The Elusive Cochlear Filter: Wave Origin of Cochlear Cross-Frequency Masking

Abstract

The mammalian cochlea achieves its remarkable sensitivity, frequency selectivity, and dynamic range by spatially segregating the different frequency components of sound via nonlinear processes that remain only partially understood. As a consequence of the wave-based nature of cochlear processing, the different frequency components of complex sounds interact spatially and nonlinearly, mutually suppressing one another as they propagate. Because understanding nonlinear wave interactions and their effects on hearing appears to require mathematically complex or computationally intensive models, theories of hearing that do not deal specifically with cochlear mechanics have often neglected the spatial nature of suppression phenomena. Here we describe a simple framework consisting of a nonlinear traveling-wave model whose spatial response properties can be estimated from basilar-membrane (BM) transfer functions. Without invoking jazzy details of organ-of-Corti mechanics, the model accounts well for the peculiar frequency-dependence of suppression found in two-tone suppression experiments. In particular, our analysis shows that near the peak of the traveling wave, the amplitude of the BM response depends primarily on the nonlinear properties of the traveling wave in more basal (high-frequency) regions. The proposed framework provides perhaps the simplest representation of cochlear signal processing that accounts for the spatially distributed effects of nonlinear wave propagation. Shifting the perspective from local filters to non-local, spatially distributed processes not only elucidates the character of cochlear signal processing, but also has important consequences for interpreting psychophysical experiments.

INTRODUCTION

The mammalian cochlea acts as an acoustic prism that maps sound frequency onto location, thereby allowing different sound components to excite distinct populations of sensory neurons. Cochlear signal processing has been likened to a bank of independent band-pass filters (e.g., oscillators), where each filter represents the frequency response of the basilar membrane (BM) at one location. Physically, however, the cochlea operates less like a bank of independent oscillators than as an acoustic waveguide (see e.g., Wegel and Lane 1924; Peterson and Bogert 1950; Zweig et al. 1976; Lighthill 1981; Neely 1985; Shera et al. 2004a). Experimental data indicate that BM motions are driven by transpartition pressure waves that travel along the cochlear duct at remarkably slow speeds (Dong and Olson 2013). In healthy preparations, and at low-to-moderate sound levels, the waves of both pressure and BM vibration increase in amplitude as they approach their best place (BP), defined as the cochlear location of maximal BM response. This location depends primarily on sound frequency and, to a lesser extent, on level. The increase in both pressure and BM motion implies that slow traveling waves undergo some form of spatial amplification (i.e., spatial accumulation of energy; see Shera 2007). The dominant view in the field is that the outer hair cells (OHCs) behave as piezo-electric actuators (Brownell et al. 1985; Frank et al. 1999; Rabbitt 2020), pumping power into the traveling wave along its journey. An alternative proposal suggests that the increase in complex power carried by the slow-traveling wave involves no actual power injection from the OHCs, but results from an exchange of energy between distinct wave modes (van der Heijden 2014). Adopting this alternative scenario has no particular consequence for the results presented here, although considering it explicitly at every stage of the argument would greatly complicate the exposition.

In the healthy cochlea, the amount of slow-wave amplification depends strongly on sound intensity (Rhode 1971; Dong and Olson 2013), causing BM responses to be nonlinear. As a consequence of this level-dependent amplification, BM and pressure responses grow compressively (i.e., less than linearly with respect to the input sound pressure). For example, intracochlear pressure measurements in the gerbil (Dong and Olson 2013) show that the gain of the cochlear pressure response at a fixed location varies by roughly 35 dB over the dynamic range of hearing; over the same range of sound levels, the gain of BM motion changes by 45 dB.

As a consequence of the cochlea’s waveguide structure, cochlear responses are primarily determined by the characteristics of wave propagation basal to the recording site (see, e.g., Siebert 1974; Zweig et al. 1976; Zweig 2015). When the cochlear partition is modeled as a cascaded array of resonators driven by the local pressure, as in transmission-line models, the level-dependent gain and tuning observed at each location (i.e., at the output of a single “filter”) primarily reflect the dramatic variation of the filter’s input rather than a significant change in its mechanical response properties. Because the “inputs” and “outputs” at each location are coupled via multiple pathways (e.g., through the fluids; see Zweig et al. 1976; Nobili et al. 1998; Shera et al. 2005), describing nonlinear cochlear processing in terms of local filters needlessly complicates the picture (Shera et al. 2004a).

A common approach to modeling nonlinear cochlear responses focuses on the dynamical properties of nonlinear oscillators (see, e.g., Camalet et al. 2000; Stoop and Kern 2004; Hudspeth et al. 2010), at the expense of the physics of wave propagation. In some cases, the global dynamics of the cochlea brought about by wave propagation do actually appear well described by the responses of simpler systems, such as isolated oscillators (see, e.g., Sisto and Moleti 1999; Altoè et al. 2017). Perhaps this is not surprising: all dynamical systems manifest correlations—such as those between bandwidth and reaction time—which are ultimately a consequence of causality and other physical constraints. As elucidated here, however, the approach of focusing foremost on the local mechanics becomes particularly problematic for understanding the physical origin of phenomena involving the nonlinear spatial interaction of waves elicited by different sound components.

Nonlinear interactions among cochlear responses to different sound components are often characterized experimentally by measuring the responses to two or more tones presented simultaneously. These “two-tone suppression” experiments determine how the presentation of one tone (the suppressor) affects the auditory-nerve (AN) (e.g., Sachs and Kiang 1968; Abbas and Sachs 1976) or mechanical (e.g., Ruggero et al. 1992; Cooper 1996) response to the other (the probe), usually assessed at its best place. Measurements consistently show that mechanical vibrations can be greatly suppressed by above-CF tones, even when the suppressor elicits little or undetectable response at the recording site (Cooper 1996; Versteegh and van der Heijden 2013; Dewey et al. 2019; Charaziak et al. 2020). This phenomenon, coined “phantom suppression” by Versteegh and van der Heijden (2013), provides a compelling argument against a filterbank view of cochlear nonlinearity. Furthermore, using wideband tone complexes as probes, Versteegh and van der Heijden (2013) found systematic effects of probe frequency on the amount and growth of suppression, which they interpreted in terms of the spatial build-up of suppression during the propagation of cochlear traveling waves.

While all these experimental findings find a natural explanation when considering the cochlea as a waveguide—where significant suppression can take place en route as waves travel through the region basal to the recording site (see Versteegh and van der Heijden 2013 for an in-depth discussion; see also Geisler et al. 1990; Pang and Guinan 1997; Dong and Olson 2016; Dewey et al. 2019; Charaziak et al. 2020)—they are not easily explained by invoking the nonlinear dynamics of isolated oscillators.

Although a number of computational models can account for specific features of two-tone suppression data (e.g., Kanis and de Boer 1994; Zhou and Nam 2019; Moleti and Sisto 2020), a simple modeling framework for quantifying the effects of cochlear suppression on the outcomes of experiments is lacking. Here, we provide a simple mathematical description of the mechanical data that yields a straightforward and intuitive representation of suppression and its relationship to other manifestations of cochlear nonlinearity. For simplicity, our description focuses on the motion of the BM as a meaningful proxy for the AN response (Narayan et al. 1998; Robles and Ruggero 2001; Temchin et al. 2008); suppressive effects on internal motions of the organ of Corti are discussed elsewhere (e.g., Dewey et al. 2019; Moleti and Sisto 2020).

