The role of scalae geometry in cochlear mechanical responses: pressure focusing in two and three dimensions

Abstract

This paper illustrates effects of model dimensionality, and of common simplifying assumptions regarding the geometry of the scalae, on the solution of cochlear mechanical models. We extend a previous theoretical framework, [Duifhuis in Auditory Function: Neurological Bases for Hearing, Wiley, New York, 1988; 189–212], to study differences between models that consider 3D and 2D fluid motion in the scalae, and how these differences depend on the assumed cochlear geometry. Our results show that, while cochlear mechanical responses obtained in 2D and 3D are nearly identical over the mid- and apical cochlear turn, they are significantly different in the base—where the basilar membrane is narrow—especially in presence of active amplification. Our analysis reveals that a narrower basilar membrane intensifies the three-dimensional shortwave “pressure-focusing” effect, which boosts the vibration of the sensory tissue at locations tuned to the stimulus frequency. Importantly, these 3D short-wave effects can be accounted for in carefully constructed 2D models, by appropriately projecting the cochlear 3D geometry in 2D. Our work shows that the cochlear 3D geometry plays a major role to high-frequency cochlear amplification, a phenomenon with a straightforward explanation and that can be included in more tractable 2D theories.

I. INTRODUCTION

This article addresses long-standing issues about the impact of the cochlea geometry—and the simplifying assumption adopted in theoretical studies—on cochlear mechanical responses. In particular, we seek to understand whether a 3D treatment of cochlear mechanics is necessary to accurately explain cochlear wave propagation, or models that use a lower number of spatial dimensions (i.e., 2D models) work sufficiently well in practice. This question is often approached ideologically; for example, it is not uncommon to hear statements along the line as “the real cochlea is three-dimensional hence realistic models should be 3D”. Such argument is easily deflected. For example, the 3D geometry of an acoustic horn is very important for its functioning; yet, in most practical cases, its functioning is precisely captured by a 1D model (Webster, 1919), i.e., a model that is concerned only with air motion over one spatial coordinate. Examples of phenomena happening in the three-dimensional space that are best understood using 1D or 2D models abound in acoustics and physics textbooks. In some cases, the simplification is achieved by neglecting motions in certain directions (e.g., the vibration of a string or of a thin membrane). In other cases, choosing alternative coordinate systems—combining rather than neglecting spatial coordinates—reduces the number of spatial variables necessary to describe the problem (e.g., propagation of a spherical wave in the free field).

The classic literature has been inconclusive in determining the impact of dimensionality on cochlear mechanical theories. It has been long known that models need to be at least 2D to capture short-wave propagation near the peak of the traveling wave (Siebert, 1974), which plays an important role for cochlear function.¹ Regarding the utility of increasing model dimensionality from two to three, different authors reached different conclusions. Some classic studies (Kolston, 2000; Steele and Taber, 1979), by comparing numerical simulations of a 3D model vs. its 2D counterpart, concluded that a 3D model is necessary to explain cochlear mechanical data with reasonable accuracy. On the other hand, theoretical arguments suggest that 2D models might work almost exactly as 3D ones, as long as they are carefully constructed (e.g., Viergever, 1978a; Zweig, 2015). Because examples and hand waving are not demonstrations, the dimensionality problem remains open to this day.

The work of de Boer (1980; 1981) is probably, to this day, the most comprehensive work on cochlear models’ spatial dimensionality. Although at a superficial level it might appear that de Boer concluded that the 2D approach works poorly, an in-depth analysis of his work reveals a less clear situation, worth re-examining. Throughout his early work, de Boer defined the 2D model rather narrowly: a box model where the basilar membrane (BM) spans the entire width of the cochlea. That specific 2D model is known to work poorly in many regards (Zweig, 2015). However, de Boer proposed two alternative 2D treatments of cochlear mechanics. One employs a 3D cylindrical model (we will analyze its function in details below), that is made 2D by constraining the fluid to move only longitudinally and radially (de Boer, 1980). The second is what de Boer named the 2D “reduced” model: a 2D model including a correction factor that accounts for the BM being narrower than the duct. de Boer (1981) stated that the cylindrical model works nearly identically to a classic 3D “box” model; unfortunately, he never compared the two models’ solutions directly, and he did not test the practical accuracy of the 2D reduced model. Careful analysis of de Boer’s work hints that the 2D approach might work.

One not fully appreciated issue in tackling the dimensionality problem, is that its effect on a model’s response largely depends on the model’s details. Indeed, the previous literature is unanimous in pointing out that the difference between 1D, 2D and 3D models depends on wavelengths relative to the dimension of the cochlear duct (de Boer, 1981; Steele and Taber, 1979). Therefore, the practical importance of model dimensionality for understanding cochlear functioning requires accounting for realistic wavelengths and dimensions, both varying substantially from base to apex. Although these aspects are essential to understand cochlear function, they have been generally glossed over—especially for what concerns wave-lengths—raising further concerns that the difference between 2D and 3D models reported in the literature could have been exacerbated or downplayed, depending on the different modeling assumptions.

Here we set to elucidate the effect of dimensionality on the solution of cochlear models, and clarify the present confusion in the literature. This is achieved by analyzing in detail the effect of different assumption regarding the model’s geometry on the 3D solution, and its difference with 2D approximations. We employ a unifying framework, obtained by building upon the work of Duifhuis (1988), which allows us to fairly compare models of different dimensionality²: this approach prevents gross (and easily avoided) distortions of the traveling wave when changing number of considered spatial dimensions, as in classic work (e.g., Steele and Taber, 1979). This framework allows us to study the effect of dimensionality and geometry on the pressure-focusing factor $\alpha$ (see e.g., Sisto et al. 2021) that represents a hydrodynamic boost to BM vibration that depends on wavelength and geometry, and which has a straightforward physical meaning.

