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.

^{ 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.

^{ 2 }high-speed cinephotography,

^{ 3 }light reflection,

^{ 4 }and the magnetic search-coil technique.

^{ 5 }

^{ 6 }

^{ 7 }

^{ 8 }

*Y*(

*t*) changes with time, satisfying the differential equation

*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.

*Y*(

*t*= 0) =

*A*and

*A*and null velocity at the initial instant) is given by

*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.

*t*is given by

*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*, since we chose

*Y*= 0 as the final position. For

*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

^{(Appendix)}. The correlation coefficients of the model (

*R*

^{2}) for all movements ranged from 0.980 to 0.999 (mean = 0.995).

*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.

^{ 6 }

^{ 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 }

^{ 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.

^{ 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 }

^{ 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 }

^{ 11 }

**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.

**Figure 1.**

**Figure 1.**

**Figure 2.**

**Figure 2.**

**Figure 3.**

**Figure 3.**

**Figure 4.**

**Figure 4.**

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 |

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 |

**Figure 5.**

**Figure 5.**

**Figure 6.**

**Figure 6.**

**Figure .**

**Figure .**