Thomas Haslwanter

Mathematics of three-dimensional eye rotations

The recording of three-dimensional eye position has become the accepted standard in oculomotor research. In this paper we review the mathematics underlying the representation of three-dimensional eye movements. Rotation matrices, rotation vectors and quaternions are presented, and their relations described. The connection between search coils and rotation matrices is explained, as well as the connection between eye position and eye velocity. While examples of applications of the formulas to vestibulo-ocular research are given, the methods and mathematical analyses are also useful for studying other motor systems.

A geometric basis for measurement of three-dimensional eye position using image processing

Polar cross correlation is commonly used for determination of ocular torsion from video images, but breaks down at eccentric positions if the spherical geometry of the eye is not considered. We have extended this method to allow three-dimensional eye position measurement over a range of +/- 20 deg by determining the correct projection of the eye onto the image plane of the camera. We also determine the orientation of the camera with respect to the eye, allowing eye position to be represented in appropriate head-fixed coordinates. These algorithms have been validated using both in vitro and in vivo measures of eye position.

Three-dimensional eye position during static roll and pitch in humans

We investigated how three-dimensional (3D) eye position is influenced by static head position relative to gravity, a reflex probably mediated by the otolith organs. In monkeys, the torsional component of eye position is modulated by gravity, but little data is available in humans. Subjects were held in different head/body tilts in roll and pitch for 35 s while we measured 3D eye position with scleral coils, and we used methods that reduced torsion artifacts produced by the eyelids pressing on the contact lens and exit wire. 3D eye positions were described by planar fits to the data (Listings plane), and changes in these planes showed how torsion varied with head position. Similar to findings in monkeys, the eyes counterrolled during roll tilts independent of horizontal and vertical eye position, reaching a maximum torsion of 4.9 degrees. Counterroll was not proportional to the shear force on the macula of the utricles: gain (torsion/sine of the head roll angle) decreased by 50% from near upright to ear down. During pitch forward, torsion increased when subjects looked right, and decreased when they looked left. However, the maximum change of torsion was only -0.06 degrees per degree of horizontal eye position, which is less than reported in monkey. Also in contrast to monkey, we found little change in torsion when subjects were pitched backwards.

Dissociation between the perception of body verticality and the visual vertical in acute peripheral vestibular disorder in humans

Estimates of the subjective visual and postural vertical were obtained from five patients with acute peripheral vestibular lesions and 20 normal subjects. The visual vertical was assessed by asking the subjects to align a target line to earth vertical by means of remote control. Postural vertical judgments were obtained by exposing them to rotational displacements in the roll plane while sitting on a motor-driven chair and requiring them to align their body to vertical using a joystick control. While the patients showed strong deviations of the visual vertical towards the lesion side, their postural vertical judgments remained veridical. We conclude that the above perceptions are not processed identically and that the participating sensory systems are differently weighted during these tasks.

Changes in the three-dimensional angular vestibulo-ocular reflex following intratympanic gentamicin for Ménière's disease

The 3-dimensional angular vestibulo-ocular reflexes (AVOR) elicited by rapid rotary head thrusts were studied in 17 subjects with unilateral Ménières disease before and 2–10 weeks after treatment with intratympanic gentamicin and in 13 subjects after surgical unilateral vestibular destruction (SUVD). Each head thrust was in the horizontal plane or in either diagonal plane of the vertical semicircular canals, so that each head thrust effectively stimulated only one pair of canals. The AVOR gains (eye velocity/head velocity during the 30 ms before peak head velocity) for the head thrusts exciting each individual canal were averaged and taken as a measure of the function of that canal. Prior to intratympanic gentamicin, gains for head thrusts that excited canals on the affected side were 0.91 ± 0.20 (horizontal canal, HC), 0.78 ± 0.20 (anterior canal, AC), and 0.83 ± 0.10 (posterior canal, PC). The asymmetries between these gain values and those for head thrusts that excited the contralateral canals were <2%. In contrast, caloric asymmetries averaged 40% ± 32%. Intratympanic gentamicin resulted in decreased gains attributable to each canal on the treated side: 0.40 ± 0.12 (HC), 0.35 ± 0.14 (AC), 0.31 ± 0.14 (PC) (p <0.01). However, the gains attributable to contralateral canals dropped only slightly, resulting in marked asymmetries between gains for excitation of ipsilateral canals versus their contralateral mates: HC: 34% ± 12%, AC: 24% ± 25%, and PC: 42% ± 13%. There was no difference in the AVOR gain for excitation of the ipsilateral HC after gentamicin in patients who received a single intratympanic injection (0.39 ± 0.11, n = 12) in comparison to those who received 2–3 injections (0.42 ± 0.15, n = 5, p = 0.7). Gain decreases attributed to the gentamicin-treated HC and AC were not as severe as those observed after SUVD. This finding suggests that intratympanic gentamicin causes a partial vestibular lesion that may involve preservation of spontaneous discharge and/or rotational sensitivity of afferents.

