Neutrally stable traveling wave solutions of nerve conduction equations

The equations studied by Hodgkin and Huxley, FitzHugh, Nagumo, and others belong to a general class of nerve conduction equations. Each has a family of periodic traveling wave solutions and typically more than one solitary pulse wave solution. Instability has been conjectured for certain of these solutions. Here stability is studied formally and neutral stability transitions are characterized parametrically. Two notions of linear stability are formulated. For initial value problems, temporal stability is applicable. Spatial stability is appropriate for boundary value problems like signal transmission along a nerve in response to a spatially localized time-dependent stimulus. Here it is shown that periodic wave trains with maximum or minimum frequency have neutral spatial stability. For solitary pulse solutions, propagation speed versus a typical parameter usually describes a double branched curve. It is shown that the speed curve knee corresponds to neutral spatial and temporal stability.