Dynamics of Freely Oscillating and Coupled Hair Cell Bundles under Mechanical Deflection

Abstract

In vitro, attachment to the overlying membrane was found to affect the resting position of the hair cell bundles of the bullfrog sacculus. To assess the effects of such a deflection on mechanically decoupled hair bundles, comparable offsets were imposed on decoupled spontaneously oscillating bundles. Strong modulation was observed in their dynamic state under deflection, with qualitative changes in the oscillation profile, amplitude, and characteristic frequency of oscillation seen in response to stimulus. Large offsets were found to arrest spontaneous oscillation, with subsequent recovery upon reversal of the stimulus. The dynamic state of the hair bundle displayed hysteresis and a dependence on the direction of the imposed offset. The coupled system of hair bundles, with the overlying membrane left on top of the preparation, also exhibited a dependence on offset position, with an increase in the linear response function observed under deflections in the inhibitory direction.

Introduction

Auditory and vestibular detection by the inner ear displays sensitivity to mechanical displacements that reaches into the subnanometer regime (1,2), while operating immersed in an aqueous environment. Thomas Gold first proposed in 1948 (3) that the auditory system contains an internal energy-consuming amplifier to sustain detection sensitivity under these overdamped conditions. Athough the cellular mechanisms underlying the active process have still not been fully determined, the phenomenon has been extensively studied and shown to be crucial to the acuity of hearing (4,5).

Amplification is performed by hair cells, the mechanical sensors of the inner ear. Active motility by the stereociliary bundles has been proposed as the amplification mechanism in nonmammalian hair cells. Electrically evoked motility in the cell soma and mechanical motion of the bundle are believed to contribute to the active process in mammalian hair cells (6–10). Spontaneous oscillations have been observed in vitro in hair bundles of the turtle papilla (11) and the bullfrog sacculus (12–17) and constitute one of the signatures of an internal amplifier. This spontaneous active motility has been shown to result from an interplay between two processes. Gating of the mechanically sensitive ion channels in the stereocilia leads to bistability in the position of the bundle. An array of molecular motors (Myosin 1c) has been proposed to be physically connected to the transduction complex, and to climb and slip along the actin filaments that form the core of the stereocilia (18). This adaptation process would continuously adjust the position of the bundle, and in conjunction with mechanical gating of the transduction channels, suffice to explain spontaneous oscillations (14). A compressive nonlinearity has been observed in the response of the auditory system, and processes that eliminate it were shown to significantly raise the threshold of detection (2,16). Active motility evoked in individual hair bundles in vitro displays a fractional power-law dependence on applied input, and has been proposed to explain the fractional power-law dependence of the basilar membrane on the intensity of applied sound (2).

Hair-bundle dynamics has therefore been modeled with systems of coupled nonlinear differential equations (19–24). Dynamic systems theory predicts the existence of two regimes—a stable one, where the system is responsive but quiescent; and an unstable regime, where spontaneous oscillations arise in the absence of external input. The two regimes are separated by a bifurcation point (25–28), where the system crosses from quiescence to a limit-cycle oscillation. A control parameter tunes the system toward or away from the critical point and thus adjusts the sensitivity of its response. A cellular analog of this control parameter could provide a means by which the hair cell can modulate its active amplification and thus tune its sensitivity to external input.

We propose mechanical offset imposed on the position of the hair bundle as a potential control parameter that can determine its dynamic state. Myosin-based adaptation serves to reduce the transduction currents evoked by steady-state stimuli by adjusting the zero-point position of the bundle. However, adaptation has been shown to be incomplete: the response is attenuated by ∼80% but not completely eliminated (29). An offset imposed on the position of a hair bundle will therefore have a residual effect on the open probability of the channel at rest and thus modulate the mechanical sensitivity of the cell.

In vivo, hair bundles of the bullfrog sacculus are coupled to the otolithic membrane, an extracellular matrix anchored to the apical surface of the sensory epithelium by a layer of columnar filaments (5–8 μm in length) (30,31). Each hair bundle protrudes into a ∼10 μm cavity or pit in the membrane. Light and electron micrographs reveal that the bundles are anchored to one wall of each pit (30,32) by filamentous fibers terminating at the kinociliary bulbs. Attachment to the otolithic membrane may therefore impose offsets in the hair bundle position with respect to their free-standing state. When connected to the overlying membrane, saccular hair bundles do not spontaneously oscillate (33), suggesting that the natural coupling tunes them into the quiescent regime.

