Effect of the cochlear microphonic on the limiting frequency of the mammalian ear

Abstract

Electromotility is a basis for cochlear amplifier, which controls the sensitivity of the mammalian ear and contributes to its frequency selectivity. Because it is driven by the receptor potential, its frequency characteristics are determined by the low-pass RC filter intrinsic to the cell, which has a corner frequency about 1∕10th of the operating frequency. This filter significantly decreases the efficiency of electromotility as an amplifier. The present paper examines a proposal that the cochlear microphonic, the voltage drop across the extracellular medium by the receptor current, contributes to overcome this problem. It is found that this effect can improve frequency dependence. However, this effect alone is too small to enhance the effectiveness of electromotility beyond 10 kHz in the mammalian ear.

INTRODUCTION

The sensitivity and frequency selectivity of the mammalian ear depend on the motile activity of outer hair cells (OHCs). The electromotility of OHCs is likely responsible for this function.1, 2, 3, 4 However, its dependence on the receptor potential (RP) may limit its efficiency because the RP is attenuated by the intrinsic low-pass RC filter of the cell, the corner frequency of which is significantly lower than the operating frequency. This is known as the RC time constant problem.5, 6 To be able to evaluate the significance of this problem, we will need a scale with which force production by electromotility is compared. In an earlier report,7 we assumed that this motility must counteract viscous drag,8 specifically the shear drag in the gap between the tectorial membrane and the reticular lamina.9 We found that this comparison leads to a frequency limit and that the limiting frequency obtained for the ear geometry of chinchillas was about 10 kHz, respectable but still short of covering the auditory frequency. This result suggests the presence of factors that enhances the effectiveness of electromotility.

To address the RC time constant problem, several mechanisms that can enhance the efficiency of electromotility have been proposed. Those mechanims include piezoelectric resonance,7, 10 modulation by a stretch-sensitive Cl⁻ current,11, 12 fast voltage-gated K channels,7, 13 and the effect of the cochlear microphonic (CM).14, 15 Of those mechanisms, a previous analysis7 showed that pure piezoelectric resonance that does not involve the mechanoelectric transducer channel is not effective and that fast voltage-gated K channels, which can counteract the capacitive current, is capable of extending the frequency limit beyond the 10 kHz limit. Now it appears possible to extend our analysis to include the effect of the CM on unstimulated outer hair cells14, 15 because recent experiments16, 17 have provided details regarding traveling waves and the CM.

Dallos and Evans14, 15 posited the following. The CM, which is a voltage drop generated by the receptor current of outer hair cells across the extracellular medium, has relatively flat frequency dependence. In addition, unstimulated outer hair cells should act as voltage dividers with relatively flat frequency dependence because at low frequencies their performance is determined by the resistance ratios of their apical and basolateral membranes and at high frequencies the performance is determined by the capacitance ratios.14, 15 Thus the oscillatory membrane potential in unstimulated OHCs induced by the CM has relatively flat frequency dependence, and this potential can be larger than the receptor potential of stimulated OHCs at high frequencies.

Here we examine whether or not the CM can contribute to overcoming the RC problem by constructing a simple theoretical model that would set an upper bound of efficiency.

First, we assume that the stimulus level is extremely low and give an analytical expression for the RP, the CM, which is locally produced by the receptor current of OHCs,18 and the oscillatory potential induced in outer hair cells by the CM. This calculation could be justified for the following reasons. Since the responses of the cochlea to stimuli with very low intensities are extremely localized,19, 20 we estimate the external potential near the locus of the characteristic frequency, assuming that the major current source is local. This assumption, which is rather approximate, appears to be compatible with the value of ∼200 μm (Ref. 16) for the wavelength of the traveling wave near the peak and estimates of >300 μm (Ref. 21) and 40 μm17 for the space constant of the CM.

Second, instead of a detailed model of cochlear mechanics, we assume that mechanical energy generated by the CM is optimally converted into mechanical energy. We further assume that mechanical energy made available from electromotility is optimally used to counteract viscous drag. Then by considering energy balance we examine whether or not this effect increases the frequency limit that is imposed by viscous drag.

