INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Downbeat nystagmus (DN) is defined as fixation nystagmus in which the fast phase is directed downward during gaze straight ahead. The drifts of the eyes show a linear increasing or decreasing velocity (Leigh and Zee 1999). Pathoanatomically, lesions of the cerebellum, mainly the vestibulo-cerebellum and seldom of the brain stem, have been shown (Baloh and Yee 1989; Büttner et al. 1995). Several explanations for the pathogenesis of DN have been considered. One hypothesis, probably the most favored one, assumes DN to be a central vestibular syndrome (Baloh and Spooner 1981; Böhmer and Straumann 1998; Brandt and Dieterich 1995; Büchele et al. 1983; Marti et al. 2001). Another hypothesis is that an imbalance of the vertical smooth pursuit tone implies a constant vertical velocity offset (Zee et al. 1974). To explain the dependence of DN on vertical eye position (vertical integrator failure), these authors additionally assumed a neural integrator deficit that results in an inability to hold the eyes in eccentric positions.

Recently Straumann et al. (2000) investigated a homogeneous group of subjects with hereditary cerebellar atrophy, most of whom had DN. With three-dimensional search-coil recordings they demonstrated that all subjects had a uniform eye movement pattern consisting of the DN component as well as a horizontal centripetal drift and a torsional drift that increased with horizontal eye eccentricity. The torsional drift component was accompanied by an increase of vertical drift velocity during lateral gaze. Straumann et al. (2000) proposed that the torsional drifts observed in DN constitute a mismatch between torsional postsaccadic position of the eye defined by the saccadic burst and the torsional resting position defined by the mechanical properties of the eye plant.

In healthy subjects, small postsaccadic torsional drifts called blips can be observed (Straumann et al. 1995). Pathologically large amplitude blips were found in a patient with a dorsolateral medulla and cerebellar lesion (Helmchen et al. 1997). Torsional drifts violate Listing's law, a fundamental property of ocular fixations. It states that the eye position has a zero torsional component when expressed as single rotations from a certain starting position, the so-called primary position. Fixations and to a lesser degree smooth pursuit and saccades in normal human subjects obey Listing's law with SDs of maximum 1° (Straumann et al. 1996; Tweed and Vilis 1990; Tweed et al. 1992). It has been shown experimentally that Listing's law represents an intrinsic property of brain stem regions such as the burst generator or the velocity-to-position integrator (Crawford 1994). In addition, peripheral structures, the pulleys of the eye plant, are possibly an important factor in implementing Listing's law (Quaia and Optican 1998; Schnabolk and Raphan 1994). Other eye movements such as the vestibuloocular reflex (VOR) or optokinetic nystagmus do not obey Listing's law. Therefore disorders of the smooth pursuit system or a leaky neural integrator are supposed to obey Listing's law, whereas nystagmus due to vestibular imbalance violates it.

The compliance of a vertical drift, as found with DN, with Listing's law in a head-fixed coordinate system can best be evaluated during different horizontal eye positions. If the axis of rotation of the eye is head-fixed, the rotation leads at all horizontal eye positions to a purely vertical angular velocity drift. In contrast, if the rotation axis is eye-fixed, a torsional component proportional to the change in horizontal eye position is introduced. For drifts related to smooth pursuit eye movements, the rotation axis of the eye is actually between eye- and head-fixed (half-angle rule) (Tweed and Vilis 1987, 1990). Thus the half-angle rule reflects Listing's law. It has been shown that during low-velocity VOR (such as oscillations used for clinical testing) the ocular rotation axis changes its orientation by one-fourth of the horizontal eye position (quarter-angle rule) (Misslisch and Tweed 2001; Misslisch et al. 1994; Palla et al. 1999). This implies a smaller torsional component of the angular velocity than in the half angle rule.

To analyze the dependence of downbeat nystagmus slow phase velocity on eye position, we recorded three-dimensional (3-D) eye movements with the search-coil technique and used multiple regression to answer the following questions: does DN have different etiologies and present with different features? Which different subtypes of DN can be distinguished? Do all types of DN violate Listing's law? Are the observed slow phases compatible with the hypothesized vestibular or smooth pursuit origins of the nystagmus? The answer to these questions might help us better understand the transformation of premotor oculomotor signals into ocular motor neuron activity.

In the following, we show that 3-D analysis of the slow phases of DN patients reveals that DN is probably neither caused by damage to central vestibular pathways carrying semicircular canal information nor by a smooth pursuit imbalance. We propose that the observed effects can be explained by partial damage of a brain stem-cerebellar loop that augments the time constant of the neural velocity to position integrators in the brain stem and neurally adjusts the orientation of Listing's plane. A preliminary account of the present data were given elsewhere (Hoshi et al. 2001).

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Subjects

Nineteen patients (ages: 29-84 yr, mean: 61.5 yr) with DN were examined with standard clinical methods and by MRI scans of the brain stem and cerebellum. Two patients had a cerebellar atrophy, three had an Arnold-Chiari malformation, two had a paraneoplastic syndrome, and three patients had lesions affecting cerebellar structures; the disorder of the remaining nine patients had no specific etiology. This distribution is generally seen in patients with DN (Leigh and Zee 1999). Eighteen healthy subjects served as controls (ages: 23-91 yr, mean: 46.9 yr). All patients and subjects gave their informed consent after explanation of the experimental procedure that was in accordance with the Declaration of Helsinki.

Experimental paradigm

A laser dot (size: 0.1°, screen distance: 140 cm) jumped to eight positions arranged on a square around the gaze straight ahead position (Fig. 1C). Positions were placed ±18° from the center on the horizontal and vertical meridians and ±25,46° on the two oblique meridians. Subjects were seated in the center of the coil frame in an otherwise dark environment and were asked to fixate the visible target, which jumped every 2.5 s between gaze straight ahead and one of the peripheral targets. The head of the subject was immobilized in an upright position by an adjustable chin rest. An additional search coil fixed to the patient's forehead monitored immobility. The conjunctiva of the left eye, in which the dual search coil was placed, was anesthetized with oxybuprocain HCl.

