Intermodulation distortions from an array of active nonlinear oscillators.

Coupling is critical in nonlinear dynamical systems. It affects the stabilities of individual oscillators as well as the characteristics of their response to external forces. In the auditory system, the mechanical coupling between sensory hair cells has been proposed as a mechanism that enhances the inner ear's sensitivity and frequency discrimination. While extensive studies investigate the effects of coupling on the detection of a sinusoidal signal, the role of coupling underlying the response to a complex tone remains elusive. In this study, we measured the acoustic intermodulation distortions (IMDs) produced by the inner ears of two frog species stimulated simultaneously by two pure tones. The distortion intensity level displayed multiple peaks across stimulus frequencies, in contrast to the generic response from a single nonlinear oscillator. The multiple-peaked pattern was altered upon varying the stimulus intensity or an application of a perturbation tone near the distortion frequency. Numerical results of IMDs from a chain of coupled active nonlinear oscillators driven by two sinusoidal forces reveal the effects of coupling on the variation profile of the distortion amplitude. When the multiple-peaked pattern is observed, the chain's motion at the distortion frequency displays both a progressive wave and a standing wave. The latter arises due to coupling and is responsible for the multiple-peaked pattern. Our results illustrate the significance of mechanical coupling between active hair cells in the generation of auditory distortions, as a mechanism underlying the formation of in vivo standing waves of distortion signals.

Dynamics of Mechanically Coupled Hair-Cell Bundles of the Inner Ear.

The high sensitivity and effective frequency discrimination of sound detection performed by the auditory system rely on the dynamics of a system of hair cells. In the inner ear, these acoustic receptors are primarily attached to an overlying structure that provides mechanical coupling between the hair bundles. Although the dynamics of individual hair bundles has been extensively investigated, the influence of mechanical coupling on the motility of the system of bundles remains underdetermined. We developed a technique of mechanically coupling two active hair bundles, enabling us to probe the dynamics of the coupled system experimentally. We demonstrated that the coupling could enhance the coherence of hair bundles' spontaneous oscillation, as well as their phase-locked response to sinusoidal stimuli, at the calcium concentration in the surrounding fluid near the physiological level. The empirical data were consistent with numerical results from a model of two coupled nonisochronous oscillators, each displaying a supercritical Hopf bifurcation. The model revealed that a weak coupling can poise the system of unstable oscillators closer to the bifurcation by a shift in the critical point. In addition, the dynamics of strongly coupled oscillators far from criticality suggested that individual hair bundles may be regarded as nonisochronous oscillators. An optimal degree of nonisochronicity was required for the observed tuning behavior in the coherence of autonomous motion of the coupled system.

Bilateral Spontaneous Otoacoustic Emissions Show Coupling between Active Oscillators in the Two Ears.

Spontaneous otoacoustic emissions (SOAEs) are weak sounds that emanate from the ears of tetrapods in the absence of acoustic stimulation. These emissions are an epiphenomenon of the inner ear's active process, which enhances the auditory system's sensitivity to weak sounds, but their mechanism of production remains a matter of debate. We recorded SOAEs simultaneously from the two ears of the tokay gecko and found that binaural emissions could be strongly correlated: some emissions occurred at the same frequency in both ears and were highly synchronized. Suppression of the emissions in one ear often changed the amplitude or shifted the frequency of emissions in the other. Decreasing the frequency of emissions from one ear by lowering its temperature usually reduced the frequency of the contralateral emissions. To understand the relationship between binaural SOAEs, we developed a mathematical model of the eardrums as noisy nonlinear oscillators coupled by the air within an animal's mouth. By according with the model, the results indicate that some SOAEs are generated bilaterally through acoustic coupling across the oral cavity. The model predicts that sound localization through the acoustic coupling between ears is influenced by the active processes of both ears.

Mechanical amplification exhibited by quiescent saccular hair bundles.

Spontaneous oscillations exhibited by free-standing hair bundles from the Bullfrog sacculus suggest the existence of an active process that might underlie the exquisite sensitivity of the sacculus to mechanical stimulation. However, this spontaneous activity is suppressed by coupling to an overlying membrane, which applies a large mechanical load on the bundle. How a quiescent hair bundle utilizes its active process is still unknown. We studied the dynamics of motion of individual hair bundles under different offsets in the bundle position, and observed the occurrence of spikes in hair-bundle motion, associated with the generation of active work. These mechanical spikes can be evoked by a sinusoidal stimulus, leading to an amplified movement of the bundle with respect to the passive response. Amplitude gain reached as high as 100-fold at small stimulus amplitudes. Amplification of motion decreased with increasing amplitude of stimulation, ceasing at ∼6-12 pN stimuli. Results from numerical simulations suggest that the adaptation process, mediated by myosin 1c, is not required for the production of mechanical spikes.