EQUIVALENT CIRCUIT FOR HAIR CELLS

Following Dallos and Evans,15 we consider two kinds of OHCs. One kind includes those cells that are stimulated at their characteristic frequency. These cells have receptor currents, which generate the RP across their plasma membrane and the CM in their vicinity in the extracellular space. Although these cells convert electrical energy made available by the RP into mechanical energy, let us refer them to sensor cells for convenience. Another kind includes unstimulated OHCs. These cells have oscillating potential not because of stimulation to their hair bundles but because of the CM. Let us call this potential “pseudo-receptor-potential” (pRP). We refer to those cells as actuator cells because their pRP drives those cells. For simplicity, we assume that these two kinds of cells, sensors and actuators, share the same electrical properties that include the apical capacitance C_a, the basal capacitance C_b, the resting apical resistance R₀, and the basal resistance R_K.

Since the resting potential of OHCs is about −70 mV,22 close to the reversal potential e_K of about 100 mV for K⁺,23 the basal conductance is dominated by K⁺ current. The apical membrane is exposed to the endolymph with endocochlear potential e_ec of about 80 mV,24 measured from the perilymph, which surrounds the basolateral membrane.

The circuit is described by

$$I_1=\frac{V_1-e_K}{R_K}+C_b\frac{d}{dt}{V}_1=\frac{V_e-V_1}{R_a}+C_a\frac{d}{dt}(V_e-V_1),$$ (1)

$$I_2=\frac{V_2-e_K}{R_K}+C_b\frac{d}{dt}{V}_2=\frac{V_e-V_2}{R_0}+C_a\frac{d}{dt}(V_e-V_2),$$ (2)

$$I_1+n{I}_2=\frac{e_{\mathrm{ec}}-V_e}{R_e},$$ (3)

where V₁ and I₁ are the membrane potential and current of sensor cells, respectively, and V₂ and I₂ are those for actuator cells. The voltage drop e_ec−V_e is produced by currents through the external resistance R_e. The factor n is the number ratio of actuator cells to sensor cells.

We placed the extracellular resistance R_e on the apical side of OHC’s (i.e., in the scala media) in the equivalent circuit diagram (Fig. 1). However, it is also reasonable to have the extracellular resistance on the basolateral side of the OHCs. These two cases give the same result for the ac voltage drop because the circuit is closed at the ground.

Figure 1.

(Color online) An equivalent electric circuit that represents the configuration in which two kinds of outer hair cells are connected. On the left is the sensor cell, which has mechanotransducer conductance represented by the resistance R_a. The transducer current induces the receptor potential V₁ in the cell as well as the cochlear microphonic V_e across the external resistance R_e. The apical capacitance is C_a; the basolateral membrane has resistance R_b and capacitance C_b. On the right is the actuator cell. The property of the actuator cell is identical to the sensor cell, with the exception that the apical resistance is constant at R₀. The cochlear microphonic induces pseudo-receptor-potential V₂ in the actuator cell.

Let us assume that the basilar membrane moves sinusoidally with a small amplitude x at an angular frequency ω(=2πf). The reason for this assumption is that the sensitivity of the basilar membrane is largest for small signals,20 and therefore examining the case of small signals must be sufficient for examining the RC time constant problem. The resistance R_a of the apical membrane, which responds to the motion of the basilar membrane without a phase delay due to fast gating of mechanotransducer channels, can be described by

$$R_a\left(t\right)=R_0(1-ϵ\exp \left[i\omega t\right]),$$ (4)

with ϵ⪡1. The relationship between x and ϵ can be given by using the sensitivity g of the mechanoelectric transducer channel to basilar membrane displacement,

$$ϵ=g{R}_{0}x.$$ (5)

The potentials V₁, V₂, and V_e can be expressed by

$$V_1\left(t\right)=V_{10}+e_K+v_1\exp \left[i\omega t\right],$$ (6)

$$V_2\left(t\right)=V_{20}+e_K+v_2\exp \left[i\omega t\right],$$ (7)

$$V_e\left(t\right)=V_{e0}+e_K+v_e\exp \left[i\omega t\right],$$ (8)

where v₁, v₂, and v_e are small complex amplitudes.

