Excitable systems with noise and delay with applications to control: renewal theory approach

We present an approach for the analytical treatment of excitable systems with noise-induced dynamics in the presence of time delay. An excitable system is modeled as a bistable system with a time delay, while another delay enters as a control term taken after Pyragas K. Pyragas, Phys. Lett. A 170, 421 1992 as a difference between the current system state and its state time units before. This approach combines the elements of renewal theory to estimate the essential features of the resulting stochastic process as functions of the parameters of the controlling term.