Kinematics of visually-guided eye movements

Abstract

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.

Figure 1. Virtual rotations required to approximate a smooth Listing-motion.

To track a target that happens to move along a direction-circle (T: left panel), eye position signals holding the eye in the required plane of rotation could theoretically be derived by the following procedure (Helmholtz 1867): To obtain a smooth motion from A to B along the associated direction-circle (white circle through pupil, T, and F), the following four virtual rotations must be compounded: A first rotation in the sagittal plane NAO through an angle η subtending the arc AO, abbreviated by , a second rotation in the frontal plane LNR through ξ subtending the arc NN′, abbreviated by , a third rotation in the meridian plane OBN′ through η′ subtending the arc OB, abbreviated by , and finally a forth rotation in the eye′s coronal plane through –ξ, abbreviated to eliminate the acquired torsion. Denoting by and the unit gaze vectors parallel to and , respectively, we have altogether . For small angles ξ, approximates a smooth Listing-motion from A to B along the direction-circle arc. Left panel, sketch of the eye: O, primary position; F, occipital position, antipodal to O; N, R, L, defining north, right and left directions in the eye’s coronal plane; direction- circle (white), circle passing through center of pupil and F. Right panel, front view onto the eye with spherical coordinate grid: ψ, meridian angle; ξ, rotation angle; ε, eccentricity relative to O along circles ψ = constant; η, rotation angle in planes ψ = constant.

Figure 2. Geometry underlying the action of the operator R_CF.

(A) Side view on the spherical field of fixations, represented by a sphere with the eye (E) in the center, gaze direction straight ahead and F, occipital fixation point. The unit gaze vector represents the direction of the line of sight, parallel to , to the fixation point T. Fixation points are parameterized by the spherical polar coordinates ψ, ε. (B) Front view on the spherical field of fixations, displaying ξ, angle of ocular rotation in the frontal plane LNR; , angle of ocular rotation in the eye’s coronal plane LN′R. For further details see text.

Figure 3. Approximation of a general Listing-motion in the spherical field of fixations.

(A) Front view on the spherical field of fixations and (B) side view on the meridian plane through OBC. Under the action of the Donders-Listing operator the fixation point A (gaze parallel to ) moves through a small angle ξ approximately along the arc AB from A to B. Similarly, under the action of the meridian operator the new fixation point B (gaze parallel to ) moves through a small angle η along the arc BC from B to C. If both the Donders-Listing and meridian operator act simultaneously the fixation point moves along the arc AC from A to C. Format similar as in Figures 1 and 2.

Figure 4. Minimizing ocular torsion during horizontal target tracking.

A, top black curve: Angular tilt of the unit gaze vector (i = 1, 2 …N) plotted against meridian angle ψ or azimuth angle . Middle black curve: Reconstructed eccentricity of ocular rotation plane plotted against meridian ψ or azimuth . Note increasing tilt of these two curves with increasing absolute azimuth . Bottom thick gray curve: Angular tilt of rotation plane of gaze line, remaining perfectly invariant during tracking motion. B: Simulated vertical (gray line) and horizontal component (black line) of 3D eye position plotted against azimuth (); onset of tracking to the left and right at straight ahead. C: Simulated torsional component (black line: 10 Hz sampling rate as in A and B; gray line: 5 Hz sampling rate). Inset: azimuth (not displayed), angle subtended by small circle arc from to ; Listing’s plane, plane through E, N and R; line segment , primary gaze direction; T⁽⁰⁾, initial target position; S⁽¹⁾, intermediate position, virtually traversed while tracking the target from T⁽⁰⁾ to T⁽¹⁾ (for more details see Results).

Figure 5. 3D angular eye position and velocity reconstruction of horizontal tracking.

A to B, left two columns: Reconstructed eye position (thick gray traces) during tracking at gaze 15° up and 15° down, superimposed on experimental 3D eye position in color (sinusoidal fits only shown for torsion). Note average torsional offset of about +1° and −1° for gaze up and down, respectively. Transformation to primary position would eliminate these offsets but not the saccadic modulation. A to B, right two columns: Reconstructed angular eye velocity (thick white traces) during tracking at gaze 15° up and 15° down, superimposed on experimental 3D angular eye velocity in color (sinusoidal fits, black traces). Torsional, vertical and horizontal experimental data in red, green and blue, respectively; for more details, see text.