Beside purely theoretical work, we will compare the solution of a 3D active model, incorporating realistic geometry and wavelength, with its 2D counterpart, by using standard assumptions to reduce the model’s dimensionality (Altoè and Shera, 2020a). This classic 2D approximation, equivalent to de Boer’s reduced 2D model (de Boer, 1981), consists of keeping the acoustic impedances and admittances that define cochlear function identical regardless of dimensionality, while keeping the height of the duct the same in 2D and 3D. In this case, while 2D and 3D solutions qualitatively agree, they quantitatively differ in the base of the cochlea, where the scala is tall and the BM narrow. Based on the insight gained by our analysis, we propose an alternative, physically justified, method to map the 3D geometry of the cochlea into a 2D “augmented” model, so to greatly improve the agreement between 2D and 3D models.

II. A FRAMEWORK THAT WORKS IN ONE, TWO AND THREE DIMENSIONS

We employ a unifying linear framework, based on Duifhuis’ (1988) averaging method for 2D models, that we expand here to work in 1,2 and 3D. Following linearity, we assume harmonic time dependence—i.e., stimuli and responses are pure tones of frequency $f$ and angular frequency $\omega =2\pi f$.

The framework is that of an uncoiled³ cochlea [Fig. 1], where the cochlear partition (the BM and overlying organ of Corti complex) divides the cochlear duct into two fluid-filled scalae (chambers): the scala tympani (ST) on bottom and the scala vestibuli (SV) on top, with cross-sectional area $S_{\text{ST}}$ and $S_{\text{SV}}$ respectively⁴ . The flexible part of the partition—identified with the BM in classic literature⁵—has width $b$, and its mechanical properties are encapsulated by the admittance function $Y$, which relates the partition’s overall velocity $V$—the partition’s transverse (along $z$) velocity⁶ averaged over $b$—with the pressure difference across it $V=P_{0}Y$.

FIG. 1.

Side and top view of a segment of an uncoiled cochlear model.

To simplify matters, we assume that the cochlea and its fluid are incompressible, we ignore effects of “fast waves” propagating at the speed of sound in water and neglect effects of viscous boundary layer as well (Puria and Allen, 1991; Sasmal and Grosh, 2019). Under these simplifying assumptions, $\overline{P}$, the average pressure difference between SV and ST —i.e., the pressure difference averaged over the scalae cross-sectional area—obeys a variant of the Webster’s horn equation (Shera et al., 2004):

$$\frac{1}{S}\frac{d}{dx}\left(S\frac{d\overline{P}}{dx}\right)+bZY\alpha \overline{P}=0,$$ (1)

where $x$ is distance from cochlear entrance, $S$ the effective cross-sectional area of the scalae:

$$S=(S_{\text{ST}}{S}_{\text{SV}})∕(S_{\text{SV}}+S_{\text{ST}}),$$ (2)

$Z$ the acoustical impedance of the cochlear duct:

$$Z=i\omega \rho ∕S,$$ (3)

with $\rho$ density of the cochlear fluids. The variable $\alpha$ represents the pressure-focusing factor:

$$\alpha =P_0∕\overline{P},$$ (4)

which is the ratio between “driving” ($P_0$, the pressure difference across the partition) and average pressure. Eq. (1) can be written as

$$\frac{1}{S}\frac{d}{dx}\left(S\frac{d\overline{P}}{dx}\right)+k^2\overline{P}=0,$$ (5)

where $k^2=bZY\alpha$. While this equation doesn’t have a closed-form solution, good numerical solutions can be obtained either numerically or analytically using perturbation theory. We use the WKB approximation (see Frost, 2024, for a modern review) because it provides not only reasonable solutions, but it also helps to get a deep insight into the cochlear physics. In our simplifying framework, the solutions of Eq. (5) are traveling waves whose wavenumber⁷ is

$$k=\pm \sqrt{-\alpha bZY},$$ (6)

i.e., two waves that are traveling in opposite directions⁸. When the cochlea is driven by sound, the solution is largely dominated by forward waves that travel towards the apical end of the cochlea—the amplitude of reverse traveling waves due to scattering is orders of magnitude smaller than the forward wave. In this case the WKB approximation for $\overline{P}$ describes a rather simple wave

$$\overline{P}(x)\approx P_{\mathrm{s}}\sqrt{\frac{S(0)}{S(x)}}\sqrt{\frac{k(0)}{k(x)}}{e}^{-i{\int }_0^{x}k(x)dx},$$ (7)

where $P_{\mathrm{s}}$ is the stimulus pressure at the cochlear entrance, typically assumed proportional to the velocity of the stapes.

The velocity of the cochlear partition is then found as

$$V_{\text{CP}}=P_{0}Y=\alpha \overline{P}Y.$$ (8)

Although the formulas to calculate BM velocity (Eqs. 6,7,8) in this framework appear 1D, considering fluid motions along one ($x$) two ($x$, $z$) or three ($x$, $y$, $z$) spatial directions produces distinct values of $\alpha$. In what follows, we assume that the physical parameters that define the model ($b$, $\rho$, $S$, $Y$) do not depend on the model’s dimensionality. In this way, models of different dimensionality differ only in their pressure-focusing factor $\alpha$ and can be fairly compared. In other words, we will study the contribution of dimensionality to cochlear mechanics, by analyzing how dimensionality affects $\alpha$. Because $\mid \alpha \mid$ represents a physical boost⁹ to the pressure driving intracochlear vibrations, our mathematical analysis has a rather straightforward physical interpretation. These are important aspects to consider when comparing our results with classic literature, and to navigate somewhat conflicting results regarding the importance of models’ dimensionality (Steele and Taber, 1979; Viergever, 1978a). For instance, 2D models have often been assumed to incorporate gross parameter distortions (de Boer, 1981; Steele and Taber, 1979), prompting hasty conclusions regarding the importance of the third dimension.

III. MATHEMATICAL ANALYSIS

A. 1D and 2D models

In 1D models, the pressure spatially depends only on the distance from the cochlear entrance—i.e., at any longitudinal location $x$, the pressure in the scala is the same, regardless of $y$ and $z$ —and the fluid moves only longitudinally (back and forth along the $x$ direction). Because the pressure is uniform there is no focusing of pressure near the BM, and therefore $\alpha =P_0∕\overline{P}=1$.

