Using the fluctuation-response relations in biological limit-cycle oscillators to interrogate active feedback and control mechanisms

Biology is replete with complex, stochastic, nonlinear systems that exhibit steady-state limit-cycle dynamics driven by energy input. Many of these systems interact with various feedback and control mechanisms necessary for adaptation and/or homeostasis. We explore fluctuation dissipation relations in these noisy limit-cycle systems focusing primarily on the hair cells of the inner ear. These endogenously driven, overdamped oscillators are essential for both the sensitivity and frequency-selectivity of the auditory system. Using this model system, we demonstrate that there are two fundamental classes of noisy limit-cycle oscillators - ones in which the power input of the drive depends on the state of the system and ones where it does not. In the former case of an adaptive drive, we show that these systems violate a particular nonequilibrium fluctuation-response relation and propose that this failure is an important indicator of the presence of adaptive, control processes in biology. We also explore how one can derive a new fluctuation theorem that takes into account this feedback between the state of the noisy oscillator and its power input. This suggests that fluctuation analyses of these oscillators may provide a new window into understanding biological control and adaptation.