Prediction of the characteristics of two types of pressure waves in the cochlea: theoretical considerations

The aim of this study was to predict the characteristics of two types of cochlear pressure waves, so-called fast and slow waves. A two-dimensional finite-element model of the organ of Corti (OC), including fluid-structure interaction with the surrounding lymph fluid, was constructed. The geometry of the OC at the basal turn was determined from morphological measurements of others in the gerbil hemicochlea. As far as mechanical properties of the materials within the OC are concerned, previously determined mechanical properties of portions within the OC were adopted, and unknown mechanical features were determined from the published measurements of static stiffness. Time advance of the fluid-structure scheme was achieved by a staggered approach. Using the model, the magnitude and phase of the fast and slow waves were predicted so as to fit the numerically obtained pressure distribution in the scala tympani with what is known about intracochlear pressure measurement. When the predicted pressure waves were applied to the model, the numerical result of the velocity of the basilar membrane showed good agreement with the experimentally obtained velocity of the basilar membrane documented by others. Thus, the predicted pressure waves appeared to be reliable. Moreover, it was found that the fluid-structure interaction considerably influences the dynamic behavior of the OC at frequencies near the characteristic frequency.     

Two-tone distortion in intracochlear pressure

Two-tone distortion was measured in the intracochlear pressure in the base of the gerbil cochlea, close to the sensory tissue, where the local motions and forces of the organ of Corti can be detected. The measurements probe both the underlying nonlinear process that generates two-tone distortion, and the filtering and spreading of the distortion products. Some of our findings are as follows: (1) The observations were consistent with previous observations of two-tone distortion in BM motion [J. Neurophysiol. 77, 2385-2399 (1997); J. Neurophysiol. 78, 261-270 (1997)]. (2) Frequency sweeps show distortion product tuning and phase-versus-frequency behavior that is similar, but not identical, to single tone tuning. (3) The decay of distortion products with distance from the basilar membrane confirms the feasibility that they could drive the stapes by a direct fluid route, as proposed by Ren [Nat. Neurosci. 7, 333-334 (2004)]. (4) The phases of the distortion products within a single family (the group of distortion products generated by a single primary pair) in some cases alternated between 0 degrees and 180 degrees when referenced to the phases of the primaries. This behavior is predicted by a simple compressive nonlinearity.     

Intracochlear pressure measurements related to cochlear tuning

Pressure in turn one of the scala tympani (s.t.) was measured close to the basilar membrane (b.m.) and at additional positions as the pressure sensor approached and/or withdrew from the b.m. The s.t. pressure measured within about 100 microm of the b.m. varied rapidly in space at frequencies around the region's best frequency. Very close to the b.m. the s.t. pressure was tuned and scaled nonlinearly with sound level. The scala vestibuli (s.v.) pressure was measured at one position close to the stapes within seconds of the s.t. pressure and served primarily as a reference pressure. The driving pressure across the organ of Corti and the b.m. velocity were derived from the pressure data. Both were tuned and nonlinear. Therefore, their ratio, the specific acoustic impedance of the organ of Corti complex, was relatively untuned, and only subtly nonlinear. The impedance was inspected specifically for negative resistance (amplification) and resonance. Both were detected in some instances; taken as a whole, the current results constrain the possibilities for these qualities.     

Effect of coiling in a cochlear model

Transformation of the three-dimensional equations of fluid motion into cylindrical coordinates allowed analysis of a coiled cochlear model by the WKB technique. The model includes a single transverse mode of basilar membrane deflection and inviscid fluid. The results calculated using realistic parameters for the guinea pig show no significant difference in the basilar membrane amplitude and phase between the straight and coiled models. Some differences exist in the fluid pressure found in the scala. The conclusion is that the macromechanical response is not significantly affected by coiling.     

