The tonotopic cochlea puzzle: A resonant transmission line with a “non-resonant” response peak

In the mammalian cochlea, a peaked frequency response at a given place (or, equivalently, in a dual scale-invariant cochlea, a peaked spatial response to a sinusoidal stimulus of given frequency) is measured, with evidence for an approximately scaling-symmetric tonotopic map, relating frequency to longitudinal position of the peaks (see, e.g., Robles and Ruggero, 2001; Rhode, 2007). This observation has naturally suggested modeling the system as a tonotopically resonant transmission line,¹ with a locally peaked admittance (see, e.g., Moleti 2009; Sisto , 2010). In that case, the high-gain, narrow-band, and stable response that is observed in mammals at low stimulus levels would require a rather strong and fine-tuned anti-damping term, associated with the active mechanism of the outer hair cells (OHCs), and capable of counteracting almost exactly the viscous damping in the resonant region. The peak of the response would coincide with that of a correspondingly peaked admittance, and its sharpness would depend on the almost exact compensation between damping and fine-tuned anti-damping terms in the region where the wave frequency matches the local resonance frequency and the admittance is approximately real. As discussed, e.g., in Sisto (2021), this is NOT the case of the real cochlea. As a general warning, although in physics a peaked function is the typical response of a resonant system, it may also be caused by the competition of different phenomena.² In other words: “A peak does not make a resonance.” On the other hand, it is still theoretically necessary, as the mechanical explanation for the growth of the wavenumber along the traveling wave (TW) path, to assume an underlying locally resonant scaling-symmetric “intrinsic” tonotopic map. The actual relation between frequency and position of the experimentally observed peaks of the cochlear response is a different one, nonlinearly dependent on the stimulus level, but still related to the intrinsic one in a scaling-symmetric way.

The resonant nature of the cochlear response is what implies that the wavenumber of a component of given frequency of the forward TW increases sharply approaching its resonant place, in the so-called short-wave region. High values of the wavenumber imply both strong pressure focusing, boosting the force driving the basilar membrane (BM) transverse motion, and strong viscous (or viscoelastic) dissipation, damping it. The forward TW grows until the viscous power loss exceeds the maximum available power from the OHCs. This condition determines the position (frequency) and the width of the experimental response peak, in a level-dependent way (see Sisto , 2021). As for each frequency component of the forward TW, the response peak is significantly basally shifted with respect to its resonant place, the local admittance is dominated by the quasi-static elastic term. Therefore, for a given frequency, the local wavenumber is almost real-valued, and grows as the reciprocal of the square of the local resonance frequency (dependent on stiffness and inertia only) over a large part of the short-wave peak region (see Tubelli , 2022). Nevertheless, a meaningful description of the mechanical response of the cochlea must describe the response of any cochlear element to any given frequency also in the regions that are not reached by the corresponding forward TW component.

Therefore, it may be useful to introduce a clear distinction between two different concepts that may be mistaken: the experimental and the theoretical definition of tonotopic map. The experimental definition is related to the measured frequency (or position) of the peak of the response at a given cochlear place (or frequency), which is more accurately named “best frequency,” BF(x) [or “best place,” BP(ω)]. In theoretical linear transmission-line models, the admittance of the Organ of Corti (OoC) is schematized by a function of position and frequency, and a characteristic frequency may be predicted at each place x as the resonance frequency CF(x) of the local oscillator (or main normal mode). Indeed, the peak frequency of the response at a given cochlear place (or, equivalently, the spatial position of the response peak to a given frequency) does not correspond to that predicted by the intrinsic “mechanical” tonotopic map, which we may call CF(x) [or “characteristic place,” CP(ω)].

