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Acoustic Boundary-Condition Dynamics and Internally Coupled Ears

By J. Leo van Hemmen

Humans can discern the direction of a sound source by sensing the time delay between the arrival of sound at each ear. However, in the case of frogs, lizards, and birds, the distance between the ears is too small to make this distinction. Since timely localization of predators and prey is essential to survival, azimuthal sound localization greatly helps animals during both day and night. 

For azimuthal sound localization, the time difference between ears is the neuronal cue determined by \(L / c,\) where \(L\) is the interaural distance and \(c\) is the sound velocity. A Tokay gecko, the world’s second-largest gecko (lizard), has an \(L\) of \(L \approx 2.2\) centimeters, which is not very large. Most frogs and birds have even smaller \(L\) values. In all cases, merely discerning whether a sound came from the left or the right is not quite what an animal likes doing neuronally. So what is nature to do?

This is where internally coupled ears—ICE for short—come in. More than half of land-living vertebrates are equipped with ICE, which are characterized by a big, air-filled cavity (grey in Figure 1) connecting the left and right eardrums. Let us put the \(x\)-axis horizontally through the eardrums in Figure 1 and assume the eardrums are at \(x=0\) and \(x=L\). For sound localization, the signals at the left and right eardrum have a time difference of \(\Delta = L \sin(\theta) /c\), where \(\theta\) is the sound-source direction, meaning that \(\theta =0\) is directly in front of the animal. The cochlea creates a frequency decomposition and the brain then evaluates \(\Delta\); we focus here on \(\Delta\), the so-called interaural time difference (ITD), which, due to ICE, is actually not what the animal hears.

Figure 1. Internally coupled ears (ICE) in (a) frog, (b) lizard, and (c) bird. Eardrums (TM) are red. Adapted from a figure by Jakob Christensen-Dalsgaard.

What knocks on the eardrums is the outside pressure, \(p_{\rm out}(x)\) at \(x=0\) and \(x=L\), due to the sound source, and it does so (practically) uniformly so that \(x=0\) and \(x=L\) suffice to specify it. Because of the air-filled interaural cavity between the left and right eardrums in Figure 1, we see coupling mediated by the internal pressure \(p\). The coupling is described [6] by the wave equation \(\partial^2 p/\partial t^2 = c^{2}\, \Delta_{(3)} p\), where \(\Delta_{(3)}\) is the three-dimensional Laplacian equipped with dynamic, time-dependent boundary conditions at the two eardrums that fluctuate under the influence of the external sound source and the internal pressure \(p\).

With a few simplifications, an eardrum can be modeled as a damped linear-elastic membrane with displacement \(u(x, r,\varphi;t)\) at either end \(x=0,L\) (see Figure 2), obeying

\[-\frac{\partial^2 u}{\partial t^2}-2\alpha\frac{\partial u}{\partial
t}+ c^2_M \Delta_{(2)} u = [p(x, r,\varphi;t) - p_{\rm out}(x, r,\varphi;t)] \, |_{x=0, \, L} \ . \quad \qquad (1) \]

Here \(\alpha\) is the damping coefficient, \(c_M\) is the membrane’s wave-propagation velocity, and \([p(x, r,\varphi;t) - p_{\rm out}(x, r,\varphi;t)]\) is the total pressure driving the membrane at either end \((\rho_Md \equiv 1)\). In lizards (see Figure 1b), an eardrum can be modeled as a circular disk with Dirichlet boundary conditions \(u(a_\mathrm{tymp}, \varphi) = 0 \) at the edge \(\{r=a_\mathrm{tymp}\}\), including a (heavy) cartilaginous sector with boundaries \(\varphi = \pm \beta\). It is shown schematically in Figure 2 and transports membrane vibrations to the cochlea.

Figure 2. Effective cylinder representing the interaural cavity. The eardrums are disks with a smaller radius atympthan acyl of the cylinder. Grey shaded surfaces are fixed. Image credit: [7].
The setup specified by Figure 2 presents a fascinating and three-fold problem. First, the membrane displacement \(u\) in \((1)\) is driven by \(p_\rm{out}\) and simultaneously also appears (see Figure 2) in \(p\)'s boundary condition for the three-dimensional wave equation \(\partial^2p/\partial t^2 = c^{2}\,\Delta_{(3)} p\) through the famous “no-slip” boundary condition \(u = v_x\) with \(\rho \partial v_{x}/\partial t = - \partial p/\partial x\), where \(\rho\) is the air’s density and \(v_{x}\) is the \(x\)-component of the eardrum’s velocity. The normal derivative vanishes elsewhere. The only given input is the outside signal \(p_\rm{out}\), and the result is a dynamical system where boundary conditions for the inside pressure p are part of the system’s dynamics, as in \((1)\). This is the only way to achieve an effective coupling between the left and right eardrums.

