Reproducing Cochlear Signals by a Minimal Electroacoustic Model

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.


Introduction
Otoacoustic Emissions (OAEs) are low-level amplitude acoustic signals generated in the inner ear and measurable in the external auditory canal.Since their discovery [1] they have been extensively used in clinical applications thanks to their reproducibility and stability.
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 [4].A first group of models, such as the gammatone model [3,5] or the electronic cochlea [6], aims to reproduce the shape of TEOAEs with no specific consideration to the involved anatomical structures.Another group of models emphasizes the correspondence between ear anatomy and model elements.The ear model developed by [7,8], for example, relies on the electro-acoustic analogy [9] and provides clues to both speech and hearing research.
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 [10].This is also important since modeling the natural inter-individual variability is by far more relevant in biomedicine than the classical reproduction of ideal cases.

Recording and Simulating TEOAEs Signals
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 [2].A schematic overview of the human ear and of the electronic model [7,8] is reported in Figure 1.
In the electro-acoustic analogy the outer ear is represented as a uniform transmission line [11], the middle ear as a complex electrical network [12] and the cochlea as a transmission line with a variable number of partitions able to simulate active processes.In particular, each partition contains a series inductor, a shunt resonant circuit and a non-linear voltage source.As for the parameters sed in the simulations, see the table Appendix.u 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 [13].The voltage source of the n-th cochlear partition used to simulate the OHC's active processes can be defined [7] as: Copyright where: Equation (2) shows the dependence of sources on the ratio between the current in the cochlear nd the voltage across .The input voltage is a rectangular pulse 80 µs wide onding to a stimulus of 80 dB SPL.

RQA-PCA of TEOAEs Signals
RQA is a technique used to q ntify the amount of deterministic structure of short, non-stationary signals [14,15] and to pick sudden phase changes possibly underlying mechanistical The RQA descriptors of each signal can be from the corresponding recurrence plot 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 A distance matrix is evaluated, whose , i j e element is the Euclidean distance between the i, j rows of EM;  If , i j e is lower or (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 [18].A detailed discussion of such applications is in the seminal paper by Webber and Zbilut [15], while a complete review of the method can be found in [19].On the basis of previous orks [3,20], the RPs of our signals have been quantified using the following descriptors:  % 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 attractors [14]. a Shannon entropy, estimated over the distribution of the length of deterministic lines, link of deterministic structure.Finally, a non redundant picture of the information provided by the RQA descriptors may be obtained by means of PCA, which mension without noticeable loss of information (see Appendix).
Using a set of 70 TEOAEs si bjects 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 [20].Moreover, since principal components based on correlation ation by construction, 96% of real signals, if taken from a homogeneous population, should fall within a circle of radius = 2 and centered in the origin of the PC1/PC2 plane (NA circle).This allows for a test of normal hearing as well as a check of similarity between simulated and natural TEOAEs.

Results
Panels (a) and (b) of Figure 2 show the output of the electronic model using 128 and 64 partitions to model the cochlea, respectively, as compar ering the valu simulated and from the initial external excitation (t = 0), to get rid of the initial ringing.In both type of signals fast oscillations last up to 20 ms, with higher frequencies having shorter latencies, in agreement with the latency-frequency relationship typical of TEOAEs [21].The signal simulated using 128 partitions (Panel a) is in complete agreement with that of a purely analytical model reported in [22]; the signal simulated by 64 partitions (Panel b), however, appears closer to the natural TEOAEs (Panel c).This is confirmed by a spectral analysis of the three signals carried out in five subsequent and identical time windows (not shown), indicating a much higher correlation with the natural TEOAEs of the 64 partitions output.A further decrease in the number of partitions, down to 32 or 16, introduces unbearable artifacts due to discontinuity related spurious reflections in the electronic circuits, while such artifacts are absent in the 64 partitions case.
Figure 3 shows the recurrence plots corresponding to the signals in Figure 2 (in the same order).RPs were calculated as follows: time course of the original signal described by 442 points on both axes (512 -70, to get rid of the initial ringing); delay in the embedding procedure (lag) = 1; embedding dimension = 10; cut-off di adius) = 15.The radius is defined as the superior threshold for a between epochs (rows of the embedding matrix) distance to be considered as recurrent.It may be noticed that even at a first sight the shape of the 64 parti-Copyright © 2012 SciRes.OJBIPHY

een signals variab
, that is by no means the only accepted meer, the point here the physical ed powerful in many biological pr gene expression profiles) the di enomenological appr 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 betw ility.

Discussion and Conclusions
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 chanism of cochlear functions.Howev is not to compare alternative explanations of 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 prov oblems ranging from protein sequence-structure relations [24] to the rationalization of gene expression dynamics in terms of attractors [25], to the synchronization of cellular activity in the form of spiral waves causing heart contraction [26].
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 fferences 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 [27].The middle layer strategy, on the other hand, allows for an analogy between single elements of the model and the corresponding elements of the real system.In our case we demonstrated that model variations due to changes in the middle ear provoked a realistic variability of the corresponding simulated signals, thus indicating the middle ear as an anatomical structure responsible for TEOAEs variations. 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 ph oaches, 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 [26], even if the heart is a three-dimensional object.In our case the clear superiority of 64 partitions model with respect to the 128 one indicates that the identification of the over 3000 functional modules present in the Organ of Corti with the partitions of the model does not hold, and that the realism (in terms of reproduction of natural variability) of the model emerges from a reduction of the active degrees of freedom of the system, possibly due to the high level of coupling of its elements.As a matter of fact, our results confirm the spectral coding of sounds in the cochlea by way of less than 40 independent filters [28].
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.

Figure 1 .
Figure 1.Simplified view of the human ear and of the electronic model.
, (b) and (c) panels show simulations by 128 and 64 cochlear partitions, and the natural signal respectively.Both in simulated and real signals, recording starts after 2.5 ms from the initial external excitation (t = 0), to get rid of the initial ringing.Fast oscillations last up to 20 ms, with higher frequencies having shorter latencies, in agreement with the latency-frequency relationship typical of TEOAEs[24].

Figure 4 (
left) reports the points in the PC1/PC2 plan orresponding to the signals simulated changing the pa eters of the middle ear according to the table the 64 partitions model.The position of parameter of the middle ear is changed, but it still reains well inside the NA circle.In terms of biological show the RPs corresponding to the signals in Figure 2, in the same order.

Figure 3 .
Figure 3. Recurrence Plots of simulated and natural signals
PCs) and correspondent to the ei 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.For the meaning of the symbols see the tex and Figure 1.The reference values (Ref) are from [7] ws 1 * , 7 * an cursion of d L and 0 L , respectively (see Figure 4 Left, Right).