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### Abstract

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

We thank Nigel Cooper, Marcel van der Heijden, Philip Joris, Ombeline de La Rochefoucauld, Elizabeth Olson, Alberto Recio-Spinoso, Tianying Ren, Mario Ruggero, and Andrei Temchin for generously sharing their data. We also thank Christopher Bergevin, Egbert de Boer, Paul Fahey, John Guinan, Ombeline de La Rochefoucauld, Jeffery Lichtenhan, Robert Withnell, and the two anonymous reviewers for valuable comments on the manuscript. This work was supported by grant R01 DC03687 from the NIDCD, National Institutes of Health.

I. INTRODUCTION

A. Propagation and gain functions

B. Overview

II. THE INVERSION PROCEDURE

A. Finding traveling waves from transfer functions

1. Generalized local scaling in the apex

B. Finding propagation and gain functions from traveling waves

C. Validation by reconstruction

III. TRAVELING-WAVE PROPAGATION AND GAIN FUNCTIONS

A. Qualitative features

1. Interpretation in terms of energy flow

B. Generalization to other locations, species, and measurements

C. Variation along the cochlea

1. Near-peak and maximum values

2. Locations of functional landmarks

3. Phase shifts and integrated power

D. Interdependence of and

IV. DISCUSSION

A. Synopsis

B. Methodological and interpretive issues

1. Generality of the wave number

2. Consistency with inverse solutions for the BM impedance

3. Behavior beyond the best place

4. Local scaling and other approximations

5. When things fall apart

6. Estimating the power gain of the cochlear amplifier

C. Interpretation as a hydromechanical laser amplifier

1. An analytic twist

2. Emergence of laser oscillations

3. Meaning of the dispersion relations

4. Interrelationship of propagation and gain

5. Biophysics of the gain medium

### Key Topics

- Hydrodynamic waves
- 32.0
- Anatomy
- 12.0
- Mechanical waves
- 12.0
- Cilia
- 10.0
- Acoustic waves
- 9.0

## Figures

Wave motion along a string and on the basilar membrane. The top panel (A) shows the transverse vibration of a uniform stretched string driven sinusoidally on the left. The propagation and gain coefficients determine the spatial pattern of the wave along the string. The change of wave amplitude reflects the sign of the string gain coefficient, . When the wave amplitude decreases with distance, as shown, the wave loses energy as it propagates (e.g., due to viscous damping in the surrounding medium), and . The bottom panel (B) shows the transverse vibration of the basilar membrane (BM) produced by sinusoidal motion of the stapes. Because the stiffness and other properties of the partition vary with position, the propagation and gain functions depend on , and the sign of the gain function cannot simply be read off from the behavior of the wave envelope. For example, the declining stiffness of the partition causes BM vibration to increase to a maximum before decreasing, even when is everywhere negative (e.g., in a passive cochlea).

Wave motion along a string and on the basilar membrane. The top panel (A) shows the transverse vibration of a uniform stretched string driven sinusoidally on the left. The propagation and gain coefficients determine the spatial pattern of the wave along the string. The change of wave amplitude reflects the sign of the string gain coefficient, . When the wave amplitude decreases with distance, as shown, the wave loses energy as it propagates (e.g., due to viscous damping in the surrounding medium), and . The bottom panel (B) shows the transverse vibration of the basilar membrane (BM) produced by sinusoidal motion of the stapes. Because the stiffness and other properties of the partition vary with position, the propagation and gain functions depend on , and the sign of the gain function cannot simply be read off from the behavior of the wave envelope. For example, the declining stiffness of the partition causes BM vibration to increase to a maximum before decreasing, even when is everywhere negative (e.g., in a passive cochlea).

(Panel A) Auditory-nerve based estimate of the BM click response, , at the cochlear location tuned to approximately in chinchilla (Recio-Spinoso *et al.*, 2005). The response estimate has been normalized by its peak value. Time, shown along the abscissa in units of the CF period, is measured relative to the approximate onset of stapes vibration by subtracting out estimates of acoustic and synaptic transmission delays amounting to a total of (Temchin *et al.*, 2005). (Panel B) The magnitude and phase of the Fourier transform of provide an estimate of the BM mechanical transfer function, , at the cochlear location . Frequency, normalized by , increases along the logarithmic abscissa. (Panel C) Application of local scaling provides an estimate of the traveling wave by reinterpreting the abscissa as a spatial axis at fixed frequency. The figure shows a snapshot of the wave whose envelope and phase are shown in Panel (B) The scale bar is based on estimates of the chinchilla cochlear map (Eldredge *et al.*, 1981; Greenwood, 1990).

