Falling off a limit cycle using phase-agnostic stimuli: Definitions and conceptual framework

Abstract

Nearly a half-century of biomedical research has revealed methods and mechanisms by which an oscillator with bistable limit cycle kinetics can be stopped using critical stimuli applied at a specific phase. Is it possible to construct a stimulus that stops oscillation regardless of the phase at which the stimulus is applied? Using a radial isochron clock model, we demonstrate the existence of such stimulus waveforms, which can take on highly complex shapes but with a surprisingly simple mechanism of rhythm suppression. The perturbation, initiated at any phase of the limit cycle, first corrals the oscillator to a narrow range of new phases, then drives the oscillator to its phase singularity. We further constructed a library of waveforms having different durations, each achieving phase-agnostic suppression of rhythm but with varying rates of phase corralling prior to amplitude suppression. The optimal stimulus energy to achieve phase-agnostic suppression of rhythm is dependent on the rate of phase corralling and the configuration of the phaseless set. We speculate that these results are generic and suggest the existence of stimulus waveforms that can stop the rhythm of more complex oscillators irrespective of the applied phase.