Stochastic Resonance: From Suprathreshold Stochastic Resonance to Stochastic Signal Quantization

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Stochastic resonance occurs when random noise provides a signal processing benefit, and has been observed in many physical and biological systems. Many aspects have been hotly debated by scientists for nearly 30 years, with one of the main questions being whether biological neurons utilize stochastic resonance. This book addresses in detail various theoretical aspects of stochastic resonance with a focus on its extension to suprathreshold stochastic resonance, in the context of stochastic signal quantization theory. Models in which suprathreshold stochastic resonance occur support significantly enhanced "noise benefits", and exploitation of the effect may prove extremely useful in the design of future engineering systems such as distributed sensor networks, nano-electronics, and biomedical prosthetics.
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Editorial Reviews

From the Publisher
"… the joint authors of this book adopt a pragmatic approach to these essentially semantic issues and point to the widespread acceptance and use of terminology as a justification for giving attention to more substantive issues associated with understanding and using stochastic resonance in its several forms. … the authors have rendered a major service to a multi-disciplinary research community which can benefit from the clarity of exposition offered in this book. It is to be hoped that that community will repay its debt to these authors by taking forward wherever possible applications of stochastic resonance."
Contemporary Physics
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Product Details

  • ISBN-13: 9780521882620
  • Publisher: Cambridge University Press
  • Publication date: 10/27/2008
  • Pages: 448
  • Product dimensions: 7.00 (w) x 9.80 (h) x 1.00 (d)

Meet the Author

Mark D. McDonnell is a Research Fellow at the Institute for Telecommunications Research, University of South Australia. Prior to this, he was at The University of Adelaide. His research interests are in the field of nonlinear signal processing, and bio-inspired engineering.

Nigel G. Stocks is a Professor in the School of Engineering at Warwick University. His research lies in stochastic nonlinear systems and biometrics.

Charles E. M. Pearce is Thomas Elder Professor of Mathematics at the School of Mathematics, University of Adelaide, and is editor of several journals.

Derek Abbott is a Professor in the School of Electrical and Electronic Engineering at the University of Adelaide. He has received several awards and has been editor or guest editor on a number of journals.

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Table of Contents

List of figures x

List of tables xiv

Preface xv

Foreword xvii

Acknowledgments xix

1 Introduction and motivation 1

1.1 Background and motivation 1

1.2 From stochastic resonance to stochastic signal quantization 3

1.3 Outline of book 4

2 Stochastic resonance: its definition, history, and debates 6

2.1 Introducing stochastic resonance 6

2.2 Questions concerning stochastic resonance 9

2.3 Defining stochastic resonance 10

2.4 A brief history of stochastic resonance 14

2.5 Paradigms of stochastic resonance 20

2.6 How should I measure thee? Let me count the ways... 30

2.7 Stochastic resonance and information theory 34

2.8 Is stochastic resonance restricted to subthreshold signals? 39

2.9 Does stochastic resonance occur in vivo in neural systems? 45

2.10 Chapter summary 46

3 Stochastic quantization 47

3.1 Information and quantization theory 47

3.2 Entropy, relative entropy, and mutual information 48

3.3 The basics of lossy source coding and quantization theory 49

3.4 Differences between stochastic quantization and dithering 52

3.5 Estimation theory 58

3.6 Chapter summary 58

4 Suprathreshold stochastic resonance: encoding 59

4.1 Introduction 59

4.2 Literature review 61

4.3 Suprathreshold stochastic resonance 67

4.4 Channel capacity for SSR 101

4.5 SSR as stochastic quantization 112

4.6 Chapter summary 117

5 Suprathreshold stochastic resonance: large N encoding 120

5.1 Introduction 120

5.2 Mutual information when fx(x) = fη(θ - x) 125

5.3 Mutual information for uniform signal and noise 131

5.4 Mutual information for arbitrary signal and noise 135

5.5 A General expression for large N channel capacity150

5.6 Channel capacity for 'matched' signal and noise 158

5.7 Chapter summary 163

6 Suprathreshold stochastic resonance: decoding 167

6.1 Introduction 167

6.2 Averaging without thresholding 172

6.3 Linear decoding theory 174

6.4 Linear decoding for SSR 179

6.5 Nonlinear decoding schemes 195

6.6 Decoding analysis 206

6.7 An estimation perspective 213

6.8 Output signal-to-noise ratio 225

6.9 Chapter summary 230

7 Suprathreshold stochastic resonance: large N decoding 233

7.1 Introduction 233

7.2 Mean square error distortion for large N 234

7.3 Large N estimation perspective 241

7.4 Discussion on stochastic resonance without tuning 244

7.5 Chapter Summary 246

8 Optimal stochastic quantization 248

8.1 Introduction 248

8.2 Optimal quantization model 252

8.3 Optimization solution algorithms 258

8.4 Optimal quantization for mutual information 260

8.5 Optimal quantization for MSE distortion 268

8.6 Discussion of results 271

8.7 Locating the final bifurcation 286

8.8 Chapter summary 289

9 SSR, neural coding, and performance tradeoffs 291

9.1 Introduction 291

9.2 Information theory and neural coding 296

9.3 Rate-distortion tradeoff 309

9.4 Chapter summary 321

10 Stochastic resonance in the auditory system 323

10.1 Introduction 323

10.2 The effect of signal distribution on stochastic resonance 324

10.3 Stochastic resonance in an auditory model 330

10.4 Stochastic resonance in cochlear implants 344

10.5 Chapter summary 356

11 The future of stochastic resonance and suprathreshold stochastic resonance 358

11.1 Putting it all together 358

11.2 Closing remarks 360

Appendix 1 Suprathreshold stochastic resonance 362

A1.1 Maximum values and modes of Py/x(n/x) 362

A1.2 A proof of Equation (4.38) 363

A1.3 Distributions 363

A1.4 Proofs that fQ (τ) is a PDF, for specific cases 370

A1.5 Calculating mutual information by numerical integration 371

Appendix 2 Large N suprathreshold stochastic resonance 373

A2.1 Proof of Eq. (5.14) 373

A2.2 Derivation of Eq. (5.18) 374

A2.3 Proof that fs(x) is a PDF 375

Appendix 3 Suprathreshold stochastic resonance decoding 377

A3.1 Conditional output moments 377

A3.2 Output moments 378

A3.3 Correlation and correlation coefficient expressions 380

A3.4 A proof of Prudnikov's integral 382

A3.5 Minimum mean square error distortion decoding 385

A3.6 Fisher information 388

A3.7 Proof of the information and Cramer-Rao bounds 390

References 392

List of abbreviations 417

Index 419

Biographies 421

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