Phase reduction theory of coherent excitable systems*

Excitable systems can exhibit large excursions under superthreshold perturbations. It has been shown that fast variable noise can induce coherent oscillations in such systems due to large timescale separation, which is named as self-induced stochastic resonance (SISR). In this work, the hybrid system reproducing the SISR oscillation is established. The phase sensitivity function, which measures the phase response to infinitesimal perturbations, is calculated by solving the corresponding adjoint system. After approximated the effective frequency and noise strength, the effective phase equation is finally obtained. Our reduced model is applied to the cases of periodic forcing and two-coupled oscillators. The predicted results are in good agreement with those of Monte Carlo simulations.