Mechanics of the Cochlea: Modeling Efforts

Finite element simulation of cochlear traveling wave under air and bone conduction hearing

Besides the normal hearing pathway known as air conduction (AC), sound can also transmit to the cochlea through the skull, known as bone conduction (BC). During BC stimulation, the cochlear walls demonstrate rigid body motion (RBM) and compressional motion (CPM), both inducing the basilar membrane traveling wave (TW). Despite numerous measuring and modeling efforts for the TW phenomenon, the mechanism remains unclear, especially in the case of BC. This paper proposes a 3D finite element cochlea model mimicking the TW under BC. The model uses a traditional "box model" form, but in a spiral shape, with two fluid chambers separated by the long and flexible BM. The cochlear fluid was enclosed by bony walls, the oval and round window membranes. Contingent boundary conditions and stimulations are introduced according to the physical basis of AC and BC. Particularly for BC, both RBM and CPM of the cochlea walls are simulated. Harmonic numerical solutions are obtained at multiple frequencies among the hearing range. The BM vibration amplitude ([Formula: see text]) and its relation with volume displacement difference between the oval and round windows [Formula: see text], as well as the pressure difference at the base of the cochlea ([Formula: see text]), are analyzed. The simulated BM response at 12 mm from the base is peaked at about 3 k Hz, which is consistent with published experimental data. The TW properties under AC and BC are the same and have a common mechanism. (1) [Formula: see text] is proportional to [Formula: see text] at low frequencies. (2) [Formula: see text] is also proportional to [Formula: see text], within 5 dB error at high frequencies such as 16 k Hz. This study partly reveals the common quantitative relations between the TW and related factors under AC and BC hearing.

Fitting pole-zero micromechanical models to cochlear response measurements

An efficient way of describing the linear micromechanical response of the cochlea is in terms of its poles and zeros. Pole-zero models with local scaling symmetry are derived for both one and two degree-of-freedom micromechanical systems. These elements are then used in a model of the coupled cochlea, which is optimised to minimise the mean square difference between its frequency response and that measured on the basilar membrane inside the mouse cochlea by Lee, Raphael, Xia, Kim, Grillet, Applegate, Ellerbee Bowden, and Oghalai [(2016) J. Neurosci. 36, 8160-8173] and Oghalai Lab [(2015). https://oghalailab.stanford.edu], at different excitation levels. A model with two degree-of-freedom micromechanics generally fits the measurements better than a model with single degree-of-freedom micromechanics, particularly at low excitations where the cochlea is active, except post-mortem conditions, when the cochlea is passive. The model with the best overall fit to the data is found to be one with two degree-of-freedom micromechanics and 3D fluid coupling. Although a unique lumped parameter network cannot be inferred from such a pole-zero description, these fitted results help indicate what properties such a network should have.

Comment on 'Are some people suffering as a result of increasing mass exposure of the public to ultrasound in air?'

A number of queries regarding the paper 'Are some people suffering as a result of increasing mass exposure of the public to ultrasound in air?' (Leighton 2016 Proc. R. Soc. A472, 20150624 (doi:10.1098/rspa.2015.0624)) have been sent in from readers, almost all based around some or all of a small set of questions. These can be grouped into issues of engineering, human factors and timeliness. Those issues (represented by the most typical wording used in queries) and my responses are summarized in this comment.

A linearly tapered box model of the cochlea

A box shape with constant area is often used to represent the complex geometry in the cochlea, although variation of the fluid chambers areas is known to be more complicated. This variation is accounted for here by an "effective area," given by the harmonic mean of upper and lower chamber area from previous measurements. The square root of this effective area varies linearly along the cochleae in the investigated mammalian species. This suggests the use of a linearly tapered box model in which the fluid chamber width and height are equal, but decrease linearly along its length. The basilar membrane (BM) width is assumed to increase linearly along the model. An analytic form of the far-field fluid pressure difference due to BM motion is derived for this tapered model. The distributions of the passive BM response are calculated using both the tapered and uniform models and compared with human and mouse measurements. The discrepancy between the models is frequency-dependent and becomes small at low frequencies. The tapered model developed here shows a reasonable fit to experimental measurements, when the cochleae are cadaver or driven at high sound pressure level, and provides a convenient way to incorporate cochlear geometrical variations.

