Development of cochlear models with high computational efficiency by using spatial and parametric transformations

Development of Cochlear Models with High

Computational Efficiency by Using Spatial and Parametric

Transformations

by

Samiya A. Alkhairy

Submitted to the Department of Electrical Engineering and Computer

Science

in partial fulfillment of the requirements for the degree of

Masters of Science in Electrical Engineering and Computer Science

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

September 2016

c

○ Massachusetts Institute of Technology 2016. All rights reserved.

Author . . . .

Department of Electrical Engineering and Computer Science

August 31, 2016

Certified by . . . .

Christopher A. Shera

Professor of Otology & Laryngology and Health Science & Technology

Thesis Supervisor

Certified by . . . .

Dennis M. Freeman

Professor of Electrical Engineering and Computer Science

Thesis Co-supervisor

Accepted by . . . .

Leslie A. Kolodziejski

Professor of Electrical Engineering and Computer Science - Chair of the

Committee on Graduate Students

Development of Cochlear Models with High Computational Efficiency

by Using Spatial and Parametric Transformations

by

Samiya A. Alkhairy

Submitted to the Department of Electrical Engineering and Computer Science on August 31, 2016, in partial fulfillment of the requirements for the degree of

Masters of Science in Electrical Engineering and Computer Science

Abstract

Purpose Our goal is to develop methods to improve the efficiency of computational models of the cochlea for applications that require the solution accurately only within a basal region of interest, specifically by decreasing the number of spatial sections needed for simulation of the problem with good accuracy.

Approach We design algebraic spatial and parametric transformations to computational models of the cochlea that are applied after the region of interest and allow for spatial preservation (spatial causality in the case study model), driven by the naturally absorptive characteristics of the cochlea.

Objectives The goal is to design, characterize and develop an understanding rather than optimization and globalization.

Scope Our scope is as follows: designing the transformations; understanding the mech-anisms by which computational load is decreased for each transformation; development of performance criteria; characterization of the results of applying each transformation to a specific physical model and discretization and solution schemes.

Case study We explore the proposed methods for a case study physical model that is a linear, passive, transmission line model in which the various abstraction layers (elec-tric parameters, filter parameters, wave parameters) are clearer than other models. This is conducted in the frequency domain for multiple frequencies using a second order finite difference scheme for discretization and direct elimination for solving the discrete system of equations.

Performance The performance is evaluated using two developed simulative criteria for each of the four transformations. For the increased dissipation transformation, we investi-gate individual deviation measures as part of constructing the corresponding transformation function. The nonsimulative process for this transformation is to stand as proof of concept for the remaining transformations.

Conclusion The developed methods serve to increase efficiency of a computational trav-eling wave cochlear model when spatial preservation can hold, while maintaining good correspondence with the solution of interest and good accuracy, for applications in which the interest is in the solution to a model in the basal region.

Thesis Supervisor: Christopher A. Shera

Title: Professor of Otology & Laryngology and Health Science & Technology Thesis Co-supervisor: Dennis M. Freeman

Title: Professor of Electrical Engineering and Computer Science

Contents

2.2 Methodology . . . 10

2.3 Mechanism . . . 11

2.4 Transformations . . . 13

3 Implementation 14 4 Performance Criteria 15 4.1 Simulated Performance Criteria . . . 15

4.2 Hypothesized Performance Criteria . . . 17

5 Transformation PD: Increased Damping 20 5.1 Transformation outline . . . 20

5.2 Transformation function details . . . 20

5.3 Performance . . . 22

5.4 Characterization and discussion . . . 23

5.5 Summary . . . 24

6 Transformation SS: Spatial Squeezing 25 6.1 Transformation outline . . . 25

6.2 Implementation details . . . 25

6.3 Transformation function details . . . 27

6.4 Performance . . . 28

6.5 Characterization and discussion . . . 28

6.6 Summary . . . 29

7 Transformation PS: Parametric Squeezing 30 7.1 Transformation outline . . . 30

7.2 Implementation details . . . 30

7.3 Transformation function details . . . 30

7.4 Performance . . . 31

7.5 Characterization and discussion . . . 32

7.6 Summary . . . 33

8 Transformation SC: Complex Spatial Transformation 34 8.1 Transformation outline . . . 34

8.2 Implementation details . . . 34

8.3 Transformation function details . . . 35

8.4 Performance . . . 36

8.5 Characterization and discussion . . . 36

9 Discussion 38

9.1 Spatial preservation requirement . . . 38

9.2 Extendibility . . . 39

10 Further Considerations 40 10.1 Applications . . . 40

10.2 Alternatives . . . 41

10.3 Considerations and future directions . . . 42

11 Conclusion 44 12 Appendix A: Abstraction Layers and Zweig Model Construction 45 12.1 Generalized structural system layer . . . 45

12.2 Circuit parameter abstraction layer . . . 46

12.3 Filter property abstraction layer . . . 46

12.4 Wave propagation abstraction layer . . . 46

13 Appendix B: Thesis Proposal 48 13.1 Introduction . . . 48

13.2 Goal and Scope . . . 49

13.3 Methods . . . 50

13.4 Sample Results . . . 55

13.5 Summary . . . 57

14 Appendix C: Posters 58

List of Figures

1 Original and Transformed Systems . . . 11

2 Schematic respresentation of two transformation mechanisms . . . 12

3 Transformation PD - Increased dampling . . . 20

4 Transformation PD function parameter - local relative |DE(a, x)| and |y(a, x)| . . . 21

5 Transformation PD function parameter - local relative |DE(b, x)| and |y(b, x)| . . . 22

6 Transformation PD function parameter set . . . 23

7 Transformation SS - Spatial squeezing . . . 26

8 Transformation PS - Parameteric squeezing . . . 31

9 Transformation SC - Complex spatial transformation . . . 34

10 Zweig model - transmission line section . . . 45

List of Tables

2 Transformation SS: Performance evaluation. Input frequency 1.5 kHz . . . 28

3 Transformation PS: Performance evaluation. Input frequency 1.5 kHz . . . 32

1

Introduction

1.1 Background

The auditory system consists of the external ear, middle ear, the cochlea, and peripheral and central nervous system components. While cochlear anatomy is complex, the classical model view of the cochlea is that of a membrane (basilar membrane) separating two symmetric fluid-filled compartments (scala vestibuli and scali tympani). More recent models incorporate other structures on the basilar membrane, as well as other fluid compartments, and can support multiple longitudinally traveling waves.

Much of the auditory pre-processing occurs in the active, nonlinear, dispersive and dissipative cochlea and hence it is of specific interest; the auditory signal is spectrally decomposed along the cochlear length before projection to higher nervous centers. This interest translates into simulations of models of the system from the middle ear to the apex of the cochlea.

1.2 Motivation

This project is motivated by a common problem encountered in computational traveling wave models of the cochlea which is that of computational efficiency. While mathematical filter-bank models of the cochlea (which are generally more computationally efficient than mechanical models) may be sufficient for studying auditory phenomenon beyond the auditory nerve (such as speech processing and higher level feedback effects), full physiological cochlear models are key to understanding the role of cochlear elements and their complex interactions in auditory phenomenon until the level of the cochlea. A real problem that arises is that simulation of full cochlear models is computationally expensive and this limitation creates a trade-off between accuracy and complexity and size.