Vibration responses of the organ of Corti and the tectorial membrane to electrical stimulation

Coupling of somatic electromechanical force from the outer hair cells (OHCs) into the organ of Corti is investigated by measuring transverse vibration patterns of the organ of Cori and tectorial membrane (TM) in response to intracochlear electrical stimulation. Measurement places at the organ of Corti extend from the inner sulcus cells to Hensen's cells and at the lower (and upper) surface of the TM from the inner sulcus to the OHC region. These locations are in the neighborhood of where electromechanical force is coupled into (1) the mechanoelectrical transducers of the stereocilia and (2) fluids of the organ of Corti. Experiments are conducted in the first, second, and third cochlear turns of an in vitro preparation of the adult guinea pig cochlea. Vibration measurements are made at functionally relevant stimulus frequencies (0.48-68 kHz) and response amplitudes (<15 nm). The experiments provide phase relations between the different structures, which, dependent on frequency range and longitudinal cochlear position, include in-phase transverse motions of the TM, counterphasic transverse motions between the inner hair cell and OHCs, as well as traveling-wave motion of Hensen's cells in the radial direction. Mechanics of sound processing in the cochlea are discussed based on these phase relationships.     

Finding the impedance of the organ of Corti

Measurements of the nonlinear response of the basilar membrane to a pure tone are shown to have a simple form for moderate membrane velocities: V(x,f;Vu)/Vu approximately [V(x,f)/Vu]v(x,f), f less than or equal to fc(x), where the response V is the velocity of the membrane at measurement position x, Vu is the umbo velocity, f is the frequency of the stimulus, and fc(x) is the local characteristic frequency. The frequency dependence of the functions v(x,f) and V(x,f) is determined from the data, and v(x,f) and ln V(x,f) are shown to be analytic functions in the lower half of the complex frequency plane, with Re [v(x,f)] a monotonically increasing function of f at fixed x. The linear limit of basilar membrane motion is characterized by a transfer function T(x,f) = (V/V1)v/(1-v), estimated by extrapolating V(x,f;Vu)/Vu to a small membrane velocity V1.T(x,f) and ln T(x,f) are shown to be analytic functions in the lower half of the complex frequency plane. The inverse of the amplitude of the transfer function, which has both a deep dip at f approximately fc(x) and a broad shoulder at lower frequencies, bears a striking resemblance to the neural threshold tuning curve. The functional form of T(x,f) is used to deduce the equation governing the motion of a section of the organ of Corti. Each section acts like a negatively damped harmonic oscillator stabilized at time t by a feedback force proportional to the velocity at the previous time t-tau. The time delay tau is proportional to the oscillator period [tau approximately 1.75/fc(x)]. Like a laser, the organ of Corti pumps energy into harmonic traveling waves. Unlike the laser, the direction of energy flow abruptly reverses as the traveling wave approaches the point of maximum membrane velocity [fc(x) approximately f]. All accumulated wave energy is then pumped back into a small section of the organ of Corti where transduction presumably occurs. Outer hair cells are conjectured to be active elements contributing to the negative damping and feedback of the cochlear amplifier.     

Supporting evidence for reverse cochlear traveling waves

As a result of the cochlea's nonlinear mechanics, stimulation by two tones results in the generation of distortion products (DPs) at frequencies flanking the primary tones. DPs are measurable in the ear canal as oto-acoustic emissions, and are used to noninvasively explore cochlear mechanics and diagnose hearing loss. Theories of DP emissions generally include both forward and reverse cochlear traveling waves. However, a recent experiment failed to detect the reverse-traveling wave and concluded that the dominant emission path was directly through the fluid as a compression pressure [Ren, 2004, Nat. Neurosc.7, 333-334]. To explore this further, we measured intracochlear DPs simultaneously with emissions over a wide frequency range, both close to and remote from the basilar membrane. Our results support the existence of the reverse-traveling wave: (1) They show spatial variation in DPs that is at odds with a compression pressure. (2) Although they confirm a forward-traveling character of intraocochlear DPs in a broad frequency region of the best frequency, this behavior does not refute the existence of reverse-traveling waves. (3) Finally, the results show that, in cases in which it can be expected, the DP emission is delayed relative to the DP in a way that supports reverse-traveling-wave theory.     

