Comparison of time stepping schemes on the cable equation

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, $$ frac{1}{r_a}frac{partial^2V}{partial x^2} = C_mfrac{partial V}{partial t} + I_{m ion}(V,t) + I_{m stim}(t) $$ where $r_a$ and $C_m$ are the axial resistance and membrane capacitance. The source term $I_{m ion}$ represents the total ionic current across the membrane, governed by the Hodgkin-Huxley or other more complicated ionic models. $I_{m stim}(t)$ is an applied stimulus current.

Keywords

• Explicit schemes
• super time stepping
• adaptive Runge Kutta
• Dufort Frankel
• action potential
• Luo-Rudy ionic models