To derive the equation for a 2D model, we consider a very short segment of the model shown in the top panel of Fig. 1, over which the dimensions of the cochlear duct can be regarded constant. When considering only two spatial coordinates, the fluids can only move along $x$ and $z$. The Laplace equation that allows, together with rigid boundary conditions at the cochlear walls, to solve the 2D model is:

$$\frac{{\partial }^{2}P}{\partial x^2}+\frac{{\partial }^{2}P}{\partial z^2}=0.$$ (9)

To solve the 2D model we only need to calculate $\alpha$; in this case its WKB approximation is well known (Duifhuis, 1988; Shera et al., 2005):

$${\alpha }_{2\mathrm{D}}\approx kh∕\tanh [kh],$$ (10)

with $h$ the height of the scala.

B. 3D box model

The typical 3D model assumes a rectangular cochlear duct and approximates the cochlear partition as a beam. The cross-section of this model is shown in Fig. 2, where the horizontal and vertical directions are $y$ and $z$, respectively. The partition is placed at $z=0$, and it divides the duct into two identical chambers of height and width $h$ and $w$, respectively. The flexible portion of the partition coincides with the BM and has width $b$. Differently from previous work (e.g., de Boer, 1981), we don’t make any assumptions regarding the BM centering along $y$. The 3D Laplace equation is

$$\frac{{\partial }^{2}P}{\partial x^2}+\frac{{\partial }^{2}P}{\partial y^2}+\frac{{\partial }^{2}P}{\partial z^2}=0,$$ (11)

whose solution can be expressed in terms of a summation of modes. To calculate $\alpha$ (and hence the model solution) we need to calculate the pressure distribution in the scala. The solution of this model is straightforward albeit somewhat cumbersome, and a detailed derivation was originally given by de Boer (1981). Employing a separation of variables, we can express the pressure as

$$P(x,y,z)=\Xi (x){\rm Y}(y)\Theta (z),$$ (12)

where $\Xi$, ${\rm Y}$, $\Theta$ express the pressure profile over $x$, $y$, $z$ respectively. We now express the pressure distribution in the cross-section as a weighted summation of modes

$$𝒫(y,z)={\rm Y}\Theta =\sum _{n=0}^{\infty }{\gamma }_n{{\rm Y}}_n{\Theta }_n,$$ (13)

where we implicitly assumed that the longitudinal profile pressure (along $x$) is well captured by a single mode of propagation, and where ${\gamma }_n$ expresses the relative amplitude (weight) of the $n$-th mode. The boundary conditions at the lateral cochlear walls require that $\frac{\partial P}{\partial y}=0$, so we have that¹⁰

$${{\rm Y}}_n=\cos (\frac{\pi ny}{w}).$$ (14)

FIG. 2.

A) Cross-section of 3D box model, with scala width and height $w$ and $h$ respectively, and with BM width $b$, centered at $y=d$. B-D Dependence of pressure-focusing factor $\alpha$, expressed as a function of $kh$, on the assumed geometry of the box model. B Dependence of $\alpha$ on $ϵ=b∕w$ [in these simulations $h=w$ and $d=w∕2$]. These simulations show that the narrower the BM relative to the scala, the stronger the pressure-focusing effect. When the BM is large enough, $\alpha$ becomes similar to that predicted by a 2D model. C) Dependence of pressure focusing on the position of the BM along the scala width [in these simulations $h=w$ and $ϵ=0.2$]. The panel shows $\alpha$ calculated assuming the BM completely to the side ($d=ϵ∕2$), perfectly centered ($d=w∕2$), and in an intermediate scenario ($d=0.35$). The three curves are all close together, illustrating that pressure focusing depends only little on the assumed position of the BM. D) Values of $\alpha$ for different “aspect ratio” of the cochlear cross-section [in these simulations $ϵ=0.2$ and $d=w∕2$]. This panel illustrates that the assumed aspect ratio plays a small role for pressure focusing (see text).

The boundary conditions impose maximal velocity near the partition ($z=0$) and zero velocity at $z=h$, leading to

$${\Theta }_n=\frac{\cosh [m_n(z-h)]}{\cosh [m_{n}h]},$$ (15)

where $m_n$ is determined by Eq. (11):

$$m_n^2=k^2+\frac{{\pi }^2{n}^2}{w^2},$$ (16)

with $k$ [Eq. (6)] the wavenumber (measured along $x$). After a few passages (see de Boer, 1981), we obtain the following expression for the pressure distribution

$$𝒫(y,z)=\sum _{n=0}^{\infty }{c}_n\frac{\cosh [m_n(z-h)]}{m_n\sinh (m_{n}h)}\cos (\frac{n\pi y}{w}),$$ (17)

with $c_n$ the Fourier coefficient of the $n$-th mode, determined by the boundary conditions at the cochlea partition at $z=0$.¹¹ We can now calculate the average pressure in the cross-section:

$$\begin{gathered} \overline{𝒫}=\frac{1}{wh}{\int }_0^w{\int }_0^h𝒫(y,z)dydz \\ =\frac{c_0}{h}{\int }_0^h\frac{\cosh [k(z-h)]}{k\sinh (kh)}dz=\frac{c_0}{k^{2}h}, \end{gathered}$$ (18)

where we exploited the fact that, due to the term $\cos (\frac{n\pi y}{w})$ in Eq. (17), only the mode with $n=0$ contributes to the average pressure. We now need to find the pressure acting on the BM (${𝒫}_{\text{BM}}$, in this case the pressure distribution at $z=0$, averaged along the BM width)¹², whose expression depends on the BM placement along the cochlear duct and on its velocity profile. For simplicity, we assume that the BM radial velocity profile $u(y)$ is a half cosine¹³ centered around $y=d$:

$$u(y)=\{\begin{array}{cc}\cos (\frac{\pi (y-d)}{b}),\hfill & \text{for}d-b∕2<y<d+b∕2,\hfill \\ 0\hfill & \text{elsewhere},\hfill \end{array}\phantom{\}}$$ (19)

