## Abstract

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.

- eye movements
- Donders' law
- Listing's law
- disconjugate saccades
- stereoscopic vision

for orientation in space our eyes constantly scan the visual surround by saccades, which are rapid, highly coordinated binocular movements. To direct gaze on objects located at different depths in space, the two eyes must rotate by different amounts. Specifically, in the horizontal plane of regard, the eyes move toward near targets by rotating through different angles in opposite directions until the visual lines cross at the target of interest. Moreover, near targets in the peripheral visual field off the horizontal plane of regard require different combinations of movements in the horizontal and vertical plane. Under these circumstances the eyes also rotate in the frontal plane for some still not fully understood reasons. Originally, this latter motion has been described as torsion of the eyes about the lines of sight (Enright 1980; Helmholtz 1867; Nakayama 1983). When we are looking, for example, at objects on the ground that require our attention during walking, the fixation eye movements are accompanied by small rotations such that the vertical retinal meridians move away from the head's midsagittal plane, which is called ex-torsion. Conversely, fixation of targets in the upper visual field, for example, for catching or hitting an approaching small ball, requires small rotations that move the vertical retinal meridians toward the head's midsagittal plane, called in-torsion. Although it has long been known that these small adjustments of eye position, which are symmetric for targets in the head's midsagittal plane, depend on both the vergence angle and the amount of elevation of the eyes (Mok et al. 1992; Porill et al. 1999), their purpose remained largely obscure. In this work we provide evidence suggesting that these movements, although small in amplitude, specifically serve stereoscopic vision.

Originally, the torsion of the eyes in near vision has been interpreted as a violation of Listing's law, according to which visually guided eye movements are controlled by two, rather than three, degrees of freedom of motion (for a review, see Hepp 1994; Hepp et al. 1997). Mok et al. (1992), however, found that under isovergence conditions, fixation eye movements still obeyed Listing's law, but with respect to torsional and vertical movement planes that were rotated in each eye in the horizontal plane of regard away from the head's midsagittal plane. The modulation of the eye's torsion in near vision thus appeared merely as an extension, rather than a violation, of Listing's law (Mok et al. 1992; Tweed 1997).

The experimental verification of the amount of rotation of the torsional and vertical movement planes of each eye turned out to be unexpectedly difficult. In the original study of Mok et al. (1992), the estimated angles were too small compared with the theoretical prediction of one-half of the half-vergence angle for symmetric vergence, whereas in the study of Van Rijn and Van den Berg (1993), it was too large. In subsequent studies, Minken and van Gisbergen (1994, 1996) reported values that were generally closer to the theoretical prediction, although they differed from each other due to different fitting models. Although these authors used various visual conditions in their first study, they could not resolve the discrepancy of their results with the earlier results of Van Rijn and Van den Berg (1993).

From a kinematics standpoint, the torsion of the eyes must change if horizontal and vertical rotations occur in mutually orthogonal planes. For example, a horizontal-vertical compounded rotation generates torsions of the right and left eye relative to a head-fixed frame whose directions are compatible with the experimentally observed torsions accompanying vertical rotations during vergence. In the following we will refer to this fact as the Helmholtz kinematic model, because it suggests that the observed torsion in near vision simply arises as a mechanical consequence of the horizontal and vertical rotations of the eyes. Indeed, the Helmholtz kinematic model could explain the kinematics of binocular fixations of targets located in as well as off the horizontal plane of regard. First, we notice that the locus of binocular targets is a circle located in a vertical plane for targets at distances smaller than (optical) infinity, which is obtained by intersecting two imaginary shells around each eye with the respective eye-to-target distances as radius (Fig. 1). This circle, which may be called the Helmholtz circle for short, contains all potentially binocularly visible targets at the given distances relative to the two eyes. Its location depends on the distance between the two eyes' rotation centers (O_{a} and O_{b} in Fig. 1) and the respective eye-to-target distances (*a* and *b* in Fig. 1).

The alternative to pure kinematic torsion is an oculomotor system that operates in default mode in a Donders-Listing mode for generating saccades across the whole visual space irrespective of changes in depth. In far vision, binocular fixations are made possible because the gaze lines and associated direction planes in which the eyes rotate are parallel to each other. This parallelism brakes down, however, when the depth plane of fixation changes relative to optical infinity. Specifically, when the eyes converge in the horizontal plane of regard, the vertical direction planes rotate toward each other (Fig. 1, white circles). For geometric reasons, the rotated planes of the right and left eye meet the Helmholtz circle of binocular positions at just one single position. Saccades that typically move the gaze lines from one fixation position to the next along direction circles will therefore no longer allow binocular fixation in near space (for far vision, see Hess 2013). In this article we will show that for pure geometric reasons, the eyes under these circumstances must also rotate in a frontal plane to fuse the dichoptic retinal target images. We will call this model the Donders-Listing model.

To test the Helmholtz kinematic model versus the Donders-Listing model, we analyzed saccades between far and near targets in the upper and lower visual half-space in three behaviorally trained rhesus monkeys. We show that the eyes redirected the gaze lines by combining rotations in horizontal and vertical direction planes with active torsion in the frontal plane. The modulation of torsion during such refixation saccades consisted of a rapidly developing, overshooting coarse torsion followed by a smaller, fine-tuned torsion at delayed onsets. Although the Helmholtz kinematic model could perfectly explain the horizontal and vertical rotations of the eyes, it failed to predict the torsion.