Although the membrane capacitance C_b depends on the membrane potential due to the membrane motor, we treat it as constant. Because the voltage changes are small, the voltage-dependent component of the capacitance contributes only to second- and higher-order terms.

Perturbation solution

The zeroth-order terms lead to

$$V_{10}=V_{20}=\frac{R_K{V}_0}{R_0+R_K+(1+n)R_e},$$ (9)

$$V_{e0}=\frac{(R_0+R_K)V_0}{R_0+R_K+(1+n)R_e},$$ (10)

where V₀=e_ec−e_K. Here V₁₀(=V₂₀) is the resting potential of OHCs. The dc component of the membrane potential of both types of cells are equal because it is determined by the time independent part of the resistance, which is equal in both.

The first-order terms lead to

$$\frac{v_1}{ϵ}=i\frac{R_0{V}_{10}}{D_n}\{n{R}_e+R_K(1+i\omega C_{b}n{R}_e)+R_0[1+i\omega C_b{R}_K+i\omega C_a(R_K+n{R}_e+i\omega C_{b}n{R}_K{R}_e)]\},$$ (11)

$$\frac{v_2}{ϵ}=-\frac{i{R}_0{R}_e{V}_{10}}{D_n}(1+i\omega C_a{R}_0)(1+i\omega C_b{R}_K),$$ (12)

$$\frac{v_e}{ϵ}=-\frac{R_0{R}_e{V}_{10}}{R_K{D}_e}(1+i\omega C_b{R}_K),$$ (13)

with

$$D_n=-i\{R_K+R_0[1+i\omega (C_a+C_b)R_K]\}{D}_e,$$ (14)

$$D_e=-(1+n)R_e-R_K[1+i\omega C_b(1+n)R_e]-R_0(1+i\omega C_b{R}_K+i\omega C_a\{(1+n)R_e+R_K[1+i\omega (1+n)C_b{R}_b]\}).$$ (15)

Here v₁ is the ac component of the RP, v₂ is the ac component of the pRP, and v_e is the CM.

In the special case of R_e=0, both CM and pRP disappear, i.e., v_e=v₂=0. The RP v₁ of the sensor cell turns into v₀ and

$$\frac{v_0}{ϵ}=\frac{R_0{V}_{10}^{\left(0\right)}}{R_0+R_K+i\omega R_0{R}_K(C_a+C_b)},$$ (16)

where $V_{10}^{\left(0\right)}$ is V₁₀ with n=0.

MAGNITUDES OF OSCILLATING POTENTIALS

Here we numerically examine the phase and the magnitude of the pRP and compare them with those of the RP. A set of typical parameter values is given in Table 1.

Table 1.

Parameter values. The values used reflect properties of basal cells.

Quantity	Used	Measured (estimated)
Ca	5 pF	(3 pF)a
Cb	10 pF	(10 pF)
Ca+Cb	15 pF	⩾15 pFf
RK	10 MΩ	10 MΩ,b 20 Mf
R0	60 MΩ	(60 MΩ)c
Re	1 MΩ	(0.7 MΩ)d
g	1 nS∕nm	1 nS∕nme

^aEstimation based on the membrane area.

^bReference 27.

^cAssumes 50% opening of the maximal hair bundle conductance, which is 9.2 nS in a perilymphatic medium (Ref. 28) multiplied by 3 to account for low Ca²⁺ endolymph (Ref. 29).

^dThe amplitude of cochlear microphonic near outer hair cells is 0.1 mV for basilar membrane motion with 1 nm amplitude (Ref. 17), which elicits 0.15 nA, given the sensitivity of 1 nS∕nm (see footnote e) and 150 mV potential drop. This gives 0.7 MΩ for R_e.

^eObtained by multiplying the steepest slope for hair bundle open probability 1∕(25 nm) multiplied by 28 nS conductance (Refs. 28, 29), assuming a 1:1 ratio of hair bundle displacement to basilar membrane displacement. This value is also consistent with the peak sensitivity of 0.7 nS∕nm (Ref. 30) obtained from the hemicochlea preparation, considering the blocking effect of high Ca²⁺ concentration in the perilymphlike medium.

^fReference 5.