View larger version (30K): [in this window] [in a new window]   Fig. 1. Original recordings from a control subject. Eye positions are expressed as rotation vector components following the right-hand rule and converted to degrees for convenience. Thus positive vertical eye position indicates downward gaze, positive horizontal eye position leftward gaze as seen from the subject, and positive torsion clockwise eye rotation, again seen from the subject's perspective. Slow phases and fixation periods are shown as big dots, saccadic eye movements are shown as small dots. A-C: data in head-fixed coil coordinates. A: eye positions viewed from above. B: eye positions viewed from the side. - - -, the fitted displacement plane (Listing's plane). The tilt of the plane of eye positions corresponds to an angle of 16.8° upward tilt of primary position. The plane has a SD of 0.44°. C: frontal view of eye positions. The 9 target positions are clearly visible as horizontal-vertical eye positions. D-F: data after rotation to Listing's plane. Note the different axis range; the 0 position is not aligned with gaze straight ahead. D and E: torsional eye position components align with the torsional axis.

Search-coil recordings

The field coil system consisted of a cubic (side length: 140 cm) aluminum frame that produced three orthogonal magnetic fields (Remmel Systems). Ocular rotation of the left eye around the horizontal (z axis), vertical (y axis), and the torsional (x axis) was recorded with a dual search coil (Skalar, Delft, Netherlands). Coil and target position data were sampled at a rate of 1 kHz.

Calibration

Immediately before each eye movement recording, the coil was moved manually within the coil frame to determine the magnitude and relative orientation of the three magnetic fields and the gain of the directional coil (see APPENDIX for detailed description). Then the offsets and the gain of the torsional coil as well as the relative orientation of the coil on the eye were determined on the basis of data recorded while the subject was wearing the eye coil and was looking at different target positions. The calibration did not depend on accurate fixation of these targets. The resulting eye positions were expressed as rotation vectors (Haustein 1989).

Data analysis

Data were analyzed off-line in MATLAB. The calibrated data were low-pass filtered with a digital Gaussian filter having a band width of 30 Hz. Saccade and fast phase were automatically detected and removed from the data using a combined velocity-acceleration criterion in interactive software so that detection errors could be corrected manually. Slow phases shorter than 25 ms were discarded. For each remaining slow phase, the median slow phase component velocity and eye position were used for analysis. Slow phases longer than 200 ms were divided into two or more parts of equal length to avoid collecting fewer data points for periods without frequent fast phases. Most of the analysis was performed after fitting Listing's plane to the data and transforming the whole data set into Listing's coordinates.

Slow phase component velocity, the temporal derivative of eye position, is not equal to angular velocity (Haslwanter 1995; Hepp 1994). In the figures, component velocity was expressed in°/s and multiplied by 2 for easy comparison with the more familiar angular velocity.

The dependence of slow phase component velocity on eye position r was analyzed by a linear least-squares fit procedure to the equation

(1)

with the free parameters T and r₀. T is a 3x3 matrix describing this dependence on eye position (see Crawford 1994), and ₀ is a component velocity vector describing an eye-position independent velocity offset. A single velocity component is, therefore, described as a linear combination of all three eye position components and a velocity offset. As an example, the equation for the vertical eye component velocity _v is

(2)

with r_t, r_v, r_h being the torsional, vertical, and horizontal eye position components, ₀ the fitted vertical velocity offset; and t_vt, t_vv, t_vh the fitted factors (elements of the matrix T), describing the influence of the respective eye position component on vertical eye velocity. In the following, t_vv is called a "principal" dependence because it describes the relationship between vertical eye velocity and vertical eye position; t_vt and t_vh are called "nonprincipal" dependencies. Therefore to describe each eye velocity component, four free parameters are computed. To describe 3-D eye velocity, 12 free parameters are fitted. During this general fit, which is equivalent to a multiple linear regression analysis, those 10% of the data points having the largest residual error were classified as outliers and removed from the data. The multiple regression results are used in the following to derive parameters such as the null position of the nystagmus.

To test for different types of eye-position dependence, three types of analysis were performed.

In the first analysis, the nonprincipal (or off-diagonal) elements of T are set to zero. In this case, the slow phase eye velocity obeys Listing's law, if the torsional offset is zero. It can thus be interpreted as a leaky integrator with vertical and/or horizontal offsets. This procedure is equivalent to a simple linear regression of each eye velocity component over the respective eye position component. Thus each eye velocity component is described by

(3)

with i = t, v, h for torsional, vertical, horizontal. Here t_ii is the slope between velocity and position and _0i is the slow phase velocity at zero eye position (primary position). In this case, each velocity component is described by only two free parameters, which gives six parameters for 3-D eye velocity. Note that _ii = 1/t_ii can be interpreted as the respective integrator time constant.

In the second analysis, best subset multiple regression was performed to validate the results from the complete regression analysis. For best subset regression, each possible combination of the parameters t_ij of the matrix T was fitted to the data, and the best combination was determined according to the minimal Mallow's Cp statistics. The minimally possible number of fitted parameters is three: the three velocity offsets.

In the third analysis, a quadratic multiple regression (21 parameters) was performed to validate clinical and experimental reports about increasing vertical drift velocity during lateral gaze (e.g., Straumann et al. 2000).

Statistical analysis

As mentioned in the preceding text, fitting Eq. 1 is equivalent to a multiple linear regression, and fitting Eq. 3 is equivalent to a simple linear regression analysis using the eye position components as independent variables and the derivatives of the eye position components as dependent variables. Thus the statistical analysis is based on tests for multiple regression analysis. A fit was considered significant if P < 0.01 as computed by the F test. As a further indicator of the goodness of fit and to compare the different types of analysis described in the preceding text, the R² value was used. It gives the percentage of variance of the data described by the respective model. For one-dimensional regression, the R² value is equivalent to the squared correlation coefficient. If the R² value was below 50%, the fit was considered to insufficiently describe the data. For further analysis of single parameter values, the SDs of each fitted parameter and the respective probability that the parameter is different from zero was computed. A parameter with a P value of P < 1-(1-0.025) ^(1/n) with n being the number of parameters is considered significantly different from zero (Bonferroni correction). Comparison of simple regression (Eq. 3), multiple regression (Eq. 1), and quadratic regression was done using F statistics, which evaluates whether adding the parameters yields a significant increase (P < 0.01) in explained variance.

Coil slippage

As has been noticed previously during recordings, torsional slippage of the search coil is frequently observed (Straumann et al. 2000; Van Rijn et al. 1994). To minimize these effects, the whole duration of the experiment lasted only 45 s. In addition, the torsional eye position at gaze straight ahead, to which the eye returned after each eccentric position, served as a control. Coil slips happened exclusively during large saccadic eye movements, presumably because a blink was performed at the same time. Coil slips were detected if the median torsion during fixation of the center target deviated more than 1.5° from the previous fixation period. In such a case, all subsequent eye position data were rotated by an eye-fixed torsional angle so that the median torsion during both fixation periods remained the same. However, the results depended only negligibly on whether slip was removed or not. For example, the SDs of Listing's plane decreased by only 0.1° for both patients and healthy subjects. Therefore the results of the original data set are reported.