Secondly, we need to know the complete solution to this linear system of coupled partial differential equations with time-varying boundary conditions, from the start of the sound signal \((t \ge 0)\) onwards. The standard solution, which is practically the only one known, is quasi-stationary and asymptotic \((\alpha >0)\), in response to a pure tone of angular frequency \(\omega\), obtained by splitting off time through the substitution \(\exp (i \omega t)\).

Thirdly, it’s highly desirable for solutions to be as exact as possible. On the basis of a solid foundation [8], solutions to all three problems have now been found [7]. The solutions are based on a novel time-dependent perturbation theory à la Paul Dirac—in the style of quantum mechanics and in conjunction with Duhamel’s principle for the first two problems and the so-called piston approximation—and are largely analytical. They yield a few surprising findings showing why ICE may facilitate sound localization in small animals.

Eardrum vibrations and their velocities are amazingly small (\(< \mu m\) and \(<\rm{mm}/\rm{s}\) for 1 kilohertz (kHz), respectively). An early idea for a simpler problem [1, 2] suggested starting with the original volume \((u \equiv 0)\) and using \(\partial p/\partial x = - \rho \partial v_{x}/\partial t\) as a boundary condition for \(p\) on both now-fixed tympana. This approach does not, however, account for eardrum fluctuation, and the enclosed volume changes as time proceeds.

It is here that the physics of our problem becomes relevant.  Let \(\mathbb{1}_{\Delta\mathbb{D}}\) be the indicator function of the interaural-cavity change and provide the restricted Laplacian \(\pm \mathbb{1}_{\Delta \mathbb{D}} \Delta_{(3)} \mathbb{1}_{\Delta \mathbb{D}}\), with a plus/minus sign when the eardrum is moving out/inwards. Then we add the time-dependent perturbation \(V(t) = \pm \mathbb{1}_{\Delta \mathbb{D}} \Delta_{(3)} \mathbb{1}_{\Delta \mathbb{D}} \) to the unperturbed Laplacian \(\Delta_{(3)},\) with \(\Delta\mathbb{D}\) as the “excess” volume. Proceed as in quantum mechanics [3], but now start with Duhamel’s formula

\[\partial \mathbf{x}/\partial t  = [\mathbb{A} + \mathcal{V}(t)] \mathbf{x} \Rightarrow \mathbf{x}(t) =T_{t}^\circ \mathbf{x}(0) +\int_{0}^t {\rm d} t' \, T_{t-t'}^\circ [\mathcal{V}(t') \mathbf{x}(t')] \quad \qquad (2) \]

while using the unperturbed semigroup \(T_{t}^\circ = \exp(t\mathbb{A})\) operating on a two-component vector \([\mathcal{V}(t') \mathbf{x}(t')]\), where \(\mathcal{V}\) contains the perturbation \(V\). The infinitesimal generator \(\mathbb{A}\) contains \(\Delta_{(3)}^\circ\) as the unperturbed and self-adjoint Laplacian with Neumann boundary conditions, i.e., the normal derivative \(\partial p/\partial n =0\) vanishing at the boundary. Start with an “intelligently” chosen zeroth-order state and iterate [4, 7]. By providing the explicit response to a time-dependent (binaural) stimulus, the acoustic boundary-condition dynamics (ABC dynamics) complements Beale’s ABC [1, 2].

Air is fairly—though not completely—incompressible; this allows it to carry sound waves. We handle vibrating eardrums approximately, and in so doing exactly, via the piston approximation [7]. We average over the left and right-hand sides of the bases \(\mathscr{S} = \{r \le a_{\rm cyl}\}\) of the cylinder in Figure 2. Once sound arrives, the integral \(\int_{\mathscr{S}} {\rm d}S u(x, r,\varphi;t) |_{x=0, \, L}\) will be nonzero, corresponding to a uniform motion of an effective piston on the left and on the right, the piston approximation. The resulting configuration can be treated exactly [7] and errors are relatively small; error estimates can be found elsewhere [4].