(Panel A) Auditory-nerve based estimate of the BM click response, , at the cochlear location tuned to approximately in chinchilla (Recio-Spinoso *et al.*, 2005). The response estimate has been normalized by its peak value. Time, shown along the abscissa in units of the CF period, is measured relative to the approximate onset of stapes vibration by subtracting out estimates of acoustic and synaptic transmission delays amounting to a total of (Temchin *et al.*, 2005). (Panel B) The magnitude and phase of the Fourier transform of provide an estimate of the BM mechanical transfer function, , at the cochlear location . Frequency, normalized by , increases along the logarithmic abscissa. (Panel C) Application of local scaling provides an estimate of the traveling wave by reinterpreting the abscissa as a spatial axis at fixed frequency. The figure shows a snapshot of the wave whose envelope and phase are shown in Panel (B) The scale bar is based on estimates of the chinchilla cochlear map (Eldredge *et al.*, 1981; Greenwood, 1990).

Spacetime slice through a symmetric, two-dimensional box model of height . The axis extends longitudinally from the base , and the axis is oriented perpendicular to the basilar membrane, which spans the entire width, , of the cochlea. The snapshot shown here has caught the BM participating in a traveling wave, whose vertical displacements have been hugely exaggerated for the purposes of illustration. The inversion procedure described in the text finds the wave’s propagation and gain functions by analyzing measurements of the spatial displacement pattern.

Spacetime slice through a symmetric, two-dimensional box model of height . The axis extends longitudinally from the base , and the axis is oriented perpendicular to the basilar membrane, which spans the entire width, , of the cochlea. The snapshot shown here has caught the BM participating in a traveling wave, whose vertical displacements have been hugely exaggerated for the purposes of illustration. The inversion procedure described in the text finds the wave’s propagation and gain functions by analyzing measurements of the spatial displacement pattern.

Derived propagation and gain functions. Plotted vs the generalized scaling variable using solid and dashed lines, respectively, the functions and were obtained by inversion from the estimate of shown in Fig. 2. Because , is nearly equivalent to the normalized frequency . Parameters for the chinchilla cochlear map were taken from Greenwood (1990). For reference, thin dashes mark the zero line. The scale bar represents a distance of .

Derived propagation and gain functions. Plotted vs the generalized scaling variable using solid and dashed lines, respectively, the functions and were obtained by inversion from the estimate of shown in Fig. 2. Because , is nearly equivalent to the normalized frequency . Parameters for the chinchilla cochlear map were taken from Greenwood (1990). For reference, thin dashes mark the zero line. The scale bar represents a distance of .

Traveling wave/transfer function reconstructed from the derived wave number using the WKB approximation. The reconstructed response (thin solid line) was obtained from the wave number in Fig. 4 using the WKB formula [Eq. (11) for ] and evaluating the integral using generalized scaling. An overall complex scale factor was determined by matching the data at the peak. For comparison, the original Wiener-kernel measurements are reproduced from Fig. 2 (thick gray line).

Traveling wave/transfer function reconstructed from the derived wave number using the WKB approximation. The reconstructed response (thin solid line) was obtained from the wave number in Fig. 4 using the WKB formula [Eq. (11) for ] and evaluating the integral using generalized scaling. An overall complex scale factor was determined by matching the data at the peak. For comparison, the original Wiener-kernel measurements are reproduced from Fig. 2 (thick gray line).

Original and reconstructed Wiener-kernel transfer functions from locations throughout the chinchilla cochlea. Transfer functions are shown normalized to the same peak amplitude.

Original and reconstructed Wiener-kernel transfer functions from locations throughout the chinchilla cochlea. Transfer functions are shown normalized to the same peak amplitude.

Propagation and gain functions derived from all nine Wiener kernels with CFs in the range . Gray lines show individual functions and ; black lines show trends obtained by loess fitting (Cleveland, 1993). For reference, thin dashes mark the zero lines and the location of the wave peak . The scale bar represents a distance of .