Our description is based on an approximation opposite to the assumptions widely adopted by filter-bank models of the cochlea. In particular, we show that representing BM motion entirely as a traveling wave—and therefore assuming that BM responses at any given location are completely determined by processes occurring basal to that place—allows one to predict, with minimal mathematical complexity, the frequencies of maximal suppression with surprising accuracy. Furthermore, we show that a simple but physically realistic 3-D cochlear model whose behavior is well captured by our traveling-wave framework produces suppression patterns in good qualitative agreement with the experimental data.

SIMPLE WAVE MODEL: THEORETICAL FRAMEWORK

General Idea and Relationship with Existing Cochlear Models

The theoretical framework proposed here can be intuitively understood by comparing its functioning with that of existing cochlear models. In particular, cochlear models can be broadly categorized based on their effective number of spatial dimensions. The first and perhaps most widely employed class of models, typically referred to as filter-bank or critical-oscillator models, are zero-dimensional (Fig. 1A). These models have no explicit representation of the traveling wave, and cochlear responses are approximated by an array of uncoupled oscillators (filters) driven directly by a proxy for the pressure at the input to the cochlea near the stapes. The absence of coupling between the oscillators implies that the large BM gain and fine frequency selectivity are necessarily obtained by employing sharply tuned filters, as the pressure driving them undergoes no form of spatial tuning.

Fig. 1.

Cartoon illustrating three different classes of cochlear models (panels A–C), and the simple traveling-wave framework (D) employed here to interpret the experimental data.

The second class consists of one-dimensional cochlear models analogous to transmission lines (Fig. 1B). Although these models assume that the cochlear fluids move only longitudinally (i.e., wavelengths are assumed significantly larger than the scala height so vertical dependencies of the pressure field are neglected), they include an explicit representation of the pressure wave traveling along the cochlear duct. When modeling the healthy cochlea, the mechanics of the partition are generally represented by active filters driven by the pressure wave. To produce realistic BM responses, these models compensate for neglecting the short-wave hydrodynamics near the peak of the response by using BM oscillators that are more sharply tuned than necessary in two or three dimensions (see Shera et al. 2005; Zweig 2015). Although not problematic for elucidating the physical origin of many cochlear response features (e.g., Zweig and Shera 1995; Shera 2001a; Shera 2001b), this necessary compensation prevents the models in this class from fully accounting for phenomena related to nonlinear wave amplification (Charaziak et al. 2020; Moleti and Sisto 2020), such as phantom suppression.

The third and final class consists of two- and three-dimensional (3-D) cochlear models (Fig. 1C). In these models, wavelengths become small compared to the height of the duct near the peak of the response, and only a tiny fraction of the fluid surrounding the organ of Corti then carries the energy of the traveling wave responsible for driving the motion of the BM. In combination with active OHC forces, this wavelength-dependent hydrodynamic “focusing” of energy to the fluid adjacent to the BM produces a dramatic spatial build-up of transpartition pressure as the wave approaches its peak. In this region, the distributed action of the cochlear amplifier both boosts the amplitude of the slow-traveling wave and simultaneously shortens its wavelength, thereby efficiently exploiting the 3-D cochlear hydrodynamics. As a consequence, the pressure wave driving the partition becomes spatially tuned and tremendously amplified, thereby allowing realistic BM responses to be obtained using relatively broad mechanical filters (see Appendix B of Altoè and Shera 2020b).

In terms of its assumptions about the relative contributions of distributed vs. local amplification to the motion of the BM, the framework we propose (Fig. 1D) can be regarded as simplification of 3-D models. In particular, the simplification consists of ignoring the distinction between transpartition pressure waves and BM mechanical waves. Thus, we assume that the waves of pressure and BM velocity grow together in constant proportion as they travel, as approximately suggested by the experimental data (Dong and Olson 2013). In contrast to existing cochlear models, this simplified framework lacks an explicit representation of the cochlear partition and its contribution to filtering and amplifying the pressure wave. Thus, there is nothing in the description that represents the mechanical properties of the BM or the piezoelectric action of the OHCs. Although the cochlear amplifier is assumed to control the local slope (i.e., spatial derivative) of the traveling-wave envelope in a level-dependent fashion, any additional contributions it may make to BM motion (e.g., local filtering) are neglected. Thus, in regions with positive spatial gain (amplification), the slope of the traveling-wave envelope is positive and the wave amplitude increases as it propagates. Similarly, in regions with negative spatial gain (attenuation), the wave amplitude decreases. Note that the spatial slope of the traveling-wave envelope is precisely zero at the peak of the wave; thus, amplification (positive gain) in this framework occurs entirely basal to the best place. In its emphasis on distributed rather than local processes, the framework is conceptually antipodal to filter-bank models on the “spherical cowchlea” (Altoè and Shera 2020b).

We apply the framework and illustrate its predictions by using measured BM transfer functions to estimate the traveling-wave envelope and thereby deduce the spatial dependence of cochlear amplification. Our analysis assumes approximate local scaling symmetry—that the peak region of BM transfer functions have the same shape, independent of CF. By determining how the spatial gain of the traveling wave depends nonlinearly on sound level, we then estimate the frequencies of maximal suppression in two-tone suppression experiments. Thus, the proposed framework provides a (largely) model-independent strategy for locating the cochlear region maximally responsible for nonlinear suppression.

Simple Wave Model

The simple traveling-wave model relates the spatial properties of the BM wave to the frequency tuning of the BM. The frequency domain representation of an harmonic traveling wave T at frequency f propagating along the x axis can be described by the equation

$$\begin{array}{c}\hfill T(x,f)=T_{0}exp(-i{\int }_0^{x}k(x^{\prime },f)d{x}^{\prime }),\end{array}$$ 1

where $i=\sqrt{-1}$, k(x, f ) is the complex wavenumber, and $T_0$ is the initial amplitude of the wave (at $x=0$). Note that the wavenumber generally depends on both location and frequency. For notational simplicity we leave the dependence on frequency implicit in most equations. Note that Eq. (1) coincides with the short-wave solution of the cochlear transpartition pressure wave near the peak of the BM transfer function (Siebert 1974). A complete physical description of the mechanical wave traveling on the BM requires the introduction of additional factors. Nevertheless, because the spatial amplification of pressure and BM waves are qualitatively similar Dong and Olson (2013), these factors play only a secondary role for the phenomena under study. For our purposes here, they introduce unnecessary complication.

By taking the magnitude of Eq. (1), the amplitude, or spatial envelope, of the wave becomes

$$\begin{array}{c}\hfill |T(x)|=|T_0|exp({\int }_0^{x}g(x^{\prime })d{x}^{\prime }),\end{array}$$ 2

where g(x), defined as the imaginary part of the wavenumber ($g\equiv \mathfrak{I}[k]$), represents the spatial gain function with units of log-gain per unit length. In this framework, spatial regions with $g(x)>0$ constitute the mathematical signature of the cochlear amplifier. It is easy to verify that

$$\begin{array}{c}\hfill g(x)=\frac{\mathrm{d}ln|\mathit{T}(\mathit{x})/{\mathit{T}}_0|}{\mathrm{d}\mathit{x}}=\frac{ln10}{20}\frac{{\mathrm{d}|\mathit{T}|}_{\mathrm{dB}}}{\mathrm{d}\mathit{x}}\propto \frac{{\mathrm{d}|\mathit{T}|}_{\mathrm{dB}}}{\mathrm{d}\mathit{x}},\end{array}$$ 3

where ${|T(x)|}_{\mathrm{dB}}$ represents the traveling-wave envelope expressed in dB relative to $|T_0|$.

