Biomedical Engineering Reference
In-Depth Information
Outside
−
−
−
R
TH
R(Ach
i
)
R(Ach
i
)
R
TH
R(Ach
i
)
R
TH
V
m
C
m
V
m
C
m
V
m
C
m
−
−
−
−
−
−
V
TH
V
TH
V
TH
V
TH
V
TH
V
TH
+
+
+
+
+
+
+
+
+
R
a
R
a
R
a
Inside
FIGURE 12.40
Equivalent circuit of series of dendritic membrane sections that includes a neurotransmitter
channel. Each section is connected together with an axial resistance,
R
a
.
A complete model of the neuron can be constructed by including as many dendritic
branches as needed, each described using Figure 12.17 and modeled by
0
...þ
C
m
dV
m
(
V
m
V
TH
Þ
R ACh
ð Þ
þ
(
V
m
V
TH
Þ
R
TH
(
V
m
V
m
Þ
dt
þ
þ
R
a
ð
12
:
59
Þ
0
0
0
0
00
þ
C
m
dV
(
V
m
V
TH
Þ
R ACh
ð Þ
(
V
m
V
TH
Þ
R
TH
(
V
m
V
m
Þ
m
dt
þ
þ
þ
þ...
R
a
Although this chapter has focused on the neuron, it is important to note that numerous
other cells have action potentials that involve signaling or triggering. Many of the principles
discussed in this chapter apply to these other cells as well, but the action potential defining
equations are different. For example, the cardiac action potential can be defined with a
DiFranceso-Noble, Luo-Rudy, or other models instead of a Hodgkin-Huxley model of the
neuron.
12.9 EXERCISES
1.
Assume a membrane is permeable to only
Ca
þ2
. (a) Derive the expression for the flow of
Ca
þ2
.
Ca
þ2
.
2.
Assume that a membrane is permeable to
(b) Find the Nernst potential for
Ca
þ2
and
Cl
but not to a large cation
R
þ
.
The inside concentrations are
RC
½ ¼
100 mM and
½
¼
200 mM, and the outside
CaCl
2
concentration is
½
CaCl
2
¼
300 mM. (a) Derive the Donnan equilibrium. (b) Find the steady-
Ca
þ2
.
3.
Assume that a membrane is permeable to
state concentration for
Cl
. The initial concentrations on
the inside are different from the outside, and these are the only ions in the solution.
(a) Write an equation for
Ca
þ2
and
J
Ca
and
J
Cl
. (b) Write an expression for the relationship between
J
Ca
and
J
Cl
. (c) Find the steady-state voltage. (d) Find the relationship between the voltage
across the membrane and
before steady-state.
4.
Assume that a membrane is permeable to only ion
½
CaCl
2
R
þ3
. The inside concentration is
RCl
½ ¼
2 mM, and the outside concentration is
RCl
½ ¼
1
:
4 mM. (a) Write an expression
R
þ3
. (b) Derive the Nernst potential for
R
þ3
at steady-state.
for the flow of
