Frequency-Dependent Properties of the Tectorial Membrane Facilitate Energy Transmission and Amplification in the Cochlea

Abstract

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.

Introduction

Sound is decomposed into individual frequency components through reciprocal, electromechanical interaction between the highly active cellular and passive noncellular elements of the mammalian cochlea (1). Cochlear frequency tuning, with high frequencies producing maximal responses close to the base and low frequencies close to the apex, is determined primarily by base-to-apex gradients in the dimensions and collagen fiber packing density of the basilar membrane (BM) (2,3). Sensitivity and frequency selectivity of the cochlea are, however, sharpened by the cochlear amplifier (4,5) in the form of electromechanical active feedback provided by the outer hair cells (OHCs) of the organ of Corti (6,7). The hallmark of amplification (8) is a compressive, nonlinear increase in BM displacement as a function of sound-pressure level. Neural measurements in the presence of discrete, acoustically induced lesions to the OHCs (9) and experiments measuring BM displacement along the length of the cochlea (10) have shown that the region of amplification is limited to a few hundred microns around the characteristic frequency (CF) of the BM. Recent modeling analysis of experiments measuring the propagation of BM traveling waves suggests that the region of amplification is just basal to the CF (11), but this conclusion has yet to be considered with respect to the earlier experimental findings (9,10).

The OHCs are located strategically, sandwiched between two extracellular matrices, the BM and tectorial membrane (TM; shown isolated in Fig. 1 A), with their stereocilia attached to the underside of the TM (12) and providing its mechanical link to the organ of Corti (Fig. 1 B). Both the BM and TM have been shown to play crucial roles (1) in mediating the gain and timing of the cochlear amplifier (4,5,13,14). Before being amplified, however, mechanical energy is transmitted to its frequency-specific location within the cochlea in the form of BM traveling waves (2). This task can be accomplished even when movements of the ear drum are less than the diameter of a hydrogen atom and the traveling wave propagates through viscoelastic biological tissues bathed in viscous cochlear fluids. The role of the TM in energy transmission within the cochlea of wild-type mice and mice lacking specific TM proteins is not yet fully understood and has become a focus of interest in auditory physiology in recent years (1,15–28).

Figure 1.

The TM in vivo and in vitro. (A) Schematic showing the whole TM isolated from other cochlear structures. The TM is a coiled extracellular matrix running from the base to the apex of the cochlea. (B) In vivo, the TM is located above the organ of Corti, in contact with the OHC stereocilia and spiral limbus, and runs parallel to the BM along the length of the cochlea. The TM itself is radially divided into three distinct regions, the limbal, middle, and marginal zones. The limbal zone forms the attachment to the spiral limbus and the underside of the marginal zone attaches to the OHC stereocilia. IHC, inner hair cell; BM, basilar membrane, STS, subtectorial space. (C) Schematic of a top-down view of the inside of the experimental chamber, which contains a mounted segment of TM attached to both vibrating and stationary supports. Stimulation was delivered by the vibrating support, which was attached to a piezoelectric actuator, and the beam of the interferometer was stepped along the TM to track amplitude and phase of radially shearing, longitudinally propagating traveling waves at different frequencies.

Both the BM and TM are composed of radial collagen fibers, but the packing density of these fibers is lower in the TM (29). TM fiber organization is maintained by a highly structured striated sheet matrix (29) composed of three noncollagenous glycoproteins that are expressed only at high levels in the inner ear and account for ∼50% of the protein present in the TM (22). In this article, we describe the frequency-dependent material properties of the isolated mouse TM using mechanical responses to physiologically realistic frequencies and amplitudes. By taking into account the fluid environment (see Materials and Methods), we reveal what we believe to be new principles of cochlear operation associated with the TM, which have important implications for amplification, frequency tuning, and the efficient transmission of energy in the mammalian cochlea. Our in vitro measurements of the mouse TM are supported by in vivo recording of neural responses from the mouse cochlea. As far as we are aware, these findings provide the first indication that the frequency-dependent material properties of the TM contribute to energy transmission in the cochlea with minimal loss.

