Interactions between hair cells shape spontaneous otoacoustic emissions in a model of the tokay gecko's cochlea

Conditions Underlying the Appearance of Spontaneous Otoacoustic Emissions in Mammals

Across the wide range of land vertebrate species, spontaneous otoacoustic emissions (SOAE) are common, but not always found. The reasons for the differences between species of the various groups in their emission patterns are often not well understood, particularly within mammals. This review examines the question as to what determines in mammals whether SOAE are emitted or not, and suggests that the coupling between hair-cell regions diminishes when the space constant of frequency distribution becomes larger. The reduced coupling is assumed to result in a greater likelihood of SOAE being emitted.

Something in Our Ears Is Oscillating, but What? A Modeller's View of Efforts to Model Spontaneous Emissions

When David Kemp discovered "spontaneous ear noise" in 1978, it opened up a whole new perspective on how the cochlea works. The continuous tonal sound emerging from most healthy human ears, now called spontaneous otoacoustic emissions or SOAEs, was an unmistakable sign that our hearing organ must be considered an active detector, not just a passive microphone, just as Thomas Gold had speculated some 30 years earlier. Clearly, something is oscillating as a byproduct of that sensitive inbuilt detector, but what exactly is it? Here, we give a chronological account of efforts to model SOAEs as some form of oscillator, and at intervals, we illustrate key concepts with numerical simulations. We find that after many decades there is still no consensus, and the debate extends to whether the oscillator is local, confined to discrete local sources on the basilar membrane, or global, in which an assembly of micro-mechanical elements and basilar membrane sections, coupled by inner ear fluid, interact over a wide region. It is also undecided whether the cochlear oscillator is best described in terms of the well-known Van der Pol oscillator or the less familiar Duffing or Hopf oscillators. We find that irregularities play a key role in generating the emissions. This paper is not a systematic review of SOAEs and their properties but more a historical survey of the way in which various oscillator configurations have been applied to modelling human ears. The conclusion is that the difference between the local and global approaches is not clear-cut, and they are probably not mutually exclusive concepts. Nevertheless, when one sees how closely human SOAEs can be matched to certain arrangements of oscillators, Gold would no doubt say we are on the right track.

Otoacoustic Emissions in Non-Mammals

Otoacoustic emissions (OAE) that were sound-induced, current-induced, or spontaneous have been measured in non-mammalian land vertebrates, including in amphibians, reptiles, and birds. There are no forms of emissions known from mammals that have not also been observed in non-mammals. In each group and species, the emission frequencies clearly lie in the range known to be processed by the hair cells of the respective hearing organs. With some notable exceptions, the patterns underlying the measured spectra, input-output functions, suppression threshold curves, etc., show strong similarities to OAE measured in mammals. These profound similarities are presumably traceable to the fact that emissions are produced by active hair-cell mechanisms that are themselves dependent upon comparable nonlinear cellular processes. The differences observed—for example, in the width of spontaneous emission peaks and delay times in interactions between peaks—should provide insights into how hair-cell activity is coupled within the organ and thus partially routed out into the middle ear.

Atypical tuning and amplification mechanisms in gecko auditory hair cells

Geckos are lizards capable of vocalization and can detect frequencies up to 5 kHz, but the mechanism of frequency discrimination is incompletely understood. The gecko’s auditory papilla has a unique arrangement over the high-frequency zone, with rows of mechanically sensitive hair bundles covered with gelatinous sallets. Lower-frequency hair cells are tuned by an electrical resonance employing Ca²⁺-activated K⁺ channels, but hair cells tuned above 1 kHz probably rely on a mechanical resonance of the sallets. The resonance may be boosted by an electromotile force from hair bundles found to be evoked by changes in hair cell membrane potential. This unusual mechanism operates independently of mechanotransduction and differs from mammals which amplify the mechanical input using the motor protein prestin.