## MATERIALS AND METHODS

The experimental data used in this study were obtained in the context of a larger project requiring three-dimensional (3D) eye movement records in nonhuman primates. We used rhesus monkeys because of the physiological proximity of the organization of their visual and oculomotor system to humans. The animals had a chronic acrylic head implant for restraining the head in the experimental sessions. Three-dimensional eye movements were recorded with the magnetic search coil technique (Robinson 1963) using a dual search coil that was implanted on both eyes under the conjunctiva as previously described (Hess 1990; Mandelli et al. 2005). All surgery was performed under aseptic conditions and general anesthesia, and postoperative pain treatment was applied for at least 3 consecutive days. All procedures were in accordance with the recommendations in the *Guide for the Care and Use of Laboratory Animals* of the U.S. National Institutes of Health. The housing, husbandry, and experimental procedures were reviewed, approved, and supervised by the Veterinary Office of the Canton of Zurich.

#### Experimental procedures.

Three-dimensional eye movement records were analyzed in three female rhesus monkeys (*Macaca mulatta*), which had been trained to refixate between a far and a near light-emitting diode. The far target was located at eye level 0.8 m straight ahead (horizontal vergence ∼2°, vertical eye position ∼0°). In most experiments, the near target was located 10° down at a distance of 0.1 m (horizontal vergence 17°). In a few experiments, it was also presented at 10° up at the same distance. To examine a larger range of azimuth and elevation angles including asymmetric vergence effects, we analyzed all saccades that landed on or in the vicinity of the near target (Table 1). All experiments were performed in dimmed light, i.e., with a background illumination inside an opaque sphere where the animal was seated upright with the head restrained in a primate chair, which completely surrounded the animal. The inner wall of the sphere was covered with randomly arranged black dots about 1 to 5 cm in size on a white background. Three-dimensional eye positions were recorded with an Eye Position Meter 3000 (Skalar, Delft, The Netherlands), calibrated as described in Hess et al. (1992), digitized at a sampling rate of 833.33 Hz, and stored on a computer for off-line analysis. To express eye positions as rotation vectors (Haustein 1989), the zero or reference positions were defined to be the eye's orientations while the monkey fixated a target 0.8 m straight ahead. In two animals (M1, M2), Listing's plane tilted less than −2° vertically (that is, backwards) and −1° horizontally (that is, leftward), and in one animal (M3) it tilted vertically about −6° and horizontally 0°. We did not correct eye positions for these deviations from primary position (see Hess and Thomassen 2014). The onset and offset of saccades were isolated by a semiautomatic procedure based on the magnitude of the jerk (derivative of angular eye acceleration), followed by applying an empirically adjusted position threshold based on the change in magnitude of the eye position vector. Specifically, after a position threshold was chosen in the one-figure percent range, the time course of the coarsely delimited saccadic events was narrowed down to stay within these provisional limits (where they exceeded the threshold). To further refine the saccade window, we computed the mean + SD of the initial rising phase up to the point of exceeding the threshold, which determined the final onset threshold. Likewise, we computed the mean + SD of the falling phase below threshold, which defined the final offset threshold of the saccade. This procedure avoids noise problems inherent with velocity or acceleration thresholds and isolates saccades by preserving their typically asymmetric time course. Saccades with amplitudes <1° were discarded.

In the following, vectors are denoted by bold characters and unit vectors by regular fonts with a caret. When referring to components, we write vectors for convenience as row vectors within round parentheses, separating the components with commas.

#### Gaze movements and representation of 3D eye position.

Three-dimensional eye positions were represented in the convenient axis-angle representation of rotation vectors, where the magnitude of rotation is expressed as the tangent of half the angle of rotation (ρ), and the axis as a vector of unity length orthogonal to the plane of rotation (denoted *n̂*): ** E** = (

*E*

_{tor},

*E*

_{ver},

*E*

_{hor}) = tan (ρ/2)

*n̂*(Haustein 1989). Torsional eye position,

*E*

_{tor}, is the rotation of the eye in the head's frontal plane (clockwise positive); vertical eye position,

*E*

_{ver}, is the rotation in the vertical plane (downward positive); and horizontal eye position,

*E*

_{hor}, is the rotation in the horizontal plane of regard containing the reference position straight ahead (leftward positive). A rotation (saccade) from straight ahead position O to position B via the intermediate position A, denoted

*E*_{BAO}, is the composition of a rotation from O to A, followed by a rotation from A to B:

*E*_{BAO}=

*E*_{BA}

*E*_{AO}:= (

*E*_{BA}+

*E*_{AO}+

*E*_{BA}×

*E*_{AO})/(1 −

*E*_{BA}·

*E*_{AO}), where “×” is vector product and “·” is dot product (Haustein 1989).

To relate gaze shifts in visual space to the underlying rotations of the eye, we represented the gaze line by a unity vector, *ĝ* = ∑_{i=1}^{3}*g*_{i}*ê*_{i} in the spherical field of fixations with coefficients *g*_{1} = cos ε, *g*_{2} = −sin ε sin ψ, and *g*_{3} = sin ε cos ψ using the polar coordinate ε, which is the angular eccentricity relative to straight ahead, and ψ, the signed dihedral angle between the plane *ê*_{1} × *ĝ* and the frontal plane represented by *ê*_{1} × *ê*_{3} (right side down positive).^{1} The unit vectors *ê*_{i} (*i* = 1, 2, and 3) represented a right-handed, head-fixed Cartesian coordinate system with *ê*_{1} pointing straight ahead, *ê*_{2} pointing along the interocular line from right to left, and *ê*_{3} pointing upward. A general rotation of the eye was described by a rotation operator *R* = *R*(*n̂*, ρ), where the unit vector describes the orientation of the rotation plane and ρ is the angle of rotation. A rotation of the gaze vector from A to B in the plane *n̂* through the angle ρ is obtained by the operation *ĝ*_{B} = *R*_{BA}*ĝ*_{A}*R*_{BA}^{−1}, where *R*_{BA}^{−1} = *R*_{AB} is the inverse of *R*_{BA}.

