Publisher Correction: On the role of ocular torsion in binocular visual matching.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

On the retinal correspondences across the binocular visual field.

We have recently reported that objects seen at near distances require adjustments of the relative torsion of the eyes to avoid blurred binocular images or double vision and ultimately to allow binocular fusion. The reason underlying these rotational adjustments is that converging eye movements undo the eyes' torsional alignment, generating disparate binocular images of objects outside the horizontal plane of regard. We show mathematically that it is the distance between the two eyes, their relative orientation in the frontal plane and the distances from each eye to the binocularly intended visual target, that determine the binocular alignment of the lines of sight. As an example, we analyze the binocular disparity field that is generated when a viewer examines objects on a planar surface whose viewing distances differ in each gaze direction. The underlying geometric computations are simple, and require no explicit knowledge of 3D eye movement kinematics.

On the role of ocular torsion in binocular visual matching.

When an observer scans the visual surround, the images cast on the two retinae are slightly different due to the different viewpoints of the two eyes. Objects in the horizontal plane of regard can be seen single by aligning the lines of sight without changing the torsional stance of the eyes. Due to the peculiar ocular kinematics this is not possible for objects above or below the horizontal plane of regard. We provide evidence that binocular fusion can be achieved independently of viewing direction by adjusting the mutual torsional orientation of the eyes in the frontal plane. We characterize the fusion positions of the eyes across the oculomotor range by deriving simple trigonometric equations for the required torsion as a function of gaze direction and compute the iso-torsion contours yielding binocular fusion. Finally, we provide experimental evidence that eye positions in far-to-near re-fixation saccades indeed converge towards the predicted positions by adjusting the torsion of the eyes. This is the first report that describes the three-dimensional orientation of the eyes at binocular fusion positions based on the three-dimensional ocular kinematics. It closes a gap between the sensory and the motor side of binocular vision and stereoscopy.

Three-dimensional ocular kinematics underlying binocular single vision.

We have analyzed the binocular coordination of the eyes during far-to-near refixation saccades based on the evaluation of distance ratios and angular directions of the projected target images relative to the eyes' rotation centers. By defining the geometric point of binocular single vision, called Helmholtz point, we found that disparities during fixations of targets at near distances were limited in the subject's three-dimensional visual field to the vertical and forward directions. These disparities collapsed to simple vertical disparities in the projective binocular image plane. Subjects were able to perfectly fuse the vertically disparate target images with respect to the projected Helmholtz point of single binocular vision, independent of the particular location relative to the horizontal plane of regard. Target image fusion was achieved by binocular torsion combined with corrective modulations of the differential half-vergence angles of the eyes in the horizontal plane. Our findings support the notion that oculomotor control combines vergence in the horizontal plane of regard with active torsion in the frontal plane to achieve fusion of the dichoptic binocular target images.

Transmitter inputs to different motoneuron subgroups in the oculomotor and trochlear nucleus in monkey.

In all vertebrates the eyes are moved by six pairs of extraocular muscles enabling horizontal, vertical and rotatory movements. Recent work showed that each extraocular muscle is controlled by two motoneuronal groups: (1) Motoneurons of singly-innervated muscle fibers (SIF) that lie within the boundaries of motonuclei mediating a fast muscle contraction; and (2) motoneurons of multiply-innervated muscle fibers (MIF) in the periphery of motonuclei mediating a tonic muscle contraction. Currently only limited data about the transmitter inputs to the SIF and MIF motoneurons are available. Here we performed a quantitative study on the transmitter inputs to SIF and MIF motoneurons of individual muscles in the oculomotor and trochlear nucleus in monkey. Pre-labeled motoneurons were immunostained for GABA, glutamate decarboxylase, GABA-A receptor, glycine transporter 2, glycine receptor 1, and vesicular glutamate transporters 1 and 2. The main findings were: (1) the inhibitory control of SIF motoneurons for horizontal and vertical eye movements differs. Unlike in previous primate studies a considerable GABAergic input was found to all SIF motoneuronal groups, whereas a glycinergic input was confined to motoneurons of the medial rectus (MR) muscle mediating horizontal eye movements and to those of the levator palpebrae (LP) muscle elevating the upper eyelid. Whereas SIF and MIF motoneurons of individual eye muscles do not differ numerically in their GABAergic, glycinergic and vGlut2 input, vGlut1 containing terminals densely covered the supraoculomotor area (SOA) targeting MR MIF motoneurons. It is reasonable to assume that the vGlut1 input affects the near response system in the SOA, which houses the preganglionic neurons mediating pupillary constriction and accommodation and the MR MIF motoneurones involved in vergence.

