Coupling a sensory hair-cell bundle to cyber clones enhances nonlinear amplification

Intermodulation distortions from an array of active nonlinear oscillators

Coupling is critical in nonlinear dynamical systems. It affects the stabilities of individual oscillators as well as the characteristics of their response to external forces. In the auditory system, the mechanical coupling between sensory hair cells has been proposed as a mechanism that enhances the inner ear's sensitivity and frequency discrimination. While extensive studies investigate the effects of coupling on the detection of a sinusoidal signal, the role of coupling underlying the response to a complex tone remains elusive. In this study, we measured the acoustic intermodulation distortions (IMDs) produced by the inner ears of two frog species stimulated simultaneously by two pure tones. The distortion intensity level displayed multiple peaks across stimulus frequencies, in contrast to the generic response from a single nonlinear oscillator. The multiple-peaked pattern was altered upon varying the stimulus intensity or an application of a perturbation tone near the distortion frequency. Numerical results of IMDs from a chain of coupled active nonlinear oscillators driven by two sinusoidal forces reveal the effects of coupling on the variation profile of the distortion amplitude. When the multiple-peaked pattern is observed, the chain's motion at the distortion frequency displays both a progressive wave and a standing wave. The latter arises due to coupling and is responsible for the multiple-peaked pattern. Our results illustrate the significance of mechanical coupling between active hair cells in the generation of auditory distortions, as a mechanism underlying the formation of in vivo standing waves of distortion signals.

Dynamics of Mechanically Coupled Hair-Cell Bundles of the Inner Ear

The high sensitivity and effective frequency discrimination of sound detection performed by the auditory system rely on the dynamics of a system of hair cells. In the inner ear, these acoustic receptors are primarily attached to an overlying structure that provides mechanical coupling between the hair bundles. Although the dynamics of individual hair bundles has been extensively investigated, the influence of mechanical coupling on the motility of the system of bundles remains underdetermined. We developed a technique of mechanically coupling two active hair bundles, enabling us to probe the dynamics of the coupled system experimentally. We demonstrated that the coupling could enhance the coherence of hair bundles’ spontaneous oscillation, as well as their phase-locked response to sinusoidal stimuli, at the calcium concentration in the surrounding fluid near the physiological level. The empirical data were consistent with numerical results from a model of two coupled nonisochronous oscillators, each displaying a supercritical Hopf bifurcation. The model revealed that a weak coupling can poise the system of unstable oscillators closer to the bifurcation by a shift in the critical point. In addition, the dynamics of strongly coupled oscillators far from criticality suggested that individual hair bundles may be regarded as nonisochronous oscillators. An optimal degree of nonisochronicity was required for the observed tuning behavior in the coherence of autonomous motion of the coupled system.

Volterra-series approach to stochastic nonlinear dynamics: Linear response of the Van der Pol oscillator driven by white noise

The Van der Pol equation is a paradigmatic model of relaxation oscillations. This remarkable nonlinear phenomenon of self-sustained oscillatory motion underlies important rhythmic processes in nature and electrical engineering. Relaxation oscillations in a real system are usually coupled to environmental noise, which further enriches their dynamics, but makes theoretical analysis of such systems and determination of the equation parameter values a difficult task. In a companion paper we have proposed an analytic approach to a similar problem for another classical nonlinear model-the bistable Duffing oscillator. Here we extend our techniques to the case of the Van der Pol equation driven by white noise. We analyze the statistics of solutions and propose a method to estimate parameter values from the oscillator's time series. We use experimental data of active oscillations in a biophysical system to demonstrate how our method applies to real observations and can be generalized for more complex models.

