Mathematics for Neuroscientists

Front cover

Mathematics for Neuroscientists

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Full Contents

Preface

Chapter 1. Introduction

1.1. How to Use This Book

1.2. Brain Facts Brief

1.3. Mathematical Preliminaries

1.4. Units

1.5. Sources

Chapter 2. The Passive Isopotential Cell

2.1. Introduction

2.2. The Nernst Potential

2.3. Membrane Conductance

2.4. Membrane Capacitance and Current Balance

2.5. Synaptic Conductance

2.6. Summary and Sources

2.7. Exercises

Chapter 3. Differential Equations

3.1. Exact Solution

3.2. Moment Methods*

3.3. The Laplace Transform*

3.4. Numerical Methods

3.5. Synaptic Input

3.6. Summary and Sources

3.7. Exercises

Chapter 4. The Active Isopotential Cell

4.1. The Delayed Rectifier Potassium Channel

4.2. The Sodium Channel

4.3. The Hodgkin–Huxley Equations

4.4. The Transient Potassium Channel*

4.5. Summary and Sources

4.6. Exercises

Chapter 5. The Quasi-Active Isopotential Cell

5.1. The Quasi-Active Model

5.2. Numerical Methods

5.3. Exact Solution via Eigenvector Expansion

5.4. A Persistent Sodium Current*

5.5. A Nonspecific Cation Current that is Activated by Hyperpolarization*

5.6. Summary and Sources

5.7. Exercises

Chapter 6. The Passive Cable

6.1. The Discrete Passive Cable Equation

6.2. Exact Solution Via Eigenvector Expansion

6.3. Numerical Methods

6.4. The Passive Cable Equation

6.5. Synaptic Input

6.6. Summary and Sources

6.7. Exercises

Chapter 7. Fourier Series and Transforms

7.1. Fourier Series

7.2. The Discrete Fourier Transform

7.3. The Continuous Fourier Transform

7.4. Reconciling the Discrete and Continuous Fourier Transforms

7.5. Summary and Sources

7.6. Exercises

Chapter 8. The Passive Dendritic Tree

8.1. The Discrete Passive Tree

8.2. Eigenvector Expansion

8.3. Numerical Methods

8.4. The Passive Dendrite Equation

8.5. The Equivalent Cylinder*

8.6. Branched Eigenfunctions*

8.7. Summary and Sources

8.8. Exercises

Chapter 9. The Active Dendritic Tree

9.1. The Active Uniform Cable

9.2. On the Interaction of Active Uniform Cables*

9.3. The Active Nonuniform Cable

9.4. The Quasi-Active Cable*

9.5. The Active Dendritic Tree

9.6. Summary and Sources

9.7. Exercises

Chapter 10. Reduced Single Neuron Models

10.1. The Leaky Integrate-and-Fire Neuron

10.2. Bursting Neurons

10.3. Simplified Models of Bursting Neurons

10.4. Summary and Sources

10.5. Exercises

Chapter 11. Probability and Random Variables

11.1. Events and Random Variables

11.2. Binomial Random Variables

11.3. Poisson Random Variables

11.4. Gaussian Random Variables

11.5. Cumulative Distribution Functions

11.6. Conditional Probabilities*

11.7. Sum of Independent Random Variables*

11.8. Transformation of Random Variables*

11.9. Random Vectors*

11.10. Exponential and Gamma Distributed Random Variables

11.11. The Homogeneous Poisson Process

11.12. Summary and Sources

11.13. Exercises

Chapter 12. Synaptic Transmission and Quantal Release

12.1. Basic Synaptic Structure and Physiology

12.2. Discovery of Quantal Release

12.3. Compound Poisson Model of Synaptic Release

12.4. Comparison with Experimental Data

12.5. Quantal Analysis at Central Synapses

12.6. Facilitation, Potentiation, and Depression of Synaptic Transmission

12.7. Models of Short-Term Synaptic Plasticity

12.8. Summary and Sources

12.9. Exercises

Chapter 13. Neuronal Calcium Signaling*

13.1. Voltage-Gated Calcium Channels

13.2. Diffusion, Buffering, and Extraction of Cytosolic Calcium

13.3. Calcium Release from the ER

13.4. Calcium in Spines

13.5. Presynaptic Calcium and Transmitter Release

13.6. Summary and Sources

13.7. Exercises

Chapter 14. The Singular Value Decomposition and Applications*

14.1. The Singular Value Decomposition

14.2. Principal Component Analysis and Spike Sorting

14.3. Synaptic Plasticity and Principal Components

14.4. Neuronal Model Reduction via Balanced Truncation