with $d\in [b∕2,w-b∕2]$. Given this velocity profile the modes coefficients are

$$c_n=\{\begin{array}{cc}\frac{2ϵ}{\pi }\hfill & \text{for}n=0,\hfill \\ \frac{4ϵ}{\pi }\frac{\cos (n\pi \frac{ϵ}{2})\cos (n\pi \frac{d}{w})}{1-{ϵ}^2{n}^2}\hfill & n>0,\hfill \end{array}\phantom{\}}$$ (20)

where $ϵ=b∕w$ indicates the ratio between BM and scala width.¹⁴

With some passage we have that:

$${𝒫}_{\text{BM}}=\sum _{n=0}^{\infty }\frac{c_n}{m_n\tanh [m_{n}h]}\cos (n\pi \frac{d}{w})\mathrm{sinc}(n\pi \frac{ϵ}{2}),$$ (21)

with

$$\mathrm{sinc}(x)=\{\begin{array}{cc}1\hfill & \text{for}x=0,\hfill \\ \frac{\sin (x)}{x}\hfill & \text{for}x\ne 0.\hfill \end{array}\phantom{\}}$$ (22)

The function $\alpha$ for the 3D model can then be written as

$$\begin{gathered} {\alpha }_{3\mathrm{D}}=\frac{{𝒫}_{\text{BM}}}{\overline{𝒫}}=\frac{kh}{\tanh kh}+\frac{1}{\overline{𝒫}}\sum _{n=1}^{\infty }\frac{c_n\mathrm{sinc}(n\pi \frac{ϵ}{2})\cos (n\pi \frac{d}{w})}{m_n\tanh [m_{n}h]}= \\ ={\alpha }_{2\mathrm{D}}+\text{higher-order 3D modes}, \end{gathered}$$ (23)

which shows that pressure-focusing is larger in 3D than in 2D ($\mid {\alpha }_{3\mathrm{D}}\mid >\mid {\alpha }_{2\mathrm{D}}\mid$), because the 3D model accounts for wave modes that increase the pressure across the BM, but without affecting the average pressure in the scala.

Dependence on box model parameters and difference with the 2D approximation.

Our mathematical analysis has revealed that the pressure-focusing effect is expected to be stronger in 3D models than 2D ones. How much this differs in 2D vs. 3D for a given wavenumber $k$ and height of the scala $h$, depends on three anatomical specifications: the placement of the BM along the scala width, the ratio between BM and duct width, and the duct width, respectively. In Fig. 2B-D we show how $\alpha$ qualitatively depends on these three aspects—the reader interested in a fined-grain numerical analysis is provided formulas and Matlab scripts available as supporting material.

First, we study how pressure-focusing depends on the ratio between BM and scala width, a geometrical factor that can be described through the variable $ϵ=b∕w$ (Fig. 2B). Differently than classic studies that assumed $ϵ=1$ in 2D (de Boer, 1981), $ϵ<1$ is partially accounted for in our 2D solution [Eq. (6)]. In our framework, the 2D solution accounts for the effect of $ϵ$ as a “geometrical gain factor” (c.f. Zweig 2015) in relating fluid velocity with BM motion, but it neglects its effect on pressure focusing. Figure 2B shows that the difference between pressure focusing in 2D vs. 3D greatly depends on $ϵ$: the narrower the BM width compared to the duct, the larger the pressure focusing in 3D than in 2D (they converge when $ϵ\to 1$).

Second, we study how pressure focusing depends on the placement of the partition along the scala, whose effect, to the author’s best knowledge, has largely been neglected to this day. Indeed, the partition is assumed perfectly centered in most cochlear mechanical treatments to our best knowledge. As it turns out, changing the centering of the BM along the scala has small effects on $\alpha$ (Fig. 2C). According to our numerical calculations—Fig. 2C shows only few examples for sake of clarity—the effect of radial BM placement becomes (barely) noticeable only when assuming that the BM is completely to one side, touching the cochlear wall. Therefore, pressure focusing depends only little on where the flexible BM is placed along $y$—as a corollary, the commonly adopted assumption of a perfectly centered partition is a good one, because it simplifies matters while not bearing particular consequences for the model’s solution.

Finally, we study how pressure focusing depends on the assumed shape of the “box” cross-section, by changing the “aspect ratio” (the ratio between height and width) of the scala while keeping $ϵ$ constant (Fig. 2D). While at short enough wavelength, the aspect ratio of the scala becomes irrelevant—the three curves in Fig. 2D converge to the same value for large enough $kh$—at slightly longer wavelengths the pressure focusing is stronger when the scala is tall and narrow, than when it is short and wide. However, these simulations come with a caveat: while narrowing the scala, we also narrowed the BM in proportion, to maintain $ϵ$ constant. Considering these caveats, we can safely conclude that the width of the scala plays an important role for pressure focusing because it determines, together with the BM width, the ratio $ϵ=b∕w$ that plays a large role for making pressure focusing stronger in 3D than in 2D (Fig. 2B). For a given value of $ϵ$ and $h$, the precise shape of the duct influences pressure focusing rather secondarily.

In a nutshell, our analysis shows that while different details of the cochlear geometry contribute to the pressure-focusing effect, the most crucial aspect is the ratio of BM to scala width. An intuitive explanation for this finding, is that the narrower the BM relative to the scala, the larger the coefficients—hence the contribution to $\alpha$—of the higher modes in Eq. (23).

C. Cylindrical model

Although the 3D box model is the de facto standard in the field (e.g., Ramamoorthy et al., 2007; Sisto et al., 2023), it has two shortcomings whose impact on the model’s quality remains unclear to this day. First, the rectangular shape of the box model’s duct does not approximate well the elliptical cross-section of the real cochlea (see also Lighthill, 1981). Second, it neglects potential effects of the finite area of the partition and its deformation on the model solution.

Leaving the deformation problem aside (see de Boer, 1990), we want to know whether these assumptions regarding the shape of the duct and the area of the partition, should be of any concern. To this purpose, we now study a model, originally proposed by de Boer (1980) to improve on 2D models of cochlear mechanics, where the cochlear duct and partition’s cross-sections are approximated by perfect circles (Fig. 3A). In this 3D, “cylindrical” model, we assume that duct and partition’s radii are $R$ and $R_0$ respectively¹⁵. Although extremely simplified, this model approximates the elliptical cross-section of the cochlear duct arguably better than a rectangle, and it accounts for the finite volume of the organ of Corti.

