Dual traveling waves in an inner ear model with two degrees of freedom

Intracochlear overdrive: Characterizing nonlinear wave amplification in the mouse apex

In this study, we explore nonlinear cochlear amplification by analyzing basilar membrane (BM) motion in the mouse apex. Through in vivo, postmortem, and mechanical suppression recordings, we estimate how the cochlear amplifier nonlinearly shapes the wavenumber of the BM traveling wave, specifically within a frequency range where the short-wave approximation holds. Our findings demonstrate that a straightforward mathematical model, depicting the cochlear amplifier as a wavenumber modifier with strength diminishing monotonically as BM displacement increases, effectively accounts for the various experimental observations. This empirically derived model is subsequently incorporated into a physics-based "overturned" framework of cochlear amplification [see Altoè, Dewey, Charaziak, Oghalai, and Shera (2022), J. Acoust. Soc. Am. 152, 2227-2239] and tested against additional experimental data. Our results demonstrate that the relationships established within the short-wave region remain valid over a much broader frequency range. Furthermore, the model, now exclusively calibrated to BM data, predicts the behavior of the opposing side of the cochlear partition, aligning well with recent experimental observations. The success in reproducing key features of the experimental data and the mathematical simplicity of the resulting model provide strong support for the "overturned" theory of cochlear amplification.

Overturning the mechanisms of cochlear amplification via area deformations of the organ of Corti

The mammalian ear embeds a cellular amplifier that boosts sound-induced hydromechanical waves as they propagate along the cochlea. The operation of this amplifier is not fully understood and is difficult to disentangle experimentally. In the prevailing view, cochlear waves are amplified by the piezo-electric action of the outer hair cells (OHCs), whose cycle-by-cycle elongations and contractions inject power into the local motion of the basilar membrane (BM). Concomitant deformations of the opposing (or "top") side of the organ of Corti are assumed to play a minor role and are generally neglected. However, analysis of intracochlear motions obtained using optical coherence tomography calls this prevailing view into question. In particular, the analysis suggests that (i) the net local power transfer from the OHCs to the BM is either negative or highly inefficient; and (ii) vibration of the top side of the organ of Corti plays a primary role in traveling-wave amplification. A phenomenological model derived from these observations manifests realistic cochlear responses and suggests that amplification arises almost entirely from OHC-induced deformations of the top side of the organ of Corti. In effect, the model turns classic assumptions about spatial impedance relations and power-flow direction within the sensory epithelium upside down.

Intracochlear distortion products are broadly generated by outer hair cells but their contributions to otoacoustic emissions are spatially restricted

Detection of low-level sounds by the mammalian cochlea requires electromechanical feedback from outer hair cells (OHCs). This feedback arises due to the electromotile response of OHCs, which is driven by the modulation of their receptor potential caused by the stimulation of mechano-sensitive ion channels. Nonlinearity in these channels distorts impinging sounds, creating distortion-products that are detectable in the ear canal as distortion-product otoacoustic emissions (DPOAEs). Ongoing efforts aim to develop DPOAEs, which reflects the ear's health, into diagnostic tools for sensory hearing loss. These efforts are hampered by limited knowledge on the cochlear extent contributing to DPOAEs. Here, we report on intracochlear distortion products (IDPs) in OHC electrical responses and intracochlear fluid pressures. Experiments and simulations with a physiologically motivated cochlear model show that widely generated electrical IDPs lead to mechanical vibrations in a frequency-dependent manner. The local cochlear impedance restricts the region from which IDPs contribute to DPOAEs at low to moderate intensity, which suggests that DPOAEs may be used clinically to provide location-specific information about cochlear damage.

