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 262

which can be used to estimate the A n once the eigenvalues and eigenfunc-
tions are known. The Fourier coecients can be estimated using the method
of residues, as shown below.

Fig. 2. Dependence of decay time ( ) on electronic length (L) in the exponential model
with a 10% origin-to-terminal capacitance change. Inset diagram summarizes the three
models developed, including the stimulus and response (I stim ; V m ) and the longitudinal
capacitance distribution in the point, step, and exponential models.

Saturation to a steady state by a constant current stimulus prior to
voltage recording implies that the initial potential distribution for (3) - (5)
is the steady state distribution V ss of the system

d 2 V ss
dX 2

V ss = 0;

0 < X < L

dV ss
dX

(L) = 0

G 1 dV ss
dX

(0) = I stim

where G 1 is the input conductance. The solution is given by

I stim cosh (LX)
G 1 sinh (L)

V ss (X) =

: (6)