Measurements show that both BM motion and transpartition pressure increase as the traveling wave approaches its best place (Dong and Olson 2013). Hence, the wave envelope |T(x)| grows and there is a region of positive spatial gain [$g(x)>0$] basal to the peak. A region where $g(x)>0$ basal to the peak location is a universal feature of 1-, 2-, and 3-D active models. Note that spatial amplification occurs not at the wave peak [located at $\widehat{x}$, where $g(\widehat{x})=0$] but at more basal locations ($x<\widehat{x}$). Indeed, the greatest amplification occurs near the point (call it $x_g<\widehat{x}$) where the gain g(x) reaches a maximum. Thus, actions that locally decrease the value of g(x) have their greatest effect (i.e., produce the largest reduction in total gain) not at $\widehat{x}$ but at locations near $x_g$. The assumption that suppressor tones act near their own best place by decreasing the local value of g(x) for the probe therefore implies that the most effective suppressors will be those that peak basal to the probe location (i.e., those whose frequencies are higher than the probe).

Finding the Traveling-Wave Envelope

The framework outlined above allows one to estimate, simply from the envelope of the traveling wave, the frequency of the suppressor tone that maximally reduces the BM response at the recording site. If measurements of the spatial wave envelope, |T(x)|, are not available, one can approximate the envelope from the BM frequency response with the help of one additional assumption. In particular, we assume that the shapes of BM transfer functions are locally identical near their peaks when expressed as a function of frequency normalized to the local CF (scaling symmetry). In the real cochlea, local scaling is only approximate. Indeed, many of the assumptions we rely on for this analysis are clear oversimplifications of cochlear physics. However, these oversimplifications turn a nearly intractable problem into one that can be solved with surprising accuracy using minimal mathematics. Scaling symmetry allows one to estimate the envelope of the traveling wave from BM transfer functions by remapping frequency into location (see Zweig et al. 1976; Zweig 1991 and Appendix A) and reinterpreting the BM frequency response measured at one location ($x_0$) as the BM spatial response (traveling wave) measured at one frequency, the CF of the recording site. CF is mapped onto location x by assuming an exponential position-frequency map [$\mathrm{CF}(\mathit{x})=\mathrm{CF}(0)e^{-x/l}$]. The location corresponding to frequency f is therefore

$$\begin{array}{c}\hfill x(f)=x_0+lln[\mathrm{CF}(x_0)/f],\end{array}$$ 4

where $x_0$ is the location of the recording site and $l=1.7$ is the space constant of the mouse cochlear map (Müller et al. 2005). Locations x can be mapped into corresponding CFs using the same position-frequency map. Note that when the cochlear map is exponential, the frequency axis f can be mapped directly into a CF axis without knowledge of the space constant of the map:

$$\begin{array}{c}\hfill \mathrm{CF}(x)={\mathrm{CF}}^2({\mathit{x}}_0)/f.\end{array}$$ 5

When frequencies are expressed in octaves relative to the CF at the recording site (so that $\nu (f)={log}_2[f/\mathrm{CF}({\mathit{x}}_0)]$), one has

$$\begin{array}{c}\hfill {\nu }_{\mathrm{CF}}(x)=-\nu (f),\end{array}$$ 6

where ${\nu }_{\mathrm{CF}}(x)$ denotes the CF at the location x in octaves re $\mathrm{CF}(x_0)$. The three abscissae of Fig. 2B illustrate the results of converting frequency into location using a typical BM transfer function measured in the mouse (Dewey et al. 2019).

Fig. 2.

(A) BM transfer-function magnitudes (i.e., BM displacement normalized by ear-canal pressure) measured using optical coherence tomography (OCT) in the apex of a mouse cochlea (9.2 kHz region) at various stimulus levels (10 dB steps). The solid lines present transfer functions that have been smoothed using a 3-point zero-phase moving average filter; the dotted lines represent raw recordings. (B) Slopes of the smoothed BM transfer-function magnitudes of panel A, that, in our framework, can be equivalently expressed in dB/Octave (left y-axis) and in dB/mm (right y-axis). Using local scaling, the frequency dependence of the BM transfer function measured at one location can be used to approximate the spatial dependence of the response to a CF tone. The two additional abscissae in panel B show the results of the frequency conversion, expressed as distance from the CF place (mm) and the corresponding CF (kHz). The dotted vertical line marks the CF of the measurement location, and the double-headed arrow indicates the frequency where the 40 dB BM transfer-function slope changes most with level (8.3 kHz). (C) Suppression tuning curve for a 40 dB probe tone, using a 3-dB suppression criterion, averaged over 6 animals. The abscissa is expressed in octaves re CF. The vertical dotted line marks the predicted frequency of maximum suppression, as estimated from the spatial location where the BM transfer-function slope experiences its largest change with level. (D) Suppressor-evoked BM displacement at the suppression threshold level of panel B. Data in C, D from Dewey et al. (2019). The BM transfer functions in panel A have been collected using the methods detailed in Dewey et al. (2019) but with an higher frequency resolution (200 Hz).

Estimating Spatial Patterns of Nonlinear Amplification

Representing frequencies in octaves relative to CF (i.e., using $\nu$) is particularly advantageous for our analysis. Not only are frequencies then mapped into CFs through a simple change of sign [Eq. (6)], but, because the exponential location-frequency map is linear when frequencies are expressed in octaves [Eq. (4)], we have

$$\begin{array}{c}\hfill g(x)\propto \frac{{\mathrm{d}|\mathit{T}|}_{\mathrm{dB}}}{\mathrm{d}\nu },\end{array}$$ 7

as demonstrated in Appendix A. That is, the spatial gain of the traveling wave can be estimated as the frequency-derivative of the BM magnitude response.

Figure 2B gives the magnitude slopes of the BM transfer functions shown in Fig. 2A. To further elucidate the proportionality between slopes in metric and log-frequency units, the left and right axis denote the slope values in dB/octaves and dB/mm, respectively.

At low and moderate sound levels, the transfer-function slope peaks at a frequency near 7.8 kHz. Reinterpreting these curves in spatial coordinates implies that the spatial amplification of a CF (9.2 kHz) tone is largest in a region located about 320 mm basal to the wave peak (or, equivalently, near the region of the cochlea tuned to 10.8 kHz). In practice, the shapes of measured transfer functions are not determined solely by spatial wave amplification. For example, their shapes can be modified by middle-ear resonances or by intracochlear wave interference (Shera and Cooper 2013); both are effects that break local scaling and are ignored in this simplified framework. Hence, the resulting estimates of the locus of maximal amplification are only approximate.