Manipulation of the Endocochlear Potential Reveals Two Distinct Types of Cochlear Nonlinearity

The mammalian hearing organ, the cochlea, contains an active amplifier to boost the vibrational response to low level sounds. Hallmarks of this active process are sharp location-dependent frequency tuning and compressive nonlinearity over a wide stimulus range. The amplifier relies on outer hair cell (OHC)-generated forces driven in part by the endocochlear potential, the ∼+80 mV potential maintained in scala media, generated by the stria vascularis. We transiently eliminated the endocochlear potential in vivo by an intravenous injection of furosemide and measured the vibrations of different layers in the cochlea's organ of Corti using optical coherence tomography. Distortion product otoacoustic emissions were also monitored. After furosemide injection, the vibrations of the basilar membrane lost the best frequency (BF) peak and showed broad tuning similar to a passive cochlea. The intra-organ of Corti vibrations measured in the region of the OHCs lost the BF peak and showed low-pass responses but retained nonlinearity. This strongly suggests that OHC electromotility was operating and being driven by nonlinear OHC current. Thus, although electromotility is presumably necessary to produce a healthy BF peak, the mere presence of electromotility is not sufficient. The BF peak recovered nearly fully within 2 h, along with the recovery of odd-order distortion product otoacoustic emissions. The recovery pattern suggests that physical shifts in operating condition are a critical step in the recovery process.

A role for tectorial membrane mechanics in activating the cochlear amplifier

The mechanical and electrical responses of the mammalian cochlea to acoustic stimuli are nonlinear and highly tuned in frequency. This is due to the electromechanical properties of cochlear outer hair cells (OHCs). At each location along the cochlear spiral, the OHCs mediate an active process in which the sensory tissue motion is enhanced at frequencies close to the most sensitive frequency (called the characteristic frequency, CF). Previous experimental results showed an approximate 0.3 cycle phase shift in the OHC-generated extracellular voltage relative the basilar membrane displacement, which was initiated at a frequency approximately one-half octave lower than the CF. Findings in the present paper reinforce that result. This shift is significant because it brings the phase of the OHC-derived electromotile force near to that of the basilar membrane velocity at frequencies above the shift, thereby enabling the transfer of electrical to mechanical power at the basilar membrane. In order to seek a candidate physical mechanism for this phenomenon, we used a comprehensive electromechanical mathematical model of the cochlear response to sound. The model predicts the phase shift in the extracellular voltage referenced to the basilar membrane at a frequency approximately one-half octave below CF, in accordance with the experimental data. In the model, this feature arises from a minimum in the radial impedance of the tectorial membrane and its limbal attachment. These experimental and theoretical results are consistent with the hypothesis that a tectorial membrane resonance introduces the correct phasing between mechanical and electrical responses for power generation, effectively turning on the cochlear amplifier.

The interplay of organ-of-Corti vibrational modes, not tectorial- membrane resonance, sets outer-hair-cell stereocilia phase to produce cochlear amplification

The mechanical motions that deflect outer-hair-cell (OHC) stereocilia and the resulting effects of OHC motility are reviewed, concentrating on high-frequency cochlear regions. It has been proposed that a tectorial-membrane (TM) resonance makes the phase of OHC stereocilia motion be appropriate to produce cochlear amplification, i.e. so that the OHC force that pushes the basilar membrane (BM) is in the same direction as BM velocity. Evidence for and against the TM-resonance hypothesis are considered, including new cochlear-motion measurements using optical coherence tomography, and it is concluded that there is no such TM resonance. The evidence points to there being an advance in the phase of reticular lamina (RL) radial motion at a frequency approximately ½ octave below the BM characteristic frequency, and that this is the main source of the phase difference between the TM and RL radial motions that produces cochlear amplification. It appears that the change in phase of RL radial motion comes about because of a transition between different organ-of-Corti (OoC) vibrational modes that changes RL motion relative to BM and TM motion. The origins and consequences of the large phase change of RL radial motion relative to BM motion are considered; differences in the reported patterns of these changes may be due to different viewing angles. Detailed motion data and new models are needed to better specify the vibrational patterns of the OoC modes and the role of the various OoC structures in producing the modes and the mode transition.

Nonlinear cochlear mechanics without direct vibration-amplification feedback

Recent in vivo recordings from the mammalian cochlea indicate that although the motion of the basilar membrane appears actively amplified and nonlinear only at frequencies relatively close to the peak of the response, the internal motions of the organ of Corti display these same features over a much wider range of frequencies. These experimental findings are not easily explained by the textbook view of cochlear mechanics, in which cochlear amplification is controlled by the motion of the basilar membrane (BM) in a tight, closed-loop feedback configuration. This study shows that a simple phenomenological model of the cochlea inspired by the work of Zweig [J. Acoust. Soc. Am. 138, 1102 (2015)] can account for recent data in mouse and gerbil. In this model, the active forces are regulated indirectly, through the effect of BM motion on the pressure field across the cochlear partition, rather than via direct coupling between active-force generation and BM vibration. The absence of strong vibration-amplification feedback in the cochlea also provides a compelling explanation for the observed intensity invariance of fine time structure in the BM response to acoustic clicks.