#### Definition of normed target space.

In the following we represent the eye movements by their rotation centers, separated by the interocular distance, and the two gaze lines pointing at the target of interest. In a spherical visual field, the distances of a single target to two rotation centers are related to the azimuth angles and the distance between the rotation centers by the following relations: *a*/*b* = cos β/cos α and *a* sin α − *b* sin β = *k*, where *a* and *b* are the distances, α and β the azimuth angles, and *k* the interocular distance (Figs. 1 and 2). From these relations one derives *a* = *k* cos β/sin (α − β) for the right and *b* = *k* cos α/sin (α − β) for the left eye-to-target distance, which simplify to *a* = *b* = *k*/(2 sin α) for equal azimuth angles. In the horizontal plane of regard, the triangle formed by the rotation centers and the point of fixation scales for fixed azimuths α and β with the interocular distance (see triangle O_{a}O_{b}A in Fig. 2). In the following we set *k* = 1 without loss of generality. By this convention, target distances scale in terms of multiples of the interocular distance. In our rhesus monkeys the interocular distances were about 3.0 cm.

#### Geometry of vertical direction circles and Helmholtz circles in near vision.

To determine the rotation of the eye during saccadic refixations from far to near targets, we decomposed the rotation into a first rotation in the horizontal plane of regard. In the Helmholtz kinematic model, this moved the gaze lines toward a point on the line of intersection of the horizontal plane of regard with the Helmholtz plane, i.e., the vertical plane of intersection of the associated target shells. A subsequent vertical rotation of the eyes in this plane then moved the gaze lines toward the brain's estimated target location.

In the alternative Donders-Listing model, the vertical rotations occurred in each eye's direction plane that intersected with the Helmholtz plane and the horizontal plane of regard at a single point (Fig. 3, position A). Thereby, the gaze lines moved along the respective vertical direction circles up to slightly divergent positions that shared the same depth plane as the final target position (Fig. 3, positions A, B_{a}, and B_{b} for a target in the upper hemifield). We then calculated the torsion of the right and left eye that was required to move the gaze lines to their intersection at the binocular target position (Fig. 3, position C). Geometrically, this position is the intersection of the two circles centered at the respective directions straight ahead and passing through the Listing positions B_{a} and B_{b} that share the same frontoparallel plane with the binocular target position C.

To determine the geometric relation between binocular Helmholtz-positions, denoted by C, and the associated Listing positions B_{a} of the right eye and B_{b} of the left eye, we first note that the *y*-coordinates are functions of the horizontal disparities of these positions and the azimuth of the respective eye (*q* and α in Fig. 4). Specifically, for the right eye we have the relations [B_{a}]_{y} = −*a* sin ψ_{B} sin ε_{B} = −*a*(*q*/*a* − sin α) and [C^{(a)}]_{y} = −*a* sin ψ_{C} sin ε_{C} = *a* sin α.^{2} From these we obtained the trigonometric relation sin ψ_{B} sin ε_{B} = sin (ψ_{B} + ω_{a}) sin ε_{B} + *q*/*a* (Fig. 4), accounting for the condition that B_{a} and C^{(a)} have the same eccentricity relative to the direction of straight ahead, namely, ε_{C} = ε_{B} = sin^{−1} (*r*_{a}/*a*), where *r*_{a} is the radius of the circle through B_{a}, and C (Fig. 4). Thus ω_{a} = sin^{−1} [sin ψ_{B} − (*q*/*a*)/sin ε_{B}] − ψ_{B}. An analogous formula holds for ω_{b}. Second, we determined the horizontal disparity *q* by *q*/*a* = (*p*/*a*) sin α/2, where *p*/*a* = (cos α − cos ε_{C})/(cos α/2), noting that
sin^{2} α/2 and
cos ε_{C}. Finally, the vertical rotation angle subtended by the line segments
and
in the direction plane (Fig. 4*C*) was calculated as a function of the ratio *p*/*a* with η_{a} = cos^{−1} [1 − (*p*/*a*)/cos α/2]. Similarly, η_{b} of the left eye was computed. In brief, the two Listing positions B_{a} and B_{b} in the neighborhood of the target position C were obtained from the estimates of C^{(a)} and C^{(b)} relative to the respective rotation centers O_{a} and O_{b}.

For simulating the trajectories of far-near refixation saccades, we applied the equation *ĝ*_{A} = *R*_{AO}*ĝ*_{0}*R*_{AO}^{−1}, which moved the gaze line from its initial position parallel to straight ahead to the line intersecting the intermediate target position A in the horizontal plane of regard. In the Helmholtz kinematic model, this motion was followed by a vertical rotation moving the tip of the gaze vector *ĝ*_{C} =
from position A to the estimated final target position C, according to *ĝ*_{C} = *R*_{CA}*ĝ*_{A}*R*_{CA}^{−1}. The compound action that moves the tip of the gaze line from O to A to C thus was obtained by computing *ĝ*_{C} = *R*_{CA}(*R*_{AO}*ĝ*_{0}*R*_{AO}^{−1})*R*_{CA}^{−1}. In the Donders-Listing model, the vertical rotation was computed in each of the direction planes, which subtended the angle α/2 in the right and β/2 in the left eye relative to the vertical plane. We simulated this motion by *ĝ*_{B} = *R*_{BA}*ĝ*_{A}*R*_{BA}^{−1}, where *R*_{BA} is a rotation operator associated to the respective eye's vertical direction plane. Finally, we computed *ĝ*_{C} = *R*_{CB}*ĝ*_{B}*R*_{CB}^{−1} for each eye, where *R*_{CB} mediated a rotation of the gaze line in the common frontoparallel plane from B_{a} to C. For details about the rotation operators *R*_{AO} and *R*_{CA} used in the Helmholtz kinematic model and *R*_{AO}, *R*_{BA}, and *R*_{CB} used in the Donders-Listing model, see appendix. In the following we refer to the compounded rotation operator *R*_{H} = *R*_{CA}*R*_{AO} as the Helmholtz operator (Helmholtz kinematics) and *R*_{DL} = *R*_{BA}*R*_{AO} as the Donders-Listing operator (Donders-Listing kinematics).

