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.


Introduction
Hair cells of the inner ear are the mechanical sensors that detect air-and ground-borne vibrations and transduce them into electrical signals (reviewed in [1][2][3][4]). A hair cell consists of a cell soma and an array of columnar structures called the stereovilli, which comprise the hair bundle. The actin-filled stereovilli are arranged in rows of increasing height and are coupled together by tip links [5]. The tips of the hair bundles are connected to an overlying membrane, termed the otolithic membrane in the bullfrog sacculus. An incoming stimulus induces a shearing motion between the overlying membrane and the tissue in which the cells are embedded, deflecting the bundle and thus increasing the tension on the tip links. Mechanically sensitive ion channels that are physically coupled to the tip links open in response and allow the inflow of cations [6,7].
When stimulated by an incoming signal, the hair bundle oscillates in a viscous fluid environment. An internal active process, which pumps energy to amplify the sound-induced vibrations, has been suggested to overcome viscous dissipation [8]. Two different mechanisms have accordance with federal and state regulations. Prior to dissection procedures, animals were euthanized while under pentobarbital anesthesia. The sacculi were excised from the inner ears of the American bullfrog (Rana catesbeiana). The preparation was mounted in a two-compartment chamber, simulating the fluid separation of the in vivo environment, with artificial perilymph solution (in mM: 110 Na + , 2 K + , 1.5 Ca 2+ , 113 Cl -, 3 d-glucose, 1 Na + pyruvate, 1 creatine, and 5 HEPES) on the basal side and artificial endolymph (2 Na + , 118 K + , 0.25 Ca 2+ , 118 Cl -, 3 d-glucose, and 5 HEPES) on the apical side. For both solutions, the pH and osmolality were adjusted to be 7.3 and 230 mmol/kg, respectively. Solutions were oxygenated immediately prior to use. The overlying otolithic membrane was digested by exposing the apical surface of the epithelium to 50μg/mL Collagenase (Sigma Aldrich) for 8 minutes and gently removed.

Artificial coupling of hair bundles
Mechanical coupling of hair bundles was provided with 50 mm diameter polystyrene microspheres (Corpuscular Inc.). The spheres were coated with concanavalin A (Sigma Aldrich), a highly charged polymer which enhances adhesion to stereovilli. After incubation in concanavalin A, the polystyrene particles were centrifuged and re-suspended in artificial endolymph, at a concentration of 3.5mg/ml. The beads were introduced into the top compartment of the recording chamber and allowed to settle onto the saccular preparation. Post deposition, fluid in the top compartment was replaced with artificial endolymph. After motion of the coupled bundles was recorded, the beads were suctioned off with a pipette.
Bundles were found to oscillate at comparable amplitudes under coupled and uncoupled conditions. Since spontaneous oscillation was shown to correlate closely with opening and closing of the transduction channels [41], the presence of the microsphere did not significantly interfere with the transduction process.

Imaging hair bundle and bead motion
Hair bundles were imaged in a top-down configuration with an upright light microscope (Olympus B51W). The image was further magnified to~500x and projected onto a Complementary Metal Oxide Semiconductor (CMOS) camera (Photron SA 1.1). Recordings were acquired at 250 to 1000 frames per second. Motion was tracked with software written in MatLab (Mathworks), which performs a center-of-mass calculation on the intensity profiles of the bundles. For each hair bundle, this calculation was averaged over at least 15 rows of pixels to extract its mean position in each frame of the recording. The position vs. time traces were smoothed by a moving average to remove higher-frequency (>150 Hz) noise.
Given the size of the beads, images were obtained at two focal planes-the equatorial plane of the sphere and the plane spanning the tips of the stereovilli. The polystyrene material was found to be sufficiently transparent to allow imaging of hair bundles through it. Also visible within the focal plane of the bundles were occasional dark spots within the bead that result from non-uniformity of the polystyrene. These spots were imaged to allow the tracking of bead motion in the same focal plane as the cell bundles. To improve the precision of the tracking, recordings of multiple dark spots within the field of view were averaged. Additionally, by tracking three spots spread over the focal plane, and measuring the distortions in the triangle defined by the points over time, we could estimate the rotation of the bead to be <4°; the bead motion was mostly confined to the x-y plane.

