Negative membrane capacitance of outer hair cells: electromechanical coupling near resonance

Abstract

Outer hair cells in the cochlea have a unique motility in their cell body based on mechanoelectric coupling, with which voltage changes generated by stimuli at their hair bundles drive the cell body and, in turn, it has been assumed, amplifies the signal. In vitro experiments show that the movement of the charges of the motile element significantly increases the membrane capacitance, contributing to the attenuation of the driving voltage. That is indeed the case in the absence of mechanical load. Here it is predicted, however, that the movement of motile charges creates negative capacitance near the condition of mechanical resonance, such as those in the cochlea, enhancing energy output.

Figure 1

Mechanical connectivity (A) and the equivalent electric circuit of the model system (B). Changes in hair bundle conductance R _a drives the system. (A) intrinsic cell stiffness k, external elastic load K, mass m, drag coefficient η. The motile element changes the cell length by x = k/(k + K)⋅anP, where P represents the fraction of the motile elements in the elongated state. The quantities a, q, and n respectively represent unitary length change, the unitary charge change, and the number of motile units. The broken line indicates the border of the OHC. The connectivity of the cell and the external load are parallel because the magnitudes of their displacements are equal. (B) the membrane potential V, the basolateral resistance R _m, the total membrane capacitance of the basolateral membrane C _m. The endocochlear potential e _ec, and the potential due to K⁺ permeability of the basolateral membrane e _K. The apical capacitance is ignored.

Figure 2

Nonlinear capacitance C _nl and power spectral density S _I(ω) of current noise. (A) Nonlinear capacitance plotted against $\overline{\omega }(=\omega /{\omega }_r)$. Nonlinear capacitance C _nl is normalized by γnq ². (B) Power spectral density of current noise is plotted against $\overline{\omega }$. $S_I(\overline{\omega })$ is normalized by $S_0\mathrm{(=4}\overline{P}\mathrm{(1}-\overline{P})n{q}^2{\omega }_r)$. Traces respectively correspond to the values of ${\overline{\omega }}_{\eta }$: 1 (black), 2 (blue), and 5 (red).

Figure 3

Power output per unit resistance change $(\hat{r}=\mathrm{1)}$. (A) Frequency dependence of power output. The reduced frequency $\overline{\omega }$ is normalized by α ² + ζ. Power output $W(\overline{\omega })$ is normalized by $W_0=\gamma \zeta a^{2}n{i}_0^2\eta k^2\mathrm{/[2}\pi {(k+K)}^2{C}_0]$. Traces correspond to the values of ${\overline{\omega }}_{\eta }$: 1, (black); 2, (blue); and 3 (red). (B) Maximum power output plotted against ${\overline{\omega }}_{\eta }(={\omega }_{\eta }/{\omega }_r)$. The scale of power output is the same as in A. Traces correspond to the values of α ² + ζ: 1, (black); 1.5, (blue); and 2 (red).

Figure 4

Membrane capacitance near resonance. The membrane capacitance C _m(=C ₀ + C _nl) normalized to the linear capacitance C ₀ is plotted against the normalized frequency $\overline{\omega }(=\omega /{\omega }_r)$. Here the ratio ζ(=γnq ²/C ₀) of nonlinear capacitance at α = 1 (load-free) and $\overline{P}\mathrm{=1/2}$ to the linear capacitance C ₀ is assumed to be unity, i.e. βnq ²/4 = C ₀ (Notice $\gamma =\beta \overline{P}\mathrm{(1}-\overline{P})$ is maximized at $\overline{P}\mathrm{=1/2}$). Filled red circles indicate frequencies and the corresponding values of the membrane capacitance that maximize the power output. Other parameter values assumed are, α ² = 1.2 and ω _η/ω _r = 6, which is smaller than more realistic ratios (See Discussion). Traces respectively correspond to the values of $\overline{P}\mathrm{(1}-\overline{P})$: 0.25 (red), 0.13 (blue), and 0.06 (black), showing the dependence on the holding potential. The dotted line indicates the level of C ₀.

Figure 5

Contour plots of H _max and ζ ² H _max for λ = 1. (A) Contour plot of H _max for λ = 1. Ordinate axis: ${\overline{\omega }}_{\eta }^2=({\overline{\omega }}_{\eta }/{\overline{\omega }}_r{)}^2$; abscissa: α ² + ζ. The values of H _max are indicated in the plot. Brighter shades indicate higher values. (B) Contour plot of (α ² + ζ)ζ ² H _max assuming α ² = 1.1, corresponding to a 10 kHz cell (see text). Ordinate axis: ${\overline{\omega }}_{\eta }^2$; abscissa: ζ. The values of (α ² + ζ)ζ ² H _max are indicated in the plot. Brighter shades indicate higher values.