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Transient-Evoked Otoacoustic Emissions (TEOAEs) were studied, with particular reference to their subject-dependent features. To this end, an electric model of the ear was implemented and validated. Simulated and natural TEOAEs were analyzed through a nonlinear analysis technique. The simulated signals were able to reproduce the dynamical features of the experimentally observed TEOAEs and, most importantly, the natural variability among individuals. The unexpected inverse relation between model complexity and adherence to the natural signals is commented.

Otoacoustic Emissions (OAEs) are low-level amplitude acoustic signals generated in the inner ear and measurable in the external auditory canal. Since their discovery [

The Transient Evoked OAE (TEOAEs) considered in this paper are signals evoked by an external stimulus. They include a passive ringing linearly dependent on the incident stimulus and with short latency, followed by a smaller, nonlinear, long latency and long duration oscillation.

Natural TEOAEs have been successfully investigated [2,3] by means of Recurrence Quantification Analysis (RQA) and simulated on the basis of different models of the human hearing function [

We implemented such a model and compared simulations and natural signals by means of RQA and Principal Components Analysis (PCA) in the aim to reproduce the natural variability of TEOAEs. Unexpectedly, a better match with natural signals was obtained by the less complicated model in terms of active cochlear modules. This is in line with the idea that reproducing complex biological phenomena, resulting from a manifold of nonlinear interactions, does not necessarily require complicated physical models [

Natural TEOAEs responses were obtained in the Audiology Department of Palermo University, Italy, from 104 healthy subjects (50 males, 54 females; age: 27.7 ± 8.2 yr). The signals were recorded by the ILO88 system (Otodynamics) according to the protocol described in [

In the electro-acoustic analogy the outer ear is represented as a uniform transmission line [

The model equations were solved using PSpice^{TM}, a standard electrical simulation tool previously used to study an entirely passive electric model of the cochlea [

where is a gain factor, , , and are the resistance, capacitance, and current of the n-th cochlear branch, is the voltage drop across, is a constant, and is the basilar membrane segment area represented by the n-th branch. Equation (1) has been reworked to make it suitable for implementation into PSpice^{TM} as:

where:

Equation (2) shows the dependence of OHC voltage sources on the ratio between the current in the cochlear branch and the voltage across. The input voltage is a rectangular pulse 80 µs wide corresponding to a stimulus of 80 dB SPL.

RQA is a technique used to quantify the amount of deterministic structure of short, non-stationary signals [14, 15] and to pick sudden phase changes possibly underlying mechanistically relevant phenomena.

The RQA descriptors of each signal can be worked out from the corresponding recurrence plot (RP). A RP is reckoned as follows:

• An embedding matrix (EM) is built, where the first column is the time series representing the signal and the following columns are time-lagged copies of it;

• A distance matrix is evaluated, whose element is the Euclidean distance between the i, j rows of EM;

• If is lower or equal to a predefined cut-off value (radius), the i, j location in the RP plot is darkened, marking a recurrent point, otherwise it is left blank.

RQA showed quite useful in the analysis of many physiological signals [16,17] as well as of spatial series like DNA and protein sequences [

• % Recurrence, fraction of the plot occupied by recurrent points, measuring the amount of periodic and auto-similar behavior of the signal;

• % Determinism, fraction of recurrent points aligned parallel to the main diagonal, indicating the degree of deterministic structure due to the presence of quasiattractors [

• a Shannon entropy, estimated over the distribution of the length of deterministic lines, linked to the richness of deterministic structure.

Finally, a non redundant picture of the information provided by the RQA descriptors may be obtained by means of PCA, which allows to reduce the data set dimension without noticeable loss of information (see Appendix).

Using a set of 70 TEOAEs signals from normoacousic subjects as a reference (training set) for the RQA-PCA analysis, the first two principal components (PC1, PC2) can explain more than 96% of the observed variability [

Panels (a) and (b) of

tions RP is closer to that of the natural signal.

relevance, such indication goes beyond the simple reproduction of a “typical” signal, which actually does not exist but as an abstraction. Avan and coworkers [

table of Appendix are more distant between each other in the left than in the right panel. This is an indication of the higher resolution power of the 64 partition model, that adds to the more natural shape of between signals variability.

In this paper the ear model developed by Giguere and Woodland [7,8] was solved using the PSpice simulator. This model is inspired to the so called travelling wave mechanism, that is by no means the only accepted mechanism of cochlear functions. However, the point here is not to compare alternative explanations of the physical mechanism at the basis of hearing: we hope to provide a useful hypotheses-generating workbench of noticeable physical appeal, which needs confirmation in the appropriate clinical context.

Our modeling approach lies somewhere in the middle between purely phenomenological models, where the main emphasis is on fitting the shape of natural signals, and models inspired by strong hypotheses on the driving forces at the basis of the observed phenomena. The middle way approach proved powerful in many biological problems ranging from protein sequence-structure relations [

The success of such models stems from their peculiar interest in reproducing the biological variability (in our case the differences among natural TEOAEs) more than the ideal functioning of the system at hand. In fact, in a set of biological elements (being protein sequences, physiological signals or gene expression profiles) the differences between elements constitute more stable and relevant observations than difficult to identify “ideal cases”. Purely phenomenological approaches appear well suited for the analysis of biological variability, thanks to their main data fitting nature. However, they are of little or no use for deriving useful hypotheses on the actual system functioning and they can only be used for empirical comparisons, e.g. the receptor binding efficiency of a drug [

Both phenomenological and physically intensive models are expected to have a monotonically increasing relation between accuracy and model complication simply due to statistical considerations. More degrees of freedom allow for a greater flexibility and consequent adaptation power in the case of phenomenological approaches, and for a higher level of detail in the case of mechanistically intensive (realistic) models. In both cases, however, the risk is that the accuracy increase will be eventually paid for in terms of over-fitting and consequent degradation of the models when applied to different data sets. After a certain accuracy, in fact, phenomenological models start to model “noise”, while mechanistically intensive approaches assume not sufficiently known (and thus unjustified) details. The situation is different for “middle layer” approaches where the optimal accuracy is usually reached at a specific detail scale. This was particularly evident in the case of heart cells synchronization where the two dimensional spiral wave model gave much more reliable results than the three dimensional scroll wave analogue [

We believe that reproducing the physiological variability instead of often unrealistic ideal cases is a desirable goal in modeling biological phenomena. In such a context, considering the non obvious correlation between model complication and heuristic power can greatly enhance the relevance of physical models.

We thank Dr. C. Parlapiano of the Palermo University (Audiology Department) for having made available the TEOAEs signals

Principal Component Analysis (PCA), is a quite common statistical technique able to project a multivariate data set into a space of orthogonal axes, called principal components (PCs) and correspondent to the eigenvectors of the covariance (or correlation) matrix between the original variables. PCs are selected, one after the other (PC1, PC2, etc.), on the basis of the maximal variance explained in the space of the original variables. The presence of nonnull correlations allows to reduce the data set dimension in the new space without noticeable loss of information.

Because PCs are, by construction, orthogonal to each other, a separation of the different and independent features characterizing the data set is possible.

TEOAEs signals were simulated by the electric model in ^{*}, 7^{*} and rows 12^{**}, 16^{**} refer to the maximal excursion of and, respectively (see