Materials and Methods

Preparation of TM samples

Data from both basal and apical locations were collected from CBA/Ca mice between 1 and 6 months of age. Mice were euthanized by CO₂ asphyxiation and dissections were performed under a light microscope in a Petri dish containing artificial endolymph (174 mmol KCl, 2.00 mmol NaCl, 0.0261 mmol CaCl₂, 3.00 mmol D-glucose, and 5.00 mmol HEPES, pH 7.3). The inner ear was removed from the skull and the cochlea was opened with forceps. The TM was detached from the spiral limbus using a tungsten probe with a tip diameter of <100 μm mounted on a syringe needle. Usually, the entire TM was removed in one piece and was cut with a scalpel blade into segments between 350 and 1000 μm long. Segments referred to as basal are from the basal third of the cochlea (∼35–60 kHz region (30)) and apical segments are from the apical third of the cochlea (∼6–11 kHz region (30)) (Fig. 1 A). The length of these segments varied depending on the region; the higher curvature in the apical regions generally limited the usable length to less than that of basal regions. Once detached and cut, a single TM segment was transferred into the pre-prepared experimental chamber by means of a glass-tipped pipette and mounted.

Traveling-wave excitation and measurements

Experiments were conducted in an experimental chamber located in a quiet room, on a vibration isolation table, and inside a Faraday cage. The experimental chamber was filled with artificial endolymph so that the prepared TM was submerged to a depth of at least 4 mm. The walls of the chamber were constructed from shaped silicone gel (732, Dow Corning, Midland, MI). A single segment of TM was suspended between and attached to a vibrating and a stationary support using Cell-Tak (BD Biosciences, San Jose, CA) (Fig. 1 C). The glass stationary support (dimensions ∼10 × 10 × 0.4 mm) was mounted on a mechanically isolated stand external to the chamber and lowered into place before the TM was added to the chamber. The vibrating support was constructed either from a small (dimensions ∼5 × 10 × 0.4 mm) piece of glass or platinum-irridium foil that was attached to a stimulation piezo (AE0203D04, Thorlabs, Newton, NJ). The stimulation piezo was mounted rigidly to the microscope slide forming the base of the chamber. A lab-built, self-mixing, homodyne laser-diode interferometer (31), the beam of which was aimed through a viewing window in the front wall of the chamber, was used to record the phase and amplitude of traveling waves at multiple points along the length of the marginal edge of the mounted segment of TM. The output signal of the interferometer was processed using a digital phase-locking algorithm, and instantaneous amplitude and phase of the wave were recorded. The viewing window was positioned so that the distance between it and the laser’s focal point on the TM (∼15 mm) was well outside the laser’s depth of focus (20 μm). The beam of the laser interferometer was focused onto the marginal edge of the TM close to the vibrating support, so that the light entering the chamber was approximately parallel to the end of the vibrating support. Measurements (three to five repetitions) started at this location and were repeated at successive locations in regular steps of 10 μm or 20 μm distal to the initial location. Measurements progressed in this way along the length of the marginal edge of the TM either to within 100 μm of the stationary support or until a segment of at least 300 μm had been covered, for each TM preparation. At every longitudinal measurement location, the vibrating support was vibrated radially in frequency steps of 1 kHz from 2 kHz to 20 kHz at amplitudes that decreased from ∼80 nm at 2 kHz to ∼50 nm at 20 kHz, as a consequence of the limited frequency response of the piezo that vibrated the support. Measurements below 2 kHz and above 20 kHz were unreliable, because the phase gradients at frequencies below 2 kHz and the vibration amplitudes above 20 kHz were too small to be detectable reliably above the measurement noise floor. Amplitude data were calibrated to control for variable reflectance at each point along the TM using the calibrated piezo on which the laser diode was mounted.