Root-mean-square errors were computed by evaluating the expression
where *X̃*_{μ}^{k} and *X*_{μ}^{k} are the *k*-th sample of the μ-th component of the experimental and reconstructed eye position, respectively (μ = tor, ver, or hor; *N* = number of samples). In the exponential least-squares fits of torsion, we computed the generalized *R*^{2} values based on the residual sum of squares of the fit and the reduced model consisting of average torsion (Anderson-Sprecher 1994).

## RESULTS

#### Reconstruction of 3D eye positions during far-near refixation saccades.

Fixation of near targets requires in general a compound rotation of the eyes consisting of a vergence movement and adjustments of the elevation of the eyes relative to the horizontal plane of regard. Such compound rotations generate torsion of the eyes, except in the case where the eyes rotate in horizontal and vertical direction planes. In earlier studies we described the suppression of ocular counter-roll during fixations of near targets in tilted head positions and also compared counter-roll before and after convergent or divergent saccades. We also reported average latencies, showing that in far-to-near refixation saccades they were shortest for the horizontal and longest for the torsional rotation component (Mandelli et al. 2005; Misslisch et al. 2001). In the following paragraphs we describe specifically the kinematics of far-near refixation saccades obtained in the same animals by comparing the Helmholtz kinematic model with the Donders-Listing model.

Using the azimuth and elevation derived from the recorded eye movement data (see materials and methods), we first estimated the torsional variation underlying the Donders-Listing kinematics. For this we estimated the rotation angles in the respective direction planes and evaluated the Donders-Listing kinematics *R*_{DL} = *R*_{BA}*R*_{AO} for each eye (see materials and methods, Fig. 4, and appendix). We found small modulations in torsion of each eye in the range of about ±0.5° during the refixation trajectory. Next, we removed this torsion from the saccade trajectory to estimate the ocular torsion caused by the active, not kinematics-related, torsion of the eye. Assuming that the experimentally measured rotation was a compounded rotation based on Donders-Listing kinematics and active torsion, we evaluated the rotation quotient *R*_{NL} = *R*_{exp}*R*_{DL}^{−1} to obtain the non-Listing rotation, from which we computed the angle of ocular torsion ξ = 2 tan^{−1} [*R*_{NL}]_{x} for each eye. Finally, we estimated the experimentally observed rotation of each eye by compounding the active rotation in the frontal plane through the angle ξ and the Donders-Listing rotation *R*_{eye} = *R*_{F}*R*_{DL}, where *R*_{F} = *R*_{F}(ξ) and *R*_{DL} = *R*_{DL}(η, ϑ) with the appropriate torsion ξ, elevation η, and azimuth angles for the right (ϑ = α, ξ = ξ_{a}, η = η_{a}) and left eye (ϑ = β, ξ = ξ_{b}, η = η_{b}), respectively. We found a close match between the experimentally measured rotations of the eyes and the so reconstructed rotations in all components (Fig. 5, *A–C*, reconstructed eye position in black superimposed on experimental eye position in gray). To highlight the spatial characteristics, we analyzed the torsional modulation that evolved during far-near refixation saccades also in the spatial domain. We found a consistent slow build-up as the gaze lines approached the near target (Fig. 5*D*, right eye torsion in black, left eye torsion in gray). Typically, the eye torsion increased in magnitude, reaching amplitudes up to about 2.5° shortly before hitting the estimated target location. Subsequently, torsion abruptly diminished such that the difference between right and left eye torsion settled to a steady position in a range of about 1–2°. To characterize the torsion of the eyes more specifically, we measured peak torsion defined as the maximum torsion of the right eye and the minimum torsion of the left eye during the saccade and compared it with the respective final torsion, which was defined as the torsion reached when the horizontal vergence movements ceased. In the three animals, we found an average maximal torsion of the right eye of 0.9 ± 0.6° (SD), 1.7 ± 1.1°, and 0.5 ± 0.5° and an average minimal torsion of the left eye of −0.9 ± 0.9°, −0.7 ± 0.6°, and −1.9 ± 0.8°. Similarly, we found an average final torsion of the right eye of 0.7 ± 0.9°, 1.6 ± 1.1°, and 0.3 ± 0.7° and an average final torsion of the left eye of −0.4 ± 1.1°, 0.1 ± 0.8°, and −1.8 ± 0.8° (*N* = 65, 297, and 202). Although, on average, peak and final torsion in the three animals were fairly close together, this was not necessarily the case for individual saccades (Fig. 5).

