Upper and Lower Eyelid Saccades Describe a Harmonic Oscillator Function

Abstract

purpose. To determine whether experimentally measured upper and lower eyelid saccades can be fitted to a mathematical function.

methods. A charge-coupled device video camera connected to a personal computer was used to record upper and lower eyelid saccades accompanying 20° and 40° of vertical eye rotation in 19 normal adult subjects. Movement analysis was performed with software that calculated the center of a blue spot in each frame. The damped harmonic oscillator model was used to fit all saccadic functions obtained.

results. All downward and upward saccades of both upper and lower eyelids were fitted with the underdamped solution of the model with correlation coefficients ranging from 0.980 to 0.999 (mean = 0.995). It was possible to measure maximum velocity at any time, amplitude, and duration of the saccade movements. For the upper eyelid, downward saccades were faster than upward saccades, a difference that was not observed for the lower eyelid. For both the upper and lower eyelids, the velocity of upward and downward movements reached a peak at approximately 0.05/0.06 second and then decreased. For both the upper and lower eyelid saccades, there was good linear correlation between amplitude and velocity. Overshoots were detected in the downward saccades of both lids.

conclusions. Normal upper and lower saccades are described by functions that are extremely well fitted by the underdamped solution of the harmonic oscillator model. Overshooting is a typical feature of normal downward saccades and can be explained by the elastic properties of the tissues.

The term eyelid saccade was introduced by Becker and Fuchs ¹ to describe the eyelid movements that accompany vertical eye saccades. The assessment of this type of lid movement is an important step in the examination of conditions that affect the eyelid function, such as blepharoptosis, cicatricial lagophthalmos, lower eyelid entropion, and Graves’ upper eyelid retraction. 

Many different methods have been used to record eyelid movements, including the use of lever arms, ² high-speed cinephotography, ³ light reflection, ⁴ and the magnetic search-coil technique. ^{5 6 7 8}  

One problem in interpreting and comparing data on eyelid saccades is the lack of a theoretical mathematical model of the dynamics of these movements. The establishment of such a model is important because it would allow the calculation of any parameters of the movement dynamics in a standard manner. 

In the present report, we describe the use of a mathematical model for analyzing data from these movements that were experimentally obtained for both upper and lower eyelids. 

Methods

The research adhered to the tenets of the Declaration of Helsinki. 

Subjects

Upper and lower eyelid saccades were recorded in 19 eyes of 19 normal adult subjects. The sample comprised 8 men and 11 women (age range, 24–30 years; mean, 27.89 ± 1.59 [SD]). The mean midpupil upper eyelid distance was 3.74 ± 0.77 mm (SD). For the lower eyelid the mean midpupil lid distance was 6.37 ± 0.72 mm. 

Saccade Recording

Eyelid motion during downward and upward eyelid saccades was recorded with a charge-coupled device (CCD) camera connected to a computer by a frame grabber (Picolo). The camera’s temporal resolution was the standard NTSC (30 Hz or 30 frames per second; National Television Systems Committee). Movement analysis was performed with software that calculated in real time the center of a blue spot in each frame. To provide the signal for the software, a small piece of blue paper (0.0113 g) was attached to the eyelashes of the central portion of the upper eyelid. A similar spot was made on the skin of the lower eyelid margin with blue makeup. 

Procedure

With the head stabilized on a chin rest, each subject faced a vertical bar located at a distance of 57 cm. This bar had two red light spots 20° above and below a central white light. To produce saccadic lid movements, the central light was aligned with the subject’s visual axis in the primary position of gaze, and the subject was instructed to look down and up at the bottom red light (20° of eye rotation in the lower hemifield) and up and down between the two red lights (40° of eye rotation across the upper and lower hemifields). 

Data Collection and Analysis

The data points of the records of downward and upward saccades were analyzed graphically and the resultant functions were fitted using the equations of the damped harmonic oscillator model, which allowed the calculation of the maximum and final amplitude, the velocity at any time, and the duration of each saccadic movement. 