The auditory papilla of geckos contains two zones of sensory hair cells, one covered by a continuous tectorial membrane affixed to the hair bundles and the other by discrete tectorial sallets each surmounting a transverse row of bundles. Gecko papillae are thought to encode sound frequencies up to 5 kHz, but little is known about the hair cell electrical properties or their role in frequency tuning. We recorded from hair cells in the isolated auditory papilla of the crested gecko, Correlophus ciliatus, and found that in both the nonsalletal region and part of the salletal region, the cells displayed electrical tuning organized tonotopically. Along the salletal zone, occupying the apical two-thirds of the papilla, hair bundle length decreased threefold and stereociliary complement increased 1.5-fold. The two morphological variations predict a 13-fold gradient in bundle stiffness, confirmed experimentally, which, when coupled with salletal mass, could provide passive mechanical resonances from 1 to 6 kHz. Sinusoidal electrical currents injected across the papilla evoked hair bundle oscillations at twice the stimulation frequency, consistent with fast electromechanical responses from hair bundles of two opposing orientations across the papilla. Evoked bundle oscillations were diminished by reducing Ca²⁺ influx, but not by blocking the mechanotransduction channels or inhibiting prestin action, thereby distinguishing them from known electromechanical mechanisms in hair cells. We suggest the phenomenon may be a manifestation of an electromechanical amplification that augments the passive mechanical tuning of the sallets over the high-frequency region.

Whistling While it Works: Spontaneous Otoacoustic Emissions and the Cochlear Amplifier

Perhaps the most striking evidence for active processes operating within the inner ears of mammals and non-mammals alike is their ability to spontaneously produce sound. Predicted by Thomas Gold in 1948, some 30 years prior to their discovery, the narrow-band sounds now known as spontaneous otoacoustic emissions (SOAEs) remain incompletely understood, their origins controversial. Without a single equation in the main text, we review the essential concepts underlying the "local-" and "global-oscillator" frameworks for understanding SOAE generation. Comparing their key assumptions and predictions, we relate the two frameworks to unresolved questions about the biophysical mechanisms of cochlear amplification.

Volterra-series approach to stochastic nonlinear dynamics: Linear response of the Van der Pol oscillator driven by white noise

The Van der Pol equation is a paradigmatic model of relaxation oscillations. This remarkable nonlinear phenomenon of self-sustained oscillatory motion underlies important rhythmic processes in nature and electrical engineering. Relaxation oscillations in a real system are usually coupled to environmental noise, which further enriches their dynamics, but makes theoretical analysis of such systems and determination of the equation parameter values a difficult task. In a companion paper we have proposed an analytic approach to a similar problem for another classical nonlinear model-the bistable Duffing oscillator. Here we extend our techniques to the case of the Van der Pol equation driven by white noise. We analyze the statistics of solutions and propose a method to estimate parameter values from the oscillator's time series. We use experimental data of active oscillations in a biophysical system to demonstrate how our method applies to real observations and can be generalized for more complex models.

Diverse Mechanisms of Sound Frequency Discrimination in the Vertebrate Cochlea

Discrimination of different sound frequencies is pivotal to recognizing and localizing friend and foe. Here, I review the various hair cell-tuning mechanisms used among vertebrates. Electrical resonance, filtering of the receptor potential by voltage-dependent ion channels, is ubiquitous in all non-mammals, but has an upper limit of ~1 kHz. The frequency range is extended by mechanical resonance of the hair bundles in frogs and lizards, but may need active hair-bundle motion to achieve sharp tuning up to 5 kHz. Tuning in mammals uses somatic motility of outer hair cells, underpinned by the membrane protein prestin, to expand the frequency range. The bird cochlea may also use prestin at high frequencies, but hair cells <1 kHz show electrical resonance.

Modeling the characteristics of spontaneous otoacoustic emissions in lizards

Lizard auditory papillae have proven to be an attractive object for modelling the production of spontaneous otoacoustic emissions (SOAE). Here we use an established model (Vilfan and Duke, 2008) and extend it by exploring the effect of varying the number of oscillating elements, the strength of the parameters that describe the coupling between oscillators, the strength of the oscillators, and additive noise. The most remarkable result is that the actual number of oscillating elements hardly influences the spectral pattern, explaining why spectra from very different papillar dimensions are similar. Furthermore, the spacing between spectral peaks primarily depends on the reactive coupling between the oscillator elements. This is consistent with observed differences between lizard species with respect to tectorial covering of hair cells and SOAE peak spacings. Thus, the model provides a basic understanding of the variation in SOAE properties across lizard species.