Constraints imposed by zero-crossing invariance on cochlear models with two mechanical degrees of freedom

The zero crossings of basilar-membrane (BM) responses to clicks are nearly independent of stimulus intensity. This work explores the constraints that this invariance imposes on one-dimensional nonlinear cochlear models with two degrees of freedom (2DoF). The locations of the poles and zeros of the BM admittance, calculated for a set of linear models in which the strength of the active force is progressively decreased, provides a playground for evaluating the behavior of a corresponding nonlinear model at increasing stimulus levels. Mathematical constraints on the model parameters are derived by requiring that the poles of the admittance move horizontally in the s-plane as the active force is varied. These constraints ensure approximate zero-crossing invariance over a wide stimulus level range in a nonlinear model in which the active force varies as a function of the local instantaneous BM displacement and velocity. Two different 2DoF models are explored, each capable of reproducing the main qualitative characteristics of the BM response to tones (i.e., the tall and broad activity pattern at low stimulus levels, the large gain dynamics, and the partial decoupling between gain and phase). In each model, the motions of the two masses are compared with response data from animal experiments.

Cochlear mechanics with fluid viscosity and compressibility

We extend the one-dimensional cochlear model to include the effect of viscosity and compressibility of the cochlear fluid. The resulting boundary-value problem is solved exactly using numerical techniques and semianalytically using the WKB approximation. Our results indicate the general trend of basilar membrane response increasing with an increase in compressibility or a decrease in viscosity. However, in the physiologically relevant range of these parameters, the change in the response is insignificant, justifying the assumption made in one-dimensional cochlear models that these effects are unimportant. Using the semianalytical WKB algorithm, we also demonstrate the simultaneous existence of forward- and backward-traveling waves on the basilar membrane.

Nonlinearity and amplification in cochlear responses to single and multi-tone stimuli

Mechanical displacements of the basilar membrane (BM) and the electrophysiological responses of the auditory outer hair cells (OHCs) are key components of the frequency tuning and cochlear amplification in the mammalian cochlea. In the work presented here, we measured the responses of (1) the extracellular voltage generated by OHCs (V_OHC) and (2) displacements within the organ of Corti complex (OCC) to a multi-tone stimulus, and to single tones. Using optical coherence tomography (OCT), we were able to measure displacements of different layers in the OCC simultaneously, in the base of the gerbil cochlea. We explored the effect of the two types of sound stimuli to the nonlinear behavior of voltage and displacement in two frequency regions: a frequency region below the BM nonlinearity (sub-BF region: f < ∼0.7 BF), and in the best frequency (BF) region. In the sub-BF region, BM motion (X_BM) had linear growth for both stimulus types, and the motion in the OHC region (X_OHC) was mildly nonlinear for single tones, and relatively strongly nonlinear for multi-tones. Sub-BF, the nonlinear character of V_OHC was similar to that of X_OHC. In the BF region X_BM, V_OHC and X_OHC all possessed the now-classic nonlinearity of the BF peak. Coupling these observations with previous findings on phasing between OHC force and traveling wave motions, we propose the following framework for cochlear nonlinearity: The BF-region nonlinearity is an amplifying nonlinearity, in which OHC forces input power into the traveling wave, allowing it to travel further apical to the region where it peaks. The sub-BF nonlinearity is a non-amplifying nonlinearity; it represents OHC electromotility, and saturates due to OHC current saturation, but the OHC forces do not possess the proper phasing to feed power into the traveling wave.

