Hair Bundle Dynamics Research Articles

Cochlear hair bundle dynamics: modeling calcium effects and row-wise interactions

External vibrations cause the eardrum to oscillate, resulting in the excitation of the sensory structures of the cochlea. In this talk, we focus on the hair bundles (HBs) of the outer hair cells situated inside the cochlea. These bundles, protruding from the cells, convert mechanical motion into electrical currents. Experimental observations reveal that the calcium concentration inside the stereocilia (hair-like structures that comprise the HB) influences current adaptation. This regulation impacts shifts of the curve, sensitivity, and active range of bundles. We aim to mechanistically understand the intracellular calcium effects by adjusting the adaptation complex in our HB model. We modified model parameters to match experimental data on current, bundle displacement, and shift trends at different calcium concentrations. This involved increasing the adaptation stiffness, stall force, stereocilia pivot, and gating stiffness. A stiffer adaptation spring reduces ion-channel reclosure, affecting the steady-state response. A higher stall force lessens the effective force on the adaptation complex, replicating experimental observations. Unlike the other properties, pivot, and gating stiffness likely do not depend on calcium concentration. Therefore, we will conduct error minimization analyses to identify adaptation complex properties while maintaining constant pivot and gating stiffness across calcium concentrations. Work supported by NIH grant NIH-NIDCD-R01 04084.

Modeling frog saccular hair-bundle dynamics using systems of differential equations

Hair cells of the inner ear are specialized cells that detect mechanical motion induced by incoming sound waves and transmit that information to innervating neurons. They are found in the auditory and vestibular systems of all vertebrate species. As these cells exhibit active, energy-consuming limit-cycle oscillations, as well as a highly nonlinear response to mechanical signals, they provide an experimental testing ground for nonlinear dynamics theory. In particular, models based on the normal form equation for the Hopf bifurcation have been shown to explain a number of experimental measurements. The nonlinear response, the amplification of low-amplitude signals, and the presence of spontaneous oscillations are all elegantly captured by this simple equation. Further, experiments have shown that the response plotted over the full physiological range of frequencies and amplitudes displays the classic Arnold Tongue, predicted by theory. In this work, we conducted a comprehensive computational study, with systems of nonlinear equations that describe the cellular process in the hair cell, including mechanotransduction, adaptation, and others, and showed that they produce the Arnold Tongue response, consistent with experiments and dynamical systems theory. The model then allowed us to vary a number of parameters in the model, and deduce the range of validity for this description. Next, we were able to explore the reverse process, to probe how the systems of ion channels in the cell body impacts the hair bundle mechanics. This numerical study was compared to measurements performed under voltage-clamp conditions, which allowed direct manipulation of resting potential of the cell in conjunction with mechanical stimuli. Our numerical model was able to reproduce the data, explore internal cellular processes not currently accessible experimentally, and suggest future experiments.

Chaotic Dynamics Enhance the Sensitivity of Inner Ear Hair Cells

Hair cells of the auditory and vestibular systems are capable of detecting sounds that induce sub-nanometer vibrations of the hair bundle, below the stochastic noise levels of the surrounding fluid. Furthermore, the auditory system exhibits a highly rapid response time, in the sub-millisecond regime. We propose that chaotic dynamics enhance the sensitivity and temporal resolution of the hair bundle response, and we provide experimental and theoretical evidence for this effect. We use the Kolmogorov entropy to measure the degree of chaos in the system and the transfer entropy to quantify the amount of stimulus information captured by the detector. By varying the viscosity and ionic composition of the surrounding fluid, we are able to experimentally modulate the degree of chaos observed in the hair bundle dynamics in vitro. We consistently find that the hair bundle is most sensitive to a stimulus of small amplitude when it is poised in the weakly chaotic regime. Further, we show that the response time to a force step decreases with increasing levels of chaos. These results agree well with our numerical simulations of a chaotic Hopf oscillator and suggest that chaos may be responsible for the high sensitivity and rapid temporal response of hair cells.

