Path integrals and large deviations in stochastic hybrid systems

Abstract

We construct a path-integral representation of solutions to a stochastic hybrid system, consisting of one or more continuous variables evolving according to a piecewise-deterministic dynamics. The differential equations for the continuous variables are coupled to a set of discrete variables that satisfy a continuous-time Markov process, which means that the differential equations are only valid between jumps in the discrete variables. Examples of stochastic hybrid systems arise in biophysical models of stochastic ion channels, motor-driven intracellular transport, gene networks, and stochastic neural networks. We use the path-integral representation to derive a large deviation action principle for a stochastic hybrid system. Minimizing the associated action functional with respect to the set of all trajectories emanating from a metastable state (assuming that such a minimization scheme exists) then determines the most probable paths of escape. Moreover, evaluating the action functional along a most probable path generates the so-called quasipotential used in the calculation of mean first passage times. We illustrate the theory by considering the optimal paths of escape from a metastable state in a bistable neural network.

Feynman-Kac formula for stochastic hybrid systems

“…n (x, 0) = 1 for all n ∈ Γ.) We thus obtain the following path-integral for a 1D stochastic hybrid system [26,27]:…”

“…[33]. Here we explore an alternative approach to the analysis of functionals of stochastic hybrid system based on our recent path-integral representation of PDMPs [26,27].…”

“…In order to derive the Feynman-Kac formula and introduce notation, we first briefly recap the construction of the path-integral representation of stochastic hybrid systems [26,27]. The first step is to discretize time and write down the path-integral representation of Q(s, t, t 0 ) given by Eq.…”

“…We illustrate the theory in section IV by considering the occupation time for a velocity jump process. Finally, in section V we relate the analysis to our recent path-integral construction of stochastic hybrid systems [26,27].…”

“…In addition, a variety of complementary techniques in applied mathematics and mathematical physics have been used to solve first passage time problems in biological applications of stochastic hybrid systems. These include WKB approximations and matched asymptotics [10,14,15,[18][19][20][21], and path-integrals [26,27].…”