Figure 2B shows that the transfer-function slope decreases at stimulus levels above 30 dB SPL, with the largest decrease occurring at frequencies near 8.3 kHz. Interpreted spatially, the data imply that at moderate levels the BM compression observed at the best place results from a reduction in the spatial gain function, g(x), at locations basal to the peak, with the region tuned to 10.2 kHz playing the leading role. If we use iso-level suppressors and assume that cochlear nonlinearity is controlled primarily by the local amplitude of the wave, maximal suppression of BM responses to a CF tone of level 30–50 dB SPL should occur when using suppressors that peak in the 10.2 kHz region (or, equivalently, 0.14 octaves above the CF of the recording site). This simple reasoning predicts that two-tone suppression experiments will find that suppressors at frequencies about 0.14 octaves above CF produce the greatest suppression of a CF probe. Note that potentially confounding linear factors—specifically, factors that do not depend upon spatial wave amplification but affect the transfer-function shape more-or-less equally at all intensities (e.g., filtering by the middle ear)—effectively cancel out in this analysis.

Figure 2C plots the 3-dB suppression tuning curve—i.e., the level of the suppressor tone that reduces the response magnitude at the probe frequency by 3 dB—for a CF-probe presented at 40 dB SPL [obtained as an average from six mice and adapted from Dewey et al. (2019)]. The measured tuning curve has a minimum for suppressors placed at about 0.15 octaves above CF, in excellent agreement with the above prediction.

Figure 2D shows the suppressor-evoked BM displacement at the 3-dB suppression threshold as a function of suppressor frequency. For suppressor frequencies below CF, the suppressor-evoked BM displacement remains roughly constant, providing a meaningful predictor of the constant suppression it produces, as anticipated and described in previous literature (Cooper 1996; Versteegh and van der Heijden 2013; Dewey et al. 2019). For suppressor frequencies above CF, however, the suppressor-evoked displacement falls rapidly, and suppression can occur in the absence of any measurable response to the suppressor at the recording site. This phantom suppression is well captured by the traveling-wave framework, in that suppression of the probe wave occurs basal to the probe’s best place (see also Geisler et al. 1990; Pang and Guinan 1997; Versteegh and van der Heijden 2013; Dewey et al. 2019; Charaziak et al. (2020).

Suppression of Broadband Stimuli

A recent study of suppression in the base of the gerbil cochlea (Charaziak et al. 2020) found that responses to acoustic clicks are maximally suppressed by tones whose frequency is higher than the CF of the recording site (tone-on-click suppression). In measurements of click-on-click suppression, the same study demonstrated that the BM response to a probe click is maximally suppressed by a suppressor click presented earlier in time. Of particular interest was the interclick time interval producing maximal suppression, which was shown to closely approximate the difference in group delay between the click response measured at the recording site and that measured at the place of maximal suppression, as estimated from the tone-on-click suppression curves. The authors argue that the nonlinear temporal dynamics of BM click responses appear dominated not by the kinetics of local micromechanical nonlinearities but by the spatial dependence of traveling-wave propagation and amplification.

Measurements of tone-on-click and click-on-click suppression have now been performed in the apex of the mouse cochlea using OCT. Figures 3A, B plot the resulting BM transfer-function magnitudes and corresponding slopes (in dB/octave). The arrow shows the frequency where the slope changes most with level (0.15 octaves below CF). The simple reasoning presented in Sec. 2.4 predicts that the BM click response should be maximally suppressed by a tone with a frequency of about 0.15 octaves above CF. As a test of this prediction, Fig. 3C shows the suppression of the BM click response measured in the same mouse using iso-level (60 dB SPL) tones of frequency indicated along the abscissa. The suppression function peaks at 0.12 octaves above CF, close to our prediction.

Fig. 3.

(A) BM transfer-function magnitudes measured in the apex of the mouse cochlea using tones of various levels (10 dB steps, CF=9.8kHz). The solid lines present transfer functions that have been smoothed using a 2-point zero-phase moving average filter; the dotted lines represent unprocessed data. (B) Slopes of the smoothed transfer-function magnitudes shown in panel A. The arrow indicates the frequency (in octaves) of maximal BM gain change with sound level. (C) Suppression (in dB rms) of the click response produced by a moderate-level tone (60 dB SPL) of varying frequency, measured in the same mouse. The vertical dotted line marks the predicted frequency of maximal suppression, as estimated from the slopes of the BM transfer-function magnitudes (see text and Fig. 2).

SUPPRESSION IN A COCHLEAR MODEL

We now apply the framework to the predictions of a simple 3-D model of the mouse cochlea. The model, detailed in Appendix B and based on the work of Zweig (2015) and Altoè and Shera (2020a, 2020b), consists of two fluid-filled chambers separated by the cochlear partition. For simplicity, the motion of the BM is assumed proportional to the motion of the center of mass of the partition. The effects of fast compressional waves, fluid viscosity, and internal reflections are assumed small and are therefore neglected. The motion of the partition is driven both by the pressure difference across it and by a nonlinear active pressure representing the piezoelectric action of the OHCs. First deduced by applying inverse methods to measured BM response data (Zweig 2015), the general form of this active pressure follows from mathematical constraints imposed by the approximate intensity-invariance of BM phase responses (Altoè and Shera 2020b). To compute nonlinear responses in the frequency domain, we neglect harmonic and intermodulation distortions, which are at least one order of magnitude smaller than the responses at the fundamental, and adopt the quasilinear perturbative method detailed in Altoè and Shera (2020b) and Appendix B (cf. Kanis and de Boer 1993, 1994). As a consequence of the perturbative approach, the reliability of our two-tone suppression calculations decreases rapidly at higher stimulus levels, and we therefore focus on model predictions obtained at moderate sound levels.

Figure 4A compares the model’s BM transfer functions magnitude with experimental data from Dewey et al. (2019). Although the model includes only a handful of adjustable parameters—the damping of the partition, the strength of the active force, and a constant that controls the activation of the nonlinearity—the model responses nicely capture both the main trends seen in the data and their dependence on level. Figure 4B shows slopes of the BM transfer-function magnitudes, which are similar to those calculated from the experimental data in Fig. 2B. The frequency of largest slope change above 30 dB occurs about 0.1 octaves below CF. Interpreted in spatial coordinates, the largest decrease in spatial gain happens in the region tuned to 0.1 octaves above CF. Interpreting model responses using the traveling-wave framework therefore predicts maximal suppression at frequencies about 0.1 octave above CF. Figure 4C shows both that the 3-dB suppression tuning curve for a 30 dB probe tone at CF qualitatively resembles the experimental data and that the frequency of maximal suppression (0.08 octaves above CF) matches well that estimated from Fig. 4B (0.1 octaves). Finally, Fig. 4D shows the suppressor-evoked displacement at the recording site, evaluated at the 3-dB suppression criterion. As in the experimental data (Fig. 2; see also Cooper 1996; Versteegh and van der Heijden 2013), suppressor-evoked BM displacement constitutes a good predictor of the amount of suppression by below-CF suppressors. Note that the compressive nonlinearity in the model depends on the amplitude of the active pressure, rather than on BM displacement; nonetheless, altering the model so that the nonlinearity depends on BM displacement leads to predictions similar to those in Fig. 4F. Although above-CF suppressors evoke BM vibrations at the recording site that can be orders of magnitude smaller than those evoked by lower frequency tones, they nevertheless greatly suppress BM responses to CF tones. Note that the model captures the experimental data only qualitatively. In particular, the model predicts excessive suppressor-evoked BM displacement near the frequency of maximal suppression, which in turn is slightly lower than in the experimental data. In part, these discrepancies reflect the simplicity of our model; increasing the number of parameters would allow the model to capture finer details of the measured BM response.