Definitions

Three-dimensional eye positions were determined according to the right-hand rule with positive rotation around the x axis referring to excyclorotation of the right and incyclorotation of the left eye, around the y axis to downward, and around the z axis to leftward movements. Eye positions are expressed as components of rotation vectors (Haustein 1989) and, for convenience, converted to Listing's angles given in degrees using _i = arctan(r_i) · 360/.

Gaze straight ahead, i.e., the midposition of the eye in the orbit, was maintained during fixation of the center position of the target. It is the reference position for analysis performed in head coordinates.

Primary position is the reference position in a 3-D coordinate system, from which ocular positions during fixation can be obtained by single rotations with all rotation axes lying in a plane (Listing's plane) and with gaze direction in primary position being orthogonal to Listing's plane.

Null position is defined as the eye position at which an eye position-dependent nystagmus reverses its direction. Gaze straight ahead, primary position, and null position do not have to coincide. The direction of nystagmus always refers to the fast phase if not stated otherwise.

"Eye velocity" refers to the temporal derivative of eye position (component velocity) rather than to angular eye velocity, if not stated otherwise.

To distinguish between different position-velocity dependencies, we define a dependence as "principal" if it relates velocity to position of the same dimension, e.g., dependence of vertical velocity on vertical position. In contrast, other dependencies such as dependence of vertical velocity on horizontal position are called "nonprincipal."

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Listing's plane

Listing's plane was determined from eye positions of slow phases for subjects with DN and fixation periods for healthy subjects.

For healthy subjects, the SD for the width of Listing's plane ranged from 0.39 to 1.33 (average: 0.79°) and the primary positions ranged from 10.0° down to 21.0° up (average: 7.0 ± 9.7° up, see Fig. 1 for an example) in the vertical and from 11.5° left to 16.3° right (average: 0.8 ± 8.0° right) in the horizontal plane. These data correspond well to previous findings (Haslwanter et al. 1994). We also computed the mean angle between gaze straight ahead and primary position. This angle quantifies the distance the eye has to be rotated from the gaze straight ahead position to primary position. For healthy subjects, it was 13.0 ± 5.3°.

Results for the patients with DN were similar (see Fig. 2 for an example). For all patients SD of Listing's plane ranged from 0.59 to 1.47° (average: 0.92°), which is on average slightly larger but not significantly different from that of healthy subjects (t-test, P = 0.12). Primary position was on average 9.5 ± 8.4° (range: 5.4° down to 23.9° up) above gaze straight ahead in the vertical plane and ranged from 25.4° left to 18.8° right (average: 2.4 ± 11.9° left) in the horizontal plane. The mean angle between straight ahead and primary position (14.5 ± 9.4°) was not different from that of healthy subjects (t-test; P = 0.58). Thus as in healthy subjects, primary position is on average above gaze straight ahead with very little horizontal deviation.

View larger version (32K): [in this window] [in a new window]   Fig. 2. Original recordings from patient V. A-F: as in Fig. 1. The tilt of the fitted displacement plane (Listing's plane, B, - - -) corresponds to an angle of 1.1° upward tilt of primary position, i.e., primary position in this patient is aligned with gaze straight ahead. The plane has a SD of 0.63°. C: the patient attempted to fixate the 9 target positions, but the eye drifted away; the drift was most pronounced for downward (positive vertical) and leftward (positive horizontal) gaze. Small torsional drift components can be seen in A and B. Because primary position was almost aligned with gaze straight ahead in this patient, there is no obvious difference between A-C and D-F.

General characteristics

To analyze DN during gaze straight ahead, a multiple linear regression (Eq. 1) was fitted to the data before rotation to Listing's plane. According to the regression, all patients had a significant upward vertical drift during fixation of the center target (DN at gaze straight ahead) with vertical slow phase velocity ranging from 0.63 to 9.40°/s. Some healthy subjects also exhibited small but significant vertical offsets, which were, however, mostly downward and smaller than 0.5°/s (range: 0.08 to 0.38°/s) except for the 91-year-old control subject (minor upbeat nystagmus, 0.60 °/s). Vertical drift in patients showed no inversion of Alexander's law (Robinson et al. 1984), i.e., the vertical drift velocity increased with downward gaze for all patients (range of difference between upward and downward gaze 0.23-8.9°/s). Six patients (31%) showed upbeat nystagmus during upward gaze (up to 2.7°/s), i.e., slow phase velocity reversed sign. In these cases, DN during downward gaze was always larger than upbeat during upward gaze.

Slow phases

For further analysis, only slow phases or fixation periods were considered. Slow phase component velocity was separated into the vertical, horizontal and torsional drift components. First, the regression results (12 parameters for multiple regression or 6 parameters for simple regression, see METHODS) will be considered, then the results will be compared with best subset regression. To evaluate possible reasons of velocity offsets, the fitted parameters will then be compared with predictions for different offsets, such as head- or eye-fixed angular velocity offsets. The results of the quadratic regression will be considered separately.

Multiple regression yielded significant fits for all subjects. The R² values of complete multiple regression after rotation to Listing's plane for patients ranged from 0.50 to 0.93, for healthy subjects from 0.06 to 0.34. Thus the statistical model explained more than 50% of the variance of the median slow phase data for all patients, while only up to 34% was explained for healthy subjects. The low R² values for healthy subjects indicate that, contrary to those for patients, drift velocities were close to noise level or resulted from noise. The root-mean-squared error (RMSE) ranged from 0.51 to 1.65°/s for patients and from 0.17 to 0.76°/s for healthy subjects. The RMSE reflects scatter in the data (see center position in Fig. 3 for an example) and measurement noise.

