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each neuron in the lattice;
z
is the presigmoidal activation level and
y
is the
postsigmoidal activation level. Adjustment of the presigmoidal activation level is
[]
(
(
[]
)
)
[ ]
IH
i
∈
z
t
=
a
F
Met
x
t
,
w
+
a
y
t
−
1
ȝ
,
(7.13)
i
1
2
j
i
j
j
N
i
[]
where
z
i
is the presigmoidal activation level for neuron
i
at time index
t
, the
input vector of the network
t
(
)
x
=
x
,
x
,
x
,
,
x
, the input weight vector
1
2
h
IN
(
)
IH
i
IH
i
IH
i
IH
h
IH
IN
IH
h
w
=
w
,
w
,
,
w
,
,
w
, and
w
is the weight associating neuron
i
1
2
i
i
with input signal
x
(
1
≤
h
≤
IN
).
h
()
()
Met
is a metric
that measures the distance between two vectors in a metric space.
N
is the
neighborhood set of neuron
i
, and
y
is the output activity of neuron
j
. The
lateral interconnection weights between neurons
i
and
j
,
F
is a positive monotonic decreasing function, and
()
Ș
, is the weight
associating neuron
i
with neuron
j
; it depends only on the relative position in
the lattice,
()
( )
Ș is the spatial impulse response function of the
network.
a
and
a
in Eq. (7.13) are two constants that are weights to inputs of
the network and outputs from other neurons in the lattice, respectively. In this
study, we used the Euclidean metric for
ȝ
=
Ș
"
−
"
.
i
j
i
j
()
()
Met
. Then
F
takes the following
form:
(
( )
)
(
)
⎠
⎛
I
N
⎞
[]
IH
i
IH
h
F
Met
x
,
w
=
f
⎜
⎝
∑
=
g
w
,
x
t
i
h
h
1
,
(7.14)
(
)
(
)
2
[]
[]
IH
h
IH
h
g
w
,
x
t
=
w
−
x
t
i
h
i
h
,
(7.15)
()
−
λ
u
f
u
=
e
.
(7.16)
The output activity of neuron
z
is
[]
[
( )
y
t
=
ı
z
t
,
(7.17)
j
i
i
()
where
i
ı is the sigmoid function.
The weight adjustment follows the Kohonen learning algorithm [3]:
for (each time index t) do
fetch source signal
[]
x
t
[] [ ] [] [ ]
select
i
:
x
t
−
w
t
−
1
≤
x
t
−
w
t
−
1
,
∀
k
∈
S
i
k
for (
∀
j
∈
N
) do
i
[] [ ] [ ]
(
)
[] [ ]
(
)
w
t
=
w
t
−
1
+
αφ
t
−
1
"
−
"
x
t
−
w
t
−
1
j
j
j
i
j
end
end
[]
[]
x
t
is the input signal of the network at time index
t
.
w
t
is the input weight
j
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