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The FitzHugh-Nagumo equation, which was derived as a simplification of theHodgkin-Huxley model for nerve impulse propagation, has been extensivelystudied as a paradigmatic activator-inhibitor system. We consider the versionof this system in which two agents diffuse at an equal rate. Using geometricsingular perturbation theory, we prove the existence and stability of fasttraveling pulses. The stability proof makes use of the Maslov index--aninvariant of symplectic geometry--to count unstable eigenvalues for thelinearization about the wave. The calculation of the Maslov index is carriedout by tracking the evolution of the unstable manifold of the rest state usingthe timescale separation. This entails a careful consideration of how thetransition from fast to slow dynamics occurs in the tangent bundle over thewave. Finally, we observe in the calculation that the Maslov index lacksmonotonicity in the spatial parameter, which distinguishes this application ofthe Maslov index from similar analyses of Hamiltonian systems.

On the Existence and Stability of Fast Traveling Waves in a Doubly-Diffusive FitzHugh-Nagumo System