development of active artificial hair cell sensors€¦ · figure 1.2. simplified schematic of the...
TRANSCRIPT
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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
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Table D.1. PID gains for input frequency for the PZT sensor. ................................................... 309
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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
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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].
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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.
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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
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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
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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.
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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.
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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
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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
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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].
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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.
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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
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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
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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
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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].
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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].
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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].
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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
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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.
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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