given a useful visual range of ±4°. Details of this procedure have been described elsewhere (Crawford and Guitton 1997b). Forty-five minute per day training sessions with the goggles then commenced. Initially, animals were trained to visually follow slow movements, and then as the speed of their gaze movements increased, targets were displaced rapidly over a wide range until they easily made rapid searching head movements to visually acquire the target. Invariably animals mastered horizontal gaze shifts first, then vertical gaze shifts, and finally oblique gaze shifts. Between training periods, the goggles were removed, and the animals were returned to their normal housing area. As judged subjectively, the total training time required for rapid, accurate, and discrete movements in all directions and throughout the entire range required ~4 wk in animal M1, and ~10 days in animal M2. After this training period, the radial and random tasks were repeated both without and then with the goggles, for a total of five experiments in animal M1 and 9 in M2. (Further behavioral experiments done with these animals/preparations were reported elsewhere) (Crawford and Guitton 1997b.)

Data analysis

The analysis of 3-D movements evokes several challenging computational problems, such as nontrivial coordinate system problems, degrees-of-freedom problems, and the noncommutativity of rotations (Haslwanter 1995; Tweed 1997). Thus a very rigorous and clearly defined quantification of the data was required. A computer was used to convert coil signals into eye position quaternions using a method described previously (Tweed et al. 1990). Quaternions were used because, unlike raw coil signals, they provide an accurate and convenient measure of 3-D eye position over a 360° range. Quaternions represent each eye position as a fixed-axis rotation from some specified reference position (Westheimer 1957). Quaternions are composed of a scalar part q₀, and a vector part q. It is the vector part that is used for representation of data. To interpret the data, one need only understand that q is parallel with the axis of eye rotation, and its length is proportional to the magnitude of this rotation. To be specific, a quaternion is related to the axis and magnitude of a rotation as follows

(1)

(2)

The angle  is the magnitude of the rotation and n is a 3-D unit vector parallel to the axis of rotation. At reference position  = zero and so q₀ = 1.0 and q = zero. The quaternion vector can be broken down into three components, ordered such that q₁ represents torsional position, q₂ represents vertical position, and q₃ represents horizontal position, with signs arranged to satisfy the right-hand rule. Throughout this paper, the Hs and Es quaternions are expressed in a right-handed coordinate system aligned with the space-fixed magnetic fields of the Helmholtz coils.

The choice of reference positions for the eye and head is potentially somewhat arbitrary in the head-free preparation because an infinite number of eye-head combinations could subserve a single gaze direction, and there may not be a rigorously definable "primary position" for Es (Glenn and Vilis 1992). To choose a convenient and behaviorally meaningful set of reference positions for each experiment, we scrutinized the spread of eye and head coil signals during fixation of the central target in the head-free radial task and then selected a point where both the 2-D pointing directions of the eye and head were near the center of their somewhat variable ranges (see Fig. 2). This resulted in a reference position for Es orientation defined in magnetic field coordinates, where gaze was directed straight ahead along the forward pointing field; a behaviorally central reference orientation for Hs orientation defined similarly in field coordinates; and a behaviorally central reference orientation for Eh, defined in head coordinates, which were themselves defined by the intersection of the magnetic fields with the head when it was at its reference position.

The position of Eh then had to be computed from both the eye coil signals (which directly gave eye position in space) and the head coil signals. In one dimension, this would be done by subtracting head position from gaze position. In three dimension, this was done by dividing the Es quaternion by the Hs quaternion (Glenn and Vilis 1992) as follows

(3)

This provides the quaternion for eye position in head coordinates. As described in the preceding text, the alignment of these coordinates with the head was initially quasiarbitrary, i.e., depending on the alignment of the magnetic fields with the head when it was at the chosen reference position. However, in some cases, these data were rotated into a new coordinate system such that the head-free analogue of Listing's plane aligned with the new coordinates, and the Eh data were measured relative to a new reference position analogous to primary position (Tweed et al. 1990). On occasion (e.g., Fig. 5) a similar procedure also was used with Hs quaternions.

2-D "pointing vectors" were computed from quaternions as described previously (Tweed et al. 1990). These unit vectors typically were projected onto the frontal plane to indicate where the eye or head was pointing. For the eye, these vectors align with the visual axis in either space coordinates, i.e., gaze, or head coordinates. For the head, they can be thought of as the aiming direction of the nose. Finally, instantaneous angular velocity vectors were computed from position quaternions using the method described previously (Crawford and Vilis 1991). These were vectors aligned with the instantaneous axis of rotation of length equal to the instantaneous angular speed of rotation. Like the quaternion vectors, the angular velocity vectors were expressed according to the right-hand rule, as described further in the following text. Because translations could not be measured, "axis constancy" in the RESULTS refers to the orientation of the velocity vector during movements.