We explore the role of steady-state mechanical deflection on the spontaneous active motility in decoupled hair bundles. We find that active oscillation is suppressed, with the transition displaying an admixture of amplitude and frequency modulation, dependent on the polarity and history of the applied offsets. Numerical simulations are presented that include an internal element with a slowly varying compliance and shown to capture the experimental observations. We further explored the effects of steady-state deflection on the collective response of the coupled system, and found that the linear response was enhanced by negative deflection.

Methods

Biological preparation

Before performing experiments, all protocols for animal care and euthanasia were approved by the UCLA Chancellor's Animal Research Committee in accordance with federal and state regulations. Adult bullfrogs were obtained from Rana Ranch (Twin Falls, Idaho). The inner ears were excised and the saccular maculae were separated from the surrounding tissue and mounted in a two-compartment chamber as described in previous publications. For experiments on free-standing hair bundles, dissections were performed in artificial perilymph (110 mM Na⁺, 2 mM K⁺, 1.5 mM Ca²⁺, 118 mM Cl⁻, 3 mM D-glucose, 1 mM sodium pyruvate, 1 mM creatine, and 5 mM HEPES).

To avoid exposing the otolithic membrane to artificially high extracellular calcium concentrations during the dissection, sacculi used in experiments in which the membrane was left intact on the epithelium were dissected in a modified artificial endolymph (117.5 mM N-methyl- D-glucamine, 2 mM Na⁺, 0.25 mM Ca²⁺, 118 mM Cl⁻, 3 mM D-glucose, and 5 mM HEPES). Once mounted in the two-compartment chamber, the apical surface of the preparation was bathed in regular artificial endolymph (117.5 mM K⁺, 2 mM Na⁺, 0.25 mM Ca²⁺, 118 mM Cl⁻, 3 mM D-glucose, and 5 mM HEPES), whereas the basolateral surface was immersed in artificial perilymph. All solutions were titrated to pH 7.3, had osmolalities adjusted with sucrose to 230 mOs/kg, and were freshly oxygenated before use. For control experiments, 20–40 μM gentamicin sulfate, a blocker of the transduction channel, was added to the artificial endolymph. The solution in the apical compartment was replaced via a fluid exchange, and measurements were repeated after a 5-min incubation period. All experiments were performed at room temperature.

Imaging

Samples were imaged with an upright B51X microscope (Olympus, Melville, NY) with a 20× water-immersion objective (XLUMPLF20XW, 0.95 N.A.; Olympus) and illuminated with an X-Cite 120 W halogenide lamp (Carl Zeiss, Peabody, MA). Images were further magnified with a double-Gauss variable-focus lens to ∼400× and projected onto a complimentary metal-oxide semiconductor camera (SA1.1; Photron, San Diego, CA). The microscope was mounted onto an optics table and placed in a sound isolation booth. High-speed video recordings were taken at 12 bit pixel depth at 500 or 1000 frames-per-second (fps). A calibrated Ronchi ruling (Edmund Optics, Barrington, NJ) was used to measure the spatial scale, which was 53 nm/pixel. For each hair bundle or other features to be tracked, we used software written in MATLAB (The MathWorks, Natick, MA) that performs a line scan through the brightest row of pixels and fits the intensity profile to a Gaussian I = I₀ exp(−(x − x₀)/2σ²). Extracting the center position x₀ for each frame in the video record determined the time-dependent trace of the motion. The root mean-square noise floor in the bundle motion was typically ∼3 nm.

