Fig 1.
Left: Frontal view of the extra-ocular muscles of the right human eye, rotated by about 20 deg leftward from the primary (+x) direction—MR (Medial Rectus), LR (Lateral Rectus), SR (Superior Rectus), IR (Inferior Rectus), SO (Superior Oblique), IO (Inferior Oblique). Right: Schematic of the muscle rotation axes, when starting from the primary direction. Right-hand (x, y, z) convention. The ML/LR muscles mainly rotate the eye horizontally (Z-direction). The IR/SR, and the IO/SO muscle pairs induce both vertical (y) and cyclotorsional (x) rotation components. The superior and inferior obliques (SO/IO) divert their pulling trajectories medially: the IO is attached to the medial side of the orbit, while the SO is diverted through a medial pulley, called the trochlea.
Fig 2.
Mechanical robotic system that serves as a biomimetic prototype for the human eye in Fig 1.
The scaled eye represented by the spherical segment (7 cm radius) with a camera in the front is driven by elastic tendons (in red) that correspond to the six extra-ocular muscles shown in Fig 1. Each pair of muscles are connected to attachment points on metal rods fixed to the motors(U1-U3), with the rotation axes shown by the green dotted lines. The SO,IO,IR and SR muscles pass through fixed pulleys between the attachment points on the eye and the motor rods to adjust their pulling directions.
Fig 3.
Listing’s law for ±3500 eye positions during spontaneous saccades in the light, of a monkey.
All eye orientations lie in a well-defined plane, which is perpendicular to the torsional direction, and defines the primary position (cross at (0,0,0)). The center of the oculomotor range is, for this animal, about 20 deg down from primary position, and the animal’s head was pitched downward by 15 deg. (A) Frontal view of LP. (B) Side view of LP. (C) Expanded side view of LP. After: [18].
Fig 4.
(A) Example velocity profiles for saccades of different amplitudes.
Note that their peak velocity is reached at the same moment. (B) The amplitude-peak velocity relation (ordinate: 0–700 deg/s) for fast and very slow (diazepam-affected and fatigued) saccades (different symbols). The red curve delineates the fastest saccades possible for this human subject. Oblique lines separate sectors where saccades have a particular skewness. The lower the slope, the slower the saccades, and the larger its skewness. (C) Tight relation for saccades between amplitude and the product of peak velocity and duration. This relation expresses the fact that typical eye-saccades have single-peaked, ‘triangular-shaped’ velocity profiles. After: [43].
Fig 5.
Schematic of the mechanical model of Fig 2.
The common driving signals, u1,2,3, are symbolized by the grey bars, that rotate around the red axes, thus providing an antagonistic contraction/relaxation force on the associated muscles (cables with same colours) from their end-insertion points at Pi (green dots). The six muscle insertion points, Qi, on the eye (red dots) form the (elliptical) blue surface; C is the center of the globe at (0,0,0). The default arrangement is nearly symmetrical with respect to the horizontal plane of the eye at z = 0.
Fig 6.
Block diagram of 3D eye saccade simulator.
Close to each block is a summary description and the equation numbers that implement the associated functions (when applicable).
Table 1.
Coordinates of the insertion points (in cm) on the mechanical model of the right eye at rest (Q0,i), and the craniocentric via points (‘pulleys’) (Xi) with respect to the (head-fixed) center of the eye.
Here, LR and MR have no pulleys; they directly connect to driver 2. The cranial insertion points, Pi,0 at rest are given in columns 8–10. Columns 11–13: the unit torque-vector components, as computed from Eq (18) with the eye in the resting orientation. Note that the SO, IO, SR and IR tendons all pull with a prominent cyclo-torsional (x) and vertical (y) component.
Fig 7.
3D oculomotor range of the model in two planar views (yz-plane, left and xz-plane, right), when the inputs ran from [-35, +35] deg for each of the three muscle pairs.
Note that the torsional range of the model is substantial, and quite comparable to the horizontal and vertical ranges.
Table 2.
Six different cost functionals, all optimized to control saccades by tuning the weights, λα for the contributing terms.
Each total cost includes the accuracy cost, but differs in the contribution from other kinematic terms. AD: accuracy and duration; AE: accuracy and energy; AED: accuracy, energy and duration; AEDL1: AED with Listing’s law at saccade offset only; AEDL2: AED with Listing’s law for the entire trajectory. The weights are the (two to four) free parameters for the optimal control.
Fig 8.