To quantify the orientation ranges of Es, Hs, and Eh, first-, second-, and third-order fits were made to Es, Hs, and Eh quaternions using the previously described procedure (Glenn and Vilis 1992; Radau et al. 1994; Tweed et al. 1990). The following formula provides the equation for a second-order fit for a generic position quaternion (q)

(4)

This is the 3-D analogue to the equation for a 1-D line in 2-D space and represents a description of torsional position as a function (through parameters a₁-a₆) of various combinations of vertical and horizontal position up to the squared terms. A first-order fit uses only the first three terms, whereas a third-order fit adds four more terms (a₇-a₁₀) for the cubic combinations of q₂ and q₃. Once such a 2-D "surface" of best fit had been computed, the standard deviation of the torsional components of all actual positions from this fit was computed using the method described previously (Tweed et al. 1990).

An additional measure of the 3-D range, called the "gimbal score" also was applied to the head-orientation data (Glenn and Vilis 1992). For this test, the head quaternion data first were rotated into alignment with the pitch-yaw axes plane using the parameters derived from Eq. 5 and then fit to the following equation to provide the single term s

(5)

The resulting number provided a convenient measure of the data's fit to a zero-torsion range at some point along the continuum from Fick coordinates to Listing coordinates to Helmholtz coordinates (described further in RESULTS).

The preceding measures of fit and variance were applied to several subranges of the data selected on the basis of various velocity criteria. For example, the "fixation" range was defined to be those positions where the velocity (more correctly the 1-D magnitude of 3-D velocity) of both Es and Eh were <10°/s. Other, more complex criteria were used to select the position ranges at peak head velocities, at the starts of Eh saccades, at the ends of Eh saccades, at the starts of slow phases, and at the ends of slow phases. Simple but robust algorithms for selection of these subranges (provided in the APPENDIX) were developed on a trial-and-error basis from the recorded 3-D data.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Figure 1 reveals most of the issues and challenges of describing eye- and head-orientation trajectories in 3-D space. These data were selected from a sequence of large, pseudorandom-gaze shifts between widely dispersed peripheral targets (i.e., the random task) in M1. The primary object of control, 2-D gaze direction in space, is shown in Fig. 1A;  show the trajectories of gaze while it was moving, whereas  indicate points of fixation between these movements (where both the eye and head were stationary). Note that these data portray gaze as the tip of a unit vector, projected onto the page as if from behind the subject (the perspective is indicated by the head caricatures in the figures). When similar representations are used below, they will be referred to as "gaze vectors" (for Es) or "direction vectors" (for Hs). This particular set of data (Fig. 1) was selected to show a sequence of very large pseudorandom gaze shifts between eccentric targets, roughly corresponding to the limit of the obtainable range in our preparation. Clearly, the animal could acquire a large range of visual targets in all directions (with the downward visual range somewhat limited because it was obscured by the primate chair).

View larger version (53K): [in this window] [in a new window]   Fig. 1. Basic two- and three-dimensional (2-D and 3-D) kinematics of randomly directed gaze shifts. Same primary data set is used throughout. A: projection of 2-D gaze vector tips onto the plane containing the horizontal and vertical magnetic fields. Such a projection results in a nonlinear (~sine of gaze angle) scale that becomes squashed toward the 90° mark. , gaze trajectories; , fixation points, where both gaze and head velocity are <10°/s. Head-and-shoulders symbol indicates that the data are viewed from behind in space coordinates. B-E: 3-D kinematics of eye and head orientation, showing "side" projections (see head/shoulders) of quaternion vectors. , orientation trajectories; , fixation data. These plots employ the right-hand rule convention, such that the vertical axis indicates positions rotated horizontally from reference (see curving arrows), and the horizontal axis indicates torsionally rotated positions. B: 3-D orientations of the eye-in-space. C: 3-D orientations of the head-in-space. D: 3-D eye-in-head orientations with trajectories of saccades shown. E: eye-in-head orientations with trajectories of slow phases shown. Initial untransformed headcentric coordinates were used in the latter 2 plots, i.e., not Listing's coordinates. Animal M1 shown.

The remainder of Fig. 1 (B-E) uses a different convention to portray 3-D orientations from the same subset of data as Fig. 1A. Again, both 3-D trajectories during movements () and orientations during eye/head fixations () are shown. However, these data now represent 3-D position vectors. For example, if one chooses any one fixation point or any one point on a trajectory, it is actually the tip of a vector emanating from the origin. Each vector is aligned with the axis of rotation that would take the eye or head from the reference position (defined in METHODS) and is stretched to a length proportionate to the angle of this relative rotation. Furthermore these vectors are oriented according to the right-hand rule, such that a vector pointing upward (thumb of right hand) represents a position rotated to the left (curl of fingers on right hand). Note that these data are portrayed in orthogonal coordinates, where the torsional, horizontal, and vertical coordinate axes are all fixed either in space or in the head. The head caricatures in Fig. 1, B-E, indicate that these coordinates and 3-D position vectors are viewed from the right of the subject throughout the figure, such that they show the horizontal and torsional components of eye and head orientations (head + torso symbols designate space coordinates, whereas head only symbols designate head coordinates).

