Development of Active Artificial Hair Cell Sensors

Bryan Steven Joyce

Dissertation submitted to the faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

Mechanical Engineering

Pablo A. Tarazaga, Chair

J. Wally Grant

Mary E. Kasarda

Donald J. Leo

Michael K. Philen

May 6, 2015

Blacksburg, Virginia

Keywords: Bioinspired, Cochlear Amplifier, Artificial Hair Cell, Biomimetic Sensor,

Nonlinear Sensor, Nonlinear Dynamics, Feedback Control

Copyright 2015, Bryan S. Joyce

Development of Active Artificial Hair Cell Sensors

Bryan Steven Joyce

ABSTRACT

The cochlea is known to exhibit a nonlinear, mechanical amplification which allows the

ear to detect faint sounds, improves frequency discrimination, and broadens the range of sound

pressure levels that can be detected. In this work, active artificial hair cells (AHC) are proposed

and developed which mimic the nonlinear cochlear amplifier. Active AHCs can be used to

transduce sound pressures, fluid flow, accelerations, or another form of dynamic input. These

nonlinear sensors consist of piezoelectric cantilever beams which utilize various feedback

control laws inspired by the living cochlea. A phenomenological control law is first examined

which exhibits similar behavior as the living cochlea. Two sets of physiological models are also

examined: one set based on outer hair cell somatic motility and the other set inspired by active

hair bundle motility. Compared to passive AHCs, simulation and experimental results for active

AHCs show an amplified response due to small stimuli, a sharpened resonance peak, and a

compressive nonlinearity between response amplitude and input level. These bio-inspired

devices could lead to new sensors with lower thresholds of sound or vibration detection,

improved frequency sensitivities, and the ability to detect a wider range of input levels. These

bio-inspired, active sensors lay the foundation for a new generation of sensors for acoustic, fluid

flow, or vibration sensing.

iii

Acknowledgments

I would like to thank my advisor, Dr. Pablo Tarazaga. This work would not be possible

without his insight, guidance, and good humor. I am also grateful for my committee members

(Dr. Wally Grant, Dr. Mary Kasarda, Dr. Donald Leo, and Dr. Michael Philen) for their

feedback, technical insight, and career advice.

I must thank the ME support staff, particularly Beth Howell, Cathy Hill, Linda Vick, and

all of the guys in the ME machine shop. Our department would grind to a halt without them. I

would also like to thank my past and current lab mates in the Center for Intelligent Materials

Systems and Structures (CIMSS); the Vibrations, Adaptive Structures, and Testing (VAST) lab;

and the Virginia Tech Smart Infrastructure Lab (VT-SIL). I have learned a great deal from the

brilliant minds around me. I would particularly like to thank Mathieu Vandaele whose help was

instrumental in the acoustic tests of artificial hair cells.

Finally, I would like to acknowledge the generous support from the U.S. Department of

Education GAANN Fellowship (Award No. P200A1000136), the Davenport Fellowship, and my

departmental research and teaching assistant positions. Their support is gratefully

acknowledged.

iv

Table of Contents

Chapter 1. Introduction and Literature Review 1

1.1. Introduction and Research Motivation............................................................................... 1

1.2. The Auditory Periphery ..................................................................................................... 2

1.2.1. Overall Structure ....................................................................................................... 3

1.2.2. Inner and Outer Hair Cells ........................................................................................ 8

1.3. The Cochlear Amplifier ................................................................................................... 12

1.3.1. History of the Cochlear Amplifier .......................................................................... 12

1.3.2. Characteristics of the Cochlear Amplifier .............................................................. 14

1.3.3. Mechanisms of Amplification................................................................................. 18

1.4. Mimicking the Cochlea through Passive Devices ........................................................... 19

1.4.1. Passive Artificial Hair Cells.................................................................................... 20

1.4.2. Passive Artificial Basilar Membranes and Cochleae .............................................. 23

1.5. Mimicking the Cochlea through Active Devices ............................................................. 24

1.6. Active Artificial Hair Cells .............................................................................................. 28

1.6.1. Basic Design of Active Artificial Hair Cells .......................................................... 28

1.6.2. Resonance Based Sensors ....................................................................................... 30

1.6.3. Feedback Control and Nonlinearity ........................................................................ 31

1.7. Dissertation Overview ..................................................................................................... 32

1.7.1. Contributions........................................................................................................... 33

1.7.2. Chapter Summary ................................................................................................... 34

Chapter 2. Passive Piezoelectric Hair Cell Models 35

2.1. Model Derivation: Distributed Parameter Model ............................................................ 35

v

2.1.1. Governing Equations .............................................................................................. 36

2.1.2. Galerkin-Finite Element Approximation Method ................................................... 40

2.1.3. Modal Decomposition and Added Damping .......................................................... 43

2.1.4. One Mode Model .................................................................................................... 47

2.1.5. One Mode Model for a Bimorph Beam .................................................................. 48

2.2. Model Derivation: System Identification Approach ........................................................ 51

2.3. Frequency Response and Tuning Curves ......................................................................... 53

2.4. Conclusions ...................................................................................................................... 55

Chapter 3. Models of Artificial Hair Cells using Cubic Damping 56

3.1. Nonlinear Oscillator at a Hopf Bifurcation ...................................................................... 56

3.2. Control Law and Closed Loop Response......................................................................... 60

3.3. Simulations of Cubic Damping Systems ......................................................................... 64

3.3.1. Harmonic Balance Method ..................................................................................... 65

3.3.3. Numerical Simulations............................................................................................ 69

3.4. Effect of Higher Modes ................................................................................................... 75

3.5. Filtering and Time Delays ............................................................................................... 79

3.5.1. Simulations with a Butterworth Filter..................................................................... 79

3.5.2. Time Delays in the Feedback Path.......................................................................... 82

3.6. Conclusions ...................................................................................................................... 85

Chapter 4. Experimental Results of Active Artificial Hair Cells using Cubic Damping 86

4.1. Proof-of-Concept Artificial Hair Cell .............................................................................. 87

4.1.1. Design ..................................................................................................................... 87

4.1.2. Model Development................................................................................................ 89

vi

4.1.3. Controller Design .................................................................................................... 92

4.1.4. Experimental Results .............................................................................................. 93

4.2. Experimental Tests for an AHC in Fluid ......................................................................... 97

4.2.1. Preliminary Tests in Water ..................................................................................... 98

4.2.2. Experimental Setup and Model Development ...................................................... 101

4.2.4. Experimental Results for the Active AHC in Water ............................................. 110

4.3. Artificial Hair Cell Accelerometer................................................................................. 112

4.3.1. Design ................................................................................................................... 113

4.3.2. Model Development.............................................................................................. 114

4.3.3. Experimental Results ............................................................................................ 116

4.3.4. Tuning Curves ....................................................................................................... 122

4.4. Split Bimorph Artificial Hair Cell Design ..................................................................... 123

4.4.1. Design ................................................................................................................... 124

4.4.2. Direct Feedthrough Coupling ............................................................................... 125

4.4.3. Frequency Response Functions............................................................................. 127

4.4.4. Numerical Simulations of Active AHC ................................................................ 131

4.5. Conclusions .................................................................................................................... 133

Chapter 5. Active Artificial Hair Cells Inspired by Outer Hair Cell Somatic Motility 134

5.1. Criteria for Implementable Cochlear Models ................................................................ 135

5.2. Sigmoidal Damping ....................................................................................................... 136

5.2.1. Model Derivation .................................................................................................. 137

5.2.2. Numerical Simulations.......................................................................................... 142

5.3. Feedback through a Tectorial Membrane System ......................................................... 144

vii

5.3.1. Overview ............................................................................................................... 145

5.3.2. First-order Tectorial Membrane in Feedback Path ............................................... 148

5.3.3. Second-order Tectorial Membrane in Feedback Path ........................................... 152

5.3.4. Numerical Simulations.......................................................................................... 157

5.5. Conclusions .................................................................................................................... 159

