Intermodulation distortions from a chain of coupled nonlinear oscillators near a hopf bifurcation

Abstract

The inner ear can produce sounds whose frequencies correspond to linear combinations of the stimulus frequencies, f1 and f2. These sounds, called Distortion Product Otoacoustic Emissions (DPOAEs), can be measured by inserting a microphone into the ear canal. In human, the intensity level of DPOAEs at frequency 2f1-f2 varies significantly with a stimulus frequency and can exhibit a quasi-periodic pattern, a feature termed a DPOAE fine structure. This distinctive characteristic has been generally accepted as a consequence of the interference between two DPOAE signals produced by two spatially separated groups of sensory hair cells, whose characteristic frequencies correspond to the stimulus frequencies, and the distortion frequency. However, a recent study revealed that a similar feature can be observed from Chinese edible frogs, whose inner ear structure does not support the two-signal interference hypothesis as in vivo amphibian hair cells are strongly coupled. In this work, we performed numerical simulations of nonlinear distortions produced by a chain of strongly coupled nonlinear oscillators, a structure that represents hair cells in a simple inner ear. Our results revealed that the amplitude of the total distortion from the oscillators, arranged in order of their characteristic frequencies, can display a quasiperiodic variation pattern under appropriate driving forces levels. Analyses of the motion of individual oscillators showed that those with low characteristic frequencies underwent a standing wave at the distortion frequency, a feature that was responsible for the quasiperiodic pattern. On the other hand, coupling between oscillators arranged in a random order of characteristic frequencies yielded a nonlinear response resembling that of a single oscillator. Results from the model agreed qualitatively with prior experimental observations. Our results thus suggest multiple mechanisms underlying DPOAEs produced by the frog’s ear, governed by the arrangement of the sensory cells.