Otogelin, otogelin-like, and stereocilin form links connecting outer hair cell stereocilia to each other and the tectorial membrane

The function of outer hair cells (OHCs), the mechanical actuators of the cochlea, involves the anchoring of their tallest stereocilia in the tectorial membrane (TM), an acellular structure overlying the sensory epithelium. Otogelin and otogelin-like are TM proteins related to secreted epithelial mucins. Defects in either cause the DFNB18B and DFNB84B genetic forms of deafness, respectively, both characterized by congenital mild-to-moderate hearing impairment. We show here that mutant mice lacking otogelin or otogelin-like have a marked OHC dysfunction, with almost no acoustic distortion products despite the persistence of some mechanoelectrical transduction. In both mutants, these cells lack the horizontal top connectors, which are fibrous links joining adjacent stereocilia, and the TM-attachment crowns coupling the tallest stereocilia to the TM. These defects are consistent with the previously unrecognized presence of otogelin and otogelin-like in the OHC hair bundle. The defective hair bundle cohesiveness and the absence of stereociliary imprints in the TM observed in these mice have also been observed in mutant mice lacking stereocilin, a model of the DFNB16 genetic form of deafness, also characterized by congenital mild-to-moderate hearing impairment. We show that the localizations of stereocilin, otogelin, and otogelin-like in the hair bundle are interdependent, indicating that these proteins interact to form the horizontal top connectors and the TM-attachment crowns. We therefore suggest that these 2 OHC-specific structures have shared mechanical properties mediating reaction forces to sound-induced shearing motion and contributing to the coordinated displacement of stereocilia.

Active Biomechanics of Sensory Hair Bundles

During the detection of sound, hair bundles perform a crucial step by responding to mechanical deflections and converting them into changes in electrical potential that subsequently lead to the release of neurotransmitter. The sensory hair bundle response is characterized by an essential nonlinearity and an energy-consuming amplification of the incoming sound. The active response has been shown to enhance the hair bundle's sensitivity and frequency selectivity of detection. The biological phenomena shown by the bundle have been extensively studied in vitro, allowing comparisons to behaviors observed in vivo. The experimental observations have been well explained by numerical simulations, which describe the cellular mechanisms operant within the bundle, as well as by more sparse theoretical models, based on dynamical systems theory.

Amphibian sacculus and the forced Kuramoto model with intrinsic noise and frequency dispersion

The amphibian sacculus (AS) is an end organ that specializes in the detection of low-frequency auditory and vestibular signals. In this paper, we propose a model for the AS in the form of an array of phase oscillators with long-range coupling, subject to a steady load that suppresses spontaneous oscillations. The array is exposed to significant levels of frequency dispersion and intrinsic noise. We show that such an array can be a sensitive and robust subthreshold detector of low-frequency stimuli, though without significant frequency selectivity. The effects of intrinsic noise and frequency dispersion are contrasted. Intermediate levels of intrinsic noise greatly enhance the sensitivity through stochastic resonance. Frequency dispersion, on the other hand, only degrades detection sensitivity. However, frequency dispersion can play a useful role in terms of the suppression of spontaneous activity. As a model for the AS, the array parameters are such that the system is poised near a saddle-node bifurcation on an invariant circle. However, by a change of array parameters, the same system also can be poised near an emergent Andronov-Hopf bifurcation and thereby function as a frequency-selective detector.

High-order synchronization of hair cell bundles

Auditory and vestibular hair cell bundles exhibit active mechanical oscillations at natural frequencies that are typically lower than the detection range of the corresponding end organs. We explore how these noisy nonlinear oscillators mode-lock to frequencies higher than their internal clocks. A nanomagnetic technique is used to stimulate the bundles without an imposed mechanical load. The evoked response shows regimes of high-order mode-locking. Exploring a broad range of stimulus frequencies and intensities, we observe regions of high-order synchronization, analogous to Arnold Tongues in dynamical systems literature. Significant areas of overlap occur between synchronization regimes, with the bundle intermittently flickering between different winding numbers. We demonstrate how an ensemble of these noisy spontaneous oscillators could be entrained to efficiently detect signals significantly above the characteristic frequencies of the individual cells.

