Figure 1

Left panel: bottom image: scanning electron microscope image of a stereociliary bundle in a top-down view. Top image: optical image of a glass fiber attached to the row of tallest stereocilia. The scale bars in both images denote $1\mu \mathrm{m}$. Right panel: mode locking at a drive amplitude of 20 nm. Bottom trace: spontaneous oscillations of a free stereociliary bundle. Middle trace: spontaneous oscillations of a stereociliary bundle with an elastic glass probe attached to its tip with a stiffness of $100\mu \mathrm{N}/\mathrm{m}$ with no imposed deflection. Top trace: displacement of the bundle is shown in red when sinusoidal deflections are imposed with a 20 nm amplitude and with increasing frequency from 17 to 23 Hz. The displacement of the base of the probe is shown overlaid in black, which has been arbitrarily scaled for visibility. Frequency increases by 1 Hz every ten cycles. Decades of drive cycles are indicated by vertical dashed lines. Events of phase slip between bundle and probe are indicated by arrows. The magnified portion shows a “phase-slip” event.Reuse & Permissions

Figure 2

Phase-locked component of the response (in nanometers) of a stereociliary bundle subject to sinusoidal deflection over a range of drive frequencies (5–50 Hz in 1 Hz increments). Each stimulus frequency is presented for ten periods. The corresponding bundle motion trace is split into ten segments and then is averaged. A single sine wave of fixed frequency is fit to the averaged response, and its amplitude is extracted to obtain the phase-locked component. Linear interpolation was applied along the stimulus amplitude direction. Stimulus was applied at 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 60, 80, 100 and 120 nm (amplitudes refer to the base of the probe).Reuse & Permissions

Figure 3

Mode locking at a drive amplitude of 120 nm. The drive frequency of the top trace is near the low-frequency boundary of the mode-locking interval (from 5 to 8 Hz). Each arrow indicates a spike like event. The bottom trace displays the progression of the mode locking with a reduction in the secondary pattern. The scale bar to the left indicates excursion of the hair bundle. The stimulus has been overlaid in black and has been arbitrarily scaled for visibility.Reuse & Permissions

Figure 4

Phase-locked amplitude as computed from Eq. (1), using scaled variables $F/{\mu }^{3/2}$ and $\omega$/$\mu$. The black lines indicate SNIC bifurcations, and the yellow lines indicate Andronov-Hopf bifurcations. They meet at a Bogdanov-Takens (BT) bifurcation. The intersection of the phase-locked region with the horizontal axis at the tip of the triangle is ${\omega }_0$/$\mu$. $\mathrm{Tr}\left[A\right]$ along the yellow dashed line and $\mathrm{Det}\left[A\right]$ along the black dashed line. These additional bifurcations are not detected experimentally as the system remains mode locked in both regions I and II.Reuse & Permissions

Figure 5

Data and numerical fits to slices through the Arnold tongue, taken along the horizontal axis. The data are indicated with blue dots, and the red lines show the numerical fits. The Bogdanov-Takens boundary point was selected to correspond to the point of inflection seen in the experimental plot and the characteristic frequency ${\omega }_0$ to match the peak in the response ($F_{bt}=40\mathrm{nm}$ and $f_{bt}=12\mathrm{Hz}$ with characteristic frequency $f_0=21\mathrm{Hz}$). The scaling factor $\mu$ for the overall response was varied to optimize the fits; no additional free parameters were introduced. [$F_{\mathrm{scale}}=24.09=F_{bt}/\left(0.53{\mu }^{3/2}\right),f_{\mathrm{scale}}=7.52=(f_0-f_{bt})/\left(0.56\mu \right),\mu =2.14,{\omega }_0/\mu =f_0/f_{\mathrm{scale}}/\mu =1.3$.] Note: We fixed the intersection between the two curves $trace=0$ and $det=0$ to be at the above points. This falls in the close vicinity rather than exactly at the Bogdanov-Takens point [27]. As these are indistinguishable experimentally, we refer to this as the BT point for conciseness.Reuse & Permissions

Figure 6

Data and numerical fits for slices through the Arnold tongue, taken along the vertical axis. The data were binned into 5 Hz intervals with blue dots (indicating the average of five neighboring frequencies) and the error bars showing the standard deviation of the phase-locked amplitude. The red lines show the numerical fits. (Scaling and fitting parameters are the same as for Fig. 5.)Reuse & Permissions