Fig. 4.

(A) BM transfer functions magnitude in a simple 3-D mouse model (see Appendix B) computed for tones of various levels (10 dB steps). For comparison, the open symbols show measured BM transfer functions from Dewey et al. (2019). (B) Slopes of the BM transfer-function magnitudes in dB/octave. The arrow indicates the frequency where the slope changes most with stimulus level above 30 dB (0.1 octaves below CF). (C) Two-tone suppression tuning curve computed for a CF probe tone (30 dB SPL) using a 3-dB suppression criterion. The vertical dashed line in panel C marks the predicted maximal suppression frequency (0.1 octaves above CF), based on the slope analysis in panel C (see text). (D) Suppressor-evoked BM displacement at the recording site for the iso-suppression contours in panel C.

According to the model, suppression at frequencies below CF appears to mirror the amplitude of the incoming suppressor wave at the measurement site. At these frequencies, pressure propagates approximately as a plane wave (Shera and Zweig 1991; Altoè and Shera 2020a), producing a nearly uniform excitation—and subsequent spatial gain reduction—in the region of active amplification. Furthermore, at frequencies far below CF, the mechanics of the partition are stiffness dominated, so that BM displacement is approximately proportional to the driving pressure. Hence, as discussed by Versteegh and van der Heijden (2013), although low- and high-side suppression arise from the same mechanism—excitation-induced reduction of spatial gain—it appears phenomenologically as if low-side suppression of BM gain is controlled by local BM vibration. A concise traveling-wave based explanation of the differences between low- and high-side suppression, along with numerical simulations, is presented in Appendix C.

At frequencies closer to CF, where the envelope of the traveling wave varies rapidly and the activation of cochlear nonlinearity is spatially non-uniform, there may be no unique way to determine from these limited BM data which quantity controls the nonlinearity at any location. Multiple different scenarios, in combination with possible systematic variations with location, may yield similar results. Recent studies suggest that the activity of the OHCs is locally regulated by a quantity with tuning similar to BM displacement (Dewey et al. 2019; Vavakou et al. 2019), but how the regulation of OHC activity correlates with the gain of the cochlear amplifier is hard to deduce without a detailed micromechanical model. Regardless of these details, both the model and the experimental data show that the relationship between two-tone suppression data and the slope of the BM transfer function is well captured by a simple mathematical depiction of the traveling wave.

DISCUSSION

The Importance of the Traveling Wave for Interpreting Cochlear Responses

Theoretical analyses of mechanical data (Neely 1980; Zweig 1991; de Boer and Nuttall 1999), the existence of otoacoustic emissions (Kemp 1978), and direct measurements of intracochlear pressure (Dong and Olson 2013) all indicate that the cochlear amplifier acts by simultaneously amplifying both the motion of the BM and the slow-traveling, transpartition pressure wave that drives it. Experiments interfering with normal OHC function (Cody 1992; Fisher et al. 2012) established that the boost provided to BM vibrations occurs over an extended region spanning more than 100 rows of OHCs ($\sim$1 mm) basal to the peak of the response. The natural physical interpretation is that cochlear amplification occurs through the spatially distributed, collective action of the OHCs. Distributed OHC forces, appropriately phased and regulated by longitudinal coupling through the fluids and other mechanisms within the partition, move the center of mass of the organ of Corti in ways still not fully understood. The result is a form of coherent amplification that boosts the wave of driving pressure and thereby the BM gain at the recording site (see Shera 2007; Altoè and Shera 2020b).

Although coupling among distributed cellular elements plays an essential role in shaping the response, attempts to understand the nonlinear dynamics of the cochlea often focus almost exclusively on the responses of uncoupled systems, such as isolated nonlinear oscillators (see e.g., Hudspeth et al. 2010). Although this approach can be mathematically justified when studying cochlear responses to pure tones in weakly nonlinear regime (i.e. near-threshold sound levels), the approach can be misleading when the responses involve the nonlinear interactions of multiple frequency components. In particular, the isolated-oscillator/filter-bank approach misses the important role of active, nonlinear cochlear mechanics in fueling the traveling wave along its journey to its tonotopic place (see also Moleti and Sisto 2020). As our results emphasize, mechanisms of spatially distributed nonlinear wave amplification—while not critical to representing cochlear responses to a single tone using a filter—appear fundamental for understanding how multiple sound components interact and mutually suppress one another along the cochlear spiral. In contrast to previous modeling studies arguing that the characteristics of two-tone suppression are chiefly dependent on micromechanical details (Stoop and Kern 2004; Meaud and Grosh 2014; Zhou and Nam 2019), our work demonstrates that a simple model of nonlinear wave amplification can account for the experimental observations, independent of detailed micromechanical implementations.

Although the traveling-wave framework described here may miss contributions from micromechanical nonlinearities local to the recording site, it nevertheless captures something rather more important: the essentially distributed nature of cochlear amplification. Unexpectedly, given the simplicity of the assumptions, the framework accurately predicts the frequencies of maximum suppression from measured BM transfer functions. But perhaps this should not be surprising. Although Eq. (1) constitutes perhaps the simplest mathematical representation of a wave traveling in a nonuniform medium, the representation is identical in form to (i) the long- and (ii) the short-wave approximations to the pressure wave in (i) the low-frequency tail and (ii) the peak region of 3-D tapered cochlear models (see Altoè and Shera 2020a; Altoè and Shera 2020b), respectively. These are precisely the regions where the two respective approximations are expected to work well. Despite its simplicity, the proposed framework therefore provides an essentially accurate representation of the traveling wave.

Traveling-Wave Suppression and Noninvasive Estimates of Cochlear Tuning

Early psychophysical studies that aimed to characterize the frequency selectivity of the auditory system often employed concepts from linear-systems (aka filter) theory (e.g., Fletcher 1940; Scharf 1959; Green 1960; Zwicker 1961). Over time, similarities between BM and behavioral frequency selectivity encouraged the notion that the psychophysically defined “auditory filters” arise predominantly through the filtering action of the cochlea (see, e.g., Moore 1986). A correlate of this view is that cochlear frequency resolution can be deduced from psychophysical experiments, fundamentally by combining linear-systems and detection theory (see, e.g., Patterson 1976).

Changing perspective from a filter-bank to a waveguide view of cochlear mechanics, however, bears important consequences for interpreting behavioral estimates of cochlear frequency selectivity. Consider, for example, applications of the power-spectrum model of masking to obtain behavioral estimates of the sharpness of cochlear tuning (i.e., of the bandwidths of the cochlear filters). A widely used masking paradigm employs simultaneous notched noise (see Patterson 1976; Glasberg and Moore 1990; Heinz et al. 2002; Oxenham and Shera 2003; Lopez-Poveda and Eustaquio-Martin 2013 for reviews and discussion). In brief, a tonal stimulus is presented simultaneously with band-reject filtered noise, with the tone frequency falling within the “notch” of the filter. The frequency response of the corresponding “cochlear filter” (i.e., the tuning at the cochlear location tuned to the tone frequency) is then estimated by measuring the psychophysical detection threshold for the tone while varying the parameters of the stimulus (e.g., the tone level, the notch bandwidth, and its placement with respect to the tone). A key assumption, recently questioned (Maxwell et al. 2020), is that detection of the tone depends on the tone-to-noise ratio at the “output” of the corresponding cochlear filter (Patterson 1976; Glasberg and Moore 1990), which is assumed to operate linearly during the experiment (see also Patterson 1986). To determine the shapes and bandwidths of the cochlear filters, the data so obtained are fit using a model derived from the basic assumption that the noise-induced masking of a tone depends on the noise power not removed by the cochlear filter tuned to the tone frequency.