Chapter 6. Active Hair Bundle-based Artificial Hair Cells 161

6.1. Overview of Active Hair Bundle Motility ..................................................................... 162

6.2. Active Hair Bundle Model without Inertia .................................................................... 165

6.3. Active Hair Bundle Model with Inertia ......................................................................... 170

6.3.1. Model Derivation .................................................................................................. 171

6.3.2. Behavior of Linearized System and Tuning to a Hopf Bifurcation ...................... 172

6.3.3. Nonlinear Response from the Harmonic Balance Method ................................... 178

6.4. Numerical Simulations for Active Hair Bundle-based AHCs ....................................... 180

6.5. Monostable Active Hair Bundle Model ......................................................................... 185

6.6. Active Hair Bundle Model Tuned to DC ....................................................................... 188

6.7. Implementation of Active Hair Bundle Model .............................................................. 190

6.7.1. Nonlinear stiffness ................................................................................................ 190

6.7.2. Controller Development........................................................................................ 191

6.7.3. System Identification by Linear System Approximation...................................... 193

6.7.4. Nonlinear System Identification through Least Squares Regression .................... 196

6.8. Conclusions .................................................................................................................... 197

Chapter 7. Comparisons of Active Artificial Hair Cell Designs 199

7.1. Cases of Active Artificial Hair Cells ............................................................................. 199

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7.1.1. Cubic Damping ..................................................................................................... 201

7.1.2. Sigmoidal Damping .............................................................................................. 202

7.1.3. Organ of Corti (OoC) with ζz = 0.1 ....................................................................... 203

7.1.4. Organ of Corti (OoC) with ζz = 0.01 ..................................................................... 204

7.1.5. Active Hair Bundle ............................................................................................... 205

7.2. Performance Metrics ...................................................................................................... 206

7.2.1. Total Harmonic Distortion .................................................................................... 206

7.2.2. Settling Time ......................................................................................................... 209

7.3. Numerical Comparisons between Active AHC Cases ................................................... 210

7.4. Sensor Recommendations and Conclusions .................................................................. 218

Chapter 8. Conclusions and Future Work 221

8.1. Brief Summary of Dissertation ...................................................................................... 221

8.2. Summary of Contributions ............................................................................................. 223

8.3. Areas for Future Work ................................................................................................... 224

8.3.1. Response to Complex Inputs and Stochastic Resonance ...................................... 224

8.3.2. Self-sensing Hair Cells ......................................................................................... 225

8.3.3. System Identification and Miniaturization............................................................ 226

8.3.4. Active Artificial Hair Cell Arrays and Active Artificial Basilar Membranes ...... 227

Bibliography 228

Appendix A. Distributed Parameter Model of an Artificial Hair Cell 251

A.1. Piezoelectric Constitutive Laws .................................................................................... 251

A.2. Mechanical Domain Equations ..................................................................................... 254

A.2.1. Free-body Diagram and Newton’s Laws ............................................................. 254

ix

A.2.2. Considerations for the Composite Beam ............................................................. 258

A.2.3. Piezoelectric Coupling Factor .............................................................................. 263

A.3. Electrical Domain Equations ........................................................................................ 265

A.3.1. Charge and Current through a Piezoelectric Actuator ......................................... 265

A.3.2. Voltage Through a Piezoelectric Sensor .............................................................. 268

A.3.3. Direct Feedthrough Term from In-plane Coupling .............................................. 269

A.4. Boundary Conditions .................................................................................................... 273

A.5. Simplifications for a Bimorph Configuration ............................................................... 273

A.6. Base Excitation ............................................................................................................. 274

Appendix B. Galerkin Method for the AHC Distributed Parameter Model 275

B.1. Overview of the Galerkin Method ................................................................................ 275

B.2. Galerkin Method for the AHC Model ........................................................................... 276

B.3. Finite Element Method .................................................................................................. 282

B.4. Base Excitation .............................................................................................................. 288

B.5. Forcing Vector from an Applied Pressure ..................................................................... 289

Appendix C. Modal Decomposition and Frequency Response Functions of the AHC Model

292

C.1. Modal Decomposition ................................................................................................... 292

C.2. Adding Damping ........................................................................................................... 296

C.3. Frequency Response Functions from General Forcing ................................................. 297

C.4. Open Circuit Voltage .................................................................................................... 299

C.5. Frequency Response Functions from Base Accelerations ............................................ 300

C.6. Frequency Response Functions from Piezoelectric Actuator ....................................... 302

x

C.7. Frequency Response Functions from Plane Wave Pressure ......................................... 303

Appendix D. Acoustic Tests of Passive Artificial Hair Cells 305

D.1. PZT Artificial Hair Cell Design and FRF ..................................................................... 305

D.2. PZT Artificial Hair Cell Tuning Curves ....................................................................... 307

D.3. PVDF Artificial Hair Cell ............................................................................................. 311

D.4. Comparison to Biology ................................................................................................. 312

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List of Figures

Figure 1.1. Diagram of the ear. From Dallos (1992),with permission of The Journal of

Neuroscience [21]. .......................................................................................................................... 4

Figure 1.2. Simplified schematic of the cochlea. Here the spiral has been “unrolled” for visual

clarity. From Dallos (1992),with permission of The Journal of Neuroscience [21]. ..................... 5

Figure 1.3. Cross-section of the organ of Corti on the basilar membrane. From Raphael and

Atlschuler (2003),with permission of Elsevier Limited [30]. ......................................................... 7

Figure 1.4. Rows of hair cell stereocilia. From Raphael and Atlschuler (2003), reproduced with

permission of Elsevier Limited [30]. .............................................................................................. 9

Figure 1.5. Diagrams of the inner and outer hair cells. From Dallos (1992),with permission of

The Journal of Neuroscience [21]. ................................................................................................ 10

Figure 1.6. Measurements showing the cochlear amplifier in a guinea pig cochlea. (a) Basilar

membrane (BM) displacement of versus frequency and sound pressure level. (b) Basilar

membrane displacement normalized by the input sound pressure level. These curves would

overlap for a linear system. Figure from Johnstone et al. (1986), reproduced with permission of

Elsevier Limited [40]. ................................................................................................................... 15

Figure 1.7. Displacement of the basilar membrane versus sound pressure level. The

displacement shows a linear trend at low and high sound pressures levels and a nonlinear

compression at intermediate sound pressure. Figure from Johnstone et al. (1986), reproduced

with permission of Elsevier Limited [40]. ................................................................................... 16

Figure 1.8. Schematic of a simple, active artificial hair cell. (a) Physical layout of the artificial

hair cell (AHC). (b) Block diagram of the closed-loop system. .................................................. 29

Figure 1.9. Frequency response function for an example sensor. ................................................ 30

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Figure 2.1. Cantilever beam with piezoelectric elements (actuators or sensors). ........................ 36

Figure 2.2. Piezoelectric bimorph beam. Arrows on the piezoceramic indicate their polarization

direction. ....................................................................................................................................... 48

Figure 2.3. First natural frequency of bimorph beam as a function of beam length. The plot uses

parameters from the small scale artificial hair cell discussed later in Chapter 4. A 1 mm change

in length changes the natural frequency by about 36 Hz (about 8% change). .............................. 52

Figure 2.4. Example response of a single degree of freedom system. (a) System response versus

input level and frequency. (b) Response amplitude versus frequency for different input levels.

(c) Response amplitude versus input amplitude at resonance. (d) Input level versus frequency for

different response levels, i.e. tuning curves. ................................................................................. 54

Figure 3.1. Sample time responses of the prototypical Hopf bifurcation system. (a) μ = 1 > 0.

(b) μ = -1 < 0. For both systems, b = 1, c = 2π 10 rad/s, and the initial condition is z(0) = 0.5. 58

Figure 3.2. Sample phase portraits of the prototypical Hopf bifurcation system. (a) μ = 1 > 0.

