Mechanical detection of the auditory system displays exquisite sensitivity, with the inner ear capable of detecting pressure waves that result in Angstrom-scale displacements. Detection in the inner ear is performed by mechanically sensitive hair cells, named for the bundles of stereocilia that protrude from their surfaces. The inner ear is also highly nonlinear in nature, exhibiting sharply tuned frequency selectivity and compression of dynamic range. Several experiments have consistently shown that the hair cells and their hair bundles are not just passive sensory detectors, rather underlying their exemplary behavior is an internal active mechanical process which is also adaptive in nature. Furthermore, this active process leads to an inherent mechanical instability manifested in the spontaneous limit cycle oscillations of the hair bundles. A number of theories based on nonlinear dynamics have described these active bundles using complex biophysical models as well as using a relatively simple two-dimensional mathematical model that exhibits a supercritical Hopf bifurcation. In this dissertation which is theoretical in nature, with the spontaneously oscillating inner ear hair bundle as our model biological system, we study the effects of stochasticity on nonlinear oscillators driven by internal active processes. First, we develop a framework for the general interpretation of such dynamical systems near a limit cycle. We demonstrate that in the presence of noise the phase of the limit cycle oscillator diffuses while fluctuations in the directions locally orthogonal to the cycle display a Lorentzian power spectrum. Further we identify two mechanisms that underlie the complex frequency dependence of the diffusive dynamics. In the subsequent chapter, we detail how the effects of stochasticity maybe observed with respect to the change in shape and size of the mean limit cycle as well. In particular, we show that the noise-induced distortion of the limit cycle generically leads to its rounding through elimination of sharp features. Conversely, using a theoretical criterion one may identify limit cycle regions most susceptible to such distortion and obtain more meaningful parametric fits of dynamical models from experimental data. In the third chapter of this thesis we study fluctuation-dissipation relations in computationally-driven, nonequilibrium limit-cycle oscillators. A computational drive is loosely analogous to the adaptive internal activity that powers the spontaneous oscillations of the hair cells. It measures the current state of the system and modifies its power input accordingly. We observe that computationally-driven systems not only violate the equilibrium fluctuation-dissipation theorem (FDT) but also a generalized FDT. Thus in turn we propose that the breakdown of this generalized theorem may be used as a tool to broadly identify the presence and effect of such drives within biological systems. Lastly, by quantifying the computing ability of these drives we seek to derive a new generalized fluctuation-dissipation theorem which can be satisfied by complex computationally-driven biological systems such as the inner ear
