Techniques for obtaining analytical solutions for Rall's model neuron.

The Green's function (G) is obtained for a cable equation with a lumped soma boundary condition at x = 0 and a sealed end at x = L infinity. The coefficients in the eigenfunction expansion of G are obtained using the calculus of residues. This expansion converges rapidly for large t. From an estimate of the higher eigenvalues, an approximate bound is obtained for the remainder after so many terms. The leading terms are also obtained in an expansion for G which converges rapidly for small t. Similarly, series expansions for the voltage are obtained which converge rapidly at small or large t when a constant current is injected at the soma and when a (synaptic input) current bte-at occurs at a point along the cable.