Biomedical Engineering Reference
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as can be veried by substitution. The sealed end boundary condition at
X = L yields
p
p
s + 1Me 2L
s+1
M + e 2L
p
D 2 = D 1
s + 1 + Me 2L
which is combined with the other boundary condition to yield a transient
at X = 0 where W = W(0) of
p s+1 + (
p
s+1Me 2L )
M+1
M+e 2L
p
W = I stim (M + 1)
G 1 s
(
s+1+Me 2L )
p
;
(9)
D
s + 1 + M
where
p
s+1 (
p
p
s+1Me 2L )
s+1M
M+1
M+e 2L
D =
s+1+Me 2L ) :
p
p
(
s+1+M
Table 1. Example of eigenvalues and Fourier coe-
cients produced by a model with a step change in mem-
brane capacitance with distance and a model with ex-
ponential spatial variation in membrane capacitance
for a voltage response at X=0 to a current pulse stim-
ulus at X=0 for a membrane cylinder with a length of
L = 1. Amplitude coecients are reported as a frac-
tion of steady state. Z is the distance from the origin
for a step change in membrane capacity (Fig. 2, in-
set) (Note: The rst two Fourier coecients were com-
puted numerically using the eigenslope method (Knis-
ley and Glenn 1997), and then veried using analytical
expressions. The eigenvalues were found using a bisec-
tion method, and the slopes at the eigenvalues were
found by the divided dierence method (see Equation
4 in Section 7.1 of Kincaid and Cheney 1991). The 0
and 1 were computed for the point capacitance model
with C s
= 0:9F , C m
= 1F , R m
= 5; 000, L = 1,
and rho = 0:5; 1; 2; 3.)
Model
0 (ms)
A 0
1 (ms)
A 1
Exponential
4.8222
0.7551
0.4334
0.1481
Z = 0:2
4.9000
0.7543
0.4355
0.1465
Z = 0:3
4.850
0.7524
0.4304
0.1497
Z = 0:4
4.801
0.7515
0.4282
0.1522
Since the eigenvalues of (9) are negative, the exponents in (9) are imag-
inary, complex exponentials and De Moivre's formula, transform (9) into
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