View larger version (34K): [in this window] [in a new window]   Fig. 3. Example of vertical slow phase velocity data () and regression predictions (+) for patient V (same patient as in Fig. 2). Median vertical slow phase component velocity is plotted against eye position. For each data point (), the predictions (+) of the simple linear regression (left), multiple linear regression (middle), and multiple quadratic regression (right) are shown. Bottom: the same plots as top but rotated so that the horizontal eye position axis is in front. Additionally, the coordinate systems are tilted by 36° so that the fitted surfaces are viewed from the side. For multiple linear and quadratic regressions, the complete eye position is taken into account for the prediction. Left: vertical velocity plotted against vertical position. The line corresponds to the simple linear regression (R2 = 0.87), which takes not into account the apparent dependence on horizontal position. Middle: vertical velocity plotted against vertical and horizontal position. The plane corresponding to the multiple linear regression (R2 = 0.90) depends on horizontal position. Right: vertical velocity plotted against vertical and horizontal position. The surface corresponds to the multiple quadratic regression (R2 = 0.93). In this patient, the quadratic multiple regression yields better fits by explaining significantly more data variance (P < 0.001) than the linear regressions, i.e., the residual distance of the data points from the fitted surface is smallest for the curved surface. The curvature of the velocity surface is significant only in the horizontal direction, thus reflecting the fact that vertical drift increased with lateral gaze.

Figure 3 gives an example for the differences between simple, multiple linear, and multiple quadratic regression. In Fig. 4, an example of original median slow phases and predicted slow phases is given for the same patient as in Fig. 3 to show how well the predictions from the multiple linear regression coincide with the data. Figure 5 further shows that prediction of positional drift directions from the fitted median slow phase velocity agrees well with the recorded data.

View larger version (31K): [in this window] [in a new window]   Fig. 4. Median values of slow phase angular velocity (torsional, vertical, horizontal) for patient V (, same data as in Fig. 3) compared with the predicted values obtained from the multiple linear regression (+, R2 = 0.90). Each data point reflects a time period of 50-200 ms (depending on nystagmus frequency, see METHODS) of slow phase velocity. Bottom: target position (solid: horizontal, dashed: vertical). All vertical slow phases had values below 0, implying large vertical offsets.  (in the torsional velocity plot), a high postsaccadic torsional velocity accompanied by a torsional-horizontal drift toward the small torsional offset.  (in the vertical velocity plot), the time that eye position is close to the vertical null position, i.e., when the subject looks to the upper-right target.

View larger version (66K): [in this window] [in a new window]   Fig. 5. Eye position during slow phases (thick lines, same data as in Figs. 2-4) and prediction of slow phase drift from multiple regression (thin lines; arrows indicate drift direction; positive vertical positions correspond to gaze down). Data are shown for torsional 0 position in Listing's coordinates. Even though the fit was performed only for slow phase median values (see Fig. 4), the predicted drift directions coincide well with the real data. According to the fit, the vertical and horizontal components of eye velocity become 0 for an eye position within Listing's plane of 25.3° up and 9.1° right from primary position. In this position, the patient would still show torsional nystagmus. The true null position at which the eye stops drifting is 2.0° clockwise (and thus out of Listing's plane), 27.0° up, and 1.4° right from primary position.

For the patient depicted in Figs. 2-5, the multiple linear regression fit assumes the following form

(4)

with the asterisks indicating parameters (units 1/s for matrix and 0.5 × rad/s for offset) significantly different from zero. For example, the lower right value in the matrix (0.2234) indicates that horizontal eye position induces horizontal drift: for 10° of lateral gaze with respect to primary position, a horizontal drift of about 2.2°/s is induced. The middle right value (0.0672) indicates that horizontal position also influences vertical drift; the lateral gaze of 10° thus not only causes the eye to move horizontally, but also vertical drift of about 0.7°/s is induced. The vertical offset (0.0520) indicates a vertical upward drift of 0.05*360/ = 5.7°/s in primary position (compare with Fig. 3).

The mean results (± SD) for all 19 patients are summarized in the following equation

(5)

with the asterisks denoting elements significantly different from zero (t-test, Bonferroni corrected). These elements indicate horizontal and vertical integrator failure (diagonal elements), violation of Listing's law by significant influence of horizontal eye position on torsional drift velocity (upper right element of the matrix), and a vertical velocity offset. In the following, it is shown that considering only mean results may give the incorrect impression of a homogeneous patient population.

Primary position and vertical drift velocity

An increasing vertical velocity on downward gaze implies an integrator failure. Ocular drifts due to an integrator failure reverse direction at the null position. Recent studies suggest that this null-position might coincide with the primary position (Crawford 1994). If a vertical drift is present at the primary position, an additional vertical velocity offset has to be assumed. Therefore a multiple linear regression (Eq. 1) was performed after rotation to Listing's plane. For two patients (11%), the null-position coincided with primary position. For one patient, the sign reverted, i.e., this patient had upbeat nystagmus in primary position. For 11 patients (58%), the vertical offset decreased with respect to the fit in head coordinates, i.e., with respect to gaze straight ahead. Thus their null position was closer to primary position than to straight ahead. The null positions computed from Eq. 1 showed considerable variation because some patients exhibited only moderate integrator leakage but strong offsets (see following text). For them, the theoretical null position was out of the vertical oculomotor range (±40°); this was the case for seven patients. For all patients, the median vertical null position was 14.9° above primary position.

Velocity dependence on nonprincipal eye position components

To determine if there are significant dependencies on nonprincipal eye position components, e.g., vertical velocity dependency on horizontal eye position, the simple regression (6 parameters, Eq. 3) and the complete regression (12 parameters, Eq. 1) were compared for each subject using an F test. For 15 patients (79%) but only 5 healthy subjects (28%), the fit was significantly better when including nonprincipal eye position dependence, i.e., when fitting all 12 rather than only 6 parameters. The data of the remaining four patients could sufficiently be described by a simple regression.

To evaluate the number of necessary parameters for each patient, a best-subset multiple regression analysis was performed. This type of analysis finds the minimal number of parameters, taking into account that more parameters always result in better fits. For patients, 8-12 parameters yielded best results (healthy subjects: 5-11 parameters; Mallow's Cp statistics). Table 1 shows the number of statistically significant parameters for this analysis. It closely resembles that of the complete multiple linear regression analysis (not shown).

