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The frequency limit of outer hair cell motility measured in vivo

  1. Anna Vavakou
  2. Nigel P Cooper
  3. Marcel van der Heijden  Is a corresponding author
  1. Erasmus MC, Netherlands
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Cite this article as: eLife 2019;8:e47667 doi: 10.7554/eLife.47667

Abstract

Outer hair cells (OHCs) in the mammalian ear exhibit electromotility, electrically driven somatic length changes that are thought to mechanically amplify sound-evoked vibrations. For this amplification to work, OHCs must respond to sounds on a cycle-by-cycle basis even at frequencies that exceed the low-pass corner frequency of their cell membranes. Using in vivo optical vibrometry we tested this theory by measuring sound-evoked motility in the 13–25 kHz region of the gerbil cochlea. OHC vibrations were strongly rectified, and motility exhibited first-order low-pass characteristics with corner frequencies around 3 kHz– more than 2.5 octaves below the frequencies the OHCs are expected to amplify. These observations lead us to suggest that the OHCs operate more like the envelope detectors in a classical gain-control scheme than like high-frequency sound amplifiers. These findings call for a fundamental reconsideration of the role of the OHCs in cochlear function and the causes of cochlear hearing loss.

https://doi.org/10.7554/eLife.47667.001

eLife digest

Our ears give us our sense of hearing. Their job is to collect sounds and pass this information on to the brain. Hair cells, a special group of cells in the ear, are responsible for detecting sound vibrations and turning them into the electrical signals that our brains can understand.

The ear contains two populations of hair cells: inner hair cells that send signals to the brain, and outer hair cells that act as a protective ‘buffer’ by modulating sound vibrations entering the innermost part of the ear. When outer hair cells are damaged, the vibrations picked up by inner hair cells are much smaller than in a healthy ear. This has led to the idea that outer hair cells actively amplify sounds before passing them on. That is, outer hair cells simultaneously act like microphones (by receiving sound from the environment) and loudspeakers (by re-emitting magnified vibrations).

One problem with this amplifier theory is that it cannot explain how some animals are able to hear extremely high-pitched sounds. If the theory is true, outer hair cells should be able to re-emit ultrasonic vibrations. However, some observations suggest that they may not vibrate fast enough to do so.

To test the amplifier theory, Vavakou et al. measured how outer hair cells in the ear of Mongolian gerbils responded to different sounds. This revealed that the motion of these cells could keep up with moderately high sounds (around the upper end of a piano’s range), but were too sluggish to amplify ultrasound despite gerbils having good ultrasonic hearing. Further experiments showed that instead of acting like amplifiers, outer hair cells seem to monitor the loudness of sound and adjust the level accordingly before passing the vibrations on to the inner hair cells.

These results shed new light on how outer hair cells help our ears work. Since damage to these cells can cause hearing loss, understanding how they work could one day guide new methods of protecting or even restoring hearing in vulnerable patients.

https://doi.org/10.7554/eLife.47667.002

Introduction

The hair bundles of auditory sensory cells are deflected by sound-driven vibrations, causing mechano-electric transduction channels to open and close. The resulting receptor current modulates the cell’s membrane potential. The mammalian cochlea contains two distinct types of hair cells. The vast majority of nerve fibers that carry the acoustic information to the brain innervate the inner hair cells (IHC). Up to a few kilohertz, IHC synapses can ‘phase-lock,’ that is, code the individual cycles of tones. At higher frequencies (>3 kHz), phase-locking rapidly declines and neural coding relies on the DC component of the IHC receptor potential generated by the asymmetric, or rectifying, nature of the IHC receptor current (Russell and Sellick, 1978).

Outer hair cells (OHC) modify the mechanical vibrations inside the organ of Corti (OoC), enabling frequency tuning and dynamic range compression. Dysfunctional and missing OHCs strongly reduce sensitivity, and this is a major cause of sensorineural hearing loss (Ryan and Dallos, 1975). The discovery of electromotility, length changes of isolated OHCs (Brownell et al., 1985) driven by variations in the membrane potential (Santos-Sacchi and Dilger, 1988), has intensified the study of OHCs and their functional significance. The membrane protein responsible for electromotility has been identified (Zheng et al., 2000), and prestin knockout mice have profound hearing loss (Liberman et al., 2002). The dominant view is that OHC electromotility drives vibrations within the OoC in a cycle-by-cycle manner (Ashmore, 2011) over the entire audible range, which extends up to 150 kHz in some species (Vater and Kössl, 2011). If this view is correct, the AC receptor potentials of OHCs must be large enough to be effective up to high frequencies, even though the membrane capacitance is expected to reduce the AC receptor potentials (and hence the OHCs' motility) at a rate of 6 dB per octave (Dallos, 1984). The functional implication of this electrical low-pass filtering is a limitation in the OHC’s ability to provide cycle-by-cycle mechanical feedback, known as the RC problem (Ashmore, 2011; Housley and Ashmore, 1992). The electrical corner frequency of OHCs has been measured electrophysiologically in vitro, with highest values ranging from 480 Hz (Mammano and Ashmore, 1996) to 1250 Hz (Johnson et al., 2011), but no systematic in vivo data exist due to technical limitations and the cochlea’s extreme vulnerability. In addition to the electrical filtering, the motile process itself may also be too slow to provide high-frequency mechanical feedback. An early in vitro report claiming a bandwidth for electromotility of at least 79 kHz (Frank et al., 1999) was recently challenged (Santos-Sacchi and Tan, 2018). Again, in vivo estimates of the corner frequency of motility are missing. Here we use optical vibrometry to measure non-linear components of the OHCs' motile response and determine the corner frequency of OHCs in the high-frequency region of the intact gerbil cochlea.

Results and discussion

In response to a tone pair (Figure 1A), vibrations in the OHC/Deiters’ cell region showed a strong envelope-following component (Figure 1B). This reveals a significant degree of rectification in OHC motion, producing 2nd-order distortions (DP2s) such as the ‘quadratic difference tone’ at f2-f1. Using multitone stimuli, we mapped the spatial profile of the DP2s inside the OoC by cross-section measurements. DP2s were concentrated in the OHC/Deiters’ cell ‘hotspot area’ (Cooper et al., 2018) (Figure 1C and D). They were observed at stimulus levels as low as 25 dB SPL (Figure 1—figure supplement 1), and disappeared post mortem (Figure 1—figure supplement 2). These observations confirm that OHCs are the source of the DP2s long known to exist from psychophysical (Zwicker, 1979), electrophysiological (Kim et al., 1980; Nuttall and Dolan, 1993), and cochlear-mechanical (Cooper and Rhode, 1997) studies. The OHC origin of DP2s is consistent with the significant rectified (‘DC’) component and 2nd harmonics observed in in vivo recordings of OHC receptor potentials (Dallos, 1986; Cody and Russell, 1987) and cochlear microphonics (Gibian and Kim, 1982).

Figure 1 with 2 supplements see all
Rectification in the mechanical response of OHCs.

(A) Two-tone stimulus with primary frequencies 4600, 5400 Hz; 70 dB SPL. Blue line, waveform; dashed black lines, envelope; red line, lowpass-filtered waveform (2000-Hz cut-off). (B) Mechanical displacement evoked by the two-tone stimulus, recorded in the gerbil OHC/Deiters’ cell region (13 kHz CF). Black line, displacement waveform; dashed black lines, envelope. Rectification is illustrated by the red line obtained by low-pass filtering (2000-Hz cut-off). Positive polarity indicates displacement away from the measurement probe, that is vertically downwards in (C) and (D). (C) OCT reflectance image (grayscale), with structural framework of Corti’s organ (yellow) superimposed for reference. Color-coded overlay: total RMS value of 2nd-order distortion products (DP2s) evoked by a 12-tone complex, 2–12 kHz; 60 dB SPL. DP2s dominate in the OHC region. Scale bar, 0.025 mm. (D) Underlying anatomical structures. BM = basilar membrane; ISL = inner spiral lamina; sm = scala media; dc = Deiters’ cells; t = tectal cells; TM = tectorial membrane; tc = tunnel of Corti; ot = outer tunnel. Hair cells, dark blue. (E) Zwuis stimulus (see text). (F) Vibration spectrum recorded in OHC region in response to the zwuis stimulus. Red diamonds, Rayleigh-significant DP2s. Dashed line, 6-dB/octave roll-off.