Recording of simultaneous suppression neural tuning curves

Mice from a mixed C57/S129 background, normally <3 months of age were anesthetized with urethane (ethyl carbamate, 2 mg/g, intraperitoneally), then tracheotomized with their core temperature maintained at 38°C. To measure the compound action potentials (CAPs) of the auditory nerve, a caudal opening was made in the ventrolateral aspect of the right bulla to reveal the round window. CAPs were measured from the round-window membrane using pipettes with tip diameters of 50–100 μm (recording bandwidth >30 kHz) filled with artificial perilymph. Sound was delivered via a probe with its tip within 1 mm of the tympanic membrane and coupled to a closed acoustic system comprising a 1-inch MK102 microphone (MicroTech Gefell, Cambridge, United Kingdom) for delivering tones and a 3135 1/4-inch microphone (Bruel & Kjaer, Naerum, Denmark) for monitoring sound pressure at the tympanum. The sound system was calibrated in situ for frequencies between 1 and 70 kHz using a laboratory-built microphone conditioning amplifier connected to a Bruel & Kjaer 3135 1/4-inch microphone, and sound level was expressed in dB SPL. CAP tuning curves were derived from simultaneous tone-on-tone masking using a 10-ms probe tone centered on a 40-ms masker tone. The probe tone was set to a level where a stable CAP appeared just above the recording noise floor. The frequency of the masker was set, and its level was adjusted until the probe-tone CAP was suppressed. The masker frequency and level were noted, a new masker frequency was set, and the process was repeated. Stimulus delivery and processing of signals from the interferometer were controlled by a DT3010/32 (Data Translation, Marlboro, MA) board by a PC running Matlab (The MathWorks, Natick, MA) at a sampling rate of 250 kHz.

All procedures involving animals were performed in accordance with United Kingdom Home Office regulations, with approval from the local ethics committee.

Model of the tectorial membrane in a fluid environment

Parameters used in the model of the TM in a fluid environment and their respective units and values are summarized in Table 1.

Table 1.

Parameters used in the model of the TM in fluid environment

Properties	Values and units
f’(x), complex shear force/unit length	N m−1
ZTM(ω), TM impedance with respect to environment	N s m−1
G′(ω), shear storage modulus	Pa
η(ω), shear viscosity	Pa · s
A, cross-sectional area of the TM	m2
WTM, width of the TM	m
TTM, thickness of the TM	base: 2 × 10−5 m
	apex: 4 × 10−5 m
vs(x), TM velocity in the shear direction	m s−1
vf(y), fluid shear velocity	m s−1
α, decay constant of the TM wave	m−1
ρ, TM density	103 kg m−3
ρf, fluid density	103 kg m−3
μ, coefficient of viscosity	7 × 10−4 kg m−1 s−1
Mi, mass/unit length due to the boundary layer	kg m−1
Ri, resistance/unit length due to the boundary layer	kg s−1 m−1

Parameter definitions with their respective units and values are shown.

During longitudinal propagation of a radially shearing wave, the complex shear force per unit length, $f^{\text{'}}\left(x\right)$ (where x is the distance along the TM), acting at frequency ω on the TM can be expressed through its impedance, Z_TM, with respect to its surrounding environment, and its internal shear modulus, $G\text{'}$, and shear viscosity, η. Hence,

$$f^{\prime }\left(x\right)=Z_{TM}{v}_s\left(x\right)-\frac{A}{i\omega }\left(G^{\prime }+i\omega \eta \right)\frac{{\partial }^2{v}_s\left(x\right)}{\partial x^2},$$ (1)

where A is the cross-sectional area of the TM and v_s is TM velocity in the shear direction.

It is assumed that each section of the TM shears radially as a whole, so that wave motion only occurs longitudinally, with wavenumber k, so that v_s(x) is proportional to e^ikx. The force per unit length is

$$f^{\prime }\left(x\right)=\left[Z_{TM}+\frac{k^{2}A}{i\omega }\left(G^{\prime }+i\omega \eta \right)\right]v_s\left(x\right).$$ (2)

If no external force is applied, the complex wavenumber is obtained by setting the term in square brackets in Eq. 2 to zero, to give

$$k=\sqrt{\left[\frac{-i\omega Z_{TM}}{A\left(G^{\text{'}}+i\omega \eta \right)}\right]},$$ (3)

which can be expressed in terms of the wave speed, c, and decay constant, α, as

$$k=\frac{\omega }{c}-i\alpha .$$ (4)