Damped Harmonic Oscillator Model

Several studies have shown that during upper eyelid downward saccades the lid transiently reaches a position lower than the final lid position (Fig. 1) . Thus, a theoretical model of the dynamics of these movements must include this oscillation, also known as overshoot, as an important aspect of the movement. 

The simplest mathematical model describing this situation corresponds to a damped harmonic oscillator. According to this model, the displacement of the eyelid from the final position Y(t) changes with time, satisfying the differential equation  

\[\frac{d^{2}Y(t)}{dt^{2}}\ {+}\ 2g\ \frac{dY(t)}{dt}\ {+}\ {\omega}^{2}Y(t)\ {=}\ 0\]

, where g is the damping coefficient and ω is the natural angular frequency—that is, the angular frequency of oscillation when no damping is present. The solutions of this equation depend on the relative values of g and ω and on the initial conditions. There are three regimens to be considered: (1) g < ω, known as underdamped oscillation; (2) g = ω, the critically damped oscillation; and (3) g > ω, corresponding to the overdamped case. 

In regimen 1, the solution satisfying the initial conditions Y(t = 0) = A and  

\[V(t\ {=}\ 0)\ {=}\ \frac{dY(t)}{dt}{\vert}\ _{t{=}0}\ {=}\ 0\]

(corresponding to displacement A and null velocity at the initial instant) is given by  

\[Y(t)\ {=}\ Ae^{{-}gt}\left(\mathrm{cos}\ {\Omega}t\ {+}\ \frac{g}{{\Omega}}\mathrm{sin}\ {\Omega}t\ \right)\]

, where  

\[{\Omega}\ {=}\ \sqrt{{\omega}^{2}\ {-}\ g^{2}}.\]

 

We see that the damping term in equation 1has two effects on the solution (equation 2) : Y(t) oscillates at a frequency of Ω/2π (<ω/2π) but with the amplitude of the oscillation being attenuated by the exponential factor over elapsed time. In the other two cases (regimens 2 and 3), the damping term fully dominates, there is no oscillatory behavior, and an exponential decay is essentially obtained. Thus, only in the case of underdamped oscillation (g < ω) is it possible to describe the observed overshoot. Figure 2schematically presents the behavior of this solution and also illustrates the critically damped and overdamped cases. By comparing the two underdamped solutions given in Figure 2 , one notices that, as g approaches ω, the overshoot becomes smaller, and the oscillation beyond it is more suppressed by the damping. 

The underdamped oscillator solution makes it possible to describe some important parameters of our model such as peak velocity, time when the velocity reached the peak, maximal amplitude, final amplitude, and duration. The velocity at an arbitrary instant t is given by  

\[V(t)\ {=}\ \frac{dY(t)}{dt}\ {=}\ {-}\ A\ \frac{{\omega}^{2}}{{\Omega}}\ e^{{-}gt}\ \mathrm{sin}\ {\Omega}t.\]

 

The velocity as a function of time for each of the underdamped solutions illustrated in Figure 2 , is plotted in Figure 3 . 

We can see that the velocity also oscillates at a frequency of Ω/2π; it starts from zero, increases in value (a negative value means that the saccade is downward), reaches a peak at the instant t _M, returning to zero at the instant of time when the maximum overshoot is reached, t ₀ = π/Ω. The amount of overshoot is given by |Y(t ₀)| = Ae ^−πg/Ω. The maximum amplitude of the saccade can therefore be written as  

\[A_{\mathrm{M}}\ {=}\ Y(0)\ {-}\ Y(t_{0})\ {=}\ A(1\ {+}\ e^{{-}{\pi}g/{\Omega}}).\]

 

Note that the final amplitude corresponds to A, since we chose Y = 0 as the final position. For  

\[\frac{2}{3}\ {\omega}\ {\leq}\ g\ {<}\ {\omega}\]

, the movement is almost finished around t > t ₀, so that t ₀ can be considered to measure the duration of the saccade. The absolute value of the mean velocity from the beginning of the saccade to the instant t ₀ is given by  

\[{\bar{V}}\ {=}\ \frac{A_{\mathrm{M}}}{t_{0}}\ {=}\ A\ \frac{{\Omega}}{{\pi}}\ (1\ {+}\ e^{{-}{\pi}g/{\Omega}}).\]