Mechanical stimulation of decoupled hair bundles

Mechanical stimuli were delivered to freestanding spontaneously oscillating hair bundles, after enzymatic removal of the otolithic membrane. Borosilicate glass capillaries were pulled with a Flaming/Brown pipette puller (Sutter Instrument, Novato, CA); the rods were then pulled at right angles to the shafts with a home-built puller. For single-bundle measurements, the probes' cantilevers were 100–300 μm in length and 0.5–1 μm in diameter. The stiffness and viscous drag coefficient were measured by recording the Brownian motion of the probe's tip in water and fitting the power spectrum to a Lorentzian distribution. The stiffness of the probes used in these experiments ranged from 100–400 μN/m. The tips of the glass fibers were dipped in Concanavalin A, a highly charged protein shown to improve adhesion to the bundles. Probes were mounted onto a piezoelectric stack actuator (PiezoJena PA 4/12; Piezosystem Jena, Jena, Germany whose amplifier was controlled with a function generator (AFG3022; Tektronix, Beaverton, OR). A motorized three-dimensional micromanipulator was used to position the probe and bring it into contact with the hair bundle. Continuous ramps were applied to the attached bundles, with displacements up to 2 μm applied to the base of the elastic probe for a duration of 10 s.

Stimulation of bundles coupled to the otolithic membrane

For experiments on the coupled system, thick probes (diameter > 5 μm) were used to deliver lateral displacements to the otolithic membrane. No measurable flexion was observed between the bases and tips of these fibers. The probes were brought into contact with the otolithic membrane 50–200 μm from the bundles that were recorded. Linear ramps with a peak amplitude of 1 μm were applied over 10 s, deflecting different sets of bundles in the excitatory direction (toward tallest stereocilia, shown to lead to preferential channel opening and conventionally defined to be positive) or the inhibitory direction (toward row of shorter stereocilia, leading to channel closure and defined to be negative). A sinusoidal modulation of 100 nm and frequency 5–200 Hz was superimposed on the ramp and sent to the base of the probe. Frequencies were chosen to span the physiological range of the sacculus. The effects of the steady-state deflection were quantified by the dimensionless response function (15): $\tilde{\Pi }\left(\omega \right)=\tilde{x}\left(\omega \right)/\tilde{\Delta }\left(\omega \right)$, in which Δ(t) is the displacement of the base of the probe, x(t) is the evoked response of the bundles, and tilde (^∼) denotes the Fourier transform.

The time-dependent traces were divided into 1 s segments, and the response $\left|\tilde{\Pi }\right|$ plotted at each mean offset position. To investigate the response of hair bundles subject to larger displacements, the micromanipulator was used to move the otolithic membrane in the negative direction in discrete 200 nm steps. At each of the steady-state offsets, the membrane was stimulated with a 100 nm sine wave over the same frequency range as above and the response function was calculated at each offset position. As a positive control, we removed the otolithic membrane from selected preparations after the measurement and imaged a 150 × 50 μm² area of the epithelium covering some 30–40 bundles. For measurements in which the bundles were driven with a sinusoidal stimulus, results were averaged on a cycle-by-cycle basis for each bundle, or over the ensemble of bundles recorded simultaneously.

Numerical simulation

We performed simulations by following the numerical model described previously (34). A hair bundle is assumed to consist of 30–50 stereocilia, exhibiting a net pivot stiffness of K_sp. Tips of the stereocilia are assumed to be connected by a fixed gating spring (K_gs). A positive deflection of the bundle tenses the gating spring and hence increases the opening probability of the transduction channels (p₀) to which the tip links are attached. Opening of the channels is accompanied by an extension of the gating spring (swing of the channel d), which releases the tension and allows further movement in the positive direction. An array of myosin motors maintains tension in the gating spring, by allowing the transduction complex to climb and slip along the actin filaments that form the core of the stereocilium. The position of the motors is denoted by X_a, and the slipping rate is dependent on the intracellular calcium concentration. In the absence of calcium, myosin motors climb along the actin filaments. As calcium enters the stereocilia, the motors slip down the actin, reducing the tension in the gating spring and thus decreasing the opening probability of the transduction channels (12,13,15).

The equation of motion of the bundle in the overdamped limit is given by

$$\xi \frac{dX}{dt}=-N\gamma K_{gs}\left(\gamma X-X_a+X_c-p_{o}d\right)-K_{sp}\left(X-X_{sp}\right)+K_f\left(\Delta -X\right)+\eta ,$$ (1)

where X_c and X_sp are offset terms, the opening probability p₀ follows a two-state Boltzmann distribution, Δ represents the displacement imposed on the base of the probe, assumed to rise linearly with time, and η denotes the noise in the bundle movement.