Two hydrodynamic phenomena–fluid focusing and viscous (or viscoelastic) damping–dominate the dynamics of the forward TW in the short-wave peak region (Sisto , 2021). The experimentally observed peaked shape of the BM frequency response is due to the interplay of these two competing effects, both proportional to the local wavenumber, which produce a peak of the frequency response at a frequency significantly lower than the local resonance frequency, or, equivalently, a peak of the spatial response at a cochlear place significantly shifted basal to the resonant place of the considered frequency. Well before reaching the mechanical resonant place, a sharp rise of the response occurs entering the short-wave region, due to the sharp increase in the forcing pressure, followed by a sharp decrease due to the even sharper growth of the viscous damping (see also Prodanovic , 2019). In this framework, the admittance is not a sharply peaked function, and the height and sharpness of the gain peak still depends on stimulus level, because at low stimulus levels, the higher effectiveness of the OHC mechanism permits approaching more closely the mechanical resonance, in a region of intrinsically larger wavenumber, which means both sharper increase due to pressure focusing and sharper decrease due to viscous damping.

As a remarkable consequence, the width and level of the peaked BM response are only indirectly related to the properties of the actual resonance, which is never reached by the forward TW wave. This observation contributes to explaining the so-called decoupling between cochlear gain and tuning, i.e., why the cochlear group delays are only weakly dependent on stimulus level whereas the cochlear gain is strongly dependent on it. This decoupling is partly due to the nonlinearity of the system (Sisto , 2015), but the main fact is that the relation between gain and tuning is not that typical of a resonant response because the resonance is not the mechanism producing the response peak. The maximal gain is still strongly dependent on the effectiveness of the OHC mechanism, which provides the power necessary to counteract the viscous losses, because a more effective anti-damping mechanism permits reaching a more basal place, where the wavenumber and the forcing pressure are significantly larger. The width of the response is only weakly dependent on it, because basally to the resonant place, the wavenumber is a rapidly varying (quadratic) function of the position. Within the resonant peak, the admittance is a slowly varying function (Dong and Olson, 2013; Altoè and Shera, 2020a,b), almost independent of the stimulus level, because the true resonant region (where the local frequency equals the TW frequency and the bandwidth would depend on the strength of the damping/anti-damping terms) is not reached yet.

From now on, we stop specifying that, in a scale-invariant cochlea, what is described as a function of frequency at a given place may be also described equivalently at a given frequency as a function of the longitudinal coordinate x (a property named duality, see e.g., de Boer , 2008). From a modeling viewpoint, the local mechanical response of the OoC has to be described as that of a (nonlinear) system whose elements have inertia, damping, and stiffness. One may consider the simplest one degree of freedom (1-DOF) nonlinear oscillator (e.g., Moleti , 2009), a two degree of freedom (2-DOF) system roughly representing the motion of the BM and of the reticular lamina (RL) (e.g., Neely and Kim, 1986; Sisto , 2019), or a finite element approximation to a continuous representation of each OoC element (e.g., Sasmal and Grosh, 2019), but any physically meaningful model must include the inertia, damping, and stiffness of such elements, along with the mechanical properties of the fluid. A set on Newton's second law equations may be written for each mechanical element coupled to the fluid and/or other mechanical elements. Stiffnesses and masses of all these elements, and their couplings, determine the intrinsic resonance frequencies and bandwidths of the normal modes of the system. Incompressible fluid dynamic equations complete the model, along with suitable boundary conditions at the fluid-membrane interfaces.

The strong dissipation associated with viscous or viscoelastic damping of the OoC motion implies that the “theoretical” resonant place cannot be reached by the TW. On the other hand, as a function of frequency, the experimental peak position function follows the theoretical tonotopic function at a constant distance, and gradually approaches it as the stimulus level decreases in nonlinear models (or, in almost equivalent linear models, as the effectiveness of the OHC system is increased as a global parameter), as discussed in Sisto (2021). Therefore, the theoretical local resonance frequency is still a necessary element of the model, scaling-symmetrically associated with the frequency of the local response peak (BF).

For this reason, recent theoretical linear models correctly neglect (see Tubelli , 2022) the inertial and damping terms of the local oscillator equations (because, for any given TW frequency, they may be neglected in the region where the forward TW propagates). Nevertheless, the intrinsic mechanical tonotopic map is still a necessary element of any model, which actually determines the spatial behavior of the physical quantity determining the local response, i.e., the local wavenumber. The experimental tonotopic map relating best place to frequency (or best frequency to place) is a scaling-symmetrically shifted version of the intrinsic mechanical map, and, although the former depends on the latter, they must not be mistaken.

