Biomedical Engineering Reference
In-Depth Information
tic spike trains. We can write it as
r
out
t
)
r
in
i
t
+
<
(
(
t
)
>
t
<
∞
0
dt
K
t
)
∑
j
w
j
r
i
j
(
t
−
t
)
r
in
i
t
+
r
in
i
=
G
(
(
t
)
>
t
−
θ
<
>
(11.19)
G
∞
0
dt
K
t
)
∑
r
in
i
t
+
r
i
j
(
t
−
t
)
>
t
−
θ
<
r
in
i
=
(
j
w
j
<
(
t
)
>.
r
in
i
t
−
r
i
j
(
t
−
t
)
>
t
t
)
If we define
<
(
t
)
to be
Q
ij
(
t
−
, then
G
∞
0
r
out
t
)
r
in
i
t
+
dt
K
t
)
∑
j
t
)
−
θ
<
r
in
i
<
(
(
t
)
>
t
=
(
w
j
Q
ij
(
−
t
−
>.
(11.20)
Let us assume that all the inputs have the same mean firing rate
r
in
and the correlation
between any two inputs
i
and
j
is
r
in
2
C
ij
e
−|
t
|/
τ
(
)=
(
+
)+
δ
(
)
.
Q
ij
t
c
1
t
r
in
(11.21)
ij
Therefore the correlation between the presynaptic and postsynaptic spike trains is
r
out
t
)
r
in
i
t
+
<
(
(
t
)
>
t
G
∞
0
dt
m
e
−
t
/
τ
C
ij
e
−|
t
+
t
|/
τ
1
j
w
j
r
in
2
=
(
+
1
)+
δ
(
t
)
r
in
δ
−
θ
rth
in
m
c
ij
=
−
θ
r
in
−
t
0
m
e
−
t
/
τ
m
e
t
−
t
)
/
τ
dt
1
j
w
j
r
in
2
C
ij
+
w
i
r
in
τ
e
t
/
τ
c
m
∞
−
t
dt
if
t
<
0
m
m
e
−
t
/
τ
m
e
t
−
t
/
τ
1
j
w
j
r
in
2
C
ij
+
∑
j
w
j
r
in
2
+
G
+
c
∞
0
dt
(11.22)
m
e
−
t
/
τ
m
e
t
−
t
/
τ
1
j
w
j
C
ij
r
in
2
j
w
j
r
in
2
+
∑
if
t
≥
0
c
=
−
θ
r
in
j
w
j
C
ij
r
in
2
e
t
/
τ
e
t
/
τ
[
m
(
−
)
c
c
m
−
τ
c
w
i
r
in
τ
m
e
t
/
τ
j
w
j
r
in
2
e
t
/
τ
+
G
+
c
]+
∑
+
if
t
<
0
m
m
c
+
τ
m
j
w
j
C
ij
r
in
2
m
e
t
/
τ
j
w
j
r
in
2
+
∑
≥
c
if
t
0
m
c
+
τ
11.4.3
Mean rate of change in synaptic strength
We can now calculated the mean rate of change in synaptic strength induced by
STDP.
∞
dw
i
(
)
dt
=
t
dt
P
t
)
r
out
r
in
t
)
,
(
(
t
)
(
t
+
(11.23)
i
−
∞
is the STDP plasticity function and
r
in
i
and
r
out
where
P
(
t
)
(
t
)
(
t
)
are the firing rates
for the ith input and the output respectively. From Section 11.2,
A
+
τ
+
e
t
/
τ
+
if
t
<
0
P
(
t
)=
(11.24)
A
e
−
t
/
τ
−
if
−
−
τ
−
t
≥
0
From Section 11.1,
G
∞
0
r
out
dt
K
t
)
∑
i
w
i
r
in
t
)
−
θ
(
)=
(
(
−
.
t
t
(11.25)
i
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