Biomedical Engineering Reference
In-Depth Information
where h (
j
) = 0 and h
0
(
j
) 6= 0; for all j = 1; 2; : : : and where g(x; p)
is also second dierentiable in x with g (x;
j
) 6= 0 for all j = 1; 2; : : : and
for all x (for example, dene g (x; p) = h (p) V (x; p) where
1
Y
p
j
e
p=
j
h (p) =
1 +
j=1
comes from Hadamard's theorem
6
). It follows that
(p +
j
) g (x; p)
h (p)
=
g (x;
j
)
h
0
(
j
)
A
i
i
(x) =
lim
p!
j
:
(2)
If g
1
(x; p) and h
1
(p) are analytic and also satisfy (1), then dene m (p)
such that h
1
(p) = m (p) h (p) for all p and m (
j
) 6= 0 for all j = 1; 2; : : : :
Since h
0
1
(
j
) = m (
j
) h
0
(
j
) ; the limit (2) holds for any ratio of the
form (1).
In particular, it follows that the A
n
are given by
1
pf (x; p)
where f (x;
j
) = 0 for all j: We say that f (x; p) is the eigenvalue function
for the problem. It follows that
V (x; p) =
1
h
i
A
i
i
(x) =
:
@f
@p
(x;
j
)
j
For each xed x; we say that f
p
(x;
j
) is the eigenslope of the eigenvalue
function.
Analytically, the eigenslope approach provides a new approach to nding
the coecients (A
i
), namely, by dierentiation of the eigenvalue equation.
Numerically, it provides a new method for determining the coecients by
nding the slope of the eigenvalue function at each eigenvalue by standard
numeric methods. Conceptually, the eigenslope shows that eigenvalues and
Fourier coecients are related by a very simple relation. Next, the eigens-
lope method is used to develop, solve, and verify a model that addresses
the signicance of non uniform cell membrane capacitance in electrotonic
signal propagation.
4. Application of the Eigenslope Method to Cellular
Models With Propagation
Cell membranes are key building blocks of all cells, and regardless of cell
type, the membranes have the hallmark electrical property of a constant
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