Propagation and gain functions derived from all nine Wiener kernels with CFs in the range . Gray lines show individual functions and ; black lines show trends obtained by loess fitting (Cleveland, 1993). For reference, thin dashes mark the zero lines and the location of the wave peak . The scale bar represents a distance of .

Propagation and gain trends throughout the cochlea. The figure shows trend functions and at seven frequencies spanning the frequency range of the chinchilla cochlea. Trend functions were computed as described for Fig. 7 by binning the 86 reconstructed Wiener kernels into seven CF groups with bin edges of . The results are plotted on a spatial axis converted to equivalent CF using an estimate of the chinchilla cochlear map (Eldredge *et al.*, 1981; Greenwood, 1990). The pair of functions is plotted at the location where , where is the geometric mean CF of the group. Short vertical lines, indexed by integers for future reference, identify the nominal frequency of each group. The trend values at the site appeared earlier in Fig. 7. The scale bar represents a distance of . For reference, thin dashes mark the zero line.

Propagation and gain trends throughout the cochlea. The figure shows trend functions and at seven frequencies spanning the frequency range of the chinchilla cochlea. Trend functions were computed as described for Fig. 7 by binning the 86 reconstructed Wiener kernels into seven CF groups with bin edges of . The results are plotted on a spatial axis converted to equivalent CF using an estimate of the chinchilla cochlear map (Eldredge *et al.*, 1981; Greenwood, 1990). The pair of functions is plotted at the location where , where is the geometric mean CF of the group. Short vertical lines, indexed by integers for future reference, identify the nominal frequency of each group. The trend values at the site appeared earlier in Fig. 7. The scale bar represents a distance of . For reference, thin dashes mark the zero line.

Superposed propagation and gain functions from throughout the cochlea. The figure shows the trend functions and from Fig. 8 superposed by plotting them versus the generalized scaling variable, . The integer labels correspond to those used in Fig. 8. The horizontal axis differs significantly from only for curve from the extreme apex. The scale bar represents a distance of . For reference, thin dashes mark the zero lines and the location of the wave peak .

Superposed propagation and gain functions from throughout the cochlea. The figure shows the trend functions and from Fig. 8 superposed by plotting them versus the generalized scaling variable, . The integer labels correspond to those used in Fig. 8. The horizontal axis differs significantly from only for curve from the extreme apex. The scale bar represents a distance of . For reference, thin dashes mark the zero lines and the location of the wave peak .

Propagation and gain functions derived from estimates of from the apex of the cat cochlea (CFs less than ) obtained by van der Heijden and Joris (2006). Gray lines show individual functions and ; black lines show corresponding trends obtained by loess fitting. The scale bar represents a distance of . For reference, thin dashes mark the zero lines and the location of the wave peak .

Propagation and gain functions derived from estimates of from the apex of the cat cochlea (CFs less than ) obtained by van der Heijden and Joris (2006). Gray lines show individual functions and ; black lines show corresponding trends obtained by loess fitting. The scale bar represents a distance of . For reference, thin dashes mark the zero lines and the location of the wave peak .

Scatterplots of and from throughout the chinchilla cochlea. The black dots give values of , defined as the value of the propagation function at the wave peak , vs characteristic frequency. Values are shown for all 86 successfully reconstructed Wiener kernels. The open squares give values of , where is the effective height of the scalae, defined as the radius of the equivalent circle (area equal to the combined areas of the scala vestibuli and tympani) as computed from measured scalae dimensions (Salt, 2001). Values imply that the hydrodynamics at the peak is short wave. The solid and dashed lines are loess trend lines (Cleveland, 1993) superposed to guide the eye. The dotted lines show trends computed from propagation functions derived using traditional rather than generalized local scaling.

Scatterplots of and from throughout the chinchilla cochlea. The black dots give values of , defined as the value of the propagation function at the wave peak , vs characteristic frequency. Values are shown for all 86 successfully reconstructed Wiener kernels. The open squares give values of , where is the effective height of the scalae, defined as the radius of the equivalent circle (area equal to the combined areas of the scala vestibuli and tympani) as computed from measured scalae dimensions (Salt, 2001). Values imply that the hydrodynamics at the peak is short wave. The solid and dashed lines are loess trend lines (Cleveland, 1993) superposed to guide the eye. The dotted lines show trends computed from propagation functions derived using traditional rather than generalized local scaling.

