  Linearization around unstable travelling waves in excitable systems can be
used to approximate strength-extent curves in the problem of initiation of
excitation waves for a family of spatially confined perturbations to the rest
state. This theory relies on the knowledge of the unstable travelling wave
solution as well as the leading left and right eigenfunctions of its
linearization. We investigate the asymptotics of these ingredients, and utility
of the resulting approximations of the strength-extent curves, in the slow-fast
limit in two-component excitable systems of FitzHugh-Nagumo type, and test
those on four illustrative models. Of these, two are with degenerate dependence
of the fast kinetic on the slow variable, a feature which is motivated by a
particular model found in the literature. In both cases, the unstable
travelling wave solution converges to a stationary "critical nucleus" of the
corresponding one-component fast subsystem. We observe that in the full system,
the asymptotics of the left and right eigenspaces are distinct. In particular,
the slow component of the left eigenfunction corresponding to the translational
symmetry does not become negligible in the asymptotic limit. This has a
significant detrimental effect on the critical curve predictions. The theory as
formulated previously uses an heuristic to address a difficulty related to the
translational invariance. We describe two alternatives to that heuristic, which
do not use the misbehaving eigenfunction component. These new heuristics show
much better predictive properties, including in the asymptotic limit, in all
four examples.
