Phase response function for oscillators with strong forcing or coupling

Abstract

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 so that the standard PRC fails. We show that in this case, the phase shift caused by each pulse depends on the history of several previous pulses. We call the corresponding function which measures this shift the phase response function (PRF). As a result of the introduction of the PRF, a variety of oscillatory systems with pulse interaction, such as neural systems, can be reduced to phase systems. The main assumption of the classical PRC model, i.e. that the effect of the stimulus vanishes before the next one arrives, is no longer a restriction in our approach. However, as a result of the phase reduction, the system acquires memory, which is not just a technical nuisance but an intrinsic property relevant to strong stimulation. We illustrate the PRF approach by its application to various systems, such as 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.

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Cited by in corpus (6)

• Global computation of phase-amplitude reduction for limit-cycle dynamics
• Generalized splay states in phase oscillator networks
• Nonlinear phase-amplitude reduction of delay-induced oscillations
• Synchronization transitions in Kuramoto networks with higher-mode interaction
• On the interpretation of Dirac $\delta$ pulses in differential equations for phase oscillators
• Describing dynamics of driven multistable oscillators with phase transfer curves