Power Dissipation in the Cochlea Can Enhance Frequency Selectivity

The cochlear cavity is filled with viscous fluids, and it is partitioned by a viscoelastic structure called the organ of Corti complex. Acoustic energy propagates toward the apex of the cochlea through vibrations of the organ of Corti complex. The dimensions of the vibrating structures range from a few hundred (e.g., the basilar membrane) to a few micrometers (e.g., the stereocilia bundle). Vibrations of microstructures in viscous fluid are subjected to energy dissipation. Because the viscous dissipation is considered to be detrimental to the function of hearing-sound amplification and frequency tuning-the cochlea uses cellular actuators to overcome the dissipation. Compared to extensive investigations on the cellular actuators, the dissipating mechanisms have not been given appropriate attention, and there is little consensus on damping models. For example, many theoretical studies use an inviscid fluid approximation and lump the viscous effect to viscous damping components. Others neglect viscous dissipation in the organ of Corti but consider fluid viscosity. We have developed a computational model of the cochlea that incorporates viscous fluid dynamics, organ of Corti microstructural mechanics, and electrophysiology of the outer hair cells. The model is validated by comparing with existing measurements, such as the viscoelastic response of the tectorial membrane, and the cochlear input impedance. Using the model, we investigated how dissipation components in the cochlea affect its function. We found that the majority of acoustic energy dissipation of the cochlea occurs within the organ of Corti complex, not in the scalar fluids. Our model suggests that an appropriate dissipation can enhance the tuning quality by reducing the spread of energy provided by the outer hair cells' somatic motility.

Forward and Reverse Waves: Modeling Distortion Products in the Intracochlear Fluid Pressure

Distortion product otoacoustic emissions are sounds that are emitted by the cochlea due to the nonlinearity of the outer hair cells. These emissions play an important role both in clinical settings and research laboratories. However, how distortion products propagate from their generation location to the middle ear remains unclear; whether distortion products propagate as a slow reverse traveling wave, or as a fast compression wave, through the cochlear fluid has been debated. In this article, we evaluate the contributions of the slow reverse wave and fast compression wave to the propagation of intracochlear distortion products using a physiologically based nonlinear model of the gerbil cochlea. This model includes a 3D two-duct model of the intracochlear fluid and a realistic model of outer hair cell biophysics. Simulations of the distortion products in the cochlear fluid pressure in response to a two-tone stimulus are compared with published in vivo experimental results. Whereas experiments have characterized distortion products at a limited number of locations, this model provides a complete description of the fluid pressure at all locations in the cochlear ducts. As in experiments, the spatial variations of the distortion products in the fluid pressure have some similarities with what is observed in response to a pure tone. Analysis of the fluid pressure demonstrates that although a fast wave component is generated, the slow wave component dominates the response. Decomposition of the model simulations into forward and reverse wave components shows that a slow forward propagating wave is generated due to the reflection of the slow reverse wave at the stapes. Wave interference between the reverse and forward components sometimes complicates the analysis of distortion products propagation using measurements at a few locations.

An elemental approach to modelling the mechanics of the cochlea

The motion along the basilar membrane in the cochlea is due to the interaction between the micromechanical behaviour of the organ of Corti and the fluid movement in the scalae. By dividing the length of the cochlea into a finite number of elements and assuming a given radial distribution of the basilar membrane motion for each element, a set of equations can be separately derived for the micromechanics and for the fluid coupling. These equations can then be combined, using matrix methods, to give the fully coupled response. This elemental approach reduces to the classical transmission line model if the micromechanics are assumed to be locally-reacting and the fluid coupling is assumed to be entirely one-dimensional, but is also valid without these assumptions. The elemental model is most easily formulated in the frequency domain, assuming quasi-linear behaviour, but a time domain formulation, using state space method, can readily incorporate local nonlinearities in the micromechanics. Examples of programs are included for the elemental model of a human cochlea that can be readily modified for other species.

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General formulation of an elemental model for cochlear mechanics.

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Reduce to the transmission line model for locally-reacting micromechanical and 1D fluid coupling.

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Incorporation of non-uniform areas, 3D fluid coupling and non locally-reacting micromechanics.

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MATLAB programs for the elemental model in the frequency domain and time domain.

