Spatiotemporal properties of microsaccades: Model predictions and experimental tests

Models for saccadic motion and postsaccadic oscillations

In a recent letter [S. Bouzat et al., Phys. Rev. Lett. 120, 178101 (2018)10.1103/PhysRevLett.120.178101], a mathematical model for eyeball and pupil motion was developed allowing for the understanding of the postsaccadic oscillations (PSO) as inertial effects. The model assumes that the inner part of the iris, which defines the pupil, moves driven by inertial forces induced by the eyeball rotation, in addition to viscous and elastic forces. Among other achievements, the model correctly reproduces eye-tracking experiments concerning PSO profiles and their dependence on the saccade size. In this paper we propose various extensions of the mentioned model, we provide analytical solutions, and we perform an exhaustive analysis of the dynamics. In particular, we consider a more general time dependence for the eyeball velocity enabling the description of saccades with vanishing initial acceleration. Moreover, we give the analytical solution in terms of hypergeometric functions for the constant parameter version of the model and we provide particular expressions for some cases of interest. We also introduce a new version of the model with inhomogeneous viscosity that can improve the fitting of the experimental results. Our analysis of the solutions explores the dependence of the PSO profiles on the system parameters for varying saccade sizes. We show that the PSO emerge in critical-like ways when parameters such as the elasticity of the iris, the global eyeball velocity, or the saccade size vary. Moreover, we find that the PSO profiles with the first overshoot smaller than the second one, which are usually observed in experiments, can be associated to parameter regions close to criticality.

Inertial Movements of the Iris as the Origin of Postsaccadic Oscillations

Recent studies on the human eye indicate that the pupil moves inside the eyeball due to deformations of the iris. Here we show that this phenomenon can be originated by inertial forces undergone by the iris during the rotation of the eyeball. Moreover, these forces affect the iris in such a way that the pupil behaves effectively as a massive particle. To show this, we develop a model based on the Newton equation on the noninertial reference frame of the eyeball. The model allows us to reproduce and interpret several important findings of recent eye-tracking experiments on saccadic movements. In particular, we get correct results for the dependence of the amplitude and period of the postsaccadic oscillations on the saccade size and also for the peak velocity. The model developed may serve as a tool for characterizing eye properties of individuals.