Finite-element model of the active organ of Corti

The cochlear amplifier that provides our hearing with its extraordinary sensitivity and selectivity is thought to be the result of an active biomechanical process within the sensory auditory organ, the organ of Corti. Although imaging techniques are developing rapidly, it is not currently possible, in a fully active cochlea, to obtain detailed measurements of the motion of individual elements within a cross section of the organ of Corti. This motion is predicted using a two-dimensional finite-element model. The various solid components are modelled using elastic elements, the outer hair cells (OHCs) as piezoelectric elements and the perilymph and endolymph as viscous and nearly incompressible fluid elements. The model is validated by comparison with existing measurements of the motions within the passive organ of Corti, calculated when it is driven either acoustically, by the fluid pressure or electrically, by excitation of the OHCs. The transverse basilar membrane (BM) motion and the shearing motion between the tectorial membrane and the reticular lamina are calculated for these two excitation modes. The fully active response of the BM to acoustic excitation is predicted using a linear superposition of the calculated responses and an assumed frequency response for the OHC feedback.

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.

Nonlinear damping and quasi-linear modelling

The mechanism of energy dissipation in mechanical systems is often nonlinear. Even though there may be other forms of nonlinearity in the dynamics, nonlinear damping is the dominant source of nonlinearity in a number of practical systems. The analysis of such systems is simplified by the fact that they show no jump or bifurcation behaviour, and indeed can often be well represented by an equivalent linear system, whose damping parameters depend on the form and amplitude of the excitation, in a 'quasi-linear' model. The diverse sources of nonlinear damping are first reviewed in this paper, before some example systems are analysed, initially for sinusoidal and then for random excitation. For simplicity, it is assumed that the system is stable and that the nonlinear damping force depends on the nth power of the velocity. For sinusoidal excitation, it is shown that the response is often also almost sinusoidal, and methods for calculating the amplitude are described based on the harmonic balance method, which is closely related to the describing function method used in control engineering. For random excitation, several methods of analysis are shown to be equivalent. In general, iterative methods need to be used to calculate the equivalent linear damper, since its value depends on the system's response, which itself depends on the value of the equivalent linear damper. The power dissipation of the equivalent linear damper, for both sinusoidal and random cases, matches that dissipated by the nonlinear damper, providing both a firm theoretical basis for this modelling approach and clear physical insight. Finally, practical examples of nonlinear damping are discussed: in microspeakers, vibration isolation, energy harvesting and the mechanical response of the cochlea.

Comparing methods of modeling near field fluid coupling in the cochlea

As well as generating the far field pressure, which allows wave propagation in the cochlea, the vibration of an individual element of the basilar membrane (BM) will also generate a near field pressure, which increases its mass and gives rise to local longitudinal coupling. This paper compares the efficiency and accuracy of a number of different methods of calculating the near field pressure distribution, and explores the connections between them. In particular it is shown that a common approximation to the wavenumber description of the near field pressure is equivalent, in the spatial domain, to an exponential decay away from the point of excitation. Two important properties of the near field pressure are its maximum amplitude, which is finite if the vibrating element has a finite length, and the value of its spatial integral, which determines the added mass on the BM due to the fluid loading. These properties are calculated as a function of the BM width relative to the width of the fluid chamber. By parameterizing the near field pressure variation in this way, it can be readily incorporated into coupled models of the cochlea, without the considerable computational expense of calculating the full three dimensional pressure field.

Conductive component after cochlear implantation in patients with residual hearing conservation

PURPOSE

Changes in auditory thresholds following cochlear implantation are generally assumed to be due to damage to neural elements. Theoretical studies have suggested that placement of a cochlear implant can cause a conductive hearing loss. Identification of a conductive component following cochlear implantation could guide improvements in surgical techniques or device designs. The purpose of this study is to characterize new-onset conductive hearing losses after cochlear implantation.

METHOD

In a prospective study, air- and bone-conduction audiometric testing were completed on cochlear implant recipients. An air-bone gap equal to or greater than 15 dB HL at 2 frequencies determined the presence of a conductive component.

RESULTS

Of the 32 patients with preoperative bone-conduction hearing, 4 patients had a new-onset conductive component resulting in a mixed hearing loss, with air-conduction thresholds ranging from moderate to profound and an average air-bone gap of 30 dB HL. One had been implanted through the round window, 2 had an extended round window, and 1 had a separate cochleostomy.

CONCLUSIONS

Loss of residual hearing following cochlear implantation may be due in part to a conductive component. Identifying the mechanism for this conductive component may help minimize hearing loss. Postoperative hearing evaluation should measure both air- and bone-conduction thresholds.