(b) μ = -1 < 0. For both systems, b = 1, c = 2π 10 rad/s, and the initial condition is z(0) = 0.5. 58

Figure 3.3. Example frequency response of the prototypical Hopf bifurcation system. Here μ =

0, b = 1, and c = 1 rad/s............................................................................................................... 60

Figure 3.4. Time response at resonance ( 1 ) of a linear versus nonlinear example. For the

linear system, 01.0 . For the nonlinear case, a3= 1x10-4

. ......................................................... 70

Figure 3.5. Example frequency response of a cubic damping system for various excitation

amplitudes. (a) Magnitude of the response. (b) Response amplitude normalized by the input

amplitude. For the nonlinear system, = 0 and a3 = 1x10-4

. ....................................................... 71

Figure 3.6. Example frequency response of a cubic damping system for various values of a3.

Here = 0 and F = 1. .................................................................................................................... 71

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Figure 3.7. Response amplitude versus input amplitude at = 1 for various values of a3. Here

= 0 for the nonlinear system. A linear system with ζ = 0.01 is shown for comparison. .............. 72

Figure 3.8. Response amplitude versus input amplitude for both the linear and nonlinear systems

at (a) = 0.999 and (b) = 0.99. For the nonlinear cases, = 0. .............................................. 74

Figure 3.9. Response amplitude versus input amplitude at = 1 for (a) = 0.001 and (b) =

0.01................................................................................................................................................ 74

Figure 3.10. Effect of multiple modes on the frequency response. ............................................. 78

Figure 3.11. Simulation results showing a limit cycle oscillation for the two mode system given

a small initial displacement. .......................................................................................................... 78

Figure 3.12. Block diagram of the closed-loop system with a filter to reduce spillover. ............ 79

Figure 3.13. Frequency response functions (FRFs) for examples Butterworth filters. The

examples use corner frequencies (c) of 3 and 5 and filter order (n) of 2 and 6. ......................... 80

Figure 3.14. Effect of the using a Butterworth filter in the feedback loop. The filter parameters

are (a) c = 3, n = 2; (b) c = 3, n = 6; (c) c = 5, n = 2; and (d) c = 5, n = 6. ...................... 81

Figure 3.15. Effect of a time delay in the feedback path. (a) Peak response versus input

excitation for different time delays T. (b) Backbone curves for different time delays. These plots

are generated from the harmonic balance method in Equations 3.46 and 3.47. For these curves,

a3 = 41.7, ζ = 0.1, and a1 = 2ζ = 0.2. The linear case (a1 = a3 = 0) is shown for reference. ........ 84

Figure 4.1. Proof-of-concept artificial hair cell design. (a) Photograph of the experimental

setup. (b) Schematic of the experiment. ....................................................................................... 88

Figure 4.2. Velocity from control voltage frequency response functions for the proof-of-concept

AHC from the data, a single degree of freedom fit (SDOF Fit) around the first mode, and a finite

element model (FE Model) with one mode and with five modes. ................................................ 92

xiv

Figure 4.3. Control algorithm for the AHC. The cantilever beam transfer function is defined in

Equation 4.2. ................................................................................................................................. 93

Figure 4.4. Frequency response of the proof-of-concept artificial hair cell for different

disturbance levels. The left column shows model predictions, while the right column shows

experimental results. ..................................................................................................................... 95

Figure 4.5. Amplitude of the tip velocity versus disturbance level at the resonance frequency

(10.8 Hz). ...................................................................................................................................... 95

Figure 4.6. Experimental setup for preliminary water tests. An aluminum cantilever beam is

partially submerged in water. ........................................................................................................ 98

Figure 4.7. Magnitude of the frequency response function for different water depths. ............ 100

Figure 4.8. Variation in natural frequency and damping of the first mode with increasing water

depth. 95% confidence intervals are also shown at each data point. ......................................... 100

Figure 4.9. Artificial hair cell in water experimental setup. (a) Photograph and (b) schematic of

the experiment. ............................................................................................................................ 101

Figure 4.10. Frequency response function (FRF) of velocity to control voltage for the beam in

air. ............................................................................................................................................... 104

Figure 4.11. Schematic of the artificial hair cell with an added tip mass and viscous damper to

account for the added inertia and damping due to the fluid damping. ........................................ 105

Figure 4.12. Velocity from control voltage frequency response function for the passive sensor

partially submerged in water. The figure shows the FRF from the data (Data), single degree of

freedom fit (SDOF fit), and the finite element (FE) model. ....................................................... 108

Figure 4.13. FRF of velocity to disturbance voltage for the beam in air and in water. ............. 109

xv

Figure 4.14. Frequency response function of velocity from disturbance voltage for the passive

artificial hair cell partially submerged in water. The figure shows the data (Data), single degree

of freedom fit (SDOF Fit), and finite element model (FE Model). ............................................ 110

Figure 4.15. Model and experimental frequency response functions (FRFs) for the active

artificial hair cell in water. The FRFs are plots of the velocity at the measurement point from

disturbance signals at different voltage levels and frequencies. ................................................. 111

Figure 4.16. (a) Input-to-output relationship for the disturbance voltage to the velocity at the

natural frequency. (b) Fit of the damping ratio versus the amplitude of the velocity. ............... 112

Figure 4.17. Artificial hair cell accelerometer. The AHC consists of a piezoelectric bimorph

beam under a base excitation. ..................................................................................................... 113

Figure 4.18. Velocity to control voltage FRFs for the passive AHC accelerometer from the data,

a single degree of freedom fit (SDOF Fit) around the first mode using the system identification

method, and a finite element model (FE Model). Here a 1 V control signal was used. ............ 115

Figure 4.19. Velocity from base acceleration FRFs of the passive AHC accelerometer. (a)

Magnitude of the velocity versus frequency for different voltages to the shaker. (b) FRFs of

velocity with respect to the base acceleration for different shaker voltages............................... 116

Figure 4.20. Velocity from base acceleration FRFs of the active AHC accelerometer. (a) and (b)

show the magnitude of the velocity versus frequency for different base accelerations. (c) and (d)

show the velocity normalized by the base acceleration for different shaker voltages. (a) and (c)

show results from the model, while (b) and (d) show results from the experiment. .................. 117

Figure 4.21. (a) Amplitude compression for the active and passive AHC accelerometer. (b)

Backbone curves for the active and passive AHC. For reference, an ideal model at the Hopf

bifurcation (μ = 0) is also shown. ............................................................................................... 119

xvi

Figure 4.22. Butterworth filter with a corner frequency of 1500 Hz and filter order n = 2 and n =

6................................................................................................................................................... 120

Figure 4.23. Experimental results for the active AHC accelerometer using a poor filter design.

(a) Magnitude of the velocity versus frequency for different voltages to the shaker. (b) Velocity

normalized by the base acceleration for different shaker voltages. ............................................ 121

Figure 4.24. Experimental input-to-output and backbone curves for the AHC using a poor filter

design. (a) Amplitude compression for the active and passive AHC. (b) Backbone curves for the

active and passive AHC. ............................................................................................................. 121

Figure 4.25. Tuning curves for the small scale artificial hair cell. The velocity threshold is set to

1.5 mm/s. ..................................................................................................................................... 123

Figure 4.26. “Split” configuration bimorph beam. (a) Schematic of the AHC. Arrows on the

piezoceramic indicate the polarization direction. (b) Test setup. .............................................. 124

Figure 4.27. Frequency response functions for the split beam sensor. (a) Tip velocity from a

base acceleration. (b) Velocity from control voltage. (c) Voltage in the sensing element from a

base acceleration. (d) Sensing voltage from the control voltage. ............................................... 130

Figure 4.28. Simulation results of the compensated voltage of the split bimorph AHC versus

frequency for different input accelerations. For comparison, the responses of the system with

and without the controller are shown. ......................................................................................... 132

Figure 4.29. Compensated voltage versus input acceleration at resonance. ............................... 132

Figure 5.1. Schematic of a simple, active artificial hair cell. (a) Physical layout of the artificial

hair cell (AHC). (b) Block diagram of the closed-loop system. ................................................ 135

Figure 5.2. (a) Diagram of a hair bundle. (b) Open channel probability (PO) as a function of the

stereocilia bundle’s deflection y. ................................................................................................. 137

xvii

Figure 5.3. Frequency response functions of an artificial hair cell with sigmoidal velocity

feedback. ..................................................................................................................................... 143

Figure 5.4. Output-input relationship at resonance for the system without a controller (a linear

system with ζ = 0.1), cubic velocity feedback tuned to the Hopf bifurcation, and the sigmoidal

velocity feedback tuned to the Hopf bifurcation. ........................................................................ 143

Figure 5.5. Cross-section of the organ of Corti on the basilar membrane. From Raphael and

Atlschuler (2003),with permission of Elsevier Limited [30]. ..................................................... 146

Figure 5.6. Simplified relationship of the components of the organ of Corti ............................. 146

Figure 5.7. Frequency response functions for various first-order tectorial membrane systems. (a)

zy . (b) zy . Parameter values and the input force to the tectorial membrane system are

given in the figure legend. .......................................................................................................... 151

Figure 5.8. Frequency response functions for various second-order tectorial membrane systems.

