The frequency limit of outer hair cell motility measured in vivo

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.


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 Kö ssl, 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 cochlearmechanical (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).
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; 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. Victor, 1979). Rectifying an N-component zwuis stimulus generates N 2 distinct DP2 components at frequencies f k ±f m , each of which can be traced back to a pair of interacting primary frequencies (f k , f m ). 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.
The first cause of scatter is combinatorial: when supplying an equal-amplitude input to a rectifier, the resulting second harmonics 2f k are 6 dB below the remaining DP2s (f k ±f m , 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 Figure 2. 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 2 nd 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 f 1 ,f 2 <<CF. They accumulate little phase and their amplitude hardly grows upon traveling. Upper clocks depict a near-CF DP2 at f 1 +f 2 . 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. DOI: https://doi.org/10.7554/eLife.47667.007 The following figure supplements are available for figure 2: 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 highfrequency asymptote ( Figure 4C) respectively. The corner frequency was 2.5 kHz, 2.7 octaves below the 16-kHz CF.
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 mV at the peak of traveling wave. At the slightly more basal location where cycle-by-cycle amplification is assumed to start, it is~1 mV. Inspection of the in vivo OHC recordings in guinea pig of Cody and Russell (1987) yield an 3.6-mV AC component at CF for a 17-kHz tone near the behavioral threshold, corresponding to~0.6 mV 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-bycycle 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
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
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 mm 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 mm FWHM and a lateral resolution (in the xy plane) of 13 mm (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 mm. 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/HHz in the cochlea down to~3 pm/HHz 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 mm in the xyplane.

Basic response analysis
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 mm in the fluid-filled spaces of the cochlea, and~3.5 mm 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
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 4figure 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 mPa (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. The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication. Data availability Source data files have been provided for Figures 1 and 3, and 4.

Author contributions
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 N 2 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 Â 11 unknowns are retrieved from 2 Â 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 H a (f), where a stands for a set of parameters that characterize the filter, Equation 1 is extended to 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 a 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 7 th -order polynomials (increasing the order did not change the results): 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 M k and M m 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. 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). 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 N 2 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 ' cycles. In the complex spectrum of the response it is represented by a component Z ¼ 10 M=20 e 2pi' ¼ e ln10=20 ð ÞMþ2pi' , or an expression that exposes the common logarithmic nature of M and ', and provides the natural conversion factor between the two in the form of the ratio Q ¼ 2p= ln10=20 ð Þ ' 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 ' (in cycles) then amounts to minimizing the sum of squares This expression was minimized to produce the fits shown as black lines in Figure 4B and C.
frequency w = 2pf. Applying Equation 3 to the values of fractional MT conductivity m, we obtain V AC » 5.7 mV at the peak of the wave and V AC » 1.0 mV at the spatial onset of its nonlinear growth.