Eigenslope Method for Second-Order Parabolic Partial Differential Equations and the Special Case of Cylindrical Cellular Structures With Spatial Gradients in Membrane Capacitance
Boundary value problems in PDEs usually require determination of the eigenvalues and Fourier coefficients for a series, the latter of which are often intractable. A method was found that simplified both analytic and numeric solutions for Fourier coefficients 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 different form, as other methods. Numerical solutions obtained by calculating 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 complex eigenvalues. As an example, the widespread assumption of uniform capacitance was tested. An analytic model of cylindrical brain cell structures with an exponential distribution of membrane capacitance was developed with the eigenslope method. The stimulus-response properties of the models were compared under different configurations and shown to fit 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 sufficient 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 physiologically significant. The widespread assumption of uniform membrane capacitance is likely to be a valid approximation.
Knisley, Jeff; Glenn, L. Lee; Joplin, Karl; and Carey, Patricia. 2007. 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 Math Tools and Statistical Technique. World Scientific Publishers. https://doi.org/10.1142/9789812772121_0012 ISBN: 9789812772121