Limit cycle oscillations in a nonlinear state space model of the human cochlea.

It is somewhat surprising that linear analysis can account for so many features of the cochlea when it is inherently nonlinear. For example, the commonly detected spacing between adjacent spontaneous otoacoustic emissions (SOAEs) is often explained by a linear theory of "coherent reflection" [Zweig and Shera (1995). J. Acoust. Soc. Am. 98, 2018-2047]. The nonlinear saturation of the cochlear amplifier is, however, believed to be responsible for stabilizing the amplitude of a SOAE. In this investigation, a state space model is used to first predict the linear instabilities that arise, given distributions of cochlear inhomogeneities, and then subsequently to simulate the time-varying spectra of the nonlinear models. By comparing nonlinear simulation results to linear predictions, it is demonstrated that nonlinear effects can have a strong impact on the steady-state response of an unstable cochlear model. Sharply tuned components that decay away exponentially within 100 ms are shown to be due to linearly resonant modes of the model generated by the cochlear inhomogeneities. Some oscillations at linearly unstable frequencies are suppressed over a longer time scale, whereas those that persist are due to linear instabilities and their distortion products.

Statistics of instabilities in a state space model of the human cochlea.

A state space model of the human cochlea is used to test Zweig and Shera's [(1995) "The origin of periodicity in the spectrum of evoked otoacoustic emissions," J. Acoust. Soc. Am. 98(4), 2018-2047 ] multiple-reflection theory of spontaneous otoacoustic emission (SOAE) generation. The state space formulation is especially well suited to this task as the unstable frequencies of an active model can be rapidly and unambiguously determined. The cochlear model includes a human middle ear boundary and matches human enhancement, tuning, and traveling wave characteristics. Linear instabilities can arise across a wide bandwidth of frequencies in the model when the smooth spatial variation of basilar membrane impedance is perturbed, though it is believed that only unstable frequencies near the middle ear's range of greatest transmissibility are detected as SOAEs in the ear canal. The salient features of Zweig and Shera's theory are observed in this active model given several classes of perturbations in the distribution of feedback gain along the cochlea. Spatially random gain variations are used to approximate what may exist in human cochleae. The statistics of the unstable frequencies for random, spatially dense variations in gain are presented; the average spacings of adjacent unstable frequencies agree with the preferred minimum distance observed in human SOAE data.

A state space model for cochlear mechanics.

The stability of a linear model of the active cochlea is difficult to determine from its calculated frequency response alone. A state space model of the cochlea is presented, which includes a discretized set of general micromechanical elements coupled via the cochlear fluid. The stability of this time domain model can be easily determined in the linear case, and the same framework used to simulate the time domain response of nonlinear models. Examples of stable and unstable behavior are illustrated using the active micromechanical model of Neely and Kim. The stability of this active cochlea is extremely sensitive to abrupt spatial inhomogeneities, while smoother inhomogeneities are less likely to cause instability. The model is a convenient tool for investigating the presence of instabilities due to random spatial inhomogeneities. The number of unstable poles is found to rise sharply with the relative amplitude of the inhomogeneities up to a few percent, but to be significantly reduced if the spatial variation is smoothed. In a saturating nonlinear model, such instabilities generate limit cycles that are thought to produce spontaneous otoacoustic emissions. An illustrative time domain simulation is presented, which shows how an unstable model evolves into a limit cycle, distributed along the cochlea.