(a) Stereocilia deflection proportional to the tectorial membrane displacement (y = z). (b)

Stereocilia deflection proportional to tectorial membrane velocity ( zy ). The legend gives

the form of the input force on the tectorial membrane system. For all of these cases, the tectorial

membrane damping ζz is 0.01. .................................................................................................... 155

Figure 5.9. Frequency response functions between displacement x and stimulus f for an active

artificial hair cell with a tectorial membrane system. Here bf = 1, ζ = 0.1, ζz = 0.1, = 1, δ = 0.1,

and a = 0.16. The parameters are chosen to achieve a Hopf bifurcation and an equivalent cubic

damping coefficient a3 is 41.7. ................................................................................................... 158

Figure 5.10. Displacement x to forcing f frequency response functions for the sensor with a

smaller bandwidth of amplification (ζz = 0.01). Here bf = 1, ζ = 0.1, ζz = 0.01, = 1, δ = 1, and

xviii

a = 0.016. The FRF for F = 1x10-1

overlaps the linear response. The parameters are chosen to

achieve a Hopf bifurcation and an equivalent cubic damping coefficient a3 is 41.7. ................. 159

Figure 6.1. Diagram of a hair bundle. ......................................................................................... 163

Figure 6.2. Data showing the nonlinear stiffness of a hair bundle from a bullfrog saccular hair

cell. Image from Martin, et al (2000). Copyright (2000) National Academy of Sciences, U.S.A

[62]. ............................................................................................................................................. 163

Figure 6.3. Diagram of a hair cell’s adaptation motor. ............................................................... 164

Figure 6.4. Frequency response functions of several example, linear, third-order systems. The

system parameters are listed in the legend key. Note bf is set to one for simplicity. ................. 177

Figure 6.5. Magnitude and phase of the FRF at resonance ( ) as a function of the third pole

ρ. Here ξ = 0.1 and = 1............................................................................................................ 178

Figure 6.6. Comparison of FRFs computed by harmonic balance method (HBM) and by ode45

numerical solution. Both methods show nearly identical results. Here β = 1, ζ = 0.1, = 0, and

F = 1x10-2

. ................................................................................................................................... 181

Figure 6.7. Harmonic balance method results of an active hair bundle. Here β = 1, ζ = 0.1, and

= 0. .............................................................................................................................................. 182

Figure 6.8. (a) Input-to-output relationship and (b) backbone curve with β = 1 and = 0. The

model shows a compressive nonlinearity with a slope of 0.33 dB/dB and a small deviation of the

resonance frequency with response amplitude. .......................................................................... 183

Figure 6.9. FRF for the active hair bundle model for varying β. The forcing input (bf F) was kept

at 1x10-3

. ..................................................................................................................................... 184

Figure 6.10. Numerical results of an active hair bundle sensor tuned to a new resonance

frequency of = 2 . Here β = 1, ζ = 0.1, and = 0. ................................................................ 185

xix

Figure 6.11. Frequency response functions of a monostable active hair bundle with = 2 and

β = 1. As before, ζ = 0.1, and = 0. ............................................................................................ 188

Figure 7.1. Example signals and their frequency spectra. (a) Example signals as a function of

time. (b) Magnitude of the Fourier transforms of the signals in (a). (c) Example signals of the

form sin(2πt)+THD sin(6πt). (d) Magnitude of the Fourier transforms of the examples in (c). 208

Figure 7.2. Example of computing the settling time. .................................................................. 210

Figure 7.3. Frequency response functions for different active AHC cases. Here the input

amplitude (bf F) was 1x10

-3. The model parameters re set such that the systems are tuned to the

bifurcation (a1 = 2ζ) and have an equivalent cubic damping coefficient (a3) of 41.7. A linear

system, Equation 7.2 with no control force, is also shown for comparison. Note the cubic

damping and sigmoidal damping cases overlap in these plots. .................................................. 211

Figure 7.4. Input-to-output relationship at resonance ( = 1) for the different active artificial hair

cell cases. The cubic damping and active hair bundle models overlap. The plots for the

sigmoidal damping and two organ of Corti (OoC) cases also overlap. ...................................... 212

Figure 7.5. Input-to-output relationship at resonance ( = 1) for active AHC models that have

been mistuned (μ ≠ 0). ................................................................................................................ 213

Figure 7.6. Displacement of the basilar membrane versus sound pressure level. Figure from

Johnstone et al. (1986), reproduced with permission of Elsevier Limited [40]. ....................... 214

Figure 7.7. Maximum control force (buU) for the active AHC cases driven at resonance ( = 1).

As with the input-to-output curve, the cases with saturating nonlinearity (sigmoidal damping and

the two OoC cases) require the same maximum output force. ................................................... 215

xx

Figure 7.8. Total harmonic distortion (THD) in decibels for the active AHC cases driven at

resonance ( = 1). The THD for the cases with saturating control forces nearly overlap. For

comparison, a line for 2% (-34 dB) distortion is also shown...................................................... 216

Figure 7.9. 1% settling time in cycles (Ns) for the active AHC cases driven at resonance ( = 1).

..................................................................................................................................................... 217

Figure 8.1. A simple circuit for self-sensing, piezoelectric actuators. ........................................ 226

Figure A.1. Cantilever beam with piezoelectric elements (actuators or sensors). ...................... 255

Figure A.2. Forces and moments acting on a differential element of a beam. ........................... 255

Figure A.3. Deformation of a differential element of a beam. ................................................... 255

Figure A.4. Cross-section of the composite beam. The arrows over the piezoelectric elements

indicate the direction of polarity. ................................................................................................ 259

Figure A.5. “Split” configuration bimorph beam. Arrows on the piezoelectric actuator and

sensor indicate the polarization direction. .................................................................................. 269

Figure B.6. Element of a beam between nodes xi and xi+1. ......................................................... 283

Figure B.7. Shape functions for an Euler-Bernoulli beam. ......................................................... 284

Figure D.1. Experimental setup for acoustic excitation of the piezoelectric bimorph. ............. 306

Figure D.2. Frequency response function (FRF) for the bimorph under acoustic excitation. (a)

FRF in dB over 100 Hz to 10,000 Hz frequency sweep. (b) FRF around the first natural

frequency (shown here in a linear scale). .................................................................................... 306

Figure D.3. (a) Schematic and (b) photograph of the experimental setup for generating tuning

curves for acoustic excitation...................................................................................................... 308

Figure D.4. PID control algorithm for generating tuning curves. ............................................... 309

xxi

Figure D.5. Tuning curves from the PZT artificial hair cell. The left column shows curves of

constant velocity, and the right column plots curves of constant voltage. The bottom row shows

the tuning curves in the top row normalized by the response level. ........................................... 310

Figure D.6. PVDF bending sensor. ............................................................................................ 311

Figure D.7. Tuning curves from the PVDF artificial hair cell. (a) Velocity tuning curve. (b)

Velocity tuning curve normalized by the velocity level. ............................................................ 312

Figure D.8. Comparison of tuning curves from (a) PZT AHC sensor and (b) biological

measurements from a guinea pig cochlea. Biological tuning curves are from Sellick et al.