Data analysis
An automated routine [42] was used to detect positive and negative transitions in the position of the hair bundles and/or overlying bead, in each low-pass filtered motion trace. It was shown [21] that these transitions correspond to the opening and closing of the transduction channels, respectively. The period between two consecutive positive transitions defines the instantaneous period of one cycle, from which we obtain the instantaneous frequency. To quantify the regularity of the oscillation frequency in each trace, the probability density of the instantaneous frequency was obtained using kernel density estimation (Matlab function ksdensity). We compute the quality factor of the distribution: Q = F peak /FWHM, where F peak is the frequency of the peak in the density function, and FWHM is the full width at half maximum. An example of this procedure is shown in the supplemental information (S1 Fig).
With the focal plane at the level of the stereovilli, our CMOS recordings provided 8-10 hair bundles in the field of view, 3-5 of which were underneath a microsphere. We obtained simultaneous motion traces for each of the bundles, as well as the dark spots within the bead. To quantify the degree of synchronization between various pairs of bundles, we computed the cross correlation function between the traces. The cross correlation function f(t) between x(τ) and y(τ) is defined as f ðtÞ ¼ <xðtÞ;yðtþtÞ> p <xðtÞ;xðtÞ><yðtÞ;yðtÞ> , where <> is the inner product of vectors. The correlation coefficient is taken to be the peak value of f(t), and is normalized so that the correlation coefficient of a function with itself is 1. Recordings of bead motion, obtained in the same focal plane, were used to calculate the correlation between the bead and each of the bundles underneath. The distribution of correlation coefficients, compiled from 8 preparations, shows a cluster of correlated bundles (panel A in S2 Fig). Based on the distribution, we selected 0.5 to be the threshold correlation coefficient, with higher values indicating synchronization. Phase lags of hair bundle motion with respect to that of the microsphere were within the time resolution of the recordings, for all synchronized hair cells.
The duration of the recordings varied from 1 to 11 seconds in length. To verify that the degree of synchronization did not vary significantly over time, we calculated the correlation coefficient in moving time windows, each 0.5 second long, for recordings longer than 5 seconds. Fluctuations in the correlation coefficient were below 0.1, for synchronized cells, and remained above 0.5 throughout the records. An example of the variation of correlation with time is shown in the supplement (panel B in S2 Fig).
To determine the instantaneous phase of oscillation, we obtained the complex form z(t) of the oscillation y(t) by letting z = −Hilbert(y) + iy, where Hilbert(y) is the Hilbert transform of y. The instantaneous phase θ is given by y ¼ arctan ImagðzÞ RealðzÞ .

Coupling strength
To estimate the coupling strength between the hair bundles and the bead, we used a glass fiber to impose lateral displacements on the bead, and recorded its response, as well as that of the bundles. Spontaneous bundle oscillations were suppressed by introducing artificial perilymph on the apical side of the preparation. Sequential bursts of sine waves were sent, each at a different frequency, and the amplitude and phase of the response were measured for each stimulus segment. The measured responses yielded values for the elastic and viscous coupling coefficients, K and ξ, between the individual bundles and the bead (detailed in S1 File). Peaks in the histogram distributions of the measured values provided estimates of the coupling coefficients; the measurement was repeated for four groups of hair bundles, yielding the average results K = 2.5 +/-1.1 mN/m and ξ = 2.8+/-0.7 μN Ã s/m.

Synchronization of active motility by artificial coupling
Prior work has shown that frequencies of spontaneous oscillation are randomly distributed throughout the saccular epithelium [40]. Consistent with this finding, the preparations studied displayed a significant dispersion in the innate frequencies of the cells. Synchronization was observed even with very disparate frequencies of the individual oscillators. Fig 1 shows an example illustrating synchronization of active hair bundle motility by the overlying bead. The power generated by the collective phase-locked motion of the bundles was found to be sufficient to drive the relatively large overlying microsphere. One set of recordings was obtained with the plane of focus at the equatorial level of the bead (Fig 1A), which revealed a robust spontaneous oscillation (Fig 1B).
To determine the degree to which individual bundles were synchronized by the coupling, we imaged the bundles through the overlying bead, with the plane of focus aimed at the tips of the stereovilli (Fig 1C). The position of each bundle was tracked separately to obtain simultaneous traces of motion ( Fig 1D). All six bundles underneath the bead exhibited spontaneous motility. The three hair cells near the rim of the microsphere showed no entrainment to neighboring cells, indicating that they were not coupled (see S1 File), as expected from the spherical shape of the bead. Three hair bundles that were more centrally located synchronized their active oscillation.
For the last set of recordings, the bead was removed by suction through a pipette. Active motility in decoupled hair cells was recorded (see Fig 1E and 1F). In the absence of the overlying structure, no innate correlation was observed in the motility of the hair bundles, consistent with prior findings.
We characterized the degree of synchronization among coupled oscillatory hair bundles by calculating the maximum in the normalized cross-correlation function for each pair (Fig 1G). To extract the motion of the bead in the same recording as the bundles, we tracked dark spots in the image of the microsphere. Overlaid traces of hair bundle oscillations and those of the bead clearly indicate that they were mode-locked and in phase ( Fig 1H). Higher correlation was typically observed between a hair bundle and the bead to which it is coupled, compared to correlations between pairs of bundles.
Synchronization was studied in seven preparations, on groups of spontaneously oscillating hair bundles with an overlying polystyrene sphere. Typically, 3-4 bundles synchronized their active motility and led to an entrained motion of the bead, yielding correlation coefficients above 0.5. Bundles near the rims of the microspheres did not synchronize their motion. Hair bundles that synchronized their oscillations were mostly located within 16 mm of the bead's center (see S1 File) due to the spherical shape of the bead and the heights of the stereovilli.
The entrained hair bundles could clearly provide sufficient power to overcome the viscous damping and drive spontaneous oscillations of the beads, with amplitudes up to 80 nm. Phase lags of bundle motion with respect to that of the microsphere were within the time resolution of tracking.

