Biomedical Engineering Reference
In-Depth Information
8.4.4.2
Membrane Potential Decomposition
Given the spike history of the network it is easy to integrate (
8.34
)andtofindan
explicit expression for the membrane potential at time
t
. It depends on the past spike
history
ω
. Denote
V
k
(
t, ω
) the membrane potential at time
t
given the spike history
before
t
.Set:
t
2
t
1
1
C
k
Γ
k
(
t
1
,t
2
,ω
)=
e
−
g
k
(
u,ω
)
du
,
(8.36)
have
V
k
(
t, ω
)=
V
(
det
)
k
corresponding to
the
flow of
(
8.34
). We
(
t, ω
)+
V
(
noise
)
k
(
t, ω
),
V
(
det
)
k
(
t, ω
)=
V
(
syn
)
k
(
t, ω
)+
V
(
ext
)
k
(
t, ω
)
(8.37)
where
W
kj
t
τ
k
(
t,ω
)
N
(
t, ω
)=
C
k
V
(
syn
)
k
Γ
k
(
t
1
,t,ω
)
α
kj
(
t
1
,ω
)
dt
1
,
(8.38)
j
=1
is the synaptic interaction term which integrates the pre-synaptic spikes from the
last time where neuron
k
has been reset. Additionally,
g
L,k
E
L
+
,
t
(
t, ω
)=
C
k
V
(
ext
)
k
i
(
ext
)
k
(
t
1
)
Γ
k
(
t
1
,t,ω
)
dt
1
(8.39)
τ
k
(
t,ω
)
contains the stimulus
i
(
ext
)
k
(
t
) influence. Thus
V
(
det
k
(
t, ω
) contains the determinis-
tic part of the membrane potential. On the opposite,
V
(
noise
k
(
t, ω
) is the stochastic
part of the membrane potential. This is a Gaussian variable with mean zero and
variance
σ
B
C
k
2
t
σ
k
(
t, ω
)=
Γ
k
(
τ
k
(
t, ω
)
,t,ω
)
σ
R
+
Γ
k
(
t
1
,t,ω
)
dt
1
.
(8.40)
τ
k
(
t,ω
)
The first term in the right-hand side comes from the reset of the membrane potential
to a random value after resetting. The second one is due the integration of synaptic
noise from
τ
k
(
t, ω
) to
t
.
8.4.4.3
Statistics of Raster Plots
Ithasbeenshownin[
7
] that the gIF model (
8.34
) has a unique Gibbs distribution
with a
non-stationary
potential:
N
φ
n
(
ω
)=
φ
n,k
(
ω
)
,
(8.41)
k
=1
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