The nonlinearity of passive extraocular muscles

Realistic 3D human saccades generated by a 6-DOF biomimetic robotic eye under optimal control

We recently developed a biomimetic robotic eye with six independent tendons, each controlled by their own rotatory motor, and with insertions on the eye ball that faithfully mimic the biomechanics of the human eye. We constructed an accurate physical computational model of this system, and learned to control its nonlinear dynamics by optimising a cost that penalised saccade inaccuracy, movement duration, and total energy expenditure of the motors. To speed up the calculations, the physical simulator was approximated by a recurrent neural network (NARX). We showed that the system can produce realistic eye movements that closely resemble human saccades in all directions: their nonlinear main-sequence dynamics (amplitude-peak eye velocity and duration relationships), cross-coupling of the horizontal and vertical movement components leading to approximately straight saccade trajectories, and the 3D kinematics that restrict 3D eye orientations to a plane (Listing's law). Interestingly, the control algorithm had organised the motors into appropriate agonist-antagonist muscle pairs, and the motor signals for the eye resembled the well-known pulse-step characteristics that have been reported for monkey motoneuronal activity. We here fully analyse the eye-movement properties produced by the computational model across the entire oculomotor range and the underlying control signals. We argue that our system may shed new light on the neural control signals and their couplings within the final neural pathways of the primate oculomotor system, and that an optimal control principle may account for a wide variety of oculomotor behaviours. The generated data are publicly available at https://data.ru.nl/collections/di/dcn/DSC_626870_0003_600.

A Novel Technique Using Paretic Superior Rectus as a Globe Suspender for Monocular Elevation Deficiency

Surgical innovations in strabismus provide opportunities to improve visual function, eye alignment, and cosmesis in rare pediatric ophthalmological conditions. Monocular elevation deficiency is a rare and multifactorial disease in which the affected eye is equally limited in terms of elevation during adduction and abduction. We aimed to present a novel procedure for the treatment of acquired monocular elevation deficiency using the paretic superior rectus muscle as a globe suspender to resolve hypotropia. We report the case of an eight-year-old girl with left eyelid ptosis and hypotropia two months after draining a left orbital abscess. Left inferior rectus muscle recession was performed at five years, with residual left hypotropia. Ophthalmological examination revealed a best-corrected visual acuity of 20/20 OD and 20/100 OS. Severe left eyelid ptosis and poor levator function were also observed. Extraocular motility showed left hypotropia of 40 prism diopters with the left superior rectus muscle under action (-4) in the adduction and abduction positions. A force duction test negative for restrictions on the inferior rectus muscle was performed intraoperatively. To reduce the risks of the Knapp procedure, the left superior rectus muscle was split into medial and temporal halves. Double-armed sutures were secured in half, and the halves were detached from the sclera. The medial and temporal halves were reattached anteriorly to the medial and lateral rectus insertions, respectively. Eight weeks after surgery, the patient had nine prism diopters of hypotropia in the primary gaze. Ten weeks after surgery, there was no change in visual acuity. In the cover test, the patient exhibited residual left hypotropia of nine prism diopters with a restriction (-4) of elevation in adduction and abduction. The parents were pleased with the satisfactory cosmetic outcomes, and postoperative clinical photographs of the patient showed improved hypotropia and persistent minimal elevation of the left eye during adduction and abduction. Superior rectus muscle splitting and vertical transposition to the medial and lateral rectus could be safer and simpler alternatives to the Knapp procedure and may offer a lower risk of anterior segment ischemia. Further studies are required to confirm these findings.

Dynamics of plant mechanics

Muscle and plant dynamics are most important during the high acceleration of saccades. Models have been developed to characterize muscle and plant dynamics. Building these models require an understanding of the length-tension (elastic) and force-velocity (viscous) relationships. Much work has been done to characterize these nonlinear functions, as they are influenced by innervation. However, the active force generator (active-state tension) in the muscle is still poorly understood. Thus, these models serve more to reveal where new studies of muscle behavior are needed than to explain what happens during a saccade.

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.

Saccades are rapid eye movements that humans and other animals perform three to four times per second to scan and perceive the environment around them. These movements orient the eye in space with high precision and in a highly stereotyped fashion. Existing studies on animal models advocate that both mechanical and neuronal functions play an important role in the control of the saccades, but some facts are still not fully understood due to difficulties in experimenting and measuring the variables in living animals. Instead, robots are computational and physical models of reality that expose all its variables and can be programmed in interpretable ways. We have built a robotic model of an artificial eye containing the basic ingredients of human eyes: full 3D rotations, viscous friction and 6 muscle-like actuators connected to the eyeball in a geometry similar to the biological system. By synthesizing robotic eye control systems we found that important characteristics of the movements become similar to human saccades when the control relies on few simple fundamental principles: the maximization of saccade accuracy and the minimization of saccade duration, energy in control, and force in the muscles during fixation.

Dynamics of abducens nucleus neurons in the awake mouse

The mechanics of the eyeball and orbital tissues (the "ocular motor plant") are a fundamental determinant of ocular motor signal processing. The mouse is used increasingly in ocular motor physiology, but little is known about its plant mechanics. One way to characterize the mechanics is to determine relationships between extraocular motoneuron firing and eye movement. We recorded abducens nucleus neurons in mice executing compensatory eye movements during 0.1- to 1.6-Hz oscillation in the light. We analyzed firing rates to extract eye position and eye velocity sensitivities, from which we determined time constants of a viscoelastic model of the plant. The majority of abducens neurons were already active with the eye in its central rest position, with only 6% recruited at more abducted positions. Firing rates exhibited largely linear relationships to eye movement, although there was a nonlinearity consisting of increasing modulation in proportion to eye movement as eye amplitudes became small (due to reduced stimulus amplitude or reduced alertness). Eye position and velocity sensitivities changed with stimulus frequency as expected for an ocular motor plant dominated by cascaded viscoelasticities. Transfer function poles lay at approximately 0.1 and 0.9 s. Compared with previously studied animal species, the mouse plant is stiffer than the rabbit but laxer than cat and rhesus. Differences between mouse and rabbit can be explained by scaling for eye size (allometry). Differences between the mouse and cat or rhesus can be explained by differing ocular motor repertoires of animals with and without a fovea or area centralis.