From Biological Cilia to Artificial Flow Sensors: Biomimetic Soft Polymer Nanosensors with High Sensing Performance

We report the development of a new class of miniature all-polymer flow sensors that closely mimic the intricate morphology of the mechanosensory ciliary bundles in biological hair cells. An artificial ciliary bundle is achieved by fabricating bundled polydimethylsiloxane (PDMS) micro-pillars with graded heights and electrospinning polyvinylidenefluoride (PVDF) piezoelectric nanofiber tip links. The piezoelectric nature of a single nanofiber tip link is confirmed by X-ray diffraction (XRD) and Fourier transform infrared spectroscopy (FTIR). Rheology and nanoindentation experiments are used to ensure that the viscous properties of the hyaluronic acid (HA)-based hydrogel are close to the biological cupula. A dome-shaped HA hydrogel cupula that encapsulates the artificial hair cell bundle is formed through precision drop-casting and swelling processes. Fluid drag force actuates the hydrogel cupula and deflects the micro-pillar bundle, stretching the nanofibers and generating electric charges. Functioning with principles analogous to the hair bundles, the sensors achieve a sensitivity and threshold detection limit of 300 mV/(m/s) and 8 μm/s, respectively. These self-powered, sensitive, flexible, biocompatibale and miniaturized sensors can find extensive applications in navigation and maneuvering of underwater robots, artificial hearing systems, biomedical and microfluidic devices.

Signal-Coupled Subthreshold Hopf-Type Systems Show a Sharpened Collective Response

Astounding properties of biological sensors can often be mapped onto a dynamical system below the occurrence of a bifurcation. For mammalian hearing, a Hopf bifurcation description has been shown to work across a whole range of scales, from individual hair bundles to whole regions of the cochlea. We reveal here the origin of this scale invariance, from a general level, applicable to all dynamics in the vicinity of a Hopf bifurcation (embracing, e.g., neuronal Hodgkin-Huxley equations). When subject to natural "signal coupling," ensembles of Hopf systems below the bifurcation threshold exhibit a collective Hopf bifurcation. This collective Hopf bifurcation occurs at parameter values substantially below where the average of the individual systems would bifurcate, with a frequency profile that is sharpened if compared to the individual systems.

Persistence, period and precision of autonomous cellular oscillators from the zebrafish segmentation clock

In vertebrate development, the sequential and rhythmic segmentation of the body axis is regulated by a “segmentation clock”. This clock is comprised of a population of coordinated oscillating cells that together produce rhythmic gene expression patterns in the embryo. Whether individual cells autonomously maintain oscillations, or whether oscillations depend on signals from neighboring cells is unknown. Using a transgenic zebrafish reporter line for the cyclic transcription factor Her1, we recorded single tailbud cells in vitro. We demonstrate that individual cells can behave as autonomous cellular oscillators. We described the observed variability in cell behavior using a theory of generic oscillators with correlated noise. Single cells have longer periods and lower precision than the tissue, highlighting the role of collective processes in the segmentation clock. Our work reveals a population of cells from the zebrafish segmentation clock that behave as self-sustained, autonomous oscillators with distinctive noisy dynamics.

DOI:http://dx.doi.org/10.7554/eLife.08438.001

The timing and pattern of gene activity in cells can be very important. For example, precise gene activity patterns in 24-hour circadian clocks help to set daily cycles of rest and activity in organisms. In such scenarios, cells often communicate with each other to coordinate the activity of their genes. To fully understand how the behavior of the population emerges, scientists must first understand the gene activity patterns in individual cells.

Rhythmic gene activity is essential for the spinal column to form in fish and other vertebrate embryos. A group of cells that switch genes on/off in a coordinated pattern act like a clock to regulate the timing of the various steps in the process of backbone formation. However, it is not clear if each cell is able to maintain a rhythm of gene expression on their own, or whether they rely on messages from neighboring cells to achieve it.

Now, Webb et al. use time-lapse videos of individual cells isolated from the tail of zebrafish embryos to show that each cell can maintain a pattern of rhythmic activity in a gene called Her1. In the experiments, individual cells were removed from zebrafish and placed under a microscope to record and track the activity of Her1 over time using fluorescent proteins. These experiments show that each cell is able to maintain a rhythmic pattern of Her1 expression on its own.

