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.
Highlights
Oscillatory behaviors and generators can be seen across all biology, from the cyclical patterns seen in certain molecular pathways and transcriptional feedback loops to the rhythms of pacemakers in the brain and the heart
We explore features of the stimulus waveform that switch off the oscillation at any phase of initial impact, by first corralling the oscillator to a narrow range of new phases and by perturbing the oscillator to its phase singularity
The aim of this study is to find an optimal stimulus waveform, u(t), which can cause every clock to transition from the stable limit cycle across the unstable limit cycle toward the stable fixed point
Summary
Oscillatory behaviors and generators can be seen across all biology, from the cyclical patterns seen in certain molecular pathways and transcriptional feedback loops to the rhythms of pacemakers in the brain and the heart. Over the past few decades, a great deal of work has been done to study the effect of stimulation on these oscillators, quantifying and modeling the dynamics and mechanisms involved.[1,2] One interesting finding is that a brief shock with a specific strength (within a narrow range) and given at a specific time (within a narrow window of phases) is capable of suppressing oscillatory behavior.[1,3] Most of this work used simple rectangular pulses as the stimulus. Previous studies have explored the use of non-rectangular waveforms to achieve efficient oscillatory suppression.[4,5] Given that non-traditional waveforms are often more energetically efficient,[6,7] the question arises regarding whether or not the use of nontraditional waveforms may open the window of successful phases such that the stimulus generated could be given at any arbitrary phase and still successfully suppress the oscillatory behavior
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