Within this interval, the maximum velocity is reached at the instant when the acceleration  

\[\frac{dV(t)}{dt}\]

equals zero,  

\[t_{\mathrm{M}}\ {=}\ \frac{1}{{\Omega}}\ \mathrm{arc}tg({\Omega}/g)\]

, and its value is given by  

\[V_{\mathrm{M}}\ {=}\ {\vert}V(t_{\mathrm{M}}){\vert}\ {=}\ {-}A{\omega}e^{{-}\ \frac{g}{{\Omega}}\ \mathrm{arc}tg({\Omega}/g)}.\]

We can see that these velocities vary linearly with the amplitude. 

Results

Typical records of downward and upward saccades of both upper and lower eyelids fitted with the underdamped solution are shown in Figure 4 . All downward and upward saccades of both the upper and lower eyelids were extremely well fitted with the underdamped solution of the model ^(Appendix). The correlation coefficients of the model (R ²) for all movements ranged from 0.980 to 0.999 (mean = 0.995). 

As shown in Figure 4 , for downward movements the lids usually reached a position lower than the final lid position. This phenomenon (the maximum amplitude is slightly greater than the final amplitude) is a characteristic finding of the downward lid saccades, known as overshoot. For the upward saccades, the difference between the maximum and final amplitude (undershoot) was less marked. For both upper and lower eyelids, the overshoots were more evident for the larger saccades that accompanied 40° of eye rotation. 

The mean values of the saccade parameters for the downward and upward saccades of both eyelids are shown in Tables 1 and 2 . 

Although there is no difference between the amplitude of upper eyelid downward and upward saccades (paired t-test = 0.14, P = 0.89), the downward saccades were faster than the upward saccades. As shown in Table 1 , for 20° of eye rotation, the maximum velocity of the upward saccades was 34.52 mm/s. For the same amount of eye rotation, the maximum velocity of downward saccades reached 43.23 mm/sec (t = 4.40, P = 0.0003). The same was observed for 40° of eye rotation (upward = 61.41 mm/s and downward = 72.49 mm/s; t = 4.08, P = 0.0007). The difference in velocity between downward and upward saccades of the upper eyelid was not observed for the lower eyelid. 

For both upper and lower eyelid saccades, there was a good linear correlation between amplitude and velocity (Fig. 5) . This relationship explains why the duration of the saccades showed little variation with increasing eye rotation. As shown in Table 1 , the mean duration of the downward upper eyelid saccades for 20° of eye rotation was 0.39 seconds, increasing only 0.08 seconds with 40° of eye rotation. For the lower lid, the duration of the saccades was essentially the same, regardless of the amount of eye rotation or the direction of the movement. 

Figure 6shows a typical example of the evolution of saccade velocity with time. For both the upper and lower eyelids, the velocity of upward and downward movements showed the same characteristics, reaching a peak at ∼0.05/0.06 second and then decreasing to zero. 

Discussion

For clinicians dealing with eyelid disease, eyelid saccades are interesting movements, because they reflect the interaction between the activity of the levator palpebrae superioris (LPS) muscle and passive forces stored in the eyelid. The upper eyelid position in alert subjects results from the tonic activity of the LPS, which stretches the tarsal plate, eyelid ligaments, and orbicularis oculi muscle. In the absence of any LPS muscle function (third nerve palsy) there is complete ptosis. When the eye rotates down, both the superior rectus and LPS are inhibited. The resultant downward movement of the upper eyelid is entirely dependent on the relaxation of the LPS muscle and the passive force generated by the stretched eyelid structures, superposed on the dissipative forces represented by friction between the lid and the eyeball and also by an intrinsic friction of the elements of the system. The upward phase of the saccades results from the contraction of the LPS muscle, which overcomes the downward passive forces to raise the lid. During both phases, the orbicularis oculi muscle remains silent. ⁶  

