Fluid focusing and viscosity allow high gain and stability of the cochlear response

Optimal Scale-Invariant Wavelet Representation and Filtering of Human Otoacoustic Emissions

Otoacoustic emissions (OAEs) are generated in the cochlea and recorded in the ear canal either as a time domain waveform or as a collection of complex responses to tones in the frequency domain (Probst et al. J Account Soc Am 89:2027-2067, 1991). They are typically represented either in their original acquisition domain or in its Fourier-conjugated domain. Round-trip excursions to the conjugated domain are often used to perform filtering operations in the computationally simplest way, exploiting the convolution theorem. OAE signals consist of the superposition of backward waves generated in different cochlear regions by different generation mechanisms, over a wide frequency range. The cochlear scaling symmetry (cochlear physics is the same at all frequency scales), which approximately holds in the human cochlea, leaves its fingerprints in the mathematical properties of OAE signals. According to a generally accepted taxonomy (Sher and Guinan Jr, J Acoust Soc Am 105:782-798, 1999), OAEs are generated either by wave-fixed sources, moving with frequency according with the cochlear scaling (as in nonlinear distortion) or by place-fixed sources (as in coherent reflection by roughness). If scaling symmetry holds, the two generation mechanisms yield OAEs with different phase gradient delay: almost null for wave-fixed sources, and long (and scaling as 1/f) for place-fixed sources. Thus, the most effective representation of OAE signals is often that respecting the cochlear scale-invariance, such as the time-frequency domain representation provided by the wavelet transform. In the time-frequency domain, the elaborate spectra or waveforms yielded by the superposition of OAE components from different generation mechanisms assume a much clearer 2-D pattern, with each component localized in a specific and predictable region. The wavelet representation of OAE signals is optimal both for visualization purposes and for designing filters that effectively separate different OAE components, improving both the specificity and the sensitivity of OAE-based applications. Indeed, different OAE components have different physiological meanings, and filtering dramatically improves the signal-to-noise ratio.

Regional differences in cochlear nonlinearity across the basal organ of Corti of gerbil: Regional differences in cochlear nonlinearity

Auditory sensation is based in nanoscale vibration of the sensory tissue of the cochlea, the organ of Corti complex (OCC). Motion within the OCC is now observable due to optical coherence tomography. In a previous study (Cooper et al., 2018), the region that includes the electro-motile outer hair cells (OHC) and Deiters cells (DC) was observed to move with larger amplitude than the basilar membrane (BM) and surrounding regions and was termed the "hotspot." In addition to this quantitative distinction, the hotspot moved qualitatively differently than the BM, in that its motion scaled nonlinearly with stimulus level at all frequencies, evincing sub-BF activity. Sub-BF activity enhances non-BF motion; thus the frequency tuning of the OHC/DC region was reduced relative to the BM. In this work we further explore the motion of the gerbil basal OCC and find that regions that lack significant sub-BF activity include the BM, the medial and lateral OCC, and the reticular lamina (RL) region. The observation that the RL region does not move actively sub-BF (already observed in Cho and Puria 2022), suggests that hair cell stereocilia are not exposed to sub-BF activity in the cochlear base. The observation that the lateral and RL regions move approximately linearly sub-BF indicates that linear forces dominate non-linear OHC-based forces on these components at sub-BF frequencies. A complex difference analysis was performed to reveal the internal motion of the OHC/DC region and showed that amplitude structure and phase shifts in the directly measured OHC/DC motion emerge due to the internal OHC/DC motion destructively interfering with BM motion.

Fluid Focusing Contributes to the BM Vibration Amplification by Boosting the Pressure

Two hydrodynamic effects are introduced in the standard transmission-line formalism, the focusing of the pressure and fluid velocity fields near the basilar membrane and the viscous damping at the fluid-basilar membrane interface, which significantly affect the cochlear response in the short-wave region. In this region, in which the wavelength is shorter than the cochlear duct height, only a layer of fluid of order of the wavelength is effectively involved in the traveling wave. This has been interpreted [8] as a reduced fluid contribution to the system inertia in the peak region, which is a viewpoint common to the 3-D FEM solutions. In this paper we propose an alternative approach, from a slightly different physical viewpoint. Invoking the fluid flux conservation along the traveling wave propagation direction, we can derive a rigorous propagation equation for the pressure integrated along the vertical axis. Consequently, the relation between the average pressure and the local pressure [4] at the fluid-BM interface can be written. The local pressure is amplified by a factor dependent on the local wavenumber with respect to the average pressure, a phenomenon we refer to as "fluid focusing", which plays a relevant role in the BM total amplification gain. This interpretation of the hydrodynamic boost to the pressure provides a physical justification to the strategy [10] of fitting the BM admittance with a polynomial containing both a conjugated pole and a zero. In the short-wave region, the sharp gradients of the velocity field yield a second important effect, a damping force on the BM motion, proportional to the local wavenumber, which stabilizes active models and shifts the peak of the response towards the base, with respect to the resonant place. This way, the peaked BM response is not that of a proper resonance, corresponding to a sharp maximum of the admittance, but rather a focusing-driven growth toward the resonant place, which is "aborted" before reaching it by the sharply increasing viscous losses. The large values of the wavenumber that ensure strong focusing are ultimately fueled, against viscosity, by the nonlinear OHC mechanism, hence the otherwise puzzling observation of a wide nonlinear gain dynamics with almost level-independent admittance.

