Phantom ducks and models of excitability

We study a system which may serve as a prototype model for excitability. Its long-term behavior is governed by a globally attracting fixed point, but it has the interesting property that an appropriate small perturbation of the equilibrium situation elicits a large (transient) response. Using singular perturbation techniques, the threshold for excitation is determined. Asymptotic expansions for the threshold are given and their validity is proved. An associated system is studied, which yields further quantitative information on the position of the threshold.