Figure 1B shows the orientations of Es that correspond to the gaze directions in A. It is important to remember that in these and all further illustrations, Fick coordinates are not used to represent the data, and thus torsion is not rotation about the visual axis. In this case, the torsional coordinate is an axis fixed in space. With these coordinates and data, it is evident that the range of eye torsion in space is relatively compact compared with the horizontal range but still shows considerable variability (~40° peak to peak), both during and between fixations. Thus a flat Listing's plane was not evident in such plots, but at this point it was not clear whether Es torsion obeyed some other nonplanar constraint, was simply much sloppier than head-restrained eye torsion (Tweed et al. 1990), or both.

The remainder of the figure shows the head and Eh orientations that contributed to the Es orientations shown in Fig. 1B. Figure 1C shows similar vector representations of head orientation in the same space-fixed, orthogonal coordinate system. The head range seems slightly more compressed torsionally and more aligned with the vertical but is otherwise similar to the Es range. However, there is still considerable variability (~30° peak to peak) in torsional head positions during gaze fixations ().

Finally, Fig. 1, D and E, shows orientations ranges of Eh, plotted in the same scale as the preceding data. The fixation points () are the same in both D and E, showing Eh orientations at the end of multiple-step gaze shifts, where both the eye and head were stationary. However, the position trajectories (underlying continuous lines) have been subdivided to show saccade trajectories (when head was also moving) in D, and VOR slow phase trajectories in E. Compared with the head and Es, which showed considerable variability in torsion both during movement and head/gaze fixation (), the Eh fixation range () formed a relatively compact range. This range was clearly most compact along the dimension indicated (- - -). However, the Eh trajectories during movements () were less restricted, extending outward from the fixation ranges both during saccades (D) and slow phases (E).

Although Fig. 1 provides an overview and some clues to the kinematics of eye-head coordination, it also shows the somewhat daunting complexity of deriving simple rules from this complex data. To derive such rules, we had to analyze the data in a more reductive fashion. The subsequent sections will adhere to the following course of analysis, noting first that Es orientation during fixations is most directly relevant to vision (the biological task of this system). Second, the head and Eh orientations observed during these fixations thus presumably will be the goals of the eye- and head-control systems. Therefore we will begin by considering the ranges of Es orientation during stable gaze fixations, and the ranges of head and Eh orientation that contribute to these positions ( in Fig. 1). After establishing the rules that govern these ranges, we then will consider the movement trajectories used to obtain these ranges.

3-D fixation ranges of Es, Hs, and Eh

Figure 2 introduces the subject by showing the ranges used to fixate a single target. These data were selected from a series of gaze fixations toward a central, straight-ahead target, reached by a radial pattern of centripetal gaze shifts (not shown) from a wide distribution of peripheral targets (the radial task). Figure 2, top (A-C), shows the horizontal and vertical components of 2-D direction vectors. The gaze vectors for Es (A) form a tight group around the zero point, showing that the target was foveated accurately after each centripetal gaze shift. In contrast, the vertical and horizontal components of the head (B) and the Eh (C) vectors show considerable variability during the same fixations. This is possible because an infinite number of combinations of head (B) and eye (C) position can give rise to the same gaze direction in space (A). Considering only gaze fixations within ±1° of the central fixation point, the range of horizontal head position (quantified as 4 SD, i.e., ~98% of the data) was 14.5° in animal M1 and 10.0° in M2, averaged across all radial task experiments. The average vertical head range was also 14.5° in animal M1 but lower at 5.2° in M2. (Note that, as follows trivially, Eh showed almost identical variations). Such 2-D variations have been described before (e.g., Freedman and Sparks 1997; Radau et al. 1994) and will not be examined further here. The point to be taken is that from this 2-D perspective, the system appeared to control gaze direction much more tightly than the precise combinations of eye and head position that contributed to gaze.

View larger version (34K): [in this window] [in a new window]   Fig. 2. Two- and 3-D distributions of gaze, eye, and head position during steady fixation of a central target. Data were selected from the radial task, where the animal arrived at the central target (A) from gaze directions in a variety of eccentric, uniformly distributed directions (trajectories not shown). Top, A-C: frontal projections of 2-D pointing vectors for gaze direction (A), head direction (B), and eye-in-head direction (C). Middle and bottom, D-I: projections of 3-D orientation vectors for eye in space, head in space, and eye in head, using the right-handed rule. Middle: horizontal position as a function of torsion (i.e., a "right-sided" perspective), whereas bottom plots vertical position as a function of torsion (requiring a slightly unorthodox perspective of the rotation axes from below the head/body to have upward positions upward and counterclockwise positions rightward on the page). Animal M1.

