Hydrodynamic analysis of a two-dimensional model for micromechanical resonance of free-standing hair bundles

doi:10.1016/0378-5955(90)90198-X

Hearing Research

Volume 48, Issues 1–2, September 1990, Pages 37-67

https://doi.org/10.1016/0378-5955(90)90198-X Get rights and content

Abstract

To investigate the role of inner ear fluids and structures on mechanical stimulation of the hair bundles of hair cells, we analyzed a two-dimensional structure that consists of: a rectangular flap (which represents a hair bundle) attached to a flat basal plate (which represents the surface of the epithelium that contains the hair cells) with a spring-loaded hinge (that represents the compliant attachment of a hair bundle to the hair cell body) and surrounded by a viscous fluid (that represents endolymph). We computed the fluid velocity as well as the forces on and motion of the flap in response to sinusoidal vibration of the plate by numerical integration of the hydrodynamic equations, and —at asymptotically low and high frequencies — by analytic methods.

The results suggest that:

(1)
the surface of the sensory epithelium, from which hair bundles project into fluid, plays an important part in the production of fluid forces on hair bundles;
(2)
both fluid inertia and viscosity play a key role in hair bundle mechanics;
(3)
passive mechanical resonances are likely to contribute to both frequency selectivity and frequency-to-place coding in the inner ear.

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Sound-induced motions of individual cochlear hair bundles
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Citation Excerpt :
The most apical hair bundles in this region are ∼30 μm tall and have best frequencies near 1 kHz, and the most basal hair bundles are ∼12 μm tall and have best frequencies above 4 kHz (Mulroy, 1974; Weiss et al., 1978; Holton and Weiss, 1983b). This correlation of best frequency with hair bundle height has prompted models in which the frequency selectivity seen at the auditory nerve arises from a mechanical resonance between the compliance of the hair bundle and the mass of fluid entrained to move with the bundle (Weiss and Leong, 1985; Freeman and Weiss, 1990a; Shatz, 2000). Other models, however, have suggested that a mechanical resonance of hair bundles is not possible (Billone and Raynor, 1973).
We present motions of individual freestanding hair bundles in an isolated cochlea in response to tonal sound stimulation. Motions were measured from images taken by strobing a light source at the tone frequency. The tips and bases of hair bundles moved a comparable amount, but with a phase difference that increased by 180° with frequency, indicating that distributed fluid properties drove hair bundle motion. Hair bundle rotation increased with frequency to a constant value, and underwent >90° of phase change. The frequency at which the phase of rotation relative to deflection of the bundle base was 60° was comparable to the expected best frequency of each hair cell, and varied inversely with the square of bundle height. The sharpness of tuning of individual hair bundles was comparable to that of hair cell receptor potentials at high sound levels. These results indicate that frequency selectivity at high sound levels in this cochlea is purely mechanical, determined by the interaction of hair bundles with the surrounding fluid. The sharper tuning of receptor potentials at lower sound levels is consistent with the presence of a negative damping, but not a negative stiffness, as an active amplifier in hair bundles.
The effect of hair bundle shape on hair bundle hydrodynamics of non-mammalian inner ear hair cells for the full frequency range
2004, Hearing Research
The effect of the size and the shape of the hair bundle of a hair cell in the inner ear of non-mammals on its motion for the full range of frequencies is determined thereby extending the results of a previous analysis of hair bundle motion for high and low frequencies [Hear Res. 141 (2000) 39–50]. A hemispheroid is used to represent the hair bundle because it can represent a full range of shapes, from thin, pencil-like shapes to wide, flat, disk-like shapes. Boundary element methods are used to approximate the solution for the hydrodynamics. For physiologically relevant parameters, an excellent match is obtained between the model's predictions and measurements of hair bundle motion in the free-standing region of the basilar papilla of the alligator lizard [Aranyosi, Measuring sound-induced motions of the alligator lizard cochlea. Massachusetts Institute of Technology, PhD Thesis, 2002]. Neither in the model's predictions nor in experimental measurements is sharp tuning observed. The model predicted the low frequency region of neural tuning curves for the alligator lizard and bobtail lizard, but could not predict the sharp tuning or the high frequency region. An element that represents an active mechanism is added to the hair bundle model to predict neural tuning curves, which are sharply tuned, and an excellent match is obtained for all the characteristics of neural tuning curves for the alligator lizard, and for the low and high frequency regions for the bobtail lizard. The model does not predict well the sharp tuning of the shorter hair bundles of the bobtail lizard, possibly because it does not represent tectorial sallets.
Clues to the cochlear amplifier from the turtle ear
2001, Trends in Neurosciences
Sound stimuli are detected in the cochlea by vibration of hair bundles on sensory hair cells, which activates mechanotransducer ion channels and generates an electrical signal. Remarkably, the process can also work in reverse with additional force being produced by the ion channels as they open and close, evoking active movements of the hair bundle. These movements could supplement the energy of the sound stimuli but to be effective they would need to be very fast. New measurements in the turtle ear have shown that such active bundle movements occur with delays of less than a millisecond, and are triggered by the entry of Ca²⁺ into the cell via the mechanotransducer channel. Furthermore, their speed depends on the frequency to which the hair cell is most sensitive, suggesting that such movements could be important in cochlear amplification and frequency discrimination.
Effect of pressure gradient on viscous shear flow past an axisymmetric depression or protuberance on a plane wall
2000, Computers and Fluids
The effect of pressure gradient on the structure of unidirectional viscous shear flow over an axisymmetric depression or protuberance on a plane wall, possibly in the presence of an upper parallel plane wall in the configuration of two-dimensional Hagen–Poiseuille flow, is considered. The problem is formulated in terms of a system of three coupled Fredholm integral equations of the first kind for the first-order Fourier coefficients of the boundary traction, and the solution is found using a boundary element method. The numerical results illustrate the effect of the pressure gradient on the streamline pattern and on the distribution of the shear stress along the plane wall and over the depression or protuberance. The topology of the streamline pattern in three-dimensional flow is found to be generally similar to that in two-dimensional flow, but important differences arise in the vicinity of stagnation points. The structure of semi-infinite parabolic flow is similar to that of the channel pressure-driven flow, when the distance between the channel walls is roughly larger than six times the characteristic size of the depression of protuberance. The force and torque exerted on a protuberance are evaluated for pure shear flow, pure parabolic flow, and pressure-driven channel flow, and a linear combination of the results for the first two cases is shown to accurately describe the results for the third case when the channel is moderately wide.
The effect of hair bundle shape on hair bundle hydrodynamics of inner ear hair cells at low and high frequencies
2000, Hearing Research
The relationship between size and shape of the hair bundle of a hair cell in the inner ear and its sensitivity at asymptotically high and low frequencies was determined, thereby extending the results of an analysis of hair bundle hydrodynamics in two dimensions (Freeman and Weiss, 1990. Hydrodynamic analysis of a two-dimensional model for micromechanical resonance of free-standing hair bundles. Hear. Res. 48, 37–68) to three dimensions. A hemispheroid was used to represent the hair bundle. The hemispheroid had a number of advantages: it could represent shapes that range from thin, pencil-like shapes, to wide, flat, disk-like shapes. Also analytic methods could be used in the high frequency range to obtain an exact solution to the equations of motion. In the low frequency range, where an approximate solution was found using boundary element methods, the sensitivity of the responses of hair cells was mainly proportional to the cube of the heights of their hair bundles, and at high frequencies, the sensitivity of the hair cells was mainly proportional to the inverse of their heights. An excellent match was obtained between measurements of sensitivity curves in the basillar papilla of the alligator and bobtail lizards and the model’s predictions. These results also suggest why hair bundles of hair cells in vestibular organs which are sensitive to low frequencies have ranges of heights that are an order of magnitude larger than the range of heights of hair bundles of hair cells found in auditory organs.
Reversed tonotopic map of the basilar papilla in Gekko gecko
1999, Hearing Research
A published model of the frequency responses of different locations on the basilar papilla of the Tokay gecko Gekko gecko (Authier and Manley, 1995. Hear. Res. 82, 1–13) had implied that (a) unlike all other amniotes studied so far, the frequency map is reversed, with the low frequencies at the base and the high frequencies at the apex, and (b) the high-frequency area is split into two parallel-lying hair cell areas covering different frequency ranges. To test these hypotheses, the frequency representation along the basilar papilla of Gekko gecko was studied by recording from single auditory afferent nerve fibers and labelling them iontophoretically with horseradish peroxidase. Successfully labelled fibers covered a range of characteristic frequencies from 0.42 to 4.9 kHz, which extended from 78% to 9% of the total papillar length, as measured from the apex. The termination sites of labelled fibers within the basilar papilla correlated with their characteristic frequency, the lowest frequencies being represented basally, and the highest apically. This confirms the first prediction of the model. The map indicates, however, that one of the two high-frequency papillar regions (the postaxial segment) represents the full high-frequency range, from about 1 to 5 kHz. No functionally identified labelling was achieved in the preaxial segment. Thus the assumptions underlying the proposed model need revision. A good mathematical description of the frequency distribution was given by an exponential regression with a mapping constant in the living state of approximately 0.4 mm/octave.

