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.

Figure 1

Pupil motion. (a) Family of saccades with constant $A=0.06$, $\gamma =0.15$, $k=0.032$, and varying $x_m$. (b) Detail of eyeball $x(t)$ and pupil $x(t)+y(t)$ positions during saccades of sizes $x_m=7$ and $x_m=15$ calculated for $A=0.05$, $\gamma =0.1$, and $k=0.035$. (c) Maximal velocity vs $x_m$. The red solid and the blue dotted lines correspond to families of saccades with constant $A$, $\gamma$, $k$ for the values indicated [the red solid line corresponds to the saccades in (a)]. The white squares are results from (single) saccades from the same observer for a left eye moving to the left, taken from experiments in [14]. The black circles are our estimations using data recovered from the mean saccades shown in Fig. 2 in [3] (pupil-corneal reflection signal) for the case of observer 3, left eye, abduction. The green dashed line corresponds to the model with force-dependent parameters presented at the end of the Letter, with parameters as in Fig. 3. The dash-dotted segment indicates the $x_m^{1/2}$ behavior.

Figure 2

Amplitude and period of the PSO. (a) PSO amplitude as a function of $x_m$. The black solid line is for $A=0.06$, $\gamma =0.15$, $k=0.032$ [set in Fig. 1], the red dashed line is for $A=0.036$, $\gamma =0.15$, $k=0.032$, and the green dotted line is for $A=0.06$, $\gamma =0.15$, $k=0.05$. The blue dash-dotted line is for the model with force-dependent parameters with the values used in Fig. 3. The arrows indicate the value of $x_m$ for which $T_F$ and $T_{\mathrm{\Omega }}$ match. (b) PSO period as a function of $x_m$ for the same calculations as in (a). The inset sketches the method for calculation of the PSO amplitude and period. (c) Inertial force $-\ddot{x}(t)$ for three values of $x_m$ from the set in the black solid line in (a). (d) $T_f$ and $T_{\mathrm{\Omega }}$ as functions of $x_m$ for the set in (c).

Figure 3

Fitting families of saccades from experiments. (a),(b) Open circles correspond to data recovered from experiments in [3] for the case indicated in Fig. 1. The solid red curves in (a) are our calculations for the model with constant parameters $A=0.04$, $k=0.032$, $\gamma =0.15$ for various values of $x_m$, while those in (b) are for the model with force-dependent parameters with $A=0.036$, $k_0=0.04$, ${\gamma }_0=0.14$, $c=0.5$, $d=3$. For the sake of completeness, in both (a) and (b), we show calculations for $x_m=16$ although there are not experimental data for such saccade size.