(1982), with permission of J. Acoust. Soc. Am. [135]. ............................................................... 313

xxii

List of Tables

Table 3.1. Parameters for the two mode example in Figure 3.10. These parameters are based on

the small scale AHC in Chapter 4. ................................................................................................ 77

Table 4.1. Properties of the proof-of-concept AHC setup. ........................................................... 88

Table 4.2. System parameters for the proof-of-concept AHC based on finite element (FE) and

system identification (ID) modeling techniques. Damping for the FE model was assumed from

the curve-fit data. .......................................................................................................................... 91

Table 4.3. Curve fit parameter values from the model and the data of the proof-of-concept AHC

results in Figure 4.5. Only data for disturbance amplitudes between 5 and 40 V were used. The

units of c are mm/s/Vk where k is the compression value in the table. ......................................... 96

Table 4.4. Properties of the artificial hair cell in water setup. ................................................... 102

Table 4.5. System identification results for the sensor in air and in water. Most of the

parameters were estimated from a single degree of freedom fit from the velocity to control FRF.

The disturbance influence term d1 was estimated from the velocity to disturbance FRF. ........ 108

Table 4.6. System parameters for the AHC accelerometer based on FE model and system

identification modeling techniques. ............................................................................................ 115

Table 4.7. Curve fit parameter values from the AHC accelerometer data in Figure 4.21a. The

units of c are mm/s/gk where k is the compression value in the table. ........................................ 119

Table 5.1. Equivalent cubic velocity coefficients for first-order tectorial membrane systems in

the feedback path. ....................................................................................................................... 152

Table 5.2. Equivalent cubic velocity coefficients for second-order tectorial membrane systems.

..................................................................................................................................................... 156

Table 7.1. Summary of active artificial hair cell (AHC) models. ............................................... 220

xxiii

Table D.1. PID gains for input frequency for the PZT sensor. ................................................... 309

1

Chapter 1. Introduction and Literature Review

This chapter begins by outlining the mechanisms behind hearing in vertebrates and the

research into mimicking these processes. Next the motivation for active artificial hair cells is

provided. This chapter concludes by highlighting the major contributions of this work and by

providing an outline of this dissertation.

1.1. Introduction and Research Motivation

Biology has provided inspiration for a number of technologies. Engineers are turning to

nature for solutions to difficult problems in locomotion, material design, signal processing,

sensor design, control, and a host of other fields [1-3]. Hearing is one biological mechanism

which has seen research interest in the past few decades. Several sophisticated components work

together to give mammals the ability to detect a remarkable range of frequencies and sound

pressure levels. Healthy human ears can detect a range of frequencies between 20 Hz and 20,000

Hz [4, 5]. Whales and some species of bats have hearing ranges as high as 200,000 Hz [6].

Humans can detect sound pressure levels as low as 0 dB (20 μPa RMS sound pressure) and up to

120 dB (20 Pa RMS) before severe pain occurs. Other mammals, such as cats, can hear as low

as -15 dB (4 μPa RMS) [4].

There has been research interest aimed at developing bio-inspired devices which could

one day replace damaged components of the ear [7-9]. The cochlea is the spiral-shaped portion

of the mammalian auditory system responsible for transducing sound into electrical signals. The

cochlea is fully developed upon birth, and its constitutive components are not repaired after they

2

are damaged [10]. Loss of cochlear hair cells can cause a profound loss in hearing [11-14]. By

replacing these hair cells with engineered, artificial hair cells could provide some recovery of

auditory function.

The cochlea and its hair cells have inspired a number of novel sensor designs. As

discussed in the next sections, the cochlea is able to detect minute, sound-induced vibrations

comparable to the Brownian motion of atoms [15]. The cochlea decomposes complex sounds

into its different frequency components like a Fourier analyzer before sending its electrical

signals to the brain. Researchers would like to produce acoustic, fluid flow, orientation, and

vibration sensors which mimic the cochlea’s incredible sensitivity to small input levels and high

frequency selectivity.

1.2. The Auditory Periphery

This section produces an overview of the anatomy and physiology of the auditory

periphery (the outer, middle, and inner ears) in mammals. These components work together to

transmit sound information to the cochlea where it is transduced into electrical signals sent to the

brain. While the central nervous system and the brain form an important component of how

mammals perceive sound, they are excluded from the discussion presented here (see Schnupp et

al. for more information [16]). More detailed information about hearing in mammals and other

animals can be found in [4-6, 16-20].

3

1.2.1. Overall Structure

The auditory periphery is often divided into three regions: the outer, the middle, and the

inner ear [4, 5]. Figure 1.1 shows a diagram of the middle and inner ear. The outer ear consists

of the pinna and the ear canal (not shown in Figure 1.1). The pinna (also known as the auricle) is

the portion of the ear on the outside of the head and is responsible for directing sound into the ear

canal. Sound waves travel down the ear canal to the tympanic membrane (also known as the ear

drum). The middle ear consists of three connected bones called the ossicles (the malleus, the

incus, and the stapes) in a space of air called the tympanic cavity. The malleus connects to the

tympanic membrane while the stapes connects to the oval window, a small membrane on the

cochlea. The ossicles and their supporting ligaments transmitted sound from the tympanic

membrane to the oval window. The middle ear structure provides a pressure amplification due to

the lever action of the ossicles and the decrease in area from the tympanic membrane to the

smaller oval window. The resulting twenty-fold increase in pressure from the tympanic

membrane to the oval window creates an acoustic impedance matching scheme [19]. This

allows for a more efficient transfer of sound waves from the air in the ear canal to the fluid in the

cochlea. In addition, ligaments connected to these bones will tension under loud sound pressure

levels [4]. This acoustic reflex attenuates sound transmission to the cochlea and offers some

protection to the cochlea from dangerous sound pressure levels. The Eustachian tube connects

the middle ear to the nasal passages and aids equalizing the pressure between the middle ear and

atmosphere and in draining mucus from the middle ear.

4

Figure 1.1. Diagram of the ear. From Dallos (1992),with permission of The Journal of

Neuroscience [21].

The inner ear consists of the cochlea and the vestibular system. The vestibular system is

responsible for determining linear and rotational accelerations. These aspects are important for

balance and sensing spatial orientation. The vestibular system consists of the semicircular canals

(which determine rotational accelerations) and the otolithic organs (which sense linear

accelerations). The spiral portion of the inner ear is the cochlea. The cochlea is responsible for

transducing the sound-induced vibrations into electrical signals.

Figure 1.2 shows a schematic of the cochlea uncoiled. In humans the cochlea makes a

little more than 2.5 turns. The human cochlea is about 35 mm long and has a radius around 1

mm [6, 19, 22]. Sound-induced vibrations of the stapes push on the oval window and generate

pressure waves in the cochlear fluid. The cochlea is divided by two membranes (the basilar

membrane and the Reissner’s membrane) to form three fluid-field chambers (the scala tympani,

the scala media, and the scala vestibuli). At the apex of the spiral the scala tympani and the scala

5

vestibuli merge at a location called the helicotrema. The scala tympani and scala vestibuli

contain an ionic fluid called perilymph which is low in potassium ions and high in sodium ions

[23]. The scala media (also called the cochlear duct) contains endolymph, another ionic fluid

which is high in potassium ions and low in sodium ions.

Figure 1.2. Simplified schematic of the cochlea. Here the spiral has been “unrolled” for

visual clarity. From Dallos (1992),with permission of The Journal of Neuroscience [21].