Finally, what is the big advantage of ICE in animals? The animals effectively perceive not the interaural difference but the internal time and level (intensity) difference, the iTD and iLD, which result from both the external signal and internal coupling through the air-filled cavity, as seen in Figures 1 and 2. Here the capital I in ITD or interaural level difference (ILD) refers to interaural; the lowercase i in iTD and iLD refers to internal, and hence to what the animals hear. It turns out that for lower frequencies the fraction iTD/ITD is a flat plateau \(\gg 1\), say typically \(3-5\), so that the interaural distance \(L\) effectively becomes much bigger and the localization precision increases as much, at least theoretically. How animals utilize this big profit is subject to hot debate among biologists, who need to interpret the results summarized in [7].

Figure 3. The eardrum’s fundamental frequency f0 segregates the low and high-frequency domains with different cues, time versus amplitude difference. iLD is a left/right amplitude fraction measured in dB. Image credit: [7].
In lizards such as the gecko, the effective volume connecting the two eardrums is much bigger than the cylinder with the tympana as endplates. That is, \(a_{\rm cyl} \approx 2 a_{\rm tymp}\); the piston approximation now makes a lot of sense. The fundamental frequency \(f_{0}\) is about 1.2 kHz. The lizards’ heads are so small that ILD \(\approx 0\) regardless of sound-source direction. Level differences are measured in decibels (dB) as the logarithm of left and right amplitude fraction. Since there is no screening, the logarithm vanishes. 

The iTD/ITD plateau occurs for \(f<0.7f_0\). For higher frequencies, the fraction iTD/ITD then quickly drops to \(0\) but simultaneously the iLD—as perceived by the animal—can reach, for a direction of \(\pm 90^{\circ}\), a level  as large as \(20\) dB (humans can just perceive \(1\) dB difference.) The maximum of the iLD occurs very near \((<5\%)\) to \(f_0\) so that it can be used to determine \(f_0\) in a live animal – a notorious problem in the past that can now be solved straightforwardly due to the present theory [7].  Figure 3 provides a summary.

What has been said about lizards also holds true for frogs and birds. Both frogs and lizards even have two kinds of cochlear hair cells; one for lower \((<f_0)\) frequencies and sensitive to time differences, the other for higher \((>f_0)\) frequencies and sensitive to amplitude differences. Though there is still hot debate among biologists as to what the animals do with their ICE, the neuronal architecture [5] corresponds well with the biophysics of ICE in having discrete pathways for low and high-frequency hair cells. 

Why do mammals refrain from ICE? They may well be small, but they have replaced the air in the air-filled tube with brains. In an evolutionary sense, brains win out over empty space.

[1] Beale, J.T., & Rosencrans, S.I. (1974). Acoustic boundary conditions. Bulletin of the American Mathematical Society, 80(6), 1276-1278. 

[2] Beale, J.T. (1976). Spectral properties of an acoustic boundary condition. Indiana University Mathematics Journal, 25(9), 895-917.

[3] Dirac, P.A.M. (1958). The Principles of Quantum Mechanics (4th ed.) New York, NY: Oxford University Press.

[4] Heider, D., Vedurmudi, A.P., & van Hemmen, J.L. (2016). Acoustic boundary-condition dynamics and internally coupled ears. TUM preprint.

[5] Szpir, M.R., Sento, S., & Ryugo, D.K. (1990). Central projections of cochlear nerve fibers in the Alligator Lizard. Journal of Comparative Neurology, 295, 530-547.

[6] Temkin S. (1981). Elements of Acoustics. New York, NY: Wiley.

[7] Vedurmudi, A.P., Goulet, J., Christensen-Dalsgaard, J., Young, B.A., Williams, R., & van Hemmen, J.L. (2016). How internally coupled ears generate temporal and amplitude cues for sound localization. Physical Review Letters, 116(2), 028101.

[8] Vossen, C., Christensen-Dalsgaard, J., & van Hemmen, J.L. (2010). An analytical model of internally coupled ears. The Journal of the Acoustical Society of America, 128(2), 909-918.

J. Leo van Hemmen is an emeritus professor of theoretical biophysics in the Physics Department at the Technical University of Munich (TUM). He also belongs to the Department of Mathematics at TUM, and advocates that a full understanding of the neuronal mechanisms of perception requires comprehension of its underlying biological physics.

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