Location of structurally similar areas in chinchilla cochleas of different lengths

In chinchillas, the length of the organ of Corti was found to vary by as much as 26% of the mean (18.31 mm). In specimens of different lengths, areas in which the basilar membrane and organ of Corti have the same physical dimensions were found to be located at the same percentage distance along the cochlear partition. Average densities of the inner and outer hair cells were noted to decrease with increasing cochlear length. However, the rate of decrease in density was less than the rate of increase in length. These findings indicate that long cochleas have a greater amount of the organ of Corti (and hence more sensory cells) in a given frequency region.     

Three-dimensional numerical modeling for global cochlear dynamics

A hybrid analytical-numerical model using Galerkin approximation to variational equations has been developed for predicting global cochlear responses. The formulation provides a flexible framework capable of incorporating morphologically based mechanical models of the cochlear partition and realistic geometry. The framework is applied for a simplified model with an emphasis on application of hybrid methods for three-dimensional modeling. The resulting formulation is modular, where matrices representing fluid and cochlear partition are constructed independently. Computational cost is reduced using two methods, a modal-finite-element method and a boundary element-finite-element method. The first uses a cross-mode expansion of fluid pressure (2.5D model) and the second uses a waveguide Green's-function-based boundary element method (BEM). A novel wave number approach to the boundary element formulation for interior problem results in efficient computation of the finite-element matrix. For the two methods a convergence study is undertaken using a simplified passive structural model of cochlear partition. It is shown that basilar membrane velocity close to best place is influenced by fluid and structural discretization. Cochlear duct pressure fields are also shown demonstrating the 3D nature of pressure near best place.     

Compliance profiles derived from a three-dimensional finite-element model of the basilar membrane

A finite-element analysis is used to explore the impact of elastic material properties, boundary conditions, and geometry, including coiling, on the spatial characteristics of the compliance of the unloaded basilar membrane (BM). It is assumed that the arcuate zone is isotropic and the pectinate zone orthotropic, and that the radial component of the effective Young's modulus in the pectinate zone decreases exponentially with distance from base to apex. The results concur with tonotopic characteristics of compliance and neural data. Moreover, whereas the maximum compliance in a radial profile is located close to the boundary between the two zones in the basal region, it shifts to the midpoint of the pectinate zone for the apical BM; the width of the profile also expands. This shift begins near the 1 kHz characteristic place for guinea pig and the 2.4 kHz place for gerbil. Shift and expansion are not observed for linear rather than exponential decrease of the radial component of Young's modulus. This spatial change of the compliance profile leads to the prediction that mechanical excitation in the apical region of the organ of Corti is different to that in the basal region.     

Harmonic distortion in intracochlear pressure and its analysis to explore the cochlear amplifier

Intracochlear pressure was measured close to the basal basilar membrane in gerbil with pure-tone stimulation. This report describes harmonic distortion in the pressure. The harmonic components were tuned in frequency and physiologically vulnerable, implying that they were related to the cell-derived force that sharpens tuning at low levels in healthy cochleae. For stimulus frequencies in the vicinity of the best frequency the harmonic distortion appeared to be produced locally, at the place of measurement. Therefore, it could be explored with a local nonlinear model. The combined model and observations demonstrate two specific points: First, the harmonics in the cell-based force were likely similar in size to the harmonics in pressure (multiplied by area) close to the basilar membrane. This is distinctly different than the situation for the fundamental component, where the cell-based force is apparently much smaller than the pressure (times area). Second, although the fundamental component of the measured pressure was much larger than its harmonic components, the harmonic and fundamental components of the active force were likely much more similar in size. This allows the harmonic components in the pressure to be used as an indirect measure of the active force.     