https://doi.org/10.7554/eLife.47667.003

Because rectification by OHC bundles produces DP2s in the receptor current, DP2s are an inevitable part of the electromotile response. More importantly, this part can be isolated through spectral analysis from recorded cochlear vibrations. This makes the DP2 spectrum ideally suited to studying the RC problem. To assess spectral trends, we employed a zwuis tone complex (Figure 1E), a stimulus designed to produce a rich DP2 spectrum upon rectification (van der Heijden and Joris, 2003; Victor, 1979). Rectifying an N-component zwuis stimulus generates N2 distinct DP2 components at frequencies fk ±fm, each of which can be traced back to a pair of interacting primary frequencies (fk,fm). The vibration spectrum obtained in the OHC region (Figure 1F) evoked by this stimulus reveals a rich family of DP2s having a systematic 6-dB/octave roll-off. This roll-off confirms the action of a low-pass filter between the hair bundle’s rectification and the motile response. Accurate estimation of the corner frequency, however, is hampered by the ~10-dB scatter of DP2 magnitudes within each frequency band. We identified three causes of this scatter (Figure 2). Their elimination reduces the scatter substantially, paving the way to accurate estimates of OHC corner frequencies.

Figure 2 with 2 supplements see all
Three causes of magnitude scatter in DP2 spectra.

(A) Combinatorial effect illustrated by feeding an equal-amplitude zwuis tone complex to a nonlinear circuit comprising a rectifier and low-pass filter (corner frequency 2.5 kHz). The 2nd harmonics (solid circles) are 6 dB weaker than the remaining DP2s (red diamonds). (B) Vector addition of DP contributions along the traveling wave (left to right). Lower row of ‘clocks’ depict amplitude and phase of the primaries f1,f2 <<CF. They accumulate little phase and their amplitude hardly grows upon traveling. Upper clocks depict a near-CF DP2 at f1+f2. Colors indicate the origin of each local contribution. Near-CF DP2 components suffer considerable phase accumulation and amplitude variation while traveling. The eventual amplitude (rightmost location) is the vector sum of multiple contributions widely differing in phase and amplitude. This interference obfuscates the spectral properties of DP2 generation investigated here. (C) Unequal primary amplitudes entering the nonlinear circuit generate a predictable scatter in DP2 magnitudes (see text). Companion phases are not affected by lack of equalization of the input. The 0.5-cycle low-frequency limit of the phase reflects the ‘negative polarity’ of the rectification.

https://doi.org/10.7554/eLife.47667.007

The first cause of scatter is combinatorial: when supplying an equal-amplitude input to a rectifier, the resulting second harmonics 2fk are 6 dB below the remaining DP2s (fk ±fm, k > m) (Figure 2A). This reflects the binomial coefficients occurring in the second-order terms of the power series describing the nonlinearity (Meenderink and van der Heijden, 2010). Since each DP2 component can be uniquely traced back to its ‘parent primaries,’ this is readily corrected. The second cause of scatter is the spatially distributed nature of DP2 generation. The primaries and DP2s propagate as traveling waves (Kim et al., 1980; Gibian and Kim, 1982), (Figure 2—figure supplement 1), so the recorded DP2s are a vector sum of contributions along the path from stapes to recording place (Schroeder, 1969). The magnitude of slowly propagating components is affected by interference across generation loci (Figure 2B), and the growth and subsequent decay of components entering their peak region further obfuscates their original magnitude. These confounding effects of propagation are eliminated by setting an upper frequency limit to the primaries and DP2s used for stimulation, analysis and iterated adjustment of the stimulus as described below. For this frequency limit we choose half the characteristic frequency (CF/2). Wave propagation below CF/2 is too fast to cause interference and magnitudes change little during fast propagation (Ren et al., 2011) (Figure 2—figure supplement 1).

The third cause of scatter in the DP2 spectrum is the unequal amplitude of the primary components entering the rectifier, that is, the effective input that deflects the OHC bundles. Possible causes of unequal amplitudes at the OHC input include imperfections in the sound calibration as well as non-flatness of middle-ear transfer and intracochlear propagation. Even a perfectly regular trend in the input spectrum such as a roll-off creates a scattered effect in the DP2 spectrum (see Appendix 1). The amplitude of a DP2 component is proportional to the product of its parents’ amplitudes (van der Heijden and Joris, 2003). This bilinearity causes a scatter in DP2 magnitude (expressed in decibels) equal to twice the range of the primary input magnitudes (Figure 2C and Appendix 1—figure 1). If the OHC input were known, it would take a simple adjustment of the stimulus spectrum to equalize the primary amplitudes at the OHC input, and thereby regularize the DP2 spectrum as in Figure 2A. Current in vivo measurement techniques lack the spatial resolution to determine OHC bundle deflection, but the effective OHC input can be retrieved from the rich DP2 spectrum by exploiting the bilinear relationship between primary and DP2 amplitudes. This computational method was previously used to retrieve the effective IHC input from auditory-nerve responses (van der Heijden and Joris, 2003). Here we use it to compute the effective OHC input (see Appendix 1) and to adjust the stimulus accordingly. The OHC input thus obtained differs from the linear component of OHC motion, and in fact resembles basilar membrane motion more closely (Figure 2—figure supplement 2). This means that motion recorded in the OHC region may not be used as a proxy for OHC input. The resemblance between basilar membrane motion and computed OHC input may shed light on the mechanisms underlying the deflection of OHC hair bundles. Within the current study, however, the OHC input is primarily of methodological interest.

Experimental equalization of the effective OHC input and compensation for the combinatorial effect had the predicted effect of markedly reducing the scatter in the DP2 spectrum (Figure 3). After equalization, the DP2 spectrum recorded from the OHCs closely resembles that of the simple nonlinear circuit of Figure 2A with its first-order low-pass characteristics. This holds for both the magnitude and the phase, having a minus 6-dB/octave high-frequency slope and a 0.25-cycle high-frequency asymptote (Figure 4C) respectively. The corner frequency was 2.5 kHz, 2.7 octaves below the 16-kHz CF.

Reducing the scatter of DP2 magnitudes by equalizing the effective OHC input during an experiment.

(A) Spectrum of the initial zwuis acoustic stimulus. (B) Spectrum of the vibrations recorded in the OHC region; CF = 16 kHz. Rayleigh-significant DP2s marked as red diamonds; 2nd harmonics corrected for the 6-dB combinatorial effect. (C) Effective OHC input computed from the DP2 magnitudes. (D) Spectrum of the adapted stimulus (4th iteration) aimed at equalizing the effective OHC input. (E) Resulting DP2 spectrum. (F) Effective OHC input. Note that the equalized input spectrum in F (compared to C) reduces the DP2 scatter in E (compared to B).

https://doi.org/10.7554/eLife.47667.010
Figure 4 with 1 supplement see all
Low-pass filtering and corner frequencies of OHCs.

(A) DP2 magnitude versus frequency measured across different cochleas at different CFs. Stimuli, 10–12 component zwuis not exceeding CF/2; component intensities 55–65 dB SPL, optimized to equalize OHC input. Individual curves are offset to avoid overlap; labeled straight lines indicate absolute displacement. CFs, see legend of panel D. Second harmonics were corrected for the 6-dB combinatorial deficiency. (B) The same magnitudes corrected post-hoc for the effects of residual magnitude inequality in the effective input (see Appendix 1—figure 3). Black lines, first-order low-pass filters fitted jointly to the magnitude and phase data of each recording. (C) Companion phase data: the difference between the recorded DP2 phases and the predictions obtained by adding or subtracting the primary phases of the OHC input (Appendix 1, Equation 3B). Black lines, phase curves of the fitted low-pass filters. (D) Corner frequencies from the fits versus CF. Symbols as in panels A-C. Open circles reproduce in vitro data from Johnson et al. (2011). Dashed line, unity line (corner frequency = CF). Explained variance of the fits (in order of increasing CF): 90%, 91%, 80%, 85%, 87%. When omitting the correction for residual scatter, and instead using the raw magnitudes from panel A, the estimates of the corner frequencies were lower by 1% to 10% (mean, 7%) compared to the estimates based on the corrected magnitudes. Captions of source Data.

https://doi.org/10.7554/eLife.47667.012

We obtained data from different cochlear regions in different animals, with CFs ranging from 13 to 25 kHz. The adjustment of the relative stimulus amplitudes always reduced the scatter of the DP2 magnitudes. First-order low-pass characteristics were consistently found across CFs (Figure 4). The corner frequencies obtained ranged from 2.1 to 3.3 kHz, showing a weak trend of increasing with CF (Figure 4D).