The adaptation motors are described by

$$\frac{d{X}_a}{dt}=-C+S\left[K_{gs}\left(\gamma X-X_a+X_c-p_{o}d\right)-K_{es}\left(X_a-X_{es}\right)\right]+\frac{{\eta }_a\gamma }{{\lambda }_a},$$ (2)

where C is the rate of climbing, assumed to be constant, S is the rate of slipping, and K_es is the stiffness of an extension spring that limits the movement of the motor complex. The rate of slipping is assumed to be linearly proportional to the calcium concentration at the motor ([Ca²⁺]_motor), and the influx of calcium through the channels follows the Goldmann-Hodgkin-Katz equation (35). The noise term η describes channel clatter and hydrodynamic friction, and η_a arises from stochastic binding and unbinding of the motors to actin filaments (22). Both noise terms are assumed to be Gaussian with zero mean.

In a recent work (34), we proposed the existence of a variable gating spring element, whose stiffness decreases upon calcium binding, the dynamics of which are slow compared to those of the myosin motors. We assume the gating stiffness (K_gs) to be linearly dependent on the probability of calcium binding to the element (p_gs),

$$K_{gs}=K_{gs0}-K_{gs1}{p}_{gs},$$ (3)

where K_gs0 and K_gs1 are constants. The probability p_gs follows the typical rate equation

$$\frac{d{p}_{gs}}{dt}=k_{gs,on}{\left[{\text{Ca}}^{2+}\right]}_{gs}\left(1-p_{gs}\right)-k_{gs,off}{p}_{gs},$$

with k_gs,on and k_gs,off denoting the rates of binding and unbinding of calcium to the variable gating spring element. The calcium concentration at the gating spring ([Ca²⁺]_gs) is assumed to be equal to that at the myosin motors [Ca²⁺]_motor ≈ [Ca²⁺]_gs, as the two are assumed to be in close proximity.

Results

Steady-state offsets under the otolithic membrane

In the sacculus, the otolithic membrane couples and imposes a load on the hair bundles. Transmission electron micrographs of cross-sectional slices reveal a dense mesh of thin filaments coupling the kinociliary bulbs to the membrane (31,32). Both light and electron micrographs have shown that the bundles are positioned adjacent to the pit boundaries (30–32). However, fixation procedures applied to those preparations could easily cause differential shrinkage of various structural elements, hence conclusions as to the native zero-point deflection of the bundles must be drawn from live and fluid-immersed epithelia.

We examined the resting position of hair bundles under conditions that mimicked more closely those in vivo, by imaging the bundles of freshly dissected epithelia through the otolithic membrane. Consistent with the results from electron microscopy, hair bundles were found to be adjacent to the edges of the ellipsoidal contours formed by the edges of the pits. Fig. 1, A and B, shows two examples of quiescent bundles with the tallest row of stereocilia adjacent to the pit boundary.

Figure 1.

Coupling to the otolithic membrane alters the hair bundles' resting position. The micrographs in panels A and B show two hair bundles that did not exhibit spontaneous oscillations while coupled to the otolithic membrane. The time-dependent traces of their positions are displayed in the panels to the right of the images. Because of light piping, the tallest row of stereocilia appears as bright linear features in each micrograph (indicated by an arrow in panel A). The kinociliary bulbs appear as circular features to the left of the stereocilia. Each hair bundle in the frog sacculus protrudes into a cavity or pit in the membrane, whose boundary appears as the elliptical feature around each bundle. Note that in panels A and B the bundles lie quite close to the boundaries. The pit boundaries are particularly clear in panels A, B, and C. The micrographs in C and D show two hair bundles that displayed significant spontaneous oscillation beneath the membrane. Note that the mean positions of the bundles are located toward the centers of their respective pits, suggesting that the bundles have been detached from the membrane. In panel C, the kinociliary bulb (indicated by the arrow) appears to have been severed from the bundle. Traces have been plotted with positive displacements in the excitatory direction, to the left in the images, and for the cells in panels B and D, we have corrected for the orientation of the bundles. All data were recorded at 500 fps. Scale bar: 5 μm.

