Biomedical Engineering Reference
In-Depth Information
from the saccade amplitude versus time graph, but it is more easily seen in the velocity ver-
sus time graph, as shown in Figure 13.5. Saccade durations can range from approximately
30 ms for saccades less than 5
and up to 100 ms for large saccades. For saccades greater than
7
, there is a linear relationship between saccade amplitude and duration. The latent period is
the time interval from when a target appears until the eyes begin to move.
Figure 13.6 shows the main sequence characteristics for a subject executing 26 sac-
cades. The subject actually executed 52 saccades in both the positive and negative direc-
tions, with only the results of the saccades in the positive direction displayed in
Figure 13.6 for simplicity. Note that saccade characteristics moving to the left are different
from those moving to the right. The solid lines in the figures include a fit to the data. Peak
velocity-saccade magnitude is basically a linear function until approximately 15
,after
whichitlevelsofftoaconstantforlargersaccades. Many researchers have fit this rela-
tionship to an exponential function. The line in graph (A) is fitted to the nonlinear
equation
e
b
v
¼
a 1
ð
13
:
1
Þ
max
where
the saccade size, and the constants a and b evaluated
to minimize the summed error squared between the model and the data. Note that a is to
represent the steady-state peak velocity-saccade magnitude curve, and b is to represent
the “time constant” for the peak velocity-saccade magnitude curve. For this data set for
positive eye movements, a equals 825, and b equals 9.3.
A similar pattern is observed with eye movements moving in the negative direction
(not shown), but the parameters are a
v
max
is the maximum velocity,
x
6.9, which are typically different from
the values computed for the positive direction. The exponential shape of the peak velocity-
saccade amplitude relationship might suggest that the system is nonlinear if a step input to
the system is assumed. A step input provides a linear peak velocity-saccade amplitude rela-
tionship. In fact, the saccade system is not driven by a step input but rather a more complex
pulse-step waveform, as discussed later. Thus, the saccade system cannot be assumed to be
nonlinear solely based on the peak velocity-saccade amplitude relationship.
Figure 13.6B shows the data depicting a linear relationship between saccade duration-
saccade magnitude. If a step input is assumed, then the dependence between saccade dura-
tion and saccade magnitude also might suggest that the system is nonlinear. A linear system
with a step input always has a constant duration. Since the input is not characterized by a
step waveform, the saccade system cannot be assumed to be nonlinear solely based on the
saccade duration-saccade magnitude relationship.
Figure 13.6C shows the latent period-saccade magnitude data. It is quite clear that the
latent period does not show any linear relationship with saccade size—that is, the latent
period's value appears independent of saccade size. However, some other investigators
have proposed a linear relationship between the latent period and saccade magnitude. This
feature is unimportant for the presentation in this topic, since in the development of the
oculomotor plant models, the latent period is implicitly assumed within the model.
Because of the complexity of the eye movement system, attention is restricted to horizontal
fast eye movements. In reality, the eyeball is capable of moving horizontally, vertically, and
torsionally. An appropriate model for this system would include a model for each muscle
¼
637 and b
¼
