Outer hair cell electromechanical properties in a nonlinear piezoelectric model

Abstract

A nonlinear piezoelectric circuit is proposed to model electromechanical properties of the outer hair cell (OHC) in mammalian cochleae. The circuit model predicts (a) that the nonlinear capacitance decreases as the stiffness of the load increases, and (b) that the axial compliance of the cell reaches a maximum at the same membrane potential for peak capacitance. The model was also designed to be integrated into macro-mechanical models to simulate cochlear wave propagation. Analytic expressions of the cochlear-partition shunt admittance and the wave propagation function are derived in terms of OHC electro-mechanical parameters. Small-signal analyses indicate that, to achieve cochlear amplification, (1) nonlinear capacitance must be sufficiently high and (2) the OHC receptor current must be sensitive to the velocity of the reticular lamina.

Figure 1

The proposed circuit model of the OHC lateral membrane. The dashed box represents a nonlinear piezoelectric component. It is connected to the electrical domain on the left and the mechanical domain on the right (symbols are defined in the text).

Figure 2

(A) Gating charge and (B) the sum of C and C_NL as a function of membrane potential. Both functions are plotted using three different values of load stiffness K_p=0, 0.05, or 0.15 N∕m. Other OHC parameters are K=0.02 N∕m, Q_max=2 pC, T=8×10⁵ m∕C, v₀=−40 mV, and v₁=28.6 mV.

Figure 3

Cell compliance B (solid line) and small-signal equivalent capacitance $\tilde{c}$ (dashed line) as a function of membrane potential. Parameter values are K=0.02 N∕m, Q_max=2 pC, v₀=−40 mV, and v₁=28.6 mV.

Figure 4

Organ of Corti micro-mechanical model (Lu et al., 2006). The RL system is characterized by parameters {K_r,R_r,M_r}, the BM system is characterized by parameters {K_b,R_b,M_b}, and OHC lateral wall (shaded area) is characterized by a contraction force f_OHC and a static stiffness K.

Figure 5

OHC frequency responses in situ. [(A) and (B)] Magnitude and phase of H_o(s). For comparison, dash-dotted curves show results for Z=0 (as in an isolated preparation). [(C) and (D)] Magnitude and phase responses, respectively, of velocities sξ_r (solid) and sξ_b (dashed) with respect to an externally applied force f_cp. Parameters used in this simulation are listed in Table 1.

Figure 6

Acoustic shunt admittance and propagation function as a function of frequency at a single location. [(A) and (B)] Real and imaginary parts of shunt admittance, respectively. [(C) and (D)] Real and imaginary parts of propagation function, respectively. Each panel shows results for three different motion-sensing conditions of the MET: α_v>0, α_d=0 (thick solid curves); α_v>0, α_d>0 (thin solid curves), and α_v=0, α_d>0 (dash-dotted curves).

Figure 7

The real part of shunt admittance y_sh as a function of frequency for different combinations of α_v and $\tilde{c}$. Panels (A), (B), (C), and (D) correspond to $\tilde{c}=2$, 5, 10, and 20 pF, respectively. In each panel, eight traces are plotted; from left to right, they correspond to α_v of 0.125, 0.25, 0.5, 1, 2, 4, 8, and 16 times the default α_v0=2.5×10⁻⁶ C∕m. Other parameters are listed in Table 1.

Figure 8

Quantifying the effect of negative damping. [(A) and (B)] Width and depth, respectively, of NDR as a function of $\tilde{c}$. Each trace is marked by a number indicating the ratio α_v∕α_v0 (0.125, 0.25, etc.). For better viewing, results for α_v∕α_v0>1 are plotted in dashed-dotted lines. Other parameters are listed in Table 1. (C) Width times depth of NDR as a function of $\tilde{c}$. (D) Definition of NDR. The dashed line marks zero damping. The crosses mark the boundaries of NDR. The lower boundary is calculated by extending the tangential line from the point of maximum negative slope (marked by a dot).

Figure 9

Analytic approximation of negative damping depth as a function of α_v. The curve shows prediction of the depth given by Eq. 30 if $\tilde{c}$ approaches ∞. The squares show numerical calculation of the depth without simplifying the expression of H_o(s) and y_sh(s).

Figure 10

An equivalent circuit to the proposed model in Fig. 1. The nonlinear component is moved from the electrical to the mechanical domain.

Figure 11

Illustration of a one-dimensional model of piezoelectric motors embedded in the OHC lateral membrane. Of the three motors shown, two are in the extended state and the middle one is in the contracted state.