Velocity dependence on principal position components and velocity offsets

As can be seen from Table 1 (white areas), 18 patients (95%) exhibited a significant dependence of vertical velocity on vertical eye position; this can be interpreted as vertical integrator failure. Thirteen patients (68%) showed a significant horizontal dependence (see Fig. 6). The negative principal dependencies can be interpreted as inverse integrator time constants. Positive principal dependencies would reflect an inversion of Alexander's law that was not found in our patients. Hence, vertical integrator failure did not imply horizontal integrator failure. Figure 6 also shows that six patients (above the diagonal line) had more horizontal than vertical integrator failure, i.e., in these patients the gaze-dependent changes of drift velocity were more prominent for the horizontal than for the vertical direction. Only seven patients (37%) had a torsional dependence significantly different from zero. For patients, the mean time constants were 4.5 s torsional, 8.8 s vertical, and 11.8 s horizontal. Only two patients had vertical time constants larger than 50 s. Healthy subjects showed 8.8 s torsional, 119 s vertical, and 374 s horizontal. Vertical time constants in the control group were larger than 50 s, except for the 91-year-old control subject, who had 15.7 s. The low time constant for the torsional integrator in both patients and healthy subjects confirms other results demonstrating that the torsional velocity-to-position integrator is also leaky in healthy subjects (Seidman et al. 1994).

View larger version (20K): [in this window] [in a new window]   Fig. 6. Dependence of vertical velocity on vertical position (1/s) plotted against dependence of horizontal velocity on horizontal position (1/s) for all patients. Parameters are estimated from complete multiple linear regression. The data of the patient shown in Figs. 2-5 are marked by the letter P; 99% confidence ranges for healthy subjects are indicated by shaded areas. These dependencies can be interpreted as inverse time constants of integrator leakage. The oblique line has slope 1 and thus indicates equal leakage of horizontal and vertical integrators. Values of horizontal-horizontal dependence, which are significantly different from 0, are indicated by an asterisk. For 6 patients, this value was not significantly different from 0; that is, they did not show damage of the horizontal integrator. This is confirmed by the fact that their values are within or close to the 99% range for control subjects. Vertical and horizontal bars indicate ±SD. The 6 patients above the diagonal show more horizontal than vertical integrator failure. The patient falling within the healthy population (top right data point) showed strong nystagmus, in which the vertical and horizontal components did not depend on eye position, i.e., this patient had no integrator failure.

Significant offsets (reflecting nystagmus at primary position) were found in 17 patients (89%) for vertical, 11 patients (58%) for horizontal, and 14 patients (74%) for torsional velocity. Mean offsets for all patients were 0.36 ± 0.63°/s torsional (range: 0.63-1.73°/s), 2.9 ± 2.7°/s vertical (range: 9.54-1.56°/s), and 0.06 ± 1.4°/s horizontal (range: 3.26-3.62°/s). Thus some patients not only had vertical but also considerable horizontal nystagmus at primary position. For healthy subjects, mean offsets were 0.05 ± 0.28°/s torsional (range: 0.72-0.54°/s), 0.13 ± 0.17°/s vertical (range: 0.11-0.46°/s), and 0.07 ± 0.09°/s horizontal (range: 0.25-0.15°/s).

Different groups of patients

The patients could be grouped according to the kind of integrator failure and offset (see Fig. 7). Basically, three groups were distinguished with respect to integrator failure (see also Fig. 6): patients with vertical and horizontal integrator failure (13, see numbers at connecting lines in Fig. 7, 13 = 1 +3 + 9), patients with only vertical integrator failure (5 = 1 +3 + 1), and one patient with normal horizontal and vertical integrator function.

View larger version (13K): [in this window] [in a new window]   Fig. 7. Grouping of patients according to significant integrator failure and offset (grouping based on best-subset multiple regression, see Table 1). All patients showed a vertical integrator failure or a vertical offset or both. The patient without vertical integrator failure also had a small but significant horizontal offset. Six patients had no horizontal integrator failure, but 2 of them still showed a significant horizontal offset.

VERTICAL NYSTAGMUS. All patients except one showed a significant dependence of vertical velocity on vertical position corresponding to vertical integrator failure (all patients: mean time constant, 8.9 s; range, 4.0-119 s). Two patients did not show a significant vertical velocity offset, i.e., they had no vertical nystagmus in primary position.

HORIZONTAL NYSTAGMUS. Thirteen patients (68%) showed horizontal velocity dependence on horizontal eye position (see Table 1 and Fig. 7) corresponding to horizontal integrator failure (these patients: mean time constant, 22.8 s; range, 3.9-77.0 s). With respect to downbeat velocity in straight ahead position, these patients were not significantly different from the remaining patients (t-test, P = 0.25). A horizontal offset was found in 11 patients (58%). Nine patients had both a significant horizontal offset and a significant horizontal integrator failure.

View larger version (21K): [in this window] [in a new window]   Fig. 8. Examples of different types of DN for 4 patients (A-D). The eyeball is shown for each of the 9 required gaze positions, including the torsional position indicated by the cross through the pupil. The arrows indicate the rotation axes of the eye (nystagmus slow phase) at each position. The length of the arrow is proportional to slow phase angular velocity, the direction of the arrow conforms to the right-hand-rule (e.g., arrows pointing rightward indicate vertical upward rotation of the eye). The radius of the eyeball corresponds to an angular velocity of 1.5°/s, thus smaller angular velocities are not visible. The angular velocity vectors are computed from multiple linear regression results of each patient. A: data for patient V (shown in Figs. 2-5). In gaze straight ahead (middle position), downbeat nystagmus is predominantly visible indicated by a horizontal rotation axis. With lateral gaze, the length and orientation of the axis change, indicating a horizontal gaze-evoked component and a dependence of the vertical component on horizontal gaze direction. With gaze up, nystagmus becomes smaller, reaching almost its null position with gaze up and right (top left). This patient also shows torsional components (not clearly visible in the figure) that violate Listing's law. B: data for patient A with vertical and horizontal integrator failure and offset. The increase of the vertical component with downward gaze is clearly visible. During lateral gaze, this patient develops a gaze-dependent horizontal component, indicated by the changing orientation of the rotation axis. In addition the vertical component changes size. Note that in this patient, nystagmus slow phases do not violate Listing's law, i.e., the torsional velocity is not significantly different from 0. C: data for patient S2 without any significant horizontal nystagmus component but a strong vertical integrator failure. During upward left gaze (right upper position), nystagmus reverts to upbeat, indicated by a small arrow pointing rightward. D: data for patient W, who showed no integrator failure. Neither vertical nor horizontal drift components depend on eye position. However, a small torsional component develops with lateral gaze indicating that the axis of slow phase eye rotation has rotated with the eye.

Violations of Listing's law

As mentioned in the preceding text, the simple regression was sufficient to describe the data for four patients. One of them showed a significant torsional offset which constitutes a violation of Listing's law in the strict sense. In the remaining three patients, slow phase velocity did not violate Listing's law.

