EIGENSLOPE METHOD FOR SECOND-ORDER PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS AND THE SPECIAL CASE OF CYLINDRICAL CELLULAR STRUCTURES WITH SPATIAL GRADIENTS IN MEMBRANE CAPACITANCE - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 266

the eigenvalue function
p p p

R m C m 1 + M tan R m C m 1 ln M + 1

p p p

R m C m 1 tan R m C m 1 ln M + 1 M

p

p p

R m C m 1 + Me 2L tan M + e 2L R m C m 1 ln

=

Me 2L :

p

p p

R m C m 1 tan R m C m 1 ln M + e 2L

p

When M is chosen such that n 1 6= M; the coecients A n are of the

form

2I stim (M + 1)
G 1 n f ( n )

p

A n =

n 1M

p

M + 1
M + e 2L

cos n 1 ln +

p p

n 1Me 2L

M + 1
M + e 2L

i

i sin n 1 ln + p ; (10)

i n 1 + Me 2L

where f ( n ) is given by
f ( n ) =
cos

p p

M+1
M+e 2L

M+1
M+e 2L

n 1 ln + i sin n 1 ln

M + 1
M + e 2L

p ln

i k j 1

M 2 e 2L n + 1 1e 2L

2M
i

p

2 :

i

p p

n 1

2

i n 1M n 1 + Me 2L

The Fourier coecients were subsequently obtained from the eigen-
value function by the eigenslope method 12 , and conrmed to numeri-
cally match the above solution. Examples of the eigenvalues and Fourier
coefficients for the above models are shown in Table 1.

4.4. Biophysical Representation

The eigenvalues in this model correspond biophysically to the decay rates
of the series of superimposed exponential decays reected by the ends of
the cylindrical cells. The Fourier coecients represent the eective (dis-
tributed) membrane capacitance. That is, the coecients correspond to
the total amount of charge storage at steady-state, for each of the reected
exponential decays. The eigenslope relation indicates that steady-state
amplitude of each reected and superimposed decay is an inverse function
of its decay rate and eective membrane capacitance.