Eighth Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 19 (2010), pp. 189-196.

Comparison of time stepping schemes on the cable equation

Chuan Li, Vasilios Alexiades

Abstract:
Electrical propagation in excitable tissue, such as nerve fibers and heart muscle, is described by a parabolic PDE for the transmembrane voltage $V(x,t)$, known as the cable equation,
$$ \frac{1}{r_a}\frac{\partial^2V}{\partial x^2} = C_m\frac{\partial V}{\partial t} + I_{ion}(V,t) + I_{stim}(t) $$
where $r_a$ and $C_m$ are the axial resistance and membrane capacitance. The source term $I_{ion}$ represents the total ionic current across the membrane, governed by the Hodgkin-Huxley or other more complicated ionic models. $I_{stim}(t)$ is an applied stimulus current.
We compare the performance of various low and high order time-stepping numerical schemes, including DuFort-Frankel and adaptive Runge-Kutta, on the 1D cable equation.

Published September 25, 2010.
Math Subject Classifications: 65M08, 35K57, 92C37.
Key Words: Explicit schemes; super time stepping; adaptive Runge Kutta; Dufort Frankel; action potential; Luo-Rudy ionic models.

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Chuan Li
Mathematics Department, University of Tennessee
Knoxville TN 37996, USA
email: li@math.utk.edu
Vasilios Alexiades
Mathematics Department, University of Tennessee
Knoxville TN 37996, USA
email: alexiades@utk.edu

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