Scatterplot of the maximum values of and versus CF. Black symbols show values of in chinchilla (circles) and cat (squares); gray symbols show corresponding values of . Loess trends for chinchilla and cat are shown with solid and dotted lines, respectively.

Scatterplot of the maximum values of and versus CF. Black symbols show values of in chinchilla (circles) and cat (squares); gray symbols show corresponding values of . Loess trends for chinchilla and cat are shown with solid and dotted lines, respectively.

Scatterplot of the maxima of and zeros of vs CF. Black symbols represent values of in chinchilla (circles) and cat (squares); gray symbols show values of . Solid lines show corresponding trends computed from the data pooled across species. For reference, a dashed line marks the wave peak . Scale bars of are shown for use when interpreting the ordinate spatially.

Scatterplot of the maxima of and zeros of vs CF. Black symbols represent values of in chinchilla (circles) and cat (squares); gray symbols show values of . Solid lines show corresponding trends computed from the data pooled across species. For reference, a dashed line marks the wave peak . Scale bars of are shown for use when interpreting the ordinate spatially.

Scatterplot of the maxima of and vs CF. Gray symbols represent values of in chinchilla (circles) and cat (squares); black symbols show values of . Solid lines show corresponding trends computed from the data pooled across species. For reference, a dashed line marks the wave peak . Scale bars of are shown for use when interpreting the ordinate spatially.

Scatterplot of the maxima of and vs CF. Gray symbols represent values of in chinchilla (circles) and cat (squares); black symbols show values of . Solid lines show corresponding trends computed from the data pooled across species. For reference, a dashed line marks the wave peak . Scale bars of are shown for use when interpreting the ordinate spatially.

Scatterplot of the relative power supplied to the wave vs CF in chinchilla (circles) and cat (squares). Net power gains were computed using Eq. (15) and are shown in arbitrary units normalized so that the trend (solid line) is unity at the highest CF. The trend was computed from the pooled data.

Scatterplot of the relative power supplied to the wave vs CF in chinchilla (circles) and cat (squares). Net power gains were computed using Eq. (15) and are shown in arbitrary units normalized so that the trend (solid line) is unity at the highest CF. The trend was computed from the pooled data.

Cochlear propagation and gain functions obey Kramers–Krönig dispersion relations. The figure shows empirical propagation and gains functions and derived from three example Wiener kernels with CFs near (thin black lines) along with their Kramers–Krönig counterparts computed one from the other (thick gray lines). The Kramers–Krönig in the top panel was computed from the empirical in the bottom panel using Eq. (16). The Kramers–Krönig in the bottom panel was computed from the empirical in the top panel using Eq. (17).

Cochlear propagation and gain functions obey Kramers–Krönig dispersion relations. The figure shows empirical propagation and gains functions and derived from three example Wiener kernels with CFs near (thin black lines) along with their Kramers–Krönig counterparts computed one from the other (thick gray lines). The Kramers–Krönig in the top panel was computed from the empirical in the bottom panel using Eq. (16). The Kramers–Krönig in the bottom panel was computed from the empirical in the top panel using Eq. (17).

Scatterplot of vs CF in chinchilla (circles) and cat (squares). Computed using Eq. (25), is the ratio of the net powers supplied to/delivered by the wave in regions basal/apical to the point where the power flow reverses near the peak of the wave. The trend (solid line) was computed from the pooled data and indicates that is fairly close to one, implying the existence of substantial power gains throughout the cochlea.

Scatterplot of vs CF in chinchilla (circles) and cat (squares). Computed using Eq. (25), is the ratio of the net powers supplied to/delivered by the wave in regions basal/apical to the point where the power flow reverses near the peak of the wave. The trend (solid line) was computed from the pooled data and indicates that is fairly close to one, implying the existence of substantial power gains throughout the cochlea.