Molding the asymmetry of localized frequency-locking waves by a generalized forcing and implications to the inner ear

Frequency locking to an external forcing frequency is a well-known phenomenon. In the auditory system, it results in a localized traveling wave, the shape of which is essential for efficient discrimination between incoming frequencies. An amplitude equation approach is used to show that the shape of the localized traveling wave depends crucially on the relative strength of additive versus parametric forcing components; the stronger the parametric forcing, the more asymmetric is the response profile and the sharper is the traveling-wave front. The analysis qualitatively captures the empirically observed regions of linear and nonlinear responses and highlights the potential significance of parametric forcing mechanisms in shaping the resonant response in the inner ear.

Frequency locking in auditory hair cells: Distinguishing between additive and parametric forcing

- The auditory system displays remarkable sensitivity and frequency discrimination, attributes shown to rely on an amplification process that involves a mechanical as well as a biochemical response. Models that display proximity to an oscillatory onset (also known as Hopf bifurcation) exhibit a resonant response to distinct frequencies of incoming sound, and can explain many features of the amplification phenomenology. To understand the dynamics of this resonance, frequency locking is examined in a system near the Hopf bifurcation and subject to two types of driving forces: additive and parametric. Derivation of a universal amplitude equation that contains both forcing terms enables a study of their relative impact on the hair cell response. In the parametric case, although the resonant solutions are 1 : 1 frequency locked, they show the coexistence of solutions obeying a phase shift of π, a feature typical of the 2 : 1 resonance. Different characteristics are predicted for the transition from unlocked to locked solutions, leading to smooth or abrupt dynamics in response to different types of forcing. The theoretical framework provides a more realistic model of the auditory system, which incorporates a direct modulation of the internal control parameter by an applied drive. The results presented here can be generalized to many other media, including Faraday waves, chemical reactions, and elastically driven cardiomyocytes, which are known to exhibit resonant behavior.

The vibrating reed frequency meter: digital investigation of an early cochlear model

The vibrating reed frequency meter, originally employed by Békésy and later by Wilson as a cochlear model, uses a set of tuned reeds to represent the cochlea’s graded bank of resonant elements and an elastic band threaded between them to provide nearest-neighbour coupling. Here the system, constructed of 21 reeds progressively tuned from 45 to 55 Hz, is simulated numerically as an elastically coupled bank of passive harmonic oscillators driven simultaneously by an external sinusoidal force. To uncover more detail, simulations were extended to 201 oscillators covering the range 1–2 kHz. Calculations mirror the results reported by Wilson and show expected characteristics such as traveling waves, phase plateaus, and a response with a broad peak at a forcing frequency just above the natural frequency. The system also displays additional fine-grain features that resemble those which have only recently been recognised in the cochlea. Thus, detailed analysis brings to light a secondary peak beyond the main peak, a set of closely spaced low-amplitude ripples, rapid rotation of phase as the driving frequency is swept, frequency plateaus, clustering, and waxing and waning of impulse responses. Further investigation shows that each reed’s vibrations are strongly localised, with small energy flow along the chain. The distinctive set of equally spaced ripples is an inherent feature which is found to be largely independent of boundary conditions. Although the vibrating reed model is functionally different to the standard transmission line, its cochlea-like properties make it an intriguing local oscillator model whose relevance to cochlear mechanics needs further investigation.

An active oscillator model describes the statistics of spontaneous otoacoustic emissions

Even in the absence of external stimulation, the cochleas of most humans emit very faint sounds below the threshold of hearing, sounds that are known as spontaneous otoacoustic emissions. They are a signature of the active amplification mechanism in the cochlea. Emissions occur at frequencies that are unique for an individual and change little over time. The statistics of a population of ears exhibit characteristic features such as a preferred relative frequency distance between emissions (interemission intervals). We propose a simplified cochlea model comprising an array of active nonlinear oscillators coupled both hydrodynamically and viscoelastically. The oscillators are subject to a weak spatial disorder that lends individuality to the simulated cochlea. Our model captures basic statistical features of the emissions: distributions of 1), emission frequencies; 2), number of emissions per ear; and 3), interemission intervals. In addition, the model reproduces systematic changes of the interemission intervals with frequency. We show that the mechanism for the preferred interemission interval in our model is the occurrence of synchronized clusters of oscillators.