Two-Tone Suppression of Simultaneous Electrical and Mechanical Responses in the Cochlea

Cochlear frequency tuning is based on a mildly tuned traveling-wave response that is enhanced in amplitude and sharpness by outer hair cell (OHC)-based forces. The nonlinear and active character of this enhancement is the fundamental manifestation of cochlear amplification. Recently, mechanical (pressure) and electrical (extracellular OHC-generated voltage) responses were simultaneously measured close to the sensory tissue's basilar membrane. Both pressure and voltage were tuned and showed traveling-wave phase accumulation, evidence that they were locally generated responses. Approximately at the frequency where nonlinearity commenced, the phase of extracellular voltage shifted up, to lead pressure by >1/4 cycle. Based on established and fundamental relationships among voltage, force, pressure, displacement, and power, the observed phase shift was identified as the activation of cochlear amplification. In this study, the operation of the cochlear amplifier was further explored, via changes in pressure and voltage responses upon delivery of a second, suppressor tone. Two different suppression paradigms were used, one with a low-frequency suppressor and a swept-frequency probe, the other with two swept-frequency tones, either of which can be considered as probe or suppressor. In the presence of a high-level low-frequency suppressor, extracellular voltage responses at probe-tone frequencies were greatly reduced, and the pressure responses were reduced nearly to their linear, passive level. On the other hand, the amplifier-activating phase shift between pressure and voltage responses was still present in probe-tone responses. These findings are consistent with low-frequency suppression being caused by the saturation of OHC electrical responses and not by a change in the power-enabling phasing of the underlying mechanics. In the two-tone swept-frequency suppression paradigm, mild suppression was apparent in the pressure responses, while deep notches could develop in the voltage responses. A simple analysis, based on a two-wave differencing scheme, was used to explore the observations.

Intracochlear Scala Media Pressure Measurement: Implications for Models of Cochlear Mechanics

Models of the active cochlea build upon the underlying passive mechanics. Passive cochlear mechanics is based on physical and geometrical properties of the cochlea and the fluid-tissue interaction between the cochlear partition and the surrounding fluid. Although the fluid-tissue interaction between the basilar membrane and the fluid in scala tympani (ST) has been explored in both active and passive cochleae, there was no experimental data on the fluid-tissue interaction on the scala media (SM) side of the partition. To this aim, we measured sound-evoked intracochlear pressure in SM close to the partition using micropressure sensors. All the SM pressure data are from passive cochleae, likely because the SM cochleostomy led to loss of endocochlear potential. Thus, these experiments are studies of passive cochlear mechanics. SM pressure close to the tissue showed a pattern of peaks and notches, which could be explained as an interaction between fast and slow (i.e., traveling wave) pressure modes. In several animals SM and ST pressure were measured in the same cochlea. Similar to previous studies, ST-pressure was dominated by a slow, traveling wave mode at stimulus frequencies in the vicinity of the best frequency of the measurement location, and by a fast mode above best frequency. Antisymmetric pressure between SM and ST supported the classic single-partition cochlear models, or a dual-partition model with tight coupling between partitions. From the SM and ST pressure we calculated slow and fast modes, and from active ST pressure we extrapolated the passive findings to the active case. The passive slow mode estimated from SM and ST data was low-pass in nature, as predicted by cochlear models.

Longitudinal spread of mechanical excitation through tectorial membrane traveling waves

The mammalian inner ear separates sounds by their frequency content, and this separation underlies important properties of human hearing, including our ability to understand speech in noisy environments. Studies of genetic disorders of hearing have demonstrated a link between frequency selectivity and wave properties of the tectorial membrane (TM). To understand these wave properties better, we developed chemical manipulations that systematically and reversibly alter TM stiffness and viscosity. Using microfabricated shear probes, we show that (i) reducing pH reduces TM stiffness with little change in TM viscosity and (ii) adding PEG increases TM viscosity with little change in TM stiffness. By applying these manipulations in measurements of TM waves, we show that TM wave speed is determined primarily by stiffness at low frequencies and by viscosity at high frequencies. Both TM viscosity and stiffness affect the longitudinal spread of mechanical excitation through the TM over a broad range of frequencies. Increasing TM viscosity or decreasing stiffness reduces longitudinal spread of mechanical excitation, thereby coupling a smaller range of best frequencies and sharpening tuning. In contrast, increasing viscous loss or decreasing stiffness would tend to broaden tuning in resonance-based TM models. Thus, TM wave and resonance mechanisms are fundamentally different in the way they control frequency selectivity.