Oscillation frequency of the coupled system
Innate frequencies of the individual bundles were determined by characterizing their oscillations in the absence of an attached bead, as illustrated in Fig 1F. Comparing the innate frequencies to those of the synchronized bundles (two examples shown in Fig 2A), we observed a consistent pattern: the synchronized oscillators converged to the group's mean frequency, which shifted very little as a result of the coupling element. The frequency of the bead We next examined the dependence of the induced frequency shift on the correlation coefficient between the bundle and the bead (Fig 2B). Within each group of hair bundles, higher correlation coefficients corresponded to smaller ΔFreq, indicating that bundles synchronized more readily when the innate frequencies matched more closely the mean frequency of the group.
Enhanced regularity of bundle oscillations in the coupled system shown in Fig 3A). Quality factors were found to be consistently higher under coupled conditions, compared to those extracted from the bundles' innate motility. As a control, the same analysis was performed for the unsynchronized edge bundles (see, for example, bundles 4-6 in Fig 1). Fig 3C shows the quality factors of spontaneous oscillations of edge bundles, with and without the presence of an overlying microsphere. For these groups of cells, quality factors of the individual bundles showed either an increase or a decrease upon the deposition and removal of the bead, with no overall trend, indicating that the regularity of the innate oscillation was not affected.
This measurement was performed for five groups of cells with recordings longer than five seconds. All of the groups showed an improvement in the regularity of spontaneous oscillation (positive ΔQ) as a result of synchronization ( Fig 3B). In comparison, the unsynchronized edge bundles showed no trend in ΔQ (Fig 3D). The synchronized system exhibited an enhanced regularity of spontaneous oscillation.

Multi-mode phase locking
Mode-locking in 1:1 ratio of frequencies, as illustrated in the traces of bundle motion shown in Fig 1, was observed in clusters of up to three hair bundles. In instances where four cells synchronized their motion, one bundle in the coupled group was found to exhibit high-order mode-locking. Fig 4 shows two examples, with overlaid traces demonstrating multi-mode phase-locking: Fig 4A shows a hair bundle whose oscillation mode-locked to that of the bead in a 3:1 ratio of frequencies, with intermittent flicker to other mode-locking ratios. Fig 4B shows an example of 2:1 mode-locking. For bundles that exhibited high-order entrainment, the correlation coefficients between their active motility and the motion of the bead were found to be lower than for 1:1 entrainment, between 0.4-0.6.
Examining the unwrapped phases of the oscillations further illustrates high-order modelocking (Fig 4C and 4D). The instantaneous phase of the bundle increased at a higher rate than that of the bead, and the two phases diverged over time. If the bead's phase was multiplied by an appropriate integer n, the time traces of the two phases were found to be parallel.