Webb et al. then compared the Her1 activity patterns in individual cells with the Her1 patterns present in a larger piece of zebrafish tissue. The experiments showed that the rhythms in the individual cells are slower and less precise in their timing than in the tissue. This suggests that groups of cells must work together to create the synchronized rhythms of gene expression with the right precision and timing needed for the spinal column to be patterned correctly.

In the future, further experiment with these cells will allow researchers to investigate the genetic basis of the rhythms in single cells, and find out how individual cells work together with their neighbors to allow tissues to work properly.

DOI:http://dx.doi.org/10.7554/eLife.08438.002

Synchronization of Spontaneous Active Motility of Hair Cell Bundles

Hair cells of the inner ear exhibit an active process, believed to be crucial for achieving the sensitivity of auditory and vestibular detection. One of the manifestations of the active process is the occurrence of spontaneous hair bundle oscillations in vitro. Hair bundles are coupled by overlying membranes in vivo; hence, explaining the potential role of innate bundle motility in the generation of otoacoustic emissions requires an understanding of the effects of coupling on the active bundle dynamics. We used microbeads to connect small groups of hair cell bundles, using in vitro preparations that maintain their innate oscillations. Our experiments demonstrate robust synchronization of spontaneous oscillations, with either 1:1 or multi-mode phase-locking. The frequency of synchronized oscillation was found to be near the mean of the innate frequencies of individual bundles. Coupling also led to an improved regularity of entrained oscillations, demonstrated by an increase in the quality factor.

Control of a hair bundle's mechanosensory function by its mechanical load

Hair cells, the sensory receptors of the internal ear, subserve different functions in various receptor organs: they detect oscillatory stimuli in the auditory system, but transduce constant and step stimuli in the vestibular and lateral-line systems. We show that a hair cell's function can be controlled experimentally by adjusting its mechanical load. By making bundles from a single organ operate as any of four distinct types of signal detector, we demonstrate that altering only a few key parameters can fundamentally change a sensory cell's role. The motions of a single hair bundle can resemble those of a bundle from the amphibian vestibular system, the reptilian auditory system, or the mammalian auditory system, demonstrating an essential similarity of bundles across species and receptor organs.

Effect of bidirectional mechanoelectrical coupling on spontaneous oscillations and sensitivity in a model of hair cells

Sensory hair cells of amphibians exhibit spontaneous activity in their hair bundles and membrane potentials, reflecting two distinct active amplification mechanisms employed in these peripheral mechanosensors. We use a two-compartment model of the bullfrog's saccular hair cell to study how the interaction between its mechanical and electrical compartments affects the emergence of distinct dynamical regimes, and the role of this interaction in shaping the response of the hair cell to weak mechanical stimuli. The model employs a Hodgkin-Huxley-type system for the basolateral electrical compartment and a nonlinear hair bundle oscillator for the mechanical compartment, which are coupled bidirectionally. In the model, forward coupling is provided by the mechanoelectrical transduction current, flowing from the hair bundle to the cell soma. Backward coupling is due to reverse electromechanical transduction, whereby variations of the membrane potential affect adaptation processes in the hair bundle. We isolate oscillation regions in the parameter space of the model and show that bidirectional coupling affects significantly the dynamics of the cell. In particular, self-sustained oscillations of the hair bundles and membrane potential can result from bidirectional coupling, and the coherence of spontaneous oscillations can be maximized by tuning the coupling strength. Consistent with previous experimental work, the model demonstrates that dynamical regimes of the hair bundle change in response to variations in the conductances of basolateral ion channels. We show that sensitivity of the hair cell to weak mechanical stimuli can be maximized by varying coupling strength, and that stochasticity of the hair bundle compartment is a limiting factor of the sensitivity.

