Biomedical Engineering Reference
In-Depth Information
Outside
R
Th
R
Th
R
Th
V
m
C
m
V
m
C
m
V
m
C
m
-
-
-
V
TH
V
TH
V
TH
+
+
+
R
a
R
a
Inside
FIGURE 12.17
Equivalent circuit of series of membrane sections connected with axial resistance, R
a
.
Current through the Membrane
I
m
FIGURE 12.18
The flow of current through a dendrite at steady-state. Since current seeks the path of least
resistance, most of the current leaves the dendrite at the injection site and becomes smaller with distance from
the injection site.
the smallest resistance
in relation to the other sections. The next largest current flow-
ing out of the membrane occurs in the next section, since it has the next smallest resistance,
R
TH
þ
R
a
R
T
ð Þ
. The change in
V
m
,
D
V
m
, from the injection site is independent of
C
m
and depends
solely on the relative values of
R
TH
and
R
a
:
The resistance seen
n
sections from the injec-
tion site is
R
TH
þ
nR
a
:
Since current decreases with distance from the injection site, then
D
V
m
also decreases with distance from the injection site because it equals the current
through that section times
R
TH
:
The change in membrane potential,
D
V
m
, decreases expo-
nentially with distance and is given by
l
D
V
m
¼
V
o
e
ð
12
:
37
Þ
q
is the membrane length constant,
R
TH
R
a
where l
¼
x
is the distance away from injection site,
V
o
is the change in membrane potential at the injection site.
1
The range of values for
and
l
is 0.1 to 1 mm. The larger the value of
l
, the greater the effect of the stimulation along
the length of the membrane.
1
Equation 12.37 is the solution of the one dimensional cable equation for the dendrite using partial
differential equations (see Keener and Sneyd for details).
