Electronic Journal of Differential Equations (Sep 2010)
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.