FIG. 3.

A) Cross-section of the 3D cylindrical model. B) Pressure-focusing factor in the cylindrical model; the different solid lines show $\alpha$ for increasing values of partition radius relative to the duct radius [$R_0∕R=(0.1,0,2,0.3)$]. The grey and dashed lines indicate $\alpha$ in the 2D and the 3D box model (with perfectly centered partition and $ϵ=0.2$), respectively. C Illustration of the partition’s mode of vibration that completely controls the pressure-focusing factor $\alpha$ in the cylindrical model: while the partition bulges out radially in one scala, it shrink towards the center in the other.

Like for the box model, the solution of the cylindrical model comprises a summation of modes—the derivation is simple and follows classic solution of the Laplace equation in cylindrical coordinates. We start by defining the cylindrical coordinate system ($x$, $r$, $\theta$) shown in Fig. 3A, where $r$ is radius and $\theta$ angle. In this coordinate system the Laplace equation becomes

$$\frac{{\partial }^{2}P}{\partial x^2}+\frac{1}{r}\frac{\partial P}{\partial r}+\frac{{\partial }^{2}P}{\partial r^2}+\frac{1}{r^2}\frac{{\partial }^{2}P}{\partial {\theta }^2}=0.$$ (24)

Solving Eq. (24) using separation of variables, and imposing rigid boundary conditions at $r=R$ and $\theta =0$, $\pi$ we find that the pressure profile in the cross-section can be expressed as a summation of modes

$$𝒫(r,\theta )=\sum _{n=0}^{\infty }{\gamma }_n\left({𝒦}_{\mathrm{n}}(kr)+g_n{ℐ}_{\mathrm{n}}(kr)\right)\cos (n\theta ),$$ (25)

with

$$g_n=\frac{{𝒦}_{\mathrm{n}-1}(kR)+{𝒦}_{\mathrm{n}+1}(kR)}{{ℐ}_{\mathrm{n}-1}(kR)+{ℐ}_{\mathrm{n}+1}(kR)},$$ (26)

where $K_n$ and $I_n$ denote modified Bessel Functions of order $n$.¹⁶

We now proceed to calculate the pressure-focusing factor $\alpha$ for the cylindrical model, by calculating the average pressure in the scala ($\overline{𝒫}$) and across the BM (${𝒫}_{\text{BM}}$):

$$\overline{𝒫}=\frac{2}{\pi (R^2-R_0^2)}{\int }_{R_0}^R{\int }_0^{\pi }𝒫(r,\theta )drd\theta$$ (27)

$${𝒫}_{\text{BM}}=\frac{1}{\pi }{\int }_0^{\pi }𝒫(R_0,\theta )d\theta .$$ (28)

Equations (27-28) reveal that only the mode with $n=0$ contributes to $\overline{𝒫}$ and ${𝒫}_{\text{BM}}$: the contribution of higher modes [due to the term $\cos (n\theta )$ in Eq. (25)] averages to 0 over the interval [$0,\pi$]. This mode of vibration, which corresponds to a radial expansion of the partition on one scala and a simultaneous contraction in the other, is illustrated in Fig. 3C. This result complements the conclusion of de Boer (1990), that only this mode of motion significantly couples to the traveling wave—it follows that the effective width of the partition in this model is the half circumference $b=R_0\pi$.¹⁷

Having noted this simplification, $\alpha$ can be directly computed from Eqs. (27-28). Although the analytic solution for $\alpha$ is not illuminating, we report it here for completeness:

$$\begin{gathered} {\alpha }_{\text{cyl}}=\frac{{𝒫}_{\text{BM}}}{\overline{𝒫}}= \\ =\frac{k(R^2-R_0^2)(g{ℐ}_0(k{R}_0)+{𝒦}_0(k{R}_0)}{2g[R{ℐ}_1(kR)-R_0{ℐ}_1(k{R}_0)]-2[R{𝒦}_1(kR)-R_0{𝒦}_2(k{R}_0)]}. \end{gathered}$$ (29)

Skipping mathematical digressions, this analysis is interesting because it reveals that the pressure focusing in the cylindrical model is accurately captured by a model where the pressure is a function of radius $r$ and location $x$, but it does not change with $\theta$. In other words, the results derived from a fully 3D cylindrical model, can be obtained in an equivalent, highly symmetric model where the fluid motions are constrained to be 2D, happening only along $x$ and $r$.¹⁸ In this equivalent model, the Laplace equation becomes

$$\frac{{\partial }^{2}P}{\partial x^2}+\frac{1}{r}\frac{\partial P}{\partial r}+\frac{{\partial }^{2}P}{\partial r^2}=0,$$ (30)

whereas in a purely 2D model is¹⁹

$$\frac{{\partial }^{2}P}{\partial x^2}+\frac{{\partial }^{2}P}{\partial r^2}=0.$$ (31)

The differences between the cylindrical and standard 2D models are brought about by the term $\frac{1}{r}\frac{\partial P}{\partial r}$ in Eq. (30), which captures the radial “fanning out” of pressure in the cochlear cross-section. This causes the pressure to decay along $r$ faster in the cylindrical than in the 2D model, causing stronger pressure focusing when waves are short, as shown in Fig. 3B. Importantly, cylindrical and 3D box model predict similar pressure focusing effect. However, whereas the explanation for why the pressure focusing is stronger in 3D vs 2D using a box model is rather technical, working in cylindrical coordinates makes the explanation very simple and intuitive: in a 2D model the pressure decays from the BM to the outer wall only along $z$, in 3D it also decays radially (along both $y$ and $z$), fanning out.

IV. SOLUTIONS IN AN ACTIVE COCHLEAR MODEL

A. Comparison of models with identical scala height

So far our analysis has revealed that at short wavelengths, the pressure-focusing factor $\alpha$ becomes larger with increasing model dimensionality. However, this analysis does not tell how much model dimensionality is of practical importance for the solution of a realistic cochlear model—and hence for understanding the physical functioning of the cochlea.

