Biomedical Engineering Reference
In-Depth Information
Table 3. Change in decay times in membrane cylinders with ex-
ponentially distributed non uniformity in C
m
. The C
m
was de-
creased across [0; L] by 2% to 20%, decreased across [0; L] from
-2% to -20%, and made constant (0%). In all cases, the average
C
m
across [0; L] remained constant (see 0%) at R
m
C
m
= 0:01 s.
Other parameters: L = 1, I
stim
= G
1
Percentage
90% Decay
50% Decay
10% Decay
Increase in C
m
Time (ms)
Time (ms)
Time (ms)
20%
0.12772
3.98
18.87
10%
0.14597
4.16
19.66
5%
0.15519
4.25
20.04
2%
0.15628
4.29
20.24
0%
0.15693
4.31
20.30
2%
0.15744
4.33
20.51
5%
0.15883
4.37
20.70
10%
0.15948
4.41
21.06
20%
0.15987
4.52
21.63
6.2. Discussion
The hypothesis tested by the mathematical models was that biologically-
signicant dierences between membrane cylinders with a homogeneous
membrane capacitance and those with a heterogeneous exponentially-
graded membrane capacitance do not aect passive responses, provided
that the variation in capacitance is within the range estimated in experi-
mental studies. The hypothesis was accepted. An exponential spatial vari-
ation in membrane capacitance at experimentally measured levels (5%)
produces an error of 1.5% in the time it takes the voltage response to a
step stimulus to decay to 50% of its initial value (Table 3), as compared
to the assumption of a uniform membrane capacitance. This is a relatively
small dierence. Moreover, although the evidence is limited (see discussion
below), the maximum that capacitance changes with distance is closer to
2%. This produces a 0.5% error in the same. Therefore, for most models
of cells with a non uniform distribution of sodium channels or other chan-
nels, we conclude that the assumption of uniform membrane capacitance is
largely a valid approximation.
The property of capacitance stems from the close proximity of two elec-
trically conducting structures surfaces with the space between them lled
by a poorly conducting medium. The membrane that envelops cells basi-
cally consists of a thin lipid bilayer that incorporates proteins, many of
which span the membrane. The membrane capacitance stems from conduc-
tive extracellular uid being separated by the inner hydrophobic layer of
the cell membrane, which forms the poorly conducting structure required
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