Analogy between laser and cochlear amplifiers. The top panel (A) shows the essentials of a laser amplifier. When excited by an optical pumping process, atoms embedded in a gain medium that supports electromagnetic wave propagation spontaneously emit incoherent light. Light of the same frequency applied at the input stimulates the atoms to radiate in phase, amplifying the applied beam. The bottom panel (B) shows corresponding features of the cochlear amplifier. When stimulated via bundle displacements induced by the traveling pressure-difference wave, hair cells create forces that couple to their mechanical environment, producing hydromechanical radiation that combines coherently with and amplifies the incident pressure wave. Although each cell may radiate symmetrically in both directions, the backward radiation combines incoherently and tends to cancel out (Shera, 2003b; Shera and Guinan, 2007); only the forward radiation is shown here. Brownian motion also drives the individual hair bundles, resulting in incoherent hydromechanical radiation whose amplitude and phase differ from cell to cell. (For the purposes of illustration, this incoherent radiation is shown propagating obliquely.) The amplification process is powered by the electrochemical potentials that drive ionic currents through the hair-cell transduction channels.

Analogy between laser and cochlear amplifiers. The top panel (A) shows the essentials of a laser amplifier. When excited by an optical pumping process, atoms embedded in a gain medium that supports electromagnetic wave propagation spontaneously emit incoherent light. Light of the same frequency applied at the input stimulates the atoms to radiate in phase, amplifying the applied beam. The bottom panel (B) shows corresponding features of the cochlear amplifier. When stimulated via bundle displacements induced by the traveling pressure-difference wave, hair cells create forces that couple to their mechanical environment, producing hydromechanical radiation that combines coherently with and amplifies the incident pressure wave. Although each cell may radiate symmetrically in both directions, the backward radiation combines incoherently and tends to cancel out (Shera, 2003b; Shera and Guinan, 2007); only the forward radiation is shown here. Brownian motion also drives the individual hair bundles, resulting in incoherent hydromechanical radiation whose amplitude and phase differ from cell to cell. (For the purposes of illustration, this incoherent radiation is shown propagating obliquely.) The amplification process is powered by the electrochemical potentials that drive ionic currents through the hair-cell transduction channels.

Propagation and gain functions for laser and cochlear amplifiers versus frequency. The top panel shows stylized propagation and gain functions for an optical laser amplifier in the frequency region about the atomic transition frequency, . Only the resonant, atomic contribution to the laser propagation function is shown; the constant wave number corresponding to background propagation in the host medium has been subtracted off. [Adapted from Figs. 2.8 and 7.3 of Siegman (1986).] The bottom panel shows stylized versions of the empirical cochlear propagation and gain functions derived here (see Figs. 8 and 9). The cochlear wave number appears isomorphic to the laser wave number rotated 90° clockwise in the complex plane. In both panels, dotted lines mark the zero along the ordinate. Note that the laser functions are substantially narrower than those in the cochlea. Had they been drawn on the same logarithmic axis used for the cochlea, the laser functions would have bandwidths smaller than the dots that compose the zero lines. In the cochlea, local scaling can be used to convert the frequency axis into a spatial one. The clockwise twist shown here is crucial to the operation of the cochlea, which must also analyze the signals it amplifies. Useful analysis requires both level-dependent amplification (to match the variance of the incoming signal to the dynamic range of the detectors) and attenuation (to clear the stage for the analysis of future sounds). The result is a region with followed by one with .

Propagation and gain functions for laser and cochlear amplifiers versus frequency. The top panel shows stylized propagation and gain functions for an optical laser amplifier in the frequency region about the atomic transition frequency, . Only the resonant, atomic contribution to the laser propagation function is shown; the constant wave number corresponding to background propagation in the host medium has been subtracted off. [Adapted from Figs. 2.8 and 7.3 of Siegman (1986).] The bottom panel shows stylized versions of the empirical cochlear propagation and gain functions derived here (see Figs. 8 and 9). The cochlear wave number appears isomorphic to the laser wave number rotated 90° clockwise in the complex plane. In both panels, dotted lines mark the zero along the ordinate. Note that the laser functions are substantially narrower than those in the cochlea. Had they been drawn on the same logarithmic axis used for the cochlea, the laser functions would have bandwidths smaller than the dots that compose the zero lines. In the cochlea, local scaling can be used to convert the frequency axis into a spatial one. The clockwise twist shown here is crucial to the operation of the cochlea, which must also analyze the signals it amplifies. Useful analysis requires both level-dependent amplification (to match the variance of the incoming signal to the dynamic range of the detectors) and attenuation (to clear the stage for the analysis of future sounds). The result is a region with followed by one with .

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