Moreover, although the most interesting aspect of the cochlear response regards the forward traveling waves fetching external acoustic signals of different frequencies to the auditory system detectors, a theoretical cochlear model must include also the physical elements that do not determine the properties of the response to forward waves of a given frequency. Indeed, nonlinear distortion phenomena generate intracochlear distortion products also at cochlear places apical to the BP of the generated wave frequency (for example, the 2f₂-f₁ intermodulation products). In such cases, the model must be able to describe the forward and backward propagation of the corresponding traveling waves in regions “prohibited” to the forward waves coming from the cochlear base, also to correctly predict the phenomenology of otoacoustic emissions (OAEs).

As we will see, the TW of given frequency ω propagates only in a region significantly basal to the place where ω_BM(x) = ω (its intrinsic resonant place). Therefore, the third quasi-static term is dominant. In other words, each local oscillator is forced at a frequency lower than the local resonance frequency. In the simple single-oscillator model, this means that the phase of the forcing term (the differential pressure) is approximately the same as that of the local oscillator displacement. Due to the low-pass characteristic of the OHC voltage buildup, their additional force is in phase with the oscillator velocity, with an anti-damping effect, as predicted by physiology-based models (see Lu , 2006; Sisto and Moleti, 2021). Interestingly, this phase is also that of the time derivative of the differential pressure, as theoretically hypothesized by Zweig (2015) with remarkable insight, mostly based on a fit to the experimental BM response functions, without the support of a physiology-based model of the OHC actuator behavior.

The local oscillator equations are coupled to the longitudinal fluid motion by the incompressibility and adhesion conditions, which imply, respectively, the propagation of a transverse slow TW for differential pressure and BM displacement, and the existence of a thin fluid layer comoving with the BM (see Sisto , 2021).

Strong anti-damping forces are still necessary to sustain the TW power flow against increasingly large viscous losses, but the high quality factor of the response peak does not need the rather unreasonable fine-tuning between passive damping and maximal anti-damping that is necessary in a classical resonant model.

The prediction of a peaked response in a region of almost constant admittance reconciles the theory with the experimental results by Dong and Olson (2013), who found similarly peaked and nonlinear response functions for the pressure near the BM and for the BM velocity, i.e., broadband and level-independent admittance.

The fact that the intrinsic resonant place is not accessible to the corresponding frequency component of the forward TW may generate confusion. Indeed, in the accessible region, one could approximate the admittance function (as well as the relation between wavenumber frequency and position) with that corresponding to the quasi-static limit, in which the mass and damping terms give a small contribution, yet the concept of intrinsic tonotopic map is still a necessary element of this description. The experimental tonotopic map BF(x) is a scaling-symmetrically shifted version (dependent also on stimulus level) of the intrinsic mechanical map ω(x). The former is the dynamic consequence of the interplay among different physical effects; the latter is an intrinsic structural element of the model.

In this study, to highlight these concepts, we consider the frequency-domain WKB solution of a simple 1-DOF transmission line cochlear model, that we name here “2-D viscous.” The two scalae are separated by a single vibrating element, the BM, on which a tonotopic map is explicitly defined as a function ω_BM(x), an anti-damping term associated with the OHC mechanism is defined as a function proportional to the BM velocity, and a viscous damping term is defined, as in Sisto (2021), as a function proportional to both the BM velocity and the local wavenumber. This way, the local admittance becomes a function of the wavenumber. The 2-D fluid focusing effect is parametrized by the function α(x), defined, e.g., in Duifhuis (2012) and Shera (2005), as a function of the local wavenumber. As in Sisto (2021), the relation between the admittance and the wavenumber is recursively used to get a self-consistent local wavenumber corrected for the fluid focusing and viscous damping effects.