A wave finite element analysis of the passive cochlea

Current models of the cochlea can be characterized as being either based on the assumed propagation of a single slow wave, which provides good insight, or involve the solution of a numerical model, such as in the finite element method, which allows the incorporation of more detailed anatomical features. In this paper it is shown how the wave finite element method can be used to decompose the results of a finite element calculation in terms of wave components, which allows the insight of the wave approach to be brought to bear on more complicated numerical models. In order to illustrate the method, a simple box model is considered, of a passive, locally reacting, basilar membrane interacting via three-dimensional fluid coupling. An analytic formulation of the dispersion equation is used initially to illustrate the types of wave one would expect in such a model. The wave finite element is then used to calculate the wavenumbers of all the waves in the finite element model. It is shown that only a single wave type dominates the response until this peaks at the best place in the cochlea, where an evanescent, higher order fluid wave can make a significant contribution.

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.

Analysis of the cochlear amplifier fluid pump hypothesis

We use analysis of a realistic three-dimensional finite-element model of the tunnel of Corti (ToC) in the middle turn of the gerbil cochlea tuned to the characteristic frequency (CF) of 4 kHz to show that the anatomical structure of the organ of Corti (OC) is consistent with the hypothesis that the cochlear amplifier functions as a fluid pump. The experimental evidence for the fluid pump is that outer hair cell (OHC) contraction and expansion induce oscillatory flow in the ToC. We show that this oscillatory flow can produce a fluid wave traveling in the ToC and that the outer pillar cells (OPC) do not present a significant barrier to fluid flow into the ToC. The wavelength of the resulting fluid wave launched into the tunnel at the CF is 1.5 mm, which is somewhat longer than the wavelength estimated for the classical traveling wave. This fluid wave propagates at least one wavelength before being significantly attenuated. We also investigated the effect of OPC spacing on fluid flow into the ToC and found that, for physiologically relevant spacing between the OPCs, the impedance estimate is similar to that of the underlying basilar membrane. We conclude that the row of OPCs does not significantly impede fluid exchange between ToC and the space between the row of OPC and the first row of OHC-Dieter's cells complex, and hence does not lead to excessive power loss. The BM displacement resulting from the fluid pumped into the ToC is significant for motion amplification. Our results support the hypothesis that there is an additional source of longitudinal coupling, provided by the ToC, as required in many non-classical models of the cochlear amplifier.

Direction of wave propagation in the cochlea for internally excited basilar membrane

Otoacoustic emissions are an indicator of a normally functioning cochlea and as such are a useful tool for non-invasive diagnosis as well as for understanding cochlear function. While these emitted waves are hypothesized to arise from active processes and exit through the cochlear fluids, neither the precise mechanism by which these emissions are generated nor the transmission pathway is completely known. With regard to the acoustic pathway, two competing hypotheses exist to explain the dominant mode of emission. One hypothesis, the backward-traveling wave hypothesis, posits that the emitted wave propagates as a coupled fluid-structure wave while the alternate hypothesis implicates a fast, compressional wave in the fluid as the main mechanism of energy transfer. In this paper, we study the acoustic pathway for transmission of energy from the inside of the cochlea to the outside through a physiologically-based theoretical model. Using a well-defined, compact source of internal excitation, we predict that the emission is dominated by a backward traveling fluid-structure wave. However, in an active model of the cochlea, a forward traveling wave basal to the location of the force is possible in a limited region around the best place. Finally, the model does predict the dominance of compressional waves under a different excitation, such as an apical excitation.

Investigating the wave-fixed and place-fixed origins of the 2f(1)-f(2) distortion product otoacoustic emission within a micromechanical cochlear model

The 2f(1)-f(2) distortion product otoacoustic emission (DPOAE) arises within the cochlea due to the nonlinear interaction of two stimulus tones (f(1) and f(2)). It is thought to comprise contributions from a wave-fixed source and a place-fixed source. The generation and transmission of the 2f(1)-f(2) DPOAE is investigated here using quasilinear solutions to an elemental model of the human cochlea with nonlinear micromechanics. The micromechanical parameters and nonlinearity are formulated to match the measured response of the cochlea to single- and two-tone stimulation. The controlled introduction of roughness into the active micromechanics of the model allows the wave- and place-fixed contributions to the DPOAE to be studied separately. It is also possible to manipulate the types of nonlinear suppression that occur within the quasilinear model to investigate the influence of stimulus parameters on DPOAE generation. The model predicts and explains a variety of 2f(1)-f(2) DPOAE phenomena: The dependence of emission amplitude on stimulus parameters, the weakness of experiments designed to quantify cochlear amplifier gain, and the predominant mechanism which gives rise to DPOAE fine structure. In addition, the model is used to investigate the properties of the wave-fixed source and how these properties are influenced by the stimulus parameters.