In summary, the rectification displayed in vivo by OHC motility provides a unique opportunity to directly measure OHC corner frequencies without opening the cochlea. When equalizing the primary amplitudes at the OHC input, the DP2 spectra reveal an unmistakable first-order low-pass character, both in terms of magnitude and phase. In the frequency range probed here (below CF/2) stiffness dominates OoC impedance rendering displacement proportional to force (Dong and Olson, 2009). Thus within the framework of models in which OHCs directly push the basilar membrane (e.g., Ramamoorthy et al., 2007), electromotile force itself suffers from the 6-dB/octave roll-off.

The corner frequencies of 2.1–3.3 kHz that we measured in vivo were 2.8 ± 0.2 octaves below the CFs of our recording locations. These corner frequencies are higher than values of membrane corner frequency from in vitro studies at lower CF: 480 Hz (guinea pig, CF ~7 kHz) (Mammano and Ashmore, 1996); 300–1250 Hz (gerbil, CF,~350–2500 Hz) (Johnson et al., 2011). The in vivo data fall considerably short of the extrapolations to higher CFs made in the in vitro gerbil study (Johnson et al., 2011) (which predict electrical corner frequencies of 6.5–11 kHz for the CFs tested here). Our observation of a simple 6-dB/octave roll-off and minus 0.25-cycle phase asymptote indicates the dominance of a single low-pass mechanism in the entire frequency range tested. Comparison with the in vitro data suggests that this dominant factor is the RC time of the cell membrane, which is fundamental to the operation of all biological cells. The somewhat higher corner frequencies of the present study (compared to the in vitro data) may be attributed the more basal location of the OHCs of the present study.

A corner frequency at 2.8 octaves below CF implies a 17-dB attenuation of the CF component. The shallow increase of OHC corner frequency with increasing CF suggests an even stronger attenuation at higher CFs than studied here. When driven by a sufficiently large electrical input, OHC motility can generate vibrations up to very high frequencies, both in vitro (Frank et al., 1999) and in vivo (Ren et al., 2016), but for acoustic stimulation the low-pass filtering of the receptor potential will limit the frequency range. Various schemes (reviewed in Johnson et al., 2011) have been proposed to push the frequency limit of electromotility beyond the corner frequency of the OHC cell membrane into the CF range. Our findings do not support such schemes, as the ~2.5-kHz corner frequency is evident in the motile response itself.

We assessed the quantitative effect of low-pass filtering by the OHCs (Appendix 2). A 16-kHz tone at the behavioral threshold of the gerbil is estimated to evoke an AC component of the OHC receptor potential of 5.7 μV at the peak of traveling wave. At the slightly more basal location where cycle-by-cycle amplification is assumed to start, it is ~1 μV. Inspection of the in vivo OHC recordings in guinea pig of Cody and Russell (1987) yield an 3.6-μV AC component at CF for a 17-kHz tone near the behavioral threshold, corresponding to ~0.6 μV at the spatial onset of the putative amplification. Even if these minute variations in the membrane potential could evoke a significant electromotile response, such a motile feedback is unlikely to improve sensitivity because of its expected poor signal-to-noise ratio (van der Heijden and Versteegh, 2015a).

Overall our data suggest that OHCs and IHCs have similar properties, namely, considerable rectification (Pappa et al., 2019) and a corner frequency not exceeding a few kilohertz. Thus, just like in high-frequency IHCs, the receptor potential of high-frequency OHCs is expected to mainly follow the envelope of the waveform that stimulates their hair bundles. In this sense both IHC and OHCs operate as envelope detectors. We therefore propose that OHC motility does not provide cycle-by-cycle feedback, but rather modulates sound-evoked vibrations (Cooper et al., 2018; van der Heijden and Versteegh, 2015b). In this scenario the dynamic range compression in the cochlea is based on an automatic gain control system (van der Heijden, 2005) in which the degree of OHC depolarization determines the gain. The spatial confinement of the motile response to the OHC/Deiters’ cell region presents another challenge to the prevailing theory that OHCs drive basilar membrane motion directly. It rather suggests that electromotility controls the local coupling between OHCs and Deiters’ cells in a parametric fashion, perhaps dynamically adjusting the amount of dissipation in the Deiters’ cell layer. This fundamentally different view of the function of OHCs has great consequences for the experimental study of their role in hearing loss and the origins of the vulnerability of cochlear sensitivity. As to theoretical work, it is important that models of cochlear function, whether invoking cycle-by-cycle feedback or not, incorporate the findings of the present study.

Materials and methods

Overview

The materials and methods employed in this study are summarized below. More extensive details are provided elsewhere (Cooper et al., 2018).

Sound evoked vibrations were recorded from the ossicles and cochlear partitions of deeply anesthetized female gerbils (n = 27, weight = 53–75 g). Spectral-domain optical coherence tomography (SD-OCT) measurements were made from the first turn of the intact cochlea, under open-bulla conditions – optical access to the partition being provided through the transparent round window membrane. The hearing thresholds of the animals were assayed using tone-evoked compound action potential (CAP) measurements from a silver electrode placed on the wall of the basal turn of the cochlea.

Animal preparation

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Animals were anesthetized using intraperitoneally injected mixtures of ketamine (80 mg/kg) and xylazine (12 mg/kg). Supplementary (1/4) doses of the same mixture were administered at intervals of 10–60 min to maintain the anesthesia at surgical levels throughout subsequent procedures. All experiments were performed in accordance with the guidelines of the Animal Care and Use Committee at Erasmus MC (protocol AVD101002015304).

OCT vibrometry

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An SD-OCT system (Thorlabs Telesto TEL320C1) was used for interferometric imaging and vibration measurements. Cross-sectional (B-scan) and axial images (A-scans and M-scans) were triggered externally using TTL pulses phase-locked to the acoustic stimulation system (Tucker Davies Technologies system III) at a sampling rate 111.6 kHz. The theoretical resolution of the OCT system was ~3.5 µm across a 3.5 mm depth-of-field (i.e., z-range), but the optics of our recording system (a Mitotoyu IR imaging lens with an NA of 0.055) introduced an axial point spread function of ~6 µm FWHM and a lateral resolution (in the xy plane) of 13 µm (all assessed in air, with a refractive index of 1; corresponding intracochlear measurements should scale inversely with the refractive index of perilymph, which we assumed to be 1.3). The linear operating range of the OCT system was >500 µm. The amount of light incident on the cochlea was ~3.7 mW. The sensitivity of the A-scan’s phase-spectra to vibration permitted measurement noise-floors that ranged from ~30 pm/√Hz in the cochlea down to ~3 pm/√Hz in the middle ear.

The OCT’s measurement beam was not aligned with any of the cochlea’s principal anatomical axes. The vibration measurements that we made should therefore be sensitive to structural movements in all three of the cochlea’s principal dimensions (radial, transverse, and longitudinal). Specifically, in all recordings used for this study, the measurement beam pointed toward scala vestibuli, toward the apex of the cochlea, and away from the modiolus.

When mapping vibrations across the width of the cochlea partition (cf. Figure 1C, Figure 1—figure supplement 2), measurements were spaced at intervals of between 6 and 12 µm in the xy-plane.

Basic response analysis

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Responses were analyzed by Fourier transformation of the vibration waveforms derived from contiguous groups of 3 pixels in each M-scan, where each pixel covers a depth of ~2.7 µm in the fluid-filled spaces of the cochlea, and ~3.5 µm in the air-filled spaces of the middle-ear. The statistical significance of each response component was assessed using Rayleigh tests of the component’s phase stability across time (Cooper et al., 2018).

