We present a new derivation of the classical action underlying a large deviation
principle (LDP) for a stochastic hybrid system, which couples a piecewise
deterministic dynamical system in Rd with a time-homogeneous Markov chain on
some discrete space Γ. We assume that the Markov chain on Γ is ergodic, and that
the discrete dynamics is much faster than the piecewise deterministic dynamics
(separation of time-scales). Using the Perron-Frobenius theorem and the calculusof-variations, we show that the resulting action Hamiltonian is given by the Perron
eigenvalue of a |Γ|-dimensional linear equation. The corresponding linear operator
depends on the transition rates of the Markov chain and the nonlinear functions of
the piecewise deterministic system. We compare the Hamiltonian to one derived
using WKB methods, and show that the latter is a reduction of the former. We
also indicate how the analysis can be extended to a multi-scale stochastic process,
in which the continuous dynamics is described by a piecewise stochastic differential
equations (SDE). Finally, we illustrate the theory by considering applications to
conductance-based models of membrane voltage fluctuations in the presence of
stochastic ion channels.