Biomedical Engineering Reference
In-Depth Information
the average of the gating variable is constant), and a fluctuating component due to
the stochastic nature of the spike trains in our model, i.e.,
s
syn
(
t
)=
s
syn
+
d
s
syn
(
t
)
.
The unitary synaptic current, Equation (15.38), now becomes
I
syn
(
t
)=
g
syn
s
syn
(
V
(
t
)
−
V
syn
)+
g
syn
d
s
syn
(
t
)(
V
(
t
)
−
V
syn
)
.
(15.44)
The main complication due to the driving force is that now the fluctuating compo-
nent of the synaptic current (second term in the right-hand side of Equation (15.44))
becomes voltage dependent. This multiplicative dependence of the fluctuations on
the membrane potential renders a rigorous treatment of the fluctuations in the current
difficult. To avoid this complication, we replace the voltage by its average
V
in the
driving force for the fluctuating component of the synaptic current, so that
V
I
syn
(
t
)
∼
g
syn
s
syn
(
V
(
t
)
−
V
syn
)+
g
syn
d
s
syn
(
t
)(
−
V
syn
)
.
(15.45)
can be dealt with easily
by noting that
g
syn
s
syn
can be absorbed in the leak conductance, and
g
syn
s
syn
V
syn
can
be absorbed in the resting membrane potential
V
L
.
The resulting effect on neuronal properties is an increase in the total effective leak
conductance of the cell
The deterministic part of the current
g
syn
s
syn
(
V
(
t
)
−
V
syn
)
g
L
→
g
L
+
g
syn
s
syn
,
(15.46)
which is equivalent to a decrease of the membrane time constant from t
m
=
C
m
/
g
L
,
to t
ef f
a
t
m
. Thus, the synaptic input makes the neuron
leakier by a factor equal to the relative increase in conductance due to synaptic input
(a
t
m
=
=
C
m
/
(
g
L
+
g
syn
s
syn
)=
t
m
/
m
1
+
s
syn
g
syn
/
g
L
).
The resting (or steady-state) membrane potential is also
re-normalized
g
L
V
L
+
g
syn
s
syn
V
syn
V
L
→
,
(15.47)
+
g
L
g
syn
s
syn
and becomes a weighted average of the different reversal potentials of the vari-
ous synaptic currents, where each current contributes proportionally to the relative
amount of conductance it carries.
Voltage-dependence of NMDA channels
. For NMDA channels to open, bind-
ing of neurotransmitter released by the pre-synaptic spike is not enough. The post-
synaptic cell must also be sufficiently depolarized to remove their blockade by mag-
nesium. It is conventional to model this using a voltage-dependent maximal conduc-
tance [66]:
g
NMDA
1
g
NMDA
(
V
)=
))
≡
g
NMDA
))
,
(15.48)
(
1
+([
Mg
2
+
]
/
g
)
exp
(
−
b
V
(
t
J
(
V
(
t
Mg
2
+
]=
with
062. To be able to incorporate this ef-
fect into the framework described in the previous sections, we linearize the voltage
dependence of the NMDA current around the average voltage
V
, obtaining
[
1mM,g
=
3
.
57 and
b
=
0
.
V
V
V
V
(
t
)
−
V
E
V
(
t
)
−
V
E
J
(
)
−
b
(
−
V
E
)(
1
−
J
(
))
V
∼
+(
V
(
t
)
−
)
V
V
J
(
V
(
t
))
J
(
)
J
2
(
)
V
2
+
O
((
V
(
t
)
−
)
)+
...
(15.49)
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