  The FitzHugh-Nagumo equation has been investigated with a wide array of
different methods in the last three decades. Recently a version of the
equations with an applied current was analyzed by Champneys, Kirk, Knobloch,
Oldeman and Sneyd using numerical continuation methods. They obtained a
complicated bifurcation diagram in parameter space featuring a C-shaped curve
of homoclinic bifurcations and a U-shaped curve of Hopf bifurcations. We use
techniques from multiple time-scale dynamics to understand the structures of
this bifurcation diagram based on geometric singular perturbation analysis of
the FitzHugh-Nagumo equation. Numerical and analytical techniques show that if
the ratio of the time-scales in the FitzHugh-Nagumo equation tends to zero,
then our singular limit analysis correctly represents the observed
CU-structure. Geometric insight from the analysis can even be used to compute
bifurcation curves which are inaccessible via continuation methods. The results
of our analysis are summarized in a singular bifurcation diagram.