However, not only patients but also healthy subjects exhibited violations of Listing's law as documented by significant nonprincipal components of the multiple regression or by statistically significant torsional offsets. We used these small violations of healthy subjects to decide whether a patient fell into the normal range of minor violations even though his slow phase velocity violated Listing's law in the strict sense. The following three parameters, derived by multiple regression after rotation to Listing's plane, indicated a violation of Listing's law: a torsional velocity offset (healthy: 0.06 ± 0.28°/s), a dependence of torsional velocity on vertical position (healthy: 0.002 ± 0.010 1/s), and a dependence of torsional velocity on horizontal position (healthy: 0.003 ± 0.007 1/s). This means that, for example, a horizontal gaze position of 18° with respect to primary position induced in healthy subjects, on average, a torsional drift velocity of 0.05°/s (18° × 0.003 1/s = 0.05°/s).

Thus for the 15 patients requiring multiple regression fits, we determined whether one or more of the three values indicating violation of Listing's law was outside of the normal range (as determined by a 99% confidence interval). This was the case for 13 patients. For 12 of them, the dependence of torsional velocity on horizontal position violated Listing's law significantly more than in healthy subjects, 6 patients had a significant torsional offset, and for 4 patients, torsional velocity depended significantly more on vertical position than in healthy subjects. This result shows that the main reason for violations of Listing's law was a dependence of torsional slow phase velocity on horizontal eye position.

When the results of comparing simple regression to multiple regression (3 patients) and specific violations of Listing's law in control subjects and patients (2 patients) were combined, five patients (26%) had slow phases that did not violate Listing's law. Note that these patients were not significantly different from the remaining 14 patients with respect to their vertical drift velocity on gaze straight ahead (t-test, P = 0.95; range: 6.03 to 1.25°/s). All five patients in whom slow phase velocity obeyed Listing's law had both vertical and horizontal integrator failure.

Possible reasons for velocity offsets

Several factors can cause vertical drift of the eyes. Nearly all patients show signs of a vertical integrator failure. However, most patients also show velocity offsets which are not expected from integrator deficits. Theoretically velocity offsets could 1) be present at the integrator level (head-fixed component velocity offset) with horizontal and vertical components resulting, e.g., from smooth-pursuit imbalance, could 2) be attributed to constant angular velocity of the eye resulting from vestibular imbalance (angular velocity offsets remaining essentially head-fixed) or could 3) be explained by constant eye-fixed velocity, i.e., a constant angular or component velocity rotating with the eye. Intermediate values are, of course, possible. Hence, while varying with eye position in one coordinate system (when expressed as head-fixed component velocity), the slow phase drifts would remain constant in another (e.g., when expressed as angular velocity in an eye-fixed coordinate system). For constant vestibular or head-fixed angular velocity, the fitted off-vertical elements of the matrix T in Eq. 1 should assume a very specific dependence on the fitted offsets.

This dependence should become clearest for the change in large torsional-vertical slow phase drifts with horizontal eye position. This change can be computed from the offset in primary position and the nonprincipal elements in the matrix T. For example, a constant eye-fixed component velocity should rotate with the eye. Its angle thus co-varies with the change in horizontal gaze. In contrast, its length stays constant over different eye positions. A constant head-fixed angular velocity translates to a component velocity that rotates halfway against the eye but does not change length. Thus by examining the change in angular position of the torsional-vertical slow phase drift with horizontal gaze, one can determine whether the drift shows an eye-fixed, head-fixed, or other rotation. Its change in length shows whether the found rotation is a true rotation in the sense that rotating a vector does not change its length. Figure 9A shows the torsional-vertical drift of patient P. In this patient, it rotated with the eye but changed length. As a theoretical example, Fig. 9B shows a purely eye-fixed constant torsional-vertical component velocity. Figure 9C shows the data of all patients by comparing changes in angle and changes in relative length for horizontal gaze positions at ±20° laterally from gaze straight ahead position. It can be seen that the torsional-vertical drift changed considerably with horizontal eye position in length and angle in most patients. Note that a torsional-vertical drift that does not change angle with horizontal position may still violate Listing's law due to a constant torsional component. Most importantly, except for one patient, none of the drifts showed the behavior expected from that caused by vestibular imbalance. Therefore we concluded that the assumption of nonprincipal dependencies caused by angular velocity offsets of vestibular, head-fixed, or eye-fixed origin has to be rejected.

View larger version (22K): [in this window] [in a new window]   Fig. 9. A: eyeball of patient V seen from above in 3 horizontal gaze positions. Thick black arrows: torsional-vertical eye velocity; dashed arrow: line of sight; gray arrows: velocity components in head coordinates. The patient shows a negative vertical velocity offset and a positive torsional offset in primary position (middle). With gaze 20° left, the torsional-vertical slow phase velocity rotates approximately with the eye and increases; with 20° left, the torsional component vanishes while the vertical component shrinks. B: theoretical effect of eye rotation on a constant eye-fixed velocity. Such eye velocity does not change length with varying horizontal gaze, and its angle co-varies with horizontal gaze angle. Comparison with A shows that for the patient, rotation of the torsional-vertical component was smaller than expected for a constant eye-fixed drift and that the patient's torsional vertical velocity changed length. C: change in relative length of the torsional-vertical component velocity plotted against change in angle for horizontal positions ±20° from primary position. Values computed from multiple regression results. Filled circles show patient data; the data of the patient shown in A are marked with the letter P. The horizontal line shows where torsional-vertical drifts that do not change in size with horizontal position are expected. These drifts stay stable in the head or rotate with the eye. The vertical line gives torsional-vertical drifts that do not change their orientation but may change their length. Such drifts fulfill Listing's law, if their torsional component is 0. The open circles show drifts which rotate with the eye, or stay stable in the head. The ellipse shows the expected values for constant torsional-vertical angular velocity caused by vestibular imbalance that should adopt a half-Listing's law strategy (quarter-angle rule) and thus change in angle and size. As can be seen, most drifts change their angle considerably with horizontal eye position, ranging from what would be expected from VOR to eye-fixed responses. However, at the same time, most drifts change considerably in size. Finally, except for 1 patient, drifts do not come close to the expectation from a vestibular angular velocity imbalance.

3-D dependence of slow phase velocity on eye position

Rather than looking at either the principal dependencies or the nonprincipal dependencies alone, one can analyze the general form of the fitted matrix T (Eq. 1), which results from the multiple regression analysis. Theoretically, if three orthogonal leaky integrators operating in the coordinate system determined by Listing's plane caused the observed slow phases, the matrix T should be diagonal, i.e., contain only principal dependencies. As reported in the preceding text, this was the case for only four patients. Such a diagonal matrix has column vectors (each column of the matrix can be understood as a 3-D vector), which are perpendicular to each other, thus each vector points along the respective axis of the coordinate system.

