INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

In normal human subjects, eye positions during steady fixation follow a mathematical specification named Listing's law: if ocular positions during fixation are expressed as single rotations from a common reference position, all rotation axes lie in a plane, the so-called Listing's plane (Helmholtz 1867). Hence, Listing's law seems to be a fundamental property of the ocular fixation system. To a lesser degree, Listing's law also applies to smooth pursuit eye movements and saccades (Haslwanter et al. 1991; Straumann et al. 1996; Tweed and Vilis 1990; Tweed et al. 1992). Listing's law also specifies Donders' law, which states that ocular torsion is unique for every direction of gaze (Donders 1848).

A fundamental question of ocular motor physiology is whether or not pathological eye movements can be identified that do not comply with Listing's law, and if these eye movements can be related to specific lesions in premotor, motor, or peripheral structures of the ocular motor system (Straumann and Zee 1995). Theoretically, four types of violations of Listing's law can be expected: I, scattering of three-dimensional eye positions on a nonplanar surface during fixations and movements; II, transient deviations from Listing's plane during and after changes in eye position; III, active torsional movements out of Listing's plane; and IV, passive torsional drift movements out of Listing's plane.

Type I violations of Listing's law can be caused by mechanical alterations of the ocular plant, e.g., orbital tumors.

Type II violations are transient and occur if, during a movement, the axis of angular velocity does not tilt in the direction of gaze by a geometrically specified angle to compensate for the noncommutativity of rotations (the so-called half-angle rule). To a small degree, such torsional blips (peak torsional deviations 2°) can be seen during and after normal saccades (Straumann et al. 1995, 1996). Large torsional blips called macroblips (peak torsional deviations ~10°) were described in a patient with a lesion involving the cerebellar vermis, its deep nuclei, and the dorsolateral medulla, suggesting that the cerebellum might be involved in implementing Listing's law (Helmchen et al. 1997).

Type III violations of Listing's law are to be expected because any active torsional movement corresponds to such a violation. They generally occur during vestibular stimulation both normally, e.g., during torsional vestibular nystagmus (Crawford and Vilis 1991), and in disease, e.g., during vertical-torsional benign paroxysmal positioning nystagmus (Fetter and Sievering 1995).

Type IV violations of Listing's law are still hypothetical. They would reflect the failure of the CNS to maintain the eyes in Listing's plane. Recent models of the three-dimensional velocity-to-position integrator assume that, in the absence of vestibular stimulation, only the horizontal and vertical components of the velocity-to-position integrator are being loaded, which implicitly results in eye positions in Listing's plane (Quaia and Optican 1998; Schnabolk and Raphan 1994; Tweed and Vilis 1987). In other words, velocity signals from the saccadic and the pursuit systems are such that their integration does not lead to a change in torsional eye position, which should always be zero if Listing's law is valid and three-dimensional eye position is encoded in a coordinate system that is equivalent to rotation or quaternion vectors. It is important to realize that this view implies that the ocular plant is constructed such that, in the absence of an eye-position-dependent torsional tonic signal, Listing's law is automatically implemented. In this case, leaky velocity-to-position integration in the presence of saccadic or pursuit velocity signals will always lead to drift movements exclusively in Listing's plane. If, however, the mechanics of the eye plant do not implement Listing's law for fixations, it is necessary that the ocular motor system produce a tonic torsional signal that is specified by the horizontal and vertical gaze direction. This tonic torsional signal must compensate the torsional error caused by the "non-Listing mechanics" of the ocular globe. It follows that a failure to map correctly the appropriate tonic torsional signal to the horizontal-vertical direction of gaze will lead to an eye movement drifting out of Listing's plane after each saccade. The amplitude of this slow torsional eye movement, in the absence of an intervening saccade, would vary with gaze direction.

It is likely that the elastic extorsional and intorsional forces acting on the ocular globe do not perfectly cancel for any direction of gaze. For instance, Seidman et al. (1995) were able to show that the time constants of torsional drift, after mechanically forcing the eyes to extorsional or intorsional offset positions, are different. Assuming such an anisotropy of elastic restoring forces in the torsional direction, tonic torsional signals become indispensable to maintain the eye in Listing's plane during fixations. Moreover, it may be that the ocular motor system holds two separate Listing's planes: one determined by the mechanical configuration of the eye plant, the other implemented by the CNS. If the orientation of the "mechanical Listing's plane" were different from the "neural Listing's plane," the absence of tonic torsional signal would lead to a drift from "central" to "mechanical" Listing's positions.

Examining patients with a cerebellar ocular motor syndrome, in which it is difficult to maintain fixation, is a good way to test if type IV violations of Listing's law exist. Typically, the eyes drift both upward and in the horizontal-centripetal direction, which results in downbeat and horizontal gaze-evoked nystagmus, respectively (Leigh and Zee 1999). We attempted to determine whether or not patients with cerebellar atrophy also have abnormal torsional drift movements and, by that, demonstrate the existence of a type IV violation of Listing's law. The following specific questions were asked: is Listing's law valid in patients with cerebellar atrophy? Is the ocular drift in these patients two-dimensional (horizontal and vertical) or three-dimensional (horizontal, vertical, and torsional)? If present, is the torsional drift independent of the drift in the vertical or horizontal direction?

We show that, in addition to the well-known horizontal gaze-evoked and upward drifts, there is a drift in the torsional direction that is dependent on the horizontal eye position. This torsional drift is independent of the upward drift and its velocity does not correlate with the velocity of horizontal drift. Thus the data suggest that the cerebellum is involved in mapping an appropriate tonic torsional signal for every direction of the line of sight.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Subjects

Eighteen patients (P1-P18, age 40-69 yr, 9 female) with horizontal gaze-evoked and downbeat nystagmus associated with cerebellar atrophy, as demonstrated by magnetic resonance imaging, were studied. The cerebellum was diffusely atrophied in each patient, i.e., both the paramedian and hemispheric cerebellar structures were affected. None of the patients had neuroradiological or clinical findings that were suggestive of extracerebellar involvement. The ocular motor findings were typical for the syndrome of the flocculus/paraflocculus (impaired smooth pursuit system; postsaccadic drift; and gaze-evoked, downbeat, and rebound nystagmus). An additional deficiency of the nodulus, ventral uvula, dorsal vermis, or fastigial nuclei was likely, but no specific ocular motor tests were performed to assess the function of these structures.