The width of the basilar membrane increases along the length of the cochlea from 0.1 mm

at the base to 0.4 mm at the apex [6, 19, 22]. Its thickness decreases from 13 μm at the base to 5

μm at the apex. Stiffness measurements of the basilar membrane also indicate that the constitute

fibers of the membrane are stiffer toward the base of the cochlea compared to the apex [22, 24,

25]. The changing geometry and fiber stiffness give the basilar membrane a spatially varying

stiffness. The result is that a tone of a particular frequency will induce a larger amplitude of

vibration in certain location than elsewhere. Thus the frequency of the stimulus can be mapped

to a particular location along the length of the cochlea. Low frequencies cause larger vibrations

near the apex of the cochlea while high frequencies induce larger vibrations near the stapes. This

6

tonotopic mapping allows the cochlea to decompose complex signals into its frequency

components and allows the ear to distinguish between different frequencies. Researchers have

shown that this mechanism is highly efficient at performing Fourier analysis on complex signals

compared to traditional discrete and fast Fourier transform algorithms [26, 27].

Another byproduct of tonotopic mapping is the appearance of a traveling wave along the

basilar membrane. When a pure tone is applied, the magnitude and phase of the basilar

membrane create the appearance of a traveling wave moving from the stapes toward the apex.

This traveling wave peaks in a region whose location is a function of the stimulus frequency.

There is still some debate whether the cochlea experiences a true traveling wave (which carries

energy along the length of the cochlea) or if this is a pseudo-traveling wave with propagation of

energy [28, 29].

Figure 1.3 shows a diagram of the interior of the cochlea. Sitting on the basilar

membrane inside the scala media is a collection of cells called the organ of Corti. The organ of

Corti and the basilar membrane together are referred to as the cochlear partition as they are the

main structural division in the cochlea. Reissner’s membrane is believed to have little influence

on the propagation of waves inside the cochlea and mainly serves to separate the endolymph in

the scala media from the perilymph of the scala vestibuli [4, 16]. The endolymph is generated by

a collection of the blood vessels in the stria vascularis.

7

Figure 1.3. Cross-section of the organ of Corti on the basilar membrane. From Raphael

and Atlschuler (2003),with permission of Elsevier Limited [30].

The organ of Corti contains a set of sound-sensing cells called hair cells (the next section

provides more details on these cells). The tectorial membrane overlays the organ of Corti. The

tips of the stereocilia of the outer hair cells are connected to the tectorial membrane, while the

stereocilia of the inner hair cells are not connected. The pillar cells, the Deiter’s cells, and the

Hensen’s cells serve various roles in supporting the other cells and giving rigidity to the overall

structure. As discussed in the next section, when fluid motion causes the stereocilia of the inner

hair cells to deflect, the inner hair cells stimulate neighboring afferent neurons which in turn

generates a neural spike, or action potential. The bodies of these afferent neurons rest in the

spiral ganglion which sits inside a conical shaped medulla at the center of the cochlea’s spiral.

These neural spikes propagate through the neurons of the vestibulocochlear nerve to the brain.

8

1.2.2. Inner and Outer Hair Cells

Mammals possess two types of hair cells: inner hair cells and outer hair cells. Humans

have around 32,000 hair cells in the two cochleae (about 8,000 inner hair cells and 24,000 outer

hair cells) [6]. The body (or soma) of outer hair cells is around 20 μm tall for cells near the basal

end of the cochlea and around 50 μm tall for cells at the apex [19]. The size of the inner hair

cells has more variation, but they are generally longer near the apex than at the basal end of the

cochlea. Both types of hair cells possess bundles of 20 to 300 stereocilia. Each stereocilia has a

diameter around 0.2 μm for most of its length, but tappers to less than 0.05 μm at its root in the

hair cell. Stereocilia lengths vary between 2 μm and 6 μm. The stereocilia are hexagonally

packed into a V-shaped pattern, and the height of the stereocilia varies linearly across the bundle

(see Figure 1.4). This geometry and the cross-links between stereocilia cause the bundle to be

more compliant along the axis of symmetry than in the orthogonal direction. Filament structures

called tip links connect the tips of adjacent stereocilia in the bundle. The tip links are connected

at one or both ends to ion channels which open when the tip links are under tension. The

basolateral side of the hair cells (the portion of the cell wall at the base and sides of the cell) is

bathed in perilymph with a resting electric potential around 0 mV [21]. The interior of cell body

has resting potential between -40 mV for the inner hair cells and -70 mV for the outer hair cells.

The stereocilia is surrounded by endolymph in the scala media with a resting potential around 80

mV to 100 mV. The result is a voltage difference between 120-160 mV between the endolymph

around the stereocilia and interior of the cell. Upon deflection of the hair cell bundle toward the

tallest stereocilia, the tip links are stretched, the ion channels open, and the voltage difference

drives positively charged potassium ions into the cell. Increasing deflection toward the largest

9

stereocilia further increases the influx of ions into the cell. Deflection away from the tallest

stereocilia causes these channels to close and decreases the flow of ions into the cell. This influx

of ions (i.e. current) increases the voltage inside the cell. When a constant displacement is

applied to the stereocilia, the current into the cell slowly decreases, or adapts, to a lower level.

This adaptation process is thought to result from the action of myosin motors which adjust the

tension in the tip links [20, 31]. Measurements show this adaptation process is dependent on the

concentration of calcium ions around the hair cells [18, 32].

Figure 1.4. Rows of hair cell stereocilia. From Raphael and Atlschuler (2003),

reproduced with permission of Elsevier Limited [30].

Mammals possess two types of hair cells: inner hair cells and outer hair cells. Figure 1.5

shows schematics of these hair cell types. The two types of hair cells are characterized by their

location in the cochlea and their function. The inner hair cells are aligned in a single row along

toward the inside of the cochlea’s spiral, while the outer hair cells are arranged in three rows

10

further away from the center of the spiral. For the inner hair cells, the depolarization caused by

hair bundle deflection opens voltage-gated calcium channels, which in turn activates a release of

glutamate, a neurotransmitter, at the base of the cell. These neurotransmitters cross the space

between the hair cell and a nearby afferent nerve cell, trigger a depolarization of the afferent

neuron, and starts an electrical nerve signal which propagates along the auditory nerve to the

brain. The resulting electrical pulses encode information about the intensity, duration, and

frequency of the resulting mechanical stimulus. Therefore, the inner hair cell serves to transduce

mechanical induced vibration into electrical signals. Increasing stereocilia deflection toward the

tallest stereocilia increases the rate of glutamate release, which increases the firing rate of the

spiral ganglion cells. If a particular section of the basilar membrane vibrates more than the rest,

then there are more neural spikes generated from that region. In this manner, the neurons encode

sound intensity at a particular frequency into a neural firing rate.

Figure 1.5. Diagrams of the inner and outer hair cells. From Dallos (1992),with

permission of The Journal of Neuroscience [21].

11

Outer hair cells do not stimulate afferent neurons like their inner hair cell counterparts.

The outer hair cell possesses a motor protein called prestin embedded in their cell walls. When

there is a voltage change across the cell’s membrane, the prestin protein changes configuration

[33]. This process causes the cell wall to deform and the body of the outer hair cell to contract.

These forces push on the basilar membrane and the tectorial membrane. The result is that the

outer hair cells function like actuators by producing mechanical forces upon application of a

voltage change. This process is referred to as somatic motility or electromotility and serves to

boost sound-induced vibration. This increased vibration in turn causes larger deflections of the

inner hair cell stereocilia and thus amplifies the perception of weak sound pressure levels. While

outer hair cells are unique to mammals, other animals have also evolved a secondary type of hair

cell [34-37]. These animals have a high sensitivity and an increased frequency range than those

with just sensing hair cells.

The inner and outer hair cells are innervated by afferent and efferent neurons. However,

the inner hair cells are predominately innervated by afferent neurons which transmit electrical

signals to the brain. The outer hair cells are innervated primarily by efferent, or motor, neurons.

Research shows that these efferent neurons cause an inhibitory effect on the outer hair cells

under the presence of loud sounds [18]. The result is that the brain can “turn down” the gain of

the cochlear amplifier when it detects high sound pressure levels.

12

1.3. The Cochlear Amplifier

This section begins by providing a brief history of the discovery and importance of the

cochlear amplifier. Next some important characteristics of the cochlear amplifier are discussed

and how they aid in sound detection.

