Two-state stochastic process with time delayed feedback: controlling noise-induced oscillations in excitable systems

We consider a two-state stochastic process with time delayed feedback which implies history-dependent residence time distributions (RTDs). The equilibrium RTD in the sense of the averaging over all possible histories is introduced. We use the path integral representation to link the equilibrium RTD to the distribution function of the history. Equilibrium RTD is determined analytically for small and intermediate delay times. Analytic results are compared with numerical simulations of a bistable system with two time delays.