Fourteen patients were tested at the Johns Hopkins Hospital in Baltimore, Maryland, and four patients were tested at Zurich University Hospital, Switzerland. The comparison group consisted of 25 normal subjects (N1-N25, age 18-54 yr, 16 female), 21 of which were tested in Baltimore, the others in Zurich. The visual acuities in the normal subjects were 20/20 corrected. In Baltimore, informed written consent from patients and normal subjects was obtained after an explanation of the experimental procedure, which was in accordance with the standards of and was approved by the Johns Hopkins Joint Committee on Clinical Investigation. In Zurich, the paradigms of this study were part of the standard ocular motor testing battery. At the time of patient recording, unwritten informed consent was sufficient.

Experimental setup

The experimental setups in Baltimore and Zurich were identical unless stated otherwise. Ocular rotation of both eyes around all three principal axes (torsional, roll, x axis; vertical, pitch, y axis; and horizontal, yaw, z axis) was simultaneously recorded with dual search coils (Skalar, Delft, Netherlands). The field coil system consisted of a cubic coil frame of welded aluminum that produced three orthogonal magnetic fields with frequencies of 55.5, 83.3, and 42.6 kHz and intensities of 0.088 gauss. Amplitude-modulated signals were extracted by synchronous detection. The bandwidth of the system was 0-90 Hz. Peak-to-peak noise signals in all three principal directions after calibration, as measured by a dual search coil placed in the center of the magnetic frame, were ~0.05° in Baltimore and ~0.1° in Zurich. The side length of the coil frame in Baltimore was 1.02 m, in Zurich 1.40 m.

Calibration procedure

The voltage offsets of the system were zeroed by placing the dual search coils in the center of a metal tube that shielded them from the magnetic fields. Thereafter, the relative gains of the three magnetic fields were determined with the search coils on a gimbal system in the center of the coil frame. The details of the calibration procedure and the computation of three-dimensional eye position are given in Straumann et al. (1995).

The three-dimensional eye position in the coil frame was expressed in rotation vectors. A rotation vector r = (r_x, r_y, r_z) describes the instantaneous orientation of a body as a single rotation from the reference position. The vector is oriented parallel to the axis of this rotation and its length is defined by tan (/2), where  is the rotation angle. The coordinate system of rotation vectors was defined by the three head-fixed orthogonal axes of the coil frame with the x axis pointing forward, the y axis leftward, and the z axis upward. The signs of rotations around these cardinal axes were determined by the right-hand rule, i.e., clockwise, leftward, and downward rotations, as seen by the subject, were positive.

For each experimental trial, a plane was fitted through the data cloud of all rotation vectors:
The offset a₀, the y slope a₁, and the z slope a₂, which determined the orientation of this best-fit plane, were used to rotate all eye positions such that this plane coincided with the y-z plane of the coordinate system, and therefore the primary position with the reference position r_ref = (0, 0, 0). The corresponding vector r_p for this rotation was given by (Haustein 1989)
From the rotation vectors that were rotated as described above, three-dimensional angular velocity vectors  were computed using the formula (Hepp 1990)
where dr denotes the derivative of r and × the cross product. Note that, because of the noncommutativity of rotations, three-dimensional angular velocity is not simply the derivative of position.

For convenience, the lengths of rotation and angular velocity vectors were given in degrees (°) and degrees per second (°/s), respectively, but the right-hand rule was maintained when describing the orientation of these vectors.

Experimental procedures

Subjects were seated inside the field coil so that the center of the interpupillary line coincided with the center of the frame. The head of the subject was immobilized with a bite bar that was oriented in an earth-horizontal plane with no rotation around the earth-vertical axis. Dual search coils were mounted on both eyes after the conjunctiva was anesthetized with proparacaine HCL 0.5% (Ophthetic).

During experiments, subjects were asked to fix on light-emitting diodes (LEDs) at different locations of an earth-vertical tangent screen that was positioned 1.24 m in front of the center of the eyes. LEDs were consecutively switched on and off such that only one LED was lit at any time in dim light or complete darkness. In Zurich, light dots were rear-projected onto a translucent tangent screen.

Voltages related to the orientation of the eye coils in the magnetic frame were digitized with a 12-bit A/D converter at 500 Hz and written to a hard disk. The data were analyzed off-line in MATLAB version 5.1.

Paradigms

The eight eccentric positions of the light dot used as the fixation stimulus were arranged on a square around the straight-ahead position. Four positions were situated ±20° from the center on the horizontal and vertical meridians, and four other positions were situated ±28.3° from the center on the two oblique meridians. Eccentric positions were consecutively lit in the counterclockwise direction. Before the appearance of each eccentric stimulus, the center position was lit so that subjects always made centrifugal and centripetal shifts of eye position. Each fixation period lasted 2.5 s. In one paradigm, the LED was on during the entire fixation period; in another paradigm, the LED was switched off after 0.5 s and switched on again for the last 0.5 s of the fixation period. Switching off the LED for some time during the fixation period did not have any effect on eye drift except that it was more difficult for the patients keep their eyes near the desired positions because of persistent centripetal and upward drift. In the analysis, we therefore used the data from the paradigms in which the LEDs were not intermittently switched off. Paradigms were repeated for binocular viewing and monocular viewing of either eye. In this study, only monocular data of the viewing eye are reported.