Consequences of Location-Dependent Organ of Corti Micro-Mechanics

The cochlea performs frequency analysis and amplification of sounds. The graded stiffness of the basilar membrane along the cochlear length underlies the frequency-location relationship of the mammalian cochlea. The somatic motility of outer hair cell is central for cochlear amplification. Despite two to three orders of magnitude change in the basilar membrane stiffness, the force capacity of the outer hair cell’s somatic motility, is nearly invariant over the cochlear length. It is puzzling how actuators with a constant force capacity can operate under such a wide stiffness range. We hypothesize that the organ of Corti sets the mechanical conditions so that the outer hair cell’s somatic motility effectively interacts with the media of traveling waves—the basilar membrane and the tectorial membrane. To test this hypothesis, a computational model of the gerbil cochlea was developed that incorporates organ of Corti structural mechanics, cochlear fluid dynamics, and hair cell electro-physiology. The model simulations showed that the micro-mechanical responses of the organ of Corti are different along the cochlear length. For example, the top surface of the organ of Corti vibrated more than the bottom surface at the basal (high frequency) location, but the amplitude ratio was reversed at the apical (low frequency) location. Unlike the basilar membrane stiffness varying by a factor of 1700 along the cochlear length, the stiffness of the organ of Corti complex felt by the outer hair cell remained between 1.5 and 0.4 times the outer hair cell stiffness. The Y-shaped structure in the organ of Corti formed by outer hair cell, Deiters cell and its phalange was the primary determinant of the elastic reactance imposed on the outer hair cells. The stiffness and geometry of the Deiters cell and its phalange affected cochlear amplification differently depending on the location.

Two-compartment passive frequency domain cochlea model allowing independent fluid coupling to the tectorial and basilar membranes

The cochlea is a spiral-shaped, liquid-filled organ in the inner ear that converts sound with high frequency selectivity over a wide pressure range to neurological signals that are eventually interpreted by the brain. The cochlear partition, consisting of the organ of Corti supported below by the basilar membrane and attached above to the tectorial membrane, plays a major role in the frequency analysis. In early fluid-structure interaction models of the cochlea, the mechanics of the cochlear partition were approximated by a series of single-degree-of-freedom systems representing the distributed stiffness and mass of the basilar membrane. Recent experiments suggest that the mechanical properties of the tectorial membrane may also be important for the cochlea frequency response and that separate waves may propagate along the basilar and tectorial membranes. Therefore, a two-dimensional two-compartment finite difference model of the cochlea was developed to investigate the independent coupling of the basilar and tectorial membranes to the surrounding liquid. Responses are presented for models using two- or three-degree-of-freedom stiffness, damping, and mass parameters derived from a physiologically based finite element model of the cochlear partition. Effects of changes in membrane and organ of Corti stiffnesses on the individual membrane responses are investigated.

Porosity controls spread of excitation in tectorial membrane traveling waves

Cochlear frequency selectivity plays a key role in our ability to understand speech, and is widely believed to be associated with cochlear amplification. However, genetic studies targeting the tectorial membrane (TM) have demonstrated both sharper and broader tuning with no obvious changes in hair bundle or somatic motility mechanisms. For example, cochlear tuning of Tectb(-/-) mice is significantly sharper than that of Tecta(Y1870C/+) mice, even though TM stiffnesses are similarly reduced relative to wild-type TMs. Here we show that differences in TM viscosity can account for these differences in tuning. In the basal cochlear turn, nanoscale pores of Tecta(Y1870C/+) TMs are significantly larger than those of Tectb(-/-) TMs. The larger pore size reduces shear viscosity (by ∼70%), thereby reducing traveling wave speed and increasing spread of excitation. These results demonstrate the previously unrecognized importance of TM porosity in cochlear and neural tuning.

