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of
p
sr
and first order for
p
suc
.Thewholetermistolongtobeshownhereand
its structure does not provide any extra information. Nevertheless, it represents
the probability of success on the explicit task of associating an object with the
rewarded choice if the object is shown in an infinite loop. So,
π
16
stat
is the upper
limit for the performance of the model and can be calculated solely based on the
parameters
p
suc
and
p
sr
. Figure 4(a) shows
π
16
stat
for variable
p
suc
and
p
sr
.The
impact of
p
sr
is much greater than
p
suc
.If
p
sr
<
1than
p
suc
only slightly affects
π
16
stat
.
0.9
1
0.8
0.9
1
0.7
0.8
0.8
1
0.7
0.6
0.6
|λ
2
...
5
|
0.8
0.4
0.6
0.5
0.6
π
16
stat
0.2
0.5
0.4
0.4
0
0
0.2
0.4
0.3
0
1
0.3
0.33
0.2
0.8
0.2
0
0.2
0.4
0.6
0.1
0.66
0.33
0.6
0.4
0.1
0.66
0.8
0.2
p
sr
p
suc
p
sr
1
1
1
p
suc
(a)
(b)
Fig. 4.
Influence of
p
suc
and
p
sr
on the learning process for a
single
object. (a) upper
limit of the success probability (
π
16
stat
). (b) absolute value of the eigenvalues (
|
λ
2
...
5
|
).
The convergence
of the learning process is also affected by the parameters.
A
=
T
T
·
C
has five non-zero eigenvalues whereas
λ
1
is dominant and
λ
2
...
5
take the
form
λ
1
=1
λ
2
=
1
28
(12
12
p
suc
)
4
−
·
(15
p
sr
−
1)
1
28
(12
12
p
suc
)
4
λ
3
=
−
−
·
(15
p
sr
−
1)
(9)
12
p
suc
)
4
λ
4
=
I
28
(12
−
·
(15
p
sr
−
1)
12
p
suc
)
4
I
28
(12
λ
5
=
−
−
·
(15
p
sr
−
1)
.
For reasonable combinations of
p
suc
,
p
sr
follows
1
15
≤
|
λ
2
...
5
|
<
1if
p
suc
,p
sr
≤
1
.
(10)
The state probability vector
π
can be written in terms of its orthogonal basis
16
π
e
i
|
e
i
|
,
π
=
e
i
,
(11)
2
i
=1
such that (7) at time
t
is
16
π
(0)
,
e
i
t
t
π
(
t
)=
A
π
(0) =
A
e
i
(12)
|
e
i
|
2
i
=1
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