Figure 6. Tilt of angular eye velocity rotation planes during straight-line tracking.

(A) Rotation plane tilt angles of total (Ω_eye) and Donders-Listing (Ω_DL) angular eye velocity during horizontal tracking in 15° gaze down, plotted across one oscillation cycle (5 response cycles superimposed, Ω_eye: black traces, Ω_DL: gray traces). (B) Rotation plane tilt of Donders-Listing angular velocity as a function of estimated target distance relative to straight ahead (data from 5 cycles). Note the small but steadily increasing deviation of tilt angles from the half-angle slope relation (dashed line) with increasing estimated target distance (least-squares fitted line through target positions: slope = 0.49, offset = 0.2°). (C) Average ratios of rotation plane tilt angles of Donders-Listing angular velocity to estimated target distance as a function of estimated target distance relative to straight ahead. Each of the data points was obtained by least-squares fitting tilt angles versus estimated target distance, collected in 5°-wide intervals across the range of 0° to 25°.

Figure 7. 3D angular eye position and velocity reconstruction of elliptic tracking.

(A) Reconstructed eye position superimposed on experimental 3D eye position in color (reconstructed torsion, thick gray trace; sinusoidal fit, black trace; reconstructed vertical and horizontal position, white traces; sinusoidal fits not shown). (B) Reconstructed angular eye velocity superimposed on experimental 3D angular eye velocity in color (reconstructed torsion, thick gray trace; sinusoidal fit, black trace; reconstructed vertical and horizontal angular velocity, white traces; sinusoidal fits not shown). Torsional, vertical and horizontal experimental data shown in red, green and blue, respectively; the minimal least-squares sinusoidal fits of vertical and horizontal eye position and angular eye velocity hardly differed from the reconstructed traces. For more details, see text.

Figure 8. Tilting of angular eye velocity rotation planes during elliptic tracking.

(A) Tracking of targets in clockwise (cw) and counterclockwise (ccw) direction along elliptic trajectories with major axes aligned with the head-horizontal and (B) head-vertical plane (ellipse major axis 20°, minor axis 10°). Traces illustrate tilt angles of rotation planes of reconstructed angular velocity (in black), Donders-Listing angular velocity (in dark gray) and sinusoidal-fitted angular velocity (in light gray). The disparity between reconstructed and Donders-Listing rotation planes is due to meridian rotations along the elliptic trajectory. The rotation planes of the sinusoidal-fitted angular velocities show large overshooting of the 5° and 10° levels predicted by the half-angle rule, particularly during tracking along horizontal elliptic trajectories; several single trials superimposed. Abscissa: phase 0°, up gaze; phase +90°, rightward gaze position (rotation sense of targets as seen from the subject).

Figure 9. Ratio of counter-roll to roll angular velocity.

Comparison of reconstructed versus experimentally estimated counter-roll to roll ratios during horizontal (A) and vertical (B) elliptic target tracking. Upper panels: Counter-roll to roll ratios obtained from single trials for three different elliptic eccentricities plotted against the angular eccentricity ε of the gaze line by superimposing reconstructed (black lines) on experimentally estimated data (gray lines). Dashed vertical lines indicate the extreme vertices of elliptic trajectories (e = 0.66, 0.87 and 0.97; a = 20°, semi-major axis, b = 15°, 10°, and 5°, semi-minor axes). The single black curve displays the curve , extending from 1 at ε = 0° to 1.15 at ε = 30°. Middle panels: Counter-roll to roll ratios as above plotted against tracking phase. Note increasing depth of modulation with increasing elliptic eccentricity, particularly during horizontal elliptic tracking. Abscissa: phase 0°, up gaze; phase +90°, rightward gaze position. Bottom panels: Torsional target velocity along the three elliptic trajectories with eccentricities 0.97, 0.87, and 0.66 plotted against phase angle. Dashed line indicates average torsional angular velocity across the three trajectories (36°/s).