Basilar membrane and tectorial membrane stiffness in the CBA/CaJ mouse

The mouse has become an important animal model in understanding cochlear function. Structures, such as the tectorial membrane or hair cells, have been changed by gene manipulation, and the resulting effect on cochlear function has been studied. To contrast those findings, physical properties of the basilar membrane (BM) and tectorial membrane (TM) in mice without gene mutation are of great importance. Using the hemicochlea of CBA/CaJ mice, we have demonstrated that tectorial membrane (TM) and basilar membrane (BM) revealed a stiffness gradient along the cochlea. While a simple spring mass resonator predicts the change in the characteristic frequency of the BM, the spring mass model does not predict the frequency change along the TM. Plateau stiffness values of the TM were 0.6 ± 0.5, 0.2 ± 0.1, and 0.09 ± 0.09 N/m for the basal, middle, and upper turns, respectively. The BM plateau stiffness values were 3.7 ± 2.2, 1.2 ± 1.2, and 0.5 ± 0.5 N/m for the basal, middle, and upper turns, respectively. Estimations of the TM Young's modulus (in kPa) revealed 24.3 ± 25.2 for the basal turns, 5.1 ± 4.5 for the middle turns, and 1.9 ± 1.6 for the apical turns. Young's modulus determined at the BM pectinate zone was 76.8 ± 72, 23.9 ± 30.6, and 9.4 ± 6.2 kPa for the basal, middle, and apical turns, respectively. The reported stiffness values of the CBA/CaJ mouse TM and BM provide basic data for the physical properties of its organ of Corti.

Frequency selectivity without resonance in a fluid waveguide

This study analyzes a waveguide consisting of two parallel fluid-filled chambers connected by a narrow slit that is spanned by two coupled elastic beams. A stiffness gradient exists in the longitudinal direction. This simple linear system, which contains no lumped mass, is shown to act as a spectral analyzer. Fluid waves traveling in the waveguide exhibit a distinct amplitude peak at a longitudinal location that varies systematically with frequency. The peaking is not based on resonance, but entirely on wave dispersion. When entering its peak region, the wave undergoes a sharp deceleration associated with a transition in which two propagation modes exchange roles. It is proposed that this mode shape swapping underlies the frequency analysis of the mammalian cochlea.

Modelling cochlear mechanics

The cochlea plays a crucial role in mammal hearing. The basic function of the cochlea is to map sounds of different frequencies onto corresponding characteristic positions on the basilar membrane (BM). Sounds enter the fluid-filled cochlea and cause deflection of the BM due to pressure differences between the cochlear fluid chambers. These deflections travel along the cochlea, increasing in amplitude, until a frequency-dependent characteristic position and then decay away rapidly. The hair cells can detect these deflections and encode them as neural signals. Modelling the mechanics of the cochlea is of help in interpreting experimental observations and also can provide predictions of the results of experiments that cannot currently be performed due to technical limitations. This paper focuses on reviewing the numerical modelling of the mechanical and electrical processes in the cochlea, which include fluid coupling, micromechanics, the cochlear amplifier, nonlinearity, and electrical coupling.

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.

An unusually powerful mode of low-frequency sound interference due to defective hair bundles of the auditory outer hair cells

A detrimental perceptive consequence of damaged auditory sensory hair cells consists in a pronounced masking effect exerted by low-frequency sounds, thought to occur when auditory threshold elevation substantially exceeds 40 dB. Here, we identified the submembrane scaffold protein Nherf1 as a hair-bundle component of the differentiating outer hair cells (OHCs). Nherf1(-/-) mice displayed OHC hair-bundle shape anomalies in the mid and basal cochlea, normally tuned to mid- and high-frequency tones, and mild (22-35 dB) hearing-threshold elevations restricted to midhigh sound frequencies. This mild decrease in hearing sensitivity was, however, discordant with almost nonresponding OHCs at the cochlear base as assessed by distortion-product otoacoustic emissions and cochlear microphonic potentials. Moreover, unlike wild-type mice, responses of Nherf1(-/-) mice to high-frequency (20-40 kHz) test tones were not masked by tones of neighboring frequencies. Instead, efficient maskers were characterized by their frequencies up to two octaves below the probe-tone frequency, unusually low intensities up to 25 dB below probe-tone level, and growth-of-masker slope (2.2 dB/dB) reflecting their compressive amplification. Together, these properties do not fit the current acknowledged features of a hypersensitivity of the basal cochlea to lower frequencies, but rather suggest a previously unidentified mechanism. Low-frequency maskers, we propose, may interact within the unaffected cochlear apical region with midhigh frequency sounds propagated there via a mode possibly using the persistent contact of misshaped OHC hair bundles with the tectorial membrane. Our findings thus reveal a source of misleading interpretations of hearing thresholds and of hypervulnerability to low-frequency sound interference.