The existence of phantom suppression, a prominent feature in both the data and the model, implies that there can be strong “off-frequency masking” by noise components that suppress (mask) the response to the tone but do not appreciably excite the corresponding cochlear filter. The existence of such off-frequency masking components violates the fundamental assumptions of the power-spectrum model of masking. Our model simulations demonstrate that the signal level at the probe location depends strongly on the bandwidth of the masking noise, even for modest activation of the cochlear nonlinearity (see Fig. 5). An obvious corollary is that the use of simultaneous masking can bias the measured filter bandwidths, making them appear broader than they really are. Indeed, Delgutte (1990) demonstrated this “broadening” side-effect of suppression in the cat auditory nerve. Note that there are known differences between BM and AN suppression when using well-below CF suppressors, whose origin is not fully understood at the present time (Cooper 1996; Nam and Guinan 2018). This unresolved issue bears no consequences for this discussion, as suppression is similar in AN and BM recordings when using near and above-CF suppressors (Cooper 1996).

Fig. 5.

Suppression produced by notched masking noise on the pressure response of a CF tone. In the spirit of the EQ-NL theorem (de Boer 1997), the effect of the noise was simulated using a quasi-linear approach. In particular, we reduced the parameter [$\tau$ in Eq. (20)] that controls active amplification in the model by 20% at all locations except those with CFs within the notch. Pressure amplification in the model occurs entirely basal to the CF place and the observed suppression therefore depends entirely on the noise above CF. These simulations show that even modest noise-induced activation of the compressive nonlinearity over a spatially distributed region can significantly reduce the pressure driving a given cochlear filter. Although these simulations elucidate the suppressive effects of noise only qualitatively—and conservatively, since the possible effects of tone and noise interaction at the micromechanical level have been neglected—the results illustrate that in notched-noise experiments the signal level at any given cochlear location (filter) depends systematically on the bandwidth of the noise. Thus, contrary to the assumptions of the power-spectrum model of masking, the signal-to-noise ratio at the output of the filter depends on much more than simply the amount of noise that passes through the filter.

Behavioral estimates of human cochlear tuning obtained using simultaneous masking are broader than those obtained with forward masking, where suppression plays no role (e.g., Moore 1978; Moore et al. 1984; Oxenham and Shera 2003). Whether methods that employ simultaneous masking make human cochlear filters appear broader than they really are, or whether forward masking methods make them appear unnaturally sharp, has been the subject of decades-long controversies (e.g., Ruggero and Temchin 2005; Shera et al. 2010). However, comparisons of behavioral and neural recordings in laboratory animals have largely demonstrated that modern forward-masking methods estimate cochlear tuning rather accurately (Joris et al. 2011; Sumner et al. 2018). In a nutshell, our framework both corroborates previous conclusions thatSuppression tuning characteristics of th filter bandwidths estimated using simultaneous masking are biased by suppression (Delgutte 1990; Heinz et al. 2002) and identifies the physical mechanism involved. This mechanism is perhaps the least controversial aspect of cochlear mechanics: the traveling wave (von Békésy 1960).

Similar considerations apply to all suppression-based methods for estimating the sharpness of cochlear tuning, including the measurement of otoacoustic-emission (OAE) suppression tuning curves (e.g., Kemp and Chum 1980a; Kummer et al. 1995; Abdala et al. 1996; Lineton and Wildgoose 2009; Gorga et al. 2011; Charaziak et al. 2013; Manley and van Dijk 2016; Rasetshwane et al. 2019). In principle, these procedures more closely assess the frequency dependence of suppression than they do the frequency dependence of excitation, and the two are by no means identical. Indeed, both stimulus-frequency (SFOAEs) and distortion-product otoacoustic emissions (DPOAEs) are most readily suppressed by tones of frequency higher than the stimulus [i.e., higher than the probe tone for SFOAEs (Lineton and Wildgoose 2009) and higher than $f_2$ for DPOAEs at $2f_1-f_2$ (Gorga et al. 2011)]. These effects presumably arise via mechanisms of spatial wave suppression analogous to those elucidated in this study. Relating patterns of OAE suppression to cochlear tuning is complicated further: For DPOAEs, by the multiple stimulus tones, source types, and cochlear locations involved in their generation (Kalluri and Shera 2001); and for SFOAEs by the fact that the suppressor not only reduces traveling-wave amplification, as intended, but also modifies the spatial distribution of impedance irregularities responsible for cochlear wave scattering (Talmadge et al. 2000; Shera et al. 2004b; Liu and Liu 2016; Moleti and Sisto 2016; Vencovskỳ et al. 2020).

Critique of the Cochlear Filter Bank

In contrast to the simple “compressive” filter commonly employed to depict cochlear signal processing (e.g., Moore et al. 1984; Dau et al. 1996; Rahman et al. 2020), state-of-the-art filter-bank models of the cochlea cannot be faulted for failing to simulate phenomena attributed here to nonlinear wave interactions (Lopez-Poveda 2005). For example, a computational filter-bank model tailored to reproduce two-tone suppression data (Heinz et al. 2002) correctly predicts the detrimental effect of mechanical suppression on estimates of cochlear tuning obtained using simultaneous masking. A noteworthy feature of the model is that the gain in any channel is not regulated directly by the response of the filter in question, but via a second, more broadly tuned filter. Indeed, zero-dimensional models that aim to reproduce mechanical and auditory-nerve data at the output of each “cochlear filter” generally employ various and sundry interconnected combinations of narrowly and broadly tuned linear and nonlinear resonators (e.g., Meddis et al. 2001; Zilany and Bruce 2006; see Lopez-Poveda 2005 for a comprehensive overview). Although they add significant complexity and often lack clear physical correspondence, the inclusion of these additional filters evidently helps compensate for the crucial spatial build-up of amplification that, although present in the cochlea, is necessarily precluded by the structure of zero-dimensional models.

Arguments for adopting a higher dimensional, wave-based framework for cochlear signal processing thus rest not on any promised improvement in reproducing the data—all useful models are necessarily imperfect (Borges 1998)—but on the framework’s fidelity to cochlear mechanics and on the relative simplicity of its conceptual foundation and construction. The presumption here is that simpler models well grounded in underlying mechanisms ultimately prove more successful at unifying disparate observations, generating fruitful hypotheses, and generalizing to the unforeseen. As Versteegh and van der Heijden (2013) noted: “Many aspects of suppression on the BM [...] appear at a first glance to be a set of disparate phenomena. Yet they become logically connected once viewed in the framework of propagating waves and the spatial buildup of suppressive effects along the propagation direction.”

