Numerical Method for Homoclinic andHeteroclinic Orbits of Neuron Models

A twisted heteroclinic cycle was proved to exist more than twenty-five years ago for the reaction-diffusion FitzHugh-Nagumo equations in theirtraveling wave moving frame. The result implies the existence of infinitelymany traveling front waves and infinitely many traveling back waves for thesystem. However efforts to numerically render the twisted cycle were not fruit-ful for the main reason that such orbits are structurally unstable. Presentedhere is a bisectional search method for the primary types of traveling wave solu-tions for the type of bistable reaction-diffusion systems the FitzHugh-Nagumoequations represent. The algorithm converges at a geometric rate and the wavespeed can be approximated to significant precision in principle. The methodis then applied for a recently obtained axon model with the conclusion thattwisted heteroclinic cycle maybe more of a theoretical artifact.