#### Orientation of rotation planes as a function of azimuth and elevation.

Comparing the estimated rotation of the eyes based on the Helmholtz kinematics and the Donders-Listing model, we found that the two approaches yielded very similar predictions of the vertical and horizontal gaze trajectories yet drastically different predictions regarding the time evolution of the spatial orientation of the rotation planes. Viewed in a frontal projection, the instantaneous horizontal and vertical rotation planes of the two models had similar orientations along the trajectory (Fig. 6*C*, *top* and *bottom*). In projections orthogonal to the frontal plane, the orientation of the vertical and horizontal rotation planes remained constant after an initial change early in the trajectory in the Helmholtz kinematic model. In contrast, both rotation planes changed their orientation continuously along the saccade trajectory in the Donders-Listing model (compare top and side views in Fig. 6, *A* and *B*, *top* and *bottom*). Whereas the torsion generated by the Helmholtz kinematics did not match the experimentally observed torsion, the Donders-Listing model reproduced the experimentally observed modulation of torsion during refixation saccades within error margins of a few percent (Table 2). Most interestingly, the rotation trajectories often made an abrupt turn toward pure torsion toward the end, which was faithfully reproduced by the Donders-Listing but not by the Helmholtz kinematic model (Fig. 6*A*, compare *top* and *bottom*).

To quantify the observed torsion, we estimated the tilt of the instantaneous rotation plane of the eyes during refixation saccades as well as the average orientation over the last 50 ms at the end of saccades. The Donders-Listing model predicted the following angular ratio between the instantaneous orientation of the torsional plane and the orientation of the vertical-horizontal rotation plane, given by *n̂*_{hv} = (*n*_{2}*ê*_{2} + *n*_{3}*ê*_{3})/(*n*_{2}^{2} + *n*_{3}^{2}) (see appendix):
where the unit vector *n̂* = (*n*_{1}, *n*_{2}, *n*_{3}) represents the normal on the rotation plane, η is the elevation, ϑ is the azimuth, and ξ is the angular excursion of the eye due to active torsion. According to this relation, the orientation of the horizontal-vertical rotation plane modulates relative to the frontal plane mainly with the torsion angle and much less with the azimuth and elevation of the eye. Indeed, refixation saccades exhibited large dynamic modulations of the orientation of the horizontal-vertical rotation plane relative to the frontal plane. Specifically, we found large transient tilts shortly before the gaze lines hit the estimated target location, which were little dependent on the azimuth and elevation of the eyes (Fig. 7). The large torsional excursions rapidly decayed to a few degrees around the equilibrium torsion at target acquisition. The formula for τ_{DL} also shows that there exists no functional relation between torsion and elevation at constant vergence or between torsion and azimuth at constant elevation. In fact, the Donders-Listing model implies that the torsion also depends on target distance (for details, see discussion).

#### Fine-tuned torsional modulation.

Although it has long been known that binocular fixations require disconjugate rotations of the eyes about the lines of sight, the quantitative aspects in terms of rotations in the frontal plane have been less clear. In contrast to the Helmholtz kinematic model, which predicts torsion as a function of azimuth and elevation, the Donders-Listing kinematics do not alter the torsion of the eyes. In this latter model, the final torsion required for binocular fusion at the time of target acquisition is determined by the geometric configuration between the Helmholtz plane and the rotations of each eye in the vertical direction planes. Evaluating the disparity between the vertical positions and the final acquired binocular target position (see materials and methods and Fig. 4), we found small but robust torsions of each eye in opposite directions in the range of 0.1° to 0.6°. These torsional rotations were often delayed by a few tens of milliseconds compared with the saccade onset (Fig. 8, *A* and *C*, and Fig. 9). To quantify these delays we compared the points in time where the torsion and ω-torsion of the eyes increased in absolute terms by more than 0.01° relative to saccade onset. In the three animals we found average delays of 4.9 ± 3.3 (SD), 4.2 ± 2.5, and 16.2 ± 9.5 ms for torsion and average delays of 35.4 ± 10.3, 33.6 ± 14.2, and 48.2 ± 7.4 ms for ω-torsion of the right eye. Similarly, we found average delays of 7.1 ± 4.3, 3.7 ± 2.0, and 10.8 ± 6.3 ms for torsion and average delays of 41.8 ± 11.5, 37.5 ± 16.0, and 45.3 ± 8.3 ms for ω-torsion of the left eye (*N* = 65, 297, and 202). The torsional rotations sharply increased in both eyes when the eye-to-target distance was less than about 10 interocular distances (Fig. 8*D*). To distinguish this from the overall torsion of the eyes, we refer to it as ω-torsion.

#### Modulation of coarse and fine-tuned cyclovergence.

The temporal evolution of cyclovergence, computed as the difference of right minus left eye torsion, exhibited in all three animals a more or less pronounced underdamped oscillation (Fig. 9, black traces). In frontal projection the retinal images thus rotated relative to each other, thereby modulating the torsional disparities. These out-of-phase rotations are likely to support the visual system's search for corresponding images in the two eyes, thereby enhancing stereopsis (see discussion). The fine-tuned cyclovergence, obtained as the difference between ω-torsion of the right minus left eye set on with a delay of about 25–50 ms relative to the onset of the coarse cyclovergence. In contrast to the coarse cyclovergence, the time evolution of the fine-tuned cyclovergence showed overdamped characteristics (Fig. 9, gray traces).

