Biomedical Engineering Reference
In-Depth Information
EXAMPLE PROBLEM 12.7
Compute I c and I l
through a cell membrane for a subthreshold clamp voltage.
Solution
Assume that the
K þ voltage-time-dependent channels are not activated because the
stimulus is below threshold. This eliminates these gates from the analysis although this is not
actually true as shown in Example Problem 12.9. The cell membrane circuit is given by
Na þ and
Outside
-
I l
R l
I C
V c
+
C m
V m
-
R s
I m
E l
+
Inside
where
R s
is the resistance of the wire. Applying Kirchhoff's current law at the cytoplasm gives
C m dV m
dt þ V m E l
þ V m V c
R s
¼
0
R l
Rearranging the terms in the previous equation yields
C m dV m
dt þ R l þ R s
V m ¼ R l V c þ R s E l
R l R s
R l R s
With the initial condition
V m (0
Þ¼ E l , the solution is given by
e R l þ R ð Þ t
V m ¼ R l V c þ R s E l
R l þ R s
þ R l E l V c
ð
Þ
R l R s C m
R l þ R s
Now
e R l þ R ð Þ t
I c ¼ C m dV m
dt ¼ E l V c
R l R s C m
R s
I l ¼ V m E l
R l
I l ¼ V c E l
R l þ R s
and
. At steady-state,
.
Reconstructing the Action Potential
By analyzing the estimated
from voltage clamp pulses of various ampli-
tudes and durations, Hodgkin and Huxley were able to obtain a complete set of non-
linear empirical equations that described the action potential. Simulations using these
G Na
and
G K
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