Nonequilibrium limit-cycle oscillators: Fluctuations in hair bundle dynamics.

We develop a framework for the general interpretation of the stochastic dynamical system near a limit cycle. Such quasiperiodic dynamics are commonly found in a variety of nonequilibrium systems, including the spontaneous oscillations of hair cells of the inner ear. We demonstrate quite generally that in the presence of noise, the phase of the limit cycle oscillator will diffuse, while deviations in the directions locally orthogonal to that limit cycle will display the Lorentzian power spectrum of a damped oscillator. We identify two mechanisms by which these stochastic dynamics can acquire a complex frequency dependence and discuss the deformation of the mean limit cycle as a function of temperature. The theoretical ideas are applied to data obtained from spontaneously oscillating hair cells of the amphibian sacculus.

Friction from Transduction Channels’ Gating Affects Spontaneous Hair-Bundle Oscillations

Hair cells of the inner ear can power spontaneous oscillations of their mechanosensory hair bundle, resulting in amplification of weak inputs near the characteristic frequency of oscillation. Recently, dynamic force measurements have revealed that delayed gating of the mechanosensitive ion channels responsible for mechanoelectrical transduction produces a friction force on the hair bundle. The significance of this intrinsic source of dissipation for the dynamical process underlying active hair-bundle motility has remained elusive. The aim of this work is to determine the role of friction in spontaneous hair-bundle oscillations. To this end, we characterized key oscillation properties over a large ensemble of individual hair cells and measured how viscosity of the endolymph that bathes the hair bundles affects these properties. We found that hair-bundle movements were too slow to be impeded by viscous drag only. Moreover, the oscillation frequency was only marginally affected by increasing endolymph viscosity by up to 30-fold. Stochastic simulations could capture the observed behaviors by adding a contribution to friction that was 3−8-fold larger than viscous drag. The extra friction could be attributed to delayed changes in tip-link tension as the result of the finite activation kinetics of the transduction channels. We exploited our analysis of hair-bundle dynamics to infer the channel activation time, which was ∼1 ms. This timescale was two orders-of-magnitude shorter than the oscillation period. However, because the channel activation time was significantly longer than the timescale of mechanical relaxation of the hair bundle, channel kinetics affected hair-bundle dynamics. Our results suggest that friction from channel gating affects the waveform of oscillation and that the channel activation time can tune the characteristic frequency of the hair cell. We conclude that the kinetics of transduction channels’ gating plays a fundamental role in the dynamic process that shapes spontaneous hair-bundle oscillations.

Subdiffusion in hair bundle dynamics: The role of protein conformational fluctuations

The detection of sound signals in vertebrates involves a complex network of different mechano-sensory elements in the inner ear. An especially important element in this network is the hair bundle, an antenna-like array of stereocilia containing gated ion channels that operate under the control of one or more adaptation motors. Deflections of the hair bundle by sound vibrations or thermal fluctuations transiently open the ion channels, allowing the flow of ions through them, and producing an electrical signal in the process, eventually causing the sensation of hearing. Recent high frequency (0.1-10 kHz) measurements by Kozlov et al. [Proc. Natl. Acad. Sci. U.S.A. 109, 2896 (2012)] of the power spectrum and the mean square displacement of the thermal fluctuations of the hair bundle suggest that in this regime the dynamics of the hair bundle are subdiffusive. This finding has been explained in terms of the simple Brownian motion of a filament connecting neighboring stereocilia (the tip link), which is modeled as a viscoelastic spring. In the present paper, the diffusive anomalies of the hair bundle are ascribed to tip link fluctuations that evolve by fractional Brownian motion, which originates in fractional Gaussian noise and is characterized by a power law memory. The predictions of this model for the power spectrum of the hair bundle and its mean square displacement are consistent with the experimental data and the known properties of the tip link.