Salient features of otoacoustic emissions are common across tetrapod groups and suggest shared properties of generation mechanisms

Otoacoustic emissions (OAEs) are faint sounds generated by healthy inner ears that provide a window into the study of auditory mechanics. All vertebrate classes exhibit OAEs to varying degrees, yet the biophysical origins are still not well understood. Here, we analyzed both spontaneous (SOAE) and stimulus-frequency (SFOAE) otoacoustic emissions from a bird (barn owl, Tyto alba) and a lizard (green anole, Anolis carolinensis). These species possess highly disparate macromorphologies of the inner ear relative to each other and to mammals, thereby allowing for novel insights into the biomechanical mechanisms underlying OAE generation. All ears exhibited robust OAE activity, and our chief observation was that SFOAE phase accumulation between adjacent SOAE peak frequencies clustered about an integral number of cycles. Being highly similar to published results from human ears, we argue that these data indicate a common underlying generator mechanism of OAEs across all vertebrates, despite the absence of morphological features thought essential to mammalian cochlear mechanics. We suggest that otoacoustic emissions originate from phase coherence in a system of coupled oscillators, which is consistent with the notion of "coherent reflection" but does not explicitly require a mammalian-type traveling wave. Furthermore, comparison between SFOAE delays and auditory nerve fiber responses for the barn owl strengthens the notion that most OAE delay can be attributed to tuning.

Effect of bidirectional mechanoelectrical coupling on spontaneous oscillations and sensitivity in a model of hair cells

Sensory hair cells of amphibians exhibit spontaneous activity in their hair bundles and membrane potentials, reflecting two distinct active amplification mechanisms employed in these peripheral mechanosensors. We use a two-compartment model of the bullfrog's saccular hair cell to study how the interaction between its mechanical and electrical compartments affects the emergence of distinct dynamical regimes, and the role of this interaction in shaping the response of the hair cell to weak mechanical stimuli. The model employs a Hodgkin-Huxley-type system for the basolateral electrical compartment and a nonlinear hair bundle oscillator for the mechanical compartment, which are coupled bidirectionally. In the model, forward coupling is provided by the mechanoelectrical transduction current, flowing from the hair bundle to the cell soma. Backward coupling is due to reverse electromechanical transduction, whereby variations of the membrane potential affect adaptation processes in the hair bundle. We isolate oscillation regions in the parameter space of the model and show that bidirectional coupling affects significantly the dynamics of the cell. In particular, self-sustained oscillations of the hair bundles and membrane potential can result from bidirectional coupling, and the coherence of spontaneous oscillations can be maximized by tuning the coupling strength. Consistent with previous experimental work, the model demonstrates that dynamical regimes of the hair bundle change in response to variations in the conductances of basolateral ion channels. We show that sensitivity of the hair cell to weak mechanical stimuli can be maximized by varying coupling strength, and that stochasticity of the hair bundle compartment is a limiting factor of the sensitivity.

Spatial irregularities of sensitivity along the organ of Corti of the cochlea

Fine structures of spatial profiles were computed from existing records of cat and chinchilla auditory-nerve fibers on the basis of their characteristic frequencies and cochlear maps. The spatial fine structures of characteristic-frequency thresholds and of "spontaneous" and driven firing rates were mutually correlated, implying the presence of sensitivity fluctuations due to spatial irregularities of presynaptic structures or processes of the inner hair cells and their input. These findings suggest that activity that appears spontaneous is not actually spontaneous and may indicate irregularities of tonotopic mapping in cochlear mechanics.

Integrating the active process of hair cells with cochlear function

Uniquely among human senses, hearing is not simply a passive response to stimulation. Our auditory system is instead enhanced by an active process in cochlear hair cells that amplifies acoustic signals several hundred-fold, sharpens frequency selectivity and broadens the ear's dynamic range. Active motility of the mechanoreceptive hair bundles underlies the active process in amphibians and some reptiles; in mammals, this mechanism operates in conjunction with prestin-based somatic motility. Both individual hair bundles and the cochlea as a whole operate near a dynamical instability, the Hopf bifurcation, which accounts for the cardinal features of the active process.