Torsional coil slippage

During search coil recordings, a long-term drift of torsional coil signals is frequently observed, which is most likely caused by a gradual slippage of the silicon annulus on the conjunctiva around the line of sight (Straumann et al. 1996; Van Rijn et al. 1994). Therefore the fixations that we used to determine Listing's plane were all inside a time interval of 60 s. When analyzing slow-phase eye movements of nystagmus at the angular velocity level, however, the very slow artificial change of torsional eye position signals was negligible because here we were interested in the three-dimensional angular drift direction of the eyes and not absolute ocular position. For this analysis, we used data from entire trials lasting 120 s, during which paradigms were performed three times by the subjects.

Pooling of binocular data

In 17 of the 18 patients, eye movement recordings from both eyes were available. In one patient, the signals of one eye were corrupted because the wires of the annulus broke during recording. Of the 25 healthy subjects, 20 were measured with annuli on one eye and the other five had annuli on both eyes. Tables and statistical values were based on pooled data whereby data from the left eyes were mirrored. For convenience, summary figures contain only data from right eyes, but statistical information in the text is based on pooled data.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Listing's law

We attempted to determine if Listing's law is obeyed in patients with cerebellar atrophy. Because these patients showed spontaneous nystagmus, especially downbeat nystagmus and horizontal gaze-evoked nystagmus, it was not a priori evident during which phase of a nystagmus cycle eye positions should be selected to test the validity of Listing's law. We chose to perform the analysis for four different data subsets extracted from single experimental trials: 1) all eye positions, 2) desaccaded eye positions (fixations and slow phases), 3) presaccadic eye positions (interval of 5-55 ms before the beginning of quick phases), and 4) postsaccadic eye positions (interval of 5-55 ms after the end of the quick phases).

Figure 1 demonstrates examples of single experimental trials. Eye positions are plotted as rotation vectors in the front (top row, y-z plane), side (middle row, x-z plane), and top (bottom row, x-y plane) views of the coordinate system. Trials consisted of 2-3 cycles. For this figure and for the entire analysis presented in this subsection, Listing's law, only single cycles of trials were selected. Such a restriction of the time interval was chosen because long-term torsional drifts of scleral annuli can lead to artificial "thickening" of Listing's plane (see METHODS).

View larger version (30K): [in this window] [in a new window]   Fig. 1. Examples of three single trials. Attempted fixations of a central and eight eccentric targets on a tangent screen. x, torsional; y, vertical; z, horizontal components of rotation vectors in °. Signs of base vector directions according to the right-hand rule. Top: front view (y-z plane). Middle: side view (x-z plane). Bottom: top view (x-y plane). A: patient P8, right eye, unrotated data (zero vector is straight ahead). B: rotated data of A (zero vector is primary position). SD of first-order linear fit: 1.09°; SD of second-order linear fit: 0.55°; twist score: 0.454. C: rotated data of normal subject N7, right eye. SD of first-order linear fit: 0.53°; SD of second-order linear fit: 0.45°; twist score: 0.154.

Figure 1, A and B, shows ocular rotation vectors of a cerebellar patient (P8, right eye) in different views. Whereas the Cartesian coordinate system in Fig. 1A is determined by the magnetic coil frame, the data in Fig. 1B are rotated such that the best-fit plane through the data cloud aligns with the y-z plane of the coordinate system (see METHODS). For comparison, Fig. 1C depicts rotated ocular rotation vectors (right eye) of a normal subject (N7).

Clearly, the cerebellar patient was not able to maintain fixation (Fig. 1, A and B). During attempted fixation along positions with no horizontal eccentricity, trajectories were oriented parallel to the y axis, which corresponded to the observed downbeat nystagmus. If the eyes were moved to positions with a horizontal eccentricity, there was an additional horizontal gaze-evoked nystagmus and, therefore, drift trajectories were oblique (Fig. 1, A and B, top). After rotation of the patient's data from the coordinate system defined by the coil frame (Fig. 1A) to the Listing's coordinate system (Fig. 1B), it became evident (middle and bottom) that the width of Listing's plane in the patient was larger than that in the normal subject (Fig. 1C). This can be seen in both the x-z (middle row) and x-y (bottom row) projections.

The standard deviations of eye positions from the best-fit plane are summarized in Table 1 (pooled data from both eyes). This so-called thickness of Listing's plane (as reflected in the mean values of the standard deviations) was significantly higher in the group of 18 patients than in the group of 25 control subjects. (Throughout the entire paper, "significant" = P < 0.05). This was the case for all four data subsets. The differences between the patients and the control group, however, were small (0.17-0.20°). The average standard deviation of data points from the plane was almost the same for pre- and postsaccadic eye positions in patients and normal subjects. Note that the slopes (y and z) of the planar fits, which are related to the primary direction, were not significantly different between the two groups, e.g., for planes fitted to presaccadic fixations, the average y slope of pooled eyes (left eyes mirrored) was 0.01 ± 0.09 SD (n = 35) in the cerebellar patients and 0.01 ± 0.07 SD (n = 30) in the normal subjects. Corresponding average z slopes were 0.09 ± 0.09 SD and 0.08 ± 0.07 SD, respectively.

Data clouds of ocular rotation vectors were also fitted with linear second-order regressions of the form
where a₃ and a₅ are measures for the curvature along the y and z axes and a₄ indicates how much the surface is twisted (Hore et al. 1992). Again, in all four subsets of data, patients differed significantly from normal subjects (Table 2). This means that the larger thickness of Listing's plane (= standard deviation from the planar fit) in the cerebellar patients was not caused by curving or twisting of the surface, on which three-dimensional eye positions lie, because otherwise the standard deviations of eye positions from the second-order fit would not be different in patients and normal subjects.

To confirm further that the increased thickness of Listing's plane in the patients was not caused by an increased twisting of Listing's surface, we computed the so-called twist factor s:
In none of the four data subsets were twist factors significantly different in patients and normal subjects, e.g., the average twist factor for presaccadic fixations was 0.33 ± 0.33 SD in the patients and 0.42 ± 0.34 SD in the normal subjects (P = 0.137).

