purpose. To establish the relationship between upper eyelid saccades and upper eyelid pursuit movements

methods. Upper eyelid saccades and periodic sinusoidal upper eyelid pursuit movements were recorded in a sample of controls and patients with Graves upper eyelid retraction. A video-computerized system was used to register both types of movements that accompanied 60° of eye rotation across the upper and lower hemifields. The forced harmonic oscillator model was used to fit saccadic and pursuit movements.

results. Mean mid-pupil eyelid distance for the Graves patients (6.6 ± 1.1 mm) was significantly higher than for the controls (4.6 ± 0.8 mm; *t* = 7.18; *P* < 0.00001). Despite the difference in the upper eyelid resting position, saccades and pursuit eyelid movements of both groups were extremely well fitted by underdamped solutions and steady forced solutions of the harmonic oscillator model, respectively. For the controls, the amplitude of the pursuit movements was well correlated with the upward and downward saccades. The amplitude of the eyelid movements of the Graves patients (saccades and pursuit) was significantly reduced compared with that of the controls.

conclusions. Saccadic and pursuit movements of the upper eyelid can be described by the harmonic oscillator model. In healthy subjects and Graves patients, the amplitude of pursuit lid movements is correlated to the saccade amplitude. Pursuit eyelid movements are more difficult to register than saccades, and their measurements do not allow clear separation of the relaxation and contraction properties of the upper eyelid retractors.

^{ 1 }These upper eyelid movements that accompany the vertical eye saccades have been well studied,

^{ 2 }

^{ 3 }and we have recently demonstrated that they represent damped harmonic oscillations.

^{ 4 }A different situation exists when the eye follows a target that moves slowly and periodically in the vertical meridian. Eye movement in this case (smooth pursuit) depends on the dynamics of the stimulus that is constantly imaged on the fovea.

^{ 5 }Upper eyelid movements generated by eye pursuit movements have not been explored in clinical settings. In the present study, we measured upper eyelid saccades and pursuit movements in healthy subjects and in a sample of patients with Graves orbitopathy. Our results indicated that upper eyelid pursuit movements correspond to forced harmonic oscillations and are closely related to eyelid saccades dynamic properties.

^{ 4 }The camera’s temporal resolution was the standard NTSC (30 Hz or 30 frames/s). Motion recording was performed with software that tracked, in real time, the center of a blue spot in each frame. This spot, which provided the localizing signal for the software, was a small piece of blue paper (0.01 g) attached to the eyelashes of the central portion of the upper eyelid.

^{ 4 }For the pursuit movements, a sinusoidal function was adjusted to the experimental data according to a forced oscillator model driven by a sinusoidal external force.

^{ 4 }In the present study, we demonstrated that both saccadic and slow pursuit movements can be described with the more general harmonic oscillator model that includes, besides elastic restoration and dissipative forces, an external force acting on the eyelid.

*g*is the damping coefficient, ζ is the natural angular frequency related to the strength of the restoration force and to the inertia of the eyelid, and

*F*(

*t*) denotes the external force per unit of mass.

*F*(

*t*) =

*F*

_{0}for

*t*≥ 0. In this case, we can redefine the origin from which the position of the eyelid is measured such that

*Z*(

*t*) =

*Y*(

*t*) −

*F*

_{0}/ζ

^{2}and, because the time derivatives of

*Y*(

*t*) and

*Z*(

*t*) are equal,

*Z*(

*t*) satisfies the homogeneous equation associated with equation 1 , that is, the equation whose right-hand side equals zero. In other words, a harmonic oscillator pushed externally by a constant force is equivalent to a damped harmonic oscillator with a displaced equilibrium position. The underdamped solutions of this homogeneous equation, which correspond to an oscillatory part enveloped by a decaying exponential function, describe adequately the eyelid saccadic movements.

^{ 4 }

*Y*

_{ h }(

*t*), with a particular solution of the inhomogeneous equation 1 ,

*Y*

_{ I }(

*t*). The function

*Y*

_{ h }(

*t*) always dies out exponentially in time, whereas the solution

*Y*

_{ I }(

*t*) remains with a steady oscillatory behavior driven by the periodic external excitation. This then is the solution of interest here, to describe eyelid pursuit movement. One expects to find the steady solution profile similar to that of the external force

*F*(

*t*).

*F*(

*t*) =

*C*cos(ω

*t*+ θ). In this case, the steady state solution of equation 1is

*A*and the phase angle β are given by

^{ 6 }

*F*(

*t*) is an arbitrary periodic function of time, we proceed by using Fourier analysis. Any periodic

*F*(

*t*) such that

*F*(

*t*+

*T*) =

*F*(

*t*) can be written as a sum of sinusoidal function in the form

_{ n }= 2π

*n/T*and with

*C*

_{ n }and θ

_{ n }constants that depend on the function

*F*(

*t*).

