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Solving the Cable Equation

A fundamental equation in mathematical biophysics is the cable equation, whose solution gives the time-dependent distribution of voltage along the length of a biological membrane, such as a neuronal axon. The voltage distribution depends upon a number of variables, such as membrane conductance , membrane capacitance , intracellular resistivity , the maximal value of the input current , the duration of this current, and the position on the cable in which the current is injected. In this Demonstration, these parameters can be varied and their effect on the spatiotemporal evolution of the voltage distribution viewed.

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The cable equation is a linear parabolic partial differential equation, in the same class as the heat and diffusion equations,
,
where is the cable diameter (50 m in this Demonstration) and is the input current. There are analytical solutions to this equation for special cases, but it is often more efficient and as accurate to break the cable into isopotential compartments so that the partial differential equation becomes a set of ordinary differential equations, one for each compartment. Such a system can be solved, with appropriate initial and boundary conditions, as a sparse-matrix-vector-valued differential equation, which is the route taken in this Demonstration.
The membrane parameters of the cable equation have different effects upon changes in the voltage distribution. This is most easily seen by considering the membrane length constant () and time constant (). Expressing these values in terms of membrane parameters,
, .
The length constant determines signal attenuation in space, where a larger value corresponds to less spatial signal attenuation. The time constant determines signal attenuation in time, where a larger value corresponds to less temporal signal attenuation.
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