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.
Author summary 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.