The kinematics of far-near re-fixation saccades.

We have analyzed the three-dimensional spatiotemporal characteristics of saccadic refixations between far and near targets in three behaviorally trained rhesus monkeys. The kinematics underlying these rapid eye movements can be accurately described by rotations of the eyes in four different planes, namely, first disconjugate rotations in the horizontal plane of regard converging the eyes toward the near target, followed by rotations in each eye's vertical direction plane, and finally, disconjugate rotations in a common frontoparallel plane. This compounded rotation of the eye was underlying an initially fast-rising variable torsion that typically overshot the final torsion, which the eyes attained at the time of target acquisition. The torsion consisted of a coarse, widely varying component of opposite polarity in the two eyes, which contained a more robust, much smaller modulation that sharply increased toward the end of saccades. The reorientation of the eyes in torsion depended on each eye's azimuth, elevation, and target distance. We conclude that refixation saccades are generated by motor commands that control ocular torsion in concert with the saccade generator, which operates in Donders-Listing kinematics underlying Listing's law.

Kinematics of visually-guided eye movements.

One of the hallmarks of an eye movement that follows Listing's law is the half-angle rule that says that the angular velocity of the eye tilts by half the angle of eccentricity of the line of sight relative to primary eye position. Since all visually-guided eye movements in the regime of far viewing follow Listing's law (with the head still and upright), the question about its origin is of considerable importance. Here, we provide theoretical and experimental evidence that Listing's law results from a unique motor strategy that allows minimizing ocular torsion while smoothly tracking objects of interest along any path in visual space. The strategy consists in compounding conventional ocular rotations in meridian planes, that is in horizontal, vertical and oblique directions (which are all torsion-free) with small linear displacements of the eye in the frontal plane. Such compound rotation-displacements of the eye can explain the kinematic paradox that the fixation point may rotate in one plane while the eye rotates in other planes. Its unique signature is the half-angle law in the position domain, which means that the rotation plane of the eye tilts by half-the angle of gaze eccentricity. We show that this law does not readily generalize to the velocity domain of visually-guided eye movements because the angular eye velocity is the sum of two terms, one associated with rotations in meridian planes and one associated with displacements of the eye in the frontal plane. While the first term does not depend on eye position the second term does depend on eye position. We show that compounded rotation - displacements perfectly predict the average smooth kinematics of the eye during steady- state pursuit in both the position and velocity domain.

Three-dimensional visuo-motor control of saccades.

Although the motion of the line of sight is a straightforward consequence of a particular rotation of the eye, it is much trickier to predict the rotation underlying a particular motion of the line of sight in accordance with Listing's law. Helmholtz's notion of the direction-circle together with the notion of primary and secondary reference directions in visual space provide an elegant solution to this reverse engineering problem, which the brain is faced with whenever generating a saccade. To test whether these notions indeed apply for saccades, we analyzed three-dimensional eye movements recorded in four rhesus monkeys. We found that on average saccade trajectories closely matched with the associated direction-circles. Torsional, vertical, and horizontal eye position of saccades scattered around the position predicted by the associated direction-circles with standard deviations of 0.5°, 0.3°, and 0.4°, respectively. Comparison of saccade trajectories with the likewise predicted fixed-axis rotations yielded mean coefficients of determinations (±SD) of 0.72 (±0.26) for torsion, 0.97 (±0.10) for vertical, and 0.96 (±0.11) for horizontal eye position. Reverse engineering of three-dimensional saccadic rotations based on visual information suggests that motor control of saccades, compatible with Listing's law, not only uses information on the fixation directions at saccade onset and offset but also relies on the computation of secondary reference positions that vary from saccade to saccade.