The Donders-Listing model implies that torsion modulates with respect to both eye's azimuth and elevation, an implication that follows by right-multiplication of the equation *R*_{eye} = *R*_{F}*R*_{DL} with *R*_{DL}^{−1}. Since the ω-torsion was obtained as part of the overall torsion, we expected a similar angular dependency. Indeed, we found that the ω-torsion of each eye depended on both the azimuth and elevation of the eye as well as target distance. To evaluate the dependency of ω-torsion on these parameters, we pooled all saccades obtained in several experimental sessions, irrespective of whether the animal perfectly hit the near target, and evaluated the ω-torsion averaged across 12 ms at the end of the saccades. In one animal we had collected enough data to be able to single out two clusters with respect to the saccades' azimuth that also covered a reasonable large range in vertical direction as well as eye-to-target distance. The two clusters showed the expected exponential dependency of ω-torsion as a function of elevation and eye-to-target distance with little or no overlap on average (Fig. 10).

## DISCUSSION

We have studied the 3D kinematics of saccadic refixations between far and near targets. We found that ocular torsion modulated in flight in a specific manner: it first increased by about 1° to 2° relative to saccade onset before it sharply decreased again around the time of target acquisition to reach constant postsaccadic values. The widely varying torsion of the eyes included a robust but much smaller component, emerging at a delay before sharply increasing up to peak values at the time of target acquisition. A detailed analysis of the underlying ocular kinematics revealed three different rotations of the eyes: one rotation in the horizontal plane of regard converging toward the near target, followed by a rotation in each eye's vertical direction plane, and finally, a rotation in the frontoparallel plane common to both eyes. Reconstruction of these rotations based on the eye excursions predicted the time course of refixation saccades in all three dimensions within error margins of a few percent. Whereas the first two rotations conformed to Listing's law, the third rotation clearly violated Listing's law even in a very limited torsional amplitude range. In the following paragraphs we first discuss these findings in light of previous studies and then address the kinematic implications in terms of binocular motor control and binocular vision.

In attempting to uncover a functional relation between ocular torsion, elevation, and vergence of the eyes, previous studies have reported conflicting results about the role of Listing's law during refixation saccades or steady fixations under isovergence conditions (Minken et al. 1995; Minken and van Gisbergen 1994, 1996; Mok et al. 1992; Van Rijn and Van den Berg 1992; Tweed 1997). Although it was found that the horizontal-vertical plane of ocular rotation tilted relative to the frontal plane, a phenomenon called “binocular extension of Listing's law,” the amount of tilt varied largely between the different studies. The reasons for these discrepancies have been sought in different visual conditions but could not be clarified to date. This was in fact not surprising, because it has been known since the time of Helmholtz that there is no single visual criterion that can define the 3D orientation of the eyes providing optimal retinal correspondences in near vision (Helmholtz 1867; Hepp 1995; Van Rijn and Van den Berg 1992). At the level of single targets, however, it is possible to define the optimal orientation of the eyes for single binocular vision. From a visuomotor standpoint, it is Donders' and Listing's laws that guarantee the existence of retinal correspondences in far vision by laying down the relations between azimuth, elevation, and torsion of each eye. Although the same kinematic principles do allow for binocular vision of near targets in the horizontal plane of regard, they fail for targets off the horizontal plane. In these circumstances the optimal binocular kinematics turns out to be more complex because of the emergence of 2D disparities in the binocular field of fixations. A key difference of the presently proposed kinematics and the more simple Helmholtz kinematics is that in a binocular setting, the Donders-Listing kinematics entail torsional rather than horizontal-vertical disparities for targets off the horizontal plane of regard. Such torsional disparities arise even under the condition of symmetric ocular vergence, which is in line with the experimentally well-established observations of disjunctive torsion of the eyes in near vision (Fig. 11).

A crucial property of torsional disparities is that they can be locally eliminated by modulating ocular torsion without affecting the intrinsic geometry of the retinal projection images. The torsion required for binocular fusion of the disjunctive Listing positions in the binocular visual field is prima facie a simple function of the estimated binocular target position and the associated Listing positions of each eye. However, the azimuth and elevation of these positions relative to each eye are linked to each other via the eye-to-target distances and the anatomically fixed interocular distance. As illustrated by the concept of normed target space (Figs. 1 and 2) and verified by experimental reconstructions (Fig. 10), four of these parameters suffice in conjunction with the anatomic interocular distance to control binocular saccades in near vision: Each point in the normed binocular space is determined by the two azimuths and the elevation of the estimated binocular target relative to either eye plus the anatomic interocular distance. Similarly, the ω-torsion required for target fusion depends on any two of the three parameters azimuth, elevation, and eye-to-target distance in a nonlinear way. Generally, the two eyes must torque by different amounts depending on the respective eye-to-target distance. More specifically, the torsion required for fusion depends exponentially on target distance. At constant vergence it depends linearly on elevation in the intermediate- and large-distance range (Fig. 12).

#### Does ocular torsion facilitate stereoscopic depth perception?

A striking observation was the steadily increasing torsion of the eyes during refixation saccades, which abruptly stopped and returned to smaller values at the time of target acquisition (Fig. 5, *C* and *D*). This behavior was observed consistently in all three animals. It was recently found that ocular torsion may enhance or oppose stereoscopic vision, suggesting that the search zones for detecting corresponding features on the retinas are retina-fixed (Schreiber et al. 2001). Disparity-selective neurons in the visual cortex V1, on the other hand, have been shown to encode a larger range of horizontal than vertical disparities (Barlow et al. 1967; Cumming 2002). On the oculomotor side, we found that horizontal and vertical disparities in peripheral viewing are in fact taken as torsional disparities driving torsional adjustments of the eyes (Figs. 3 and 11; ω-torsion in Figs. 8–10). Taking these findings together suggests that torsional disparity represents a crucial stimulus in stereoscopic vision. If so, it is plausible that the brain actively modulates ocular torsion in search for corresponding features during refixation saccades: In fact, ocular torsion sweeps the epipolar lines (relative to either eye) across the frontal projection of the retinae with the result that the two retinal projection images can be correlated with each other without affecting their intrinsic geometry. Thereby, the predominantly horizontally organized disparity zones acquire a vertical dimension. In this view, the observed active modulation of ocular torsion might represent an important mechanism that helps search for corresponding retinal features localizing a small target in near space. Although we have studied refixation saccades in a restricted field of view, the principals of ocular kinematics outlined in this report may also shed some new light on the debate about the role of vertical disparity detection in stereoscopic depth perception (Read et al. 2006; for a review, see DeAngelis 2000).