Phase of shear vibrations within cochlear partition leads to activation of the cochlear amplifier

Since Georg von Bekesy laid out the place theory of the hearing, researchers have been working to understand the remarkable properties of mammalian hearing. Because access to the cochlea is restricted in live animals, and important aspects of hearing are destroyed in dead ones, models play a key role in interpreting local measurements. Wentzel-Kramers-Brillouin (WKB) models are attractive because they are analytically tractable, appropriate to the oblong geometry of the cochlea, and can predict wave behavior over a large span of the cochlea. Interest in the role the tectorial membrane (TM) plays in cochlear tuning led us to develop models that directly interface the TM with the cochlear fluid. In this work we add an angled shear between the TM and reticular lamina (RL), which serves as an input to a nonlinear active force. This feature plus a novel combination of previous work gives us a model with TM-fluid interaction, TM-RL shear, a nonlinear active force and a second wave mode. The behavior we get leads to the conclusion the phase between the shear and basilar membrane (BM) vibration is critical for amplification. We show there is a transition in this phase that occurs at a frequency below the cutoff, which is strongly influenced by TM stiffness. We describe this mechanism of sharpened BM velocity profile, which demonstrates the importance of the TM in overall cochlear tuning and offers an explanation for the response characteristics of the Tectb mutant mouse.

Stimulated acoustic emissions from coupled strings

We consider traveling transverse waves on two identical uniform taut strings that are elastically coupled through springs that gradually decrease their stiffness over a region of finite length. The wave system can be decomposed into two modes: an in-phase mode ([Formula: see text]) that is transparent to the coupling springs, and an out-of-phase mode ([Formula: see text]) that engages the coupling springs and can resonate at a particular location depending on the excitation frequency. The system exhibits linear mode conversion whereby an incoming ([Formula: see text]) wave is reflected back from the resonance location both as a propagating ([Formula: see text]) wave and an evanescent ([Formula: see text]) wave, while both types emerge as propagating forward through the resonance location. We match a local transition layer expansion to the WKB expansion to obtain estimates of the reflection and transmission coefficients. The reflected waves may be an analog for stimulated emissions from the ear.

Frequency-dependent properties of the tectorial membrane facilitate energy transmission and amplification in the cochlea

The remarkable sensitivity, frequency selectivity, and dynamic range of the mammalian cochlea relies on longitudinal transmission of minuscule amounts of energy as passive, pressure-driven, basilar membrane (BM) traveling waves. These waves are actively amplified at frequency-specific locations by a mechanism that involves interaction between the BM and another extracellular matrix, the tectorial membrane (TM). From mechanical measurements of isolated segments of the TM, we made the important new (to our knowledge) discovery that the stiffness of the TM is reduced when it is mechanically stimulated at physiologically relevant magnitudes and at frequencies below their frequency place in the cochlea. The reduction in stiffness functionally uncouples the TM from the organ of Corti, thereby minimizing energy losses during passive traveling-wave propagation. Stiffening and decreased viscosity of the TM at high stimulus frequencies can potentially facilitate active amplification, especially in the high-frequency, basal turn, where energy loss due to internal friction within the TM is less than in the apex. This prediction is confirmed by neural recordings from several frequency regions of the cochlea.

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.

Instrumentation for studies of cochlear mechanics: from von Békésy forward

Georg von Békésy designed the instruments needed for his research. He also created physical models of the cochlea allowing him to manipulate the parameters (such as volume elasticity) that could be involved in controlling traveling waves. This review is about the specific devices that he used to study the motion of the basilar membrane thus allowing the analysis that lead to his Nobel Prize Award. The review moves forward in time mentioning the subsequent use of von Békésy's methods and later technologies important for motion studies of the organ of Corti. Some of the seminal findings and the controversies of cochlear mechanics are mentioned in relation to the technical developments.