In summary, analysis of three-dimensional eye movements based on the positions of the eye revealed that Listing's law was less valid in the cerebellar patients than in the control group. Yet the differences in plane or surface thickness (standard deviations from best-fit first- or second-order linear fits) were small between the two groups. One explanation for this relatively small increase may be that quick phases effectively compensate for drift movements that do not comply with Listing's (and Donders') law (Lee et al. 1998; Van Opstal et al. 1996). This hypothesis is supported by the fact that the correlation between the thickness of Listing's plane and the drift velocity at 20° horizontal eccentricities (combined mean velocity in abduction and adduction) was weak (R = 0.36; P = 0.029; n = 18 right eyes + 17 left eyes) and that there was no significant correlation between the thickness of Listing's plane and the time constants of horizontal (R < 0.01, P = 0.857) or vertical (R = 0.32, P = 0.057) ocular drift (scatter plots not shown). At this point of analysis, however, we could not conclude that the drift movements in cerebellar patients violated Listing's law because, in the cerebellar patients, the eyes were constantly moving and it was therefore impossible to determine the "true" Listing's plane. As a result, we also could not determine in which period(s) of nystagmus the deviations from Listing's plane occurred or at which moment of the nystagmus cycle they were corrected.

To decide whether or not the drift movements complied with Listing's law, trajectories had to be analyzed at the angular velocity level. It follows from the definition of an angular velocity vector (see METHODS) that Listing's law specifies the value of the torsional vector component (_x) by an equation that includes the values of the two other components (_y and _z) and the vertical (r_y) and horizontal (r_z) components of the instantaneous rotation vector (Van den Berg 1995):
Solving this equation for a particular angular velocity vector that moves orthogonal to a radial line passing through the reference position, the angular velocity axis tilts out of Listing's plane by half the gaze angle in the direction of gaze. Accordingly, the equation is also know as the "half-angle rule."

In Three-dimensional angular drift, we present each of the three angular velocity components (vertical, horizontal, and torsional) of ocular drift separately. Then we correlate the drift components to see if they are independent from each other. Finally we attempt to determine if the ocular drift in the patients follows the half-angle rule, i.e., complies with Listing's law.

Three-dimensional angular drift

Figure 2 depicts an example of how the data were further processed. The vertical rotation vector component r_y of the same data shown in Fig. 1B is plotted as a function of time (top). Recall that the set of rotation vectors was rotated such that the best-fit plane through the data cloud aligned with the y-z plane of the coordinate system. The letters above the eye position trace indicate which of the eight eccentric target lights was switched on to be fixed by the subject (R, right; RU, right-up; L, left; etc.). As described in Paradigms, subjects always had to look at the central target between eccentric fixations. In this example, there was a persistent drift of the eye in the upward (negative vertical) direction with compensatory downbeating quick phases.

View larger version (42K): [in this window] [in a new window]   Fig. 2. Same data as in Fig. 1B (rotated data, patient P8, right eye), but only the vertical eye movement component is plotted as a function of time. Top: vertical rotation vector component. Letters indicate directions of attempted eccentric fixations after central fixation. R, right; RU, right-up; U, up; LU, left-up; L, left; LD, left-down; D, down; RD, right-down. Middle: vertical angular velocity vector component. Bottom: median vertical angular velocity component for each nystagmus slow phase.

Fig. 2, middle, contains the vertical component of the angular velocity vectors. Note that we always computed the angular velocity vectors from those rotation vectors that were rotated into the y-z plane, as described in Calibration procedure, because a tilt of Listing's plane with respect to the y-z plane of the coordinate system would introduce a "false reference-position-dependent torsion" (Suzuki et al. 1994).

For each slow phase of nystagmus, the median values of the horizontal, vertical, and torsional angular velocity components were computed. The corresponding vertical medians in the example are plotted in Fig. 2, bottom. The medians of the corresponding eye rotation vectors during each slow phase were also determined (not shown) so that each data point representing one slow phase consisted of three rotation and three angular velocity vector components. The data points were then assigned to one of the eight targets of attempted eccentric fixation. After selecting subsets of data according to the horizontal and vertical eye positions, components of rotation and angular velocity vectors were plotted against each other.

Each component of three-dimensional eye movements was specifically tested for the validity of Alexander's law (Alexander 1912; Robinson et al. 1984). This law describes the relation between the slow-phase eye velocity and the change in eye position along the velocity vector: as the eye moves in the direction of the slow phase, slow-phase velocity decreases. If the relation between eye position (independent variable) and eye velocity (dependent variable) forms a line, the negative slope is reciprocal to the time constant of the "velocity-to-position integrator" (Hess et al. 1984; Robinson et al. 1984).

Figure 3 depicts two examples of the relation between vertical eye position component r_y and vertical angular velocity component _y. Only data with the eye in center-up and center-down positions are plotted. The regression line was computed by fitting the data to the equation
The following procedure was used: first-order linear regressions were iterated 20 times. After each regression, the data subset of points that were farthest away from the best-fit line was discarded. If the initial number of data points was >100, 1% of data points was discarded after each regression; if the initial number of data points was <100, only one data point was discarded after each regression. The procedure was repeated until 20% of the data were excluded from the fit. The offset (a), slope (b), and significance (P) of the last regression were further analyzed. If P < 0.05 and the slope "eye position divided by velocity" was negative, Alexander's law was said to be valid. If the slope was positive but P was still significant, this relation was named "inverse Alexander's law." If P was not significant, Alexander's law and inverse Alexander's law were rejected.

View larger version (15K): [in this window] [in a new window]   Fig. 3. Vertical angular velocity components of nystagmus slow phases as a function of gaze elevation during attempted fixations along the vertical meridian. First-order linear fits were computed by the fitting procedure described in Three-dimensional angular drift. A: patient P8, right eye (same trial as in Figs. 1 and 2); slope = 0.041, offset = 4.0, P = 0.017. B: patient P6, right eye; slope = 0.072, offset = 2.2, P < 0.001.