In a previous publication, we reported that spontaneous oscillations were suppressed by the presence of the otolithic membrane (33). Only two exceptions were observed from over 400 cells measured for that experiment. Fig. 1, C and D, shows micrographs of the two cells that did sustain oscillation beneath the membrane. As indicated in the figure, the two bundles' mean positions were near the center of their respective pits, indicating detachment from the otolithic membrane. The apparent shift in the resting positions upon detachment from the membrane suggests that intact hair bundles may be pretensed by this loading. This finding indicates a native offset in the average position of the bundles coupled to the otolithic membrane.

Offsets modulate spontaneous oscillation

To explore the effects of offset on the dynamics of individual hair bundles, we imposed slow deflections on their position via glass probes. By convention, positive displacements are defined to be toward the kinociliary bulb, in a direction that promotes channel opening, whereas negative displacements are in the opposite direction, promoting channel closure. Fig. 2 displays records from spontaneously oscillating hair bundles subjected to slow ramps (one cycle of a triangular waveform). As can be seen in all records, the temporal characteristics of spontaneous oscillations are strongly affected by the applied deflection.

Figure 2.

(A) Spontaneous hair bundle oscillation from a single hair cell subjected to slow ramps, imposed with an attached probe of 100 μN/m. One cycle of a triangle wave was sent to the base of the probe with a 30-s period and amplitudes 0.4 μm (top trace), 1.2 μm (middle trace), and 2.0 μm (bottom trace). The portion of the ramps shown moves from positive deflection, through zero deflection (at the 15 s mark), and ends at negative deflection. Transitions between single and multimode oscillation can be seen as well as velocity-dependent suppression dynamics. (B) Spontaneous hair bundle oscillations from two different cells (top traces are from one cell, bottom traces are from another), subjected to slow ramps, imposed with an attached probe of 100 μN/m. One cycle of a triangle wave with amplitude 1.2 μm and period 30 s was sent to the base of the probe. Nine seconds of negative deflection, during which the oscillation remained suppressed, have been removed (as indicated by the dashed line) from the figure to elucidate a more-interesting dynamics. The differences between an increasing (left traces) and decreasing (right traces) negative ramp can be seen. (C) Spontaneous hair bundle oscillations from two different cells, subjected to slow ramps, imposed with an attached probe of 400 μN/m. One cycle of a triangle wave was applied at an amplitude 0.4 μm to the base of a 400 μN/m probe over a period of 20 s. The figure shows only the first half of the record, where the bundles were subject to positive displacements. Positive suppression and recovery can be seen in both cells. (D) Spontaneous hair bundle oscillation, subjected to slow ramps. One half-cycle of a triangular waveform was applied over 28 s with amplitude 0.8 μm and a probe of 100 μN/m in stiffness. From the trace, one can see slow adaptation within the system. At the initial offset, the bundle is oscillating, but upon returning to the same offset after positive displacements, the bundle crosses into the quiescent state. (Note: In figures with more than one trace, offsets are added for clarity.)

Applying an initial positive offset to the bundle causes an increase in the main frequency of the oscillation, with the bundle spending an increasing fraction of time in the positively deflected state, corresponding to preferential channel opening, and exhibiting noisy and spikelike closures, whereas return from positive deflection led to a recovery of the native spontaneous oscillation. Deflecting the bundle in the negative direction leads to a decrease in the frequency and eventual suppression of spontaneous oscillation, with the bundle remaining in the closed-channel state. This suppression is likewise reversible, with robust oscillation regained upon reduction of the negative offset. We consistently observed that the oscillation is much more readily suppressed in the negative direction than the positive. In both directions, changes in the oscillation profile during the reduction of applied offset are not simply time-reversals of the behavior observed under initial deflection but show qualitatively distinct effects.

