Biomedical Engineering Reference
In-Depth Information
EXAMPLE PROBLEM 12.8
Calculate
G
Na
at resting potential for the squid giant axon using the Hodgkin-Huxley
model. Parameter values are
G
K
¼
G
K
and
10
-3
S and
G
Na
¼
10
-3
S.
36
120
Solution
At resting potential,
G
K
and
G
Na
are constant with values dependent on
n
,
m,
and
h
. Since the
membrane is at steady state,
dn
0,
dm
0, and
dh
dt
¼
dt
¼
dt
¼
0. Using Eqs. (12.43), (12.45), and (12.46),at
resting potential and steady state
a
0
n
n
¼
n
þb
0
a
0
n
a
0
m
m
¼
b
0
m
a
0
m
þ
a
0
h
h
¼
h
þb
0
a
0
h
where a
0
i
, and b
0
i
is a at
V
¼
0 for
i
¼
n
,
m
, and
h
is b at
V
¼
0 for
i
¼
n
,
m
, and
h
. Calculations
yield a
0
0.0582, b
0
0.31769, a
0
0.2236, b
0
0.05294, a
0
n
¼
n
¼
0.125,
n
¼
m
¼
m
¼
4,
m
¼
h
¼
0.07,
b
0
h
¼
0.04742, and
h
¼
0.59615. Therefore, at resting potential and steady state
G
K
¼
G
K
n
4
4
10
3
(0
10
3
S
¼
36
:
0
:
31769
Þ
¼
0
:
3667
and
G
Na
¼
G
Na
m
3
3
10
3
(0
10
3
S
h
¼
120
:
0
:
05294
Þ
0
:
59615
¼
0
:
010614
12.6.3 Equation for the Time Dependence of the Membrane Potential
Figure 12.28 shows a model of the cell membrane that is stimulated via an external stim-
ulus,
, which is appropriate for simulating action potentials. Applying Kirchhoff's current
law at the cytoplasm yields
I
m
Þþ
V
m
E
l
ð
Þ
þ
C
m
dV
m
dt
I
m
¼
G
K
V
m
E
K
ð
Þþ
G
Na
V
m
E
Na
ð
ð
12
:
47
Þ
R
l
where
G
K
and
G
Na
are the voltage-time dependent conductances given by Eqs. (12.42) and (12.44).
EXAMPLE PROBLEM 12.9
For the squid giant axon, compute the size of the current pulse (magnitude and pulse width)
necessary to raise the membrane potential from its resting value of
60 mV to
40 mV and then
K
þ
and
Na
þ
conductances from resting poten-
back to its resting potential. Neglect any changes in
tial, but include
G
K
and
G
Na
at resting potential in the analysis. Hodgkin-Huxley parameter values
Continued