Wave propagation patterns in a "classical" three-dimensional model of the cochlea

The generation mechanisms of cochlear waves, in particular those that give rise to otoacoustic emissions (OAEs), are often complex. This makes it difficult to analyze wave propagation. In this paper two unusual excitation methods are applied to a three-dimensional stylized classical nonlinear model of the cochlea. The model used is constructed on the basis of data from an experimental animal selected to yield a smooth basilar-membrane impedance function. Waves going in two directions can be elicited by exciting the model locally instead of via the stapes. Production of DPOAEs was simulated by presenting the model with two relatively strong primary tones, with frequencies f1 and f2, estimating the driving pressure for the distortion product (DP) with frequency 2f1 - f2, and computing the resulting DP response pattern - as a function of distance along the basilar membrane. For wide as well as narrow frequency separations the resulting DP wave pattern in the model invariably showed that a reverse wave is dominant in nearly the entire region from the peak of the f2-tone to the stapes. The computed DP wave pattern was further analyzed as to its constituent components with the aim to isolate their properties.     

Mechanical passive and active models of the human basilar membrane

Simple three-dimensional passive and active models of the human basilar membrane were built, solved using the Finite Element Method and tested. In the active model an active mechanism connected with electromotility of outer hair cells was included. In the active model the active mechanism was incorporated in the form of additional, local pressure load. In the passive model the active mechanism was neglected. Hydrodynamic coupling between the cochlear partition and cochlear fluid was excluded in both models. Geometrical and physical parameters of the model were chosen to be adequate to those of humans in the best possible way. However, some of these parameters had to be estimated. The models were tested by calculation of typical curves known from cochlear measurements performed mostly on animals. For the passive model a linear input-output function and very small values of the basilar membrane velocities were obtained. This behaviour is to be expected for the passive model and for the basilar membrane in the poor physiological condition. For the active model the compressed input-output functions, tuning curves, isointensity curves and reasonable BM velocities were obtained. Possible inadequacies, which could explain the differences between numerical results and measurements were described.     

A symmetry suppresses the cochlear catastrophe

When the independent spatial variable is defined appropriately, the empirical finding that the phase of the cochlear input impedance is small [Lynch et al., J. Acoust. Soc. Am. 72, 108-130 (1982)] is shown to imply that the wavelength of the pressure wave in the cochlea changes slowly with position near the stapes. As a result, waves traveling in either direction through the basal turn undergo little reflection, and the transfer of energy between the middle and inner ears remains efficient at low frequencies. The slow variation of the wavelength implies that the series impedance Z and shunt admittance Y of the cochlear transmission line are approximately proportional at low frequencies and thus requires that the width of the basilar membrane and the cross-sectional areas of the cochlear scalae taper in opposite directions. Maintenance of the symmetry between Z and Y is both necessary and sufficient to ensure that the spatial derivative of the wavelength, and hence the phase of the cochlear input impedance, remains small. Although introduced in another context, the model of Zweig ["Finding the impedance of the organ of Corti," J. Acoust. Soc. Am. 89, 1229-1254 (1991)] manifests the symmetry between Z and Y. In other transmission-line models of cochlear mechanics, however, that symmetry is absent, and the spatial derivative of the wavelength diverges at low frequencies--the "cochlear catastrophe." Those models therefore contradict the impedance measurements and predict little transfer of energy between the middle and inner ears.     

Cochlear mechanics: analysis for a pure tone

A three-dimensional hydroelastic model of the cochlea is analyzed, in which the fluid is viscous and the basilar membrane is an inhomogeneous orthotropic elastic plate. After the solution is obtained using a multiple-scale approximation, comparison is made with experiment for the human cochlea.     