Specific examples of the above effects are displayed in Fig. 2. Measurements were obtained from preparations in which the otolithic membrane had been removed, thus decoupling the hair cells. Fig. 2 A shows traces of spontaneous motility recorded from an individual bundle subjected to ramps of increasing amplitude (top to bottom: 0.4 μm,1.2 μm, and 2.0 μm) with a period of 30 s applied via a probe of stiffness 100 μN/m. The portion of the ramp displayed in the figure shows the change in oscillation dynamics as the base of the stimulus probe moved from positive to negative deflection, with the zero at 15 s. As the imposed offset moved toward the negative direction, the oscillation frequency decreased, leading to eventual suppression of the limit-cycle oscillation at higher deflections. The dynamics of the transition from oscillatory behavior to quiescence was seen to be dependent upon the rate of the applied ramp. During the imposition of slower ramps, the suppression of oscillation proceeded primarily via a divergence of the period of oscillation (top two ramps). This is consistent with effects of steady-state deflection (34), which have been shown to suppress active motility primarily through slowing down of the oscillation. At increasing ramp speeds, an admixture of amplitude modulation was clearly observed (bottom ramp). In the figure, ramps have been offset from each other by 110 nm for visual clarity.

The majority of hair bundles displayed a complex oscillation profile, with oscillatory behavior interspersed with quiescent intervals. During these nonoscillatory periods, the bundles still exhibited slow movement in the negative direction. According to models of spontaneous oscillation, this corresponds to the closed-channel state during which the myosin motors steadily climb, increasing the tension in the tip links and thus deflecting the bundle. Negative offsets increased the occurrence and prolonged the duration of these quiescent intervals. Two examples of this effect, obtained from different cells, are shown in the left part of Fig. 2 B. Increases in the negative offset reduced the amplitude of the innate movement, eventually entirely suppressing the oscillation. Interestingly, the dynamics of this transition between oscillatory and quiescent behavior was dependent on history and direction of the stimulus and not simply on offset value. The traces for the return from negative suppression for the same two cells are shown in the right part of Fig. 2 B. Oscillation recovery proceeded via distinct dynamics and at larger negative offsets than oscillation suppression. The dashed line denotes a discontinuity in the figure's time axis where a portion of each trace, during which the bundle was quiescent, was removed for better visualization of transition dynamics. The stimulus applied to the base of the probe was 1.2 μm in amplitude with a period of 30 s and a probe stiffness of 100 μN/m. Negative displacements above 1 μm applied to the base of a 100-μN/m probe, corresponding to ∼400 nm of mean bundle deflection, generally led to full suppression of the spontaneous oscillation even at slowest ramps.

Deflections in the positive direction likewise suppressed spontaneous bundle oscillation but with distinct dynamics (Fig. 2 C). For cells exhibiting multimode oscillation, small positive offsets reduced the occurrence of quiescent intervals, rendering the oscillation more periodic (left part of Fig. 2 A). Further displacements increased the frequency of oscillation, with the appearance of irregular and noisy negative twitches at larger deflections, until active motility was fully suppressed in the open-channel state. Shown in Fig. 2 C are examples of positive suppression recorded from two different hair cell bundles. As with negative suppression, bundle behavior was dependent on the history of applied ramps, with recovery exhibiting distinct features from suppression. The cells were found to be more adaptive in the positive direction, with higher displacement forces and/or faster ramps required to suppress spontaneous motility.

As noted previously, hair bundles consistently displayed long-term dependence on the history of applied offset, as illustrated in Fig. 2 D. The positive-half of an 800-nm amplitude triangle wave was applied to the base of a 100 μN/m probe for a duration of 28 s, with the bundle at zero offset both at the beginning and at the end of the stimulus. As can be seen from the trace, the return ramp from positive deflection evoked behavior reminiscent of that induced by negative deflection from zero offset. Note that at the onset of the stimulus ramp, the hair bundle exhibited robust spontaneous oscillation, whereas it was rendered quiescent as it approached zero offset upon termination of the stimulus.

Numerical analysis

In Fig. 3, we present the results of a numerical simulation showing the effects of a linear ramp imposed on the position of a hair bundle. The model is based on a variable-stiffness element inside the stereocilia, whose elastic properties are modulated by calcium binding (34). The simulation was shown to reproduce the multimode oscillation pattern exhibited by the majority of recorded cells, provided that the dynamics of calcium attachment and detachment are slow with respect to other timescales of the system.

Figure 3.