The question for the remaining patients, for whom nonprincipal dependencies have been found, is whether the matrix is still composed of vectors perpendicular to each other. If this was the case, it would mean that the three integrators are independent of each other but that their coordinate system is rotated with respect to Listing's plane. If it is not the case, then the three integrators do not operate independently of each other; in other words, the integrator coordinate system would be distorted and not perpendicular. We therefore plotted the vertical and horizontal column vectors of the fitted matrices, i.e., the vectors describing the effect of a vertical or horizontal eye position on drift velocity (Fig. 10). The torsional vector showed too much scatter to be useful because the effect of torsional position on slow phase velocity could be determined less reliably due to the small range of adopted torsional eye positions. For the sample patient (shown in Figs. 3-5 and 10, top row), it can be seen (data in Eq. 5) that vertical position significantly contributed only to vertical velocity (significant value in the 2nd column of the matrix). This corresponds to the alignment of the vector with the vertical y axis. In contrast, torsional, vertical, and horizontal velocity significantly depend on horizontal position (3rd column); i.e., the corresponding vector is not aligned with the horizontal z-axis but is tilted in the vertical-torsional direction.

View larger version (28K): [in this window] [in a new window]   Fig. 10. Axes of integrator coordinate systems derived from best-subset multiple regression. The 3-D plots (left: torsional x axis points out of the paper, right: view from an oblique direction, axes shown in gray) show the horizontal (light gray) and vertical (black) components of the integrator feedback matrix. Top: data for sample patient shown in Figs. 3-5. The large size of both vectors indicates that nystagmus drift is severely influenced by eye position. In other words, this patient shows integrator failure for both vertical and horizontal eye movements. The vertical vector (black) is almost aligned with the vertical y axis, indicating that changes in vertical eye position only affect vertical drift velocity. The horizontal vector (light gray) is tilted in the vertical-torsional direction, thus changes in horizontal eye position affect not only horizontal, but also vertical and torsional slow phase eye velocity. Bottom: data for all patients. The vertical vectors (pointing rightward) almost align with the vertical y axis while most horizontal vectors (pointing downward) tilt in the vertical direction (best seen in left view) and/or the torsional direction (x axis, best seen in right view). The tilt in the torsional direction is equivalent to a violation of Listing's law in that horizontal eye position causes torsional nystagmus. Note that small vectors correspond to patients with small eye position dependence of slow phase velocity. These patients may, nevertheless, have severe nystagmus due to an eye-position independent component.

As can be seen from Fig. 10, most vectors for the vertical component (y axis, black arrows) aligned well with the vertical y axis. Thus vertical eye position had only a minor effect on torsional or horizontal drifts. The vectors for the horizontal axis show much larger tilts away from the horizontal z axis, both in the vertical and torsional directions. The tilts in the torsional directions indicate violation of Listing's law. In conclusion, different vertical eye position mainly caused changes in vertical slow phase drift (corresponding to the alignment of the vertical vector with the y axis), while different horizontal eye positions caused changes of drift in all three components of slow phase velocity. One possible explanation for this is that, apart from the horizontal integrator failure, projections from the horizontal integrator to the torsional-vertical velocity-to-position integrator are damaged or incorrectly weighted and thus produce eye-position dependent drifts. Conversely, projections from the vertical-torsional integrator to the horizontal integrator seem to be less affected because vertical eye position has almost no effect on horizontal eye velocity.

Increase of vertical drift with eccentric gaze

An increase of vertical drift velocity with lateral gaze is often reported clinically (Leigh and Zee 1999; Straumann et al. 2000). The multiple linear regression analysis can, so far, not explain this due to its linear model. We therefore applied a quadratic regression to the data. For nine patients, but none of the healthy subjects, the quadratic regression yielded significantly better results. All nine patients showed a significant negative quadratic dependence of vertical drift velocity on horizontal position (for an example see Fig. 3). In eight of these patients, vertical drift velocity with both right- and leftward lateral gaze was larger than with gaze straight ahead. We assume that these side-dependent changes in vertical drift velocity are caused by asymmetric damage to the torsional-vertical integrator (see DISCUSSION).

Pre- and postsaccadic eye positions

Inspection of the slow phase eye velocity data revealed that torsional eye velocity often had increased values at the beginning of an attempted fixation period and decreased to zero or to a constant velocity offset over time (see Fig. 4 for an example). This suggests that torsional eye position shortly after a saccade to a new target position is different from that at the same gaze position before the next saccade started. Such transient torsional deviations, called blips, have been found in healthy subjects (Straumann et al. 1995), but also with much larger amplitude in a patient (Helmchen et al. 1997). Therefore the orientation of Listing's plane was examined before and after large saccades. Depending on the amount of nystagmus, 10-26 saccades larger than 10° were found. For each saccade, eye positions 100 ms immediately before and after the saccade were selected. In patients, the angle between primary positions corresponding to the planes fitted to the eye positions before and after saccades was 6.2 ± 3.7° (see Fig. 11 for an example), in healthy subjects, 3.5 ± 1.6°. The difference was statistically significant (t-test, P = 0.007). In patients, the angle between pre- and postsaccadic data correlated with the amount of DN during gaze straight ahead (r = 0.50, P = 0.03), i.e., patients with larger slow phase velocities showed a larger angle between pre- and postsaccadic Listing's planes.

View larger version (17K): [in this window] [in a new window]   Fig. 11. Eye positions of pre- () and postsaccadic () data in a patient shown in Listing's coordinates (left: side view; right: view from above). The lines represent planes fitted to pre- or postsaccadic data ( or - - -), i.e., how Listing's plane would be oriented if only pre- or postsaccadic data were analyzed. Note that the planes differ considerably when viewed from the side (left), the 2 corresponding primary positions are 13.3° apart. The torsional deviations between both planes are transient and correspond to blips.

If the same analysis was performed for all saccades, taking also small saccades and nystagmus beats into account, the angle was no longer significantly different between patients and healthy subjects. This is possibly due to the fact that saccadic corrections for torsional deviations from Listing's plane depend on the amplitude of the saccade (Lee et al. 2000): while large saccades bring the eye close to the torsional position defined by the burst generator, small nystagmus beats cannot achieve the necessary torsional correction and, therefore the torsional position of the eye remains close to the torsional null position defined by the neural integrator.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Multiple linear regression sufficiently described 3-D slow phase component velocity of eye position data in patients with DN. This method considerably simplified the quantitative analysis and statistical evaluation of the data and allowed a physiologically meaningful interpretation of the fitted parameters.

