Temporal stability of solitary impulse solutions of a nerve equation.

We study a differential equation that models nerve impulse transmission. The nonlinearity is simplified to be piecewise linear in order to allow explicit solution. In general, two solitary impulse solutions are exhibited. The temporal stability of these solutions is analyzed by a technique that identifies the number of unstable modes. These results extend the results of Rinzel and Keller (1973, Biophys. J. 13:1313) by showing that the slower unstable solution has only one unstable mode, and that the fast solution, as conjectured, has no unstable modes and is therefore stable.