Quick phases control ocular torsion during smooth pursuit.

One of the open questions in oculomotor control of visually guided eye movements is whether it is possible to smoothly track a target along a curvilinear path across the visual field without changing the torsional stance of the eye. We show in an experimental study of three-dimensional eye movements in subhuman primates (Macaca mulatta) that although the pursuit system is able to smoothly change the orbital orientation of the eye's rotation axis, the smooth ocular motion was interrupted every few hundred milliseconds by a small quick phase with amplitude <1.5° while the animal tracked a target along a circle or ellipse. Specifically, during circular pursuit of targets moving at different angular eccentricities (5°, 10°, and 15°) relative to straight ahead at spatial frequencies of 0.067 and 0.1 Hz, the torsional amplitude of the intervening quick phases was typically around 1° or smaller and changed direction for clockwise vs. counterclockwise tracking. Reverse computations of the eye rotation based on the recorded angular eye velocity showed that the quick phases facilitate the overall control of ocular orientation in the roll plane, thereby minimizing torsional disturbances of the visual field. On the basis of a detailed kinematic analysis, we suggest that quick phases during curvilinear smooth tracking serve to minimize deviations from Donders' law, which are inevitable due to the spherical configuration space of smooth eye movements.

Processing of angular motion and gravity information through an internal model.

The vestibular organs in the base of the skull provide important information about head orientation and motion in space. Previous studies have suggested that both angular velocity information from the semicircular canals and information about head orientation and translation from the otolith organs are centrally processed in an internal model of head motion, using the principles of optimal estimation. This concept has been successfully applied to model behavioral responses to classical vestibular motion paradigms. This study measured the dynamic of the vestibuloocular reflex during postrotatory tilt, tilt during the optokinetic afternystagmus, and off-vertical axis rotation. The influence of otolith signal on the VOR was systematically varied by using a series of tilt angles. We found that the time constants of responses varied almost identically as a function of gravity in these paradigms. We show that Bayesian modeling could predict the experimental results in an accurate and consistent manner. In contrast to other approaches, the Bayesian model also provides a plausible explanation of why these vestibulooculo motor responses occur as a consequence of an internal process of optimal motion estimation.

Decoding 3D search coil signals in a non-homogeneous magnetic field.

We present a method for recording eye-head movements with the magnetic search coil technique in a small external magnetic field. Since magnetic fields are typically non-linear, except in a relative small region in the center small field frames have not been used for head-unrestrained experiments in oculomotor studies. Here we present a method for recording 3D eye movements by accounting for the magnetic non-linearities using the Biot-Savart law. We show that the recording errors can be significantly reduced by monitoring current head position and thereby taking the location of the eye in the external magnetic field into account.

Control of ocular torsion in the rotational vestibulo-ocular reflexes.

Visual stabilization of the retina during rotational head movements requires that in far vision the eyes rotate about the same axis as the head but in opposite direction with a gain close to unity (optimal strategy). To achieve this goal the vestibulo-oculomotor system must be able to independently control all three rotational degrees of freedom of the eye. Studies of the human rotational vestibulo-ocular reflexes (VOR) have shown that its spatial characteristics are best explained by a strategy that lies halfway between the optimal image stabilization and perfect compliance with Listing's law. Here we argue that these spatial characteristics are fully compatible with an optimal strategy under the condition of a restrained gain of the torsional velocity-to-position integration. One implication of this finding is that the rotational VORs must override the default operation mode of the ocular plant that, according to recent findings, mechanically favours movements obeying Listing's law.

Geometrical considerations on canal-otolith interactions during OVAR and Bayesian modelling.