1.3.1. History of the Cochlear Amplifier

In 1928, Georg von Békésy performed some of the first experiments on the cochleae of

human cadavers [24, 38]. He applied a silver speckle pattern to the basilar membrane and

observed its motion using a strobe light. While he played tones through a loud speaker, von

Békésy observed a traveling wave moving longitudinally along the basilar membrane. The

amplitude of the traveling wave reached a maximum at a position along the length of the basilar

membrane dependent upon the frequency of the tone. Because of the limited sensitivity of

optical measurements from that era, von Békésy had to apply sound pressures greater than 100

dB in order to detect the displacement of the basilar membrane [4]. von Békésy’s work earned

him the 1961 Nobel Prize in Physiology or Medicine.

However, the early work of von Békésy and others with cadavers could not explain the

high pressure sensitivity and sharp frequency tuning seen in responses from auditory nerve fibers

of living mammals [39]. In addition to the “first filter” of BM’s traveling wave, a “second filter”

was proposed to aid in frequency selectivity of the cochlea. While the first filter was

mechanical, the second filter was thought to be electrical in nature. However theories of how the

proposed second filter works were based on passive filtering and could not account for the

13

observed responses of the healthy cochlea. Measurements inside the cochlea suggested an active

mechanical feedback was at work [40].

The idea of an active mechanism inside the cochlea was first suggested by T. Gold in

1948 [41]. He proposed the “regeneration hypothesis” in which an active process was present in

the cochlea to counteract viscous damping from the cochlear fluids. The idea is named after

“regenerative receivers” used in radio engineering to create positive feedback to counteract

resistive losses which would normally limit frequency selectivity. However Gold’s work was

largely forgotten until experimental work in the 1970’s demonstrated the passive “second filter”

was inadequate for describing the neural responses in mammals [39]. While an electrical

filtering process is known to exist in the cochleae of turtles, it is widely believed not to be

present in mammals [4, 18].

As technology advanced, it became possible to make measurements inside the cochlea.

In 1971 Rhode was able to use the Mössbauer effect to measure basilar membrane vibrations of a

living squirrel monkey [42, 43]. The results showed the vibrations underwent compressive

nonlinearity, i.e. vibrations grew with increasing sound pressure at a rate of less than 1 dB/dB.

This nonlinearity occurred only for frequencies near the characteristic frequency and disappeared

upon the death of the monkey. In Rhode’s 1978 work, he indicated that the basilar membrane

was sharply tuned at low sound pressure levels and was poorly tuned at high sound pressures [4,

44].

In the late 1970’s, Kemp was the first to record acoustic emissions from the human ear

[45]. He detected these acoustic emissions around 10 milliseconds after an impulsive acoustic

excitation (a “click”). These sounds are now referred to as “transient-evoked otoacoustic

emissions” (TEOAE) [19]. Kemp noted that the acoustic emissions demonstrated compressive

14

nonlinearity characteristics (the response amplitude did not scale linearly with input level), and

these emissions were absent in deafened ears. He theorized these acoustic emissions were

byproducts of an active mechanism in the cochlea. In 1979, Kemp made measurements of tones

emitted from the ear without external stimulus [4, 45]. These spontaneous otoacoustic emissions

(SOAE) are also considered to be byproducts of the feedback processes. In 1980, Mountain

showed that the distortion product form of the evoked otoacoustic emissions could be shifted by

changing the endocochlear potential or by stimulating the crossed olivocochlear bundle in the

brain stem [4]. Thus the OHC’s were linked to frequency tuning in the neural and mechanical

responses of the cochlea.

1.3.2. Characteristics of the Cochlear Amplifier

The cochlear amplifier has four important characteristics: amplification, frequency

sensitivity, compressive nonlinearity, and spontaneous oscillations [20]. The first three

characteristics can be summarized in data obtained shown in Figure 1.6 and Figure 1.7. Figure

1.6a plots the basilar membrane displacement in a guinea pig as a function of driving frequency

and sound pressure level. Figure 1.6b the basilar membrane displacement from Figure 1.6a

divided by the driving sound pressure level. Figure 1.7 shows the basilar membrane amplitude at

the characteristic frequency (14 kHz for this data) as a function of sound pressure level. These

plots show the basilar membrane responds linearly except near the resonance or characteristic

frequency. For low sound pressure levels, the cochlear amplifier boosts the response over a

narrow range of frequencies, creating a sharp resonance peak at the characteristic frequency.

This amplification of small inputs is evident in Figure 1.6b and Figure 1.7. The characteristic

15

frequency depends on the measurement location along the cochlea. For a given location along

the basilar membrane, the frequency which causes the largest deflection is called the

characteristic frequency. Because of the tonotopic nature of the cochlea, one can also define the

characteristic place as the location of largest deflection for a given stimulus frequency.

(a) (b)

Figure 1.6. Measurements showing the cochlear amplifier in a guinea pig cochlea. (a)

Basilar membrane (BM) displacement of versus frequency and sound pressure level. (b)

Basilar membrane displacement normalized by the input sound pressure level. These

curves would overlap for a linear system. Figure from Johnstone et al. (1986),

reproduced with permission of Elsevier Limited [40].

16

Figure 1.7. Displacement of the basilar membrane versus sound pressure level. The

displacement shows a linear trend at low and high sound pressures levels and a nonlinear

compression at intermediate sound pressure. Figure from Johnstone et al. (1986),

reproduced with permission of Elsevier Limited [40].

This amplification allows the ear to detect low sound pressures. At the threshold of

auditory deflection (0 dB or 20 μPa RMS sound pressure level), the deflection of the basilar

membrane is around 0.3 nm [40, 44, 46]. For comparison, the diameter of a hydrogen atom is on

the order of 0.1 nm; thus mammals are able to detect vibrations comparable to the thermal noise

[20]. This amplification only occurs in living creatures. A few minutes after death, the threshold

of auditory response increases by 40 to 60 dB (i.e. sensitivity falls to less than 1% of that of the

living cochlea) [47]. The narrow bandwidth of the peak allows the cochlea to detect small

changes in frequency, which gives the cochlea an improved frequency selectivity. This

frequency discrimination is vital for comprehension for speech and music [48].

17

It is important to note this amplification is nonlinear in nature. As also shown in Figure

1.6 and in Figure 1.7, the displacement at the characteristic frequency increases at a less-than-

proportional rate with increasing sound pressure [18]. Humans have a threshold of sound

detection around 0 dB or 20 μPa RMS sound pressure level. This corresponds to a basilar

membrane oscillation around 0.3 nm in amplitude. At the threshold of pain (120 dB or 20 Pa

RMS), the basilar membrane of many mammals oscillates at an amplitude on the order of 100

nm [18, 49]. Thus the nonlinearity compresses a large sound pressure range into a smaller range

of response amplitudes. This compression can be seen in Figure 1.6a and Figure 1.7.

Compressive growth rates between 0.12 to 0.5 dB/dB have been seen in measurements from

various mammals [18, 49, 50]. This compressive nonlinearity allows the cochlea to shrink a 120

dB dynamic range of sound pressure levels into a range around 30 to 40 dB of basilar membrane

displacement. Similar amplitude compression is observed in neuron firing rates and in

recordings of inter-cochlear voltage changes [19, 51]. In addition to the nonlinear growth of the

peak response, the bandwidth of amplification also increases as sound pressure level increases.

The final characteristic of the cochlear amplifier is spontaneous otoacoustic emissions.

Studies from several different animals show the ear can actively emit tones of distinct

frequencies [4, 45, 52, 53]. These tones are considered by many to be byproducts of the active

feedback process. In addition to spontaneous oscillations, when the cochlea is stimulated with

two or more tones, additional tones of different frequencies can be measured emanating from the

ear and can be heard by the listener [4, 54]. These distortion-product otoacoustic emissions are

also indicators of a nonlinear process. Spontaneous and distortion-product otoacoustic emissions

are altered or absent in damaged cochleae [45]. Thus these emissions can used be used as a

diagnostic tool to test hearing in newborns [54].

