Abstract
The spontaneously oscillating hair bundle of sensory cells in the inner ear is an example of a stochastic, nonlinear oscillator driven by internal active processes. Moreover, this internal activity is adaptive -- its power input depends on the current state of the system. We study fluctuation dissipation relations in such adaptively-driven, nonequilibrium limit-cycle oscillators. We observe the expected violation of the well-known, equilibrium fluctuation-dissipation theorem (FDT), and verify the existence of a generalized fluctuation-dissipation theorem (GFDT) in the non-adaptively driven model of the hair cell oscillator. This generalized fluctuation theorem requires the system to be analyzed in the co-moving frame associated with the mean limit cycle of the stochastic oscillator. We then demonstrate, via numerical simulations and analytic calculations, that the adaptively-driven dynamical hair cell model violates both the FDT and the GFDT. We go on to show, using stochastic, finite-state, dynamical models, that such a feedback-controlled drive in stochastic limit cycle oscillators generically violates both the FDT and GFDT. We propose that one may in fact use the breakdown of the GFDT as a tool to more broadly look for and quantify the effect of adaptive, feedback mechanisms associated with driven (nonequilibrium) biological dynamics.
Highlights
Biology is replete with nonequilibrium systems that expend energy to maintain cyclic steady-state dynamics
In addition to the equilibrium fluctuation-dissipation theorem (FDT) [27], we focus on the generalized FDT (GFDT) of Speck and Seifert [29,31]
While neither the time derivative of the correlation function nor the response function in a system with a constant internal probability current agrees with predictions based on the equilibrium system, their agreement with each other shows that a generalized fluctuation-dissipation theorem holds in such a system, as expected based on the work of Seifert and collaborators [29,31]
Summary
Biology is replete with nonequilibrium systems that expend energy to maintain cyclic steady-state dynamics. The limit cycle oscillations of the hair cell bundle are innately noisy and provide a window into the basic nonequilibrium statistical mechanics of a noisy limit cycle oscillator They operate under adaptive control where the internal drive maintaining the nonequilibrium steady state responds to the oscillator’s state. The simplest model that captures the essential phenomena is a two-dimensional dynamical system that undergoes a supercritical Hopf bifurcation to the limit cycle (oscillatory) state [24,25,26] In this manuscript, we use the stochastic normal form equation for the Hopf bifurcation to model the spontaneously oscillating state of the hair bundle, in order to study fluctuation-dissipation theorems associated with noisy nonequilibrium systems. We propose that just as the violation of the original FDT in biological systems is an important quantitative measure of nonequilibrium dynamics [38], violation of the nonequilibrium GFDT provides a quantitative indicator of the presence of an internal state-dependent drive in biological dynamical systems
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