Figure 3A shows traces from the same experimental trial as in Figs. 1 and 2 (patient P8, right eye), but now the data is derived from the entire length of the trial, i.e., the sequence of eight targets appeared three times. The data points are widely scattered along the ordinate, but the negative correlation between vertical angular velocity and vertical eye position is still significant (P < 0.05). Alexander's law is therefore valid in this example (values given in the legend of Fig. 3). In Fig. 3B, which gives another example (patient P6, right eye), the correlation is even better (P < 0.001). The offset of the regression line in Fig. 3A is more negative than in Fig. 3B. The offset values represent an additional drift velocity that is unexplained by Alexander's law and that could be, for example, of vestibular origin. The presence of this additional drift velocity leads to a shift of the intercept of the regression line through the eye position axis (abscissa).

In the subsequent analysis, we name the slopes and offsets of the first-order regression between eye position and eye velocity according to the directions of the two variables, e.g., the _x-r_y slope is the slope of the regression between torsional angular velocity (_x, ordinate) and vertical eye position (r_y, abscissa). Vertical drift, Horizontal drift, and Torsional drift summarize vertical, horizontal, and torsional angular eye drifts as a function of horizontal-vertical gaze directions, thus describing _y-r_y, _z-r_z, _x-r_y, and _x-r_z slopes and offsets of the eyes of the cerebellar patients. As already stated in METHODS, the figures depict data from right eyes only, but in the text we present the data from both eyes.

Vertical drift

Figure 4A shows all _y-r_y slopes that were significant in the group of cerebellar patients (right eyes only). The first-order linear regressions were computed with gaze directed at three different horizontal target positions (center, 20° right, and 20° left). Dashed lines connect the data points of individual cerebellar patients. With gaze at zero horizontal eccentricity, 12 patients showed Alexander's law and six patients showed inverse Alexander's law (Fig. 4A, center data points). To visualize the differences in the _y-r_y slope caused by changing horizontal gaze direction, we subtracted the _y-r_y slope values for each subject obtained during fixations along the vertical meridian (Fig. 4B). No systematic change was observed as a function of horizontal eye position. For the pooled data (n = 18 right + 17 mirrored left eyes), the differences of _y-r_y slopes between the three horizontal gaze directions were not significant [t-tests: P (right-center) = 0.119; P (right-left) = 0.445; P (center-left) = 0.710].

View larger version (26K): [in this window] [in a new window]   Fig. 4. Cerebellar patients, right eyes. Attempted fixations along vertical lines. Abscissa: 20, 0, 20° horizontal eccentricities. Data points of individual patients are connected by dashed lines. A: y-ry slopes of significant fits. B: differences of y-ry slopes from the values at 0° eccentricity. C: y-ry offsets. D: differences of y-ry offsets from the values at 0° eccentricity.

For the _y-r_y offsets, the same two types of plots are shown in Fig. 4, C and D. Of the 18 patients, 15 showed negative _y-r_y offsets at zero horizontal gaze eccentricity, i.e., they had an upward drift when they attempted to look at the center target (Fig. 4C). There was a significant increase of _y-r_y offsets as patients moved their line of sight left or right (Fig. 4D). On average, this increase was similar both in adduction and abduction. For the pooled data, the differences of _y-r_y offsets between the two eccentric horizontal gaze directions and the central gaze direction were highly significant [t-tests: P (right-center) < 0.001; P (left-center) < 0.001], but the difference between the two eccentric positions was not significant [t-test: P (right-left) = 0.086].

Thus Fig. 4 demonstrates that the increase of downbeat nystagmus with horizontal gaze eccentricity is not caused by a further decrease of the time constant of the integrator, but is instead a result of an increased velocity bias and therefore independent of velocity-to-position integration. Note that in none of the patients was there a significant correlation between the _y-r_y offset (i.e., the vertical velocity bias) and the _y-r_y slope (i.e., the reciprocal of the vertical integrator time constant) for any of the three horizontal gaze directions.

Horizontal drift

The same steps of analysis used for vertical drift were also applied to horizontal drift. Figure 5, A and B, shows the _z-r_z slopes of the patients at three different vertical gaze eccentricities (20° up, center, and 20° down). Most data points were negative, consistent with Alexander's law (Fig. 5A). No differences were found between the _z-r_z slopes at different vertical eccentricities (Fig. 5B). This was confirmed for the pooled data [t-tests: P (up-center) = 0.076; P (up-down) = 0.575; P (center-down) = 0.129].

View larger version (20K): [in this window] [in a new window]   Fig. 5. Cerebellar patients, right eyes. Attempted fixations along horizontal lines. Abscissa: 20, 0, 20° vertical eccentricities. Data points of individual patients are connected by dashed lines. A: z-rz slopes of significant fits. B: differences of z-rz slopes from the values at 0° eccentricity. C: z-rz offsets. D: differences of z-rz offsets from the values at 0° eccentricity.

Velocity biases, i.e., _z-r_z offsets, were scattered around zero (Fig. 5C) with no tendency to shift in the positive or negative direction with changing vertical eccentricity (Fig. 5D). Indeed, for the pooled data, the differences between the three data sets were statistically not different [t-tests: P (up-center) = 0.269; P (up-down) = 0.380; P (center-down) = 0.970].

Thus Fig. 5 demonstrates that the time constant of the horizontal velocity-to-position integrator did not change when patients were looking up or down and that there was no consistent velocity bias in the horizontal direction. In other words, there was no spontaneous horizontal nystagmus.

Torsional drift

In theory, it should be possible to determine if Alexander's law is valid in the torsional direction if torsional nystagmus is present. Because the range of torsional eye positions in which this nystagmus takes place is very limited, we were not able to fit significant first-order regressions between the torsional eye position and the torsional component of angular velocity. We also tried to determine if torsional drift velocity decreases exponentially with time during slow phases, which would be expected if the drift is caused by a leaky torsional velocity-to-position integrator. The limited torsional range, however, made such an analysis impossible.

