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| Numerical Method for Homoclinic andHeteroclinic Orbits of Neuron Models |
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| KeyWord:FitzHugh-Nagumo equations twisted heteroclinic loop bifurca-
tion singular perturbation bisection method |
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| Abstract: |
| A twisted heteroclinic cycle was proved to exist more than twenty-
five years ago for the reaction-diffusion FitzHugh-Nagumo equations in their
traveling wave moving frame. The result implies the existence of infinitely
many traveling front waves and infinitely many traveling back waves for the
system. However efforts to numerically render the twisted cycle were not fruit-
ful for the main reason that such orbits are structurally unstable. Presented
here is a bisectional search method for the primary types of traveling wave solu-
tions for the type of bistable reaction-diffusion systems the FitzHugh-Nagumo
equations represent. The algorithm converges at a geometric rate and the wave
speed can be approximated to significant precision in principle. The method
is then applied for a recently obtained axon model with the conclusion that
twisted heteroclinic cycle maybe more of a theoretical artifact. |