*Y*

_{1}(

*t*) and

*Y*

_{2}(

*t*) are solutions of equation 2with stimuli

*F*

_{1}(

*t*) and

*F*

_{2}(

*t*)

**,**respectively, then

*Y*

_{1}(

*t*) +

*Y*

_{2}(

*t*) is a solution of equation 2with the inhomogeneous term

*F*

_{ 1 }(

*t*) +

*F*

_{2}(

*t*). This permits us to write the steady state solution for the case of the general periodic stimulus as

*t*-tests. Linear correlations were determined using least squares regression and expressed by the Pearson (

*r*) coefficient of correlation. The goodness-of-fit for the nonlinear models used in the study was measured by the calculation of coefficient of determination (

*R*

^{2}). Variability was indicated by the symbol ± and always expressed standard deviations of the mean.

*t*= 7.18;

*P*< 0.00001).

*g*, for upward and downward saccades. For both groups of subjects, the mean damping coefficient of upward saccades was higher than of downward movements. Paired

*t*-tests indicated that this difference was significant only for the controls (

*t*= 2.52;

*P*= 0.02).

*t*= 2.67,

*P*= 0.01; downward

*t*= 3.01,

*P*= 0.005). The difference between control subjects and Graves patients was also verified when pursuit movements were compared (mean controls = 6.42. ± 1.10 mm, Graves = 5.13 ± 1.53 mm;

*t*= 2.96,

*P*= 0.05).

*r*= 0.85) though the correlation was slightly worse for the Graves patients (

*r*= 0.64), reflecting the inhomogeneous nature of this population.

^{ 1 }Mathematically, both movements follow a forced harmonic oscillator model. For the saccades, a suddenly established constant force acts on the eyelid that is equivalent to an unforced harmonic oscillator with a displaced equilibrium position. As we demonstrated in a previous work, the lid saccades are well described as underdamped harmonic oscillations.

^{ 4 }For the pursuit movements, the eye continuously fixates a slowly moving target. When the eye movement is periodic, the external force that acts on the upper eyelid is also periodic. In the case of a sinusoidal eye movement, the resultant eyelid movement is also sinusoidal.

^{ 7 }The retraction is clearly multifactorial, and distinct neural and restrictive factors have been implicated in the genesis of the retraction.

^{ 8 }A well-known neural mechanism is LPS overaction associated with inferior rectus muscle restriction. Because the LPS action is linked to the superior rectus muscle activity, any effort to obtain vertical eye alignment in the presence of inferior rectus muscle restriction will cause an increased innervation of the superior and LPS muscles.

^{ 9 }This type of retraction is not associated with intrinsic abnormalities within the LPS muscle and is corrected with squint surgery. Another cause of retraction that is not associated with LPS abnormalities is Müller muscle hyperaction. This mechanism appears to operate on selected patients with retractions that are highly variable and fully corrected with the use of guanethidine drops.

^{ 8 }In other cases, eyelid retraction is clearly associated with LPS enlargement

^{ 10 }and lid motion restriction on down gaze (the so-called von Graefe sign).

**Figure 1.**

**Figure 1.**

Group | Upward (s^{−1}) | Downward (s^{−1}) | ||||
---|---|---|---|---|---|---|

Natural Angular Frequency | Damping Coefficient | Natural Angular Frequency | Damping Coefficient | |||

Control subjects | 15.5 ± 2.27 | 14.7 ± 2.81 | 14.3 ± 3.22 | 11.8 ± 4.16 | ||

Graves patients | 15.4 ± 4.26 | 14.07 ± 4.70 | 16.2 ± 4.66 | 12.28 ± 6.25 |

**Figure 2.**

**Figure 2.**

**Figure 3.**

**Figure 3.**

*Invest Ophthalmol Vis Sci*. 1991;32(2)387–400. [PubMed]

*Invest Ophthalmol Vis Sci*. 2001;42(3)620–625. [PubMed]

*Invest Ophthalmol Vis Sci*. 1995;36(13)2686–2694. [PubMed]

*Invest Ophthalmol Vis Sci*. 2005;46(3)857–862. [CrossRef] [PubMed]

*J Physiol*. 2007;584(1)11–23. [CrossRef] [PubMed]

*Mechanics*. 1971; 3rd ed.Addison-Wesley Publishing Reading, MA.

*Am J Ophthalmol*. 1995;119(6)792–795. [CrossRef] [PubMed]

*Ophthal Plast Reconstr Surg*. 2001;17(5)309–315. [CrossRef] [PubMed]

*Ophthalmology*. 1994;101(9)1608–1613. [CrossRef] [PubMed]

*Radiology*. 1993;188(1)115–118. [CrossRef] [PubMed]