Next we tried to determine if torsional drift depends on the horizontal and vertical eye position components. This was not a test for Alexander's law in the torsional direction, but a test to determine if the tendency of the eye to drift out of Listing's plane is a function of horizontal and vertical eye position.

Using the same data on which Fig. 3A is based (patient P8, right eye), Fig. 6 depicts the torsional angular drift velocity as a function of eye position along the vertical and horizontal meridians. While the negative _x-r_y slope was very small (Fig. 6A), there was a clear relation between _x and r_z (_x-r_z slope): in abduction (negative horizontal position), there was an intorsional drift and in adduction (positive horizontal position), an extorsional drift (Fig. 6B). The offsets in Fig. 6, A and B, are small, meaning that, in this patient, there was almost no torsional spontaneous nystagmus.

View larger version (14K): [in this window] [in a new window]   Fig. 6. Example of the relation between torsional components of angular velocity vectors (ordinate) and gaze eccentricity along the meridians (abscissa) (patient P8, right eye, same data as in Fig. 3A). A: torsional velocity vs. vertical eye position; slope = 0.015, offset = 0.1, P < 0.001. B: torsional velocity versus horizontal eye position; slope = 0.060, offset = 0.1, P < 0.001.

Figure 7, A and B, shows the _x-r_y slopes in the usual format for all patients. There was a considerable overlap between the data points at the three horizontal directions (Fig. 7A). This is better visualized by subtracting the values obtained during fixations along the vertical meridian (Fig. 7B). For the pooled data, the _x-r_y slopes at the two horizontal gaze eccentricities were not significantly different from the slope at 0° [t-tests: P (right-center) = 0.552; P (left-center) = 0.075], but the differences between _x-r_y slopes at the two eccentric positions were significant [t-test: P (right-left) = 0.001], with the slopes being more positive in adduction.

View larger version (19K): [in this window] [in a new window]   Fig. 7. Cerebellar patients, right eyes. Attempted fixations along vertical lines. Abscissa: 20, 0, 20° horizontal eccentricities. Data points of individual patients are connected by dashed lines. A: x-ry slopes of significant fits. B: differences of x-ry slopes from the values at 0° eccentricity. C: x-ry offsets. D: differences of x-ry offsets from the values at 0° eccentricity.

Figure 7, C and D, shows the _x-r_y offsets at the different horizontal gaze eccentricities. There was a clear gradient of increasing extorsional velocity as the eyes moved left, i.e., toward adduction; on average, there was already an extorsional velocity bias at zero horizontal eccentricity, which corresponds to an extorsional spontaneous nystagmus (Fig. 7C). The gradient of _x-r_y offsets becomes even clearer when plotting the differences to the offsets at central fixation (Fig. 7D). For the pooled data, the differences between the three data sets were highly significant [t-tests: P (right-center) < 0.001; P (left-center) < 0.001; P (right-left) < 0.001].

In Fig. 8, A and B, _x-r_z slopes are plotted with the line of sight pointing up, straight ahead, and down. As expected from Fig. 7, C and D, which shows a gradient of the _x-r_y offsets toward a drift in the extorsional direction when the eyes moved to the left (adduction), almost all eyes had positive _x-r_z slopes (Fig. 8A). There was no gradient of _x-r_z slopes as a function of vertical eccentricity (Fig. 8B). Accordingly, the differences between the three data sets were not significant for the pooled data [t-tests: P (up-center) = 0.106; P (up-down) = 0.597; P (center-down) = 0.858].

View larger version (20K): [in this window] [in a new window]   Fig. 8. Cerebellar patients, right eyes. Attempted fixations along horizontal lines. Abscissa: 20, 0, 20° vertical eccentricities. Data points of individual patients are connected by dashed lines. A: x-rz slopes of significant fits. B: differences of x-rz slopes from the values at 0° eccentricity. C: x-rz offsets. D: differences of x-rz offsets from the values at 0° eccentricity.

The _x-r_z offsets at different elevations are shown in Fig. 8, C and D. Data points were scattered around zero with averages slightly above zero for all three different vertical directions of the line of sight (Fig. 8C). There was no gradient of _x-r_z offsets as a function of vertical gaze eccentricity (Fig. 8D). Statistically, this was confirmed in that there were no significant differences between the three data sets [t-tests: P (up-center) = 0.388; P (up-down) = 0.591; P (center-down) = 0.921].

In summary, we found a significant gradient of torsional drift velocity in the horizontal direction: as the eyes adducted, extorsional drift increased. In many patients, a small extorsional drift was already present when the eyes looked straight ahead. There were no torsional drift gradients as a function of vertical position.

Is torsional drift independent from vertical drift?

In theory, torsional angular drift may simply be a result of cross-coupling from vertical angular drift. For instance, if the ocular rotation axis of a vertical drift changes with eye position, we expect an increasing torsional component of drift as a function of horizontal gaze eccentricity. Therefore, in each patient, we looked for correlations between torsional and vertical angular drift velocities. Note again that rotation and angular velocity vectors are, by definition, given in a head-fixed coordinate system (see METHODS), i.e., if a vector has a torsional component, the vector tilts out of the frontal head-fixed plane spanned by the vertical and horizontal axes.

We investigated possible cross-coupling effects between vertical and torsional drifts. Because the torsional drift in the straight-ahead gaze position is low, but increases as a function of horizontal gaze eccentricity, we studied the correlation between torsional and vertical drifts in 20° abduction and 20° adduction during attempted fixations along a vertical line. Figure 9A shows data from the same example given in Figs. 1, 2, and 3A (patient P8, right eye): torsional drift velocity is plotted against vertical drift velocity during 20° abduction and attempted vertical fixations (20° up, 20° down). There was a good correlation between torsional and vertical velocity in that the intorsional velocity decreased as upward velocity decreased. In this patient, however, the correlation between torsional and vertical velocity in adduction was not significant, but there was a tendency of extorsional velocity to decrease as upward velocity decreased (Fig. 9C). In the whole study population, 26 of 35 eyes showed a significant positive _x-_y slope during abduction, and 21 of 35 during adduction.

