Large deviations of piecewise-deterministic Markov processes with application to stochastic calcium waves

We prove a large deviation principle for piecewise deterministic Markov processes (PDMPs). This is an asymptotic estimate for the probability of a trajectory in the large size limit. Explicit Euler-Lagrange equations are determined for computing optimal first-hitting-time trajectories. The results are applied to a model of stochastic calcium dynamics. It is widely conjectured that the mechanism of calcium puff generation is a multiscale process: with microscopic stochastic fluctuations in the opening and closing of individual channels generating cell-wide waves via the diffusion of calcium and other signaling molecules. We model this system as a PDMP, with $N\gg 1$ stochastic calcium channels that are coupled via the ambient calcium concentration. We employ the large deviations theory to estimate the probability of cell-wide calcium waves being produced through microscopic stochasticity.