Acoustic stimulation

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Acoustic stimuli were tailored to fit the nature of each experiment, as described below. Each stimulus was coupled into the exposed ear-canal using a pre-calibrated, closed field sound-system. Stimuli were generally presented for 12 s, with inter-stimulus intervals ~ 1 min.

Broad-band multi-tone ‘zwuis’ complexes (van der Heijden and Joris, 2003) were used to determine the characteristic frequency and sensitivity functions of each recording site (e.g. see Figure 4—figure supplement 1). Each broad-band stimulus had 43 spectral components, spanning from 0.4 to 30 kHz with an average spacing of 705 Hz. The components all had equal amplitudes, with levels expressed in decibels re: 20 μPa (i.e., dB SPL), but stimulus phase was randomized across frequency.

The unique property of a zwuis stimulus is that the frequencies of all of its primary components, and all of its potential inter-modulation distortion products up to the third order, are unambiguous. This means that all of the second-order distortion products (i.e. DP2s) studied in this paper can readily be attributed to a unique pair of spectral ‘parents’ (see Appendix 1).

Narrow-band zwuis stimuli were used to simplify the analysis and interpretation of DP2 spectra. They consisted from 10 to 15 components, ranging from few hundred hertz to at least one octave below the characteristic frequency of the recording side. The first presentation of each narrow-band stimulus had equal primary amplitudes, but their relative amplitudes were adjusted during subsequent presentations (fixing the average magnitude in dB SPL) in order to equalize the input to the OHCs. This procedure is described in the Appendix 1.

Appendix 1

Reduction of DP2 scatter by equalizing the effective OHC input

Here we describe the computational procedure leading to the reduction of magnitude scatter in the DP2 spectra (Figures 3 and 4). The computations are illustrated using the example primary spectrum of Figure 2C with its unequal primary magnitudes.

In a zwuis tone complex the primary frequencies f1fN are chosen such that all possible combinations fkm+=fk+fm (k ≥ m) and fkm=fkfm (k > m) are distinct and never coincide with any primaries. This means that all the N2 second-order distortion products (DP2s) generated by rectifying the stimulus are different, and can be uniquely traced back to their ‘parent’ primaries. Figure 2C of the main text shows the DP2 spectrum obtained by half-wave rectification of a 12-tone zwuis stimulus with unequal linear amplitudes A1A12. To good approximation, and up to a common scale factor, the amplitude of a DP2 component at frequency fk ±fm (k > m) is equal to the product AkAm of the parent amplitude (van der Heijden and Joris, 2003) whereas the amplitude of second harmonics at 2fk equals ½A2k (Meenderink and van der Heijden, 2010)

Appendix 1—figure 1A shows a numerical test of the amplitude approximation. The unequal-amplitude stimulus shown in Figure 2C of the main text, was rectified (but not low-pass filtered) and the DP2 spectrum was extracted, corrected for the 6-dB combinatorial effect, and compared against the bilinear prediction. The predictions are accurate with a fraction of a dB except for the very weakest DP2s, which are up to 1.2 dB lower than predicted. (This deviation stems from the imperfect approximation of half-wave rectification in terms of a second-order distortion.) The prediction for the phase of a DP2 at fk ±fm is φk±φm, where φ1φN are the primary phases (Meenderink and van der Heijden, 2010). Appendix 1—figure 1B shows the test of the prediction of DP2 phases from the primary phases using the same stimulus as in Appendix 1—figure 1A. The phase predictions are accurate to within 0.0025 cycle.

Appendix 1—figure 1
Predicting the DP2 spectrum from a primary spectrum and vice versa; no low-pass filtering.

(A) Actual DP2 magnitudes obtained by rectifying (not followed by filtering) the tone complex with unequal primary magnitudes shown in Figure 2C, left panel, plotted against the approximation of Equation 1. (B) As in (A), but now for the phase. (C) Retrieving the primary magnitudes from the DP2 spectrum by inverting (fitting) Equation 1. Actual primary magnitudes are plotted versus computed magnitude. (D) As in (C), but now for the phase.

https://doi.org/10.7554/eLife.47667.017

The ‘forward prediction’ (DP2s from primaries) illustrated in Appendix 1—figures 1A,1B is of less importance; it is the reverse procedure that is used to reconstruct from the measured DP2 data the underlying primary amplitudes and phases. Specifically, denoting the primary magnitudes by Mk=20logAk, and denoting DP2 magnitudes of sum tones at fkm+=fk+fm and difference tones at fkm=fkfm by Mkm+ and Mkm, respectively, and analogously for the phase, we obtain

(1A) Mkm±=Mk+Mm
(1B) φkm±=φk±φm

Note that scatter in the DP2 magnitudes results from any variation of primary magnitudes results.This includes regular trends in the primary magnitudes such as the deviation of a single primary component (e.g. due to a dip in the middle-ear transfer) or a systematic roll-off. Such regular trends in the input give scatter throughout the DP2 spectrum because each primary component affects DP2 components at multiple frequencies.

Retrieving the N primary magnitudes and phases (up to overall offsets) from the N2 DP2 magnitudes and phases amounts to solving the overdetermined set of Equation 1 in a least squares sense. Because Equation 1 is linear this leads to a unique and stable solution, and the numerical implementation is straightforward and efficient (e.g. MATLAB left matrix division). For N = 12 (as in Appendix 1—figure 1), 2 × 11 unknowns are retrieved from 2 × 144 knowns. The overdetermined character makes the procedure accurate and robust. A test of the retrieval of the relative primary magnitudes and phases from the DP2 spectrum is shown in Appendix 1—figures 1C, 1D. The deviations in the retrieved magnitude are systematic but small (≤0.25 dB); the phase is retrieved to within 0.00025 cycle.

When the rectifier is followed by a filter (whether low-pass or not) having complex transfer function Hα (f), where α stands for a set of parameters that characterize the filter, Equation 1 is extended to

(2A) Mkm±=Mk+Mm+20log|Hα(fkm±)|
(2B) φkm±=φk±φm+arg(Hα(fkm±))

In principle, Equation 2 can be used to fit the DP2 data in a least-squares sense as before, now also incorporating the new parameters α to the fit (in addition to the primary magnitudes and phases). When fitting experimental data, however, it is unclear a priori what type of filter to anticipate. To accommodate a variety of possible filter shapes, we extended Equation 1 by adding 7th-order polynomials (increasing the order did not change the results):

(3A) Mkm±=Mk+Mm+n=17βn(fkm±)n
(3B) φkm±=φk±φm+n=17γn(fkm±)n

Like Equation 1, this model is linear in its fit parameters, so it leads to a unique solution (in a least squares sense). In Equation 3A the primary magnitudes Mk and Mm describe the ‘within-band’ scatter of DP2 magnitudes, and the polynomial describes the post-rectifier filter. Fitting Equation 3 to the DP2 spectrum of Figure 2C (which includes the low-pass filtering) reproduces the primary spectrum accurately (Appendix 1—figure 2). The largest deviations are 0.3 dB and 0.07 cycle.

Appendix 1—figure 2
Retrieving the primary spectrum from the low-pass filtered DP2 spectrum.

(A) A tone complex with non-equalized primary spectrum (green circles) was rectified and low-pass filtered at 2.5 kHz. From the resulting DP2 spectrum, the primary spectrum was reconstructed (black Xs). (B) Scatter plot comparing the actual primary spectrum entering the rectifier + low-pass filter scheme to the primary spectrum reconstructed from the DP2 spectrum at the output of the low-pass filter.

https://doi.org/10.7554/eLife.47667.018

Having retrieved the primary spectrum (whose lack of flatness causes the scatter of DP2s), we can assess the post-rectifier filter. This is illustrated in Appendix 1—figure 3 for the magnitudes; the phase analysis is analogous. The actual DP2 spectrum of the rectified + low-pass filtered waveform is shown in Appendix 1—figure 3A. Inserting the retrieved primary magnitudes (shown in Appendix 1—figure 2A) into Equation 1 yields the predicted unfiltered DP2 spectrum (Appendix 1—figure 3B). This isolates the scatter. Subtracting the scatter from the actual DP2 spectrum retrieves the effect of the filter (Appendix 1—figure 3C). This is the DP2 spectrum ‘corrected for scatter’ (for the experimental data this scatter-corrected version of the magnitudes is shown in Figure 4B of the main text). It clearly reproduces the first-order low-pass filter used to generate the DP2 spectrum, which is shown in Appendix 1—figure 3C for reference. Note that up to this point nothing in the fitting procedure has presumed a low-pass filter; the polynomial terms of Equation 3 are agnostic in this respect.