Inverted direction of wave propagation (IDWP) in the cochlea

The "classical" view on wave propagation is that propagating waves are possible in both directions along the length of the basilar membrane and that they have identical properties. Results of several recently executed experiments [T. Ren, Nat. Neurosci. 2, 333-334 (2004) and W. X. He, A. L. Nuttall, and T. Ren, Hear. Res., 228, 112-122 (2007)] appear to contradict this view. In the current work measurements were made of the velocity of the guinea-pig basilar membrane (BM). Distortion products (DPs) were produced by presenting two primary tones, with frequencies below the characteristic frequency f(0) of the BM location at which the BM measurements were made, with a constant frequency ratio. In each experiment the phase of the principal DP, with frequency 2f(1)-f(2), was recorded as a function of the DP frequency. The results indicate that the DP wave going from the two-tone interaction region toward the stapes is not everywhere traveling in the reverse direction, but also in the forward direction. The extent of the region in which the forward wave occurs appears larger than is accounted for by classical theory. This property has been termed "inverted direction of wave propagation." The results of this study confirm the wave propagation findings of other authors. The experimental data are compared to theoretical predictions for a classical three-dimensional model of the cochlea that is based on noise-response data of the same animal. Possible physical mechanisms underlying the findings are discussed.     

Cochlear micromechanics--a mechanism for transforming mechanical to neural tuning within the cochlea.

A linear mathematical model is proposed which will account for the differences observed between mechanically measured data of Rhode (1971) for basilar membrane motion, and the responses of neural tuning curves (Kiang et al., 1974). We show that theoretical tuning curves may be derived from mechanical responses by forming the difference between the pressure across the basilar membrane and its displacement. Some ramifications of this proposal are discussed. We then propose a hypothetical physical model which could perform such a function.     

The mechanical waveform of the basilar membrane. II. From data to models--and back

Mechanical responses in the basal turn of the guinea-pig cochlea are measured with low-level broad-band noise as the acoustical stimulus [for details see de Boer and Nuttall, J. Acoust. Soc. Am. 101, 3583-3592 (1997)]. Results are interpreted within the framework of a classical three-dimensional model of the cochlea that belongs to a very wide class of nonlinear models. The use of linear-systems analysis for this class of nonlinear models has been justified earlier [de Boer, Audit. Neurosci. 3, 377-388 (1997)]. The data are subjected to inverse analysis with the aim to recover the "effective basilar-membrane impedance." This is a parameter function that, when inserted into the model, produces a model response, the "resynthesized" response, that is similar to the measured response. With present-day solution methods, resynthesis leads back to an almost perfect replica of the original response in the spatial domain. It is demonstrated in this paper that this also applies to the response in the frequency domain and in the time domain. This paper further reports details with regard to geometrical properties of the model employed. Two three-dimensional models are studied; one has its dimensions close to that of the real cochlea, the other is a stylized model which has homogeneous geometry over its length. In spite of the geometric differences the recovered impedance functions are very similar. An impedance function computed for one model can be used in resynthesis of the response in the other one, and this leads to global amplitude deviations between original and resynthesized response functions not exceeding 8 dB. Discrepancies are much larger (particularly in the phase) when a two-dimensional model is compared with a three-dimensional model. It is concluded that a stylized three-dimensional model with homogeneous geometric parameters will give sufficient information in further work on unraveling cochlear function via inverse analysis. In all cases of a sensitive cochlea stimulated by a signal with a stimulus level of 50 dB SPL per octave or less, the resulting basilar-membrane impedance is found to be locally active, that is, the impedance function shows a region where the basilar membrane is able to amplify acoustic power or to reduce dissipation of power by the organ of Corti. Finally, the influence of deliberate errors added to the data is discussed in order to judge the accuracy of the results.     

Finite element cochlear models and their steady state response

Numerical cochlear models are constructed by means of a finite element approach and their frequency and spatial responses are calculated. The cochlea is modelled as a coupled fluid-membrane system, for which both two- and three-dimensional models are considered. The fluid in the scala canals is assumed to be incompressible and the basilar membrane is assumed to be a locally reactive impedance wall or a lossy elastic membrane. With the three-dimensional models, the effects are examined of the spiral configuration of the cochlea, of the presence of the lamina and the ligament that narrows the coupling area between the two fluid canals (scala vestibuli and scala tympani), and of the extended reaction of the basilar membrane which cannot be included in case of the two-dimensional models. The conclusion is that these effects on the cochlear response and the inherent mechanism governing the cochlear behaviour are found to be rather secondary.