Biomedical Engineering Reference
In-Depth Information
Whether a synapse increases or decreases in strength depends on the sign of
A
for
that particular weight value. The diffusion term of the random walk can be calculated
as
(
w
)
∞
dt
P
t
)
2
r
out
r
in
i
t
)
.
D
(
w
)=
(
(
t
)
(
t
+
(11.36)
−
∞
A full derivation of A(w) and D(w) for the spiking model can be found in [16].
Because we have imposed both a lower bound of zero and upper bound of
g
max
on
the synaptic strengths, this equation has the following boundary conditions.
(
,
)=
(
,
)=
,
J
0
t
J
g
max
t
0
(11.37)
where
J
(
w
,
t
)
is the probability current defined by
1
2
J
(
w
,
t
)=
A
(
w
)
P
(
w
,
t
)
−
w
(
D
(
w
)
P
(
w
,
t
))
.
(11.38)
According to Fokker-Planck theory, the equilibrium density
P
(
w
)
can be described
as a Gibbs distribution with a plasticity potential
U
(
w
)
, where in the limit of small
step size
2
w
0
dw
A
w
)
/
w
)
.
U
(
w
)
≈−
(
D
(
(11.39)
(
)
(
)
So,
P
and have a spread that
is proportional the drift term and thus the step size. The minimum can be located at
an interior point where the drift term vanishes or at the boundaries. In the situations
described in this chapter, the equilibrium is typically located at the boundaries( [68]).
If
A
−
/
w
will be concentrated near the global minima of
U
w
A
+
is set to be slightly larger than 1, the weak negative bias in the plasticity
function can balance the positive correlations. Under these conditions, the diffusion
constant
D
is approximately constant and given by
Gr
in
2
∑
j
=
i
w
j
−
θ
A
2
A
2
D
≈
(
r
in
)((
−
/
τ
−
+
+
/
τ
−
)
/
2
.
(11.40)
The potential is given by
r
in
A
+
τ
+
+
τ
Hr
in
2
C
ii
−
(
r
in
2
w
i
U
(
w
i
)
≈
2
(
−
G
(
A
−
−
A
+
)
+
m
)
τ
+
τ
m
GHr
in
2
r
in
2
−
2
(
i
C
ij
w
j
−
G
(
A
−
−
A
+
)
i
w
j
+
θ
r
in
(
A
−
−
A
+
)
w
i
)
(11.41)
j
=
j
=
Gr
in
2
A
2
A
2
/
(
i
w
j
−
θ
r
in
)((
−
/
τ
−
+
+
/
τ
−
))
.
j
=
P
(
w
i
)
is given by
P
(
w
i
)=
Ne
−
U
(
w
i
)
, where N is a normalization factor. The expected
value of
w
i
is then
g
max
dw
w
P
w
)
,
E
i
=
(
(11.42)
0
Since the distribution of synaptic strength for each synapse depends on the strength
of other synapses(interpreted as the expected value), these equations need to be
solved for self-consistency.
Search WWH ::
Custom Search