Nonlinear models of development, amplification and compression in the mammalian cochlea

This paper reviews current understanding and presents new results on some of the nonlinear processes that underlie the function of the mammalian cochlea. These processes occur within mechano-sensory hair cells that form part of the organ of Corti. After a general overview of cochlear physiology, mathematical modelling results are presented in three parts. First, the dynamic interplay between ion channels within the sensory inner hair cells is used to explain some new electrophysiological recordings from early development. Next, the state of the art is reviewed in modelling the electro-motility present within the outer hair cells (OHCs), including the current debate concerning the role of cell body motility versus active hair bundle dynamics. A simplified model is introduced that combines both effects in order to explain observed amplification and compression in experiments. Finally, new modelling evidence is presented that structural longitudinal coupling between OHCs may be necessary in order to capture all features of the observed mechanical responses.

Enhancement of sensitivity gain and frequency tuning by coupling of active hair bundles

The vertebrate inner ear possesses an active process that provides nonlinear amplification of mechanical stimuli. A candidate for this process is active hair bundle mechanics observed, for instance, for hair cells of the bullfrog's sacculus. Hair bundles in various inner ear organs are coupled by overlying membranes. Using a stochastic description of active hair bundle dynamics, we study the consequences of an elastic coupling on the properties of amplification. We report that collective effects in arrays of hair bundles can enhance the amplification gain and the sharpness of frequency tuning as compared with the performance of an isolated hair bundle. We also discuss the transient response elicited by the sudden onset of a periodic stimulus and its relation to temporal integration curves. Simulations of systems with a gradient of intrinsic frequencies show an enhanced amplification gain while preserving a frequency gradient, provided the coupling strength is similar to the hair bundle stiffness. We relate our findings to the situation in the bullfrog's sacculus and the mammalian cochlea.

Two-state approach to stochastic hair bundle dynamics

Hair cells perform the mechanoelectrical transduction of sound signals in the auditory and vestibular systems of vertebrates. The part of the hair cell essential for this transduction is the so-called hair bundle. In vitro experiments on hair cells from the sacculus of the American bullfrog have shown that the hair bundle comprises active elements capable of producing periodic deflections like a relaxation oscillator. Recently, a continuous nonlinear stochastic model of the hair bundle motion [Nadrowski, Proc. Natl. Acad. Sci. U.S.A. 101, 12195 (2004)] has been shown to reproduce the experimental data in stochastic simulations faithfully. Here, we demonstrate that a binary filtering of the hair bundle's deflection (experimental data and continuous hair bundle model) does not change significantly the spectral statistics of the spontaneous as well as the periodically driven hair bundle motion. We map the continuous hair bundle model to the FitzHugh-Nagumo model of neural excitability and discuss the bifurcations between different regimes of the system in terms of the latter model. Linearizing the nullclines and assuming perfect time-scale separation between the variables we can map the FitzHugh-Nagumo system to a simple two-state model in which each of the states corresponds to the two possible values of the binary-filtered hair bundle trajectory. For the two-state model, analytical expressions for the power spectrum and the susceptibility can be calculated [Lindner and Schimansky-Geier, Phys. Rev. E 61, 6103 (2000)] and show the same features as seen in the experimental data as well as in simulations of the continuous hair bundle model.

Chapter 12 - Hair Cell Mechanotransduction: The Dynamic Interplay Between Structure and Function

Hair cells are capable of detecting mechanical vibrations of molecular dimensions at frequencies in the 10s to 100s of kHz. This remarkable feat is accomplished by the interplay of mechanically gated ion channels located near the top of a complex and dynamic sensory hair bundle. The hair bundle is composed of a series of actin-filled stereocilia that has both active and passive mechanical components as well as a highly active turnover process, whereby the components of the hair bundle are rapidly and continually recycled. Hair bundle mechanical properties have significant impact on the gating of the mechanically activated channels, and delineating between attributes intrinsic to the ion channel and those imposed by the channel's microenvironment is often difficult. This chapter describes what is known and accepted regarding hair-cell mechanotransduction and what remains to be explored, particularly, in relation to the interplay between hair bundle properties and mechanotransducer channel response. The interplay between hair bundle dynamics and mechanotransduction are discussed.