Biomedical Engineering Reference
In-Depth Information
dimensionless units are denoted by uppercase variables)
52
. Note that C
m
is a function of X and varies with distance from origin, which is the end of
the cylinder. We assume sealed-end boundary conditions
@V
@X
(L; t) = 0
(4)
@V
@X
(0; t) = 0:
(5)
Typically, the cell is saturated to a steady state using a somatic cur-
rent source with a constant magnitude of I
stim
, and then the current source
is switched o (actually, it is switched to a voltage recorder). Thus, the ini-
tial condition for (3) - (5) is the steady state of equations (3) and (5) subject
to the modied boundary condition
@V
@X
(L; t) = I
stim
:
Separation of variables with V (X; t) = (X)T (t) yields
00
+ (R
m
C
m
(X)1) = 0
and
T
0
=T:
The solution is of the form
8;9;10;11
X
1
A
n
n
(X) e
n
t
V (X; t) =
n=0
where the
n
> 0 are the eigenvalues, the A
n
are the Fourier coecients
and the
n
are the separated solutions or eigenfunctions of equations (3)
- (5). We normalize the eigenfunctions so that
n
(0) = 1. At the proximal
end of the cylinder,
1
X
A
n
e
n
t
:
V (0; t) =
n=0
OnL
2
[0; L] ; we dene the inner product with weight (X) by
Z
L
hf; gi=
f (X) g (X) (X) dX:
0
From Sturm-Liouville theory
2
we have orthogonality of the eigenfunctions
and
hV (; t) ;
n
i
A
n
e
n
t
=
h
n
;
n
i
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