An application of a weak solution of the cable equation to the Rall model of a nerve cell

The Rall model of a nerve cell with the lumped soma and natural termination boundary conditions at the nerve ends is considered as a particular case of the cable equation with oblique linear boundary conditions. A variational approach is taken to derive a weak solution of the problem. This extends classical results and removes a major obstacle of nonorthogonal boundary conditions in standard analysis of the Rall model. Additionally, new estimates of nerve cell time constants are obtained. The weak solution is applied to describe the nerve membrane potential following current injection at the soma, synaptic input, and instant unit current charge.