Numerical examination shows that the magnitude of the pRP is less than that of the RP if n⩾1 for the entire frequency range. However, the pRP increases with decreasing n, and a reversal takes place at a high frequency range if n<1 (Fig. 2). The reason for the reversal is the phase. The phase of the pRP is similar to that of the CM, which differs in phase from the RP by about π. Namely, the RP suffers destructive interference from the CM.

Figure 2.

(Color online) The magnitude (A) and phase (B) of the RP, CM, and pRP. The values of the parameters are shown in Table 1. The value for n is 0.5.

These observations are consistent with the high frequency asymptotes. For high frequency, Eqs. 11, 12 yield

$$\frac{v_1}{ϵV_{10}}\approx -i\frac{n}{n+1}\frac{1}{\omega (C_a+C_b)},$$ (17)

$$\frac{v_2}{ϵV_{10}}\approx i\frac{1}{n+1}\frac{1}{\omega (C_a+C_b)}.$$ (18)

These equations show that at high frequencies the inequality ∣v₁∣>∣v₂∣ holds for n>1 and ∣v₁∣<∣v₂∣ for n<1. They also show the phase difference of π.

COLOCALIZATION OF TWO TYPES OF OHCs

If actuator cells are colocalized with sensor cells, what is expected? Since electromotility is proportional to the oscillating potential of OHCs, the effect of CM can be assessed by the ratio of the weighted mean of the receptor potential and the pRP ∣v₁+nv₂∣∕(1+n) to the RP ∣v₁∣. Since the phase of v₂ differs from that of v₁ approximately by π, the RP and the CM have destructive interference. Thus we expect

$$\mid v_1+n{v}_2\mid ∕(1+n)<\mid v_1\mid ,$$ (19)

which is illustrated in Fig. 3. This inequality is obvious for the high frequency asymptotes. For this reason, having both actuator cells in the immediate neighborhood of the sensor cells is not only inconsistent with morphological observations, but also detrimental to the effectiveness of electromotility.

Figure 3.

The dependence of the mean amplitude (∣v₁+nv₂∣)∕(n+1) on the number ratio n of the actuator cells to the sensor cells. The n-dependence of the mean amplitude depends on the frequency f. The top trace is for 10 kHz and the bottom trace is for 70 kHz, with frequency incremented in 20 kHz steps.

UPPER BOUND OF CM UTILIZATION

Following Dallos and Evans,15 here we assume that the sensor cells are in the characteristic location of frequency ω and the cells that uses the CM are more basal to the characteristic location.

Because we are interested in finding the frequency limit of the ear, we consider energy balance at the most basal part of the cochlea. We examine an upper bound of the efficiency with which actuator cells use the pRP by considering energy balance alone. Specifically, the CM generated by OHCs in the location of peak amplitude is used exclusively by those cells in a slightly more basal location to convert electrical energy into mechanical energy to be sent back to those cells that generate the CM.

Where are of the actuator cells located in this hypothesis? The conditions for optimal location for the actuator cells would be as follows:

• (1)The mechanical energy output of the actuator cells depends on cell displacement, which is associated with basilar membrane movement. For this reason, a higher efficiency of actuators in producing mechanical energy requires a larger amplitude of basilar membrane motion at the site of these cells.
• (2)Displacements of the basilar membrane generate receptor currents that contribute to the CM. For this reason, the amplitude of basilar membrane motion at the location of the actuator cells must be small enough not to diminish the CM by destructive interference.
• (3)The RP of actuator cells must have constructive interference with the CM. The zone of constructive interference is where phase advance is between π∕2 and 3π∕2, optimally π. Without constructive interference, all the electrical energy of a cell is spent in counteracting the viscous drag at the location of the cell.
• (4)The efficiency of energy transmission decreases with the distance between sensors and actuators. Transmission of electrical energy depends on a space constant, which is ∼40 μm (Ref. 17) or <300 μm,21 perhaps somewhat shorter than the wavelength of about 200 μm.16

Condition (3) on the phase requires that the actuator cells be located between 1∕4 and 3∕4 wavelength basal to the characteristic location of the frequency. The most effective location is between 1∕4 and 1∕2 wavelength basal, where the amplitude is larger [condition (1)] and closer to the sensors [condition (4)].

