Biomedical Engineering Reference
In-Depth Information
V
TH
¼
R
Na
R
Cl
E
K
þ
R
K
R
Cl
E
Na
þ
R
K
R
Na
E
Cl
R
Na
R
Cl
þ
R
K
R
Cl
þ
R
K
R
Na
ð
12
:
36
Þ
Thetimeconstantforthemembranecircuitmodelist
¼
R
TH
C
m
,andat5t the response
is within 1 percent of steady-state. The range for t is from 1 to 20 ms in a typical neuron.
In addition, at steady-state the capacitor acts as an open circuit and
V
TH
¼
V
m
,asit
should.
EXAMPLE PROBLEM 12.6
Compute the change in
V
m
due to a current pulse through the cell membrane.
Outside
I
i
R
TH
I
C
I
m
(t)
I
m
C
m
V
m
K
-
V
TH
+
0
t
0
Time
Inside
Solution
Experimentally, the stimulus current is a pulse passed through the membrane from
an intracellular electrode to an extracellular electrode, as depicted in the circuit diagram
above. The membrane potential,
V
m
, due to a current pulse,
I
m
, with amplitude
K
and duration
t
0
applied at
t
¼
0, is found by applying Kirchhoff's current law at the cytoplasm, yielding
I
m
þ
V
m
V
TH
R
TH
þ
C
m
dV
m
dt
¼
0
The Laplace transform of the node equation is
s
Þþ
V
m
(
R
TH
V
TH
s
Þ
s
Þ
C
m
V
m
0ðÞ¼
I
m
(
sR
TH
þ
sC
m
V
m
(
0
Combining common terms gives
V
m
(
1
C
m
R
TH
s
Þ¼
V
m
0ðÞþ
I
m
ðÞ
C
m
þ
V
TH
s
þ
sC
m
R
TH