Phase-locked spiking of inner ear hair cells and the driven noisy Adler equation.

The inner ear constitutes a remarkably sensitive mechanical detector. This detection occurs in a noisy and highly viscous environment, as the sensory cells-the hair cells-are immersed in a fluid-filled compartment and operate at room or higher temperatures. We model the active motility of hair cell bundles of the vestibular system with the Adler equation, which describes the phase degree of freedom of bundle motion. We explore both analytically and numerically the response of the system to external signals, in the presence of white noise. The theoretical model predicts that hair bundles poised in the quiescent regime can exhibit sporadic spikes-sudden excursions in the position of the bundle. In this spiking regime, the system exhibits stochastic resonance, with the spiking rate peaking at an optimal level of noise. Upon the application of a very weak signal, the spikes occur at a preferential phase of the stimulus cycle. We compare the theoretical predictions of our model to experimental measurements obtained in vitro from individual hair cells. Finally, we show that an array of uncoupled hair cells could provide a sensitive detector that encodes the frequency of the applied signal.

Phase slips in oscillatory hair bundles.

Hair cells of the inner ear contain an active amplifier that allows them to detect extremely weak signals. As one of the manifestations of an active process, spontaneous oscillations arise in fluid immersed hair bundles of in vitro preparations of selected auditory and vestibular organs. We measure the phase-locking dynamics of oscillatory bundles exposed to low-amplitude sinusoidal signals, a transition that can be described by a saddle-node bifurcation on an invariant circle. The transition is characterized by the occurrence of phase slips, at a rate that is dependent on the amplitude and detuning of the applied drive. The resultant staircase structure in the phase of the oscillation can be described by the stochastic Adler equation, which reproduces the statistics of phase slip production.

Dynamics of freely oscillating and coupled hair cell bundles under mechanical deflection.

In vitro, attachment to the overlying membrane was found to affect the resting position of the hair cell bundles of the bullfrog sacculus. To assess the effects of such a deflection on mechanically decoupled hair bundles, comparable offsets were imposed on decoupled spontaneously oscillating bundles. Strong modulation was observed in their dynamic state under deflection, with qualitative changes in the oscillation profile, amplitude, and characteristic frequency of oscillation seen in response to stimulus. Large offsets were found to arrest spontaneous oscillation, with subsequent recovery upon reversal of the stimulus. The dynamic state of the hair bundle displayed hysteresis and a dependence on the direction of the imposed offset. The coupled system of hair bundles, with the overlying membrane left on top of the preparation, also exhibited a dependence on offset position, with an increase in the linear response function observed under deflections in the inhibitory direction.

Magnetic actuation of hair cells.

The bullfrog sacculus contains mechanically sensitive hair cells whose stereociliary bundles oscillate spontaneously when decoupled from the overlying membrane. Steady-state offsets on the resting position of a hair bundle can suppress or modulate this native motility. To probe the dynamics of spontaneous oscillation in the proximity of the critical point, we describe here a method for mechanical actuation that avoids loading the bundles or contributing to the viscous drag. Magnetite beads were attached to the tips of the stereocilia, and a magnetic probe was used to impose deflections. This technique allowed us to observe the transition from multi-mode to single-mode state in freely oscillating bundles, as well as the crossover from the oscillatory to the quiescent state.

Multiple-timescale dynamics underlying spontaneous oscillations of saccular hair bundles.

Spontaneous oscillations displayed by hair bundles of the bullfrog sacculus have complex temporal profiles, not fully captured by single limit-cycle descriptions. Quiescent intervals are typically interspersed with oscillations, leading to a bursting-type behavior. Temporal characteristics of the oscillation are strongly affected by imposing a mechanical load or by the application of a steady-state deflection to the resting position of the bundle. Separate spectral components of the spontaneous motility are differently affected by increases in the external calcium concentration. We use numerical modeling to explore the effects of internal parameters on the oscillatory profiles, and to reproduce the experimental modulation induced by mechanical or ionic manipulation.