System of coupled nonlinear equations
We also explore coupled non-linear oscillators theoretically. We describe the dynamics of each individual hair bundle by the normal form equation of the Andronov-Hopf bifurcation. In the complex form, the equation is given by: ,where the ω is the frequency of spontaneous oscillation and the μ is the control parameter. This model undergoes the Andronov-Hopf bifurcation when μ = 0. We define the real part of the equation to be the lateral deflection of the bundle, and the imaginary part to reflect the internal dynamics of the hair cell. The parameters m and w represent the negative stiffness and the characteristic frequency of the bundle, respectively.  This simple model has been shown to capture the main characteristics of hair bundle response [43,44].
The coupled system is modeled with three nonlinear oscillators, connected to an overlying sphere (see schematic in Fig 5A). We introduce dimensions into our model:  Table A in S1 File). We assume that the internal dynamics of a hair cell are not directly coupled to the bead; we hence introduce elastic and viscous coupling only between the sphere (X) and the real components of the motion of the oscillators (x i ).

Synchronization
Non-linear oscillators can be synchronized by either elastic or viscous coupling. In Fig 5B, we define the oscillations to be synchronized, if all four peak frequencies (three oscillators and the bead) are within a 5% range. The result shows the K and ξ values required to yield synchronization of the bundles. Synchronization occurs when the elastic coupling strength is comparable to or higher than the bundle stiffness, or when the viscous coupling constant is comparable to or higher than the friction coefficient of the bundle.

Phase lags
In our model, the hair bundles are assumed to be coupled only via the overlying spherical mass, rather than through any direct coupling among the bundles. As the spherical mass is not attached to any external structures, it moves in phase with the viscous force and exhibits a phase lag with respect to the elastic force. Hence, the two modes of coupling show different phase delays. A purely viscous force will lead to synchronization with zero phase differences among the bundles. On the other hand, purely elastic coupling can synchronize oscillations with a non-zero phase differences. S5 Fig panel (D) and (E) show the phase differences between the three oscillators as K or ξ is varied, demonstrating the difference between the two types of coupling. When the bundles are coupled by a weak elastic force, phase differences arise among   Table A in S1 File). (A) Elastic coupling (Blue: λ = 2.8 μN*s/m, K = 1000 μN/m, Red: λ = 0.28 μN*s/m, the oscillators (S5A and S5D Fig), whereas phase differences are effectively zero when the oscillators are synchronized by even weak viscous coupling (S5 Fig panels (B) and (E)). However, the phase differences decrease to zero as the elastic coupling strength increases, and the response becomes indistinguishable from viscous coupling. Synchronization can be induced by various combinations of elastic and viscous coupling. S5 Fig panel (C) shows the phase differences when the system is coupled by both viscous and elastic elements. The plot clearly shows that the phase differences can be reduced by elastic or viscous couplings, but the viscous coupling is more effective.

Multi-Mode Locking
Nonlinear systems can exhibit multi-mode phase-locking to an external signal, with the order of the mode dependent on the frequency of the imposed stimulus. A plot of winding number vs. detuning typically shows the "Devil's staircase" structure [45]. For a system of coupled oscillators, the synchronization mode will depend not only on the detuning parameter, but also on the strength of the coupling coefficients.
In Fig 7A and 7B, we plot traces of motion for one of the three coupled oscillators and the spherical mass, with the coupling strength of one of the oscillators assumed to be weaker than the other two. Both purely elastic and viscous coupling produce clear multi-mode phase-locking. Variations in the frequency of the oscillator with the weaker coupling coefficient lead to the devil's staircase (Fig 7C and 7D). The experimentally observed multi-mode locking is readily reproduced by the numerical simulation, indicating that the nonlinearity of the system is well described by the model.

Synchronization of innate motility
The auditory system detects mechanical signals as weak as 0 dB SPL. The system is also robust, with the dynamic range of detectable sound pressures spanning over 6 orders of magnitude [5,46]. It has been shown that the sensitivity and robustness of the inner ear require a nonlinear response. Many studies further indicate the presence of an underlying active mechanism that amplifies the mechanical response. SOAEs in vivo and active hair bundle oscillations in vitro are two of the signatures indicating the presence of an energy-consuming process. Connections between the two phenomena have however not yet been established.
Our results demonstrate synchronization between spontaneously oscillating hair cell bundles of the inner ear. These experiments confirm theoretical predictions for synchronization under coupled conditions [33], in a biological preparation that maintains the functional integrity of the hair cells. Coherent active motility of the bundles was clearly sufficient to drive the oscillations of the overlying bead in a viscous fluid environment. Synchronization was observed in the systems studied, despite significant dispersion of the characteristic frequencies of the constituent oscillators.

