^{1}

^{1}

^{2}

^{2}

^{3}

^{1}

^{2}

^{*}

Conceived and designed the experiments: RS. Performed the experiments: CH. Contributed reagents/materials/analysis tools: W-HS NS. Wrote the paper: RS. Designed theory: AK.

The authors have declared that no competing interests exist.

A majority of hearing defects are due to malfunction of the outer hair cells (OHCs), those cells within the mammalian hearing sensor (the cochlea) that provide an active amplification of the incoming signal. Malformation of the hearing sensor, ototoxic drugs, acoustical trauma, infections, or the effect of aging affect often a whole frequency interval, which leads to a substantial loss of speech intelligibility. Using an energy-based biophysical model of the passive cochlea, we obtain an explicit description of the dependence of the tonotopic map on the biophysical parameters of the cochlea. Our findings indicate the possibility that by suitable local modifications of the biophysical parameters by microsurgery, even very salient gaps of the tonotopic map could be bridged.

The cochlea, the mammalian hearing sensor, is a formidable biophysical construct in many respects. Its task is to pick up environmental auditory information, which provides us with a sensory communication channel without which we experience great problems in our every day life. In its extreme form, the lack of hearing capability often leads to social isolation. Mending hearing deficits—increasingly important in societies of growing average age—is difficult, not least because of a delicate interplay between the brain and the sensor. Here, we investigate to what extent the hearing sensor could be tuned in such a way that regions of malfunction are circumvented by relaying the signal to areas of normal functionality. The means by which we envisage achieving this goal is through local changes of the biophysical parameters of the cochlea. By investigation of a detailed biophysical model of the cochlea, we find that nature indeed appears to offer such a possibility.

The heart of the mammalian auditory system is the hearing sensor, the cochlea. Acoustic signals arriving in the form of sound pressure waves are funneled by the outer ear towards the tympanic membrane, forcing the latter to oscillate. These oscillations are transmitted to the oval window, a membrane-covered oval opening to the cochlea, by the middle ear ossicles. Within the cochlea, the oscillating membrane elicits incompressible and inviscid hydrodynamic surface waves, comprising the basilar membrane (BM) that partitions the tube containing the cochlear fluid (see

Emergence of a travelling wave on the BM due to a sinusoidally varying sound wave with stapes amplitude _{st}

Using coordinate _{0} e^{−αx}, where ^{−1} _{c}_{c}_{G}_{c}_{c}

The position

Measured OHC-amplification profiles can be modelled by driven Hopf oscillators _{c}

In this contribution, we investigate on biophysical grounds whether the TM could theoretically be modified in order to overcome this problem, by a modification of the biophysical parameters of the cochlea. In particular, we study whether modifications can be identified that send incoming frequency information originally targeted to handicapped OHC, to regions of intact OHC. In this case, all salient properties of the cochlea, such as its high input-dependent sensitivity, two-tone-suppression, combination-tone-generation and the ability to tune in into sound sources, would be preserved, and could be used for information processing. It is the aim of this contribution to show that by suitable alterations of the biophysical parameters of the cochlea, such a remapping is, in principle, possible.

We use an energy-based passive modeling approach that, on the desired level of description, has previously reliably described the physical processes within the cochlea _{G}_{1},_{2} of the cochlear duct as_{1}(_{2}(

From the equipartition theorem, we can express the traveling wave amplitude _{0}, the passive traveling wave amplitude has the expression_{0}(_{0}.

Based on a two-dimensional surface-wave analogy of the cochlea _{G}_{c}

Analytical prediction (dashed) together with the corresponding numerical evaluation (solid) for space dependence a) ^{2} and b) _{0}′ e^{−α′g(x)}, using _{0}′ = 10^{2}_{0} and ^{2} and b)

In order to estimate the effects introduced by changed biophysical parameters, the space derivative of the travelling wave amplitude,

This equation provides the desired description of the impact of parameter variations to the TM. We are unaware that with classical, not explicitly energy-based approaches

Whereas Equation 11 is strongly nonlinear in the space coordinate _{c}^{2}, for _{c}_{c}^{2}, and _{c}^{2} are positive (otherwise for any value of _{c}^{2} are of the same order of magnitude. For Equation 11 to hold, _{1} needs to balance the two subtractive terms _{2} and _{3}. Owing to the different powers associated with the parameters in _{1,2,3}, the TM is affected more by changes in

(I) Space-dependent or -independent increase/decrease of

(II) A change of the transversal stiffness exponent from

TM obtained from variations of BM transversal stiffness, evaluated numerically from Equation 6 and predicted analytically by Equation 12, are compared in ^{−2} cm.

