Mathematics for Neuroscientists (eBook)

Steven James Cox, Fabrizio Gabbiani (Autoren)

eBook Download: PDF

2010 | 1. Auflage
498 Seiten
Elsevier Science (Verlag)
978-0-08-089049-4 (ISBN)

Infos

This book provides a grounded introduction to the fundamental concepts of mathematics, neuroscience and their combined use, thus providing the reader with a springboard to cutting-edge research topics and fostering a tighter integration of mathematics and neuroscience for future generations of students. The book alternates between mathematical chapters, introducing important concepts and numerical methods, and neurobiological chapters, applying these concepts and methods to specific topics. It covers topics ranging from classical cellular biophysics and proceeding up to systems level neuroscience. Starting at an introductory mathematical level, presuming no more than calculus through elementary differential equations, the level will build up as increasingly complex techniques are introduced and combined with earlier ones. Each chapter includes a comprehensive series of exercises with solutions, taken from the set developed by the authors in their course lectures. MATLAB code is included for each computational figure, to allow the reader to reproduce them. Biographical notes referring the reader to more specialized literature and additional mathematical material that may be needed either to deepen the reader's understanding or to introduce basic concepts for less mathematically inclined readers completes each chapter.

• A very didactic and systematic introduction to mathematical concepts of importance for the analysis of data and the formulation of concepts based on experimental data in neuroscience


• Provides introductions to linear algebra, ordinary and partial differential equations, Fourier transforms, probabilities and stochastic processes


• Introduces numerical methods used to implement algorithms related to each mathematical concept


• Illustrates numerical methods by applying them to specific topics in
  neuroscience, including Hodgkin-Huxley equations, probabilities to describe stochastic release, stochastic processes to describe noise in neurons, Fourier transforms to describe the receptive fields of visual neurons


• Provides implementation examples in MATLAB code, also included for download on the accompanying support website (which will be updated with additional code and in line with major MATLAB releases)


• Allows the mathematical novice to analyze their results in more sophisticated ways, and consider them in a broader theoretical framework
  
  
  
  

---

Virtually all scientific problems in neuroscience require mathematical analysis, and all neuroscientists are increasingly required to have a significant understanding of mathematical methods. There is currently no comprehensive, integrated introductory book on the use of mathematics in neuroscience; existing books either concentrate solely on theoretical modeling or discuss mathematical concepts for the treatment of very specific problems. This book fills this need by systematically introducing mathematical and computational tools in precisely the contexts that first established their importance for neuroscience. All mathematical concepts will be introduced from the simple to complex using the most widely used computing environment, Matlab. This book will provide a grounded introduction to the fundamental concepts of mathematics, neuroscience and their combined use, thus providing the reader with a springboard to cutting-edge research topics and fostering a tighter integration of mathematics and neuroscience for future generations of students. - A very didactic and systematic introduction to mathematical concepts of importance for the analysis of data and the formulation of concepts based on experimental data in neuroscience- Provides introductions to linear algebra, ordinary and partial differential equations, Fourier transforms, probabilities and stochastic processes- Introduces numerical methods used to implement algorithms related to each mathematical concept- Illustrates numerical methods by applying them to specific topics in neuroscience, including Hodgkin-Huxley equations, probabilities to describe stochastic release, stochastic processes to describe noise in neurons, Fourier transforms to describe the receptive fields of visual neurons- Allows the mathematical novice to analyze their results in more sophisticated ways, and consider them in a broader theoretical framework