18

1.3.3. Mechanisms of Amplification

The exact processes behind the cochlear amplifier are still an issue of debate in the

auditory community [4, 55-59]. Several theories have been proposed over the years, but current

research focuses on two experimentally observed mechanisms: hair bundle motility and somatic

motility.

Hair bundle motility theories advocate that the stereocilia bundles protruding from the

hair cells contribute to the amplification of incoming sounds. Measurements of hair cells from

insects, turtles, and bullfrogs show compressive nonlinearity to an applied stimulus and

spontaneous oscillations [20, 60-63]. Hair bundle motility is believed to be created by a

combination of a nonlinear stiffness of the hair bundles (which has a negative stiffness region

around the vertical position of the hair bundle) and an adaptation process which readjusts the

tension in the tip links to force the hair bundles toward an unstable vertical position [62, 64-66].

These effects have also been seen in mammalian hair cells [67]. While this process likely

underlies active hearing in lower vertebrates, critics argue that active hair bundle motility lacks

the power required to drive the amplifications recorded in mammals [4, 56].

In mammals there is strong evidence that somatic motility of the outer hair cells

(previously mentioned in Section 1.2.2.) plays an important role in amplification. A change in

voltage across the cell membrane causes the cell body (the soma) to contract. In this manner, the

outer hair cell acts like a piezoelectric actuator producing a force under an applied voltage

change [68]. Also like a piezoelectric material, the outer hair cells show a measureable charge

displacement under an applied force [69]. These measurements indicate that isolated outer hair

cells have an effective piezoelectric coefficient around 20 μC/N, which is four orders of

19

magnitude larger than any man-made piezoelectric materials [69-71]. While this effect is

powerful enough to drive amplification in the mammalian cochlea, critics argue that somatic

motility behaves linearly for physiologically relevant voltage changes, and therefore somatic

motility alone cannot explain the nonlinear oscillations observed in the cochlea [56]. In addition,

the capacitance of the cell membrane and the conductance of the ionic fluid inside the cochlea

create a low-pass filter with a corner frequency around 1,000 Hz (significantly lower than the

20,000 Hz upper limit of human hearing) [21, 34, 72, 73]. This low pass filter on its own should

significantly attenuate high frequency oscillations.

Some researchers are now advocating that both hair bundle motility and somatic motility

are needed to explain the observed characteristics of the mammalian cochlea [20, 39, 56, 74].

The hair bundle motility creates a nonlinear input to the otherwise linearly behaved

electromotility of the outer hair cell. This in turn creates the nonlinear behavior and frequency

sensitivity of the cochlear amplifier. The somatic motility serves to boost amplification and aids

in detecting higher frequencies. Some propose that the membrane filtering problem which

plagues somatic motility alone could be overcome by its coupling with an active hair bundle [56,

75, 76].

1.4. Mimicking the Cochlea through Passive Devices

The cochlea has severed as inspiration for many researchers looking to develop new

types of sensors or auditory prosthetics. This section will examine several passive devices which

mimic the cochlea and its sound transduction. The next section will examine systems which

attempt to reproduce the cochlea’s nonlinear amplification.

20

1.4.1. Passive Artificial Hair Cells

A common, bio-inspired approach to designing a flow or acoustic sensor is to mimic the

cochlear hair cells. One common approach to creating an artificial hair cell is to treat the hair

bundle like a single cantilevered beam. Flow- or acoustic-induced vibration of a cantilever beam

can be transduced into an electrical signal through a variety of methods. The natural frequency

of a beam is dependent on the material properties, length, width, and tip mass. An array of

beams of varying geometries and tip masses can mechanically filter the inducing force into a set

of frequency sub-bands much like the basilar membrane. Simplicity of design, low power

requirements, and their small size make these artificial hair cells an appealing approach for many

researchers.

Researchers have used a variety of transduction methods to transform the mechanical

vibration of these hair cells into an electrical signal. One method is to use piezoelectric

materials. Piezoelectric materials produce an electrical charge when subjected to an applied

mechanical stress. These materials are attractive to many researchers because of the possibility

of developing a self-powered sensor which does not need an external power source.

In late 1990s and early 2000s, Mukherjee and colleagues examined the feasibly of using

polyvinylidene fluoride (PVDF) cantilever beams to build cochlear implants [7, 77-79]. For

acoustic sensing, piezoelectric polymers like PVDF offer several advantages over ceramic

piezoelectric materials, such as a greater range of motion, higher voltage sensitivity, higher

dielectric breakdown voltages, better acoustic impedance matching with air and water, and easier

manufacturing [78, 80]. Experimental work of their beams showed a sensitivity up to 447 μV/Pa

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for underwater testing, which is comparable to some commercially available hydrophones [81].

However the authors were skeptical whether an unamplified signal from an acoustically excited

piezoelectric material could generate a sufficient electrical impulse to stimulate the neurons in

the spiral ganglion.

Hur et al. fabricated and characterized an array of cantilever beams constructed from

single crystal lead magnesium niobate-lead titanate (PMN-PT) [82, 83]. The single crystal

PMN-PT is a newer type of piezoelectric material which offers higher sensitivities and lower

dielectric losses than PVDF or the traditional lead zirconium titanate (PZT) piezoceramic

material. The authors applied a 94 dB (1 Pa) constant sound pressure to the array of these PMN-

PT cantilever beams with lengths varying from 3000 μm to 500 μm. The maximum voltage

output from a single cantilever was 80.4 mV for the longest beam, while the smaller beams

produced less voltage. At higher frequencies, the higher natural frequencies of the longer beams

could be observed. These frequencies were lower than the first natural frequencies of the shorter

beams. Without filtering the signals, excitation of higher modes would complicate the

frequency-spatial decomposition. This problem occurs for other arrays of beam-like sensors

throughout the literature.

Kim et al. examined several arrays of piezoelectric beams with narrow supports [8].

Each array consisted of 16 beams that were 200 μm wide and spaced 400 μm apart. The beam

lengths were varied between 305 and 3200 μm, and the array showed frequency selectivity

between 2-20 kHz. The system was excited electrically and acoustically and the response was

measured using a scanning laser-Doppler vibrometer. Sound pressures between 105-110 dB

were used to acoustically excite the system, however the output voltage levels from the beams

during this testing was not stated.

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Other researchers have developed MEMS-scale hair sensors which utilize piezoresistance

[3, 84-90]. Piezoresistance is the property of a material to change resistance when strained. This

method is commonly used in developing strain gauges. For these artificial hair cells, the

deflection of a hair-like structure deformed a piezoresistor and created a change in its resistance.

Like their larger strain gauge counterparts, these devices required an external current source in

order to measure the change in resistance of the piezoesistor under flow-induced strain.

Several sensors have been constructed using a variable capacitor design [91-93]. Here a

hair-like structure was connected to one side of a parallel plate capacitor. Deflection of the hair

changed the separation distance of the electrode plates. The result was a time-varying

capacitance which under a fixed voltage resulted in a change in current through the capacitor.

This change in current was translated into an applied deflection on the hair structure.

Leo, Sarles, and colleagues have examined artificial cell membranes formed by lipid

bilayers [94-96]. A hair-like structure was inserted into a water-swollen, lipid-encased hydrogel

droplet. A liquid bilayer was formed with a neighboring lipid-encased aqueous volume. Air

flow forced the hair and the bilayer to vibrate. This vibration varied the curvature of the lipid

bilayer and resulted in a change in the membrane’s capacitance. This time-varying capacitance

generated several picoamperes of current across the membrane. These sensors used soft

materials commonly found in most cells as opposed to the artificial materials found in other

designs. This allows for a more biocompatible artificial hair cell. While the transduction

mechanism is different than that of the natural hair cells, the composition of their artificial

membranes allows for the possibility of adding ion channels or other molecules to modify or

enhance transduction in ways similar to mammalian hair cells.

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1.4.2. Passive Artificial Basilar Membr