Modelling 3D saccade generation by feedforward optimal control

An interesting problem for the human saccadic eye-movement system is how to deal with the degrees-of-freedom problem: the six extra-ocular muscles provide three rotational degrees of freedom, while only two are needed to point gaze at any direction. Measurements show that 3D eye orientations during head-fixed saccades in far-viewing conditions lie in Listing’s plane (LP), in which the eye’s cyclotorsion is zero (Listing’s law, LL). Moreover, while saccades are executed as single-axis rotations around a stable eye-angular velocity axis, they follow straight trajectories in LP. Another distinctive saccade property is their nonlinear main-sequence dynamics: the affine relationship between saccade size and movement duration, and the saturation of peak velocity with amplitude. To explain all these properties, we developed a computational model, based on a simplified and upscaled robotic prototype of an eye with 3 degrees of freedom, driven by three independent motor commands, coupled to three antagonistic elastic muscle pairs. As the robotic prototype was not intended to faithfully mimic the detailed biomechanics of the human eye, we did not impose specific prior mechanical constraints on the ocular plant that could, by themselves, generate Listing’s law and the main-sequence. Instead, our goal was to study how these properties can emerge from the application of optimal control principles to simplified eye models. We performed a numerical linearization of the nonlinear system dynamics around the origin using system identification techniques, and developed open-loop controllers for 3D saccade generation. Applying optimal control to the simulated model, could reproduce both Listing’s law and and the main-sequence. We verified the contribution of different terms in the cost optimization functional to realistic 3D saccade behavior, and identified four essential terms: total energy expenditure by the motors, movement duration, gaze accuracy, and the total static force exerted by the muscles during fixation. Our findings suggest that Listing’s law, as well as the saccade dynamics and their trajectories, may all emerge from the same common mechanism that aims to optimize speed-accuracy trade-off for saccades, while minimizing the total muscle force during eccentric fixation.

V1 neurons encode the perceptual compensation of false torsion arising from Listing’s law

We try to deploy the retinal fovea to optimally scrutinize an object of interest by directing our eyes to it. The horizontal and vertical components of eye positions acquired by goal-directed saccades are determined by the object’s location. However, the eccentric eye positions also involve a torsional component, which according to Donder’s law is fully determined by the two-dimensional (2D) eye position acquired. According to von Helmholtz, knowledge of the amount of torsion provided by Listing’s law, an extension of Donder’s law, alleviates the perceptual interpretation of the image tilt that changes with 2D eye position, a view supported by psychophysical experiments he pioneered. We address the question of where and how Listing’s law is implemented in the visual system and we show that neurons in monkey area V1 use knowledge of eye torsion to compensate the image tilt associated with specific eye positions as set by Listing’s law.

V1 neurons encode the perceptual compensation of “false torsion” arising from Listing’s law

We try to deploy the retinal fovea to optimally scrutinize an object of interest by directing our eyes to it. Horizontal and vertical components of these fixation eye movements are determined by the object’s location. However, fixation eye movements also involve a torsional component, which according to Listing’s law is fully determined by the 2D eye position acquired. According to Von Helmholtz knowledge of the torsion provided by this law alleviates the perceptual interpretation of the image tilt that changes with fixation, a view supported by psychophysical experiments he pioneered. We address the question where and how Listing’s law is implemented in the visual system and we show that neurons in monkey area V1 use knowledge of torsion to compensate the image tilt associated with specific eye positions as set by Listing’s law.

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.

Revealing the Kinematics of the Oculomotor Plant With Tertiary Eye Positions and Ocular Counterroll

Retinal information is two-dimensional, whereas eye movements are three-dimensional. The oculomotor system solves this degrees-of-freedom problem by constraining eye positions to zero torsion (Listing's law) and determining how eye velocities change with eye position (half-angle rule). Here we test whether the oculomotor plant, in the absence of well-defined neural commands, can implement these constrains mechanically, not just in a primary position but for all eye and head orientations. We stimulated the abducens nerve at tertiary eye positions and when ocular counterroll was induced at tilted head orientations. Stimulation-induced eye velocities follow the half-angle rule, even for tertiary eye positions, and microstimulation at tilted head orientations elicits eye positions that adhere to torsionally shifted planes, similar to naturally occurring eye movements. These results support the notion that oculomotor plant can continuously apply these three-dimensional rules correctly and appropriately for all eye and head orientations that obey Listing's law, demonstrating a major role of peripheral biomechanics in motor control.

Understanding the Extraocular Muscles and Oculomotor, Trochlear, and Abducens Nerves Through a Simulation in Physical Examination Training: An Innovative Approach*

Purpose: The purpose of this study was to investigate the effect of an innovative exhibitory eye model simulation in a physical examination laboratory format on explaining Listing's Law concerning the individual extraocular muscle action and the rationale for cranial nerve testing. Methods: Participants were 71 volunteers in the third quarter of a chiropractic training program. The study involved a specially designed eyeball model used to explain the movements of individual extraocular muscles based on Listing's law and their cranial innervations in conjunction with the physical examination. Pre- and post-written tests were used to assess participants' understanding of the subjects taught. The test results were compared with those of nonparticipants who also took the same pre- and posttests. Results: An independent samples t-test of the posttest showed a significant difference between the groups. The study group students achieved higher scores than their counterparts in the control group. Conclusion: Using an innovative approach to explain Listing's law and rationale for cranial nerve tests can improve physical examination skill and help produce more effective written test results.