Inhalt

Front cover 1
Mathematics for Neuroscientists 4
Copyright page 5
Full Contents 8
Preface 12
Chapter 1. Introduction 14
1.1. How to Use This Book 15
1.2. Brain Facts Brief 15
1.3. Mathematical Preliminaries 17
1.4. Units 20
1.5. Sources 21
Chapter 2. The Passive Isopotential Cell 22
2.1. Introduction 22
2.2. The Nernst Potential 24
2.3. Membrane Conductance 25
2.4. Membrane Capacitance and Current Balance 25
2.5. Synaptic Conductance 27
2.6. Summary and Sources 28
2.7. Exercises 29
Chapter 3. Differential Equations 34
3.1. Exact Solution 34
3.2. Moment Methods* 36
3.3. The Laplace Transform* 38
3.4. Numerical Methods 40
3.5. Synaptic Input 41
3.6. Summary and Sources 42
3.7. Exercises 42
Chapter 4. The Active Isopotential Cell 46
4.1. The Delayed Rectifier Potassium Channel 47
4.2. The Sodium Channel 49
4.3. The Hodgkin–Huxley Equations 50
4.4. The Transient Potassium Channel* 53
4.5. Summary and Sources 56
4.6. Exercises 56
Chapter 5. The Quasi-Active Isopotential Cell 62
5.1. The Quasi-Active Model 62
5.2. Numerical Methods 64
5.3. Exact Solution via Eigenvector Expansion 67
5.4. A Persistent Sodium Current* 71
5.5. A Nonspecific Cation Current that is Activated by Hyperpolarization* 72
5.6. Summary and Sources 73
5.7. Exercises 74
Chapter 6. The Passive Cable 80
6.1. The Discrete Passive Cable Equation 80
6.2. Exact Solution Via Eigenvector Expansion 82
6.3. Numerical Methods 84
6.4. The Passive Cable Equation 86
6.5. Synaptic Input 91
6.6. Summary and Sources 94
6.7. Exercises 95
Chapter 7. Fourier Series and Transforms 100
7.1. Fourier Series 100
7.2. The Discrete Fourier Transform 102
7.3. The Continuous Fourier Transform 107
7.4. Reconciling the Discrete and Continuous Fourier Transforms 108
7.5. Summary and Sources 111
7.6. Exercises 111
Chapter 8. The Passive Dendritic Tree 116
8.1. The Discrete Passive Tree 116
8.2. Eigenvector Expansion 118
8.3. Numerical Methods 120
8.4. The Passive Dendrite Equation 123
8.5. The Equivalent Cylinder* 124
8.6. Branched Eigenfunctions* 126
8.7. Summary and Sources 128
8.8. Exercises 128
Chapter 9. The Active Dendritic Tree 132
9.1. The Active Uniform Cable 133
9.2. On the Interaction of Active Uniform Cables* 135
9.3. The Active Nonuniform Cable 138
9.4. The Quasi-Active Cable* 143
9.5. The Active Dendritic Tree 147
9.6. Summary and Sources 149
9.7. Exercises 149
Chapter 10. Reduced Single Neuron Models 156
10.1. The Leaky Integrate-and-Fire Neuron 156
10.2. Bursting Neurons 159
10.3. Simplified Models of Bursting Neurons 160
10.4. Summary and Sources 165
10.5. Exercises 166
Chapter 11. Probability and Random Variables 168
11.1. Events and Random Variables 168
11.2. Binomial Random Variables 170
11.3. Poisson Random Variables 172
11.4. Gaussian Random Variables 172
11.5. Cumulative Distribution Functions 173
11.6. Conditional Probabilities* 174
11.7. Sum of Independent Random Variables* 175
11.8. Transformation of Random Variables* 176
11.9. Random Vectors* 177
11.10. Exponential and Gamma Distributed Random Variables 180
11.11. The Homogeneous Poisson Process 181
11.12. Summary and Sources 183
11.13. Exercises 183
Chapter 12. Synaptic Transmission and Quantal Release 188
12.1. Basic Synaptic Structure and Physiology 188
12.2. Discovery of Quantal Release 190
12.3. Compound Poisson Model of Synaptic Release 191
12.4. Comparison with Experimental Data 193
12.5. Quantal Analysis at Central Synapses 194
12.6. Facilitation, Potentiation, and Depression of Synaptic Transmission 196
12.7. Models of Short-Term Synaptic Plasticity 199
12.8. Summary and Sources 202
12.9. Exercises 203
Chapter 13. Neuronal Calcium Signaling* 206
13.1. Voltage-Gated Calcium Channels 208
13.2. Diffusion, Buffering, and Extraction of Cytosolic Calcium 211
13.3. Calcium Release from the ER 214
13.4. Calcium in Spines 222
13.5. Presynaptic Calcium and Transmitter Release 226
13.6. Summary and Sources 230
13.7. Exercises 230
Chapter 14. The Singular Value Decomposition and Applications* 236
14.1. The Singular Value Decomposition 236
14.2. Principal Component Analysis and Spike Sorting 239
14.3. Synaptic Plasticity and Principal Components 241
14.4. Neuronal Model Reduction via Balanced Truncation 243
14.5. Summary and Sources 246
14.6. Exercises 246
Chapter 15. Quantification of Spike Train Variability 250
15.1. Interspike Interval Histograms and Coefficient of Variation 251
15.2. Refractory Period 252
15.3. Spike Count Distribution and Fano Factor 253
15.4. Renewal Processes 253