#### Donders' law in binocular fixation space.

Donders' law states that the ocular orientation of each eye during fixation of a target does not depend on the location of the previously fixated target whatever path the eye may take in the configuration space of rotations (Donders 1848). In far vision, this fundamental law in oculomotor control holds in the monocular as well as in the binocular visual fixation space because of the normally strong yoking of saccades. In near visual space, the question arises how Donders' law might be realized, because now the ocular motion required for fixation of a binocular target is a compounded rotation in three rather than just two rotation planes. What does the third rotational degree of motion freedom determine when the eye saccades between binocular targets in near visual space? The answer lies in the particular geometry of disjunctive eye positions on the vertical direction circles in relation to the neighboring binocular positions on the associated Helmholtz circle. For fusion of the dichoptic target images, these positions must share the same frontoparallel depth plane, which guarantees that there is one and only one path in the configuration space of rotations that connects those disjunctive positions with the binocular position in question. Because the two possible points of intersection of the respective vertical direction circle and the respective frontoparallel plane are in symmetric position relative to the horizontal plane of regard (Fig. 3),^{3} there is no ambiguity about which path to choose.

For saccades between the monocular and binocular field or outside the binocular field of fixations, this singular connectivity obviously does not exist, because the Helmholtz circle of binocular positions is not a direction circle for eye movements. Saccade thus may start from or land on a Listing position in near vision that cannot be fused for physical reasons, in which case Donders' law just holds for each eye separately. In contrast to far vision, the binocular extension of Donders' law requires an extra effort of the brain to fuse the images of the two eyes to establish stereoscopic vision.

#### Possible role of Donders-Listing kinematics on binocular perception.

The Donders-Listing kinematics of the eyes implies that vertical rotations typically occur in planes that are tilted about the vertical axis toward the Helmholtz plane of binocular positions in the visual field. Viewed in frontal projection, the tangential lines to Listing positions on the vertical direction circles thus increasingly incline toward the Helmholtz circle with increasing eccentricity of visual lines (Fig. 3), thus describing a systematic torsional gradient of disparities that ultimately drives the torsion of the eyes. Since ocular torsion can only locally eliminate these disparities, the fusion of the disparate monocular target images in near vision is bound to interfere with the perception of the visual field in specific ways. One well-studied example is the perception of verticality, which has been shown to deviate from the physical vertical in near vision as might be expected from the distribution of torsional disparities along the Helmholtz circle in symmetric vergence (Fig. 11). The possible influence of torsional disparity gradients across the binocular visual field on perception remains to be investigated (for a review on binocular vision, see Howard and Rogers 1995).

## GRANTS

This work was supported by the Swiss National Science Foundation and the Betty and David Koetser Foundation for Brain Research.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

## AUTHOR CONTRIBUTIONS

B.J.M.H. conception and design of research; B.J.M.H. analyzed data; B.J.M.H. interpreted results of experiments; B.J.M.H. prepared figures; B.J.M.H. drafted manuscript; B.J.M.H. edited and revised manuscript; B.J.M.H. approved final version of manuscript; H.M. performed experiments.

## ACKNOWLEDGMENTS

We thank Carla Bettoni, Elena Buffone, and Urs Scheifele for technical assistance.

Present address of H. Misslisch: German Aerospace Center, Bonn, Germany.

## Appendix

For computing rotations in 3D Euclidean space, we used the associated Clifford algebra that is generated by the three numbers γ̂_{1}, γ̂_{2}, γ̂_{3} and unity *I*, which are defined by the properties (γ̂_{i})^{2} = *I* (identity) and γ̂_{j}γ̂_{k} + γ̂_{k}γ̂_{j} = 2δ_{jk}*I*, with δ_{jk} = 1 for *j* = *k* and δ_{jk} = 0 if *j* ≠ *k* (Snygg 1997). Using these numbers, we represented the unit gaze vector *ĝ* as *ĝ* = ∑_{i=1}^{3}*g*_{i}γ̂_{i} by replacing the Cartesian basis vectors *ê*_{i} with the basis 1-vectors γ̂_{i}, which can be conveniently represented by 4 × 4 Dirac matrices. In this basis the 2-vectors γ̂_{23} := γ̂_{2}γ̂_{3}, γ̂_{31} := γ̂_{3}γ̂_{1}, and γ̂_{12} := γ̂_{1}γ̂_{2} represent the frontal, vertical, and horizontal plane, respectively. A rotation of a 1-vector *x* through angle χ in the plane Â = γ̂_{αβ} spanned by the two 1-vectors γ̂_{α} and γ̂_{β} with |Â| = 1 is obtained by the conjugation *R*_{A}(χ)*xR*_{A}^{−1}(χ) = *x′* with the operator *R*_{A}(χ) = *I* cos(χ/2) − sin(χ/2)γ̂_{αβ}. The inverse of *R*_{A} is *R*_{A}^{−1}(χ) = *R*_{A}(−χ).