Appendix 1—figure 3
Computational separation of DP2 scatter and filter effect.

(A) DP2 spectrum obtained by rectifying and low-pass filtering a zwuis multitone waveform having unequal primary amplitudes. (B) Magnitude scatter in the DP2 spectrum of panel (A), computed by inserting the retrieved primary magnitudes into Equation 1. (C) The effect of the low-pass filter isolated by subtracting the scatter contribution of panel (B) from the DP2 spectrum in (A). For reference, the gain curve of actual filter that was used to generate the DP2 spectrum (first-order low-pass; corner frequency 2.5 kHz) is also shown (black line).

https://doi.org/10.7554/eLife.47667.019

Theoretically, the above computational procedure is all that is needed to isolate and estimate the filter contribution to the DP2 spectrum (Appendix 1—figure 3C). In the experiments, we went further and used the retrieved OHC input to adapt the acoustic stimulus aimed at equalizing the OHC input and reducing the DP2 scatter (this step was iterated if necessary). We had two reasons for doing so. (1) Equalized OHC input spectra yield higher numbers of Rayleigh-significant DP2 components, thus richer datasets. This can be understood from the limited dynamic range of the measurements: if the DP2 magnitude scatter is too large, the weaker DP2s will drop below the noise floor. (2) Scatter reduction by adapting the stimulus is a powerful way of interrogating the system. The simple rectifier + filter scheme predicts that an adjustment of the relative SPLs of N primaries (i.e. using N-1 degrees of freedom) reduces the scatter in as many as N2 distortion components in the raw data. This is a strong prediction (recall that N ≥ 10 in our recordings), and to see it happening in the real data (Figures 3 and 4) confirms that the simple scheme is an adequate description of the DP2 spectra generated by the OHCs.

Fitting a first-order low-pass filter to the scatter-corrected DP2 spectrum (the last analysis step) was done jointly to the magnitude (Figure 4B) and phase data (Figure 4C). Consider a DP2 having a magnitude of M dB and a phase of φ cycles. In the complex spectrum of the response it is represented by a component Z=10M/20e2πiφ=e(ln10/20)M+2πiφ, or

(4) lnZ=(ln10/20)M+2πiφ

an expression that exposes the common logarithmic nature of M and φ, and provides the natural conversion factor between the two in the form of the ratio Θ=2π/(ln10/20)54.6 dB/cycle (this conversion factor would be unity if magnitude and phase were expressed in nepers and radians, respectively). The joint fit of magnitude data M (in dB) and phase data φ (in cycles) then amounts to minimizing the sum of squares

(5) χ2=(MdataMmodel)2+Θ2(φdataφmodel)2

This expression was minimized to produce the fits shown as black lines in Figure 4B and C.

https://doi.org/10.7554/eLife.47667.016

Appendix 2

Estimate of the AC receptor potential of OHCs near threshold

In this Appendix estimates are made of the AC receptor potential in OHCs in response to high-frequency tones near behavior threshold. Specifically, the AC receptor potential evoked by a 17-kHz tone at 5 dB SPL in guinea pig are derived from the in vivo OHC recordings of Cody and Russell (1987), and the AC receptor potential evoked by a 16-kHz, 5-dB-SPL tone in gerbil is estimated by combining recordings of basilar-membrane vibrations with electrophysiological data.

Figure 6B of Cody and Russell (1987) shows a 160-μV AC receptor potential for a 17-kHz tone at 15 dB SPL. These data were corrected for the membrane time constant, for which the authors used 6 dB per octave above 1200 Hz. Thus the raw AC component VAC, corrected for equipment filtering but not for the membrane time constant, was 160 × 1200/17000 = 11.3 μV. In this low-SPL regime the growth is linear, leading to an estimate of VAC = 3.6 μV for a 5-dB-SPL tone. These recordings originate from the 17-kHz location, where the wave is assumed to have been amplified. At the spatial onset of the putative amplification, basal to the best 17-kHz location, the excitation is smaller by ~15 dB (see below), leading to VAC = 0.6 μV at the spatial onset of compressive growth.

The AC receptor potential of OHCs measured in vivo saturates with low-frequency (200 Hz) stimulation at ~95 dB SPL (Fig. 11 of Dallos, 1986) and this matches the saturation of the cochlear microphonic potential at the round window, which is dominated by basal OHCs (Dallos and Cheatham, 1976), observed with 95-dB-SPL tones at 200 Hz (Fig. 4 of Patuzzi et al., 1989). In gerbil, a 95-dB-SPL tone at 200 Hz evokes a basilar membrane displacement of ~300 nm. Combining these two findings, the Boltzmann function relating BM displacement dBM to the mechano-transducer channel conductivity GMT,

(1) GMT(dBM)=Gmax/(1+eβdBM)

has β ≈ 0.01 nm−1. (Gmax is the maximum value with all transducer channels open.) Assuming that the resting value GMT,0 of the MT conductivity equals Gmax/2 (this is where the Boltzmann function is steepest, maximizing the AC component of the receptor current), the fractional conductivity variation (modulation m) for small BM displacements equals

(2) m=ΔGMT/GMT,0=β2dBM

The behavior threshold for 16-kHz tones in gerbil is 5 dB SPL (Ryan, 1976), and at the 16-kHz location of the sensitive gerbil cochlea, this evokes a BM displacement of dBM ≈ 0.1 nm, corresponding to m ≈ 5 × 10−4. This is the value at the peak excitation following the putative amplification. The actual SPL-dependent (physiologically vulnerable) amplitude growth occurs basal to the peak. In the 16-kHz region the amplitude growth amounts to ~15 dB at the lowest SPLs (Fig. 3A of Ren, 2002). Thus at the spatial onset of the putative amplification of the near-threshold 16-kHz tone, dBM ≈ 0.018 nm, corresponding to m ≈ 9 × 10−5.

The AC receptor potential follows from a straightforward linearization of the equivalent OHC circuit (e.g., Fig. 6B of Johnson et al., 2011),

(3) VAC=m(EK+EEP)/[(1/GK+1/GMT,0)(GK+GMT,0+iωC)]

using a total basolateral membrane capacitance C = 5 pF (neglecting the apical membrane capacitance); deriving the basolateral membrane resting conductance GK,0 from our corner frequency: GK,0 = Cω0 =Cfcorner, with fcorner = 2200 Hz (our Figure 4); the K+ reversal potential of EK = −75 mV; the resting value of MT conductivity GMT,0 = 75 nS (Fig S1 of Johnson et al., 2011); an endocochlear potential EEP = 90 mV; and the angular stimulus frequency ω = 2πf. Applying Equation 3 to the values of fractional MT conductivity m, we obtain VAC ≈ 5.7 μV at the peak of the wave and VAC ≈ 1.0 μV at the spatial onset of its nonlinear growth.

https://doi.org/10.7554/eLife.47667.020

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Decision letter

  1. Andrew J King
    Senior Editor; University of Oxford, United Kingdom
  2. Tobias Reichenbach
    Reviewing Editor; Imperial College London, United Kingdom
  3. Tobias Reichenbach
    Reviewer; Imperial College London, United Kingdom
  4. Elizabeth Olson
    Reviewer; Columbia University, United States
  5. Thomas Risler
    Reviewer; Institut Curie, France

In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.

Thank you for submitting your article "The frequency limit of outer hair cell motility measured in vivo" for consideration by eLife. Your article has been reviewed by three peer reviewers, including Tobias Reichenbach as the Reviewing Editor and Reviewer #1, and the evaluation has been overseen by Andrew King as the Senior Editor. The following individuals involved in review of your submission have agreed to reveal their identity: Elizabeth Olson (Reviewer #2); Thomas Risler (Reviewer #3).

The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.