Numerical simulations of spontaneous hair bundle oscillation, with a slow ramp imposed on the resting position with a probe of 100 μN/m stiffness. Half a cycle of a triangle wave was applied in the (A) positive and (B) negative directions. The model required higher deflections in the negative direction to reach quiescence. The oscillatory behavior near suppression in the positive direction showed both amplitude and frequency modulation. The transition to quiescence under negative offset exhibited frequency modulation. Significant hysteresis was observed under both directions. Parameters used in this simulation are listed in Table S1 in the Supporting Material.

The transition from multi- to single-mode oscillation and the suppression of active motility were readily captured by the numerical simulation. As shown in Fig. 3, the return ramps induced different dynamics in the simulated cells than simple time-reversals of the patterns of bundle oscillation under initial ramps. This dependence of the transition from oscillatory to quiescent behavior on the history of applied deflections was consistent with that seen in the experimental data. Simulations that included only myosin-mediated adaptation likewise captured a hysteresis, but one that only reproduced the frequency but not the amplitude modulation of spontaneous oscillation (see Fig. S1 in the Supporting Material). The inclusion of the slowly varying calcium-dependent stiffness element therefore captured the salient features observed in the experimental data.

Offsets affect the response function of hair bundles coupled to the otolithic membrane

Fig. 4 shows the ensemble-average response function versus mean displacement for four neighboring cells recorded in the same field of view. We recorded evoked responses at frequencies spanning the physiological range of the sacculus: 5 Hz (N = 14 cells on three preparations), 25 Hz (seven cells on two preparations), 50 Hz (nine cells on three preparations), 100 Hz (nine cells on two preparations), and 200 Hz (nine cells on two preparations) simultaneously subjected to slow deflections up to ±200–300 nm. Offset deflections in the negative direction increased the response to sinusoidal stimuli, whereas deflections in the positive direction had the opposite effect. At lower frequencies, up to 50 Hz, all cells displayed an increase in response with negative deflections whereas the decrease in response upon positive deflections was weaker on some preparations. At frequencies of 100 and 200 Hz, near the upper limit of the range of the sacculus, the response function showed less modulation as a function of the bundles' mean positions. To verify the presence of a robust active process, for two preparations, we removed the otolithic membrane at the conclusion of the measurement and imaged a 150 × 50 μm² area of the epithelium spanning some 30–40 cells. On these samples, the majority of bundles showed spontaneous oscillations, indicating that the active process was maintained throughout the measurement.

Figure 4.

Slow and steady-state offsets alter the response of hair bundles coupled to the otolithic membrane. (A) Ensemble average (N = 4 cells) response function versus average offset under slow (10 s) ramps in the excitatory (positive) and inhibitory (negative) directions. An uncompliant probe was used to deliver a linear ramp with a sinusoidal modulation to the otolithic membrane. When deflected in the inhibitory direction, the coupled bundles' response function at each frequency increased. Deflections in the excitatory direction had the opposite effect. (Open circles) 50 Hz, (solid circles) 100 Hz, and (shaded circles) 200 Hz. Recorded at 1000 fps. The schematic diagram below shows the command signal sent to the base of the probe. The large and small scale bars indicate, respectively, the magnitude of the linear ramp and the sinusoidal signal while the frequency is schematic. (B) Response function of a hair bundle under large steady-state deflections in the inhibitory direction. To explore the effects of larger offsets in the inhibitory direction, a micromanipulator and a glass fiber were used to apply ∼200-nm step deflections to the otolithic membrane. The membrane was then stimulated sinusoidally at each offset position. For this bundle, the response function reached a peak value at deflections ∼1.4 μm in the negative direction at each frequency. (Open circles) 50 Hz, (solid circles) 100 Hz, and (shaded circles) 200 Hz. Recorded at 1000 fps.

To explore the effect of larger (up to 2.5 μm) steady-state offsets, we imposed discrete 200-nm step deflections in the negative direction, followed by sinusoidal stimulation at the same frequencies: 5, 25, 50, 100, and 200 Hz. Fig. 4 B shows the response function of a selected hair bundle imaged through the otolithic membrane. At all frequencies, the linear response function of this bundle peaked at deflections ∼1.4 μm in the negative direction. At large deflections, significant variation was observed in the linear response function, both across cells within a preparation and across different preparations. We recorded 14 hair bundles on three preparations; nine showed one or more peaks in their response function at offsets between 0.3 μm and 1.4 μm in the inhibitory direction. (Note that under natural circumstances, the hair bundles are offset in the positive direction by attachment to the otolithic membrane. Externally imposed displacement in the negative direction therefore removes some of this offset, bringing them closer to the average position they display when decoupled.)