Analytical prediction (dashed) and corresponding numerical evaluation (solid). From left bottom to right top: _{0} e^{−α′x}. Analytical prediction (dashed) together with the corresponding numerical evaluation (solid). From right top to left bottom:

Changes in the initial transversal stiffness _{0} lead to a shift of the TM parallel to the original TM (see

These observations are sufficient to construct TM profiles according to will and need. In particular, by suitable modifications of the transversal stiffness (or the surface tension ^{−5} ^{2}

Assume, as an example, that the outer hair cells are damaged across the frequency range

Dotted lines indicate the defect region in frequency space (

As a second example, we consider how a local variation of the mass of the BM can be used to bridge gaps. Suppose that the mass distribution is modified from Π_{0}(_{0} (_{0}(

Our study is based on a passive cochlea model that has already reliably served as the blueprint for a biomorphic cochlea model and its electronic implementation

Moreover, a paradigm has been derived that allows one to evaluate the modifications of the local transversal stiffness, mass density, fluid density viscosity, required to bridge defective regions on the BM. We provide here but an short overview of why we believe that the obtained results could have medical relevance. An in-depth discussion of this topic would clearly lead beyond the scope of this research. Although the working BM is extremely sharply tuned, where this tuning comes from is still a matter of debate

One of the classical references for modeling the passive membrane behavior is Zwislocki's seminal paper

A model that achieved this and accordingly allowed to include active amplification by outer hair cells in a transparent way was introduced in _{G}

The model differential equation for the _{1}(_{2}(_{i}_{G}_{i}_{G}_{2})_{2})−_{G}_{1})_{1})>0, i.e., if more energy leaves the interval [_{1}(_{2}(_{0}(_{G}

For the evaluation of the the traveling wave amplitude _{pot}(_{kin}(_{0}, the passive traveling wave amplitude is described by_{0}(_{0}. In order to render this expression computationally seizable, we need to specify the functions _{G}

For the evaluation of these functions, we proceed similarly to what is explained to more detail in _{G}_{cc}

Whereas for the investigation in mind we can dispose of the active amplification contribution _{I}^{2} (as originally derived by Stokes), where _{c}_{cc}

Numerical evaluation of the TM from Equation 20 also requires the values of the remaining constants and the form of ^{2} s^{−1}, ^{−3}, ^{−2}, ^{−3} m. The transversal stiffness, finally, has the form _{0} e^{−αx}, with the decay exponent ^{4} m^{−1}. The surface tension contribution _{0} e^{−αx}+^{2}^{2})_{0} e^{−αx}, with the proportionality constant ^{−9} m^{2}, see

In order to estimate the effects introduced by changed biophysical parameters, the space derivative of the travelling wave amplitude, ^{2}^{2} changes in a complex manner in the neighborhood of the peak. Whereas an analytical treatment appears nearly impossible, the problem stated in this form is accessible to numerical approaches, via Equation 20. Since we will be interested in the location of the response peak alone, we expect that the surface tension can be neglected. _{c}_{c}_{c}

Since we are interested in the close neighborhood of the peak only, where _{c}_{0} exp(−_{G}_{G}^{2})^{4} for convenience, and remembering that we required _{c}

Whereas Equation 35 is strongly nonlinear in the space coordinate _{c}^{2}, for _{c}_{c}^{2}, and _{c}^{2} are positive (otherwise for any value of _{c}^{2} are of the same order of magnitude. For Equation 35 to hold, _{1} needs to balance the two subtractive terms _{2} and _{3}. Due to the different powers associated with the parameters in _{1,2,3}, the TM is affected more by changes in

For _{0} e^{−αx}, we obtain

For ^{3} in _{1} reacts stronger upon changes in ^{2} in _{2}. Therefore, we obtain_{2} and _{3} by an increase of _{1}. Such an increase, however, again implies_{0}, the inequalities invert.

Inserting into Equation 35 a modified transversal stiffness _{0}′ e^{−α′g(x′)} (where _{c}_{c}_{c}_{c}

The second part of the theorem is new and can be seen as the direct benefit of our energy-based description. Equation 35 predicts numerically the effects generated by variations in the transversal stiffness ^{−2} cm. The figures reveal that changes in the initial transversal stiffness _{0} lead to a shift of the TM parallel to the original TM (