Phase Slips in Oscillatory Hair Bundles

Hair cells of the inner ear contain an active amplifier that allows them to detect extremely weak signals. As one of the manifestations of an active process, spontaneous oscillations arise in fluid immersed hair bundles of in vitro preparations of selected auditory and vestibular organs. We measure the phase-locking dynamics of oscillatory bundles exposed to low-amplitude sinusoidal signals, a transition that can be described by a saddle-node bifurcation on an invariant circle. The transition is characterized by the occurrence of phase slips, at a rate that is dependent on the amplitude and detuning of the applied drive. The resultant staircase structure in the phase of the oscillation can be described by the stochastic Adler equation, which reproduces the statistics of phase slip production.

Figure 1

Contraction of the phase distribution induced by a weak stimulus. (a) An example of hair bundle spontaneous oscillations with 1.7 nm stimulus applied from 2.5 to 7.5 s. ($\omega =20\mathrm{Hz}$, ${\omega }_0\sim 21.5\mathrm{Hz}$). (b) The averaged response of the bundle (top), performed over 40 presentations, and the stimulus profile (bottom). (c) Time-dependent histogram of $\Delta \varphi (t)$. The color-coded scale indicates the height of the histogram in arbitrary units. (d) Time-averaged histogram, in arbitrary units, of $\Delta \varphi (t)$ before (left), during (middle), and after the stimulation (right). Histograms of freely oscillating and driven intervals are averaged over 2.5 and 5 s, respectively.

Figure 2

Time evolution of the histogram of the unwrapped $\Delta \varphi (t)$ before (a), during (b), and after the stimulation (c). (a), (c) Without stimulus, $\Delta \varphi (t)$ is diffusive. (b) During stimulation, plateaus can be observed in the histogram indicating phase locking in the ensemble response. The color-coded scale indicates the height of the histogram in arbitrary units.

Figure 3

Time evolution of $\Delta \varphi (t)$ at different stimulus amplitudes. ($\omega =10\mathrm{Hz}$, ${\omega }_0\sim 14\mathrm{Hz}$). From (a) to (h), $f_0=0.2$, 0.35, 0.5, 0.6, 0.7, 0.8, 1.0, and 1.2 pN, respectively. Phase-locking intervals can be observed for $f_0$ above $\sim 0.5\mathrm{pN}$ (c) and extend in duration as the stimulus amplitude increases.

Figure 4

(a) Characteristic dynamics of hair bundle motion during a phase slip. The superimposed gray lines indicate the averaged stimulus during the phase slip. Top, middle: Phase slip associated with positive detuning ($\omega =40\mathrm{Hz}$, ${\omega }_0=30\mathrm{Hz}$, and $\omega =15\mathrm{Hz}$, ${\omega }_0=10\mathrm{Hz}$, respectively). Bottom: A phase slip associated with negative detuning ($\omega =5\mathrm{Hz}$, ${\omega }_0=6.5\mathrm{Hz}$). (b) Correlation between the appearance of the phase slips and the compressive nonlinearity. Black dots indicate the stimulus amplitudes at which the phase slips occur. Blue and green dots indicate linear regimes corresponding to low and high amplitudes, respectively.

Figure 5

Comparison with the Adler equation. Analytic solutions (solid line) are superposed on experimental data (dots). (a) Phase-locked amplitude as a function of $f_0$ ($\omega =5\mathrm{Hz}$, ${\omega }_0\sim 4.5\mathrm{Hz}$). The measured $r=28\mathrm{nm}$. The proportionality constant $ϵ/T=0.66f_0$ is obtained from the fit. (b) Rate of phase slip production as a function of $f_0$. The fit of the analytic solution is plotted with the same fitting parameter as in (a), with $\Delta \omega =4.7\mathrm{rad}/\mathrm{s}$. (c) $D_{\mathrm{eff}}$ predicted by Eq. (1) and extracted from the data. The proportionality constant $ϵ/T=0.63f_0$ and $D_0=40{\mathrm{s}}^{-1}$ are obtained from the fit, with $\Delta \omega$ fixed at the measured value of $60\mathrm{rad}/\mathrm{s}$.