The 3-D orientations of the same eye and head data revealed a somewhat different picture. The remainder of Fig. 2 shows the torsional orientations of the eye and head plotted against horizontal position (middle) and vertical position (bottom). In the case of Es (left), the torsional orientation is considerably more variable than either horizontal (D) or vertical (G) position. In this particular task, the space-fixed torsional coordinate happens to align with the visual axis. Most interestingly, whereas horizontal and vertical position were controlled tightly to land gaze on target (A), the torsional orientation of Es around the line of gaze varied by 20° for this same single target direction (Fig. 2, D and G). Using the 4 SD definition from the preceding paragraph, the average torsional range of Es was 9.3° in animal M1 and 13.7° in M2, considerably larger than the ±1° gaze window used to compute these values. (Animal M3 refused to fixate the central target over enough repetitions to perform this measurement but showed qualitatively similar results).

In contrast, Eh torsion formed a very compact range much smaller than the horizontal (F) or vertical (I) range of Eh positions used for this one target. Indeed, even for a single target these ranges give the appearance of a tilted Listing's plane, suggesting that eye orientation was only allowed to vary within a quasiplanar range. Because torsional variance previously has been quantified using the single standard deviation (e.g., Glenn and Vilis 1992; Radau et al. 1994), we henceforth will follow the same convention. The average (across experiments) torsional SD of Eh was only 0.99° in M1 and 0.72° in M2 in this task (where gaze was within ±1° of center), considerably less than those of Es (2.32 and 3.43°, respectively). The distribution of head orientations (Fig. 2, E and H) was intermediate between these extremes, showing roughly equal ranges of torsion, vertical, and horizontal variance (2.99, 2.46, and 3.07° SD, respectively; averaged across experiments and then between animals).

Thus from this data, it would appear that the 2-D contributions of the eye and head to gaze direction were controlled rather loosely compared with gaze in space (top) but that Eh torsion was controlled much more tightly than Es torsion (middle and bottom). However, the rule governing head position is not yet clear from Fig. 2. To more completely characterize these rules, we next examined orientations over a much wider distribution of gaze fixations toward multiple target directions.

The first two columns of Fig. 3 show 3-D orientation vectors during fixations toward a wide range of target directions, as best seen in A. (The data range in the previous Fig. 2 corresponds roughly to a small subset of the data near the coordinate origins in Fig. 3.) In Fig. 3, Es (top), Hs (middle), and Eh (bottom) orientations are viewed from the behind perspective (1st column), i.e., vertical versus horizontal, and side perspective (2nd column), i.e., torsion versus horizontal. From the behind perspective, it is evident that the Hs fixation range (B) was elongated in the horizontal dimension, whereas the Eh range (C) was elongated in the vertical dimension. Because the Es range (A) roughly corresponds to the sum of the latter two ranges, the head thus contributed relatively more to the horizontal Es fixation range, whereas Eh contributed relatively more to the vertical Es range. This was typical in our animals and matched previous reports of head-free primate behavior (Freedman and Sparks 1997; Glenn and Vilis 1992) but was not further quantified. From the side perspective, it is evident that for a large range of fixation directions, torsional Es (D), Hs (E), and Eh (F) ranges were somewhat restricted, but the precise nature of this restriction remains incomplete in these plots.

View larger version (50K): [in this window] [in a new window]   Fig. 3. Fixation ranges of eye in space (top), head in space (middle). and eye in head (bottom) during random-gaze shifts. A-F: quaternion vectors are plotted according to the right-hand rule from the behind perspective (A-C) and side perspective (D-F), as indicated by the caricatures. Eye-in-head data have not been rotated into Listing's coordinates. Same data set is used throughout. G-L: 2nd-order surface fits to the fixation data in previous columns. Each grid indicates 10° vertical/horizontal across the rectangular surface as it would be seen from a "behind view" (not shown), with limits corresponding approximately to the range of data used in the fit. (Because the 2-D data ranges are somewhat rounded or oval shaped, whereas the illustrated fits are square, they overextend the data range at the corners.) Thickened lines correspond to the upper and leftward (in terms of gaze direction) edges of the illustrated fit, and the corners of the fit are labeled (DL, down-left; UL, up-left; UR, up-left; DR, down-right) according to gaze direction. G-I: side perspective, corresponding directly to D-F. J-L: below perspective of the same data, showing the dependence of torsion on vertical position. Animal M1.