Integrating the active process of hair cells with cochlear function

Uniquely among human senses, hearing is not simply a passive response to stimulation. Our auditory system is instead enhanced by an active process in cochlear hair cells that amplifies acoustic signals several hundred-fold, sharpens frequency selectivity and broadens the ear's dynamic range. Active motility of the mechanoreceptive hair bundles underlies the active process in amphibians and some reptiles; in mammals, this mechanism operates in conjunction with prestin-based somatic motility. Both individual hair bundles and the cochlea as a whole operate near a dynamical instability, the Hopf bifurcation, which accounts for the cardinal features of the active process.

Active phase and amplitude fluctuations of flagellar beating

The eukaryotic flagellum beats periodically, driven by the oscillatory dynamics of molecular motors, to propel cells and pump fluids. Small but perceivable fluctuations in the beat of individual flagella have physiological implications for synchronization in collections of flagella as well as for hydrodynamic interactions between flagellated swimmers. Here, we characterize phase and amplitude fluctuations of flagellar bending waves using shape mode analysis and limit-cycle reconstruction. We report a quality factor of flagellar oscillations Q = 38.0 ± 16.7 (mean ± s.e.). Our analysis shows that flagellar fluctuations are dominantly of active origin. Using a minimal model of collective motor oscillations, we demonstrate how the stochastic dynamics of individual motors can give rise to active small-number fluctuations in motor-cytoskeleton systems.

The physics of hearing: fluid mechanics and the active process of the inner ear

Most sounds of interest consist of complex, time-dependent admixtures of tones of diverse frequencies and variable amplitudes. To detect and process these signals, the ear employs a highly nonlinear, adaptive, real-time spectral analyzer: the cochlea. Sound excites vibration of the eardrum and the three miniscule bones of the middle ear, the last of which acts as a piston to initiate oscillatory pressure changes within the liquid-filled chambers of the cochlea. The basilar membrane, an elastic band spiraling along the cochlea between two of these chambers, responds to these pressures by conducting a largely independent traveling wave for each frequency component of the input. Because the basilar membrane is graded in mass and stiffness along its length, however, each traveling wave grows in magnitude and decreases in wavelength until it peaks at a specific, frequency-dependent position: low frequencies propagate to the cochlear apex, whereas high frequencies culminate at the base. The oscillations of the basilar membrane deflect hair bundles, the mechanically sensitive organelles of the ear's sensory receptors, the hair cells. As mechanically sensitive ion channels open and close, each hair cell responds with an electrical signal that is chemically transmitted to an afferent nerve fiber and thence into the brain. In addition to transducing mechanical inputs, hair cells amplify them by two means. Channel gating endows a hair bundle with negative stiffness, an instability that interacts with the motor protein myosin-1c to produce a mechanical amplifier and oscillator. Acting through the piezoelectric membrane protein prestin, electrical responses also cause outer hair cells to elongate and shorten, thus pumping energy into the basilar membrane's movements. The two forms of motility constitute an active process that amplifies mechanical inputs, sharpens frequency discrimination, and confers a compressive nonlinearity on responsiveness. These features arise because the active process operates near a Hopf bifurcation, the generic properties of which explain several key features of hearing. Moreover, when the gain of the active process rises sufficiently in ultraquiet circumstances, the system traverses the bifurcation and even a normal ear actually emits sound. The remarkable properties of hearing thus stem from the propagation of traveling waves on a nonlinear and excitable medium.

Transduction channels' gating can control friction on vibrating hair-cell bundles in the ear

