Synchronization of Spontaneous Active Motility of Hair Cell Bundles

<div><p>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 <i>in vitro</i>. Hair bundles are coupled by overlying membranes <i>in vivo</i>; 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 <i>in vitro</i> 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.</p></div

Frequency of the synchronized system.

<p><b>(A)</b> Each panel represents a group of synchronized hair cell bundles. For each group, the frequency values of bundle oscillations, with and without an overlying bead, is plotted versus the correlation coefficient of the bundle motion with respect to that of the bead. For synchronized bundles, their frequencies of oscillation converge to the mean frequency of the group. <b>(B)</b> Change in the bundle frequency (ΔFreq) versus the correlation coefficient. ΔFreq is defined as the absolute value of the difference between a bundle’s oscillation frequency in the synchronized and unsynchronized state. Each color represents a synchronized group (8 groups total) of hair bundles, and each point represents a bundle in the group. ΔFreq shows a decreasing trend with the correlation coefficient.</p

Enhanced regularity of spontaneous oscillations.

<p><b>(A)</b> Each panel represents a synchronized group of oscillators. For each group, the quality factor of the oscillations exhibited by hair bundles, when synchronized by the bead and upon its removal, is plotted against the bundle’s coefficient of correlation with the bead. Synchronization increases the regularity of the oscillations. <b>(B)</b> ΔQ versus the correlation coefficient. ΔQ, defined as Q_synchronized-Q_{unsynchronized}, is measured for each bundle. Each color represents a synchronized group (5 groups total), and each point represents a bundle in the group. ΔQ is always positive, indicating that the synchronized system exhibits an enhanced regularity of oscillation. <b>(C)</b> Each panel shows a group of unsynchronized bundles, positioned near the rim of the bead. For each group, we compare the quality factors of the bundles with and without the bead present. The quality factor either increases or decreases, showing no consistent trend. <b>(D)</b> ΔQ, obtained for groups of unsynchronized bundles near the rims of the beads. ΔQ was either positive or negative, showing no consistent trends.</p

Multi-mode phase-locking by elastic or viscous coupling.

<p><b>(A)</b> Multi-mode locking due to elastic coupling (Winding Number = 1.98). The red trace shows the bead motion, and the blue trace shows the motion of the hair bundle with the weaker coupling coefficient. The innate frequencies of the three oscillators are Ω₁ = 7Hz, Ω₂ = 17Hz and Ω₃ = 25Hz; the coupling strength is K₁ = 1000 μN /m, K₂ = 1000 μN /m and K₃ = 300 μN /m. <b>(B)</b> Multi-mode locking due to viscous coupling (Winding Number = 2.06). The red trace shows the bead motion, and the blue trace shows the motion of the hair bundle with the weaker coupling coefficient. The innate frequencies of the three oscillators are Ω₁ = 7Hz, Ω₂ = 17Hz and Ω₃ = 25 Hz; the coupling strength is ξ₁ = 40 μN*s/m, ξ₂ = 40 μN*s/m and ξ₃ = 2 μN*s/m. Both forms of coupling lead to multi-mode phase-locking. <b>(C-D)</b> Winding Number vs. frequency of one of the oscillators. Both forms of coupling show the devil’s staircase. <b>(C)</b> K₁ = K₂ = 1000 μN /m and K₃ = 300 μN /m, Ω₁ = 7 Hz, Ω₃ = 17 Hz. <b>(D)</b> ξ₁ = ξ₂ = 40 μN*s/m, and ξ₃ = 5 μN *s/m, Ω₁ = 7 Hz, Ω₃ = 17 Hz.</p

Synchronized frequency versus the average frequency of the bundles.

