Biomedical Engineering Reference
In-Depth Information
in membrane capacitance, and thus meets the minimum requirement for
biologically realism of continuity.
Unfortunately, the cable equation (3) cannot be solved in general.
The same waveform can be well-approximated by more than one multi-
exponential
11
, so numerical solutions to (3) - (5) are limited in their ap-
plicability to the problem of parameter identication. There are certain
choices for C
m
(X) for which closed form solutions are possible, but for
such choices of C
m
(X), it has not been possible to nd closed form so-
lutions for the eigenvalues and Fourier coecients except in special cases.
However, many choices for C
m
(X) lead to closed form expressions for the
Laplace transform of the solution. From the Laplace transform solutions,
the eigenvalues and Fourier coecients can be determined using the theory
of residues from complex analysis
1;6;31
.
In particular, such a solution is possible if C
m
(X) represents the expo-
nentially graded membrane capacitance given by
(1 + Me
2X
)
2
C
m
(X) =
where for the ratio parameter " = C=C
m
we have
1
p
"
M =
p
"e
2L
= C (1 + M)
2
:
In Fig. 2, the relation between and L is shown for " = 0:9; R
m
C
m
=
0:005. Note how cylinders with an electrotonic length greater than one, and
a 10% exponential gradient in capacitance produce a decay time that is very
close to that of the uniform capacitance model (in which = 0:005 ms).
The Laplace transform of (3) is
V
00
(sRC(X) + 1) V =C
m
(X)V
ss
(8)
which has a solution of the form
V (X) = W (X)
V
ss
(X)
s
;
where the transient W satises the homogeneous equation associated with
(8). The Laplace transform of the transient is given by
D
1
p
p
p
M + e
2x
s+1=2
+D
2
s+1=2
M + e
2x
W (X) =
Me
2x
+ 1
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