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.