If the TM dynamics is dominated by its mass, so that Z_TM = iωρA (where ρ is the density of the TM), the wavenumber takes the simple form

$$k=\sqrt{\left(\frac{{\omega }^2\rho }{G^{\text{'}}+i\omega \eta }\right)}.$$ (5)

If the loss is small, so that ωη can be ignored compared to G′, then k is real and equal to ω/c, where c is the shear wave speed given by $\sqrt{G^{\prime }\text{/}\rho }$. Assuming that G′ and η are independent of frequency, Eq. 5 can be used to express the wave speed as a function of these parameters, as used by Ghaffari et al. (20). More generally, Eq. 3 can be used to express G′ and η as a function of both wave speed and decay constant, as the real and imaginary components of k, as

$$G^{\prime }\left(\omega \right)+i\omega \eta \left(\omega \right)=\frac{-i\omega Z_{TM}\left(\omega \right)}{k^2\left(\omega \right)A},$$ (6)

so that the frequency variation of these quantities can be calculated.

We can also consider an additional term in Z_TM, which is due to the effect of the frequency-dependent viscous boundary layer of the fluid acting on the TM in the experimental setup. The complex fluid shear velocity at a distance y from a moving surface is approximately (32)

$$v_f\left(y\right)=v_s{e}^{\frac{-\left(1+i\right)y}{\sqrt{2}\delta }},$$ (7)

where the boundary-layer thickness, δ, is

$$\delta =\sqrt{\frac{\mu }{\omega {\rho }_f}},$$ (8)

where μ is the coefficient of viscosity and is assumed to be 7 × 10⁻⁴ kg m⁻¹ s⁻¹ and ρ_f is the fluid density, assumed to be equal to that of the TM, ρ, at 10³ kg m⁻³.

The shear force per unit length is given by

$$f^{\prime }\left(y\right)=-\mu W_{TM}{\frac{\partial v_f}{\partial y}|}_{y=0},$$ (9)

where W_TM is the TM’s width. On the surface of the TM, where y = 0, this force is

$$f^{\prime }=\frac{v_s\left(1+i\right)\mu W_{TM}}{\sqrt{2}\delta },$$ (10)

and since μ = δ² ω ρ_f, then

$$f^{\prime }=(\frac{1+i}{\sqrt{2}})\delta \omega {\rho }_f{W}_{TM}{v}_s.$$ (11)

If M_i and R_i are the mass and resistance, respectively, per unit length due to the viscous boundary layer on one side of the TM, then with the viscous boundary layer on both sides, the total added mass and damping per unit length are

$$M_{2i}=2M_i=\sqrt{2}{\rho }_f{W}_{TM}\delta ,$$ (12)

$$R_{2i}=2R_i=\sqrt{2}\omega {\rho }_f{W}_{TM}\delta .$$ (13)

Hence, in terms of the environment surrounding the TM,

$$Z_{TM}=i\omega {\rho }_f{W}_{TM}\left(T_{TM}+\sqrt{2}\delta \right)+\sqrt{2}\omega {\rho }_f{W}_{TM}\delta .$$ (14)

Substituting this into Eq. 6, and using the fact that A = W_TMT_TM, gives G′ and η in terms of quantities that can be calculated or measured, as

$$G^{\prime }\left(\omega \right)+i\omega \eta \left(\omega \right)=\frac{{\omega }^2{\rho }_f\left(T_{TM}+\sqrt{2}\delta \right)-i\sqrt{2}{\omega }^2{\rho }_f\delta }{k^2\left(\omega \right)T_{TM}}.$$ (15)

Results

Frequency-dependent propagation of longitudinal traveling waves in the TM was investigated in segments isolated from apical (low-frequency) and basal (high-frequency) regions of the mouse cochlea (Fig. 1 A). Longitudinally propagating traveling waves were excited in the TM by sinusoidal vibration of the piezoelectric actuator (Fig. 1 C). The magnitude and phase of the traveling waves as functions of distance from the source of excitation were measured with a laser-diode interferometer (31) (Fig. 1 C) and provided the basis for deriving the dynamic material properties of the TM.