Hence, it is desirable to decrease the computational load of these models based on their uses. Cochlear

model responses to transient stimuli are often only desired over a restricted range (for x < x0) (e.g., in

comparing with experimental data, like responses to clicks, where the data is mostly from the basal region of the cochlea; or in the mechanical and parametric model construction stages). Hence, while the whole cochlea is simulated accurately, only a portion of the output is used for analysis.

1.3 Objective

Our goal is to decrease the computational load of models if the application only requires the solution

for x < x0. This would be especially beneficial for 3D nonlinear time domain models which are very

required for simulation of the discretized model, which is generally determined by both the cochlear length and the spatial resolution-accuracy relationship.

The thesis is of foundational nature and hence the goal is to design, test, characterize and build an understanding rather than to optimize and globally extend the methods to other models. We develop methods, and present two of these in sections 5, 6, 7, 8 and study their applicability, and go further in depth for a single method in section 5 to be taken as a proof of concept for the other methods.

In conducting a literature survey, we have found few other studies [3] that take advantage of the form

of the solution in the cochlea to increase computational efficiency of cochlear models1_.

1.4 Approach Framework

In development of these methods, we are motivated by cochlear characteristics and inspired by the perfectly

matched layer (PML) approach for the simple (constant coefficient) wave equation [7]. Since the interest

is in the solution for x < x0, and using our underlying assumptions (as described in section 2.1), we

hypothesize that altering x > x0 (using one of the proposed methods) may allow for retaining the solution

in the region x < x0 while improving computational efficiency.

1.5 Scope

The scope of this project is as follows: • Design transformations

• Hypothesize and understand mechanism by which transformations lead to an increase in computa-tional efficiency

• Develop performance criteria

• We use the Zweig model (as is organized into relevant abstraction layers in section 12) as a case study

model [11]. We characterize each of the methods using the chosen physical Zweig model and a chosen

discretization scheme.

2

Approach

2.1 Underlying assumptions of cochlear nature

In designing the method, we take major hints from the natural characteristics of the cochlea, specifically absorptive behavior, and from some cochlear models, specifically a subset of spatial preservation behavior - as explained 2.2. Driven by a natural stimulus, the cochlear motion along its longitudinal axis can be modeled as waves traveling primarily in one direction (from base to apex). This is a consequence of approximate spatial causality. Thinking in terms of differential fluid pressure (across the basilar membrane), a given frequency component would decay after its characteristic frequency (CF) place very rapidly. Right after this position, the cochlea acts as a very powerful absorbing boundary layer (ABL) for the corresponding frequency. This is exemplified for a relatively low frequency in the CF range by the solid line in figure 2. This is the case for all frequencies in the CF-range of the cochlea where high frequencies decay more towards the base and the lower frequencies more towards the apex, such that the pressure decays almost entirely before the apex, and hence, the reverse traveling waves resulting from apical boundary reflections are negligible in the cochlea and most cochlear models.

Reflections can generally be terminal reflections or internal reflections from within the media. It is experimentally seen in the cochlea, that terminal reflections are negligible for frequencies roughly within the CF range. For a class of cochlear models (e.g., models with WKB solutions), we observe that the solution is spatially causal (i.e. the response at earlier sections is independent of later section characteristics) for frequencies within the CF range of the cochlea. While lack of internal reflections implies spatial causality, the reverse is not true. Specifically, if we are interested in some point A, we consider internal reflections that may arise only at some apical point B but they decay while they propagate in reverse, such that this component is negligible as seen by point A.

2.2 Methodology

For cochlear models that are spatially causal, we hypothesize that can transform the model after the section of interest to potentially decrease total computational load (specifically by decreasing the total number of sections required for simulation). For an appropriately chosen transformation, approximate spatial causality is retained in the region of interest, x < x0, preserving the solution for that region.

Underlying assumption More generally, all that is required is spatial preservation (SP): that the

original solution in x < x0 is preserved with the transformation. In other words, the effect of the original

way of this occuring is that spatial independence (SI): the solution within x < x0 be independant of the

characteristics of x > x0 in the original and transformed models. In other words, spatial causality across

x0. A subset of this is spatial causality within the region of interest, which is realized in cochlear models

where WKB approximates the original model solution (as is the case with the case study model used here), and may still apply for the transformed model depending on the tranformation function. When considering extendibility of the methods developed here to other cochlear models, it is necessary to consider SP.

Figure 1: Original and Transformed Systems Top: System A is original system. Bottom: System

B is transformed system. For the application of interest (solution for x < x0), the two systems have the

same behavior. Simulation of system B is with greater computational efficiency. The objective is finding candidates for a system B that is a spatially and/or parametrically transformed version of system A, where the transformations are preferably non-specific to the cochlear models.

Cochlear models can be represented as f(t, ~X, ~Θ)where ~Xis the set of spatial coordinates and ~Θ is set of parameters. Hence the transformation after the region of interest may be applied to the spatial coordinates or the parameter set. Here, we apply and test algebraic parametric transformations and algebraic spatial transformations, that we design taking into account the ABL nature of the cochlea and assuming that approximate (see section 9) spatial causality may be retained for the sections of interest.

The method has two aspects

• Tranformed layer (TL): part of the cochlear length after the region of interest (RI)

• Computational box: which may have a computational end (xcomp) before the actual apex (xapex)

2.3 Mechanism

We hypothesize that the goal may be achieved for the transformations in section 2.4 through one or both

of the mechanisms outlined below. Figure 2 shows a schematic with the region of interest (x < x0), the

transformed region which ends with a computational apex (x0 < x < xcomp), the original apex (xapex), and

decays most apically) in the CF range as a function of distance from the stapes for the two mechanisms as well as the regular cochlear model.

Figure 2: Schematic respresentation of two transformation mechanisms Top: Mechanism A; Bot-tom: Mechanism B. Solid line: no transformation; Dashed line: with transformation. y is differential pressure across the basilar membrane

• Mechanism A: The solution (differential pressure) decays faster in the transformed model than the regular model. Therefore, the solution will have decayed entirely before the apex - allowing for imposition of a computational end before the actual apex xcomp < xapex, and hence the total cochlear

length required for simulation is reduced. If this reduction in simulated length is not outweighted by a reduction in the accuracy-spatial resolution relationship (accuracy of the lowest accuracy frequency in the RI; x < x0, and resolution in the TL; x > x0), then - if using a uniform step size, fewer total

spatial sections would be needed for simulation. Using this mechanism, we would only simulate the length minimally necessary to get the results that really pertain to the uses of the simulations, and not for x > x0+ ; ( = xcomp− x0).

• Mechanism B: The solution to the transformed model does not decay much faster than in the original,

therefore the transformation does not change the effective apex (xcomp < xapex), or cochlear length

needed for simulation. However, the transformation provides improved accuracy-spatial resolution relationship due to decreased solution/parameter gradients. Thereby, use of the transformed model would achieve the same accuracy (in the RI) as that of the regular model but by using larger step sizes in the TL. Here, the total number of sections required for simulation for the same accuracy in the region of interest would be decreased if one allows for a nonuniform step size for multi-frequency component stimuli.

2.4 Transformations

In designing the methods, we would preserve the model in the region of interest (RI: x < x0) and then apply

one of the following transformations after the RI to produce the TL in order to decrease the total number of space sections required for simulation while retaining the original solution in RI (assuming approximate spatial causality is retained) and accuracy in the RI. Each method and its solution is described, tested and characterized in sections 5, 6, 7, 8. We note that most of the transformations are not mutually exclusive, and that other transformations are possible.