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Hydrodynamic analysis of a two-dimensional model for micromechanical resonance of free-standing hair bundles

Abstract

Hear. Res.

Hear. Res.

Hear. Res.

Hear. Res.

Hear. Res.

Hear. Res.

Neuron

Hear. Res.

Hear. Res

Hear. Res.

Hear. Res.

Hear. Res.

Hear. Res.

Variation of membrane properties in hair cells isolated from the turtle cochlea

J. Physiol.

A fast motile response in guinea-pig outer hair cells: The cellular basis of the cochlear amplifier

J. Physiol.

Transducer motor coupling in cochlear outer hair cells

Models for electrical tuning in hair cells

Sensory and effector functions of vertebrate hair cells

J. Submicrosc. Cytol.

Quantum noise and the threshold of hearing

Phys. Rev. Let.

Transmission of radial shear forces to cochlear hair cells

J. Acoust. Soc. Am.

Evoked mechanical responses of isolated cochlear outer hair cells

Science

Acoustic stimulation causes tonotopic alterations in the length of isolated outer hair cells from guinea pig hearing organ

An electrical tuning mechanism in turtle cochlear hair cells

J. Physiol.

The mechanical properties of ciliary bundles of turtle cochlear hair cells

J. Physiol.

Note on the slow motion of fluid

The coding of sound pressure and frequency in cochlear hair cells of the terrapin