During constant-velocity rotation about a tilted axis (OVAR), the VOR and the rotation perception last indefinitely, but show a striking dependency on tilt angle. We show that, during OVAR, a variety of motions can account for the head motion relative to gravity. Some of these are in conflict with canal signals, but correspond to a lower angular velocity; we suggest that the brain performs a trade-off in order to select the best motion. We show that this theory explains the effect of tilt angle on velocity estimation during OVAR.

Dynamic effects on the subjective visual vertical after roll rotation.

We investigated in normal human subjects how semicircular canal and otolith signals interact in the estimation of the subjective visual vertical after constant velocity or constant acceleration roll tilt. In the constant velocity paradigm, subjects were rotated in darkness at +/-60 degrees/s for five complete cycles before being stopped in one of seven orientations ranging from 0 to +/-90 degrees (right/left ear down). In the constant acceleration paradigm, subjects were rotated with an acceleration of +30 or -30 degrees/s2 to the same seven end positions between -90 and +90 degrees , by way of passing once through the upside-down position. The subjective visual vertical was assessed by measuring the setting of a luminous line that appeared at different test delays after stop rotation in otherwise complete darkness. The data suggest that gravitational jerk signals generated by otolith-semicircular canal interactions and/or carried by phasic otolith signals are responsible for the observed transient bias in the estimation of the subjective visual vertical. This transient bias depended on both rotation and tilt direction after constant velocity rotations, but was almost abolished following constant acceleration rotations.

Human visuospatial updating after passive translations in three-dimensional space.

To maintain a stable representation of the visual environment as we move, the brain must update the locations of targets in space using extra-retinal signals. Humans can accurately update after intervening active whole-body translations. But can they also update for passive translations (i.e., without efference copy signals of an outgoing motor command)? We asked six head-fixed subjects to remember the location of a briefly flashed target (five possible targets were located at depths of 23, 33, 43, 63, and 150 cm in front of the cyclopean eye) as they moved 10 cm left, right, up, down, forward, or backward while fixating a head-fixed target at 53 cm. After the movement, the subjects made a saccade to the remembered location of the flash with a combination of version and vergence eye movements. We computed an updating ratio where 0 indicates no updating and 1 indicates perfect updating. For lateral and vertical whole-body motion, where updating performance is judged by the size of the version movement, the updating ratios were similar for leftward and rightward translations, averaging 0.84 +/- 0.28 (mean +/- SD) as compared with 0.51 +/- 0.33 for downward and 1.05 +/- 0.50 for upward translations. For forward/backward movements, where updating performance is judged by the size of the vergence movement, the average updating ratio was 1.12 +/- 0.45. Updating ratios tended to be larger for far targets than near targets, although both intra- and intersubject variabilities were smallest for near targets. Thus in addition to self-generated movements, extra-retinal signals involving otolith and proprioceptive cues can also be used for spatial constancy.

Noncommutative control in the rotational vestibuloocular reflex.

To investigate the role of noncommutative computations in the oculomotor system, three-dimensional (3D) eye movements were measured in seven healthy subjects using a memory-contingent vestibulooculomotor paradigm. Subjects had to fixate a luminous point target that appeared briefly at an eccentricity of 20 degrees in one of four diagonal directions in otherwise complete darkness. After a fixation period of approximately 1 s, the subject was moved through a sequence of two rotations about mutually orthogonal axes in one of two orders (30 degrees yaw followed by 30 degrees pitch and vice versa in upright and 30 degrees yaw followed by 20 degrees roll and vice versa in both upright and supine orientations). We found that the change in ocular torsion induced by consecutive rotations about the yaw and the pitch axis depended on the order of rotations as predicted by 3D rotation kinematics. Similarly, after rotations about the yaw and roll axis, torsion depended on the order of rotations but now due to the change in final head orientation relative to gravity. Quantitative analyses of these ocular responses revealed that the rotational vestibuloocular reflexes (VORs) in far vision closely matched the predictions of 3D rotation kinematics. We conclude that the brain uses an optimal VOR strategy with the restriction of a reduced torsional position gain. This restriction implies a limited oculomotor range in torsion and systematic tilts of the angular eye velocity as a function of gaze direction.