Velocity of the TM traveling waves

The accumulated phase measured as a function of distance from the source of generation (vibrating platform) was used to calculate the propagation velocity, c, of the evoked traveling waves for different frequencies of stimulation. For both apical and basal segments, progressive phase lag was generally observed (Fig. 2, A and B) with increasing longitudinal distance (Fig. 1 C, x) from the stimulation place (x = 0 μm).

Figure 2.

Phase data collected from the basal and apical segments of the TM. (A and B) Average phase lag as a function of longitudinal distance along the TM segments, for each stimulus frequency (for clarity, error bars are not shown). (C) Average wave propagation velocity, calculated from the full longitudinal distance obtained in each experiment (basal, n = 18; apical, n = 6). The range of frequencies between the two vertical dashed lines indicates cochlear-specific frequencies for the apical TM segments.

The velocity of the traveling wave was calculated at each frequency, ω, as $c=\omega \times x\text{/}\Delta \phi$, where Δφ is the overall change in phase over distance x. To calculate the propagation velocity, Δφ and x were taken from the longest segments of phase data available for each animal. This means in effect that an average velocity was calculated over the entire segment for any single frequency. The mean and standard error of c at each frequency are shown in Fig. 2 C. The propagation velocity was lower for the apical segments for all frequencies except those around 5 kHz, where the propagation velocity of the basal TM has a local minimum that is comparable to propagation velocities measured in the apical TM. The local minimum and propagation velocities presented here are comparable to those obtained by Ghaffari et al. (20) using a similar stimulating technique but with displacements approximately an order of magnitude larger and measured with an optical technique rather than with an interferometer. The growth of the propagation velocity for frequencies above the minimum is somewhat higher in our preparation than that reported by Ghaffari and colleagues for the same frequency range. This difference may arise because the TM sections were taken from slightly different places in the cochlea.

Decay of the TM traveling waves

The amplitude of the traveling wave decays with distance along the TM. The decay constant, α, was derived by fitting an exponential decay to the wave amplitude, Y(x) as a function of longitudinal distance x, namely, $Y\left(x\right)=Y\left(0\right)e^{-\propto x}$, where Y(0) is the wave amplitude at the stimulation place.

Overall, the decay constant was larger in the apical TM segments for all but the lowest frequencies studied (2 kHz) (Fig. 3). In the apical region, the decay constant, α, increased significantly for frequencies below 10 kHz and then decreased slightly with increasing stimulus frequency above this (Fig. 3, red curve). By contrast, the decay constant measured from the basal turn TM segments declined slightly as a function of stimulus frequency throughout the frequency range (Fig. 3, blue line).

Figure 3.

Amplitude decay and space constants of the traveling wave as a function of frequency for basal and apical TM segments. Values are shown as the mean ± SD. Solid lines show a polynomial fit to the data points (α = 2967 − 0.053f for the apical and α = 1939 + 0.4879f − 0.00001869f² for the basal segments). Polynomial regression analysis was used to determine the significance criterion of the fit parameters with p < 0.05. The criterion for the best fit was the significance of the respective approximation calculated in a t-test. Range of frequencies between two vertical dashed lines indicates cochlea-specific frequencies for the apical TM segments.

Examples of reconstructed traveling waves for different frequencies using amplitude and phase data collected from a single TM segment are given in Fig. 4.

Figure 4.

Instantaneous traveling-wave displacement for different frequencies reconstructed from amplitude and phase data collected from a single TM segment. Smooth lines represent fit of the data point with equation Y(x) = Y(0)exp(−αx)sin(Ax + B), where A and B are fitting parameters.