(A) Example of minimizing the total movement cost, JAED (purple curve), as function of the simulated saccade duration for a 10 deg rightward saccade. In this case, the optimum is reached at a duration of 50 ms (circular symbol). (B) Illustration of the first 150 goals for the saccade planner in degrees. The simulator started at the straight-ahead fixation point at (0,0), with the eye in the (assumed) primary position, at r0 = 0, in a random direction, and any subsequent saccade started (black dot) from the end point of its predecessor (red symbol).
Table 3.
Results for the different optimization strategies specified in Table 2.
n: normal to the best-fit plane through the rotation-vector data, and absolute angle between n and the straight-ahead direction, P = [1, 0, 0]; Width LP: standard deviation of the data around the best-fit plane. MS: nonlinear main-sequence data fit on all saccade data: V0: asymptotic velocity (deg/s); α: angular constant (in deg); rCS: component stretching correlation of vpk,ΔH=8o(Φ) vs. cos(Φ) (cf. Fig 11B).
Fig 9.
3D oculomotor behavior and saccade dynamics, resulting from minimizing the AED cost.
(A,B) 3D trajectories in the (y,z) and (x,z) views. Although the torsional range is constrained to a width of σx = 4.88 deg, the eye is not confined to Listing’s Plane. (C) Saccade dynamics. Black dots: data from 1500 saccades starting from randomized initial positions in randomized directions. When starting from straight-ahead, saccades follow a clear saturating amplitude-peak velocity relation (red dots: purely horizontal saccades, blue dots: purely vertical saccades; green dots: 45 deg oblique saccades.) Note that vertical saccades are slowest and horizontal saccades are fastest. (D) All trajectories were nearly straight, as most correlations between the horizontal and vertical velocity profiles were near 1.0.
Fig 10.
Same data as in Fig 9, but rotated by Rp (Eq 32), so that it is expressed in the reference frame in which the primary position (the normal to the best-fit plane) coincides with P = [1, 0, 0].
This is defined as Listing’s reference frame. Compare to Fig 9B.
Fig 11.
Component stretching in oblique saccades (after JAED minimization).
In these examples, the horizontal component was kept fixed at 8.0 deg, while the vertical component varied from 0 to 30 deg in 5 deg steps. (A) Horizontal and (inverted) vertical velocity profiles of the oblique saccades. Note that the velocity profiles approximately match in duration. This is especially clear for the larger (≥15 deg) vertical components (green, red, cyan and yellow traces). (B) Peak velocity of the horizontal component depends on the saccade direction. Dashed line: cosine prediction for a common-source vectorial drive ([47] see also Table 3).
Fig 12.
(A,B) 3D oculomotor behavior and (C,D) saccade dynamics, resulting from minimizing the AED cost and penalizing deviations from Listing’s Plane at saccade offset.
Now, the torsional range is constrained to a width of only σx = 1.7 deg (B), which keeps the eye close to Listing’s Plane, also during the saccade trajectories. (C) The saccade dynamics still follow the same saturating amplitude-peak velocity relation as in Fig 9C, and nearly straight trajectories (D), as there is clear component stretching.
Fig 13.
(A,B) 3D oculomotor behavior and (C,D) saccade dynamics, after minimizing the AED cost and the total force on the eye during fixation.
Default case: Δz = 0. Now, the torsional range is constrained to a width of only σx = 1.25 deg (B), which is closely in line with Listing’s Plane. The saccade dynamics follow a saturating amplitude-peak velocity relation (C). Also, oblique saccades show similar component stretching as in the AED and 3D-target optimizations (D).
Fig 14.
The model produces a tight relationship (r2 = 0.89) between saccade amplitude and the product between peak eye velocity and saccade duration (quantified at 10 ms resolution) for all saccades shown in Fig 13 (cf. Fig 4C).
Fig 15.
Eye orientations in the xz-plane for all saccades, for three different vertical shifts, Δz, of the cranial muscle insertion points with respect to the horizontal midplane of the eye.
Note that the orientation of LP tilts as a function of the shift.
Fig 16.
The pitch angle of Listing’s plane varies systematically with the craniocentric insertion points of the muscles re. eye’s horizontal plane.
Negative values: downward pitch angle.
Fig 17.
The total model encompasses two linked (nonlinear) components: the sequence of motor commands, Ut, and the 3D eye plant, H.
The plant’s output (described by 3D eye orientation, rt, and angular velocity, ωt) is used to tune the cost function, JForce(r, ω), that drives the motor outputs. The system’s behavior follows the dynamics and kinematics of real saccades (insets, right). Any change in the plant’s parameters will require corresponding changes in the motor outputs, ut. rD: 2D target location. The thin feedback arrow represents the training phase during which eye movements recurrently train the motor outputs to minimize the imposed cost.