Discussion

The temporal profile of spontaneous oscillation was shown to be strongly affected by mechanical deflections imposed on the position of the hair bundle. Slow offsets were found to evoke transitions from a multimode oscillation to a single limit cycle, and from spontaneous oscillation to quiescence. These changes in the dynamic state of the bundle indicate modulation of an internal control parameter.

Our results indicate that the system can support a supercritical Hopf bifurcation, an infinite period bifurcation, or both. The suppression of spontaneous oscillation by imposed ramps occurs primarily by frequency modulation, indicating proximity to an infinite-period bifurcation (known in dynamical systems literature as a “saddle-node bifurcation on invariant cycle” (25)). Under ramps of increasing speed, an admixture of amplitude modulation was observed at the approach to quiescence, indicating a multicritical point (Bogdanov-Takens bifurcation (27)). Finally, recovery ramps typically exhibited strong modulation in the amplitude of oscillation, indicative of a supercritical Hopf. Mechanical deflection may therefore poise the hair bundles in the proximity of different bifurcations, characterized by different response dynamics.

We hence propose offset on the resting position of the stereociliary bundle to be a control parameter that affects the dynamic state of a hair cell. Although the molecular machinery for this process is outside the scope of this work, a number of possible mechanisms have been proposed whereby a bundle could adjust the tension of its internal constituents and thus fine-tune its resting position. The incompleteness of the myosin-based adaptation process (29) has been modeled numerically by introducing a putative extension spring in parallel with the motors that limits their range of motion. If the as-yet-unidentified extension spring were modified by biochemical pathways within the cell, it could effect a change in the steady-state position of the stereociliary bundle.

A possible candidate for modulation of internal stiffness elements in the hair bundle is calcium. Experimental findings consistently showed it to modulate frequency and amplitude of spontaneous oscillations and to enhance the mechanical twitch in response to transient stimuli (14,17). Calcium affects myosin-motor activity (18), thus controlling the speed of adaptation in stereociliary bundles. Calcium feedback has been theoretically proposed to maintain self-tuned critical oscillations (21). Further, a variable gating spring element in series with the tip link has been hypothesized to underlie the fast adaptation process in hair cells (36). Under resting conditions, tip links maintain the stereocilia under tension, and severing them was shown to induce a small positive displacement in the position of the bundle (37). Hence, any modification of a variable gating spring element in series with the tip link would change the bundle's resting-state position.

In a recent study (34), we had proposed a variable gating spring that was likewise modulated by calcium attachment but which exhibited slow dynamics. In the present work, we demonstrate:

• First, that the effects of offset on the oscillatory dynamics of a bundle are consistent with the variable gating-spring model. In particular, transitions from multimode to single-mode oscillation are reproduced by the model.

• Second, the simulation captures the approach to quiescence, accompanied by both frequency and amplitude modulation.

• Third, hystereses evoked by slow ramps in opposite directions are consistently seen in both the experimental and theoretical traces.

As the stiffness of the variable gating element is dependent on the internal calcium concentration, cellular control of internal calcium levels could thus modulate the internal stiffness of the bundle. Pretensing of the bundle by the otolithic membrane (as seen in Fig. 1) would in turn lead to an offset in its resting position, as the stiffness of the gating spring was varied. Hence, offset in the bundle position could provide the mechanism by which calcium feedback could exert control over the dynamic state of the hair bundle. Under natural conditions, hair bundles are coupled by the otolithic membrane and operate in the quiescent regime. We probed the sensitivity of the coupled system and found it to be enhanced by negative deflection. Our experiments performed on individual hair bundles showed that offsets can induce changes in the characteristic frequency of a cell, particularly in the vicinity of the critical point. Hence, any slight modulation of the resting positions of individual hair bundles could affect the number of cells that respond optimally to particular input frequencies. In the sacculus, which specializes in low-frequency broadband detection, this control mechanism could serve to enhance sensitivity and to extend the low-frequency range.