To evaluate an upper bound of the efficiency with which actuator cells convert electrical energy associated with the CM, we make the following simplifications (Fig. 4). First, the sensor cells are OHCs in the 1∕2 wavelength zone centered at the location of the characteristic frequency. Second, the actuator cells are OHCs in the 1∕2 wavelength zone centered at the location 1∕2 wavelength more basal to the characteristic location. Third, the CM spreads $3∕4(=\frac{1}{2}\cdot \frac{1}{2}+\frac{1}{2})$ wavelength in the basal direction from the characteristic location to cover the actuator cells and abruptly diminishes in a step. Because a space constant does not have directionality, the CM should spread in the apical direction symmetrically as it does in the basal direction. These assumptions lead to a value of 2 for the ratio n, which determines the attenuation of the CM. The exact value for n is not critical for our analysis, as will be shown later. Fourth, the displacement amplitude of the actuator is equal to that of sensors. This condition to maximize the actuator output (the condition 1) would reduce the CM [condition (2)]. However, we assume that the CM is not affected. These assumptions, each of which would be too optimistic, are again intended to give an upper bound of efficiency.

Figure 4.

(Color online) A simplified model showing lateral energy transfer. The bold line represents a traveling wave that propagates from the basal end (left). The sensor cells have the RP V₁ and are in the zone of 1∕2 wavelength (the shaded band on the right-hand side) centered at the characteristic location of the frequency, where the envelope (the thin line) peaks. The actuator cells have the pRP V₂ and are at a more basal zone of 1∕2 wavelength (the shaded band on the left-hand side) centered at the point where traveling wave’s phase is advanced by π. The spread of the CM is represented by the box centered at the sensor cells and extends 3∕4 $(=\frac{1}{2}\cdot \frac{1}{2}+\frac{1}{2})$ wavelength in both directions.

With these assumptions, an upper bound of the energy generated by an OHC per cycle to counteract viscous drag is represented by

$${\overline{E}}_{\mathrm{OHC}}={\int }_0^{2\pi ∕\omega }dt\varphi {\cos }^2\left(\omega t\right)(\mid v_1\mid +\mid v_2\mid )x,$$ (20)

where ϕ is the force generation per unit voltage change and x is the displacement of the basilar membrane at the characteristic location.

Since the CM is the product of the transducer current of OHCs through the extracellular space, its effect on enhancing electromotility requires introducing an extracellular resistance R_e. However, the RP increases as R_e→0. As illustrated in Fig. 5, the reduction of the RP from v₀ to v₁ due to the external resistance is somewhat compensated by the effect of utilizing the CM under the most favorable condition. However, the combined amplitude ∣v₁∣+∣v₂∣ still remains less than ∣v₀∣, i.e., the RP without external resistance, throughout the frequency range.

Figure 5.

(Color online) Relative magnitude of electromotility output, which is proportional to the electric potential plotted against the frequency. The combined magnitude (∣v₁∣+∣v₂∣)∕(ϵV₁₀) is compared with ∣v₁∣∕(ϵV₁₀) and ∣v₀∣∕(ϵV₁₀) (dashed line), the magnitude of the sensor cell’s receptor potential with and without (dashed line) the external resistance R_e, respectively. The asymptotic form of ∣v₀∣∕(ϵV₁₀) for high frequencies 1∕[ωR_b(C_a+C_b)] (dotted line) was used in the previous treatment (Ref. 7).

With Eqs. 5, 17, 18, the asymptotic form for high frequencies is

$${\overline{E}}_{\mathrm{OHC}}\approx \pi \varphi \frac{g{R}_0{V}_{10}}{R_K\omega (C_a+C_b)}{x}^2.$$ (21)

LIMITING FREQUENCY REVISITED

In a previous analysis,7 viscous drag in the gap between the tectorial membrane and the reticular lamina has been addressed. It showed that the drag is proportional to the velocity of shear motion and that the friction coefficient γ is expressed by γ=ηS∕d, where η is the viscosity of the fluid, S the area of the gap per OHC, and d the gap. The viscous loss per cycle per OHC at the characteristic location is then

$$E_{\text{drag}}=\pi \gamma \omega x^2.$$ (22)

