  We use geometric singular perturbation techniques combined with an action
functional approach to study traveling pulse solutions in a three-component
FitzHugh--Nagumo model. First, we derive the profile of traveling $1$-pulse
solutions with undetermined width and propagating speed. Next, we compute the
associated action functional for this profile from which we derive the
conditions for existence and a saddle-node bifurcation as the zeros of the
action functional and its derivatives. We obtain the same conditions by using a
different analytical approach that exploits the singular limit of the problem.
We also apply this methodology of the action functional to the problem for
traveling $2$-pulse solutions and derive the explicit conditions for existence
and a saddle-node bifurcation. From these we deduce a necessary condition for
the existence of traveling $2$-pulse solutions. We end this article with a
discussion related to Hopf bifurcations near the saddle-node bifurcation.
