Solution of the nerve cable equation using Chebyshev approximations

Abstract

The propagation of excitation along the dendrites and the axon of a neurone is described by a partial differential equation which is nonlinear when voltage-gated conductances are present. In this case, numerical methods are employed to obtain a solution: the evolution of the membrane potential in space and time. Even when the membrane is passive (linear), numerical methods might still be preferred to analytical ones that are often too cumbersome to obtain. In this paper, we present the Chebyshev pseudospectral or collocation method as an alternative to the hitherto commonly used finite difference schemes (compartmental models) that are based on sufficiently fine equidistant subdivisions of the spatial structure (dendrites or axon). In the Chebyshev method, solutions are approximated by finite Chebyshev series. The solutions have uniform, usually high, numerical accuracy at any spatial point, not only at the original collocation points. Often, truncation errors become negligible, hence, the total error is essentially the rounding error of the computations. Furthermore, quantities involving spatial derivatives, and in particular the axial current, can be computed exactly from the solution, i.e. the membrane potential. Space-dependent parameter distributions (channel densities, non-uniform dendritic geometries), as well as mixed linear boundary conditions can easily be implemented, and can be chosen from the large class of piecewise smooth functions.