A theoretical analysis of three-dimensional eye position measurement using polar cross-correlation

Polar cross-correlation is a commonly used technique for determination of torsional eye position from video images. At eccentric eye positions, the projection of the sampling window onto the image plane of the camera is translated and deformed due to the spherical shape of the eyeball. Here, the authors extend the polar cross-correlation technique by developing the formulas required to determine the correct location and shape of the sampling window at all eye positions. These formulas also allow the representation of three-dimensional eye position in Fick-angles, which are commonly used in oculomotor research. A numerical simulation shows the size of the errors in ocular torsion if the spherical geometry of the eye is not considered. Other effects which can affect the accuracy of video-based eye position measurements are also discussed.<<ETX>>

Hyperventilation-induced nystagmus in patients with vestibular schwannoma

Objective: To analyze the nystagmus evoked by hyperventilation in patients with unilateral vestibular schwannoma and to use this information to predict the effects of hyperventilation on individual ampullary nerves. Methods: Three-dimensional scleral search coil eye movement recording techniques were used to record the magnitude and time course of eye movements in six patients with unilateral vestibular schwannoma and hyperventilation-induced nystagmus. The presenting complaints in five of these patients were vertigo or dysequilibrium. Results: The eye movement response to hyperventilation was a “recovery” nystagmus with slow-phase components corresponding to excitation of the affected vestibular nerve. Projection of the eye velocity vector into the plane of the semicircular canals revealed that fibers arising from the ampulla of the horizontal canal were most affected by hyperventilation with lesser activation of fibers to the superior canal and smaller, more variable responses from posterior canal fibers. Conclusions: The three-dimensional characteristics of the nystagmus evoked by hyperventilation in patients with vestibular schwannoma provide insight into the vestibular end organs affected by the tumor and the mechanism responsible for the nystagmus. This finding indicates that hyperventilation resulted in a transient increase in activity from these partially demyelinated axons.

Non-commutativity in the brain

In non-commutative algebra, order makes a difference to multiplication, so that a × b ≠ b × a (refs 1, 2). This feature is necessary for computing rotary motion, because order makes a difference to the combined effect of two rotations. It has therefore been proposed that there are non-commutative operators in the brain circuits that deal with rotations, including motor circuits that steer the eyes, head and limbs,,, and sensory circuits that handle spatial information,. This idea is controversial,,: studies of eye and head control have revealed behaviours that are consistent with non-commutativity in the brain,, but none that clearly rules out all commutative models. Here we demonstrate non-commutative computation in the vestibulo-ocular reflex. We show that subjects rotated in darkness can hold their gaze points stable in space, correctly computing different final eye-position commands when put through the same two rotations in different orders, in a way that is unattainable by any commutative system.

Contribution of the Vertical Semicircular Canals to the Caloric Nystagmus

Modulation of the caloric nystagmus in response to repositioning the plane of one vertical semicircular canal from gravitational horizontal to vertical during continuous caloric stimulation was used to measure the vertical canals contribution to the nystagmus. The rationale was to examine the thermovective response from one vertical canal at a time, after a temperature gradient had been established across its two limbs. The nystagmus was measured and analysed in three dimensions using orthogonal head-referenced coordinates. The magnitude of each semicircular canals contribution to the overall caloric response, the canal vector, was determined in non-orthogonal, contravariant semicircular canal plane coordinates. By using the canal plane reorientation technique and contravariant canal plane coordinates, we were able to measure the proportional thermovective response magnitude generated by each vertical canal during caloric stimulation. We found that the anterior canal contributed about one-third and the posterior canal about one-tenth as much as the lateral canal did to the overall caloric response magnitude when it was reoriented from horizontal to vertical. Comparison of the eye rotation axis before and after each vertical canal plane reorientation, with the geometry of the stimulated semicircular canals, also showed directional modulation of the caloric nystagmus by the vertical canal response. When one vertical canal plane was horizontal during caloric stimulation, the eye rotation axis aligned with the resultant of the other vertical canal and the lateral canal response axes. After vertical canal plane reorientation, the eye rotation axis realigned towards the resultant of the maximally stimulated vertical canal and the lateral canal, by 55.2+/-33.9 degrees (mean+/-SD) after anterior canal plane reorientation and by 32.3+/-21.2 degrees after posterior canal reorientation.

Listing's plane rotation with convergence : role of disparity, accommodation, and depth perception

Abstract Earlier studies have reported temporal rotation of Listing’s plane with convergence of the eyes causing torsion, which is dependent on eye elevation. The amount by which the planes rotate differs from study to study. To gain insight into the functional significance of the temporal tilt of Listing’s plane for vision, we examined whether the rotation of the plane depends on the visual conditions, namely on the stimuli driving vergence. In different conditions, accommodative vergence, disparity-vergence, combinations of disparity with accommodation or depth perception were used and the resulting rotation of Listing’s plane was measured. Our findings show, for the first time, that the relationship between convergence and Listing’s-plane temporal rotation depends on the stimuli driving vergence. When the stimulus contains only disparity cues, vergence and Listing’s plane rotate immediately and consistently among subjects. Accommodative vergence, the mutual couplings between vergence and accommodation, can influence the orientation of Listing’s plane, but they do so in a idiosyncratic way. The largest rotation was elicited by stereograms combining disparity-vergence with depth perception. These findings support the idea of a functional role of Listing’s plane rotation for binocular vision, perhaps for depth perception.