Viewed from the wave perspective, the inner workings of complex filter-bank models seem reminiscent of the deferents, equants, and epicycles that abound in Ptolemaic models of the solar system (Kuhn 1957). Just as there are no three-body problems in the Ptolemaic universe, filter-bank models appear artfully fine-tuned to represent the data with accuracy and computational ease. But they build upon an unphysical and highly localized (zero-dimensional) foundation that creates unnecessary complication. Phenomena that emerge spontaneously via the traveling wave while requiring ad-hoc processes in the filter model range from the asymmetric filter shape itself (Wegel and Lane 1924; Peterson and Bogert 1950; von Békésy 1960) to the complex temporal dynamics observed in responses to consecutive acoustic clicks (Goblick and Pfeiffer 1969; Kemp and Chum 1980b; Charaziak et al. 2020). Just as all shadows whisper of the sun, very little in cochlear processing makes sense except in light of the traveling wave.

To conclude, we have presented perhaps the simplest representation of cochlear mechanical data [Eq. (1) and Fig. 1D] that explicitly accounts for the traveling wave. This framework, whose application requires no complex computational or mathematical tools, can be employed to assess the impact of spatially-distributed nonlinear wave amplification on cochlear signal processing.

APPENDIX A: APPROXIMATION OF SPATIAL GAIN FROM MEASURED BM TRANSFER FUNCTIONS

The assumption of local scaling symmetry means that BM transfer functions are identical near their peaks when the frequency axis is normalized to the local characteristic frequency (CF), that is, when frequencies are expressed as a function of the normalized variable $\beta =f/\mathrm{CF}(\mathit{x})$. In other words,

$$\begin{array}{c}\hfill T(x_1,f_1)=T(x_0,f_0)\text{(peak region)}\end{array}$$ 8

when

$$\begin{array}{cc}\hfill \beta (x_1,f_1)& =f_1/\mathrm{CF}({\mathit{x}}_1)\hfill \end{array}$$ 9

$$\begin{array}{cc}& =f_0/\mathrm{CF}({\mathit{x}}_0)=\beta ({\mathit{x}}_0,f_0).\hfill \end{array}$$ 10

Scaling enables one to estimate the cochlear response to a specific frequency at any location by knowing (i) the BM transfer function at one location and (ii) the cochlear position-frequency map. Assuming that the tonotopic map is approximately exponential [$\mathrm{CF}(\mathit{x})=\mathrm{CF}(0)e^{-x/l}$, with l species dependent space constant] leads to

$$\begin{array}{c}\hfill g(x)\propto \frac{{\partial |T|}_{\mathrm{dB}}}{\partial x}=\frac{\beta }{l}\frac{{\partial |T|}_{\mathrm{dB}}}{\partial \beta }=\frac{1}{l}\frac{{\partial |T|}_{\mathrm{dB}}}{\partial (ln\beta )}\propto \frac{{\partial |T|}_{\mathrm{dB}}}{\partial \nu },\end{array}$$ 11

where $\nu ={log}_2\beta$ is the normalized frequency expressed in octaves.

APPENDIX B: 3-D MODEL OF THE MOUSE COCHLEA

The model of the mouse cochlea used in this paper is based upon the work of Zweig (2015) and Altoè and Shera (2020a, 2020b). The model incorporates a highly simplified description of the cochlear 3-D hydrodynamics in conjunction with a scaling symmetric admittance of the cochlear partition (CP, the BM and organ of Corti). In this appendix, we present an alternate derivation of the WKB solution presented in Altoè and Shera (2020a), obtained using a different set of simplifying assumptions and geometry in order to highlight the generality of the model’s solution character.

Figure 6A, B presents the longitudinal and cross-sectional view of the model. The cochlear duct and the CP are assumed, for simplicity, of circular cross section of radii h and d respectively. The effective, acoustic width of the BM is $b=2d$. For simplicity, the CP is assumed centered with respect to the scalae. Because we focus on the slow-propagating wave component—that represents the pressure difference across the cochlear partition and the motion of the CP center of mass—we neglect effects of possible CP deformations. That is, we consider the mode of vibration consisting of the CP moving up and down “en bloc”. A general solution for this model is given by de Boer (1980) in terms of Bessel functions.

Fig. 6.

A) Longitudinal and B) cross-sectional geometry of the mouse model. (C) Acoustic area of the scalae of the mouse cochlea, extrapolated from Fig. 5 of Burda et al. (1988). The values are peak-normalized. The solid line represents the data, while the dotted line represents a curve with slope $e^{-x/2l}$, where x is the distance from the cochlear base and l the space constant of the mouse frequency map. (D) Radius of the scalae, extrapolated by fitting a circle to the total cross-sectional area of the scalae (from Fig. 5 of Burda et al. 1988). The dotted line represents a linear fitting function $h=0.13(1-x/L)+0.12$, with L the length of the cochlea (assumed here of 5.2 mm) used to compute the model’s solution. The fitting functions in panels C and D are used to compute the model’s solution [Eqs. (18,19)]

In the mouse model, the cochlea is assumed 5.2 mm long, with an exponential location-frequency map

$$\begin{array}{c}\hfill \mathrm{CF}(\mathit{x})=\mathrm{CF}(0)e^{-\mathit{x}/l},\end{array}$$ 12

with space constant $l=1.7$ mm and highest CF [$\mathrm{CF}(0)$] of 78 kHz, based upon Müller et al. (2005).

Assuming harmonic time dependence, the relationship between pressure difference across the organ of Corti ($P_d$) and BM velocity ($V_{\mathrm{BM}}$) is expressed through an effective acoustic admittance

$$\begin{array}{c}\hfill Y_{\mathrm{CP}}(\omega ,x)=V_{\mathrm{BM}}(\omega ,x)/P_d(\omega ,x),\end{array}$$ 13

which is a function of location x and angular frequency $\omega$. Since the cochlea is nonlinear, $Y_{\mathrm{CP}}$ represents a sinusoidal describing function. For notational simplicity, dependencies on frequency and location are left implicit in what follows.

The longitudinal fluid flow in the cochlea can be conveniently described by standard transmission-line theory (Shera et al. 2005). This requires introducing the variable $\overline{P}$ representing the pressure difference across the scalae averaged over their cross-sectional area (Duifhuis 1988). The function $\alpha =P_d/\overline{P}$ quantifies the ratio between driving and average pressure. With this stratagem, Newton’s second law and mass conservation imply that

$$\begin{array}{c}\hfill \frac{\text{d}\overline{P}}{\text{d}x}=-i\omega \frac{\rho }{S(x)}U(x),\end{array}$$ 14

$$\begin{array}{cccc}\hfill \text{and}& & \hfill \frac{\text{d}\mathit{U}}{\text{d}x}=& -b(x)V_{\text{BM}}(\mathit{x})\hfill \\ \hfill & & \hfill =& -\alpha (x)b(x)Y_{\text{CP}}(x)\overline{P}(x),\hfill \end{array}$$ 15

respectively. In these equations U represents the (longitudinal) volume velocity, $\rho$ the density of the fluids, b the width of the BM, and S the acoustic area of the scalae [$S=S_{\mathrm{st}}{S}_{\mathrm{sv}}/(S_{\mathrm{st}}+S_{\mathrm{sv}}$)]. Combining Eqs. (14-15) leads to the Webster horn equation familiar from acoustics (Peterson and Bogert 1950)

$$\begin{array}{c}\hfill \frac{1}{S}\frac{\text{d}}{\text{d}x}\left(S\frac{\text{d}\overline{P}}{\text{d}x}\right)+k^2\overline{P}=0,\end{array}$$ 16

where k, the complex wavenumber, is (Duifhuis 1988)

$$\begin{array}{c}\hfill k=\sqrt{-\alpha \frac{i\omega \rho b}{S}{Y}_{\text{CP}}}.\end{array}$$ 17

