# 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.

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## Acknowledgements

We thank Michael G. Heinz, Enrique A. Lopez-Poveda, and two anonymous reviewers for their comments and constructive criticism. Supported by NIH/NIDCD grants R01 DC003687 (CAS), K99 DC016906-01A1(KKC), F32 DC016211 and R21 DC019209 (JBD), R01 DC017741 and R01 DC014450 (JSO), and INAIL grant BRIC 2019 ID-9 (RS).

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Correspondence to Alessandro Altoè.

## Appendices

### 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 $$\beta =f/\mathrm{CF}(\textit{x})$$. In other words,

\begin{aligned} T(x_1,f_1) = T(x_0,f_0) \quad \hbox {(peak region)} \end{aligned}
(8)

when

\begin{aligned} \beta(x_1,f_1)&= f_1/\mathrm{CF}(\textit{x}_1) \end{aligned}
(9)
\begin{aligned}&=f_0/\mathrm{CF}(\textit{x}_0)=\beta (\textit{x}_0,f_0)\;. \end{aligned}
(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 [$$\mathrm{CF}(\textit{x})=\mathrm{CF}(0)e^{-x/l}$$, with l species dependent space constant] leads to

\begin{aligned} g(x)\propto {\partial |T|_{\mathrm{dB}}\over \partial x}={\beta \over l}{\partial |T|_{\mathrm{dB}}\over \partial \beta }={1\over l}{\partial |T|_{\mathrm{dB}}\over \partial {(\ln {\beta })}} \propto {\partial |T|_{\mathrm{dB}}\over \partial {\nu }}\;, \end{aligned}
(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 (2020a2020b). 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.

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

\begin{aligned} \mathrm{CF}(\textit{x})=\mathrm{CF}(0)e^{-\textit{x}/l}\;, \end{aligned}
(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{aligned} Y_{\mathrm{CP}}(\omega ,x)=V_{\mathrm{BM}}(\omega ,x) / P_d(\omega ,x)\;, \end{aligned}
(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 $$\bar{P}$$ representing the pressure difference across the scalae averaged over their cross-sectional area (Duifhuis 1988). The function $$\alpha =P_d/\bar{P}$$ quantifies the ratio between driving and average pressure. With this stratagem, Newton’s second law and mass conservation imply that

\begin{aligned}{{\text{d}}\bar{P}\over{\text{d}}x} =-i\omega {\rho \over S(x)} {U}(x), \end{aligned}
(14)
\begin{aligned} \text{and} & & {\text{d}{\textit{U}}\over\text{d}x} =&-b(x)V_{\text{BM}}(\textit{x}) \nonumber \\ & & =&-\alpha (x) b(x) Y_{\text{CP}}(x) \bar{P}(x)\;, \end{aligned}
(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{aligned} \frac{1}{S} \frac{{\text{d}}}{{\text{d}}x}\left( {\!S\,\frac{{\text{d}}\bar{P}}{{\text{d}}x}}\right) +k^2 \bar{P}= 0\;, \end{aligned}
(16)

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

\begin{aligned} k=\sqrt{-\alpha \frac{i\omega \rho b}{S}Y_{\text{CP}}}. \end{aligned}
(17)

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

\begin{aligned} P_d(x,\omega )=\alpha (x,\omega )P_0\sqrt{{S(0)}\over {S(x)}}\sqrt{\frac{k(0,\omega )}{k(x,\omega )}} e^{-i\int _0^xk(x',\omega )\text{d}\textit{x}'}, \end{aligned}
(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(xyz)] is conveniently described in cylindrical coordinates as a function of location x and radial position [distance $$r'$$ from the CP center and angle $$\phi$$ : $$P(x,r',\phi )$$]. When the CP is small enough, gradients of the pressure field along $$\phi$$ 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 [$$P(x,r',\phi )\approx P(x,r')$$]. With these simplification we can adopt the 2-D approximation for $$\alpha$$ given by Duifhuis (1988)

\begin{aligned} \alpha \approx \frac{k h}{\tanh (k h)}. \end{aligned}
(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{aligned} Y_{\mathrm {CP}}=i\beta \frac{1+i\tau \beta }{m(1+2i\zeta \beta -\beta ^2)}\;, \end{aligned}
(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{aligned} \tau =\tau _0[1-\tanh \Big ({\frac{|\beta P|}{P_{\mathrm{sat}}}}\Big )], \end{aligned}
(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{aligned} \tau =\tau _0[1-\tanh \Big ({\frac{\beta _1|P_1|+\beta _2|P_2|}{P_{\mathrm{sat}}}\Big )}]. \end{aligned}
(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).
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