Summary:

This is a well-presented and illustrated paper that addresses an important issue that has been controversial in the field of cochlear mechanics for years: is OHC prestin-based motility capable of amplifying the input signals on a cycle-by-cycle basis at auditory frequencies, up to several tens of kHz and above 100 kHz in some species? As the authors correctly point out, accurate in-vivo data acquired at higher frequencies are crucial to establish a clear cut answer to that essential question. The authors present convincing in vivo data that show that the corner frequencies of the outer hair cells are comparatively low, about a few kHz.

Essential revisions:

1) A major issue concerns the interpretation of the results, in particular the authors' conclusion that the low corner frequencies rule out cycle-by-cycle amplification by OHCs in the cochlea. For example, in Figure 3 the stimulus level is ~ 55 dB and the DPs are ~ 0.3 nm at 2 kHz (a frequency below the corner frequency) and ~ 0.1 nm at 10 kHz. From the literature, the response to primaries would be ~ 0.1-1 nm at this level (Cooper et al., 2018, Figure 6 panels c-f). Presumably the electromotility at the primaries is larger than the DP electromotility (because the primaries are bigger than the DPs), which means the expected primary electromotility response is roughly equal to the actual primary responses even with the corner frequency taken into account. This example is meant to emphasize that the question of what is "big enough" is a quantitative issue. How big does electromotility need to be in order to work effectively on a cycle-by-cycle basis? Computational models show that amplification at high frequencies can potentially work with comparatively slow OHC electromotility, in particular when combined with active hair-bundle motility (see, e.g., Maoileidigh and Jülicher, JASA 2010; Meaud and Gross, Biophysics. J. 2011). The low corner frequencies that the authors find here are clearly relevant for modelling, but the specific implications are less clear. Please discuss these issues.

2) Could the authors explain and justify their choice of restricting the range of both primaries and DP2s to frequencies not exceeding half of the characteristic frequency (why not another fraction)?

3) Please show at least one full frequency response to illustrate that these were healthy cochleae. Please show these data on a nm scale (ie, not normalized to stapes). As you note, you don't know what the input to the OHC stereocilia is (where the nonlinearity is that will produce the DPs), but a first approximation will be the motion of the primaries so please show that – show the parent f1,f2 in addition to the DPs.

4) The phase reference is not obvious, please include that info when plotting phase. (We assume that it is ph2-ph1, ph2+ph1 of the OHC primary motion response.) Also, depending on which side of the IO curve the system is operating on, the sign of the DP would be different, which would result in a 180 phase difference. Was evidence of that ever seen in preparations? (For example, see Figure 2 of Brown et al., JASA 125, 2009.)

5) Where does the scatter in the amplitudes of the primary components of the zwuis stimulus come from? Is it already present in the sound stimulus, or does it reflect irregularities in how the sound propagates into the cochlea? It would also be good to see if this scatter is robust for a given ear, that is, whether one obtains the same scatter when the sound system is detached and then re-attached to a given ear.

6) Is correction 3 actually required? We think that it would be good to compare the results obtained from the corrected data to those obtained without correction 3, since the latter results might be similarly clean. In particular, it appears that the phase is not affected by the primary scatter and that is favorable to your argument, and in many of your examples the scatter is not so large as to obscure the low-pass trend of the amplitude even without correction 3.

[Editors' note: further revisions were requested prior to acceptance, as described below.]

Thank you for resubmitting your work entitled "The frequency limit of outer hair cell motility measured in vivo" for further consideration at eLife. Your revised article has been favorably evaluated by Andrew King (Senior Editor) and three reviewers, one of whom is a member of our Board of Reviewing Editors.

The manuscript has been improved but there are some remaining issues that need to be addressed before acceptance, as outlined below:

Abstract: "exceeds the electrical limits" is vague and sounds worse than the reality. It's better to be quantitative and write something like "are attenuated by a factor of ~ 7".

Introduction section: “Again, in vivo data are missing” Please replace "missing" with "minimal" since in vivo data do exist and are referred to later.

Results and Discussion paragraph four: For clarity, please write "BM motions and calculated OHC input"

Results and Discussion paragraph seven: A reference to "In-vivo impedance of the gerbil organ of Corti at auditory frequencies." Biophysical Journal 97: 1233 – 1243" would provide support for the statement that the OOC impedance is stiffness dominated.

Results and Discussion paragraph eight: Johnson et al. predicted the electrical corner frequency, whereas your measurements are of the corner frequency due to electrical + any mechanical low-pass filtering. (You alluded to the possibility of mechanical filtering in the Introduction.) Thus a direct comparison with Johnson et al. needs qualification here.

Paragraph nine: Please either say "much stronger at much higher" or "stronger at higher" (we suggest the latter.)

Final paragraph: Please delete "clear-cut" – the reader should decide this on their own.

Figure 2—figure supplement 2. To make this figure easier to access, we suggest adding to the caption after the first sentence something along the lines of, "In A all the DP components relating to the lowest primary are shown. In B, that lowest primary is excluded and all the DP components relating to the new lowest primary are shown, and so on through panel K."

Results and Discussion paragraph three: This relates to our previous comment on the first version. The authors correctly addressed the combinatorial effect, but kept the larger or equal k > = m in the parenthesis. It seems that it should be k>m (and not equal), otherwise we fall back to the 2fk case.

Appendix 1—figure 2: Regarding our previous comment on this figure, the authors acknowledged that there was a typo referring to Figure 3C and not Figure 2C (after Equation 3B). They have however omitted to correct for that in the revised manuscript. Please correct this.

https://doi.org/10.7554/eLife.47667.022

Author response

Summary:

This is a well-presented and illustrated paper that addresses an important issue that has been controversial in the field of cochlear mechanics for years: is OHC prestin-based motility capable of amplifying the input signals on a cycle-by-cycle basis at auditory frequencies, up to several tens of kHz and above 100 kHz in some species? As the authors correctly point out, accurate in-vivo data acquired at higher frequencies are crucial to establish a clear cut answer to that essential question. The authors present convincing in vivo data that show that the corner frequencies of the outer hair cells are comparatively low, about a few kHz.

We thank the Editor and reviewers for their careful comments and helpful suggestions. Before addressing the individual comments, we would like to clarify a general methodological aspect that is relevant to several of them. The central results of our study (Figure 4) were obtained exclusively from the distortion (DP2) spectrum. Only the DP2 part of the response could be ascribed with confidence to electromotility. In contrast, the linear response part recorded in the OHC region is an unknown mix of passive (non-vulnerable) and motile contributions. Thus a priori one may not expect the linear component of OHC motion to be a good proxy for the input to the OHCs that shapes its motile response. In fact, as shown below, BM motion resembles OHC input more than does OHC motion itself.

Essential revisions:

1) A major issue concerns the interpretation of the results, in particular the authors' conclusion that the low corner frequencies rule out cycle-by-cycle amplification by OHCs in the cochlea. For example, in Figure 3 the stimulus level is ~ 55 dB and the DPs are ~ 0.3 nm at 2 kHz (a frequency below the corner frequency) and ~ 0.1 nm at 10 kHz. From the literature, the response to primaries would be ~ 0.1-1 nm at this level (Cooper et al., 2018, Figure 6 panels c-f). Presumably the electromotility at the primaries is larger than the DP electromotility (because the primaries are bigger than the DPs), which means the expected primary electromotility response is roughly equal to the actual primary responses even with the corner frequency taken into account. This example is meant to emphasize that the question of what is "big enough" is a quantitative issue. How big does electromotility need to be in order to work effectively on a cycle-by-cycle basis? Computational models show that amplification at high frequencies can potentially work with comparatively slow OHC electromotility, in particular when combined with active hair-bundle motility (see, e.g., Maoileidigh and Jülicher, JASA 2010; Meaud and Gross, Biophysics. J. 2011). The low corner frequencies that the authors find here are clearly relevant for modelling, but the specific implications are less clear. Please discuss these issues.

For the relative magnitudes of DP2s and linear response components, see the reply to comment #3 and the new Figure 2—figure supplement 2. Please note that the vibration amplitudes recorded in the OHC/Deiters’ cell region in response to 55-dB-SPL tones in Figure 6d of Cooper et al., 2018, are much higher than the ~0.1-1 nm mentioned in the comment: they are 4.2 nm at 2.3 kHz and 1.4 nm at 9.3 kHz. Even so, the direct comparison of amplitudes between linear components and distortion products is of limited use because overall DP2 magnitude (but not the relative DP2 amplitudes) depends on the unknown degree of nonlinearity. In the above comment and elsewhere in the review the assumption seems to be made that OHC vibrations are primarily motility-driven in the first place. As shown below this assumption is contradicted by the data, but in any case we object (on logical grounds) against using this premise when reasoning about the potency of motility to drive motion in the cochlea.