To reveal the 3-D shape of these flattened position ranges, we employed the same methods used in similar human studies (Glenn and Vilis 1992; Melis 1996; Radau et al. 1994), applying second order "surface" fits (Eq. 5). The remaining rightward columns of Fig. 3 graphically depict the surfaces fit to the data shown in the first two leftward columns. These plots show the dependence of torsional position on the other components of position, where each grid mark indicates 10° of horizontal/vertical excursion. The third column of Fig. 3 shows the side view corresponding to the second column, whereas the rightmost column of Fig. 3 depicts the "below" view, showing the dependence of torsional orientations on vertical positions in a way that orients upward position upward on the page. (Imagine that the plots in the 3rd column were rotated 90° about the torsional axis, bringing their bottom parts out of the page, to obtain the "below" perspective in the 4th row). The rectangular corners of these fits, which usually extended beyond the more rounded range of the actual data, are labeled according to the corresponding 2-D pointing directions (DL, UL, UR, DR) to facilitate interpretation.

The fits to the Es and Hs data show the same striking feature: the ranges are twisted into a flattened 2-D bow-tie-like shape. This is most evident from the side views (note that the apparently greater "fanning out" of the Es bow-tie compared with the head bow-tie is largely due to the wider distribution of vertical gaze positions rather than a greater twist to the bow-tie). The same bow-tie like twist is evident in the below views of this data as well (right column), but the vertex of the twist was shifted upward on the graph (i.e., toward an upward position). The latter is simply a perspective effect: because the bow tie is tilted backward (i.e., upward about the horizontal axis) from the side perspective, its vertex automatically looks shifted from the "below" view. In concrete terms, this bow-tie shape signifies that Es/Hs assumes relatively (compared with its neighboring corner on the illustrated plot) counterclockwise orientations in down-left (DL) and up-right (UR) positions and assumes relatively clockwise orientations in up-left (UL) and down-right (DR) positions. (Note that this description is only meaningful in the current orthogonal, space-fixed coordinate system).

To be consistent, we also have plotted the second-order surfaces fit to the Eh data (bottom). These also show a similar bow-tie twist in this particular example but were highly variable. For example, the "twist score" (described in more detail in the following text) of the Eh range across experiments was 3.3 times as variable as the Es twist score [quantified as the average ratio of the standard deviations of parameter a₅ (Eq. 5) across all random-task experiments, further averaged across all 3 animals]. This variability may have been exaggerated because the Eh ranges were too small (Fig. 3E) for this method to provide a highly meaningful fit (as pointed out by Radau et al. 1994). Therefore we did not further investigate the subtleties of the shape of the Eh fixation range but rather focused on its torsional "thickness."

Quantifying the torsional thickness of the fixation ranges

In quantifying the 3-D ranges of eye and head orientation in humans, two measures have proven useful: the parameters used to form the fits illustrated in Fig. 3, and the standard deviation of torsional eye position (in the actual data) from these ideal 2-D fits (Glenn and Vilis 1992; Radau et al. 1994). We employed the same approach to quantitatively address two questions: which of the three segments (Es, Hs, and Eh) are constrained most tightly by Donders' law, and what form does this law take in each of these segments?

Figure 4 shows torsional standard deviations from first-, second-, and third-order fits to all three segments, computed from random-task fixation data such as those illustrated in Fig. 3. Data from all three animals are plotted separately in A-C, whereas each bar represents a value averaged across all random-task measurements in one particular animal. In each case, the bars are grouped according to the order of the fit used to compute the relative torsional standard deviation. The first point to be taken from this figure is that as the order of the fit increased, the torsional variance from this fit decreased. For example, note how the average torsional SD for Es () dropped dramatically between the first- and second-order fits and dropped further at the third-order fit. However, note (in Fig. 4) that the jump from first- to second-order fits produced the greatest effect in Es and Hs variance (compared with the 2nd-3rd interval), whereas the torsional SD of Eh showed a steady, less dramatic decline toward higher orders of fit (perhaps due to the problems in fitting the head-free Eh range mentioned earlier). Nevertheless, statistically significant (P < 0.001) decreases in variance occurred across both intervals (i.e., 1st-2nd and 2nd-3rd) for each and every case for Es, Hs, and Eh, in all three animals. In summary, although each increase in the order of fit decreased the torsional variance of Es, Hs, and Eh in the statistical sense, visual inspection of the data (Fig. 4, see particularly animal M3) suggested that the jump from first to second order gave the most important improvement in the Es and Hs fits, whereas second- and third-order terms appeared to have little effect on the Eh fit.