Unification and extension of monolithic state space and iterative cochlear models

Time domain cochlear models have primarily followed a method introduced by Allen and Sondhi [J. Acoust. Soc. Am. 66, 123-132 (1979)]. Recently the "state space formalism" proposed by Elliott et al. [J. Acoust. Soc. Am. 122, 2759-2771 (2007)] has been used to simulate a wide range of nonlinear cochlear models. It used a one-dimensional approach that is extended to two dimensions in this paper, using the finite element method. The recently developed "state space formalism" in fact shares a close relationship to the earlier approach. Working from Diependaal et al. [J. Acoust. Soc. Am. 82, 1655-1666 (1987)] the two approaches are compared and the relationship formalized. Understanding this relationship allows models to be converted from one to the other in order to utilize each of their strengths. A second method to derive the state space matrices required for the "state space formalism" is also presented. This method offers improved numerical properties because it uses the information available about the model more effectively. Numerical results support the claims regarding fluid dimension and the underlying similarity of the two approaches. Finally, the recent advances in the state space formalism [Bertaccini and Sisto, J. Comp. Phys. 230, 2575-2587 (2011)] are discussed in terms of this relationship.

Reply to "on cochlear impedances and the miscomputation of power gain" by Shera et Al. J. Assoc. Re. Otolaryngol

Using a scanning laser interferometer, we recently measured the volume velocity of the basilar membrane vibration in the sensitive gerbil cochlea and estimated that the cochlear power gain is ~100 at low sound pressure levels (Ren et al., Nat Commun 2:216-223, 2011a). We thank Shera et al. for recognizing the technical challenges of our experiments and appreciating the beauty of our data in their comment (Shera et al., J Assoc Res Otolaryngol (in press), 2011). These authors argue that our analysis is inappropriate, invalidating our conclusion; moreover, they suggest that our finding of a power gain of >1 could arise from a passive structure or cochlea. While our analysis and interpretation remain to be verified, they are justified according to commonly accepted assumptions and theories in cochlear mechanics. Here, we also show that the mathematical demonstration of a power gain of >1 in a passive cochlea by Shera et al. is inconsistent with our data, which show that the volume velocity and power gain decrease and become <1 as the sound level increases.

Two measures of temporal resolution in brown-headed cowbirds (Molothrus ater)

Studies of auditory temporal resolution in birds have traditionally examined processing capabilities by assessing behavioral discrimination of sounds varying in temporal structure. Here, temporal resolution of the brown-headed cowbird (Molothrus ater) was measured using two auditory evoked potential (AEP)-based methods: auditory brainstem responses (ABRs) to paired clicks and envelope following responses (EFRs) to amplitude-modulated tones. The basic patterns observed in cowbirds were similar to those found in other songbird species, suggesting similar temporal processing capabilities. The amplitude of the ABR to the second click was less than that of the first click at inter-click intervals less than 10 ms, and decreased to 30% at an interval of 1 ms. EFR amplitude was generally greatest at modulation frequencies from 335 to 635 Hz and decreased at higher and lower modulation frequencies. Compared to data from terrestrial mammals these results support recent behavioral findings of enhanced temporal resolution in birds. General agreement between these AEP results and behaviorally based studies suggests that AEPs can provide a useful assessment of temporal resolution in wild bird species.

Fluid coupling in a discrete model of cochlear mechanics

A discrete model of cochlear mechanics is introduced that includes a full, three-dimensional, description of fluid coupling. This formulation allows the fluid coupling and basilar membrane dynamics to be analyzed separately and then coupled together with a simple piece of linear algebra. The fluid coupling is initially analyzed using a wavenumber formulation and is separated into one component due to one-dimensional fluid coupling and one comprising all the other contributions. Using the theory of acoustic waves in a duct, however, these two components of the pressure can also be associated with a far field, due to the plane wave, and a near field, due to the evanescent, higher order, modes. The near field components are then seen as one of a number of sources of additional longitudinal coupling in the cochlea. The effects of non-uniformity and asymmetry in the fluid chamber areas can also be taken into account, to predict both the pressure difference between the chambers and the mean pressure. This allows the calculation, for example, of the effect of a short cochlear implant on the coupled response of the cochlea.