Our findings agree well with data reported in the previous studies, which have described the metrics of upper eyelid saccades in normal subjects. ^{5 6 7} Downward saccades are faster than upward movements and exhibit an overshoot that takes the eyelid to a position lower than the final one. These oscillations are typically less marked for the upward movements. ⁶  

A point that deserves some comment is the units used in the present study (millimeters and millimeters per second). Vertical eyelid displacements are rotational movements, ⁶ and thus it seems more appropriate to express lid saccades in degrees and degrees per second. However, rotational movements are mathematically related to linear displacements ^(Appendix). Clinicians have always recorded lid movements in terms of linear displacements. All data on levator function are based on millimeters and not degrees. It is doubtful that expressing lid movements as curvilinear displacements increases the precision of the measurement. A study with search coils has demonstrated that coil position is a critical parameter for recording eyelid saccades, suggesting that different parts of the upper eyelid have different rotational axes. ⁷ If it is assumed that 2.4 mm of lid displacement equals 10° of lid rotation, ⁵ our results of downward upper eyelid peak velocities (43.29 mm ± 1.66 for 20° and 72.49 mm ± 3.21 for 40° of eye rotation) are slower than reported by other authors. ^{5 6 7} However, when results of maximum velocities are pooled from different studies, there is a large variation in the data, which suggests that specific experimental and recording conditions strongly influence the absolute velocity values. Therefore, we agree with Guitton et al. ⁵ that absolute values of velocity must be interpreted with caution if they are used as normative data. More interesting is the analysis of the shape of the functions described by the saccadic movements and the study of the relation between the parameters that characterize these functions. 

In the studies published thus far on eyelid saccades, the metrics of the movements were not mathematically modeled, and typically only the maximum amplitude and peak velocity were quantified. ^{1 5 6 7} Overshoots have been identified in several of those studies but were explicitly mentioned for the first time by Evinger et al. ⁶ In one study, the presence of the overshoots was taken as an indicator that the orbicularis muscle was being activated at the end of the downward saccadic movement. ⁷ Our results show that normal upper eyelid saccades are described by functions that are extremely well fitted by the underdamped solution of the harmonic oscillator model. As discussed in detail in the ^Appendix, this model assumes that the overshoot is a normal characteristic of the dynamics of the lid saccades. It reflects an oscillation of the elastic eyelid components that can be large or small, depending on the magnitude of the damped coefficient. The idea that the dynamics of the upper eyelid saccadic movements are strongly influenced by the elastic components of the eyelid tissues was clearly elaborated by Evinger et al. ⁶ The mathematics used in the present study is merely a formal description of the data shown in Figure 13 of Evinger et al. ⁶  

The metrics of the lower eyelid saccades have never been measured. The results of the few old studies that have been performed using millimeter rules were extremely variable. Shore ⁹ measured the vertical excursion of the lower eyelid from extreme infraduction to full upgaze in 188 normal subjects of different ages and found a range of amplitude of 0 to 8 mm. Similar results were published in the 1960s by Fox. ¹⁰  

Our results demonstrated that the lower eyelid saccade metrics can also be described by the underdamped solution of the model. Their peak velocities are also linearly correlated with movement amplitude. Overshoots are detected in the downward movements, and their velocity–time function is similar to the functions obtained for the upper eyelid saccades. 

Lower eyelid saccades are small (almost four times smaller than the upward saccades), stereotyped movements. Their duration does not change when the eye rotation increases from 20° to 40°, and they do not show the asymmetry in velocity displayed by the upper eyelid saccades. Their velocity is essentially the same for upward and downward movements. 

The characteristics of the lower eyelid saccades dynamics can be explained by the fact that these movements are entirely passive. They do not result from the contraction or relaxation of an independent muscle such as the levator. During vertical eye rotations, it is the transmission of forces through the capsulopalpebral fascia that produces the vertical lower eyelid motion. ¹¹  

A fine analysis of the eyelid saccade metrics may be clinically important. It may detect early abnormalities of the levator muscle in Graves’ upper eyelid retraction and also assess the transmission of forces to the lower eyelid in cases of entropion. 