View larger version (38K): [in this window] [in a new window]   Fig. 4. Quantitative comparison between torsional thickness of eye-in-space (Es), head-in-space (Hs), and eye-in-head (Eh) fixation ranges at each order of fit. Major bars show mean values of torsional standard deviations (from 1st-, 2nd-, and 3rd-order surface fits) averaged across all random task experiments, and error bars provide SEs across experiments. A: animal M1. B: animal M2. C: animal M3.

The second point to be gleaned from Fig. 4 is that there were consistent differences between the torsional variance of Es, Hs, and Es, regardless of which order of fit was used. First, the torsional SD of the Es range was always greater than the torsional variance of the Hs range. This was true in all animals, no matter how complex were the parameters of the fit (1st, 2nd, or 3rd order). In each and every individual case, this difference was statistically significant (range of P  0.0005-0.007 in M1 and M2, P  0.04 in M3). Similarly, in each and every case in all animals, for all levels of fit, the torsional SD of the Hs range was significantly larger than the torsional SD of the Eh range (P  0.0005). Thus even at a third-order fit the torsional SD of Es (mean across animals: 3.55°) was greater than the torsional variance of Hs (mean 3.10°), which in turn was greater than that of Eh (mean 1.87°). This relative ranking agrees with the qualitative descriptions provided above (Figs. 1-3), as well as observations in humans (Radau et al. 1994). According to the arguments employed by Radau et al. (1994), this suggests that Donders' law was only enforced by the individual control systems of Hs and Eh, whereas the Donders' law for Es was simply a geometric byproduct of the other two laws.

From the preceding, data we draw three interim conclusions necessary for the further development of our analysis: first, the motor implementation of Donders' law in Es during head-free gaze fixations can be attributed to relatively tight 3-D control of Hs and Eh. Second, the rule governing head orientation during fixation gives rise to a decidedly nonplanar, twisted range. Third, Eh orientation vectors were restricted to a relatively thin quasiplanar (or indeterminately twisted) range. These observations are in close agreement with the human data (Glenn and Vilis 1992; Radau et al. 1994). On the basis of these observations, we then analyzed the segmental trajectories that give rise to these fixation constraints.

3-D eye-in-head trajectories

On the basis of head-fixed data (Ferman et al. 1987b; Straumann et al. 1995; Tweed and Vilis 1990; Tweed et al. 1994), it might seem reasonable to hypothesize that Eh would obey Donders' law (i.e., remain in Listing's plane) at all times throughout a head-free gaze shift, with only minor and pseudorandom "torsional transients." However, this hypothesis is problematic because head-free gaze shifts include substantial periods where gaze is stabilized by VOR slow phases. Slow phases are known to violate Listing's law in that they tend to rotate the eye about almost the same axis as the head (Crawford and Vilis 1991; Misslisch et al. 1994a). Thus if the head rotation axis does not align with Listing's plane, a slow phase will tend to drive eye torsion out of Listing's plane. Moreover, even if the head axis does align with Listing's plane, the complex kinematics of eye rotation will cause the eye to deviate torsionally as a function of eye position (Crawford and Vilis 1991). How does the head-free gaze control system deal with this problem?

Figure 5 provides several clues to the solution to this problem. Figure 5, left, shows the familiar 2-D plot of Eh direction trajectories, where up is up and left is left, etc., whereas Fig. 5, right, shows the right-side view of Eh quaternion vectors. The latter have been rotated into a coordinate system that aligns with the first-order fit to the associated fixation range. From this data it is evident that the primary position at the origin (where the 2-D Eh direction vector is orthogonal to the fixation range) did not necessarily correspond to the center of the head-free Eh range. Figure 5, top, shows Eh saccade trajectories (where  indicate saccade endpoints) during random head-free gaze shifts, whereas bottom shows similar 2- and 3-D views of the slow phase eye movements from the same data set, and their endpoints (). Note that, as shown more explicitly in the next figure, most head movements were associated with several Eh saccades and slow phases. Each and all of these sequential eye movements have been included in Fig. 5.

View larger version (45K): [in this window] [in a new window]   Fig. 5. Two- and 3-D trajectories of saccades and quick phases during head-free gaze shifts. Top, A and C: saccades, where , trajectories; , saccade endpoints. Bottom, B and D: slow phases, where , trajectories; , slow phase endpoints. Note that these were mostly multiple-step gaze shifts, as shown in Fig. 6. Therefore some of the slow phases would be the final eye movements preceding stable eye/head fixation, but most would be interim slow phases followed by 1 further saccade in the multiple-step gaze shift. Left: 2-D gaze projected onto frontal plane in magnetic filed coordinates. Origin is the preferred eye position used to look straight ahead and is not necessarily the center of the mechanical oculomotor range. Right: side view of 3-D quaternion vectors, plotted relative to the head-free primary position in Listing's coordinates. Animal M1.

