Starting, stopping, and resetting biological oscillators: in search of optimum perturbations

Biological oscillators are commonly subjected to a single brief stimulus to perturb the ongoing rhythm. After stimulation, the oscillator can recover although its phase may be advanced or delayed relative to the original cycle. A single transient perturbation can also stop the ongoing rhythm. Arthur Winfree systematically classified these responses to brief perturbations, as determined by the strength of the stimulus, and the phase within the cycle at which the stimulus was given. A natural question arises from Winfree's work. Are certain stimulus shapes better than others at achieving a desired effect? The present study explores this question using: (1) analysis of phase space geometry, (2) calculus of variations, and (3) analysis of responses to noisy perturbations. Methods 1 and 2 can yield exact solutions, but have limited applicability in biology because they require a detailed mathematical description of the oscillator. Method 3 is applicable to any oscillator without mathematical prerequisites. We validate this method by finding optimum stimuli that start and stop repetitive firing in a neural pacemaker model, and the optimum light stimulus for resetting the circadian rhythm in a model of the human circadian clock. We propose that the elucidation of optimum stimulus shapes may be useful for studying many periodic phenomena in biology and medicine. Optimum stimuli can be used to induce a desired behavior without producing undesirable side effects.