##### Helmholtz kinematic model.

The Helmholtz kinematic model as defined in this study consisted of a compounded rotation of the eye, which in the case of the right eye moved the gaze line in the horizontal plane of regard from
to
through the azimuth α (denoted *R*_{AO}) and subsequently in the vertical plane from
to
, where C is a binocular position on the Helmholtz circle, through the elevation angle η_{a} (denoted *R*_{CA}). An analogous compounded rotation was used for the left eye (see Figs. 1–4). Gaze trajectories were reconstructed step by step using the compounded rotation operator *R*_{CAO} (for notational simplicity, we use “η” instead of “η_{a}” in the following equations):
With the rotation angle χ/2 = cos^{−1} (cos η/2 cos α/2), we can write this compounded rotation in standard format *R*_{CAO} = *I* cos χ/2 − sin χ/2 γ̂_{CAO}, where γ̂_{CAO} = ∑_{i}*n*_{i}γ̂_{jk} is the rotation plane. The unit vector *n̂* = (1/)(sin η/2 sin α/2*ê*_{1} + sin η/2 cos α/2*ê*_{2} + cos η/2 sin α/2*ê*_{3}) describes the orientation of the rotation plane in the Cartesian frame *ê*_{1} (straight ahead), *ê*_{2} (leftward, parallel to the interocular line), and *ê*_{3} (upward). Note that in this model torsion is generated proportional to sin η/2 sin α/2 Analogous formulas hold for the left eye with azimuth β and elevation η = η_{b}.

##### Donders-Listing kinematics.

The Donders-Listing kinematics used in this study consisted of a rotation in the horizontal plane of regard, exerted by a rotation operator *R*_{AO}, followed by a rotation in the vertical direction plane, exerted by a rotation operator denoted *R*_{BA}. The direction planes of the eye, denoted by γ̂_{DL} in the following, are in general linear combinations of the frontal plane (γ̂_{23}), the vertical plane (γ̂_{31}), and the horizontal plane of regard (γ̂_{12}), that is, γ̂_{DL} = ∑_{i}*d*_{i}γ̂_{jk} (*i* ≠ *j* ≠ *k*). In the horizontal plane of regard we have for the right eye with azimuth α (Fig. 4):
An analogous relation holds for the left eye with azimuth β.

Rotations of the right eye (and similarly, those of the left eye) moving the gaze line in the horizontal plane of regard from
to
through the azimuth α and further in the vertical direction plane from
to
through the elevation angle η_{a} were simulated with the compound rotation operator *R*_{BAO} (for notional simplicity, we use “B” instead of “B_{a}” and “η” instead of “η_{a}” for the right eye as shown in Fig. 4):
With the rotation angle χ/2 = cos^{−1} (cos η/2 cos α/2), we can write this compound rotation in standard format *R*_{BAO} = *I* cos χ/2 − sin χ/2 γ̂_{BAO}, where γ̂_{BAO} = ∑_{i}*n*_{i}γ̂_{jk} is the rotation plane. The unit vector *n̂* = (1/)(sin η/2*ê*_{2} + cos η/2 sin α/2*ê*_{3}) describes the orientation of the rotation plane in the Cartesian frame *ê*_{1} (straight ahead), *ê*_{2} (leftward, parallel to the interocular line), and *ê*_{3} (upward). Note that *R*_{BAO} does not generate torsion.

##### Donders-Listing model.

The Donders-Listing model combines the Donders-Listing kinematics with rotations of the eye in a common frontoparallel plane, which fuse the disparate target images of the right and left eye. The rotation that moves the gaze line of the right eye through the angle ω_{a} from
to
is *R*_{CB}(ω) = (*I* cos ω/2 − sin ω/2 γ̂_{23}), where again we have simplified the notation by writing “B” instead of “B_{a}” and “ω” instead of “ω_{a}.” For the compounded rotation we obtain
where χ_{ω}/2 = cos^{−1} (cos ω/2 cos η/2 cos α/2) is the rotation angle and γ̂_{CBAO} = ∑_{i}*n*_{i}γ̂_{jk} is the rotation plane. The unit vector *n̂*_{ω} = (1/) ∑_{i}*n*_{i}*ê*_{i} has the Cartesian coefficients *n*_{1} = sin ω/2 cos η/2 cos α/2, *n*_{2} = cos ω/2 sin η/2 − sin ω/2 cos η/2 sin α/2, and *n*_{3} = cos ω/2 cos η/2 sin α/2 + sin ω/2 sin η/2. The tilt angle of the rotation plane relative to the vertical-horizontal rotation plane is
Note that the tilt is zero for zero elevation because the torsion required for binocular vision vanishes as the elevation approaches zero. The same is true for version movements in far vision where α = β.

## Footnotes

↵1 In geometric terms the cross product of two unit vectors

*ê*_{i}×*ê*_{j}, (*i*≠*j*) represents an oriented unit plane. Indeed, the exterior product*ê*_{ij}=*ê*_{i}^*ê*_{j}(bivector) is related to the cross product by*ê*_{i}×*ê*_{j}= −(*ê*_{i}^*ê*_{j})*ê*_{123}using the 3D pseudoscalar*ê*_{123}=*ê*_{1}*ê*_{2}*ê*_{3}.↵2 Superscripts

*a*and*b*in parentheses indicate that the coordinates of a particular binocular position*C*are taken relative to the rotation center*O*_{a}and*O*_{b}, respectively.↵3 There is a pair of Listing-positions that share the same frontoparallel depth plane with a binocular target of interest, one in the upper and one in the lower visual plane. Only one of these positions is close to the target.

- Copyright © 2015 the American Physiological Society