Hearing starts when sound-evoked mechanical vibrations of the hair-cell bundle activate mechanosensitive ion channels, giving birth to an electrical signal. As for any mechanical system, friction impedes movements of the hair bundle and thus constrains the sensitivity and frequency selectivity of auditory transduction. Friction is generally thought to result mainly from viscous drag by the surrounding fluid. We demonstrate here that the opening and closing of the transduction channels produce internal frictional forces that can dominate viscous drag on the micrometer-sized hair bundle. We characterized friction by analyzing hysteresis in the force-displacement relation of single hair-cell bundles in response to periodic triangular stimuli. For bundle velocities high enough to outrun adaptation, we found that frictional forces were maximal within the narrow region of deflections that elicited significant channel gating, plummeted upon application of a channel blocker, and displayed a sublinear growth for increasing bundle velocity. At low velocity, the slope of the relation between the frictional force and velocity was nearly fivefold larger than the hydrodynamic friction coefficient that was measured when the transduction machinery was decoupled from bundle motion by severing tip links. A theoretical analysis reveals that channel friction arises from coupling the dynamics of the conformational change associated with channel gating to tip-link tension. Varying channel properties affects friction, with faster channels producing smaller friction. We propose that this intrinsic source of friction may contribute to the process that sets the hair cell's characteristic frequency of responsiveness.

Phase-noise-induced resonance in arrays of coupled excitable neural models

Recently, it is observed that, in a single neural model, phase noise (time-varying signal phase) arising from an external stimulating signal can induce regular spiking activities even if the signal is subthreshold. In addition, it is also uncovered that there exists an optimal phase noise intensity at which the spiking rhythm coincides with the frequency of the subthreshold signal, resulting in a phase-noise-induced resonance phenomenon. However, neurons usually do not work alone, but are connected in the form of arrays or blocks. Therefore, we study the spiking activity induced by phase noise in arrays of globally and locally coupled excitable neural models. We find that there also exists an optimal phase noise intensity for generating large neural response and such an optimal value is significantly decreased compared to an isolated single neuron case, which means the detectability in response to the subthreshold signal of neurons is sharply improved because of the coupling. In addition, we reveal two new resonance behaviors in the neuron ensemble with the presence of phase noise: there exist optimal values of both coupling strength and system size, where the coupled neurons generate regular spikes under subthreshold stimulations, which are called as coupling strength and system size resonance, respectively. Finally, the dependence of phase-noise-induced resonance on signal frequency is also examined.

Enhanced signal-to-noise ratios in frog hearing can be achieved through amplitude death

In the ear, hair cells transform mechanical stimuli into neuronal signals with great sensitivity, relying on certain active processes. Individual hair cell bundles of non-mammals such as frogs and turtles are known to show spontaneous oscillation. However, hair bundles in vivo must be quiet in the absence of stimuli, otherwise the signal is drowned in intrinsic noise. Thus, a certain mechanism is required in order to suppress intrinsic noise. Here, through a model study of elastically coupled hair bundles of bullfrog sacculi, we show that a low stimulus threshold and a high signal-to-noise ratio (SNR) can be achieved through the amplitude death phenomenon (the cessation of spontaneous oscillations by coupling). This phenomenon occurs only when the coupled hair bundles have inhomogeneous distribution, which is likely to be the case in biological systems. We show that the SNR has non-monotonic dependence on the mass of the overlying membrane, and find out that the SNR has maximum value in the region of amplitude death. The low threshold of stimulus through amplitude death may account for the experimentally observed high sensitivity of frog sacculi in detecting vibration. The hair bundles' amplitude death mechanism provides a smart engineering design for low-noise amplification.

A mechanical model of stereocilia that demonstrates a shift in the high-sensitivity region due to the interplay of a negative stiffness and an adaptation mechanism

Stereocilia are the basic sensory units of nature's inertial sensors and are highly sensitive over broad dynamic ranges, which is a major challenge in the design of conventional engineering sensors. The high sensitivity that is maintained by stereocilia was hypothesized to exist due to a combination of adaptation and negative stiffness mechanisms, which shift the region of highest sensitivity toward the active operation range of the stereocilia bundle. To examine the adaptation hypothesis in terms of its potential applicability to future applications regarding the design of inertial sensors, we developed a mechanical mimicry of the interplay between negative stiffness and the adaptation of the stereocilia that produces spontaneous oscillation of the hair bundle. The mechanical model consists of an inverted pendulum and a fixed T-bar that mimic the interaction of two adjacent stereocilia. To focus on the interaction of one gating spring and the corresponding adaptation motor without the effect of coupling from the other gating springs attached to the neighboring stereocilia, we fixed one bar that contains the adaptation motor. To emulate the negative resistance of the tip-link due to the transient stiffness softening by the gating ion channel, a magnet pair was attached to the top of the inverted pendulum and the fixed T-bar. Readjustment of the tip-link tension by the 'slipping down and climbing up' motion of the adaptation molecular motors was demonstrated by the side-to-side movement of the magnet by a step motor. The negative stiffness region was observed near the equilibrium position and shifted with the activation of the adaptation motor. The temporal demonstration of the stiffness shift was measured as a spontaneous oscillation. The results showed that the interplay between the negative stiffness and the adaptation mechanism was mechanically produced by the combination of a repulsive force and its continuous readjustment and is better understood through a parameter study of a biomimetic mechanical system.