TM becomes stiffer and less viscous at high stimulus frequencies

In this section, we calculated the frequency dependence of two dynamic material properties of the TM that have significance for frequency tuning and propagation of the traveling wave in the cochlea. These properties are the shear storage modulus, (G′(ω) (kPa)), which characterizes shear stiffness of the TM, and shear viscosity (η(ω) (Pa ⋅ s)), which determines the energy dissipation during shear displacements of the TM. These two material properties of the TM can be obtained from the relationships between the velocity, c(ω), and decay constant, α(ω), of the traveling wave and the stimulation frequency (Figs. 2 C and 3). The relationships were used to calculate specific values of the complex wavenumber, k(ω), of the wave propagating along the TM as $k\left(\omega \right)=\omega /c\left(\omega \right)-i\alpha \left(\omega \right)$. The wavenumber was used to calculate the corresponding frequency dependence of shear storage modulus and shear viscosity (Eq. 15 in Materials and Methods), shown in Fig. 5, A and B.

Figure 5.

Frequency dependence of dynamic material properties of the TM. Equation 15 and the experimental data presented in Fig. 2C and Fig. 3 were used for calculations. (A) Shear storage modulus, G′. (B) Shear viscosity, η. (C and D) Loss tangent, tan(δ). (C), and reciprocal of the loss tangent, 1/tan(δ) (D), which is proportional to the quality factor Q. TM thickness, T_TM, was taken as 20 μm and 40 μm for the basal and apical segments, respectively.

The shear storage modulus, which characterizes shear stiffness of the TM, of both the basal and apical segments of the TM increased with increasing stimulus frequency, although growth of the shear storage modulus was notably more pronounced at the base of the cochlea (Fig. 5 A). The storage modulus was similar when measured at 2 kHz in both apical and basal segments but it was >2.5 times larger in basal segments at 20 kHz. The shear viscosity of the TM, in turn, has similar levels in both basal and apical segments at high stimulus frequency (Fig. 5 B), but the rate of its change with frequency is initially much larger in the basal segments, where the shear viscosity is substantially higher at low frequencies.

In contrast to our findings for a frequency dependence of the shear storage modulus and shear viscosity, Ghaffari et al. (20) reported frequency-independent values of the shear storage modulus and shear viscosity (47 ± 12 kPa and 0.19 ± 0.07 Pa ⋅ s, respectively, for basal TM segments, and 17 ± 5 kPa and 0.15 ± 0.04 Pa ⋅ s, respectively, for apical TM segments). These values are within the range of frequency-dependent values reported here (see Fig. 5, A and B), although our estimates of η are higher for the lower stimulus frequencies. Direct comparisons of our data to those in other reports on the mechanical properties of the TM (16,24,33–38) cannot be made. The earlier measurements used different techniques and were made for static or low-frequency deformations of the TM, and where comparable frequencies were used, frequency dependence of the material properties was not considered (20,24). Due to the anisotropy and frequency dependence of the TM’s properties (16,33,37,39), model-dependent assumptions would be needed to compare the data presented here to earlier data.

Neural correlates of the TM resonance

Simultaneous-suppression neural tuning curves (Fig. 6) (40), which closely resemble the tuning properties of single auditory nerve fibers (41,42), have a minimum of sensitivity or change in the slope of the curve at frequencies below the probe-tone frequency (Fig. 6, vertical dashed lines). This slope discontinuity has been attributed to the TM resonance and corresponding drop in TM impedance at this frequency (15,21,43–45). It is most pronounced for tuning curves obtained with high-frequency probe tones and disappears completely for low-frequency probe tones. This difference in the shape of the high- and low-frequency tuning curves is likely to be due to different mechanical properties of the TM in the corresponding regions of the cochlea (see Discussion).

Figure 6.

Simultaneous-suppression neural tuning curves for different probe-tone frequencies. Values are shown as the mean ± SD (n = 30). Frequency and level of the probe tone for each curve are indicated by an asterisk. Vertical dashed lines indicate notches and slope discontinuities at frequencies below the probe tone. Data for figures were obtained, with permission, from those used to produce Fig. 5A (21).