The WKB solution of Eq. (16) is (Altoè and Shera 2020a)

$$\begin{array}{c}\hfill P_d(x,\omega )=\alpha (x,\omega )P_0\sqrt{\frac{S(0)}{S(x)}}\sqrt{\frac{k(0,\omega )}{k(x,\omega )}}{e}^{-i{\int }_0^{x}k(x^{\prime },\omega )\text{d}{\mathit{x}}^{\prime }},\end{array}$$ 18

where $P_0$ is the pressure at the cochlear entrance. Due to the radial symmetry of the model, the 3-D pressure field [P(x, y, z)] is conveniently described in cylindrical coordinates as a function of location x and radial position [distance $r^{\prime }$ from the CP center and angle $\varphi$ : $P(x,r^{\prime },\varphi )$]. When the CP is small enough, gradients of the pressure field along $\varphi$ can be regarded to play a secondary role for the model’s solution (see also de Boer 1980, 1981), and hence the 3-D field is, for simplicity, approximated by a 2-D field [$P(x,r^{\prime },\varphi )\approx P(x,r^{\prime })$]. With these simplification we can adopt the 2-D approximation for $\alpha$ given by Duifhuis (1988)

$$\begin{array}{c}\hfill \alpha \approx \frac{\mathit{kh}}{tanh(kh)}.\end{array}$$ 19

The model solution depends on how S and h vary along the cochlea; their values for the mouse cochlea are obtained by fitting simple functions to published morphometric data (Burda et al. 1988) in Fig. 6C, D. Although the pressure field is nearly 2-D, the model effectively has three spatial dimensions because the BM width is shorter than the scalae radius [Eq. 15]. The narrow BM produces large differences between 2- and 3-D models (de Boer 1981; Zweig 2015). Alternative and more general solutions to 3-D models fundamentally rely on describing the pressure field as a superposition of distinct modes that satisfy the boundary condition at the wall separating the two scalae (see, e.g., Steele and Taber 1979; Lighthill 1981; de Boer 1981).

The mechanical admittance of the CP is taken to be (Zweig 2015; Altoè and Shera 2020b)

$$\begin{array}{c}\hfill Y_{\mathrm{CP}}=i\beta \frac{1+i\tau \beta }{m(1+2i\zeta \beta -{\beta }^2)},\end{array}$$ 20

where $\beta =f/\mathrm{CF}$ is the ratio between frequency and characteristic frequency ($\mathrm{CF}$) at the specific location, m is the acoustic mass of the CP, $\zeta$ is the damping coefficient, and $\tau$ is a level-dependent parameter that controls the magnitude of the active term, $i\tau \beta$, which represents an active pressure proportional to the time-derivative of the driving pressure. The value of $\tau$ at each cochlear location is determined by

$$\begin{array}{c}\hfill \tau ={\tau }_0[1-tanh(\frac{|\beta P|}{P_{\mathrm{sat}}})],\end{array}$$ 21

where ${\tau }_0$ is the coefficient of the active force term at the lowest sound levels, and $P_{\mathrm{sat}}$ a constant, independent of location, that controls the level-dependence of $\tau$. For simulations using two tones of normalized frequencies ${\beta }_1$ and ${\beta }_2$, eliciting pressures $P_1$ and $P_2$ respectively, the value of $\tau$ is calculated at each location as

$$\begin{array}{c}\hfill \tau ={\tau }_0[1-tanh(\frac{{\beta }_1|P_1|+{\beta }_2|P_2|}{P_{\mathrm{sat}}})].\end{array}$$ 22

The model solution is determined iteratively starting from a linear solution consisting of the linear superposition of the responses to the two tones separately. The value of $\tau$ at each cochlear location is then computed and used to update the values of k and $\alpha$. This procedure is iterated 10 times; within each step of the solution, the values of k and $\alpha$ are determined by iterating Eqs. (17) and (19) (also 10 times).

APPENDIX C: LOW- AND HIGH-SIDE SUPPRESSION

This appendix elucidates the differences between high-side and low-side suppression using the simple traveling-wave framework. The explanation is conceptually equivalent to that given by Versteegh and van der Heijden (2013). Cooper (1996) and Versteegh and van der Heijden (2013) noted that both above-CF and below-CF suppressors can greatly suppress the BM response to a near-CF tone. However, the total BM response (i.e., the superposition of suppressor and probe response) increases with suppressor level when using below-CF suppressors (low-side suppression) whereas it decreases when using above-CF suppressors (high-side suppression). In light of the simple traveling-wave framework, the explanation for this difference is straightforward. Consider, for example, the case when the suppressor frequency is well below CF. The BM response to this low-frequency suppressor will be nearly linear because at tail frequencies the BM wave undergoes little spatial amplification. Nonetheless, this suppressor, by exciting the BM in the active region where a near-CF probe tone is spatially amplified, will activate the cochlear nonlinearity and hence attenuate the BM probe response. As a result, the probe response is attenuated, but not the suppressor response. Therefore, the overall BM response will increase with suppressor level, despite the significant suppressive effects at play.

Conversely, above-CF suppressors will attenuate the near-CF probe without eliciting a significant response on the BM (phantom suppression). As a consequence, increasing the level of above-CF suppressors will reduce the overall BM response, as this is dominated by the probe response. Figure 7A,B shows the model’s two-tone responses as a function of suppressor level, showing that the model captures the differences between low- and high-side suppression via the mechanisms of nonlinear wave amplification elucidated throughout this study. Note that accounting for differences between low- and high-side suppression is particularly straightforward using the traveling-wave framework because the activation of the cochlear nonlinearity is rather frequency-independent—it depends on local excitation level, without additional “filtering”—whereas its effects (i.e., the reduction of spatial amplification) are strongly frequency dependent (e.g., they affect the response to near-CF but not tail frequency tones). This dichotomy between the causes and the effects of cochlear nonlinearity is readily observed in the experimental data (e.g., Dong and Olson 2013; Vavakou et al. 2019; Fallah et al. 2019; Dewey et al. 2019), and naturally emerges from the long- and short-wave hydrodynamics at tail and peak frequencies, respectively (Altoè and Shera 2020b).

Fig. 7.

BM model responses to suppressor and probe tones as a function of suppressor level. The probe is a 40 dB CF-tone (9.3 kHz), while the suppressor level is indicated in the abscissa. The colored lines indicate results obtained using suppressors of different frequencies. A) Overall BM response magnitude, i.e., sum of magnitudes of suppressor and probe frequency components. B) BM magnitude response at the probe frequency. Panel B highlights that a near-CF probe response monotonically decreases with increasing suppressor level—regardless of suppressor frequency. Panel A shows that the overall excitation level of the BM either decreases or increases with suppressor level depending on whether the suppressor frequency is above or below CF, respectively.