To answer this question, we calculate responses using one-, two- and three-dimensional solutions for an active model of the gerbil cochlea. We begin by using the cylindrical model as the 3D reference—it has less geometrical parameter than the 3D box model. The model, largely based on Altoè and Shera (2020a), includes a symmetric, tapered geometry of the cochlea, shown in Fig. 4B. This symmetric and idealized model is expected to approximate well the cochlear acoustic at some distance from the stapes—in the basal end, in the so-called hook region, the anatomy of the real cochlea is strongly asymmetric, and its acoustic properties complicated by various factors such as the presence of the oval and round window.²⁰

FIG. 4.

Cylindrical model of the gerbil cochlea. A) Comparison of predicted cochlear frequency map with that determined experimentally by Müller (1996). B) Predicted near-CF wavelength along the cochlea. The inset in this panel shows the side view of the model’s geometry, where the radius of the duct ($R$) and of the partition ($R_0$) are chosen so the model’s impedance $Z$ and BM width $b$ match those calculated from the morphometric measurements of Plassmann et al. (1987). C) Comparison of measured and predicted BM response (normalized by stimulus level) in vivo and postmortem in the ≈ 16 kHZ region of the gerbil. The data—previously presented in Altoè and Shera (2020b)—was collected at 40 dB SPL in vivo, and 80 dB SPL postmortem, and it was kindly provided to us by Dr. Wei Dong. The measured phase response was corrected for 25 μs middle ear delay (de La Rochefoucauld et al., 2010).

The radii of the duct and the partition are chosen so that $S$ [Eq. (2)] and $b$ in the model match the morphometric measurements of Plassmann et al. (1987). The model embeds an idealized “scaling symmetric” (Zweig, 1991) partition’s mechanics—i.e., $Y$ is identical every-where, when frequencies are expressed as a function of the local CF. The partition’s admittance includes a linear active term, which produces traveling wave amplification through a mechanism we previously described in Altoè et al. (2022), and whose details are unimportant for the analysis that follows. The active term can be “turned off” to simulate postmortem “passive” cochlear responses. The adjustable parameters of the model (the amount of damping and amplification) are adjusted so that the 3D solution matches the reference data. The interested reader can find parameters and details of the calculation in the Matlab scripts provided as supporting material.

The cylindrical model predicts a realistic frequency map (Fig. 4A) and realistic wave-lengths (Fig. 4B), in the order of ~ 250 μm near CF in the base of the gerbil (Ren, 2002; van der Heijden et al., 2024), and in the order of 1 mm at more apical locations where CF~2 kHz (Meenderink and Dong, 2022). Realistic wavelengths and cochlear dimensions are essential to evaluate the effect of dimensionality on cochlear responses, because pressure focusing depends on wavelength and duct radius. Figures 4C,D show that the model replicates well the BM frequency response measured in vivo in a sensitive cochlea, and postmortem.²¹

We obtain 1D and 2D solutions, by performing calculations using models with the same exact parameters ($S$, $Z$, $Y$, $b$) but where the value of $\alpha$ is calculated using 1D and 2D approximations, respectively. For the 1D case we use the long-wave approximation and impose $\alpha =1$; for the 2D case we use Eq. (10) with $h=R$, i.e., we keep the “height” of the scala the same in 2D and 3D.

Figure 5 shows the BM response along the cochlea, in both active and passive models, in response to tones of different frequencies, for models of different dimensionality. 2D and 3D models predict pronounced apical/basal differences in the BM response, a feature that is brought about by gradients in gross anatomical dimensions (opposite tapering of the duct radius and BM width), as explained previously (Altoè and Shera, 2020a). In both active and passive models, the dimensionality is largely irrelevant for responses to tones below 4 kHz, where solutions obtained in 1D, 2D and 3D models are virtually identical. Indeed, in the region of the cochlea tuned to low frequencies (toward the cochlear apex) wavelengths are large and the duct is narrow (Fig. 4B): the 1D long-wave approximation works well in the apex.

FIG. 5.

BM velocity, magnitude and phase along the cochlea calculated in a linear model of the gerbil cochlea with varying dimensionality (1, 2 and 3D), in response to tones of different frequencies (indicated in the top-panel). Top: active model of the sensitive gerbil cochlea. Bottom: passive model of a dead cochlea. In this figure, the 3D solutions are calculated using the cylindrical model. Curves are normalized (scaled by a constant factor) so that the peak in vivo BM velocity for the lowest frequency is 0 dB.

The difference between the models’ solutions becomes significant at higher frequencies, especially in the active case—active amplification not only increases vibration magnitude, but it also shortens wavelengths (Ren, 2002; Shera, 2007), in turn increasing pressure focusing. In particular, 2D and 3D responses show a much more peaked response near the CF place than in 1D, due to pressure focusing. This focusing causes also a steeper phase gradient near the CF location in 2D and 3D, than in 1D²². As the frequency increases and the CF location moves closer to the cochlear entrance, the pressure-focusing effect is visibly stronger in 3D than in 2D. This is explained by that the BM-to-scala width ratio narrows significantly moving towards the cochlear entrance (Fig. 4B), making pressure focusing significantly stronger in 3D than in 2D (Fig. 3B). For what concerns the passive case, responses are qualitatively similar regardless of dimensionality, the main difference being that the traveling wave peaks more basally in 2D and 3D models than in 1D.

B. Comparison of area-matched models

Our mathematical analysis and numerical simulations agree in that the pressure focusing effect, which boosts BM vibration when waves are short, is significantly stronger in 3D than 2D models with the same scala height. On one hand, this result seems to suggest that the three-dimensionality of the cochlear duct plays a vital role for cochlear wave propagation. On the other hand, the pressure focusing present in the 3D cylindrical model is calculated using a mathematical model that considers only two spatial coordinates. Furthermore, by construction, 3D and 2D models that we have employed have the same scala impedance ($Z$), but different volumes of fluid involved in the calculation of pressure focusing effect. In the 3D model, all the fluid in the scala’s cross-section is involved in the calculation of pressure focusing; the 2D model considers the decay of pressure along a column of fluid on top and bottom of the BM. A physical implementation of this 2D model—where calculations of $Z$ and $\alpha$ involve different amounts of fluid—is perhaps unfeasible.