The physics of hearing: fluid mechanics and the active process of the inner ear

Most sounds of interest consist of complex, time-dependent admixtures of tones of diverse frequencies and variable amplitudes. To detect and process these signals, the ear employs a highly nonlinear, adaptive, real-time spectral analyzer: the cochlea. Sound excites vibration of the eardrum and the three miniscule bones of the middle ear, the last of which acts as a piston to initiate oscillatory pressure changes within the liquid-filled chambers of the cochlea. The basilar membrane, an elastic band spiraling along the cochlea between two of these chambers, responds to these pressures by conducting a largely independent traveling wave for each frequency component of the input. Because the basilar membrane is graded in mass and stiffness along its length, however, each traveling wave grows in magnitude and decreases in wavelength until it peaks at a specific, frequency-dependent position: low frequencies propagate to the cochlear apex, whereas high frequencies culminate at the base. The oscillations of the basilar membrane deflect hair bundles, the mechanically sensitive organelles of the ear's sensory receptors, the hair cells. As mechanically sensitive ion channels open and close, each hair cell responds with an electrical signal that is chemically transmitted to an afferent nerve fiber and thence into the brain. In addition to transducing mechanical inputs, hair cells amplify them by two means. Channel gating endows a hair bundle with negative stiffness, an instability that interacts with the motor protein myosin-1c to produce a mechanical amplifier and oscillator. Acting through the piezoelectric membrane protein prestin, electrical responses also cause outer hair cells to elongate and shorten, thus pumping energy into the basilar membrane's movements. The two forms of motility constitute an active process that amplifies mechanical inputs, sharpens frequency discrimination, and confers a compressive nonlinearity on responsiveness. These features arise because the active process operates near a Hopf bifurcation, the generic properties of which explain several key features of hearing. Moreover, when the gain of the active process rises sufficiently in ultraquiet circumstances, the system traverses the bifurcation and even a normal ear actually emits sound. The remarkable properties of hearing thus stem from the propagation of traveling waves on a nonlinear and excitable medium.

A resonance approach to cochlear mechanics

BACKGROUND

How does the cochlea analyse sound into its component frequencies? In the 1850s Helmholtz thought it occurred by resonance, whereas a century later Békésy's work indicated a travelling wave. The latter answer seemed to settle the question, but with the discovery in 1978 that the cochlea emits sound, the mechanics of the cochlea was back on the drawing board. Recent studies have raised questions about whether the travelling wave, as currently understood, is adequate to explain observations.

APPROACH

Applying basic resonance principles, this paper revisits the question. A graded bank of harmonic oscillators with cochlear-like frequencies and quality factors is simultaneously excited, and it is found that resonance gives rise to similar frequency responses, group delays, and travelling wave velocities as observed by experiment. The overall effect of the group delay gradient is to produce a decelerating wave of peak displacement moving from base to apex at characteristic travelling wave speeds. The extensive literature on chains of coupled oscillators is considered, and the occurrence of travelling waves, pseudowaves, phase plateaus, and forced resonance in such systems is noted.

CONCLUSION AND SIGNIFICANCE

This alternative approach to cochlear mechanics shows that a travelling wave can simply arise as an apparently moving amplitude peak which passes along a bank of resonators without carrying energy. This highlights the possible role of the fast pressure wave and indicates how phase delays and group delays of a set of driven harmonic oscillators can generate an apparent travelling wave. It is possible to view the cochlea as a chain of globally forced coupled oscillators, and this model incorporates fundamental aspects of both the resonance and travelling wave theories.

A model for the relation between stimulus frequency and spontaneous otoacoustic emissions in lizard papillae

Spontaneous otoacoustic emissions (SOAEs) and stimulus frequency otoacoustic emissions (SFOAEs) have been described from lizard ears. Although there are several models for these systems, none has modeled the characteristics of both of these types of otoacoustic emissions based upon their being derived from hair cells as active oscillators. Data from the ears of two lizard species, one lacking a tectorial membrane and one with a chain of tectorial sallets, as described by Bergevin et al. ["Coupled, active oscillators and lizard otoacoustic emissions," AIP Conf. Proc. 1403, 453 (2008)], are modeled as an array of coupled self-sustained oscillators. The model, originally developed by Vilfan and Duke ["Frequency clustering in spontaneous otoacoustic emissions from a lizard's ear," Biophys. J. 95, 4622-4630 (2008)], well describes both the amplitude and phase characteristics of SFOAEs and the relation between SFOAEs and SOAEs.

Are human spontaneous otoacoustic emissions generated by a chain of coupled nonlinear oscillators?

Spontaneous otoacoustic emissions (SOAEs) are generated by self-sustained cochlear oscillators. Properties of a computational model for a linear array of active oscillators with nearest neighbor coupling are investigated. The model can produce many experimentally well-established properties of SOAEs.

