  This paper investigates travelling wave solutions of the FitzHugh-Nagumo
equation from the viewpoint of fast-slow dynamical systems. These solutions are
homoclinic orbits of a three dimensional vector field depending upon system
parameters of the FitzHugh-Nagumo model and the wave speed. Champneys et al.
[A.R. Champneys, V. Kirk, E. Knobloch, B.E. Oldeman, and J. Sneyd, When
Shilnikov meets Hopf in excitable systems, SIAM Journal of Applied Dynamical
Systems, 6(4), 2007] observed sharp turns in the curves of homoclinic
bifurcations in a two dimensional parameter space. This paper demonstrates
numerically that these turns are located close to the intersection of two
curves in the parameter space that locate non-transversal intersections of
invariant manifolds of the three dimensional vector field. The relevant
invariant manifolds in phase space are visualized. A geometrical model inspired
by the numerical studies displays the sharp turns of the homoclinic
bifurcations curves and yields quantitative predictions about multi-pulse and
homoclinic orbits and periodic orbits that have not been resolved in the
FitzHugh-Nagumo model. Further observations address the existence of canard
explosions and mixed-mode oscillations.