We agree that a quantitative analysis may be helpful. Using a refinement of the derivation in Versteegh and Van der Heijden (2015), we estimated the AC receptor potential evoked by a 16-kHz tone at 5 dB SPL, i.e., at the behavioral threshold of the gerbil. The calculations are presented in the new Appendix 2. In the penultimate paragraph of the main text we now briefly discuss the outcome in the context of cycle-by-cycle feedback.

Apart from these quantitative aspects, there is the scientific context created by the 35-year-long debate on this topic. That corner frequencies much smaller than CF “spell trouble” for somatic cycle-by-cycle amplification is a view expressed in a large body of literature. For instance, Ashmore, 2011, writes “In the intact cochlea, however, the electrical filtering effect of the cell membrane, effectively possessing an electrical time constant = RmCm, would reduce potential changes to negligible levels at any significant acoustic frequencies.” It is against this background that we discuss our result as being problematic for cycle-by-cycle amplification, and propose potential alternatives for the role of somatic motility. In the new text in the penultimate paragraph we touch upon this context by addressing previous proposals aimed at circumventing the RC problem.

2) Could the authors explain and justify their choice of restricting the range of both primaries and DP2s to frequencies not exceeding half of the characteristic frequency (why not another fraction)?

Explanation: In order to isolate the effect of DP generation one must stay well away from the effects of slow propagation of the primaries and the DPs. Since any deviation in primary amplitude and phase will be doubled in the DP2 components, it is wise to be on the safe side. The iterative equalization method is based on the computation of the effective input from the previous recording. This computation depends on the choice of the frequency range of both primaries and DP2s, so we had to fix the choice prior to collecting the bulk of the data. From the pilot experiments we observed that CF/2 was certainly safe enough and we stuck to it for consistency. Note also that using primaries above CF/2 results in their sum DP2s being above CF, which typically makes them too weak to record. Such an a priori waste of DP2 components spoils the redundancy needed for computing the effective OHC input.

Justification: DP2s do propagate and in doing so accumulate phase and amplitude changes that are unrelated to the local parent primaries (Figure 2—figure supplement 1). The choice of CF/2 led to an accurate and reproducible determination of the effective input spectrum, which in turn enabled us to iteratively reduce the DP2 scatter. The frequency range is large enough by a considerable margin to observe the first-order filtering and to fit the data (Figure 4). In all this, an important practical consideration is the time it takes to record, transfer the raw data from the measurement computer, perform the basic processing steps and compute the spectrum of the next stimulus in the adaptive process leading to equalization of the DP2 spectrum. The cycle is 15-20 minutes and we typically need 3 cycles. In vivo experiments that require sensitive cochleae allow little room for non-critical fine tuning.

In the text we clarified the CF/2 choice and its relation to the iterative method, including an explicit reference to Figure 2—figure supplement 1.

3) Please show at least one full frequency response to illustrate that these were healthy cochleae. Please show these data on a nm scale (ie, not normalized to stapes). As you note, you don't know what the input to the OHC stereocilia is (where the nonlinearity is that will produce the DPs), but a first approximation will be the motion of the primaries so please show that – show the parent f1,f2 in addition to the DPs.

This comment has two different aspects: (A) show the relation between DPs and linear response components; (B) show the sensitivity of the cochleae.

A) As requested, we provided a new figure (Figure 2—figure supplement 2) that combines linear components and DPs on a nm scale. These are responses to the main stimuli of the study (Figure 4, highest CF). While we indeed note in the manuscript that we cannot directly record the input to the OHCs, we do know it accurately from the analysis of DP2s (up to an overall scaling factor, see Appendix 1). The OHC input is also shown in Figure 2—figure supplement 2. Contrary to the expectation expressed in the reviewers’ comments, the vibrations in the OHC region are not a good approximation of the input of to the OHCs. In fact, BM motion (also shown in Figure 2—figure supplement 2) is a better proxy for the OHC input. This fits with Ter Kuile’s mechanism for hair-bundle excitation by the tilting of the tunnel of Corti following BM deflection. It is also consistent with the fact that suppression, which is mediated by OHCs, is accurately predicted by BM displacement over a wide frequency range (Cooper, 1996; Versteegh and van der Heijden, 2013). It also agrees with the observation made and discussed in Cooper et al., 2018, that inner hair cell excitation (as reflected by tuning in the auditory nerve) resembles basilar membrane motion more than hotspot motion. The new Figure 2—figure supplement 2 is cited, when introducing the computation of OHC input. The contrast between OHC input and OHC motion is mentioned, as is the resemblance of BM motion and OHC input. We do not elaborate on these findings here because they do not directly relate to the OHC corner frequency.

The expectation that OHC motion would reflect OHC input is expressed multiple times in the review. We also in Author response image 1 address this question in a complementary way. It compares two different equalization strategies: (1) equalizing linear OHC motion; (2) equalizing OHC input computed from the DP2 spectrum (as in the rest of the study). The scatter in the DP2 spectrum is much larger with (1) than with (2), confirming that OHC motion is a poor predictor of OHC input. This graph is not included in the manuscript because it is not directly related to the OHC corner frequency.

B) Cochlear sensitivity cannot be assessed using the main stimulus used in this study since it has no near-CF components. We therefore provide (Figure 4—figure supplement 1A) a set of responses to a wide range of frequencies (including CF) at multiple SPLs, recorded from the CF=18 kHz cochlea of Figure 4. They display the familiar compressive behavior of sensitive cochleae. For comparison Figure 4—figure supplement 1B shows 30-dB-SPL recordings of all five cochleae of Figure 4. The new Figure 4—figure supplement 1 is called out in the Materials and methods section.

Author response image 1

4) The phase reference is not obvious, please include that info when plotting phase. (We assume that it is ph2-ph1, ph2+ph1 of the OHC primary motion response.) Also, depending on which side of the IO curve the system is operating on, the sign of the DP would be different, which would result in a 180 phase difference. Was evidence of that ever seen in preparations? (For example, see Figure 2 of Brown et al., JASA 125, 2009.)

Phase reference. Just like the magnitude data, the phase data shown in Figure 4 are exclusively derived from the DP2 spectrum. What is shown in Figure 4C (and explained in Appendix 1) is the difference between the measured DP2 phase and the DP2 phase predicted to occur after rectification (without filtering) of the OHC input having the computed phases. This prediction is indeed ph2-ph1, ph2+ph1, but the primary phases ph1, ph2 are taken from the effective OHC input. The phase reference is now mentioned in the caption of Figure 4C, including a reference to Appendix 1, Equation 3B.

Direction of rectification. The waveform in Figure 1B is representative of the “polarity” of the rectification in all the data in this study. We now mention in the caption what the positive direction in Figure 1B means in terms of the measurement beam, and in Materials and methods we now mention the relation between beam direction and each of the anatomical axes of the OoC. Again, however, we caution against overinterpretation. Information regarding IO curves can only be obtained by direct comparison of the input and the output, and as explained under #1 and #3, recorded OHC motion may not be identified with the output of electromotility. From the perspective of the desired IO analysis, OHC motion is “polluted” by non-motile-driven motion.

The link between phase offset and polarity of rectification is now pointed out in the caption of Figure 3C. In the caption of Figure 2C we now clarify low-frequency limit of the phase offset in terms of the direction of the rectifier.

5) Where does the scatter in the amplitudes of the primary components of the zwuis stimulus come from? Is it already present in the sound stimulus, or does it reflect irregularities in how the sound propagates into the cochlea? It would also be good to see if this scatter is robust for a given ear, that is, whether one obtains the same scatter when the sound system is detached and then re-attached to a given ear.

