Biomedical Engineering Reference
In-Depth Information
1
R
l
¼
G
K
¼
G
Na
¼
10
3
10
3
10
3
S,
for the squid giant axon are
G
l
¼
0
:
3
S,
36
S,
120
10
3
V,
10
3
V,
10
3
V, and
10
6
F
E
K
¼
72
E
l
¼
49
:
4
E
Na
¼
55
C
m
¼
1
:
Solution
Let current
I
m
be given by
I
m
¼
Ku
ð
(
t
Þ
u
(
t
t
0
Þ
Þ
. In Example Problem 12.8, the conductances
10
3
S and
10
3
S
at resting potential were calculated as
G
K
¼
0
:
3667
G
Na
¼
0
:
010614
:
Since
G
K
and
G
Na
remain constant for a subthreshold current stimulus in this problem, the circuit in
Figure 12.28 reduces to the following circuit. For ease in analysis, this circuit is replaced by the
following Thevenin's equivalent circuit with
1
G
Na
þ
G
K
þ
G
l
¼
R
Th
¼
1
:
4764
k
O
and
V
TH
¼
60 mV
:
Outside
G
K
G
l
G
Na
I
m
C
m
V
m
-
-
-
E
Na
E
K
E
l
+
+
+
Inside
Since the solution in Example Problem 12.6 is the same as the solution in this problem,
we have
u
u
t
R
TH
C
m
e
t
t
o
e
V
m
(
t
Þ¼
V
TH
þ
R
TH
K
1
(
t
Þ
R
TH
K
1
(
t
t
0
Þ
R
TH
C
m
For convenience, assume the current pulse
t
0
>
5t:
Therefore, for 5t <
t
t
o
,
V
m
¼
40 mV accord-
ing to the problem statement at steady state, and from the preceding equal,
V
m
reduces to
V
m
¼
0
:
040
¼
V
TH
þ
K
R
TH
¼
0
:
06
þ
K
1476
:
4
which yields
K
¼
13
:
6 mA
:
Since t ¼
R
TH
C
m
¼
1
:
47 ms, any value for
t
0
greater than 5t ¼
7
:
35 ms
brings
V
m
to
40 mV with
K
¼
13
:
6 mA
:
Naturally, a larger current pulse magnitude is needed for
an action potential because as
V
m
exponentially approaches threshold (reaching it with a duration
Na
þ
conductance channels become inactive and shut down.
of infinity), the
