Amplitude death of coupled hair bundles with stochastic channel noise

Hair cells conduct auditory transduction in vertebrates. In lower vertebrates such as frogs and turtles, due to the active mechanism in hair cells, hair bundles (stereocilia) can be spontaneously oscillating or quiescent. Recently an amplitude death phenomenon has been proposed [K.-H. Ahn, J. R. Soc. Interface, 10, 20130525 (2013)] as a mechanism for auditory transduction in frog hair-cell bundles, where sudden cessation of the oscillations arises due to the coupling between nonidentical hair bundles. The gating of the ion channel is intrinsically stochastic due to the stochastic nature of the configuration change of the channel. The strength of the noise due to the channel gating can be comparable to the thermal Brownian noise of hair bundles. Thus, we perform stochastic simulations of the elastically coupled hair bundles. In spite of stray noisy fluctuations due to its stochastic dynamics, our simulation shows the transition from collective oscillation to amplitude death as interbundle coupling strength increases. In its stochastic dynamics, the formation of the amplitude death state of coupled hair bundles can be seen as a sudden suppression of the displacement fluctuation of the hair bundles as the coupling strength increases. The enhancement of the signal-to-noise ratio through the amplitude death phenomenon is clearly seen in the stochastic dynamics. Our numerical results demonstrate that the multiple number of transduction channels per hair bundle is an important factor to the amplitude death phenomenon, because the phenomenon may disappear for a small number of transduction channels due to strong gating noise.

Figure 1

(a) A model for coupled hair bundles. (b) Schematic drawing for Markov analysis of stochastic simulation for the coupled hair bundles dynamics.

Figure 2

(a) The parameter distribution for the pivotal stiffness of hair bundles $k_{sp}$ and the maximum forces $f_{max}$ of molecular motors. We use this distribution for all simulations in this paper. (b)–(g) The histograms show the number of events which are counted every 1 ms for 2 s for (b) incoherent ($k$ = 0), (c)–(f) locking (or collectively oscillating) ($k$ = 1 pN/nm, $k$ = 2 pN/nm, $k$ = 3 pN/nm, and $k$ = 4 pN/nm), and (g) amplitude death region ($k$ = 5 pN/nm) of 10 hair bundles.

Figure 3

The velocity distribution for amplitude death ($k$ = 6 pN/nm) and locking ($k$ = 2 pN/nm) of 10 coupled hair bundles. The green dashed line is a fitting curve for the velocity distribution for a locking oscillation (black solid line), which is a half-Lorentzian curve. The blue dashed line is a Gaussian fitting curve for the velocity distribution for the amplitude death state (red solid line). (Inset) The averaged displacement of the coupled hair bundles at $k$ = 2 pN/nm as a function of time.

Figure 4

The averaged variance for the displacement of the coupled hair bundles as a function of the interband coupling strength $k$.

Figure 5

(a) The average displacements $X=\frac{1}{N}{\sum }_i{X}_i$ for amplitude death($k$ = 6 pN/nm), collectively oscillating ($k$ = 2 pN/nm), and uncoupled ($k$ = 0) states of 10 coupled hair bundles. The top of the figure is the external force acting on the hair bundles. (b) The open channel ratio ${\tilde{G}}_o$ of 10 coupled hair bundles.

Figure 6

(a–c) The power spectra of mechanical displacement $S_X\left(f\right)$ as a function of frequency $f$ when an external force is applied. The coupling strengths are (a) $k=0$, (b) $k=3$ pN/nm, (c) $k=9$ pN/nm. (d) SNR as a function of the coupling strength $k$. We used a 0.2 pN external force with frequency of 12 Hz in our simulations (a–e). We obtained the power spectra averaged over 50 different trials.

Figure 7

(a) The variance ${\sigma }_x$ of the hair bundles' displacement as a function of the coupling strength $k$ (b) The variance ${\sigma }_G$ of the number of the open transduction channels as a function of the coupling strength. The black square dots are for the case when only the thermal noise exists. When only channel gating noise exists, we plot the variances for the various number of channels. The error bars are obtained by 50 different trials. We used a correlation time ${\tau }_c=$ 0.14 ms for the thermal noise and a relaxation rate $\gamma =$ 10 ms${}^{-1}$ for the stochastic channel gating noise. In both cases, as the number of ion channels per hair bundle increases, the variance decreases.