Songbirds tradeoff auditory frequency resolution and temporal resolution

Physical tradeoffs may in some cases constrain the evolution of sensory systems. The peripheral auditory system, for example, performs a spectral decomposition of sound that should result in a tradeoff between frequency resolution and temporal resolution. We assessed temporal resolution in three songbird species using auditory brainstem responses to paired click stimuli. Temporal resolution was greater in house sparrows (Passer domesticus) than Carolina chickadees (Poecile carolinensis) and white-breasted nuthatches (Sitta carolinensis), as predicted based on previous observations of broader auditory filters (lower frequency resolution) in house sparrows. Furthermore, within chickadees, individuals with broader auditory filters had greater temporal resolution. In contrast to predictions however, temporal resolution was similar between chickadees and nuthatches despite broader auditory filters in chickadees. These results and the results of a model simulation exploring the effect of broadened auditory filter bandwidth on temporal resolution in the auditory periphery strongly suggest that frequency resolution constrains temporal resolution in songbirds. Furthermore, our results suggest that songbirds have greater temporal resolution than some mammals, in agreement with recent behavioral studies. Species differences in temporal resolution may reflect adaptations for efficient processing of species-specific vocalizations, while individual differences within species may reflect experience-based developmental plasticity or hormonal effects.

The interplay between active hair bundle motility and electromotility in the cochlea

The cochlear amplifier is a nonlinear active process providing the mammalian ear with its extraordinary sensitivity, large dynamic range and sharp frequency tuning. While there is much evidence that amplification results from active force generation by mechanosensory hair cells, there is debate about the cellular processes behind nonlinear amplification. Outer hair cell electromotility has been suggested to underlie the cochlear amplifier. However, it has been shown in frog and turtle that spontaneous movements of hair bundles endow them with a nonlinear response with increased sensitivity that could be the basis of amplification. The present work shows that the properties of the cochlear amplifier could be understood as resulting from the combination of both hair bundle motility and electromotility in an integrated system that couples these processes through the geometric arrangement of hair cells embedded in the cochlear partition. In this scenario, the cochlear partition can become a dynamic oscillator which in the vicinity of a Hopf bifurcation exhibits all the key properties of the cochlear amplifier. The oscillatory behavior and the nonlinearity are provided by active hair bundles. Electromotility is largely linear but produces an additional feedback that allows hair bundle movements to couple to basilar membrane vibrations.

Reverse cochlear propagation in the intact cochlea of the gerbil: evidence for slow traveling waves

The inner ear can produce sounds, but how these otoacoustic emissions back-propagate through the cochlea is currently debated. Two opposing views exist: fast pressure waves in the cochlear fluids and slow traveling waves involving the basilar membrane. Resolving this issue requires measuring the travel times of emissions from their cochlear origin to the ear canal. This is problematic because the exact intracochlear location of emission generation is unknown and because the cochlea is vulnerable to invasive measurements. We employed a multi-tone stimulus optimized to measure reverse travel times. By exploiting the dispersive nature of the cochlea and by combining acoustic measurements in the ear canal with recordings of the cochlear-microphonic potential, we were able to determine the group delay between intracochlear emission-generation and their recording in the ear canal. These delays remained significant after compensating for middle-ear delay. The results contradict the hypothesis that the reverse propagation of emissions is exclusively by direct pressure waves.

Inverse-solution method for a class of non-classical cochlear models

Measurements of distortion-product (DP) waves inside the cochlea have led to a conception of wave propagation that is at variance with the "classical" attitude. Of the several alternatives that have been proposed to remedy this situation, the feed-forward model could be a promising one. This paper describes a method to apply the inverse solution with the aim to attain a feed-forward model that accurately reproduces a measured response. It is demonstrated that the computation method is highly successful. Subsequently, it is shown that in a feed-forward model a DP wave generated by a two-tone stimulus is almost exclusively a forward-traveling wave which property agrees with the nature of the experimental findings. However, the amplitude of the computed DP wave is only substantial in the region where the stimulation patterns of the two primary tones overlap. In addition, the model developed cannot explain coherent reflection for single tones. It has been suggested that a forward transversal DP wave induced by a (retrograde) compression wave could be involved in DP wave generation. This topic is critically evaluated.

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.

Testing coherent reflection in chinchilla: Auditory-nerve responses predict stimulus-frequency emissions

Coherent-reflection theory explains the generation of stimulus-frequency and transient-evoked otoacoustic emissions by showing how they emerge from the coherent "backscattering" of forward-traveling waves by mechanical irregularities in the cochlear partition. Recent published measurements of stimulus-frequency otoacoustic emissions (SFOAEs) and estimates of near-threshold basilar-membrane (BM) responses derived from Wiener-kernel analysis of auditory-nerve responses allow for comprehensive tests of the theory in chinchilla. Model predictions are based on (1) an approximate analytic expression for the SFOAE signal in terms of the BM traveling wave and its complex wave number, (2) an inversion procedure that derives the wave number from BM traveling waves, and (3) estimates of BM traveling waves obtained from the Wiener-kernel data and local scaling assumptions. At frequencies above 4 kHz, predicted median SFOAE phase-gradient delays and the general shapes of SFOAE magnitude-versus-frequency curves are in excellent agreement with the measurements. At frequencies below 4 kHz, both the magnitude and the phase of chinchilla SFOAEs show strong evidence of interference between short- and long-latency components. Approximate unmixing of these components, and association of the long-latency component with the predicted SFOAE, yields close agreement throughout the cochlea. Possible candidates for the short-latency SFOAE component, including wave-fixed distortion, are considered. Both empirical and predicted delay ratios (long-latency SFOAE delay/BM delay) are significantly less than 2 but greater than 1. Although these delay ratios contradict models in which SFOAE generators couple primarily into cochlear compression waves, they are consistent with the notion that forward and reverse energy propagation in the cochlea occurs predominantly by means of traveling pressure-difference waves. The compelling overall agreement between measured and predicted delays suggests that the coherent-reflection model captures the dominant mechanisms responsible for the generation of reflection-source otoacoustic emissions.

From Hearing to Listening: Design and Properties of an Actively Tunable Electronic Hearing Sensor

An important step towards understanding the working principles of the mammalian hearing sensor is the concept of an active cochlear amplifier. Theoretical arguments and physiological measurements suggest that the active cochlear amplifiers originate from systems close to a Hopf bifurcation. Efforts to model the mammalian hearing sensor on these grounds have, however, either had problems in reproducing sufficiently close essential aspects of the biological example (Magnasco, M.O. Phys. Rev. Lett. 90, 058101 (2003); Duke, T. & Jülicher, F. Phys. Rev. Lett. 90, 158101 (2003)), or required complicated spatially coupled differential equation systems that are unfeasible for transient signals (Kern, A. & Stoop, R. Phys. Rev. Lett. 91, 128101 (2003)). Here, we demonstrate a simple system of electronically coupled Hopf amplifiers that not only leads to the desired biological response behavior, but also has real-time capacity. The obtained electronic Hopf cochlea shares all salient signal processing features exhibited by the mammalian cochlea and thus provides a simple and efficient design of an artificial mammalian hearing sensor.

Laser amplification with a twist: traveling-wave propagation and gain functions from throughout the cochlea

Except at the handful of sites explored by the inverse method, the characteristics-indeed, the very existence-of traveling-wave amplification in the mammalian cochlea remain largely unknown. Uncertainties are especially pronounced in the apex, where mechanical and electrical measurements lack the independent controls necessary for assessing damage to the preparation. At a functional level, the form and amplification of cochlear traveling waves are described by quantities known as propagation and gain functions. A method for deriving propagation and gain functions from basilar-membrane mechanical transfer functions is presented and validated by response reconstruction. Empirical propagation and gain functions from locations throughout the cochlea are obtained in mechanically undamaged preparations by applying the method to published estimates of near-threshold basilar membrane responses derived from Wiener-kernel (chinchilla) and zwuis analysis (cat) of auditory-nerve responses to broadband stimuli. The properties of these functions, and their variation along the length of the cochlea, are described. In both species, and at all locations examined, the gain functions reveal a region of positive power gain basal to the wave peak. The results establish the existence of traveling-wave amplification throughout the cochlea, including the apex. The derived propagation and gain functions resemble those characteristic of an active optical medium but rotated by 90 degrees in the complex plane. Rotation of the propagation and gain functions enables the mammalian cochlea to operate as a wideband, hydromechanical laser analyzer.

Longitudinally propagating traveling waves of the mammalian tectorial membrane

Sound-evoked vibrations transmitted into the mammalian cochlea produce traveling waves that provide the mechanical tuning necessary for spectral decomposition of sound. These traveling waves of motion that have been observed to propagate longitudinally along the basilar membrane (BM) ultimately stimulate the mechano-sensory receptors. The tectorial membrane (TM) plays a key role in this process, but its mechanical function remains unclear. Here we show that the TM supports traveling waves that are an intrinsic feature of its visco-elastic structure. Radial forces applied at audio frequencies (2-20 kHz) to isolated TM segments generate longitudinally propagating waves on the TM with velocities similar to those of the BM traveling wave near its best frequency place. We compute the dynamic shear storage modulus and shear viscosity of the TM from the propagation velocity of the waves and show that segments of the TM from the basal turn are stiffer than apical segments are. Analysis of loading effects of hair bundle stiffness, the limbal attachment of the TM, and viscous damping in the subtectorial space suggests that TM traveling waves can occur in vivo. Our results show the presence of a traveling wave mechanism through the TM that can functionally couple a significant longitudinal extent of the cochlea and may interact with the BM wave to greatly enhance cochlear sensitivity and tuning.

Semirealistic models of the cochlea

The aim of this paper is the introduction and comparison of consistent albeit passive mechanical models for the whole cochlea. A widely used transmission line filterbank, which hydrodynamically speaking is a long wave approximation (L model), suffers from a well-known inconsistency: its main modeling assumption is not valid within the resonance region, where most of the overall excitation takes place. In the present paper two approaches to overcome this inconsistency are discussed. One model is the average pressure (AP) model by Duifhuis, the other is obtained by a combination of a long and a short wave approximation (LS model). Considerable differences between the L and the LS model are observed. All models are compared by inserting them into the full integral equation obtained from the hydrodynamic equations and the boundary conditions. Here the LS model fares better than the AP model for small damping, whereas the opposite is true for higher damping. As expected, the L model fails badly in the resonance region.

A generalization of the van-der-Pol oscillator underlies active signal amplification in Drosophila hearing

The antennal hearing organs of the fruit fly Drosophila melanogaster boost their sensitivity by an active mechanical process that, analogous to the cochlear amplifier of vertebrates, resides in the motility of mechanosensory cells. This process nonlinearly improves the sensitivity of hearing and occasionally gives rise to self-sustained oscillations in the absence of sound. Time series analysis of self-sustained oscillations now unveils that the underlying dynamical system is well described by a generalization of the van-der-Pol oscillator. From the dynamic equations, the underlying amplification dynamics can explicitly be derived. According to the model, oscillations emerge from a combination of negative damping, which reflects active amplification, and a nonlinear restoring force that dictates the amplitude of the oscillations. Hence, active amplification in fly hearing seems to rely on the negative damping mechanism initially proposed for the cochlear amplifier of vertebrates.

Motility-associated hair-bundle motion in mammalian outer hair cells

Mammalian hearing owes its remarkable sensitivity and frequency selectivity to a local mechanical feedback process within the cochlea. Cochlear outer hair cells (OHCs) function as the key elements in the feedback loop in which the fast somatic motility of OHCs is thought to be the source of cochlear amplification. An alternative view is that amplification arises from active hair-bundle movement, similar to that seen in nonmammalian hair cells. We measured voltage-evoked hair-bundle motions in the gerbil cochlea to determine if such movements were also present in mammalian OHCs. The OHCs showed bundle movement with peak responses of up to 830 nm. The movement was insensitive to manipulations that would normally block mechanotransduction in the stereocilia, and it was absent in neonatal OHCs and prestin-knockout OHCs. These findings suggest that the bundle movement originated in somatic motility and that somatic motility has a central role in cochlear amplification in mammals.

Dynamic material properties of the tectorial membrane: a summary

Dynamic material properties of the tectorial membrane (TM) have been measured at audio frequencies in TMs excised from the apical portions of mouse cochleae. We review, integrate, and interpret recent findings. The mechanical point impedance of the TM in the radial, longitudinal, and transverse directions is viscoelastic and has a frequency dependence of the form 1/(K(j2pif)(alpha)) for 10<or=f<or=4000 Hz, where f is frequency, K is a constant, j=-1, and alpha approximately 0.66. Comparison with other connective tissues shows that the TM is a relatively lossy viscoelastic material. The median magnitudes of the point impedance at 10 Hz in the radial, longitudinal, and transverse directions are 4.6 x 10(-3) N.s/m, 1.8 x 10(-3) N.s/m, and 2.7 x 10(-3) N.s/m. Consistent with osmotic responses (Freeman et al., 2003), the TM point impedance is anisotropic - the TM is stiffer in the radial than in the longitudinal and transverse directions. The mechanical space constant of the TM is approximately 20 microm. Comparisons reveal that in the apical region of the mouse cochlea, the TM dynamic stiffness at 10 Hz is 10 times larger than the static stiffness of the aggregate hair cells in a mechanical space constant and roughly comparable to the stiffness of the basilar membrane. We conclude that the TM provides a mechanical load on the basilar membrane and that the lability of the TM to changes in endolymph composition may well be reflected in changes in basilar membrane motion.

The short-wave model and waves in two directions

In the region where a sinusoidal wave in the cochlea reaches its maximum amplitude, the long-wave (or one-dimensional) model of the cochlea is deficient. In this region a short-wave model is more appropriate. However, in its current form, the short-wave model supports only waves in one direction. Therefore, it cannot cope with reflection effects associated with, e.g., inhomogeneities. Theoretical explorations of creation and internal reflection of otoacoustic emissions have almost exclusively been based on the long-wave model. In this article the road is paved for future explorations on a generalized form of the short-wave model, one that supports forward as well as backward waves, and thus can include internal reflections.

The importance of phase data and model dimensionality to cochlear mechanics

The vulnerability of the mammalian cochlear amplifier to surgical trauma hinders observations of its behaviour in vivo. This produces a greater need for realistic models to aid the interpretation of the experimental observations. The emphasis in most modelling studies has been to simulate the gain of the response of the basilar membrane. This paper argues that matching the phase behaviour of the response should be given at least equal importance. When it is, many of the models used to justify hypotheses regarding the operation of the cochlear amplifier cannot simulate the response even of the dead cochlea. This discrepancy is due to oversimplification of the mechanics of the cochlear fluids. It is argued that three-dimensional fluid behaviour should be regarded as a bare minimum in any quantitative description of cochlear mechanics. Furthermore, it is shown that a three-dimensional model is consistent with experimental data from a healthy cochlea only when the main effect of the cochlear amplifier is to inject mechanical energy into the basilar membrane. The injection of mechanical energy is fundamentally different to modifying the stiffness of the basilar membrane. This means that existing models which possess cochlear amplifiers that effect large changes on the stiffness of the basilar membrane may not be accurate representations of the real organ.

Deformations of the isolated mouse tectorial membrane produced by oscillatory forces

Mechanical properties of the isolated tectorial membrane (TM) of the mouse were measured by applying oscillatory shear forces to the TM with a magnetic bead (radius approximately 10 mcm). Sinusoidal forces at 10 Hz with amplitudes from 5 to 33 nN were applied tangentially to the surfaces of 11 TMs. The ratio of force to bead displacement ranged from 0.04 to 0.98 N/m (median: 0.18 N/m, interquartile range: 0.11-0.30 N/m, n=90). Increasing frequency from 10 to 100 Hz decreased the magnitude of the displacement of the magnetic bead by 6-7.3 dB/decade. The phase of the displacement lagged that of the stimulus current by approximately 27-44 degrees across frequencies. Displacement of the adjacent tissue decreased as the distance from the magnetic bead increased. Space constants were of the order of tens of micrometers. Forces with equal amplitude and frequency were applied radially and longitudinally. Longitudinal displacements in response to longitudinal forces were 1-10 times as large as radial displacements in response to radial forces in 85% of 560 paired measurements. These results suggest that the following mechanical properties of the TM are important. (1) Viscoelasticity: The frequency dependence of TM displacement lies between that of a purely viscous and a purely elastic material, suggesting that both are important. (2) Mechanical coupling: Space constants indicate that hair bundles could interact mechanically with adjacent hair bundles via the TM. (3) Anisotropy: The mechanical impedance is greater in the radial direction than it is in the longitudinal direction. This mechanical anisotropy correlates with anatomical anisotropies, such as the radially oriented fibrillar structure of the TM.

A micromechanical model of the cochlea with radial movement of the tectorial membrane

The function of the tectorial membrane in the cochlear micromechanics is uncertain. In modeling approaches some models have assumed it to be a resonator that participates in the sharp tuning mechanisms of the cochlea with its mass coupled to the ciliary stiffness of outer hair cells, being driven by the shear force between the reticular lamina and itself. This paper presents a different type of micromechanical model which assumes that the tectorial membrane is driven by a lymphatic fluid flow that can be shown to have a substantial radial component. It also assumes that the reticular lamina is relatively stiff and thereby restrains the top end of outer hair cells that exert a force to the basilar membrane via Deiters cells. When combined with a three-dimensional block model, it can simulate the sharp tuning mechanisms of the cochlea well.

Measurements of the stiffness map challenge a basic tenet of cochlear theories

The cochlear frequency map is believed to depend on the progressive decrease in partition stiffness from base to apex. Measurements on cochleae from human cadavers by von Békésy (1960) suggested that the elasticity of the partition increases by a factor of 100 from the stapes to the helicotrema. However, conventional models require a factor of nearly 10,000 to support the frequency range of normal hearing if entirely determined by partition stiffness. To test this assumption, we measured point stiffness along the width and length of the partition in the gerbil cochlea. Two major findings result from this study: (1) contrary to von Békésy's results, both cellular and extracellular elements of the sensory epithelium exhibit stiffness gradients; and (2) the stiffness changes by only a factor of 100 over the whole cochlea. Our results imply that present ideas regarding partition vibration need to be significantly revised.