Noisy driven oscillators: Adaptive drives break the fluctuation-dissipation theorem

The steady-state dynamics of complex nonlinear systems include limit cycles in which the dynamic variables trace a closed path in phase space. Biological systems are replete with examples of such driven oscillators in a diverse range of systems including circadian rhythms, neuronal central pattern generators, and the active mechanics of hearing. These biological systems are inherently noisy, and they are typically controlled by active feedback. We explore the fluctuations and response functions of intrinsically noisy limit-cycle oscillators starting with models of stereocilium dynamics in the inner ear. We show that one can obtain a generalized fluctuations-dissipation theorem (GFDT) for the system in a reference frame comoving with the mean dynamical state moving about the limit cycle. However, in the presence of adaptive drives where there is feedback so that the energy input driving the oscillator depends on the state of the system, as in the driven stereocilium, even these generalized fluctuation theorems fail. We further explore the essential role of these feedback mechanisms in breaking GFDTs in noisy driven systems using a combination of simple computational models, analytical calculations, and stereocilium dynamics data.