The analysis of slow phases in patients with DN revealed two important results. First, while a vertical integrator failure could be found for all except one patient, only about half of the patients also showed a horizontal integrator failure. This result, which agreed with previous findings (Leigh and Zee 1999), suggests different etiologies of DN. Second, about one-fourth of the patients showed slow phases that essentially obeyed Listing's law, i.e., in the range of observed eye positions, the slow phase component velocity of nystagmus did not show significantly more torsion than in healthy subjects. These patients did not show torsional offsets, and the (nonprincipal) dependence of torsional velocity on horizontal or vertical eye position was negligibly small. This result casts doubt on one of the major hypotheses of DN, i.e., that DN is caused by a disruption of central vestibular pathways. In such a case, the resulting vestibular imbalance would cause a nystagmus offset that would violate Listing's law, as does any vestibular nystagmus. Furthermore, a closer inspection of the torsional-vertical offsets in the remaining patients further corroborated our finding that these offsets do not agree with the hypothesis of a vestibular or head-fixed angular velocity offset.

Previous studies

While the literature provides a wealth of studies on patients with DN (e.g., Baloh and Spooner 1981; Baloh and Yee 1989; Gresty et al. 1986; Yee 1989; Zee et al. 1974), only one study employed a detailed analysis of 3-D eye movement recordings in patients with DN. This recent investigation of patients with cerebellar atrophy (Straumann et al. 2000) reported that these patients show upward drift (DN), horizontal centripetal drift during lateral gaze (gaze-evoked nystagmus), and torsional drift depending on horizontal position which violated Listing's law. The vertical nystagmus component was composed of a gaze-evoked component and a velocity bias. Vertical nystagmus increased with lateral gaze. Although our study confirmed these results for about 50% of our 19 patients (see significant parameters in Eq. 5), we found that two of our DN patients did not show a bias component (offset) if slow phase velocity was analyzed with respect to Listing's plane (corresponding to a vertical integrator failure without offset), six patients did not show gaze-evoked horizontal slow phase components (no horizontal integrator failure), slow phases in five patients did not violate Listing's law, and only nine of our patients showed significantly increased vertical nystagmus with right and left lateral gaze.

Different etiologies

Most hypotheses on DN assume that there is one type of DN. Our results show, however, that three distinct groups of patients can be distinguished: DN patients with vertical-horizontal integrator failure, with vertical integrator failure only, and without integrator failure (1 patient). A plausible explanation for the first two groups is that, physiologically, the horizontal and vertical-torsional integrators are at separate locations in the brain stem. The first is located in the nucleus prepositus hypoglossi and the medial vestibular nucleus (Cannon and Robinson 1987), while the other is located in the interstitial nucleus of Cajal (Crawford 1994). Even though these structures may not be damaged themselves, it is reasonable to assume that cerebellar projections to the two integrators, which are assumed to improve the integrator time constants (Zee et al. 1980, 1981), are affected differently by cerebellar lesions. The third distinct type, DN without integrator failure, appeared to be the exception.

Main hypotheses about DN

Two hypotheses about DN focus on the vertical bias component of slow phase velocity, i.e., the vertical slow phase velocity that cannot be explained by the vertical integrator failure found in all except one of our patients.

The first hypothesis, vestibular imbalance due to central damage of semicircular canal pathways (Baloh and Spooner 1981; Böhmer and Straumann 1998) can be excluded in our patients because the nystagmus slow phase velocity did not agree with vestibular origin (Fig. 9). Furthermore, this hypothesis requires that two distinct functions of the brain stem ocular-motor circuitry are affected: the vertical integrator that causes the gaze-dependent vertical nystagmus and the cerebellar structures that project to the vestibular nuclei to suppress the vertical offset (Böhmer and Straumann 1998).

The alternative hypothesis of smooth pursuit imbalance (Zee et al. 1974) poses a similar problem: it requires that both pursuit pathways and the vertical integrator are damaged. This is still more appealing because lesions of floccular regions are known to cause both gaze-evoked nystagmus and smooth-pursuit deficits (Waespe 1992; Zee et al. 1981). Another study showing that horizontal pursuit deficits and horizontal gaze-evoked nystagmus usually occur in combination further supports this hypothesis (Büttner and Grundei 1995). However, the pursuit theory only explains why DN patients usually have pursuit deficits and why they cannot suppress their nystagmus in light. It does not explain the various findings associated with DN slow phase velocity described in the preceding text, specifically it does not explain why slow phases do not obey Listing's law in the majority of our patients.

The most recent hypothesis by Straumann et al. (2000) specifically addresses the violations of Listing's law, suggesting that the impaired cerebellum can no longer hold torsional eye positions, which then drift back to a mechanically determined resting position. Thus Straumann et al. propose a mismatch between two different Listing's planes, a neural plane and a mechanical plane. This proposal does not explain why torsional drifts, if present at all, show time constants larger (mean 4.5 s) than that of the eye plant (about 200 ms) (Seidman et al. 1994). It also requires separate explanations for the gaze-dependent drifts (leaky integration) and for the vertical velocity bias.

Integrator function

Because the most common finding in our patients, except for DN in gaze straight ahead, was gaze-evoked vertical nystagmus, we first consider velocity-to-position integrator function.

As Cannon and Robinson (1987) showed, loss of the neural integrator in the brain stem leads to gaze-evoked nystagmus. However, this nystagmus does not necessarily drift toward the zero position of the integrator but may have its null position at different points if combined with an offset. Such an offset, caused, for example, by a difference in push-pull firing rate, may be the result of a vestibular imbalance, but it may also have its origin in any other imbalance, for example, a smooth-pursuit imbalance. Pharmacological lesions of the horizontal oculomotor integrator in monkeys induced different types of nystagmus (Arnold et al. 1999): slow phases showed exponential eye position dependent drifts with null positions not coinciding with gaze straight ahead, but also linear eye position-independent drifts. These results suggest that integrator lesions, and not only lesions of integrator input structures, may cause shifts of the null position of nystagmus: in other words, the zero position of the integrator may be altered intrinsically. To summarize, in the simple one-dimensional case integrator function can be described by a gain (not important for the current analysis), a time constant, and an offset that shifts t