Dynamics of Eye Movements Under Time Varying Stimuli

Abstract

In this paper we study the pure-slow and pure-fast dynamics of the disparity convergence of the eye movements second-order linear dynamic mathematical model under time varying stimuli. Performing simulation of the isolated pure-slow and pure-fast dynamics, it has been observed that the pure-fast component corresponding to the eye angular velocity displays abrupt and very fast changes in a very broad range of values. The result obtained is specific for the considered second-order mathematical model that does not include any saturation elements nor time-delay elements. The importance of presented results is in their mathematical simplicity and exactness. More complex mathematical models can be built starting with the presented pure-slow and pure-fast first-order models by appropriately adding saturation and time-delay elements independently to the identified isolated pure-slow and pure-fast first-order models.

Introduction

Studying dynamics of eye movements plays an important role in the development of various eye therapies (Alvarez, 2015) and provides useful information about understanding of neurological processes and the human brain function, (Kennard & Leigh, 2008; Leigh & Zee, 2006). Modeling of the disparity convergence has been studied in several papers (Alvarez, Bhavsar, Semmlow, Bergen, & Pedrono, 2005; Alvarez, Semmlow, & Pedrono, 2005; Alvarez, Semmlow, & Yuan, 1998; Alvarez, Semmlow, Yuan, & Munoz, 1999; Horng, Semmlow, Hung, & Ciuffreda, 1998; Hung, 1998; Hung, Semmlow, & Ciufferda, 1986; Jiang, Hung, & Ciuffreda, 2002; Khosroyani & Hung, 2002). In some studies (Alvarez, Jaswal, Gohel, & Biswal, 2014; Kim, Vicci, Granger-Donetti, & Alvarez, 2011; Lee, Semmlow, & Alvarez, 2012; Radisavljevic-Gajic, 2006) different problem formulations are used. Some of the papers have observed experimentally and analytically the presence of the slow and fast eye movement dynamics, (Alvarez et al., 1998; Hung et al., 1986; Jiang et al., 2002; Khosroyani & Hung, 2002; Lee et al., 2012; Radisavljevic-Gajic, 2006). The analytical observation was made using the corresponding secondorder mathematical model (Alvarez et al., 1999; Horng et al., 1998).

This paper is a continuation of our previous paper (Radisavljevic-Gajic, 2006), originally done for the constant eye stimuli using the second-order dynamic mathematical model derived in Alvarez et al. (1999). For the model of Alvarez et al. (1999), we perform exact mathematical analysis with the goal to isolate the slow and fast components, and present simulation results for the case of time varying eye stimuli since they produce some interesting phenomena not previously observed for the case of constant eye stimuli (Radisavljevic-Gajic, 2006). The im-

The simulation results of (15)–(16) and (21)–(22), assuming zero initial conditions, that is, x₁(0) = 0 and x₂(0) = 0, are presented in Figure 1, Figure 2, Figure 3 and Figure 4.

Discussion of the Obtained Simulation Results

The dynamic responses of the pure-slow x_s(t) and pure-fast x_f(t) variables in the time interval of 10 seconds are presented in Figure 1. It can be observed from this picture that the eye stimuli in the range of 10^° to 30^°, due to amplification at steady state as given by (20) generate the pure-fast component in the range from 4.4645 × 10^° = 44.645^° to 4.4645 × 30^° = 133.9353^°. The same figure shows that the pure-slow component remains in the same range as the input signal, that is, from 10^° to 30^°, due to the fact that the pure-slow subsystem steady state gain is G_s = 1.

The eye position in the original coordinates x₁(t), and the eye original coordinates angular velocity (the time rate of the position change) x₂(t) are plotted in Figure 2. It should be observed that the first peak of x₂(t) is around 116^°/s and that the follow up peaks are around 79^°/s. This is caused due to different initial conditions at t = 0 and t = 4,8. We started simulation with zero initial conditions, that is, for the first period the initial conditions are x₁(0) = 0^° and x₂(0) = 0^°. For the second period, the initial conditions obtained from formulas (21) and (22) are non-zero and given by

In addition, due to the input signal decrease from 30^° to 10^°, the fast component takes a large negative value of ≈−67^°/s. Hence, during the half period of two seconds, the eye angular velocity changes very drastically, from positive 79^°/s to negative ≈−67^°/s, producing the absolute angular velocity change of ≈146^°/s.

The slow variable (eye position) x₁(t) and its pureslow and pure-fast components are presented in Figure 3. Due to the fact that the slow variable is dominated by its pure-slow component and that it has a negligible contribution of the pure-fast component, the figure shows that practically x₁(t) ≈ x_1s(t), could have been also verified analytically using formula (21).

Much more interesting situation is with the fast variable x₂(t) that represents the eye angular velocity, see Figure 4. It can be seen from this figure that its pure-fast component x_2f(t) very quickly, in several milliseconds, reaches steady state with a maximum value of around $x_{2f}^{\max }$ = 140^°/s. When the stimuli changes instantly from 30^° to 10^°, the variable x_2f(t) drops within several milliseconds to a little bit below $x_{2f}^{\min }$ = 50^°/sec. On the other hand, the pure-slow component x_2s(t) goes in the opposite direction and reaches in less than a second $x_{2s}^{\min }$ = −130^°/s. These two components form x₂(t) = x_2s(t)+ x_2f(t) and together produce at steady state ≈10^°/s for the eye angular velocity. Without the pure-slow/pure-fast decomposition, one would not be able to see these violent component in the disparity convergence of the eye movement dynamics.

Conclusions

It was shown that the fast component of the eye dynamics displays very fast and abrupt changes due to considered time varying stimuli as demonstrated in Figure 2 and Figure 4. The angular velocity, due to the change of the stimuli force of 20 degrees (from to 30° to 10°), displays large variations of more than 140°/s, as shown in Figure 4. This large change could have been restricted by introduction of a saturation element. However, that will lead to a new nonlinear mathematical model different than the linear second-order mathematical model considered in this paper. Such nonlinear models are not the subject of this paper, and they will be interesting for future research.