Optimal signal amplification in weighted scale-free networks

It has been revealed that un-weighted scale-free (SF) networks have an effect of amplifying weak signals [Acebrón et al., Phys. Rev. Lett. 99, 128701 (2007)]. Such a property has potential applications in neural networks and artificial signaling devices. However, many real and artificial networks, including the neural networks, are weighted ones with adaptive and plastic couplings. For this reason, here we study how the weak signal can be amplified in weighted SF networks by introducing a parameter to self-tune the coupling weights. We find that the adaptive weights can significantly extend the range of coupling strength for signal amplification, in contrast to the relatively narrow range in un-weighted SF networks. As a consequence, the effect of finite network size occurred in un-weighted SF networks can be overcome. Finally, a theory is provided to confirm the numerical results.

Phantom tones and suppressive masking by active nonlinear oscillation of the hair-cell bundle

Processing of two-tone stimuli by the auditory system introduces prominent masking of one frequency component by the other as well as additional "phantom" tones that are absent in the sound input. Mechanical correlates of these psychophysical phenomena have been observed in sound-evoked mechanical vibrations of the mammalian cochlea and are thought to originate in sensory hair cells from the intrinsic nonlinearity associated with mechano-electrical transduction by ion channels. However, nonlinearity of the transducer is not sufficient to explain the rich phenomenology of two-tone interferences in hearing. Here we show that active oscillatory movements of single hair-cell bundles elicit two-tone suppression and distortions that are shaped by nonlinear amplification of periodic stimuli near the characteristic frequency of spontaneous oscillations. When both stimulus frequencies enter the bandwidth of the hair-bundle amplifier, two-tone interferences display level functions that are characteristic both of human psychoacoustics and of in vivo mechanical measurements in auditory organs. Our work distinguishes the frequency-dependent nonlinearity that emerges from the active process that drives the hair bundle into spontaneous oscillations from the passive nonlinear compliance associated with the direct gating of transduction channels by mechanical force. Numerical simulations based on a generic description of an active dynamical system poised near an oscillatory instability--a Hopf bifurcation--account quantitatively for our experimental observations. In return, we conclude that the properties of two-tone interferences in hearing betray the workings of self-sustained "critical" oscillators, which function as nonlinear amplifying elements in the inner ear.

A mean-field approach to elastically coupled hair bundles

We study the dynamics of oscillatory hair bundles which are coupled elastically in their deflection variable and are subject to noise. We present a stochastic description capturing the dynamics of the hair bundles' mean field. In particular, the presented derivation elucidates the origin of the previously described noise reduction by coupling. By comparison of simulations of the approximate dynamics and the full system, we verify our results. Furthermore, we demonstrate that the specific type of coupling considered implies coupling-induced changes in the dynamics beyond mere noise reduction.