Foundations of the Wentzel-Kramers-Brillouin approximation for models of cochlear mechanics in 1- and 2-D

The Wentzel-Kramers-Brillouin (WKB) approximation is frequently used to explore the mechanics of the cochlea. As opposed to numerical strategies, the WKB approximation facilitates analysis of model results through interpretable closed-form equations and can be implemented with relative ease. As a result, it has maintained relevance in the study of cochlear mechanics for half of a century. Over this time, it has been employed to study a variety of phenomena, including the limits of frequency tuning, active displacement amplification within the organ of Corti, feedforward mechanisms in the cochlea, and otoacoustic emissions. Despite this ubiquity, it is challenging to find rigorous exposition of the WKB approximation's formulation, derivation, and implementation in cochlear mechanics literature. In this tutorial, the foundations of the WKB approximation are discussed in application to models of one- and two-dimensional cochlear macromechanics. This includes mathematical background, rigorous derivation and details of its implementation in software.

A frame and a hotspot in cochlear mechanics

Auditory sensation is based in nanoscale vibration of the sensory tissue of the cochlea, the organ of Corti complex (OCC). Motion within the OCC is now observable due to optical coherence tomography. In the cochlear base, in response to sound stimulation, the region that includes the electro-motile outer hair cells (OHC) was observed to move with larger amplitude than the basilar membrane (BM) and surrounding regions. The intense motion is based in active cell mechanics, and the region was termed the "hotspot" (Cooper et al., 2018, Nature comm). In addition to this quantitative distinction, the hotspot moved qualitatively differently than the BM, in that its motion scaled nonlinearly with stimulus level at all frequencies, evincing sub-BF activity. Sub-BF activity enhances non-BF motion; thus the frequency tuning of the hotspot was reduced relative to the BM. Regions that did not exhibit sub-BF activity are here defined as the OCC "frame". By this definition the frame includes the BM, the medial and lateral OCC, and most significantly, the reticular lamina (RL). The frame concept groups the majority OCC as a structure that is largely shielded from sub-BF activity. This shielding, and how it is achieved, are key to the active frequency tuning of the cochlea. The observation that the RL does not move actively sub-BF indicates that hair cell stereocilia are not exposed to sub-BF activity. A complex difference analysis reveals the motion of the hotspot relative to the frame.

Crucial 3-D viscous hydrodynamic contributions to the theoretical modeling of the cochlear response

This study uses a 3-D representation of the cochlear fluid to extend the results of a recent paper [Sisto, Belardinelli, and Moleti (2021b). J. Acoust. Soc. Am. 150, 4283-4296] in which two hydrodynamic effects, pressure focusing and viscous damping of the BM motion, both associated with the sharp increase in the wavenumber in the peak region, were analyzed for a 2-D fluid, coupled to a standard 1-D transmission-line WKB approach to cochlear modeling. The propagation equation is obtained from a 3-D fluid volume conservation equation, yielding the focusing effect, and the effect of viscosity is represented as a correction to the local 1-D admittance. In particular, pressure focusing amplifies the BM response without modifying the peak admittance, and viscous damping determines the position of the response peak counteracting focusing, as sharp gradients of the velocity field develop. The full 3-D WKB formalism is necessary to represent satisfactorily the behavior of the fluid velocity field near the BM-fluid interface, strictly related to viscous losses. As in finite element models, a thin layer of fluid is effectively attached to the BM due to viscosity, and the viscous force associated with the vertical gradient of the fluid vertical velocity acts on the BM through this layer.

Overturning the mechanisms of cochlear amplification via area deformations of the organ of Corti

The mammalian ear embeds a cellular amplifier that boosts sound-induced hydromechanical waves as they propagate along the cochlea. The operation of this amplifier is not fully understood and is difficult to disentangle experimentally. In the prevailing view, cochlear waves are amplified by the piezo-electric action of the outer hair cells (OHCs), whose cycle-by-cycle elongations and contractions inject power into the local motion of the basilar membrane (BM). Concomitant deformations of the opposing (or "top") side of the organ of Corti are assumed to play a minor role and are generally neglected. However, analysis of intracochlear motions obtained using optical coherence tomography calls this prevailing view into question. In particular, the analysis suggests that (i) the net local power transfer from the OHCs to the BM is either negative or highly inefficient; and (ii) vibration of the top side of the organ of Corti plays a primary role in traveling-wave amplification. A phenomenological model derived from these observations manifests realistic cochlear responses and suggests that amplification arises almost entirely from OHC-induced deformations of the top side of the organ of Corti. In effect, the model turns classic assumptions about spatial impedance relations and power-flow direction within the sensory epithelium upside down.