A mean-field approach to elastically coupled hair bundles

We study the dynamics of oscillatory hair bundles which are coupled elastically in their deflection variable and are subject to noise. We present a stochastic description capturing the dynamics of the hair bundles' mean field. In particular, the presented derivation elucidates the origin of the previously described noise reduction by coupling. By comparison of simulations of the approximate dynamics and the full system, we verify our results. Furthermore, we demonstrate that the specific type of coupling considered implies coupling-induced changes in the dynamics beyond mere noise reduction.

MDMA (ecstasy) enhances loud noise-induced morphofunctional alterations in heart and adrenal gland

Noise is an environmental stressor increasingly more present in modern life and, in particular, in a variety of recreational contexts. The aim of this work is to show the effects of noise on the myocardium and adrenal gland, through a careful review of the literature dealing with the peripheral effects of noise exposure in experimental and clinical studies. Noise induces adverse effects in human health, principally involving the cardiovascular and autonomic nervous systems, and the endocrine apparatus. Several factors in recreational environments potentially worsen the effects induced by loud noise. Among these, the intake of 3,4-methylenedioxymethamphetamine (MDMA) is frequently associated with noise exposure in recreational situations, because of its high compliance within social and relaxation settings. For this reason, MDMA is defined as a club drug--as its intake by young people often occurs in association with other factors, such as aggregation, high temperatures, and noise. It is known that self-administration of MDMA by humans causes severe toxicity. In particular, the myocardium is affected early after MDMA intake--resulting in tachycardia, hypertension, and arrhythmia. Furthermore, MDMA alters the activity of the adrenal glands by elevating catecholamines and corticosterone levels. This review shows that combining MDMA and loud noise exposure potentiates the effects that are produced by each single stimulant alone as seen in experimental animal models. The convergence of the effects of prolonged loud noise exposure and the consumption of MDMA on the same system might explain the sudden fatal events that happen in recreational situations.

Comparison of otoacoustic emissions within gecko subfamilies: morphological implications for auditory function in lizards

Otoacoustic emissions (OAEs) are sounds emitted by the ear and provide a non-invasive probe into mechanisms underlying peripheral auditory transduction. This study focuses upon a comparison of emission properties in two phylogenetically similar pairs of gecko: Gekko gecko and Hemidactylus turcicus and Eublepharis macularius and Coleonyx variegatus. Each pair consists of two closely related species within the same subfamily, with quantitatively known morphological properties at the level of the auditory sensory organ (basilar papilla) in the inner ear. Essentially, the comparison boils down to an issue of size: how does overall body size, as well as the inner-ear dimensions (e.g., papilla length and number of hair cells), affect peripheral auditory function as inferred from OAEs? Estimates of frequency selectivity derived from stimulus-frequency emissions (emissions evoked by a single low-level tone) indicate that tuning is broader in the species with fewer hair cells/shorter papilla. Furthermore, emissions extend outwards to higher frequencies (for similar body temperatures) in the species with the smaller body size/narrower interaural spacing. This observation suggests the smaller species have relatively improved high-frequency sensitivity, possibly related to vocalizations and/or aiding azimuthal sound localization. For one species (Eublepharis), emissions were also examined in both juveniles and adults. Qualitatively similar emission properties in both suggests that inner-ear function is adult like soon after hatching and that external body size (e.g., middle-ear dimensions and interaural spacing) has a relatively small impact upon emission properties within a species.

A critique of the critical cochlea: Hopf--a bifurcation--is better than none

The sense of hearing achieves its striking sensitivity, frequency selectivity, and dynamic range through an active process mediated by the inner ear's mechanoreceptive hair cells. Although the active process renders hearing highly nonlinear and produces a wealth of complex behaviors, these various characteristics may be understood as consequences of a simple phenomenon: the Hopf bifurcation. Any critical oscillator operating near this dynamic instability manifests the properties demonstrated for hearing: amplification with a specific form of compressive nonlinearity and frequency tuning whose sharpness depends on the degree of amplification. Critical oscillation also explains spontaneous otoacoustic emissions as well as the spectrum and level dependence of the ear's distortion products. Although this has not been realized, several valuable theories of cochlear function have achieved their success by incorporating critical oscillators.