We also consider, for comparison, a “1-D classical” anti-damping model with neither pressure focusing nor fluid viscous damping, and a “2-D classical” model with focusing and without fluid viscous damping. As a function of the strength of the OHC amplifier term, we get three sets of BM responses which can be compared, to show the effect of the different physical assumptions.

In Fig. 1, we show the BM velocity response of the three cochlear models, “1-D classical,” “2-D classical,” and “2-D-viscous” at a given frequency ω₀ as a function of the cochlear longitudinal coordinate x, expressed in scaling units as the dimensionless ratio y(x) = ω₀/ω_BM(x). In the model, the “intrinsic” resonant place is known, and corresponds in the plot to y = 1.

Thanks to duality, in a scaling-symmetric cochlea, this is almost equivalent to the outcome of a more practical experiment, in which the frequency response is measured at a fixed position x₀ as a function of the frequency ω.

The viscous coefficient was increased by a factor 10 with respect to that of water, to account for the additional viscous losses within the OoC, not schematized in the simple 1-DOF model. The gain parameter G, varied as a global one (independent of x) in the range [0.19–1.19] at 0.2 steps for all models, represents the dimensionless ratio between the amplitude of the anti-damping term associated with the OHC amplifier and that of the passive damping term of a scale-invariant model with passive Q = 1, which represents the underlying low gain and low tuning of a post-mortem cochlea for G = 0.

In the viscous 2-D model, a number of noteworthy features are immediately visible in Fig. 1:

1. The BM response peak is shifted basally with respect to the intrinsic tonotopic place CP(x), increasingly with decreasing effectiveness G of the OHC anti-damping mechanism.

2. The viscous model remains stable for G significantly larger than unity.

3. The admittance function is not peaked and weakly dependent on the strength of the cochlear amplifier, because the pressure and BM velocity functions grow in a similar way over a wide short-wave region.

4. The focusing factor α grows (as the wavenumber) as the square of the local resonance frequency in the short-wave region (y > 0.2) and remains large and almost constant, and real-valued, over a wide peak region.

5. Due to the faster increase of the wavenumber, the phase response is steeper in the 2-D models, and it does not change significantly by introducing viscosity (red and green curves are almost coincident), because the region in which the wavenumber would be different in the two models is not reached by the TW in the viscous model. One may also note that in the 2-D models, the slope (and therefore the delay) is less dependent on the gain of the cochlear amplifier than in a 1-D resonant model.

All these features agree with the observed cochlear phenomenology. The models without viscosity, either including focusing (2-D classical) or not (1-D classical) show instability for G > 1, admittance function peaked at the intrinsic resonant place, and a basal peak shift due, in this case, to the increase in the negative imaginary part of the wavenumber. High gain and stable BM response require fine-tuned G in both cases. The instability of the linear models could be considered as a curable one, because, in a fully nonlinear version of the model, the effectiveness of the anti-damping term in the peak region would decrease with increasing BM displacement regaining stability. This is only partly true, because, in these conditions, if the nonlinear anti-damping function saturates at a given displacement (or velocity) threshold level, as, e.g., in Sisto (2010), the system is stable, but the peak response to very different stimulus levels tends to almost the same saturation level (near the threshold for nonlinear saturation), which is not the experimentally observed nonlinear behavior. Note also that both the non-viscous models need unnaturally fine-tuned maximal gain (G = 0.99 in Fig. 1) to get both a stable response and a sufficiently large gain dynamics (40 dB), and yield in that case a too sharply tuned admittance. On the other hand, the faster growth of the focusing factor alpha is a feature common to the 2-D models. Comparing the response of the two 2-D models for similar peak gain, one may also appreciate how viscosity, quite paradoxically, improves cochlear tuning, as noted by Prodanovic (2019).

In simple 2-D transmission-line models, the seeming paradox of a peaked cochlear response without most of the typical features of a resonant response is easily explained as due to two important hydrodynamic effects, pressure focusing and viscous damping, both proportional to the wavenumber.

The authors have no conflicts to disclose.

The Matlab codes (The MathWorks, Natick, MA) used in this work will be made available upon request.