• Transformation PS - Parametric squeezing: After some x0 of interest, squeeze all cochlear parameters

(i.e. change their spatial variation with x) to get the same range of CFs in a shorter space. We demonstrate that this transformation decreases spatial resolution requirements through mechanism A.

• Transformation PD - Increase damping: After some x0of interest, increase the damping parameter(s)

-eg. the resistance in the parallel element of the Zweig model. We demonstrate that this transformation decreases spatial resolution requirements through mechanism B.

• Transformation SS - Spatial squeezing: After some x0 of interest, squeeze cochlear space to get the

same range of CFs in a shorter space. We demonstrate that this transformation decreases spatial resolution requirements through mechanism A.

• Transformation SC - Complex spatial transformation: After some x0 of interest, squeeze the cochlear

space and extend it into the complex domian. For the particular transformation function we choose for this transformation, we demonstrate a decrease in spatial resolution requirements through both mechanisms A and B .

3

Implementation

The case study model (Zweig model) is approximately spatially causal for all x and for most frequencies of interest- specifically, it’s well-approximated using WKB for a wide range of frequencies within the CF range. The Zweig model is a 1D linear, passive, classical transmission line model.

For simplicity of application, performance evaluation and characterization, the discretization scheme used is a finite difference scheme with second order accuracy even at the boundaries, and direct elimination using LU decomposition is used to solve the resultant system of equations. Also for simplicity, we are first testing the methods in the frequency domain to deal with ODEs rather than PDEs. To account for this, we examine the solution using the methods, for multiple frequencies within the natural range of CF for which the solution is well approximated using WKB. The model can be implemented in the time domain with a filtered impulse input to better show results for the major applications of interest in the future.

The spatial step size is chosen to be uniform for simplicity. As mentioned in section 2.2, depending on the mechanism of achieving the goal, a uniform or nonuniform step size would ideally be used. While we use a uniform step size for single frequencies in our characterization, we explain how this carries to nonuniform spatial discretization for multi-frequency simulations in section 5.

The details of the model and discretization scheme are as follows

• The frequency domain governing equation in terms of y - the spatial component of the differential pressure P is2_,

d2y

dx2 + α(ω, x)y = 0 (1)

Where α(ω, x) = k2_{(ω, x). With a fixed apical boundary condition (based on the cochlear nature}

in section 2.1, the choice for apical BC should be inconsequential), and driven by stapes velocity

(proportional to dP

dx|x=0 for a given frequency) at the base.

• The parameter values are taken from [11] and are of the squirrel monkey species

• The resultant discretized system of equations is written in the form My = f, exemplified below for three spatial nodes, with a uniform step size h,

[ 1 h2 −2 2 1 −2 1 1 −2
+ α1 α2 α3
]y = f (2)

Where f is a zero vector except at base (where BC is nonzero, and driven by stapes velocity).

4

Performance Criteria

In this section we develop the guiding performance criteria for transformation design and the performance evaluation criteria. The simulated performance criteria and hypothesized performance criteria detailed in the following sections may serves both as evaluation criteria and as part of the guiding transformation design criteria, in the design-evaluate-characterize loop. In this project, we make full use of the simulated criteria, but only partially used the hypothesized criteria (for design rather than evaluation) as we have not provided validation for the hypothesized criteria assumptions in this thesis. Also, we only investigate the hypothesized measures for one of the proposed transformations (transformation PD, section 5), to stand as an example for the remaining transformations.

4.1 Simulated Performance Criteria

In this section, we discuss the evaluation criteria for the transformations as carried out in this project. Given a desired RI, we evaluate the performance of each transformation by simulating the entire cochlear

length (0 to xapex) for both the original model and the modified model. We compare the differential fluid

pressure for both models.

Simulation of 0 to xcomp would be more intuitive (and ideal as a future implementation), though it

would require knowledge (or an expected value) of xcomp. While the values for xcomp for transformations

SS and PS are as we intended, and the values for xcomp for transformations PD and SC are as we expect

assuming a solution form, we shall not detail appropriateness of these values for the simulation here. As

a result, we will instead simulate both the original and transformed models for x = 0 : xapex, rather than

x = 0 : xcomp(and view the results taking this into account). This is appropriate as the behavior of 0 : xcomp

is approximately independant of the xcomp: xapex region.

The performance criteria (accuracy and convergence as will be explained below) depend on the

trans-formation function, x0, xcomp, input frequency, spatial step size - h. Hence a more comprehensive view

of performance measures would take these into account: for example, accuracy would be considered as a function of number of sections, n, rather than considering accuracy at a single value of n or considering the required n for a desired accuracy value. Nonetheless, in this foundational work, we shall take (a few) point values indicators (though not representative) for performance evaluation. For all transformations, the performance will be evaluated using the following criteria and the assigned measures, which are derived entirely from the simulated responses within the RI for frequencies within the CF range.

the solution to the continuous transformed model, ˆy(x), within RI for frequencies within the CF range. The measure for this criteria may be any of the following,

• We may choose to measure correspondence by the 2−norm of the relative deviation between the solution to the transformed model and the solution to the regular model within RI for a ’continuous’ solution, as shown in equation 3.

• We may choose to alternatively measure the deviation between the solution to the ’continuous’ transformed model and the WKB solution within RI. This would be a less direct measure of deviation but would make characterization easier by having an analytic approximation to the solution of the transformed model in TL.

We take the former option as a measure for criteria I. Equation 3 measures the estimated continuous deviation between y(x) and ˆy(x). The continuous solution is approximated using n = 7000 (h = 2.9e − 4). C₁∗= ( j=j0 X j=1 |y(x = jh) − ˆy(x = jh) y(x = jh) | 2₎1/2 ₍₃₎

2. Criteria II: Good accuracy-resolution relationship of the solution to the discrete transformed model compared to the solution to the discrete original model. We measure accuracy based on the 2−norm of the relative deviation error between the solution within RI for some number of sections n and the solution with n = 7000, with the assumption that n = 7000 approximates the continuous solution well. The accuracy of both the original and transformed solutions are measured using equations 4 and 5, to be compared. C2 = ( j=j0 X j=1 |y(x = jdx) − y[j] y(x = jdx) | 2₎1/2 ₍₄₎ ˆ C2 = ( j=j0 X j=1 |y(x = jdx) − ˆˆ y[j] ˆ y(x = jdx) | 2₎1/2 ₍₅₎

Comparitively good accuracy-resolution relationship may be measured based on the relationship

be-tween accuracy (A, ˆA), step size (h, ˆh) and number of sections (n, ˆn). We choose the following

measures to be used,

accuracy is improved for a set number of spatial sections. In other words, given ˆn = n (ˆh = h assuming xcomp = xapex as done in PD section 5), we want ˆA > A.

• For transformations that improves computational efficiency through mechanism A: we must be carful that the number of sections decreased due to having shorter cochlear length is not rendered useless because of an increase in DE / resolution constraints. In other words, given n → A, then ˆ

n < n → ˆA, where ˆA is either equivalent to A, or is more suitable than A. The term more

suitable does not dictate ˆA(ω) ≥ A(ω), but rather, the transformed solution may have the same

or better accuracy than that of the frequency component with lowest accuracy (see section 7.5)

in the original model - ˆA(ω) ≥ A(ωhigh). Note that here, ˆn is not chosen based on equivalence

of A, ˆAbut rather based on the amount of squeezing, as is clarified in section 6, 7, 8.

