Efficient stochastic simulation of piecewise-deterministic Markov processes and its application to the Morris-Lecar model of neural dynamics

Piecewise-deterministic Markov processes combine continuous in time dynamics with jump events, the rates of which generally depend on the continuous variables and thus are not constants. This leads to a problem in a Monte-Carlo simulation of such a system, where, at each step, one must find the time instant of the next event. The latter is determined by an integral equation and usually is rather slow in numerical implementation. We suggest a reformulation of the next event problem as an ordinary differential equation where the independent variable is not the time but the cumulative rate. This reformulation is similar to the H & eacute;non approach to efficiently constructing the Poincar & eacute; map in deterministic dynamics. The problem is then reduced to a standard numerical task of solving a system of ordinary differential equations with given initial conditions on a prescribed interval. We illustrate the method with a stochastic Morris-Lecar model of neuron spiking with stochasticity in the opening and closing of voltage-gated ion channels.