From Maxwell's equations to the cable equation and beyond

Maxwell's equations are taken as the starting point for the development of a mathematical model of a dendrite. The three-dimensional model of the evolution of the dendritic membrane potential based on these equations gives rise to a hierarchy of one-dimensional membrane equations. Under sufficiently strong assumptions, the first membrane equation is identical to the conventional cable equation. The second membrane equation explicitly includes the influence of dendritic taper and non-axial gradients in the intra-cellular potential. The procedure of starting from a three-dimensional model and extracting from it a one-dimensional approximation provides a prescription of how to incorporate three-dimensional properties of a dendrite in a one-dimensional representation, by contrast with an approach which aims to modify the traditional cable equation to take account of three-dimensional structure. Finite element methods are used to solve the membrane equations. An example based on a simple model of a tapered dendrite with differently placed distributions of synaptic input suggests that the effect of taper on the spike train output from the model is more important for distal synapses than those closer to the soma.