Coupling and elastic loading affect the active response by the inner ear hair cell bundles

Active hair bundle motility has been proposed to underlie the amplification mechanism in the auditory endorgans of non-mammals and in the vestibular systems of all vertebrates, and to constitute a crucial component of cochlear amplification in mammals. We used semi-intact in vitro preparations of the bullfrog sacculus to study the effects of elastic mechanical loading on both natively coupled and freely oscillating hair bundles. For the latter, we attached glass fibers of different stiffness to the stereocilia and observed the induced changes in the spontaneous bundle movement. When driven with sinusoidal deflections, hair bundles displayed phase-locked response indicative of an Arnold Tongue, with the frequency selectivity highest at low amplitudes and decreasing under stronger stimulation. A striking broadening of the mode-locked response was seen with increasing stiffness of the load, until approximate impedance matching, where the phase-locked response remained flat over the physiological range of frequencies. When the otolithic membrane was left intact atop the preparation, the natural loading of the bundles likewise decreased their frequency selectivity with respect to that observed in freely oscillating bundles. To probe for signatures of the active process under natural loading and coupling conditions, we applied transient mechanical stimuli to the otolithic membrane. Following the pulses, the underlying bundles displayed active movement in the opposite direction, analogous to the twitches observed in individual cells. Tracking features in the otolithic membrane indicated that it moved in phase with the bundles. Hence, synchronous active motility evoked in the system of coupled hair bundles by external input is sufficient to displace large overlying structures.

Spontaneous voltage oscillations and response dynamics of a Hodgkin-Huxley type model of sensory hair cells

We employ a Hodgkin-Huxley type model of basolateral ionic currents in bullfrog saccular hair cells to study the genesis of spontaneous voltage oscillations and their role in shaping the response of the hair cell to external mechanical stimuli. Consistent with recent experimental reports, we find that the spontaneous dynamics of the model can be categorized using conductance parameters of calcium activated potassium, inward rectifier potassium, and mechano-electrical transduction ionic currents. The model is demonstrated to exhibit a broad spectrum of autonomous rhythmic activity, including periodic and quasiperiodic oscillations with two independent frequencies as well as various regular and chaotic bursting patterns. Complex patterns of spontaneous oscillations in the model emerge at small values of the conductance of Ca(2+) activated potassium currents. These patterns are significantly affected by thermal fluctuations of the mechano-electrical transduction current. We show that self-sustained regular voltage oscillations lead to enhanced and sharply tuned sensitivity of the hair cell to weak mechanical periodic stimuli. While regimes of chaotic oscillations are argued to result in poor tuning to sinusoidal driving, chaotically oscillating cells do provide a high sensitivity to low-frequency variations of external stimuli.

A differentially amplified motion in the ear for near-threshold sound detection

The ear is a remarkably sensitive pressure fluctuation detector. In guinea pigs, behavioral measurements indicate a minimum detectable sound pressure of ∼20 μPa at 16 kHz. Such faint sounds produce 0.1-nm basilar membrane displacements, a distance smaller than conformational transitions in ion channels. It seems that noise within the auditory system would swamp such tiny motions, making weak sounds imperceptible. Here we propose a new mechanism contributing to a resolution of this problem and validate it through direct measurement. We hypothesized that vibration at the apical side of hair cells is enhanced compared with that at the commonly measured basilar membrane side. Using in vivo optical coherence tomography, we demonstrated that apical-side vibrations peaked at a higher frequency, had different timing and were enhanced compared with those at the basilar membrane. These effects depend nonlinearly on the stimulus sound pressure level. The timing difference and enhancement of vibrations are important for explaining how the noise problem is circumvented.

A critique of the critical cochlea: Hopf--a bifurcation--is better than none

The sense of hearing achieves its striking sensitivity, frequency selectivity, and dynamic range through an active process mediated by the inner ear's mechanoreceptive hair cells. Although the active process renders hearing highly nonlinear and produces a wealth of complex behaviors, these various characteristics may be understood as consequences of a simple phenomenon: the Hopf bifurcation. Any critical oscillator operating near this dynamic instability manifests the properties demonstrated for hearing: amplification with a specific form of compressive nonlinearity and frequency tuning whose sharpness depends on the degree of amplification. Critical oscillation also explains spontaneous otoacoustic emissions as well as the spectrum and level dependence of the ear's distortion products. Although this has not been realized, several valuable theories of cochlear function have achieved their success by incorporating critical oscillators.