15.5. Return Maps and Empirical Correlation Coefficient 256
15.6. Summary and Sources 258
15.7. Exercises 259
Chapter 16. Stochastic Processes 264
16.1. Definition and General Properties 264
16.2. Gaussian Processes 265
16.3. Point Processes 267
16.4. The Inhomogeneous Poisson Process 270
16.5. Spectral Analysis 272
16.6. Summary and Sources 275
16.7. Exercises 275
Chapter 17. Membrane Noise* 280
17.1. Two-State Channel Model 280
17.2. Multistate Channel Models 283
17.3. The Ornstein–Uhlenbeck Process 284
17.4. Synaptic Noise 285
17.5. Summary and Sources 288
17.6. Exercises 288
Chapter 18. Power and Cross Spectra 292
18.1. Cross Correlation and Coherence 292
18.2. Estimator Bias and Variance 293
18.3. Numerical Estimate of the Power Spectrum* 295
18.4. Summary and Sources 299
18.5. Exercises 299
Chapter 19. Natural Light Signals and Phototransduction 304
19.1. Wavelength and Intensity 304
19.2. Spatial Properties of Natural Light Signals 306
19.3. Temporal Properties of Natural Light Signals 306
19.4. A Model of Phototransduction 307
19.5. Summary and Sources 310
19.6. Exercises 311
Chapter 20. Firing Rate Codes and Early Vision 312
20.1. Definition of Mean Instantaneous Firing Rate 312
20.2. Visual System and Visual Stimuli 313
20.3. Spatial Receptive Field of Retinal Ganglion Cells 314
20.4. Characterization of Receptive Field Structure 316
20.5. Spatio-Temporal Receptive Fields 319
20.6. Static Nonlinearities* 321
20.7. Summary and Sources 321
20.8. Exercises 322
Chapter 21. Models of Simple and Complex Cells 324
21.1. Simple Cell Models 324
21.2. Nonseparable Receptive Fields 331
21.3. Receptive Fields of Complex Cells 333
21.4. Motion-Energy Model 334
21.5. Hubel–Wiesel Model 334
21.6. Multiscale Representation of Visual Information 335
21.7. Summary and Sources 336
21.8. Exercises 336
Chapter 22. Stochastic Estimation Theory 340
22.1. Minimum Mean Square Error Estimation 340
22.2. Estimation of Gaussian Signals* 342
22.3. Linear Nonlinear (LN) Models* 344
22.4. Summary and Sources 345
22.5. Exercises 345
Chapter 23. Reverse-Correlation and Spike Train Decoding 348
23.1. Reverse-Correlation 348
23.2. Stimulus Reconstruction 351
23.3. Summary and Sources 353
23.4. Exercises 353
Chapter 24. Signal Detection Theory 356
24.1. Testing Hypotheses 356
24.2. Ideal Decision Rules 359
24.3. ROC Curves* 361
24.4. Multidimensional Gaussian Signals* 361
24.5. Fisher Linear Discriminant* 364
24.6. Summary and Sources 367
24.7. Exercises 367
Chapter 25. Relating Neuronal Responses and Psychophysics 368
25.1. Single Photon Detection 368
25.2. Signal Detection Theory and Psychophysics 372
25.3. Motion Detection 374
25.4. Summary and Sources 376
25.5. Exercises 377
Chapter 26. Population Codes* 380
26.1. Cartesian Coordinate Systems 380
26.2. Overcomplete Representations 382
26.3. Frames 383
26.4. Maximum Likelihood 385
26.5. Estimation Error and the Cramer–Rao Bound* 387
26.6. Population Coding in the Superior Colliculus 388
26.7. Summary and Sources 389
26.8. Exercises 391
Chapter 27. Neuronal Networks 394
27.1. Hopfield Networks 395
27.2. Leaky Integrate-and-Fire Networks 396
27.3. Leaky Integrate-and-Fire Networks with Plastic Synapses 402
27.4. Hodgkin–Huxley Based Networks 405
27.5. Hodgkin–Huxley Based Networks with Plastic Synapses 410
27.6. Rate Based Networks 411
27.7. Brain Maps and Self-Organizing Maps 414
27.8. Summary and Sources 416
27.9. Exercises 417
Chapter 28. Solutions to Selected Exercises 422
28.1. Chapter 2 422
28.2. Chapter 3 424
28.3. Chapter 4 426
28.4. Chapter 5 427
28.5. Chapter 6 429
28.6. Chapter 7 432
28.7. Chapter 8 434
28.8. Chapter 9 435
28.9. Chapter 10 435
28.10. Chapter 11 436
28.11. Chapter 12 441
28.12. Chapter 13 443
28.13. Chapter 14 444
28.14. Chapter 15 446
28.15. Chapter 16 449
28.16. Chapter 17 455
28.17. Chapter 18 458
28.18. Chapter 19 465
28.19. Chapter 20 466
28.20. Chapter 21 466
28.21. Chapter 22 468
28.22. Chapter 23 471
28.23. Chapter 24 472
28.24. Chapter 25 477
28.25. Chapter 26 479
28.26. Chapter 27 483
References 486
Index 496

Produkt-Details

• Erscheint lt. Verlag: 16.9.2010
• Sprache: englisch
• Themenwelt: Geisteswissenschaften ► Psychologie ► Allgemeine Psychologie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Medizin / Pharmazie ► Medizinische Fachgebiete ► Neurologie
• Themenwelt: Naturwissenschaften ► Biologie ► Humanbiologie
• Themenwelt: Naturwissenschaften ► Biologie ► Zoologie
• Themenwelt: Technik
• ISBN-10: 0-08-089049-0 / 0080890490
• ISBN-13: 978-0-08-089049-4 / 9780080890494