<p><b>(A-B)</b> Numerical calculations of the synchronized frequency (<i>f</i>_<i>sync</i>) of three bundles versus the average of their characteristic frequencies (<i>f</i>_<i>ave</i>). Each data point represents a group of bundles, with a specific distribution of frequencies. The color coding represents different sets of parameter values, which include negative stiffness (μ) of the bundle, friction coefficient (λ), and coupling strength, while all other parameters in the model are fixed (see Table A in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141764#pone.0141764.s006" target="_blank">S1 File</a>). <b>(A)</b> Elastic coupling (Blue: λ = 2.8 μN*s/m, K = 1000 μN/m, Red: λ = 0.28 μN*s/m, K = 100 μN/m, Green: λ = 0.28 μN*s/m, K = 1000 μN/m, Magenta: λ = 28 μN*s/m, K = 10000μN/m). <b>(B)</b> Viscous coupling (Blue: λ = 2.8 μN*s/m, ξ = 40 μN*s/m, Red: λ = 0.28 μN*s/m, ξ = 4 μN*s/m, Green: λ = 0.28 μN*s/m, ξ = 40 μN*s/m, Magenta: λ = 28 μN*s/m, ξ = 400μN*s/m). The simulations indicate that the synchronized frequencies are clustered near the average frequency values. However, precise values of the synchronized frequency depend on the characteristics of the bundles. <b>(C)</b> Experimental data. The error bars indicate the standard deviation for the three synchronized bundle frequencies.</p

High-order mode-locking.

<p><b>(A)</b> Traces of motion for a hair bundle and bead pair, showing 3:1 mode locking. Scale bar x = 200 ms, y = 30 nm. <b>(B)</b> Traces of motion for a bundle and bead pair, with 2:1 mode locking. Scale bar x = 100 ms, y = 30 nm. <b>(C)</b> The unwrapped phase of the pair shown in part A. Instantaneous phase of the bundle (Φ_bundle) increases faster than that of the overlying bead (Φ_bead). Multiplying Φ_bead by 3 leads to a largely parallel evolution of the phases with time. Scale bar x = 300 ms, y = 50 rad. <b>(D)</b> The unwrapped phase of the pair shown in part B. Multiplying Φ_bead by 2 leads to a largely parallel evolution of the two phases with time. Scale bar x = 200 ms, y = 20 rad.</p

Experimental setup.

<p>(A) Top-down view of a freestanding hair bundle with a glass probe (black line positioned vertically across the image) attached to the tallest row of stereocillia. Because of light piping, the stereocillia appear much brighter than the background. Scale bar: 2.5 m. (B) Schematic diagram showing a side view of a hair bundle with a probe attached. The piezoelectric actuator displaces the probe's base in the direction of the bundle's axis of sensitivity indicated by the arrow in the figure. (C) Top-down view of hair bundles coupled to the otolithic membrane. The pits in the membrane into which the bundles protrude give rise to the bright ellipses around each bundle. The shadow of the probe's tip has been highlighted with the dashed line. Scale bar: 5 m. (D) Schematic diagram of a side view of hair bundles coupled to the otolithic membrane and stimulated with a probe. In the actual experiment, the probe's tip is embedded a few microns into the otolithic membrane. Note than in C and D, the probes have not been drawn to scale; in particular, the cantilever arms are typically a few hundred microns in length.</p

Offset in the position of the hair bundle position does not determine its dynamic state.

<p>For a series of traces obtained from a single cell, the bundle’s position, with respect to that at the end of the recording, is represented in the form of a contour plot (A). Different rows correspond to different stimulus durations, with the recording order displayed from top to bottom. (B) from 17 different cells for which recordings at a minimum of two different stimulus durations were obtained. Cells are represented by different symbols. The red line corresponds to the cell shown in (A). Five cells are displayed in color to guide the eye. The remaining cells are shown in gray with different symbols. (C) Same as (B), but plotted on a logarithmic abscissa.</p

Post-stimulus adaptation of the resting position of the bundle exhibits at least two time scales.

<p>The slow component of the bundle movement following a prolonged deflection was fitted with the sum of two exponentials, yielding two time constants and , with >. Good fits were obtained ( & record duration) for 282 recordings (of ). The two time constants are represented in the histogram by different shadings of gray.</p

The time required by the bundle to return to its steady-state position depends on the stimulus duration.

<p>For a series of traces from a single cell, each recording was fitted with a sum of two exponentials. Derivatives of the fitted functions are represented in the form of a contour plot with slopes given in the legend (A). Rows correspond to different stimulus durations, with the recording order from top to bottom. We arbitrarily chose nm/ms as the threshold slope defining the steady-state. Similarity between the different contours indicates that the results do not critically depend on the selected criterion. (B) The time to reach steady-state () as a function of stimulus duration was determined for 17 different cells. These cells each had recordings for at least two different stimulus durations. Cells are represented by different symbols. The red line corresponds to the cell shown in (A). Five cells are displayed in color to guide the eye. The remaining cells are shown in gray with different symbols. (C) Semi-log plot of the data shown in (B).</p