View larger version (28K): [in this window] [in a new window]   Fig. 9. Vertical and torsional components of nystagmus slow phases. A: patient P8, right eye (same trial as in Figs. 1, 2, and 3A). Torsional vs. vertical angular velocity components of nystagmus slow phases during 20° abduction and attempted vertical fixations (20° upward, 20° downward); slope = 0.246, offset = 0.7861, P < 0.001. B: x-ry slopes vs. y-ry slopes in all patients (right eyes) during 20° abduction and attempted vertical fixations; regression not significant (P = 0.532). C: patient P8, right eye (same trial as in Figs. 1, 2, and 3A). Torsional vs. vertical angular velocity components of nystagmus slow phases during 20° adduction and attempted vertical fixations; regression not significant (P = 0.094). D: x-ry slopes vs. y-ry slopes in all patients (right eyes) during 20° adduction and attempted vertical fixations; regression not significant (P = 0.890).

To test if this correlation between torsional and vertical angular velocity during fixations along vertical lines with eccentric horizontal gaze (abduction or adduction) was indeed caused by a cross-coupling of the two velocity components, or if it was only a result of individually different slopes between vertical eye position and torsional or vertical drift velocity, we plotted the _x-r_y and _y-r_y slopes of all patients against each other (Fig. 9, B and D). Both in abduction (Fig. 9B) and adduction (Fig. 9D), the slopes did not correlate significantly; _x-r_y slopes were scattered around zero whereas _y-r_y slopes were negative (Alexander's law) or positive (inverse Alexander's law). Therefore, torsional drift is not simply caused by a fixed cross-coupling of vertical drift, and the correlation between _x and _y in individual patients only reflects the fact that torsional velocity is modulated slightly by vertical eye position. If the velocity of torsional drift would mathematically depend on the velocity of vertical drift, e.g., via the half-angle rule, we would expect a similar correlation between _x and _y at a specific horizontal gaze eccentricity in all patients.

Another argument against torsional drift velocity being a result of a fixed cross-coupling with vertical drift velocity is shown in Fig. 10. Here we plotted the vertical-torsional drift direction of a patient's right eye (patient P8) as a function of horizontal eye position in upward (Fig. 10A) and in downward (Fig. 10B) gaze. The 0° vertical-torsional drift direction corresponds to a purely vertical drift whereas the +90° vertical-torsional drift is purely extorsional. In the following analyses, we call the slope of the linear regression between horizontal direction of gaze and vertical-torsional drift direction the tilt angle coefficient because it describes by how much the ocular rotation axis tilts in the horizontal plane as a function of the horizontal direction of the line of sight. If the tilt angle coefficient is zero, the ocular rotation axis stays head-fixed independent of the horizontal direction of gaze. A coefficient of 1 corresponds to an eye-fixed axis and a coefficient of 0.5 is necessary to follow Listing's law and, therefore, the half-angle rule. Clearly, in upward gaze, the tilt angle coefficient was much higher, and therefore close to eye-fixed (0.89), than in downward gaze, where it was close to head-fixed (0.19). This means that the torsional drift component relative to the vertical drift component was stronger in upward gaze than in downward gaze. Because for this patient Alexander's law was valid along the vertical direction, vertical velocity changed as a function of vertical eye position. On average, torsional drift velocity also changed somewhat, but less. Therefore the drift direction tilted toward more torsion as the eye was elevated.

View larger version (29K): [in this window] [in a new window]   Fig. 10. Vertical-torsional drift directions and tilt angle coefficients. A: patient P8, right eye (same trial as in Figs. 1; 2; 3A; and 9, A and C). Vertical-torsional drift direction as a function of horizontal position in 20° upward gaze. The regression line (solid) was determined by an iterative linear fit whereby the most distant point from the line was discarded each time; the iteration was stopped after 20% of the data points were rejected. Slope ("tilt angle coefficient") = 0.89, offset = 9.2. B: same as in A but in 20° downward gaze. Slope (tilt angle coefficient) = 0.19, offset = 0.5. C: tilt angle coefficients in 5 patients (right eyes) that showed Alexander's law of vertical drift. Solid lines connect data points of individuals in upward (Up) and downward (Down) gaze. D: same as in C but with data from 5 other patients (right eyes) with inverse Alexander's law of vertical drift.

Figure 10, C and D, summarizes the tilt angle coefficients of 10 of the 18 right eyes in which the slopes between horizontal eye position and vertical-torsional drift direction were significant in both upward and downward gaze. Five right eyes demonstrated Alexander's law (Fig. 10C), the remaining five demonstrated inverse Alexander's law (Fig. 10D) in the vertical direction. Clearly, the ocular rotation axis became more head-fixed as vertical gaze was moved in the direction in which vertical velocity increased. Some patients did not show this pattern, but in these cases the regressions were not significant, mainly because drift velocities were low and, therefore, vertical-torsional drift directions were noisy.

Is torsional drift independent from horizontal drift?

This subsection follows the same logic as Is torsional drift independent from vertical drift? We attempted to determine if torsional angular drift is a result of a fixed cross-coupling with horizontal angular drift. Because both torsional and horizontal angular drift increased with horizontal gaze eccentricity, the correlation between the two velocity components was, as expected, typically significant in 15 of the 18 right eyes and 14 of the 17 left eyes. Figure 11A depicts the correlation between these two parameters for patient P8 (right eye).