3. Criteria III: If we are constrained to a particular h, the criteria of interest would be correspondence between the discrete solutions y[j] and ˆy[j]. This would give a mixed criteria of both continuous correspondence as well as accuracy differences between the transformed and original models. While it does not dissociate the errors and hence may not be ideal in the design stage, it can be used in the application stage as a single direct parameter of interest rather than using both performance criteria individually. This mixed criteria is measured using the compound measure in equation 6, by taking the 2-norm of the relative deviation between the discrete solutions within RI.

C_m∗ = ( j=j0 X j=1 |y[j] − ˆy[j] y[j] | 2₎1/2 ₍₆₎

It is important to note that all of the above criteria measures, for the various transformations, are for the simulated solution only within RI, and for frequencies within the CF range.

4.2 Hypothesized Performance Criteria

As mentioned earlier, the hypothesized performance criteria may be used as both a guide for the transforma-tion functransforma-tion constructransforma-tion, as well as to evaluate performance of transformatransforma-tions. However, in this thesis, we do not show that these hypothesized performance measures are reflective of the simulated performance measures. Hence, we will only used these as a guide for transformation design of transformation PD, in addition to the simulated criteria.

Choosing the Zweig model as a case study is also beneficial in performance evaluation and design: we can compute estimates for the performance criteria above even without simulating both the transformed

and regular model. Instead, for the Zweig model, we may instead approach the criteria above based on the following ideas:

1. For criteria I - continuous solution convergence: Since the solution to the Zweig model is well

approx-imated by the WKB equation, we may test the global continuous correspondence for RI (x < x0) by

using a global WKB-deviation measure that accounts for cumulative effect by more apical sections on sections in RI.

The local continuous error (CE) approximated by WKB deviation [8] is as follows

1 α(x)( 1 4 α00 α − 5 16( α0 α) 2_{) (7)}

2. For criteria II - discrete solution accuracy: Discretization error arises as a result of local and cumulative residual error. Since the system of equations for the Zweig model can be collapsed into a single governing equation (equation 1), an expression for the local residual error may be derived using Taylor series expansion of the discrete equation terms (equation 8).

DE = dx 2 12 y 0000_{+ O(dx}4_{) = −}dx2 12 [(α 00_{− α}2_{)y + 2α}0_y0_{] + O(dx}4_{) (8)}

If the forward WKB estimate holds,

y+= α−14e−jR √αdx (9)

We may write a relative local DE measure in terms of only α and x 3

DE_{W KB}+ y = dx2 12 [−α 00 + α2+ 2α0(α 0 4α+ j √ α)] + O(dx4) (10)

This is an expression for relative local residual error, and a global relative error at each point (reflective

of local and cumulative effects) can be computed using the appropriate (original or transformed) M−1

; M is shown for the original model is equation 2. However, in this thesis, we will not compute (or validate that it is reflective of C2 and ˆC2) the global relative error. For transformation PD, in section

5, we take the maximum local relative discretization error to be reflective of glocal discretization error at a particular point in RI because the local relative discretization error at a particular point within

the RI is unchanged. We do not discuss the validity of this assumption 4_.

4_{In making this assumption, we are motivated by coherent reflection theory that suggests that reflections arise}

5

Transformation PD: Increased Damping

5.1 Transformation outline

In the Zweig model, the damping parameter is the parallel resistance. Here, we explore increasing the

damping parameter after x0 to decrease spatial gradients.

Figure 3: Transformation PD - Increased dampling Left: The dissipative parameter, R(x), is in-creased. This occurs after the beginning of the boundary layer (at 0.5 cm) as indicated by the dashed red line. Middle top: |P | for the regular and transformed cochlear models for step size = 1.1e-2 cm (n = 175). Middle bottom: Magnitude of difference vector between original and transformed model solutions using n = 175. Right top:|P | for the regular and transformed cochlear models for step size = 2.9e-3 cm (n = 700). Right bottom: Magnitude of difference vector between original and transformed model solutions using n = 700. Stimulus frequency = 1.5 kHz

5.2 Transformation function details

We choose the following simple polynomial to scale the transformed parameter as follows with the RI, ˜

R(x)

R(x) = 1 + a(x − x0)

b ₍₁₁₎

This transformation is anticipated to improve computational efficiency through mechanism B. The choice of values of the transformation function parameter set (a, b) is based on the continuous solution deviation within RI, the deviation within RI attributable to discretization, and the decay of the solution by the apex, as detailed in section 4. For this transformation, we detail the process of choosing the transformation function - while maintaining an foundational level, through the simulated or hypothesized performance evaluation criteria. This process is exemplified here, and while not repeated for other transformations, can stand as a proof of concept to extend to other transformations.

choosing the transformation function parameters. Specifically, the continuous relative deviation is negligible for a large set of parameter value set which allow for sufficient decay by the apex. This can be estimated from large n (as we have done to get values around 3.5e − 3) or using a global WKB deviation measure. Therefore, we only need to focus attention on the DE and the solution decay by the apex. We explore the local DE (using equation 10 under the assumption of having only a forward traveling wave) within RI and the solution decay for a few parameter value sets, as shown in figures 4 and 5 for the lowest frequency, 1.1 kHz, within the CF range of the cochlea (the corresponding CF place is most apical). The figures are well representative for a wide set of (a, b).

Figure 4: Transformation PD function parameter - local relative |DE(a, x)| and |y(a, x)| First: Relative local |DE| is estimated using equation 10 and, second: |y| is estimated using equation 9 to explore

in the a domain; Third: maximum relative local |DE| as a function of a; Fourth: |yapex|as a function of a.

n = 700. Stimulus frequency = 1.1 kHz. b = 3

We note that the reverse waves (if well approximated by the WKB solution) should decay as traveling

from apex to RI, thereby allowing for a range of Papex. This range would ideally be estimated from the

reverse WKB (until the RI) and the apical BC and given a margin of error; to then be used to choose (a, b).

For simplicity, we may also have limited the growth of R(x) by imposing xcomp ≤ xapex to avoid extending

the cochlea. However, as this work is foundational, rather than optimizationary, we are shall not address this.

While local DE at any point within the RI is unchanged, the DE projected back from the TL to the RI is altered with the transformation. Based on the relative local DE equation (equation 10) from which we can estimate the local relative DE at any point within the RI (and hence the global DE if desired as

Figure 5: Transformation PD function parameter - local relative |DE(b, x)| and |y(b, x)| First: Relative local |DE| is estimated using equation 10 and, second: |y| is estimated using equation 9 to explore

in the b domain; Third: maximum relative local |DE| as a function of b; Fourth: |yapex|as a function of b.

n = 700. Stimulus frequency = 1.1 kHz. a = 200

detailed in section 4), the R(x) transformation in TL is limited by this. For parameter a > 0, based on the pattern seen in figure 4, we take the maximum local relative DE within TL to be representative of the global relative discrete deviation within RI (as assumed in section 4), comparatively to find an appropriate

(a, b). This is also the case for b5 _{for large a - as in figure 5. We will not discuss the appropriateness of}

this representation in this thesis.

We plot the maximum local DE and the Papex for the different (a, b) set values in figure 6 as a guide to

choose an appropriate set. Based on this, we choose (a, b) = (200, 4) for the transformation function. The remainder of this section discusses results using this chosen transformation function. The results are shown in figure 3.

5.3 Performance

Table 1: Transformation PD: Performance evaluation. Input frequency 1.5 kHz

C₁∗ C2 Cˆ2 C2 Cˆ2 Cm∗

n≈continuous= 7000 n = 175 n = 175 n = 700 n = 700 n = 700

3.8e-3 1.06 0.06 0.01 0.01 5e-3

Figure 3 is for stimulus frequency = 1.5 kHz. Based on C2,175 and ˆC2,175, an h = 1.1e − 2 cm is not

Figure 6: Transformation PD function parameter set Top: maximum relative local DE(a, b) ; Bottom:

|yapex(a, b)|. n = 700. Stimulus frequency = 1.1 kHz. a = 200

suitable for the regular cochlea, but is more suitable for transformed cochlea. A step size of h = 2.9e − 3 cm may instead be normally used for the regular cochlea and the solution for transformed cochlea has

similar accuracy. Based on C∗

1, there is good correspondence between the "continuous" (approximated with

h = 2.9e − 4) y(x) and ˆy(x) within RI. The compound measure, C_m∗, shows good correspondence between

the discrete solutions y[j] and ˆy[j] for h = 2.9e − 3.

5.4 Characterization and discussion

We apply the transformation and solve the system of equations (equation 2) after modifying αj’s in TL for

the differential pressure given some frequency component that naturally decays after the RI. The amplitude of the pressure is plotted in figure 3 for both the transformed and the regular models, and investigated for the performance criteria. There is almost perfect correspondence between the solution to the transformed model and the solution to the regular model within RI (for the approximately continuous and the discrete cases). With increased damping, solution starts decaying before original but does not decay as quickly.

Hence, approximately the same cochlear length is required for simulation. However, fewer sections are required for appropriate accuracy: since the solution gradients are reduced, discretization constraints are

reduced for the frequencies that decay after x0, as in DE equation 8. However that step size would not be

suitable for frequencies that decay in the nontransformed region. Hence, in multi-frequency simulations, if using a non-uniform step size (or an effective nonuniform step size - eg. combining with transformations

SS, PS), this may allow for a smaller spatial step size (fewer sections) for x > x0 for the same accuracy.

5.5 Summary

We conclude that this transformation is appropriate for multi-frequency stimuli if using a non-uniform step size. The transformation allows for decreased computational load through mechanism B. We note that the RI accuracy of y and ˆy and deviation between them are a function of transformation function parameter set values, frequency, step size. However, in construction of the transformation function, we have made multiple assumptions in this foundational work, in order to assign simple measures to finding the appropriate transformation function.

6

Transformation SS: Spatial Squeezing

6.1 Transformation outline

Given a uniform step size, the solution within the RI to high frequency components is less accurate than that of the low frequency components. This is explained in section 7.5. We propose methods to decrease the number of sections required while retaining the high frequency accuracy (lowest accuracy) more uniformly

accross frequencies 6_{. One method would be using a nonuniform mesh. The SS method proposed in this}

section, is to use an effective nonuniform step size by simulating a modified governing equation (where the independent variable x is transformed) in terms of another independent variable with uniform spacing. Another method (transformation PS) we propose in section 7 is a parametric equivalent to transformation SS given some constraints on the model. We expect these three approaches to be equivalent. Also, note that the SS and the PS transformations share many aspects. (section 7).

We hypothesize that âĂŞ for an appropriately chosen transformation, the solution to the model for

x < x0 remains unaltered, and the solution beyond that would decay faster than in the original model as

expected by the spatial transformation map to allow for xcomp < xapexand hence, possibly fewer TL spatial

sections with the same RI accuracy.

After x0, we extend the modified independant variable (˜x) as shown in figure 7 in an attempt to have

the same CF range in a shorter length (1.7 cm rather than 2 cm), and all spatially varying parameters squeeze accordingly. The transformation is continuous and smooth at the interface between the region of interest and the transformed region. The discretization is carried out in terms of the equivalent modified governing equation in the original space, for simplicity of comparison with a uniform spatial step size.

6.2 Implementation details

We will use the following notation for the independant variables for the spatial transformations - shared with those for section 8. These are not unique and can be equivalent, but are presented independantly here to serve a generalization for spatial transformations as is clarified by the transformation in section 8. Note that for the parametric transformations x = ˜x = u, and hence the distinction are not made for transformations PS and PD. Independant variables :           

x distance from stapes in original model

˜

x transformed

u simulated (with uniform mesh)

6_{In this thesis, we do not choose the transformation function to achieve uniform accuracy based on the lowest accuracy}

Figure 7: Transformation SS - Spatial squeezing Left: The transformed spatial variable is squeezed as a function of the simulated variable, which is uniformly discretized. The squeezing occurs after the beginning of the boundary layer (at 0.5 cm) as indicated by the dashed red line. Right top: |P | for the regular and transformed cochlear models as a function of the simulated variable. Right bottom: Magnitude of difference vector between original and transformed model solutions. Input frequency = 1.5 kHz; step size = 2.9e-3 cm

Transformation The transformation can be denoted by x → ˜x(x) or an equivalent dx → ˜dx(x). We modify the governing equation based on the spatial transformation to write a modified governing equation with ˜x as the independant variable.

d2y

d˜x2 + α(ω, ˜x)y = 0 (12)

The equation above shows that the solution as a function of the corresponding independant variable is unchanged for the transformed model when compared to the original model. This is a feature of spatial (and not parametric) transformations. The transformation function is chosen based on the relationship with the simulated variable.

Simulation We discretize the modified and original governing equations in terms of a simulative

inde-pendant variable, u. To do so, we choose x = u and ˜dx = S(u)du. We use a uniform mesh for u, which

equivalent to using a uniform mesh for x and a nonuniform mesh for ˜x. The equivalent modified governing equation (in terms of u) is as follows,

1 S2_(u) d2y du2 − 1 S3_(u) dS(u) du dy du + α(ω, Z u 0 S(u0)du0)y = 0 (13)

The ODE is discretized with a uniform step size for u, denoted as h, to give the following example system of equations in the fourth quadrant of the mass matrix, to be equivalent with the regular governing

equation at the RI-TL interface. Here, we write α(ω, Pj

i=1S(i)h) as βj for simplicity

[ 1 h2 S−2_n−2 S−2_n−1 S−2n
−2 1 1 −2 1 1 −2
− 1 2h S−3_n−2 S−3_n−1 S−3n
dS du|n−2 dS du|n−1 dS du|n
1 −1 1 −1
+ βn−2 βn−1 βn
]y = f (14)

6.3 Transformation function details

To be compared with transformation PS in section 7 (the constrained parametric counterpart of this trans-formation), we choose S(u) such that,

˜

xSS(u) =

Z u

0

S(u0)du0 = zP S(u) (15)

Hence, ˜ x(u) =      u if u ≤ u0 u + a(u − u0)b if u ≥ u0 (16) Where u0≡ u(x0) = x0. S(u) =      1 if u ≤ u0 1 + ab(u − u0)b−1 if u ≥ u0 (17) As in transformation PS in section 7, we choose the parameters for the transformation function S(u)

so that b = 3 and a so that ˜x(u(xcomp)) = xapex as in equation 21.