Coupling elements in auditory and vestibular systems
Saccular hair bundles in vivo are coupled by the overlying otolithic membrane, a 25-30 mm thick matrix of densely packed randomly cross-linked filaments [47]. Hair bundles were shown to constitute the dominant elastic component of the lateral shearing of the otolithic membrane [40,48]. Consistent with these findings, we observed in a prior study that a localized mechanical perturbation elicited a coherent response across hundreds of cells [40]. Comparable results were obtained in other species: a patch of tectorial membrane isolated from the mouse cochlea was~10 times stiffer than the aggregate of the spanned bundles [49]. In the current study, the compliance of the polystyrene sphere was negligible with respect to the bundles, approximating the properties of coupling structures in vivo.
Comparatively little is known about the viscoelastic properties of the connections between the bundles and the overlying membranes. In the sacculus, thin linkages were observed to connect the kinocilia to pits in the otolithic membrane [47]. A comparative study of these connecting elements across the species, and their effects on synchronization, has not been performed.
Our artificial coupling element allowed us to experimentally study a specific range of coupling coefficients, weaker than those observed in the semi-intact saccular preparation [50]. The numerical model indicated that, for small groups of coupled hair cells, synchronization should arise over a broad range of coupling coefficients. This implies that synchronized active motility could be observed in a number of different species.

Frequency clustering
Frequency clustering of coupled oscillators was explored in a theoretical study of spontaneous otoacoustic emissions [34]. Different modes of coupling were shown to lead to different patterns in the observed frequencies of oscillation of the synchronized clusters. With connections between the individual elements primarily elastic, frequency of the entrained group of oscillators coincided with the highest frequency within the cluster. In contrast, with dissipative coupling between the oscillators, entrained motion was shown to occur at the mean frequency of the cells.
To estimate the effects of coupling coefficients on the synchronized frequency, we modeled our system with three oscillatory hair bundles, exhibiting different innate frequencies, coupled by an overlying sphere. We found that either elastic or viscous coupling could lead to synchronization frequencies close to the mean of the individual oscillators. The lack of a significant phase lag indicated a significant viscous component in the coupling coefficient.

Quality factors
Hair bundles oscillated more regularly when coupled to the overlying bead. The quality factors of the coupled bundles were~1.2-1.8x larger than those of the individual bundles. Slightly higher enhancement (2x) was observed in the system where one hair bundle was coupled to its cyber clone [36]. The theoretical model for a coupled system of 2x2 identical oscillators showed a 4-5x increase in the quality factor under conditions of strong coupling [33]. The enhancement was therefore higher than what was observed in our hybrid preparations. We ascribe the discrepancy to the frequency dispersion of the biological system. In both of the prior studies, the coupled bundles were chosen to exhibit identical innate frequencies, whereas the saccular epithelium displays broad frequency dispersion [40]. The enhancement of the quality factor, present despite dispersion in the frequencies of constituent oscillators, indicates the importance of coupling in shaping the frequency tuning of the whole system.
In addition to synchronizing the spontaneous oscillations, the overlying microsphere also imposed a mass load on the hair bundles. The 50 mm polystyrene sphere yields~60 ng of mass; for comparison, the mass of a hair bundle is estimated at~60 pg [21]. The effects of mass on the nonlinear response has not been extensively addressed in theoretical studies of the coupled system, but is likely to play a role in determining the overall quality factor of naturally coupled hair bundles in vivo.

Potential implications for in vivo phenomena
Spontaneous oscillation [21] exhibited by a hair cell's stereovillar bundle constitutes a potential cellular mechanism underlying the phenomenon of spontaneous otoacoustic emission [30][31][32]. However, these active oscillations have been mostly studied in the uncoupled bundles of the amphibian sacculus. The presence of the overlying otolithic membrane, which strongly loads and couples the bundles across the full epithelium, was shown to inhibit innate oscillations [38]. The question of whether different coupling conditions could lead to synchronized active bundle motility, thus providing a potential mechanism for in vivo emissions, has remained open. The use of a hybrid preparation, in which a small number of hair cells are artificially connected, provides us with a model system, wherein we can mimic coupling in other species. In particular, many species of lizards have been shown to have robust emissions [31]; in a number of lizard papillae, small numbers of hair cells are connected to patches of membrane known as sallets. Coupling between hair bundles is ubiquitous in inner ear end organs of many other species as well, including mammalian cochleae, where an overlying tectorial membrane couples hair bundles over various spatial scales [32]. Our findings demonstrate that synchronized active motility of a small number of hair cell bundles could power oscillations in significantly larger overlying structures. Spontaneous bundle motility hence constitutes a plausible mechanism for the generation of sound, which could be emitted via the reverse auditory pathway.