  The Fitzhugh-Nagumo equations have been used as a caricature of the
Hodgkin-Huxley equations of neuron firing to better understand the essential
dynamics of the interaction of the membrane potential and the restoring force
and to capture, qualitatively, the general properties of an excitable membrane.
Even though its simplicity allows very valuable insight to be gained, the
accuracy of reproducing real experimental results is limited. In this paper, we
utilize a modified version of the Fitzhugh-Nagumo equations to model the
spatial propagation of neuron firing; we assume that this propagation is (at
least, partially) caused by the cross-diffusion connection between the
potential and recovery variables. We show that the cross-diffusion version of
the model, besides giving rise to the typical fast traveling wave solution
exhibited in the original diffusion Fitzhugh-Nagumo equations, also gives rise
to a slow traveling wave solution. We analyze all possible traveling wave
solutions of the Fitzhugh-Nagumo equations with this cross-diffusion term and
show that there exists a threshold of the cross-diffusion coefficient (the
maximum value for a given speed of propagation), which bounds the area where
normal impulse propagation is possible.