14.5. Summary and Sources

14.6. Exercises

Chapter 15. Quantification of Spike Train Variability

15.1. Interspike Interval Histograms and Coefficient of Variation

15.2. Refractory Period

15.3. Spike Count Distribution and Fano Factor

15.4. Renewal Processes

15.5. Return Maps and Empirical Correlation Coefficient

15.6. Summary and Sources

15.7. Exercises

Chapter 16. Stochastic Processes

16.1. Definition and General Properties

16.2. Gaussian Processes

16.3. Point Processes

16.4. The Inhomogeneous Poisson Process

16.5. Spectral Analysis

16.6. Summary and Sources

16.7. Exercises

Chapter 17. Membrane Noise*

17.1. Two-State Channel Model

17.2. Multistate Channel Models

17.3. The Ornstein–Uhlenbeck Process

17.4. Synaptic Noise

17.5. Summary and Sources

17.6. Exercises

Chapter 18. Power and Cross Spectra

18.1. Cross Correlation and Coherence

18.2. Estimator Bias and Variance

18.3. Numerical Estimate of the Power Spectrum*

18.4. Summary and Sources

18.5. Exercises

Chapter 19. Natural Light Signals and Phototransduction

19.1. Wavelength and Intensity

19.2. Spatial Properties of Natural Light Signals

19.3. Temporal Properties of Natural Light Signals

19.4. A Model of Phototransduction

19.5. Summary and Sources

19.6. Exercises

Chapter 20. Firing Rate Codes and Early Vision

20.1. Definition of Mean Instantaneous Firing Rate

20.2. Visual System and Visual Stimuli

20.3. Spatial Receptive Field of Retinal Ganglion Cells

20.4. Characterization of Receptive Field Structure

20.5. Spatio-Temporal Receptive Fields

20.6. Static Nonlinearities*

20.7. Summary and Sources

20.8. Exercises

Chapter 21. Models of Simple and Complex Cells

21.1. Simple Cell Models

21.2. Nonseparable Receptive Fields

21.3. Receptive Fields of Complex Cells

21.4. Motion-Energy Model

21.5. Hubel–Wiesel Model

21.6. Multiscale Representation of Visual Information

21.7. Summary and Sources

21.8. Exercises

Chapter 22. Stochastic Estimation Theory

22.1. Minimum Mean Square Error Estimation

22.2. Estimation of Gaussian Signals*

22.3. Linear Nonlinear (LN) Models*

22.4. Summary and Sources

22.5. Exercises

Chapter 23. Reverse-Correlation and Spike Train Decoding

23.1. Reverse-Correlation

23.2. Stimulus Reconstruction

23.3. Summary and Sources

23.4. Exercises

Chapter 24. Signal Detection Theory

24.1. Testing Hypotheses

24.2. Ideal Decision Rules

24.3. ROC Curves*

24.4. Multidimensional Gaussian Signals*

24.5. Fisher Linear Discriminant*

24.6. Summary and Sources

24.7. Exercises

Chapter 25. Relating Neuronal Responses and Psychophysics

25.1. Single Photon Detection

25.2. Signal Detection Theory and Psychophysics

25.3. Motion Detection

25.4. Summary and Sources

25.5. Exercises

Chapter 26. Population Codes*

26.1. Cartesian Coordinate Systems

26.2. Overcomplete Representations

26.3. Frames

26.4. Maximum Likelihood

26.5. Estimation Error and the Cramer–Rao Bound*

26.6. Population Coding in the Superior Colliculus

26.7. Summary and Sources

26.8. Exercises

Chapter 27. Neuronal Networks

27.1. Hopfield Networks

27.2. Leaky Integrate-and-Fire Networks

27.3. Leaky Integrate-and-Fire Networks with Plastic Synapses

27.4. Hodgkin–Huxley Based Networks

27.5. Hodgkin–Huxley Based Networks with Plastic Synapses

27.6. Rate Based Networks

27.7. Brain Maps and Self-Organizing Maps

27.8. Summary and Sources

27.9. Exercises

Chapter 28. Solutions to Selected Exercises

28.1. Chapter 2

28.2. Chapter 3

28.3. Chapter 4

28.4. Chapter 5

28.5. Chapter 6

28.6. Chapter 7

28.7. Chapter 8

28.8. Chapter 9

28.9. Chapter 10

28.10. Chapter 11

28.11. Chapter 12

28.12. Chapter 13

28.13. Chapter 14

28.14. Chapter 15

28.15. Chapter 16

28.16. Chapter 17

28.17. Chapter 18

28.18. Chapter 19

28.19. Chapter 20

28.20. Chapter 21

28.21. Chapter 22

28.22. Chapter 23

28.23. Chapter 24

28.24. Chapter 25

28.25. Chapter 26

28.26. Chapter 27

References

Index