Biomedical Engineering Reference
In-Depth Information
CHAPTER 12
EIGENSLOPE METHOD FOR SECOND-ORDER
PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS AND
THE SPECIAL CASE OF CYLINDRICAL CELLULAR
STRUCTURES WITH SPATIAL GRADIENTS IN
MEMBRANE CAPACITANCE
Lloyd Lee Glenn a and Je Knisley
The Institute for Quantitative Biology,
East Tennessee State University,
P.O. Box 70658, Johnson City, TN 37614, USA
E-mail: a glennl@etsu.edu
Boundary value problems in PDEs usually require determination of the
eigenvalues and Fourier coecients for a series, the latter of which are
often intractable. A method was found that simplied both analytic
and numeric solutions for Fourier coecients based on the slope of the
eigenvalue function at each eigenvalue (eigenslope). Analytic solutions
by the eigenslope method resulted in the same solutions, albeit in dif-
ferent form, as other methods. Numerical solutions obtained by calcu-
lating the slope of the eigenvalue function at each root (hand graphing,
Euler's, Runge-Kutta, and others) also matched. The method applied
to all classes of separable PDEs (parabolic, hyperbolic, and elliptical),
orthogonal (Sturm-Liouville) or non orthogonal expansions, and to com-
plex eigenvalues. As an example, the widespread assumption of uniform
capacitance was tested. An analytic model of cylindrical brain cell struc-
tures with an exponential distribution of membrane capacitance was de-
veloped with the eigenslope method. The stimulus-response properties
of the models were compared under dierent congurations and shown
to t to experimental data from dendritic neurons. The long-standing
question was addressed of whether the amount of variation of membrane
capacitance measured in experimental studies is sucient to markedly
alter the vital neuron characteristic of passive signal propagation. We
concluded that the degree of membrane capacitance variation measured
in cells does not alter electrical responses at levels that are physiologi-
cally signicant. The widespread assumption of uniform membrane ca-
pacitance is likely to be a valid approximation.
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