6.4 Performance

Table 2: Transformation SS: Performance evaluation. Input frequency 1.5 kHz

C₁∗ C2 Cˆ2 Cm∗

n≈continuous= 7000 n = 700 n = 700 n = 700

2e-8 0.01 0.023 0.03

Figure 7 is for input frequency = 1.5 kHz, and illustrative of the pattern for other frequencies. C∗

1 shows

that in the continuous domain y(x) and ˆy(˜x) have excellent correspondence within the RI7_{. The discrete}

ˆ

y has a slightly lower accuracy than y[j], for h = 2.9e − 3. However, this error norm is much less than that

of higher frequencies in the regular cochlea - which decay before the transformed region, with the same uniform step size. For comparision, a higher frequencies (32 kHz) that decays more towards the base (not shown in figure), has a relative residual error norm = 0.23 for both the regular and transformed models.

The cumulative measure C∗

m shows that for h = 2.9e − 3, the discrete solutions are comparable.

6.5 Characterization and discussion

Qualitatively, the figures for transformation SS and PS are similar - and hence much of the secondary characterization that is shared with transformation PS and will not be repeated here, but quantitatively somewhat different. We would expect an equivalent solution for both transformations. Criteria II is approximately the same, but criteria I is very different. We will not discuss this further in this thesis. Another issue of interest is comparing the solution to the regular model and the transformed model as a function of the respective independant variable, in both RI and TL as we would expect them to be

equivalent. While they converge for the approximately continuous similation (n = 7000) there is some deviation (unclear in the figure) for the discrete simulation (n = 700). We do not further discuss this here. The applicability of a nonuniform mesh, or transformation PS, or transformation SS would even be better suited for more realistic models of the cochlea in which wave number scaling symmetry is not applicable, and instead the peak of the κ function decreases with frequency (wavelengths larger for apical frequencies).

6.6 Summary

Transformation SS makes use of the frequency dependance of accuracy. The transformation is appropriate for multi-frequency simulations limited by the accuracy of the lowest accuracy frequency component (the highest frequency). Details are left for discussion with its constrained parametric counterpart in section 7.

7

Transformation PS: Parametric Squeezing

7.1 Transformation outline

This transformation is a constraint parametric counterpart to SS and to using a nonuniform grid. Using the parametric squeezing transformation we attempt to preserve the content of the model (for future easier characterization), but in a squeezed manner. In other word, plotting the transformed model solution as a function of a different independant variable in the transformed model, should collapse the solution to that of the original model. Hence, it requires that all spatially varying parameters have the same underlying

parameter (i.e. depend on the same function of x, in the Zweig model, this is ωr(x) as in section 12)8 so

that, once squeezed, their relationship/local unloaded filter characteristics may be preserved and simply shifted in space. In the Zweig model, the underlying parameter is a pre-estimated CF(x) which is set to be

equivalent to the undamped unloaded ωr(x)of the parallel element of each section as described in section

12.

7.2 Implementation details

After x0 we squeeze the CF as shown in figure 8 in an attempt to have the same CF range in a shorter

length (xcomp =1.7 cm rather than 2 cm), and all spatially varying parameters squeeze accordingly. The

transformation is continuous and smooth at the interface between the region of interest and the transformed region. While this would shift the wave number in equation 35, it would retain the wave number scaling symmetry as in equation 36. This allows for easier characterization in the future.

We apply the transformation denoted by ωr(x, ω) → ˜ωr(x, ω)or equivalently in the governing equation,

k(x, ω) → ˜k(x, ω). The modified governing equation is discretized and results in a system of equations

similar to equation 2, but with the modified kj’s within the TL. We solve the modified system of equations

for the differential pressure given some frequency component that naturally decays at some length after the RI. The amplitude of the pressure is plotted in figure 8 for both the transformed and the regular models, and investigated for the performance criteria9_.

7.3 Transformation function details

The original ωr(x)has a form as in equation 18, where d is a space constant. In the transformed cochlea,

ωr(x) → ˜ωr(x) which has the form in equation 19. ˜ωr(x) is continuous and smooth across x0, and is

8_{Having an underlying parameter would also be useful more generally in construction of scaling symmetric cochlear models,}

in which specific properties have the same characteristics but shifted CF or equivalently, x. This could occur at the wavenumber level or until the P level or until the BM motion level

Figure 8: Transformation PS - Parameteric squeezing Left: Cochlear parameters (underlying pre-estimated CF(x) shown) are squeezed so that the artificial apex is at 1.7 cm of the original cochlea. The squeezing occurs after the beginning of the boundary layer (at 0.5 cm) as indicated by the dashed red line. Right top: |P | for the regular and transformed cochlear models. Right bottom: Magnitude of difference vector between original and transformed model solutions. Input frequency = 1.5 kHz; step size = 2.9e-3 cm.

chosen to have two parameters, b and a. After setting b = 3 (we only go in depth for the transformation function choice process in transformation PD, and hence, we shall not discuss the parameter choice here)

10_{, parameter a is chosen so that ˜ω}

r(xcomp) = ωr(xapex), as in equation 21.

ωr(x) = ωr(0)e−x/d (18) ˜ ωr(x) = ωr(x = 0)e−z(x)/d (19) z(x) =      x if x ≤ x0 x + a(x − x0)b if x ≥ x0 (20) a = xapex− xcomp (xcomp− x0)b (21) 7.4 Performance

Figure 8 is for input frequency = 1.5 kHz. C∗

1 shows that in the continuous domain y(x) and ˆy(x) have

decent correspondence within the RI. In the discrete domain, for h = 2.9e − 3, ˆy[j] has lower accuracy than

10_{A more systematic way of constructing an appropriate transformation function would be to follow the process for}

trans-formation PD. However, as mentioned earlier, in terms of construction we only go in depth for transtrans-formation PD to stand as a proof of concept for other transformations

Table 3: Transformation PS: Performance evaluation. Input frequency 1.5 kHz

C₁∗ C2 Cˆ2 Cm∗

n≈continuous= 7000 n = 700 n = 700 n = 700

0.01 0.01 0.025 0.02

y[j]when compared with their respective ’continuous’ solutions (approximated by h = 2.9e−4) as indicated

by C2 and ˆC2. However, this error norm is much less than that of higher frequencies in the regular cochlea

- which decay before the transformed region, with the same uniform step size. For comparision, a higher frequencies (32 kHz) that decays more towards the base (not shown in figure), has a relative residual error

norm = 0.23 for both the regular and transformed models. The cumulative measure C∗

m shows that for

h = 2.9e − 3, the discrete solutions ˆy[j] and y[j] are comparable.

While figure 8 (right) is only for a single frequency, it is illustrative of the qualitative pattern of trans-fomation effect for frequencies within the CF range for which the solution is well approximated with WKB.

The continuous and discrete correspondence between y and ˆy (C∗

1 and Cm∗) improves with increased

fre-quency until approximately 1e-10 for the highest frefre-quency within the CF range. However, as mentioned earlier, the accuracy of both within RI decrease with input frequency.

7.5 Characterization and discussion

In the TL, we find an increased pressure decay rate in the transformed model than in the regular model,

which allows for truncation (xcomp < xapex). In the RI, the solutions to the transformed model and to the

regular model have decent correspondence. In the RI, the accuracy of the solution to the transformed model is less than the accuracy of the solution to the original model for the input frequency. However, it still has much better accuracy than the solution to the original model for higher frequencies (that decay in the base before the TL). Hence, for muti-frequency component stimuli as an impulse in the time domain, the transformation retains the order of accuracy of the highest frequency in the region of interest. Therefore, we find that with this transformation, the decrease in cochlear length needed for simulation corresponds to a decrease in total number of sections if using a uniform step size.

A natural question that arises is whether or not a decrease is number of sections of the original model (with no transformation) would yield results with similar accuracy. We do expect that this transformation is a parametric counterpart to having some nonuniform step size function. However, as long as the simulation is done using a uniform spatial step size, these two approaches differ in a fundamental manner since accuracy is dependant on the input frequency component. Decreasing the number of sections in the original model decreases the accuracy for both high and low frequency components of a solution. On the other hand, for

a given n, transformation PS retains high frequency accuracy (which is originally lower than that for lower frequencies) but reduces low frequency accuracy, thereby it is appropriate as long as the accuracy of the lowest frequency for the transformed case is not less than the accuracy of the highest frequency components for the original model.

It is important to note (for application to other models) that in the Zweig model, reverse waves dissipate as they travel from apex to base. This contributes to the reason the local RI DE for the high frequencies is larger than the reverse propagated (TL to RI) DE as seen by the RI from lower frequencies for the same step size. The local DE within RI for low frequencies is less than that for higher frequencies because the wavelength is shorter. This pattern should also be taken into account when considering extendibility to other models that may be active or have a positive growth function in some region.

7.6 Summary

We conclude that this transformation is appropriate for multi-frequency stimuli if using a uniform step size and limited by the maximal DE of the lowest accuracy frequency component in the original model. The transformation allows for decreased computational load through mechanism A. This transformation is a constrained parametric counterpart to transformation SS.

To retain the same maximal DE (across frequencies) as in the original model, the degree of squeezing in this transformation is limited by the error of high frequencies that have a high spatial gradient within the nontransformed region. The restriction of CF squeezing is that we can squeeze until the cumulative DE seen at the RI of lower frequencies (that naturally decay after the RI) approximates the DE of the higher frequencies (that naturally decay within the RI) for the same step size.

8

Transformation SC: Complex Spatial Transformation

Note - A more complete discussion regarding this transformation has appeared in [14]

8.1 Transformation outline

In this transformation, after x0, we extend x in the complex domain. The transformation is purely spatial

but the function parameters is chosen based on the wave number. This transformation is directly motivated by [7].

Figure 9: Transformation SC - Complex spatial transformation Left top: Wavenumber before transformation. Left bottom: Wavenumber after transformation. Right top: spatial extension into the complex domain occurs after the beginning of the boundary layer (at 0.5 cm) as indicated by the dashed red line. Right middle: |P (x)| for original and transformed models. Right bottom: Magnitude of difference vector between original and transformed model solutions. Input frequency = 2.4 kHz, step size = 2.9e-3 cm.

8.2 Implementation details

The major implementation details follow that described in the real spatial transformation - section 6, and hence will not be repeated here. For a wide frequency range within the CF range, the forward WKB

solution is dominated by the e−jR k(ω,x)dx _{portion. The spatial transformation can be noted as x → ˜x(x)}

function is based on the relationship to the simulated variable. x = u, and ˜dx = S(u)du.

S(u) = ˜ dx(u)

du = r(u) + jm(u) (22)

Where r(u < u0) = 1, m(u < u0) = 0

For simplicity and based on the points below, we set r(u) = 1. As u = x, bookkeeping is no longer necessary and we will henceforth use them interchangably.

The transformation is purely spatial and there are no modifications made to the wavenumber, but we may think of an equivalent parametric transformation to guide the choice of r(x) and m(x). The wavenumber has a form in equation 35, and shown in figure 9. We may write equivalent propagation and gain function based on the dominant portion of the WKB form. The equivalent propagation function is

κ0(x, ω) = κ(x, ω) − γ(x, ω)m(x), and the equivalent gain function is γ0(x, ω) = m(x)κ(x, ω) + γ(x, ω).

The solution originally had the increase in oscillation rate before the increase in decay rate. We wish to increase the decay rate but limit an increase in the oscillation rate while the amplitude of the solution is

still large to avoid the additional discretization error. Hence, we choose to use the fact that the κ peaks

before γ and attempt to switch this pattern. So, we choose m(x > x0) < 0, and rewrite in terms of the

positive parameters κ, G = −γ and M = −m, as in the equations below,

κ0= κ − GM (23)

γ0 = −M κ − G (24)

Given the ’flipped across CF place, scaled’ wavenumber, the κ0 _{will peak before the γ}0_{. Ideally, we}

would have a transformation that allows for delinking the growth of γ0 _{from κ}0_{. For that, r(x) 6= 1 would be}

utilized. However, for the transformation function parameters chosen in the section below, an m = m(x) is sufficient.

8.3 Transformation function details

Assuming the WKB approximation still holds, one can estimate the expected xcomp and choose the

trans-formation function parameters accordingly and then test that errors are negligible for that parameter set. Though, as mentioned earlier, we will not go into depth for this transformation. The governing equation transformation and derivation of the discrete system of equations is the same as that in equation 13 but

and b = 3

S(x) = 1 − ja(x − x0)b (25)

8.4 Performance

Table 4: Transformation SC: Performance evaluation. Input frequency 2.4 kHz

C₁∗ C2 Cˆ2 Cm∗

n≈continuous= 7000 n = 700 n = 700 n = 700

0.05 0.01 0.01 0.05

Figure 9 is for input frequency = 2.4 kHz 11_{. C}∗

1 shows that in the continuous domain y(x) and ˆy(˜x)

have decent correspondence within the RI. In the discrete domain, for h = 2.9e−3, y[j] and ˆy[j] have similar accuracy when compared with their respective ’continuous’ solutions (approximated by h = 2.9e − 4) as

indicated by C2 and ˆC2. The cumulative measure Cm∗ shows that for h = 2.9e − 3, the discrete solutions

are comparable.

While figure 9 (bottom right) is only for a single frequency, it is illustrative of the qualitative pattern of transfomation effect for frequencies within the CF range for which the solution is well approximated by the dominant portion of WKB. Outside of this range, the deviation error increases (though the solution follows the same qualitative pattern). We attribute this to the process of constructing the transformation function which we based on wavenumber from an approximate solution. The wavenumber computed may not have the same interpretation for frequencies outside of the WKB range.

8.5 Characterization and discussion

We neglect the frequencies for which the dominant portion of WKB is not an appropriate estimate (on the low frequency side of the CF range). The pressure in the transformed cochlea decays before it does in the regular cochlea allowing for imposition of a computational boundary at the artificial apex. Input

frequency = 2.4 kHz. As noted in 8.4, the deviation before x0 between the solutions to the regular and

transformed models still exists in the (approximately) continuous domain. We rule out DE, and hypothesize that it occurs due to internal reflections from the large gradients of the effective material properties, and may be measurable using a cumulative WKB deviation measure. We hypothesize allowing r(x) 6= 1 could potentially help in overcoming this.

11_{this frequency is different than that shown for other transformations because 1.5 kHz is outside the range in which the}

The construction and hence extendability of this transformation is dependant on the shape of the wavenumber(s). Hence, for some models, either the transformation function needs to be reconstructed based on the wavenumber(s), or the same transformation function can be used after allowing a small layer (before applying the transformation) for converging the model wavenumber to a form similar in shape to figure 9 - top left. For example, the extra layer may be used to turn off an amplifier for an active model in which the constructed transformation function is inappropriate.

Another matter of extendibility is to address models in which the longitudinal wavenumber are not estimated, or models in which there are multiple dominant longitudinal wavenumbers and for which there is no single transformation function that is for all of them. This transformation cannot be applied to such models.

8.6 Summary

We conclude that this transformation with the above specific constructed transformation function is ap-propriate for the Zweig model for multi-frequency stimuli with frequency components that fall within the

range of frequencies for which eiR kdx_{, the dominant portion of WKB for those frequencies, is an appropriate}

approximation to the solution. More generally, the construction of the appropriate transformation function is dependant on longitudinal wavenumber(s), and is appropriate for frequency components that fall within a range of frequencies based on the analytic solution form (eg. WKB here) used for transformation function construction.

The method is applied here with a uniform stepsize, but we anticipate that using a nonuniform stepsize would be equivalent to combining transformation SC with transformation SS. Within the appropriate frequency range, the transformation function is appropriate with some deviation error. Outside of this range, (if using the same constructed transformation function based on an analytic form for the solution), spatial causality across the RI-TL interface is lost (and hence SP does not hold) and hence, deviation error increases. Accuracy is not an issue.

Another difficulty arises in extendibility due to the possible need of transformation function reconstruc-tion. In the case of reconstruction, the following is required: an estimate for the longitudinal wavenumber(s), an approximate analytic form to understand the dependance of the solution on the wavenumber, and if there are multiple dominant mode shapes then an a single construction must be appropriate for all of them.

9

Discussion

9.1 Spatial preservation requirement

In this section, we discuss the SP requirement for application of the methods developed in this thesis, and

an example with a constraint version of this requirement. After that, we discuss results from [9] that the

real cochlea (and hence reflective cochlear models) is spatially causal and hence, more generally, fulfills the SP assumption.

While the original case study model is approximately spatially causal for all x (for frequencies within the CF range of the cochlea), we have mentioned in section 2.2 that this is a constrained version of the SP requirement (spatially causal ⊂ SI ⊂ SP). While general SP generally determined by both the physical model and transformation, SI can be determined based on the physical model only, and retained or not for the transformed model. We note that existence of these criteria in the continuous domain does not necessitate their existence in the discrete domain.

The methods developed in this thesis can be applied either to constructed models and transformations that fulfill SP (in which case a simple global nonsimulative method of determinging SP - or the more constraint spatial causality, would be very useful), or to models in the construction phase for which the parameter value sets can be limited only to those sets that are appropriate for SP.

We discuss an example, in which spatial causality does not hold, but SI does. Such a situation would arise for originally spatially causal models that are altered using a micromechanical perturbation approach to simulate high stimulus frequency otoacoustic emissions (SFOAEs - internal reflections in response to single frequency stimuli). Based on the coherent reflection theory, the cumulative measured reflections arise mainly from places very close to the CF-place corresponding to the stimulus frequency. Hence, the effective coupling is mainly very local around the CF (and so only that small region needs to be perturbated), which lies within the RI for high frequencies. As a result, while spatial causality within the RI does not hold, the

solution remains independant of the characteristics of TL. 12

While our observation of spatial causality holds true in the Zweig model and other WKB models, the phenomenon of spatial causality does not seem to be discussed in the context of cochlear models, or cochlear theory. Hence, it is not certain whether it is a characteristic of the cochlea and for what range of cochlear models this holds true. We propose to approach the question of SP in the cochlea by determining spatial causality as follows (a similar derivation can be made for cochlear models). Degree of spatial causality

12_{Though in the single high stimulus frequency case, the TL is only necessary while transients exists, and after they decay}

sufficiently, the model may be truncated (with no transformations required) shortly after the CF place for the stimulus frequency as explained in section 10.2

may be measured by the degree of propagated internal reflections. Measures of the propagation and gain

function of the dominant longitudinal mode in the real cochlea have been estimated in [9] by collapsing the

problem into a governing equation in the form of equation 1 - for which the solution may be approximated using the WKB estimate. The cumulative deviation from the WKB solution may be estimated as used

as a degree of spatial causality measure. The results of [9] show that the reconstructed basilar membrane

velocity, vBM, from k using the WKB (spatially causal) form works very well for some cochlea (and hence

spatial causality is a good assumption), but fails for others (this does not necessarily mean that spatial causality fails for these cochleas). We note negative result of spatial causality does not imply a negative result for SP, which is the criteria of interest.

9.2 Extendibility

We do not expect great impediments in extending the methods to different discretization methods or different methods of solving systems of equations. There are two major impediments to extending the methods proposed to use in physical cochlear models other than Zweig’s. Firstly, The underlying assumption of SP must be addressed for each model individually. This has been discussed in section 9.1. Secondly, application of some of the transformations require additional assumptions (beyond SP). Transformation PS in section 7 requires a shared underlying parameter for all spatial dependant parameters; transformaton SC in section 8 while entirely spatial in nature, has a transformation function constructed based on the wavenumber - a problem arises in this case when the wavenumber is not known or there exists multiple longitudinal mode shapes.

10

Further Considerations

10.1 Applications

The methods would be useful to efficiently compute the RI response not just for impulse and click stimuli, but also for single high frequency stimuli in the time domain where transients are part of the solution and hence, one cannot simply truncate the cochlea beyond the CF-place until they die out.

The parametric construction phase of building a full model is often done using mostly unguided trial and error rather than a guided search or constrained optimization (as would be ideal if one can neglect com-putational inefficiency issues). Further improvements to the designed methods to better improve efficiency

would allow for finding mutliple appropriate parameter value sets13 _{for a narrow basal region, which can}

then be extrapolated using an underlying parameter function to produce the mechanical parameter value sets for the entire model length if it is of interest. These parameter value sets would only yield appropriate results if the corresponding system is spatially independant (SI). Hence, if this assumption can be made or the corresponding limitation imposed, the construction phase of model development would be made more systematic.

In addition to improving the computational efficiency, transformations PS, SS, SC may also be used to • Eliminate apical boundary reflections: Apical boundary reflections arise if a frequency component of the forward traveling wave has not decayed entirely before the apex. This may be the case for models that are not perfect portrayals of the apical experimental cochlea (but may be appropriate models for basal behavior in which the mechanics are different). For the transformations that speeds up the absorbance of the pressure, the solution will decay before the apex, thereby eliminating the apical reflections given the same range of input frequencies. This would be more efficient than the natural approach of extending the cochlea.

• Allow for wider-band stimuli: In general, cochlear models are designed to process a specific range of frequencies (roughly, the CF range). If the stimulus is an impulse, the solution to frequency components outside that range on the low frequency side may introduce standing waves and other effects that may be undesirable and complicate the solution characterization. Hence, the stimulus should generally be filtered to the range of frequencies that the model can handle. However, as the filtering tails are not steep, this may lead to either incorporating frequencies outside the model range

13_{Sets rather than set, because the mapping from the mechanical parameters space to to that of wave propagation parameters,}

filter characteristics, and other phenomenon is not distinct and hence mutiple parameter value sets may produce the same results for a given phenomenon. The mapping is also not redundant and hence, while mutiple parameter sets may produce the same results for one phenomenon, they may differ in another