Focusing first on 2-D direction (Fig. 5, A and B), it is evident that during the random task, Eh saccades (A) were directed in a pseudorandom pattern that tended to land final horizontal and vertical eye position over a roughly homogenous range including relatively eccentric positions (). In contrast, slow phases (B) tended to center eye position, driving it more exclusively toward less eccentric final positions (). This is not surprising, given the known kinematics of head-free gaze control (Bizzi et al. 1971). However, how should one expect torsional eye position to behave? From the known kinematics of saccades and the VOR, one might expect the latter to drive the eye out of Listing's plane, whereas each saccade might correct any accumulated deviation from Listing's plane. In fact, the opposite pattern appeared to occur: Eh saccades (C) tended to drive final torsional eye position () away from the fixation plane (centered on the vertical axis), whereas VOR slow phases (D) tended to land final torsional eye position () closer to the center of the fixation plane. Thus the same rule applied to all three dimensions of Eh: saccades tended to drive the eye eccentrically, whereas slow phases tended to center the eye.

The detailed kinematics of this behavior are more clear in Fig. 6, which plots the three components of eye position relative to the head as a function of time during a typical large gaze shift, selected from data recorded during the random paradigm. For reference, the components of head position also are plotted, rotated to align with the vertical axis of the Fick range, such that the horizontal dotted line corresponds approximately to zero torsion in Fick coordinates. As described in the METHODS, the animals were not constrained by time or reward to use any particular strategy to obtain visual targets, and they invariably chose to implement large multiple-step gaze shifts (i.e., with several interim saccades and fixations). This is evident in Fig. 6 where the animal made four consecutively smaller vertical/horizontal Eh saccades () during the single head movement, roughly in the same direction as the single smoother head movement profile. Between these saccades, the VOR evidently was engaged, producing slow phases () that rotated the eye counter to the head and thus briefly stabilized gaze whenever possible during the gaze shift. Considered from the perspective of 1-or 2-D gaze control, the temporal coordination of this pattern seemed relatively trivial, i.e., saccades shifted gaze toward the target, and then the VOR trivially caused the eye to roll back toward a central position.

View larger version (18K): [in this window] [in a new window]   Fig. 6. Temporal kinematics of a typical multisaccade/single head movement gaze shift. Top: 3 components of eye position plotted (in headcentric "Listing's coordinates") as a function of time. , saccades; , VOR slow phases. Also shown is extrapolated torsional eye position excursion ({) if quick phases are removed, and the remaining slow phase trajectories are fit to a continuous spline curve. - - -, 0 torsion in Listing's plane. Bottom: 3 components of head position, plotted in space coordinates but rotated to align with the vertical Fick coordinate. · · · , approximates 0 torsion in Fick coordinates. Animal M1.

From a 3-D perspective, the temporal coordination of this pattern was much more complex. A similar nystagmus-like pattern also occurred in the torsional component of Eh position ({). Note that this Eh data has been rotated into Listing's coordinates, such that - - - represents zero torsion in Listing's plane. The slow phases thus had a considerable torsional component relative to Listing's plane even though the head did not rotate torsionally. As explained in more detail elsewhere (Crawford and Vilis 1991; Smith and Crawford 1998), this accumulation of torsion was primarily a kinematic consequence of rotating the eye about the same axis as the head independent of eye position. In this case, slow phase rotation of Eh in the rightward direction, from an upward eye position, caused torsional eye position to deviate in the counterclockwise direction, as predicted from the laws of rotational kinematics (Smith and Crawford 1998). However, these slow phases did not take the eye away from Listing's plane but rather toward it. This resembled a similar effect described for torsional slow phases during passive head rotation (Crawford and Vilis 1991) except that passively induced slow phases tended to overshoot Listing's plane, whereas these slow phases appeared to land the eye quite accurately within the fixation range (as quantified in the next figure).

The latter effect occurred because each preceding saccade also had a torsional component, but in the opposite direction, such that the overall torsional excursion was negated by the subsequent slow phase. Indeed without such torsional saccades the eye would presumably accumulate a huge Eh torsion, as extrapolated on Fig. 6. Furthermore because the torsional components of these quick phases did not contribute to shifting gaze toward the target and because the torsional direction of the subsequent slow phase could only be predicted by taking both head velocity and eye position into account (Crawford and Vilis 1991), the temporal coordination of this 3-D pattern did not appear to arise accidentally. Instead, it appeared that the system produced specific torsional components in Eh saccades to anticipate subsequent vestibular-driven torsional movements, such that ocular torsion always returned to Listing's plane at the end of each slow phase.