These observations suggest that the difference between 3D and 2D model’s response observed in Fig. 5 has been exacerbated by the näive—rather “mathematical” than “physical”—way that the 2D model is constructed. Indeed, we hypothesize that the large difference between 2D and 3D models is caused by underestimating the area of fluid involved in the pressure-focusing effect in 2D. We test this hypothesis by comparing responses of the 3D cylindrical model, with those obtained in a 3D box model—the 3D box helps deducing how to map the 3D cochlear geometry in a 2D model—and a purely 2D model, whose geometries and relations between dimensions are illustrated in Fig. 6A. We map the 3D cylindrical geometry in a nearly-equivalent 3D box model, where the duct cross-section is a square $w\times w$—i.e., in both lower and upper scala the width is double the height ($h=w∕2$), and impose that in the two models the volume of fluids in the scalae is identical. We also increase the BM width in the 3D model relative to that of the cylindrical one, of a factor $\pi ∕2$. This “fudge factor” helps the two models behave similarly, and it has been chosen so that the BM, given its peak velocity, displaces the same amount of fluid in the scala in the two models.²³

FIG. 6.

Models with different geometries but similar response. A) Illustration of the scala cross-sectional geometry of 3D cylindrical, 3D box and 2D “augmented” box models with identical scala-fluid area. The formulae next to each illustration explain the relationships between models’ dimensions. B) Comparison of BM responses across the different models in vivo (top) and postmortem (bottom). C) Comparison of pressure-focusing factor $\alpha$ between 3D box and 2D augmented model, for different values of the ratio between BM and scala width [$ϵ=(0.2,0.4,0.8)$]. D) Responses in a classic 2D “reduced” model, where the cochlear partition’s damping factor is made location dependent to mimic the location-dependent tuning sharpness observed in the 3D model. In this 2D model the damping coefficient is made proportional to the square root of BM-to-scala width ratio $ϵ$, to make response more sharply tuned at the base of the cochlea, where $ϵ$ is small.

We then map the 3D box model in a 2D model that physically represents a box cochlea with the BM in a “wall-to-wall” carpet configuration. In this 2D model, the height of the scala is increased so that its cross-sectional area remains invariant when reducing dimensionality from three to two. In this manner, the quantities that describe wave propagation in Eq. (1) remain the same in 2D and 3D, except $\alpha$.²⁴ We call this 2D model “augmented” as the height of the scala is increased to correctly account for the acoustic impedance of the scala; this approach contrasts with that of the “reduced” 2D model (de Boer, 1981), where the compensation is obtained mathematically, by reducing the scala impedance by a factor $b∕w$, but without manipulating the scala height accordingly.

Numerical simulations of the augmented model are shown in Fig. 6B. Although necessarily non-identical, the solutions of the different models are now in good agreement. The largest disagreement is between 3D models and the augmented 2D model in the “hook region” at the extreme base of the cochlea. In this region, the 2D model predicts elevated responses to high frequency sounds, in both active and passive models. This discrepancy is caused by the 2D augmented model pressure-focusing factor $\alpha$, that, while now matches well that of a 3D model when wavelengths are short (when $kh$ is large), it is elevated when wavelengths are longer, especially when the BM is narrow (Fig. 6C). Although certainly more refined and elaborate ways to map parameters across models are possible, this simple mapping indeed demonstrates that i) solutions in cylindrical and box models of comparable dimensions are indeed very similar and ii) carefully constructed 2D and 3D box models predict very similar responses.

Finally, we want to point out that the response of a 2D “reduced” model—i.e., the 2D model with height matching that of the 3D model, whose simulations were presented in Section IV A—can also be made comparable to that of a 3D model by a simple alteration of the model’s parameters—for example by decreasing damping in the cochlear base as illustrated in Fig. 6D.

V. DISCUSSION

Our analysis shows that, when 2D models are carefully constructed, the difference between 2D and 3D models’ prediction is modest. Our results show that a full 3D treatment is not strictly necessary to understand cochlear wave propagation accurately—at least to what concerns propagation of the “classic” pressure-difference wave²⁵. In this regard, our analysis shows that the pressure focusing present in a fully 3D cylindrical model, can be precisely captured by its simplified 2D version (de Boer, 1981), where fluid motions are constrained to happen only longitudinally and radially²⁶. Furthermore, the response predicted by the cylindrical model is nearly indistinguishable to that of a classic box model whose solution requires a truly 3D mathematical treatment (Fig. 6B, see also de Boer 1981). These results show that, by choosing a highly symmetric model and adequate coordinate system, cochlear macromechanics can be more simply—but equally accurately—studied using two spatial coordinates instead of three.

By mapping the 3D cochlear geometry to a 2D box model in a physically informed manner—i.e., preserving the scala cross-sectional area when the 2D model represents a box with the BM in a wall-to-wall carpet configuration—makes the 2D “augmented” model behave very similar to its 3D counterpart, except very close to the cochlear entrance.²⁷ To the author’s best knowledge, increasing the height of a 2D model in order to preserve the fundamental parameters describing wave propagation of a 3D model (i.e., the acoustical impedance of the scala), is an original proposal of this study, never tested before. As it turns out, this method of mapping the 3D cochlea geometry into a 2D model, works rather well in practical settings (Fig. 6B). Taken all together our results show that, when carefully accounting for the 3D geometry of the cochlea—either geometrically by “combining” dimensions in the cylindrical model or more approximately by matching the scala area (and hence the fluid volume) in the augmented 2D model—the 2D approach works well. Indeed, this approach avoids gross distortion of cochlear wave propagation that happens when reducing model’s dimensionality by simply neglecting one or more spatial coordinates, as in some classic studies (e.g., Kolston, 2000; Steele and Taber, 1979).

