Violation of generalized fluctuation-dissipation theorem in biological limit cycle oscillators with state-dependent internal drives: Applications to hair cell oscillations

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 state dependent in nature—it measures the current state of the system and changes its power input accordingly. We study the breakdown of two fluctuation-dissipation relations in these nonequilibrium limit cycle oscillators with and without state-dependent drives. First, in the simple model of the hair cell oscillator where the internal drive of the system is state independent, 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). This generalized theorem is analogous to one proposed earlier by Seifert and Speck. It 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 in the presence of a state-dependent drive, the dynamical hair cell model violates both the FDT and this GFDT. We go on to show, using stochastic, finite-state, dynamical models, that such a drive in stochastic limit cycle oscillators generically violates both the FDT and GFDT. We propose that one may in fact use the breakdown of this particular GFDT as a tool to more broadly look for and quantify the effect of state-dependent drives within (nonequilibrium) biological dynamics.