Discussion

Propagation velocities of the TM traveling waves presented in this study are comparable to those obtained by Ghaffari et al. (20), who used a CCD camera instead of an interferometer to measure TM vibrations. Ghaffari et al. (20) used a stimulation technique similar to those reported here but delivered TM displacements that were an order of magnitude larger. The similarity in propagation velocities obtained at significantly different displacement levels in the two studies may indicate that the TM displays linear mechanical properties over a large range of stimulus amplitudes. A major difference between the study reported here and that reported by Ghaffari et al. (20) is the development here of a model of the TM in a fluid environment. The vital feature of this model is that constant values for the shear storage modulus and shear viscosity are not assumed, and therefore, the true frequency dependence of these material properties can be calculated.

Material properties of the TM minimize energy loss and facilitate energy transmission along the cochlea

The frequency dependence of the material properties of the TM reported here is important, because it facilitates transmission of energy along the cochlea and ensures that the 1000-fold gain in the basal, high-frequency region of the cochlea, is sufficient to provide sensitivity to high-frequency tones (8). The frequency dependence of the shear storage modulus and shear viscosity are used to calculate the loss tangent, tan(δ) = G″/G′, (the loss modulus G″ is calculated as G″ = ωη(ω) using data for η(ω) in Fig. 5 B), which is defined as the ratio of energy dissipated to energy stored within a unit TM volume. The loss tangent is a crucial material property of the TM, because it reflects the loss in energy transmitted along the length of the cochlea as a consequence of the mechanical coupling of the TM to the organ of Corti. The loss tangent is relatively large at low stimulus frequencies for the basal TM segments (Fig. 5 C), because at these frequencies, G′ is relatively small (Fig. 5 A). This large loss tangent does not, however, lead to higher energy losses in vivo. This is because the TM in the basal region does not experience significant radial shear during the propagation of low-frequency BM traveling waves. The waves peak closer to the cochlear apex and do not show significant phase change in the basal region (Fig. 7 A). In vivo, the TM in the basal region of the cochlea experiences significant shear at higher frequencies that are close to the CFs of that region (Fig. 7 B). This significant shear is, however, not associated with large energy losses because the TM becomes stiffer at higher stimulus frequencies (Fig. 5 A), with a corresponding decrease in the loss tangent (Fig. 5 C).

Figure 7.

BM and TM waves in vivo for a single frequency. Color coding for the TM wave: red stiff; blue, compliant. STS, subtectorial space. (A) Lower-frequency peaks closer to the apex of the cochlea. There is no significant radial shearing motion of the TM for the extended region basal to the peak. Hence, relatively high loss tangent does not result in large energy losses. (B) Higher frequency peaks closer to the cochlear base. Higher TM stiffness at this frequency allows coupling of a larger number of the OHCs, which provide active feedback and higher gain of the cochlear amplifier specific for the base of the cochlea.

The decrease in the TM stiffness at low stimulus frequencies ensures that energy is transmitted by BM-pressure-driven traveling waves along the cochlear partition to their CF place with minimum energy loss, even when the partition is loaded with the elasticity of the TM. Basal to the peak of the traveling wave, mechanics of the cochlea is governed by elastic forces (46,47). Since the TM is less stiff for frequencies corresponding to these positions, compared with those corresponding to the characteristic place, the TM will play a smaller part in the overall response of the organ of Corti and, hence, the propagation of the traveling wave, and so will dissipate less energy. At the same time, a more elastic TM would be more readily displaced by stiff hair bundles at the base of the cochlea. The consequence is less shear movement between the TM and the organ of Corti. Because fluid damping in the subtectorial space (Fig. 1 B) is hypothesized to be the major source of energy dissipation in the cochlea (48,49), reduction in shear displacement between the TM and organ of Corti should lead to lesser energy dissipation and to more efficient transmission of energy to the CF place. However, at frequencies that correspond to the CFs at the base of the cochlea (Fig. 7 B), the basal TM becomes stiffer, resulting in more efficient coupling to the other elements of the organ of Corti, especially the hair bundles of the OHCs, thereby enabling them to more effectively transmit active forces to amplify the motion of the cochlear partition.

