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- Bilinear Control Systems : Matrices in Action - kometdownblosbel.ga
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- Bilinear Control Systems: Matrices in Action
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The in vivo data fall considerably short of the extrapolations to higher CFs made in the in vitro gerbil study Johnson et al. 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.

## Bilinear Control Systems : Matrices in Action - kometdownblosbel.ga

A corner frequency at 2. When driven by a sufficiently large electrical input, OHC motility can generate vibrations up to very high frequencies, both in vitro Frank et al. Various schemes reviewed in Johnson et al. We assessed the quantitative effect of low-pass filtering by the OHCs Appendix 2. A kHz tone at the behavioral threshold of the gerbil is estimated to evoke an AC component of the OHC receptor potential of 5.

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, a. 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.

We therefore propose that OHC motility does not provide cycle-by-cycle feedback, but rather modulates sound-evoked vibrations Cooper et al. In this scenario the dynamic range compression in the cochlea is based on an automatic gain control system van der Heijden, in which the degree of OHC depolarization determines the gain.

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.

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

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.

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When mapping vibrations across the width of the cochlea partition cf. Acoustic stimuli were tailored to fit the nature of each experiment, as described below.

## Bilinear Control Systems: Matrices in Action

Each stimulus was coupled into the exposed ear-canal using a pre-calibrated, closed field sound-system. Each broad-band stimulus had 43 spectral components, spanning from 0. 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.

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.

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. Figure 2C of the main text shows the DP2 spectrum obtained by half-wave rectification of a tone zwuis stimulus with unequal linear amplitudes A 1 … A 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. This deviation stems from the imperfect approximation of half-wave rectification in terms of a second-order distortion. 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. 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. 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. 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.

## Python fem 2d code

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. 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.

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.

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A A tone complex with non-equalized primary spectrum green circles was rectified and low-pass filtered at 2. From the resulting DP2 spectrum, the primary spectrum was reconstructed black Xs. 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.

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. 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. 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. 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. 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. 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.

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This expression was minimized to produce the fits shown as black lines in Figure 4B and C. In this Appendix estimates are made of the AC receptor potential in OHCs in response to high-frequency tones near behavior threshold. These data were corrected for the membrane time constant, for which the authors used 6 dB per octave above Hz. These recordings originate from the kHz location, where the wave is assumed to have been amplified.