Phase response function for oscillators with strong forcing or coupling

Phase response curve (PRC) is an extremely useful tool for studying the response of oscillatory systems, e.g., neurons, to sparse or weak stimulation. Here we develop a framework for studying the response to a series of pulses which are frequent or/and strong. In this case the effect of a stimulus does not vanish before the next one arrives, so the standard PRC fails. In the present letter, we introduce the so-called phase response function (PRF) that measures the phase response even when the system is far from the limit cycle. The PRF uses the history of several previous pulses and does not need the information about the distance from the limit cycle. As a result, an oscillator with pulse input is reduced to a phase system. We illustrate our approach by its application to various systems, such as the Morris-Lecar, Hodgkin-Huxley neuron models, and others. We show that the PRF allows predicting the dynamics of forced and coupled oscillators even when the PRC fails. Thus, the PRF provides an effective tool that may be used for simulation of neural, chemical, optic and other oscillatory systems.