As shown in Author response image 2, detaching and re-attaching the ear coupler resulted in minor variations in stapes vibration (RMS deviation, 1 dB). We realize that our use of the term “scatter” has been a bit imprecise. If we define scatter as “distributed deviations from a regular pattern” then scatter in the DP2 spectrum results from any magnitude differences among primaries of the OHC input. This includes regular (“non-scattered”) trends in the OHC input such as the deviation of a single component (e.g. from a notch in the middle-ear transfer) as well as systematic roll-offs. Such regular trends in the input give scatter throughout the DP2 spectrum because each primary component affects DP2 components at multiple frequencies. We now point this out in Appendix 1 (following Equation 1), and replaced the term “scatter” by “unequal magnitudes” or equivalent whenever referring to the OHC input. We can think of three causes of non-equalized spectrum at the OHC input when using an equal-amplitude acoustic stimulus: (1) imperfections in the calibration; (2) middle ear transfer characteristics; (3) non-flatness of the intracochlear transfer from stapes to OHC bundle deflection. In each case, irregularities as well as systematic trends will result in scatter of the DP2 spectrum, and the range of the Dp2 scatter will be twice the range of the input magnitudes (see Appendix 1—figure 1). We now briefly mention these potential causes in the main text (description of Figure 2B) and Appendix 1 (following Equation 1).

Author response image 2

6) Is correction 3 actually required? We think that it would be good to compare the results obtained from the corrected data to those obtained without correction 3, since the latter results might be similarly clean. In particular, it appears that the phase is not affected by the primary scatter and that is favorable to your argument, and in many of your examples the scatter is not so large as to obscure the low-pass trend of the amplitude even without correction 3.

When using the raw data of Figure 4A (instead of the corrected ones of Figure 4B), corner estimates are slightly lower than the reported ones, namely by 1 to 10% (mean, 7%). This is now mentioned in the caption. Still, the correction is necessary. Our method is valid on the grounds that any trend that cannot be explained by variation in the primaries magnitude is attributed to filter properties of the OHCs. Thus any scatter introduced by primary magnitude variation should be removed from our data.

[Editors' note: further revisions were requested prior to acceptance, as described below.]

The manuscript has been improved but there are some remaining issues that need to be addressed before acceptance, as outlined below:

Abstract: "exceeds the electrical limits" is vague and sounds worse than the reality. It's better to be quantitative and write something like "are attenuated by a factor of ~ 7".

At this introductory stage of the Abstract, there is no factor of 7 yet (it is part of our findings). The sentence merely introduces the problem to be addressed; it contains no claims as to its gravity. We chose the phrasing “frequencies that exceed the electrical limits (i.e. corner frequencies) of their cell membranes” in order to briefly explain what a corner frequency is. We did so because the reviewers had asked us to make the Abstract more accessible to non-specialist readers.

We have kept the formulation as is, but we would be happy to change it to “frequencies that exceed the corner frequency of their cell membranes” if the Editor prefers this.

Introduction section: “Again, in vivo data are missing” Please replace "missing" with "minimal" since in vivo data do exist and are referred to later.

In vivo studies that used high-frequency current injection to evoke mechanical responses (e.g. Ren et al., 2016) have not yielded estimates of corner frequencies, the topic being reviewed here. We clarified this by replacing “in vivo data” by “in vivo estimates of the corner frequency of motility”

Results and Discussion paragraph four: For clarity, please write "BM motions and calculated OHC input"

Done.

Results and Discussion paragraph seven: A reference to "In-vivo impedance of the gerbil organ of Corti at auditory frequencies." Biophysical Journal 97: 1233 – 1243" would provide support for the statement that the OOC impedance is stiffness dominated.

We agree and have inserted the reference.

Results and Discussion paragraph eight: Johnson et al. predicted the electrical corner frequency, whereas your measurements are of the corner frequency due to electrical + any mechanical low-pass filtering. (You alluded to the possibility of mechanical filtering in the Introduction.) Thus a direct comparison with Johnson et al. needs qualification here.

We added “electrical” in “which predict electrical corner frequencies.” In the subsequent sentence we added “and minus 0.25-cycle phase asymptote” when addressing the first order lowpass character of our data. The next sentence is now expanded to “Comparison with the in vitrodata suggests that this dominant factor is the RC time of the cell membrane, which is fundamental to the operation of all biological cells.” We feel that the alternative explanation (mechanical filtering being the dominant factor), although theoretically possible, is too farfetched to warrant discussion. It would amount to denying well-known electrical properties of cells (even beyond Johnson et al’s bold extrapolations) with little functional implication or relevance, as it does not push the corner frequency of motility itself.

Paragraph nine: Please either say "much stronger at much higher" or "stronger at higher" (we suggest the latter.)

We changed it to “an even stronger attenuation at higher CFs than studied here”, underscoring that 17 dB is already a considerable attenuation (and alluding to the much higher CFs found in some species mentioned in the Introduction).

Final paragraph: Please delete "clear-cut" – the reader should decide this on their own.

Deleted.

Figure 2—figure supplement 2. To make this figure easier to access, we suggest adding to the caption after the first sentence something along the lines of, "In A all the DP components relating to the lowest primary are shown. In B, that lowest primary is excluded and all the DP components relating to the new lowest primary are shown, and so on through panel K."

We inserted this very helpful description exactly as suggested.

Results and Discussion paragraph three: This relates to our previous comment on the first version. The authors correctly addressed the combinatorial effect, but kept the larger or equal k > = m in the parenthesis. It seems that it should be k>m (and not equal), otherwise we fall back to the 2fk case.

Apologies – now corrected.

Appendix 1—figure 2: Regarding our previous comment on this figure, the authors acknowledged that there was a typo referring to Figure 3C and not Figure 2C (after Equation 3B). They have however omitted to correct for that in the revised manuscript. Please correct this.

Apologies – now corrected.

To strengthen the requested quantitative analysis (Appendix 2), we added a rather straightforward estimate of the AC receptor potential at 17 kHz based on Cody and Russell’s, 1987, in vivo OHC recordings. The estimate is briefly mentioned in the penultimate paragraph of the main text.

We also spotted and corrected an error in the references (van der Heijden and Versteegh, 2015a).

https://doi.org/10.7554/eLife.47667.023

Article and author information

Author details

  1. Anna Vavakou

    Department of Neuroscience, Erasmus MC, Rotterdam, Netherlands
    Contribution
    Data curation, Software, Validation, Investigation, Visualization, Methodology, Writing—original draft
    Competing interests
    No competing interests declared
  2. Nigel P Cooper

    Department of Neuroscience, Erasmus MC, Rotterdam, Netherlands
    Contribution
    Supervision, Validation, Investigation, Methodology, Writing—review and editing
    Competing interests
    No competing interests declared
  3. Marcel van der Heijden

    Department of Neuroscience, Erasmus MC, Rotterdam, Netherlands
    Contribution
    Conceptualization, Resources, Data curation, Software, Formal analysis, Supervision, Funding acquisition, Investigation, Visualization, Methodology, Project administration, Writing—review and editing
    For correspondence
    m.vanderheyden@erasmusmc.nl
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-3876-1257

Funding

Horizon 2020 (H2020- MSCA-ITN-2016 [LISTEN - 722098])

  • Anna Vavakou

Nederlandse Organisatie voor Wetenschappelijk Onderzoek (ALW 823.02.018)

  • Nigel P Cooper

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Acknowledgements

This work was supported by the Netherlands Organization for Scientific Research, ALW 823.02.018, and an EU Horizon 2020 Marie Skłodowska-Curie Action Innovative Training network, H2020- MSCA-ITN-2016 [LISTEN - 722098].

Ethics

Animal experimentation: Experiments were performed in accordance with the guidelines of the Animal Care and Use Committee at Erasmus MC, which approved all protocols. Protocol number: AVD101002015304.

Senior Editor

  1. Andrew J King, University of Oxford, United Kingdom

Reviewing Editor

  1. Tobias Reichenbach, Imperial College London, United Kingdom

Reviewers

  1. Tobias Reichenbach, Imperial College London, United Kingdom
  2. Elizabeth Olson, Columbia University, United States
  3. Thomas Risler, Institut Curie, France

Publication history

  1. Received: April 13, 2019
  2. Accepted: August 27, 2019
  3. Version of Record published: September 24, 2019 (version 1)

Copyright

© 2019, Vavakou et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

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