Appendix 1

Linear and rotational displacements are geometrically related. As shown in , if the radius of the sphere is known, the segment of line bc can be related to the arc of curve ba. If the eye is assumed to be a sphere, bc = sinα and ba = Rα, where R is the radius of the eye, with α measured in radians; recall that α(deg) = 180 α(rad)/π. 

 Supported by Consellho Nacional de Pesquisa (CNPq), Brazil.

 Submitted for publication September 14, 2004; revised October 26, 2004; accepted November 7, 2004.

 Disclosure: J.M.C. Malbouisson, None; A.A.V. Cruz, None; A. Messias, None; L.V.O. Leite, None; G.D. Rios, None

 The publication costs of this article were defrayed in part by page charge payment. This article must therefore be marked “advertisement” in accordance with 18 U.S.C. §1734 solely to indicate this fact.

 

Figure 1.

 

Eyelid saccades. Left: Primary position of gaze (t0 and t3). When the eye rotates down (t1) and up (t2), the upper and lower eyelids move synchronically (solid lines) following the pattern shown on the right.

Figure 2.

 

Solutions of the damped harmonic oscillation model. In all cases, we fixed ω at 3 and took as initial conditions Y(0) = A = 1 and V(0) = 0. Dashed and the solid curves: underdamped oscillations with g = 1 and g = 2, respectively; dotted curve: the solution for the critically damped situation (g = ω = 3); dashed-dotted curve: the overdamped solution with g = 6. The units of both g and ω are the inverse of time and are arbitrarily taken as it is for the length unit of A.

Figure 3.

 

Velocity as a function of time corresponding to the two cases of underdamped solutions shown in Figure 1 : ω = 3, Y(0) = A = 1, with g = 1 (dashed line) and g = 2 (solid line).

Figure 4.

 

Typical recordings of upward (left) and downward (right) eyelid saccades accompanying 40° of vertical eye rotation fitted with the underdamped solution of the damped harmonic oscillator model. Top: upper eyelid; bottom: lower eyelid.

Table 1.

 

Upper Eyelid Saccades Parameters

Parameter	20°			40°
		Downward	Upward	Downward	Upward
Maximum amplitude (mm)	5.31 ± 0.29	5.15 ± 0.31	9.71 ± 0.40	9.60 ± 0.45
Final amplitude (mm)	5.20 ± 0.27	5.09 ± 0.30	9.55 ± 0.38	9.58 ± 0.44
Peak velocity (mm/s)	43.29 ± 1.66	34.82 ± 2.76	72.49 ± 3.21	61.41 ± 2.14
Time of maximum velocity (s)	0.054 ± 0.002	0.063 ± 0.003	0.059 ± 0.003	0.062 ± 0.003
Duration (s)	0.39 ± 0.07	0.60 ± 0.08	0.47 ± 0.08	0.84 ± 0.13

 Data are expressed as the mean ± SEM.

Table 2.

 

Lower Eyelid Saccade Parameters

Parameter	20°			40°
		Downward	Upward	Downward	Upward
Maximum amplitude (mm)	1.39 ± 0.12	1.35 ± 0.10	2.50 ± 0.15	2.50 ± 0.15
Final amplitude (mm)	1.37 ± 0.11	1.33 ± 0.10	2.42 ± 0.13	2.47 ± 0.15
Peak velocity (mm/s)	11.22 ± 0.94	11.03 ± 0.76	16.66 ± 1.00	17.30 ± 1.01
Time of maximum velocity (s)	0.053 ± 0.003	0.052 ± 0.003	0.066 ± 0.002	0.063 ± 0.003
Duration (s)	0.44 ± 0.06	0.46 ± 0.07	0.41 ± 0.04	0.45 ± 0.006

 Data are expressed as the mean ± SEM.

Figure 5.

 

Linear relationship between amplitude and velocity for upward (left) and downward (right) eyelid saccades. Top: upper eyelid; bottom: lower eyelid.

Figure 6.

 

Saccade velocity as a function of time for upward (left) and downward (right) eyelid saccades. Top: upper eyelid; bottom: lower eyelid.

Figure .

 

Geometric relations between linear and rotational displacements.