To quantify these observations, it was necessary to further subdivide and analyze the Eh position range during multiple-step gaze shifts. Figure 7A shows examples of each of these subranges, derived from a single random-task measurement with the use of the algorithms described in the APPENDIX. The 3-D data are viewed from the side (i.e., torsional position vs. horizontal position) in coordinates rotated to align with the head-free Eh fixation range. The first example (Fig. 7A, left) shows the latter fixation range explicitly, revealing a torsionally compact group of Eh positions. The remaining examples show Eh positions at various start/endpoints of the saccades and slow phases from the same data set. The rest of Fig. 7 (B-D) shows the SDs of torsion from each of these ranges, relative to a second-order fit made to the fixation range at gaze shift ends. The individual bars show average values across all head-free random paradigm measurements (with error bars showing SEs across measurements) in animals M1 (B), M2 (C), and M3 (D). These bars have been aligned in vertical columns with the example ranges in Fig. 7A, to facilitate comparison between these columns, where the fixation data (column 1) establishes the baseline (- - -) for such comparisons.

View larger version (37K): [in this window] [in a new window]   Fig. 7. Quantitative torsional eye-in-head ranges during different stages of the gaze shift. A: examples of 3-D eye-in-head position ranges during (1) fixation at gaze shift ends, (2) the starting positions of quick phases throughout gaze shift, (3) the endpoints of all quick phases, (4) the endpoints of quick phases where head velocity >150°/s, and (5) the endpoints of all slow phases during gaze shifts. Positions selected automatically according to the velocity criteria described in text from one random-gaze shift sequence (animal M1) and plotted in head-free Listing's coordinates fit the fixation data. B: torsional standard deviations of the same 5 position ranges (aligned as already mentioned), as compared with 2nd-order fit to the fixation data for each experiment. Data are averaged across all random task experiments in animal M1, with error bars showing SEs across experiments. C: same as B but computed for animal M2. D: animal M3.

Figure 7, second column, shows eye positions at the initial points of Eh saccades (labeled as quick phases) sampled throughout the multiple-step gaze shift. Compared with the baseline, this range was only slightly, but significantly, expanded (P  0.001, all animals). However, at the ends of these saccades (column 3), the torsional variance was increased further (P  0.001, all animals) and by a larger step. If the function of these saccades components is to anticipate subsequent torsional components in slow phases, as hypothesized above, one might expect saccades during high head velocities (which tend to produce larger violations of Listing's law in the typical intersaccadic interval) to push torsion even more eccentrically. As shown in the fourth column, the subset of saccade ends where current head velocity was >150°/s did tend to land the eye at positions even more eccentric to the zero torsion zone (P  0.003, all animals). However, after the subsequent slow phases (column 5), the eye was brought back to a torsional range almost as tight as the fixation range. Not surprisingly, the slow phase end range was very similar to the saccade start range because the two overlapped considerably during multiple-step gaze shifts (although final fixation points with the head motionless were included only in the former and starting points of the 1st Eh saccade with the head moving were only included in the latter). More importantly, the torsional variance at the end of slow phases was reduced (highly significantly) compared with saccade ends in all three animals (P < 2 × 10⁴ in M1, P < 5 × 10⁹ in M2, P < 6 × 10⁶ in M3). Thus unlike the head-fixed Listing's law, Donders' law of Eh was not obeyed throughout the overall gaze shift but only held during steady fixation (column 1) and transiently approximated toward the ends of interim slow phases (column 5).

3-D head trajectories

Figures 3 and 4 suggested that during steady fixations, the head-orientation range formed a pseudoplanar, twisted surface that required at least a second-order fit for adequate quantification. The latter is provided explicitly in Fig. 8, which shows the average parameters of second-order fits to the head-fixation range (averaged ± SD across all random-task measurements) for animals M1 (A), M2 (B), and M3 (C).

View larger version (21K): [in this window] [in a new window]   Fig. 8. Average parameters (a1-a6) of 2nd-order fits to 3-D head-orientation range in random saccade task. A: means and SDs across experiments in animal M1. B: means and SDs across experiments in animal M2. C: means and SDs across experiments in animal M3.

Each parameter provides the dependence of Hs torsion on various combinations of horizontal and vertical position. The first parameter (a₁) measures the torsional shift of the range from the reference position, which was (not surprisingly) small (0.025, averaged across animals). The second parameter (a₂) indicates the dependence of torsion on vertical eye position, which was also small (0.022, averaged across animals) and rather inconsistent. In comparison, the third parameter (a₃) was relatively large (0.274, averaged across animals) and consistent both within and between animals. This term was significantly less than zero in all animals (P < 3 × 10⁸ in M1, P < 2 × 10¹⁴ in M2, P < 10⁶ in M3). This value represents the dependence of torsional head position on horizontal head position, and the illustrated negative value describes the upward (backward) tilt observable in the side view of the head fixation range illustrated in Fig. 3 (E and H), such that rightward positions tended to be more clockwise compared with lef