Biomedical Engineering Reference
In-Depth Information
scribed by individual conductances
g
j
AMPA
and
g
j
NMDA
,
j
=
1
,
2
, ...,
C
E
;
g
j
GABA
,
j
=
1
,
2
, ...
C
I
). In the presence of these inputs, Equation (15.1) now reads
C
m
dV
)
dt
=
−
(
t
g
L
(
V
(
t
)
−
V
L
)
−
C
Â
j
=
1
g
j
NMDA
s
j
NMDA
(
t
)
−
g
j
AMPA
s
j
AMPA
(
t
)+
(
V
(
t
)
−
V
E
)
−
J
(
V
(
t
))
C
Â
j
−
g
j
GABA
s
j
GABA
(
t
)
(
V
(
t
)
−
V
I
)
.
(15.53)
=
1
For simplicity, we again assume that the synaptic conductances and the firing rates
of all pre-synaptic inputs from the same sub-population are identical.
Using the
approximations described in the previous sections, this equation becomes
C
m
dV
(
t
)
=
−
(
(
)
−
)
−
g
L
V
t
V
L
dt
−
C
E
[
g
AMPA
s
AMPA
](
V
(
t
)
−
V
E
)
−
C
E
g
ef f
NMDA
s
NMDA
(
V
ef f
E
−
V
(
t
)
−
)
−
−
C
I
[
g
GABA
s
GABA
](
V
(
t
)
−
V
I
)+
d
I
(
t
)
,
(15.54)
where
s
AMPA
=
where the function y is
defined in Equation (15.43), and the fluctuations are described by
n
E
t
AMPA
,
s
GABA
A
=
n
I
t
GABA
and
s
NMDA
=
y
(
n
E
)
d
dt
t
AMPA
d
I
(
t
)=
−
d
I
(
t
)+
s
ef f
h
(
t
)
(15.55)
V
s
ef f
=
g
AMPA
(
2
C
E
s
AMPA
t
AMPA
.
−
V
E
)
(15.56)
Since all the deterministic components of the current are now linear in the voltage,
the equations describing the membrane potential dynamics can be expressed as
dV
)
dt
=
−
(
(
t
d
I
)
g
ef f
L
(
t
t
ef f
m
V
(
t
)
−
V
ss
)+
(15.57)
d
dt
t
AMPA
d
I
(
t
)=
−
d
I
(
t
)+
s
ef f
h
(
t
)
.
(15.58)
The effective membrane time constant is
C
m
g
ef
L
=
g
L
g
ef f
L
t
ef f
m
=
t
m
,
(15.59)
and the effective leak conductance of the cell is the sum of the passive leak conduc-
tance plus the increase in the conductances associated to all the synaptic inputs to
the cell
g
ef f
L
g
ef f
=
g
L
+
g
AMPA
C
E
s
AMPA
+
NMDA
C
E
s
NMDA
+
g
GABA
C
I
s
GABA
.
(15.60)
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