More recently, Zweig (2015) assumed that Siebert’s 1974 formula for the short-wave approximation (deduced in a 2D framework) works for a 3D model. Our results in Fig. 6C, that compare the wavelength-dependence of the 2D augmented model with its 3D counterpart, corroborate the conjecture of Zweig. In our framework, Siebert’s conditions for the short-wave approximation are satisfied when wavelengths are short enough that $\alpha \propto kh$ (see Sisto et al., 2021). Figure 6C, shows that at short enough wavelength (equivalently, large enough $kh$), each 3D model solution converges to that of a 2D model. Hence, our results demonstrate that Siebert’s short-wave approximation works in the 3D model as well. The main difference between 2D and 3D models, is in the transition between long- (where $\alpha \approx 1$) and short-wave behavior (when $\alpha$ increases linearly with $kh$). This transition happens more abruptly in 3D than in 2D; furthermore, in 3D the transition does not depend only on the wavelength relative to the height of the scala, but also on additional geometrical aspects of the scala (mainly on the ratio between BM and scala width, see Fig. 2B-D).

Our simulations (Fig. 6D) show that the discrepancies between a 3D model and a more classic “reduced” 2D approximation—where the height of the scala is assumed the same in the two models—can be greatly reduced by a slight change in the cochlear partition’s parameters such as damping. This corroborates the idea that the difference between the classic 2D and 3D approach is rather modest in comparison to that between 1D and 2D models. Indeed, to obtain a 1D model whose response is comparable to its 2D counterpart, on needs to introduce additional oscillating elements in the cochlear partition’s mechanics to compensate for the lack of pressure focusing (Shera et al., 2005; Zweig, 2015). However our analysis does not support the conjecture (c.f. Viergever, 1978a) that the solution of a 2D box model is simply equivalent to the solution of its 3D counterpart, averaged over the radial direction.

Our results also show that the actual geometry and shape of the cochlea is important for the cochlear response, especially for what concerns amplification and tonotopic variation of tuning. While various factors, such as the placement of the partition along the scala, play some role for the BM response, our results show that the scala height, and the ratio between BM and scala width play a major role for cochlear wave propagation (Fig. 2 and Fig. 5). While the contribution of scalae dimensions to apical/basal differences have been previously discussed (Altoè and Shera, 2020a; Sasmal and Grosh, 2019) and are accounted for in the “standard” 2D treatment (Shera et al., 2005), our results point out that the narrower the BM relative to the scala, the stronger the pressure focusing when waves are short (Fig. 2B). Because the BM-to-scala width ratio gets small in proximity to the cochlear entrance, 3D pressure focusing boosts and sharpens cochlear basal responses in vivo, rendering high-frequency cochlear amplification more efficient than in a standard 2D model (Fig. 5A). This mechanism facilitates high-frequency cochlear amplification, potentially compensating for intrinsic limitations of the outer-hair-cell electromotility machinery (see Altoè and Shera, 2023, for a critical review).

While the standard 2D approximation predicts tonotopic gradients of cochlear tuning and apical/basal differences that are consistent with the experimental data (Altoè and Shera, 2020a), accounting more precisely for the cochlear 3D geometry appears to make these apical/basal differences more prominent (Fig. 5). Indeed, in our previous work we have shown that a 2D model with scaling-symmetric partition can predict qualitatively well the trends of tonotopic variation of cochlear tuning observed experimentally in various species—albeit the predicted gradients of tuning with location were not the steepest observed (Altoè and Shera, 2020a; Shera and Altoè, 2023)—and that this gradient was caused primarily by near-CF waves getting longer and scalae getting shorter when moving from base to apex. Better accounting for the 3D cochlear geometry in the calculation of pressure focusing, makes high-frequency cochlear responses more localized (i.e., more sharply tuned) than in the 2D treatment, without affecting the apical, low-frequency behavior (Fig. 5). That is, our work suggests that the tapering of the BM width significantly contributes to known apical/basal differences, by increasing 3D pressure focusing in the base of the cochlea (Fig. 5).

By highlighting the importance of the shape of the cochlear duct for wave propagation, our work cautions against too simplistic evaluation of cochlear mechanical models. Even small differences regarding cochlear dimension or wavelength, can cause significant variation in pressure focusing and BM response, in the order of several dB (Fig. 2). That is, any model with pretense of realism must ensure that at least cochlear dimensions and wavelength agree with the experimental data. Unfortunately wavelengths are not commonly shown or discussed in modeling papers—but phase responses that are at odd with the experiment hint that these are likely “off” by a fair amount— and the geometry is often oversimplified and glossed over.

VI. CONCLUSION

Besides many important technical details, our analysis highlights the primary role of hydrodynamics for cochlear function and how these depend on the cochlear 3D geometry. The most interesting aspect revealed by our work, from a functional perspective, is that the increase of duct radius and narrowing of the basilar membrane towards the cochlear entrance contribute, through pressure focusing, to amplify high-frequency basilar membrane vibration. The estimated high-frequency hydrodynamic boost surpasses 20 dB in the base of the gerbil in vivo while it is small postmortem. That is, our calculations suggest that a large fraction of cochlear “active” gain in the base of the cochlea can be attributed to the amplifier “beaming” the traveling wave energy around the sensory tissue. While this idea is not new and such energy-beaming effect is generally significant in 2D and 3D models (see e.g., Kolston, 2000; Sisto et al., 2023; van der Heijden, 2014; Wang et al., 2016), the magnitude of the effect reported here is not merely theoretical speculation. In our framework, the effect of such energy beaming is encapsulated by the pressure-focusing factor, which describes the short-wave hydrodynamic boost to cochlear vibration. Such boost is entirely the consequence of the incompressibility of cochlear fluids and walls, and it is primarily determined by wavelength and cochlear dimensions—both measurable quantities that are accounted for realistically in our treatment. This large hydrodynamic effect cautions against the excessive focus on the mechanics of a single row of outer hair cells, and reliance upon models with unrealistic phase (wavelength) response to understand how cochlear amplification works.

Data Availability

The Matlab scripts used for numerical simulation are available at https://github.com/AuditoryPhysicsGroup/Models.