View larger version (15K): [in this window] [in a new window]   Fig. 11. Horizontal and torsional components of nystagmus slow phases. A: patient P8, right eye (same trial as in Figs. 1; 2; 3A; 9, A and C; and 10, A and B). Torsional vs. horizontal angular velocity components of nystagmus slow phases during attempted eccentric fixations along the horizontal meridian (20° rightward, 20° leftward). Slope = 0.385, offset = 0.111, P < 0.001. B: x-rz slopes vs. z-rz slopes in all patients (right eyes) during attempted eccentric fixations along the horizontal meridian; regression not significant (P = 0.214).

To investigate whether or not these significant individual correlations between torsional and horizontal angular velocity during fixations along the horizontal meridian are caused by a fixed cross-coupling of the two velocity components, or are only a result of individually different slopes between horizontal eye position and torsional or horizontal drift velocity, we plotted the _x-r_z and _z-r_z slopes of all right eyes against each other (Fig. 11B). Because the adducting horizontal drift (positive sign) increased as the eye abducted (negative sign), and the extorsional drift increased (positive sign) as the eye adducted (positive sign), data points are scattered in the upper left quadrant. However, there was no significant correlation between the slopes, which indicates that the torsional drift was independent from the horizontal drift, even though they correlated in individual patients, because of the horizontal-eye-position dependence of both drifts. Therefore, the torsional drift is not caused by a fixed cross-coupling from the horizontal drift, and the correlation between _x and _z in individual patients only reflects the fact that both torsional and horizontal velocity were modulated by horizontal eye position.

We also investigated whether or not there were consistent gradients between the horizontal-torsional drift direction and the gaze elevation in abduction and adduction. Significant gradients could be seen in most eyes (33 of 35 eyes in abduction, 34 of 35 in adduction), but the magnitudes and signs of these gradients showed no pattern. In only a minority of eyes (12 of 35), the gradients between horizontal-torsional drift direction and vertical gaze direction had the same sign in abduction and adduction.

Does vertical drift obey Listing's law?

To interpret further the patterns of vertical drift in our patients, we developed a theoretical scheme of the predicted effects of different types of abnormalities on the directions and eye-position dependence of vertical-torsional drift. Figure 12 depicts four possible relations between torsional and vertical drift angular velocities. Figure 12, A, C, E, and G, shows vertical angular drift velocity as a function of torsional angular drift velocity; Fig. 12, B, D, F, and H, shows vertical-torsional drift direction (in polar coordinates) as a function of horizontal eye position. In each panel, four data points are depicted: when the patient is looking 20° right and 20° up (), when the patient is looking 20° right and 20° down (), when the patient is looking 20° left and 20° up (), and when the patient is looking 20° left and 20° down (). In some instances (Fig. 12, B-D), data points during upward and downward viewing overlap (producing hexagrams). In the left panels, we connected the points with the same horizontal eccentricity; in the right panels, the points with the same vertical eccentricity.

View larger version (21K): [in this window] [in a new window]   Fig. 12. Theoretical scheme of possible relations between torsional and vertical drift angular velocities. , eye position 20° right/20° up. , eye position 20° right/20° down. , 20° left/20° up. , 20° left/20° down. Overlying upward and downward data points produce hexagrams. A, C, E, G: vertical angular drift velocity as a function of torsional angular drift velocity. Data points with the same horizontal eccentricity are connected. B, D, F, H: vertical-torsional drift direction in polar coordinates as a function of horizontal eye position. Data points with the same vertical eccentricity are connected. A, B: vertical drift in Listing's plane caused by leaky vertical velocity-to-position integration. C, D: vertical drift with no eye position dependence. E, F: summation of the vertical drifts in (A, B) and (C, D). G, H: summation of the vertical drift in (E, F) and horizontal-eye-position-dependent torsional drift.

Figure 12A shows vertical drift caused by a leaky vertical velocity-to-position integration; this drift, we assume, obeys Listing's law, i.e., the half-angle rule. Consequently, there is a torsional drift component that depends on the vertical drift velocity (3°/s at 20° vertical gaze eccentricity) and the horizontal gaze eccentricity (20°). The lines connecting the drift directions during rightward and leftward viewing superimpose in upward and downward gaze; the slope of both lines is 0.5 according to the half-angle rule (Fig. 12B).

Figure 12C depicts a situation with an upward vertical drift (5°/s) of purely vestibular origin that, we assume, shows no eye position dependence (i.e., is head-fixed). Therefore, in no direction of gaze is there a torsional drift component (Fig. 12D). The slope between drift directions during rightward and leftward viewing is 0 and the lines superimpose in upward and downward gaze.

Figure 12E combines the situations from Fig. 12, A and C, i.e., we added together the vertical leaky integrator drift, which obeys Listing's law, and the upward bias drift, which is independent of vertical eye position and does not include a torsional component in any direction of gaze. As a consequence, the slopes between the vertical-torsional drift directions during abduction and adduction are different between upward and downward gaze (Fig. 12F).

In Fig. 12G we added the type of torsional drift that we found in the patients. This torsional drift depends on horizontal eye position and is intorsional in abduction (in this example, 1°/s during 20° abduction) and extorsional in adduction (in this example, 1°/s during 20° adduction). This leads to an additional change of the slopes of the lines between vertical-torsional drift direction during abduction and adduction in both upward and downward gaze.

This theoretical example demonstrates that we cannot simply examine the vertical-torsional drift directions to decide whether or not leaky velocity-to-position integration in the vertical direction leads to drift movements following Listing's law. To test this question correctly, the slopes between vertical position and torsional or vertical angular drift (_x-r_y slope and _y-r_y slope) must be analyzed, both of which reflect the drift induced by leaky integration when the eye is moved in a vertical direction. Assuming that Listing's law is adhered to by the drift caused by leaky vertical velocity-to-position integration alone (i.e., there is no vertical velocity bias), the coefficient c_xy = (_x-r_y slope/_y-r_y slope) will increase as a function of horizontal eccentricity (abduction and adduction). It can easily be shown that c_xy coincides with the horizontal component r_z of the current ocular rotation vector, provided Listing's law is valid: