Biomedical Engineering Reference
In-Depth Information
and
⎛
⎞
0
⎝
⎠
a
d
U
=
l
(
t
)
dt
+
l
(
t
)
dW
t
t
1
t
2
(
)
>
with
W
t
being Brownian motion, l
t
is the control signal, and a
0 (see Section
1.3). Basically, under the rate coding assumption, a
=
1
/
2 corresponds to Poisson
inputs [57, 77] (also see Section 1.3).
We call
a
<
1
/
2
sub-Poisson inputs
,and
a
>
2
supra-Poisson inputs
.
Here is the control problem.
1
/
2a
•
(
)
∈ L
[
,
+
]
For a fixed, deterministic time
T
,findl
s
0
T
R
which minimizes
T
+
R
var
(
x
1
(
t
))
dt
(1.19)
T
subject to the constraint
E
[
x
1
(
T
)] =
D
for
t
∈
[
T
,
T
+
R
]
(1.20)
The meaning is clear. Equation (1.20) ensures that at time
T
, the saccadic movement
stops at the position
D
and keeps there for a while, i.e., from time
T
to
T
+
R
.During
the time interval
[
T
,
T
+
R
]
, it is required that the fluctuation is as small as possible
(Equation (1.19)).
When a
>
1
/
2, define
⎧
⎨
exp
exp
t
1
t
2
t
2
−
t
2
(
t
1
(
b
12
(
t
−
s
)=
−
t
−
s
)
−
−
t
−
s
)
t
1
t
1
exp
t
2
exp
t
1
t
2
t
2
−
t
1
(
t
2
(
⎩
b
22
(
t
−
s
)=
−
t
−
s
)
−
−
t
−
s
)
,
t
1
so the solution of the optimal control problem is
m
2
exp
s
t
2
m
1
exp
t
1
)
sgn
m
1
exp
s
t
1
m
2
exp
t
2
/
(
−
1
2a
1
+
+
l
(
s
)=
dt
1
/
(
2a
−
1
)
T
+
R
b
12
(
t
−
s
)
0
(1.21)
with m
1
,
m
2
being given by
⎧
⎨
D
t
2
exp
t
2
exp
t
2
T
A
(
s
)
=
·
ds
B
(
s
)
D
t
1
exp
t
1
0
(1.22)
exp
t
1
T
A
(
s
)
⎩
=
·
ds
.
B
(
s
)
0
with
m
1
exp
t
1
m
2
exp
s
t
2
)
sgn
m
1
exp
s
t
1
m
2
exp
t
2
1
/
(
2a
−
1
A
(
s
)=
+
+
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