Human visuospatial updating after noncommutative rotations.

As we move our bodies in space, we often undergo head and body rotations about different axes-yaw, pitch, and roll. The order in which we rotate about these axes is an important factor in determining the final position of our bodies in space because rotations, unlike translations, do not commute. Does our brain keep track of the noncommutativity of rotations when computing changes in head and body orientation and then use this information when planning subsequent motor commands? We used a visuospatial updating task to investigate whether saccades to remembered visual targets are accurate after intervening, whole-body rotational sequences. The sequences were reversed, either yaw then roll or roll then yaw, such that the final required eye movements to reach the same space-fixed target were different in each case. While each subject performed consistently irrespective of target location and rotational combination, we found great intersubject variability in their capacity to update. The distance between the noncommutative endpoints was, on average, half of that predicted by perfect noncommutativity. Nevertheless, most subjects did make eye movements to distinct final endpoint locations and not to one unique location in space as predicted by a commutative model. In addition, their noncommutative performance significantly improved when their less than ideal updating performance was taken into account. Thus the brain can produce movements that are consistent with the processing of noncommutative rotations, although it is often poor in using internal estimates of rotation for updating.

Differences in the accuracy of human visuospatial memory after yaw and roll rotations.

Our ability to keep track of objects in the environment, even as we move, has been attributed to various cues including efference copies, vestibular signals, proprioception, and gravitational cues. However, the presence of some cues, such as gravity, may not be used to the same extent by different axes of motion (e.g., yaw vs. roll). We tested whether changes in gravitational cues can be used to improve visuospatial updating performance for yaw rotations as previously shown for roll. We found differences in updating for yaw and roll rotations in that yaw updating is not only associated with larger systematic errors but is also not facilitated by gravity in the same way as roll updating.

Self-motion-induced eye movements: effects on visual acuity and navigation.

Self-motion disturbs the stability of retinal images by inducing optic flow. Objects of interest need to be fixated or tracked, yet these eye movements can infringe on the experienced retinal flow that is important for visual navigation. Separating the components of optic flow caused by an eye movement from those due to self-motion, as well as using optic flow for visual navigation while simultaneously maintaining visual acuity on near targets, represent key challenges for the visual system. Here we summarize recent advances in our understanding of how the visuomotor and vestibulomotor systems function and interact, given the complex task of compensating for instabilities of retinal images, which typically vary as a function of retinal location and differ for each eye.

Roles of gravitational cues and efference copy signals in the rotational updating of memory saccades.

Primates are able to localize a briefly flashed target despite intervening movements of the eyes, head, or body. This ability, often referred to as updating, requires extraretinal signals related to the intervening movement. With active roll rotations of the head from an upright position it has been shown that the updating mechanism is 3-dimensional, robust, and geometrically sophisticated. Here we examine whether such a rotational updating mechanism operates during passive motion both with and without inertial cues about head/body position in space. Subjects were rotated from either an upright or supine position, about a nasal-occipital axis, briefly shown a world-fixed target, rotated back to their original position, and then asked to saccade to the remembered target location. Using this paradigm, we tested subjects' abilities to update from various tilt angles (0, +/-30, +/-45, +/-90 degrees), to 8 target directions and 2 target eccentricities. In the upright condition, subjects accurately updated the remembered locations from all tilt angles independent of target direction or eccentricity. Slopes of directional errors versus tilt angle ranged from -0.011 to 0.15, and were significantly different from a slope of 1 (no compensation for head-in-space roll) and a slope of 0.9 (no compensation for eye-in-space roll). Because the eyes, head, and body were fixed throughout these passive movements, subjects could not use efference copies or neck proprioceptive cues to assess the amount of tilt, suggesting that vestibular signals and/or body proprioceptive cues suffice for updating. In the supine condition, where gravitational signals could not contribute, slopes ranged from 0.60 to 0.82, indicating poor updating performance. Thus information specifying the body's orientation relative to gravity is critical for maintaining spatial constancy and for distinguishing body-fixed versus world-fixed reference frames.