Stiffening of the TM maximizes cochlear amplifier gain at high frequencies

Our finding that the TM in the basal region of the cochlea becomes stiffer with increasing stimulus frequency has important implications for the amplification of cochlear responses. Increased stiffening of the TM should enhance longitudinal coupling between adjacent OHCs (21,50), thereby increasing the number of OHCs contributing to the amplification of the responses at the CF place. This frequency-dependent material property of the TM may be an important factor contributing to the typically high gain of the basal region of the cochlea (8). In this respect, longitudinal elastic coupling, and hence mechanical coupling between longitudinally adjacent OHCs, is reduced (24) in the TM of mouse mutants lacking the protein β-tectorin (21), with the result that the cochleae of β-tectorin mice have a slightly reduced sensitivity compared with those of their wild-type littermates. Frequency tuning is, however, sharper, which might be expected as a consequence of weaker longitudinal coupling between the OHCs (21).

Cochlear amplification is also influenced through constructive interaction between BM and TM traveling waves, which should occur only in the region where their phase velocities are similar (20,51) (Fig. 7). The higher velocity of an elastically coupled traveling wave in a stiffer TM should, therefore, match the wave velocity of the stiff basal BM (20), thereby also resulting in a smaller phase roll-off over the length of the TM. Constructive interaction between the BM and TM traveling waves over longer stretches of the cochlea would boost cochlear amplification through increasing the number of OHCs contributing to amplifying a single frequency.

Stiffening causes a sharper TM resonance in the basal cochlea

The discovery that the TM in the basal region of the cochlea becomes stiffer with increasing frequency also has important consequences for cochlear frequency tuning. By becoming stiffer at high frequencies, the TM resonance should become sharper, because the resonance quality factor, Q, is proportional to the reciprocal of the loss tangent Q ∝ 1/tan(δ) = G′/G″ (Fig. 5 D), which increases with the growth of G′. Indeed, the TM resonance in the basal part of the cochlea in vivo is more sharply tuned, as observed in neural tuning curves (Fig. 6). The discontinuity below the probe-tone frequency (Fig. 6, vertical dashed lines) would be expected to disappear if the TM resonance was strongly damped, with lower Q, as it appears to be in the apex of the cochlea (Fig. 5 D), where the internal friction within the TM is relatively large. It is unlikely that the discontinuity is due to the negative damping (4) action of the cochlear amplifier, because the amplifier makes a negligible contribution to cochlear responses for frequencies well below the probe-tone CF (8) where the discontinuity is observed. It should be noted that because the TM is mechanically coupled to the organ of Corti and the BM, the resonances of the entire system do not directly indicate the resonances of the individual components (52). Another consequence of this coupling is that the TM resonance can be reflected in other measurements (52). For example, the secondary maximum in sensitivity, observed at frequencies below the CF in BM frequency-tuning curves recorded in the extreme basal part of the mouse cochlea, is predicted to be due to the sharp TM resonance in the basal region of the cochlea (15,19,21,53,54).

TM internal damping dissipates energy in the cochlea

Our findings also have important significance for the identification of the source of damping in the cochlea. Ever since it was first proposed by Gold (55) that the micromechanics of the cochlea is likely to be dominated by viscous damping, it has been widely held that the main source of the damping in the cochlea is fluid, possibly in the subtectorial space (48,49). The close correspondence between the decrease in the relative amount of energy dissipated due to internal friction within the TM (Fig. 5 C) and the manifestation of a sharper TM resonance in the base of the cochlea in vivo (Fig. 6) lead us to suggest that the internal damping within the TM contributes significantly to overall energy dissipation in the cochlea.

Conclusions

Measurements from TM segments isolated from the mouse cochlea reveal that the dynamic material properties of the TM are frequency-dependent. The TM becomes stiffer and less viscous at high stimulus frequencies. This frequency dependence is more pronounced in the basal region of the cochlea, where the TM becomes significantly stiffer than in the apex and considerably less viscous than at low frequencies. This kind of frequency dependence of the material properties of the TM facilitates amplification and the efficient transmission of energy in the cochlea and requires a revision of our view on the micromechanics of the cochlea, which provides the basis for the first signal-processing stage in the auditory system.