Biomedical Engineering Reference
In-Depth Information
and
tan f
¼
tan c
Substituting for tan o
d
t
þ
ð
c
Þ
from Eq. (13.17) into Eq. (13.16) gives
tanc ¼
1
2z
2
tan o
d
ðÞþ
tan c
1
ð
13
:
18
Þ
1
tan o
d
ðÞ
tanc
Multiplying both sides of Eq. (13.18) by
ð
1
tano
d
t
tan cÞ
gives
tan c
tan o
d
ðÞ
1
2z
2
1
2z
2
2
ð
tan c
Þ
¼
tan o
d
ðÞþ
tan c
ð
13
:
19
Þ
1
1
Collecting like terms in Eq. (13.19) gives
1
2z
2
¼
tan c
2z
2
2
tan o
d
ðÞ
1
þ
ð
tan c
Þ
1
ð
13
:
20
Þ
Dividing both sides of Eq. (13.20) by the term multiplying tan o
d
ðÞ
gives
tanc
tan c
tan o
d
ðÞ¼
¼
ð
13
:
21
Þ
2
2z
2
2
2z
2
2z
2
1
þ
ð
tanc
Þ
1
2z
2
1
þ
ð
tan c
Þ
2z
2
z
1
With tan c ¼
tanf ¼
p
, we have
z
2
z
1
z
1
z
1
p
p
p
0
@
1
A
z
2
z
2
z
2
z
2
1
0
@
1
A
0
@
1
A
tan o
d
ðÞ¼
¼
¼
z
2
z
2
1
2z
2
2z
2
1
1
2z
2
2z
2
þ
1
z
2
z
2
2
þ
2
þ
z
2
1
z
2
z
2
1
1
p
1
p
1
z
2
z
2
¼
þ
¼
z
2
2z
2
z
z 21
1
!
tan
1
p
z
2
1
w
d
1
Taking the inverse tangent of the previous equation gives
T
mv
¼
:
z
Westheimer noted the differences between saccade duration-saccade magnitude and
peak velocity-saccade magnitude in the model and the experimental data and inferred that
the saccade system was not linear because the peak velocity-saccade magnitude plot was
nonlinear. He also noted that the input was not an abrupt step function. Overall, this model
provided a satisfactory fit to the eye position data for a saccade of 20
but not for saccades
of other magnitudes. Interestingly, Westheimer's second-order model proves to be an
adequate model for saccades of all sizes if a different input function, as described in the
next section, is assumed. Due to its simplicity, the Westheimer model of the oculomotor
plant is still popular today.