The condition for OHC electromotility to counteract viscous drag is ${\overline{E}}_{\mathrm{OHC}}⩾E_{\mathrm{drag}}$. This leads to a condition for the frequency ω,

$${\omega }^2⩽\alpha \left(n\right)\cdot \frac{\varphi g{V}_0}{g(C_a+C_b)},$$ (23)

with a factor

$$\alpha \left(n\right)\equiv \frac{V_{10}{R}_0}{V_0{R}_K}=\frac{R_0}{R_0+R_K+(1+n)R_e},$$ (24)

which is smaller than unity.

In the absence of extracellular resistance, the limiting frequency is somewhat higher because α(0)>α(n) for n>0.

Because α(n)<1, inequality 23 gives a frequency limit lower than that given by

$${\omega }^2⩽\frac{\varphi g{V}_0}{\gamma (C_a+C_b)},$$ (25)

which was obtained assuming R₀⪢R_K and R_e=0⁷. With ϕ=0.1 nN∕mV (Ref. 25) and γ=1.4×10⁻⁷ N s∕m,7 condition 25 leads to the frequency limit of about 10 kHz for guinea pigs and chinchilla.7 It should be noted that no single factor has a decisive effect on the frequency limit because the limit depends on the square root of each factor. Although the value for the friction coefficient γ of the subtectorial space, for example, is determined based on electron microscopy data that may need correction for shrinkage, such a correction has a relatively modest effect on the frequency limit because of the square root dependence.

DISCUSSION

Our model is intended to obtain an upper bound of the CM’s effect on enhancing electromotility, and it is much simpler than the configuration in which most experimental data are obtained. Specifically, we evaluated the CM generated by the transducer current of OHCs in one location of the cochlea associated with basilar membrane motion with small amplitudes. However, in most experimental conditions, the transducer current that contributes to the CM passes through OHCs in a much wider area, and the contributions of these OHCs are determined by a space constant. This explains why the calculated frequency dependence of the potential (Fig. 2) differs from that of the experimentally observed one.17

In an experimental condition where the cochlea is stimulated via stapes and an electrode for recording the CM is placed at a location in the organ of Corti, an increase in the frequency ω with constant stimulation intensity at the stapes shifts the location of maximum amplitude basally and changes the amplitude x of the basilar membrane motion at a given location, where the recording electrode is placed.

If the stimulation frequency is increased from below the best frequency of the recording locus, the peak of the traveling wave moves closer to the recording location from a more apical location. This basal shift of the current source advances the phase of the CM closer to π, from 130° to 164°in Fridberger et al.,17 as the frequency approaches the best frequency. The phase of CM further advances when the frequency further increases beyond the best frequency for stimulation intensities below 40 dB sound pressure level.17 This explanation is consistent with the interpretation given by Fridberger et al.17

Our sensor cells are at the location of the highest characteristic frequency, and we are only interested in high frequency response. We do not have to consider basal shifts of the current source as frequency increases. In addition, the phase of the CM experimentally observed near the best frequency is close to π,17 consistent with our model. For these reasons, our model does not contradict the experimental data for our purpose. The phase of CM only decreases with increasing frequency, reflecting the RC filter of the circuit.

Finally, our result that extracellular potential can improve the effectiveness of electromotility is consistent with a recent model evaluation26 and an experimental study.17 Our result is further consistent with a conclusion of the latter17 in that this factor alone is insufficient to make electromotility as the basis of cochlear amplifier.

CONCLUSIONS

We confirmed that the oscillatory potential in unstimulated OHCs induced by the extracellular potential can be greater than the receptor potential and that this effect can increase the efficiency of electromotility, as proposed by Dallos and Evans.15 However, we do not find that this effect can extend the frequency limit of about 10 kHz, which was previously estimated.7 It turned out that the introduction of an extracellular resistance, which is needed to produce the CM, reduces the RP. Since the significance of prestin and electromotility has been established,4 our result indicates that there must be another factor, such as fast voltage-gated K currents, in the basal turn where the characteristic frequencies exceed 10 kHz.