Eighth Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
Electron. J. Diff. Eqns., Conference 19 (2010), pp. 189-196.
Title: Comparison of time stepping schemes on the cable equation
Authors: Chuan Li (Univ. of Tennessee, Knoxville TN, USA)
Vasilios Alexiades (Univ. of Tennessee, Knoxville TN, USA)
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,
$$
\frac{1}{r_a}\frac{\partial^2V}{\partial x^2} =
C_m\frac{\partial V}{\partial t} + I_{\rm ion}(V,t)
+ I_{\rm stim}(t)
$$
where $r_a$ and $C_m$ are the axial resistance and membrane
capacitance. The source term $I_{\rm ion}$ represents the total ionic
current across the membrane, governed by the Hodgkin-Huxley or other
more complicated ionic models. $I_{\rm stim}(t)$ 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.