Chapter 1: Introduction

• Abstract
• 1.1. How to Use This Book
• 1.2. Brain Facts Brief
• 1.3. Mathematical Preliminaries
• 1.4. Units
• 1.5. Sources
• Bibliography

Chapter 2: The Passive Isopotential Cell

• Abstract
• 2.1. Introduction
• 2.2. The Nernst Potential
• 2.3. Membrane Conductance
• 2.4. Membrane Capacitance & Current Balance
• 2.5. Synaptic Conductance
• 2.6. Summary and Sources
• 2.7. Exercises
• Bibliography

Chapter 3: Differential Equations

• Abstract
• 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
• Bibliography

Chapter 4: The Active Isopotential Cell

• Abstract
• 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. The Sodium–Potassium Pump?
• 4.6. Summary and Sources
• 4.7. Exercises
• Bibliography

Chapter 5: The Quasi-Active Isopotential Cell

• Abstract
• 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. Linearization of the Sodium–Potassium Pump?
• 5.7. Summary and Sources
• 5.8. Exercises
• Bibliography

Chapter 6: The Passive Cable

• Abstract
• 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
• Bibliography

Chapter 7: Fourier Series and Transforms

• Abstract
• 7.1. Fourier Series
• 7.2. The Discrete Fourier Transform
• 7.3. The Fourier Transform
• 7.4. Reconciling the Discrete and Continuous Fourier Transforms
• 7.5. Summary and Sources
• 7.6. Exercises
• Bibliography

Chapter 8: The Passive Dendritic Tree

• Abstract
• 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
• Bibliography

Chapter 9: The Active Dendritic Tree

• Abstract
• 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
• Bibliography

Chapter 10: Extracellular Potential

• Abstract
• 10.1. Maxwell's Equations
• 10.2. The Wave Equation
• 10.3. From Maxwell to Laplace
• 10.4. The Solution to Laplace's Equation
• 10.5. Extracellular Potential Near a Passive Cable
• 10.6. Extracellular Potential Near Active Cables
• 10.7. Summary and Sources
• 10.8. Exercises
• Bibliography

Chapter 11: Reduced Single Neuron Models

• Abstract
• 11.1. The Leaky Integrate-and-Fire Neuron
• 11.2. Bursting Neurons
• 11.3. Simplified Models of Bursting Neurons
• 11.4. Summary and Sources
• 11.5. Exercises
• Bibliography

Chapter 12: Probability and Random Variables

• Abstract
• 12.1. Events and Random Variables
• 12.2. Binomial Random Variables
• 12.3. Poisson Random Variables
• 12.4. Gaussian Random Variables
• 12.5. Cumulative Distribution Functions
• 12.6. Conditional Probabilities?
• 12.7. Sum of Independent Random Variables?
• 12.8. Transformation of Random Variables?
• 12.9. Random Vectors?
• 12.10. Exponential and Gamma Distributed Random Variables
• 12.11. The Homogeneous Poisson Process
• 12.12. Summary and Sources
• 12.13. Exercises
• Bibliography

Chapter 13: Synaptic Transmission and Quantal Release

• Abstract
• 13.1. Basic Synaptic Structure and Physiology
• 13.2. Discovery of Quantal Release
• 13.3. Compound Poisson Model of Synaptic Release
• 13.4. Comparison with Experimental Data
• 13.5. Quantal Analysis at Central Synapses
• 13.6. Facilitation, Potentiation and Depression of Synaptic Transmission
• 13.7. Models of Short-Term Synaptic Plasticity
• 13.8. Summary and Sources
• 13.9. Exercises
• Bibliography

Chapter 14: Neuronal Calcium SignalingNeuronal Calcium Signaling?

• Abstract
• 14.1. Voltage Gated Calcium Channels
• 14.2. Diffusion, Buffering and Extraction of Cytosolic Calcium
• 14.3. Calcium Release from the Endoplasmic Reticulum
• 14.4. Regulation of Calcium in Spines
• 14.5. Spinal Calcium and Bidirectional Synaptic Plasticity
• 14.6. Presynaptic Calcium and Transmitter Release
• 14.7. Summary and Sources
• 14.8. Exercises
• Bibliography

Chapter 15: Neurovascular Coupling, the BOLD Signal and MRI

• Abstract
• 15.1. The Metabolic Cost of Neural Signaling
• 15.2. Astrocytes
• 15.3. Smooth Muscle
• 15.4. Endothelium
• 15.5. The Neurovascular Unit
• 15.6. How Blood Distorts an Applied Magnetic Field
• 15.7. Nuclear Magnetic Resonance and the BOLD Signal
• 15.8. The Hemodynamic Response
• 15.9. Magnetic Resonance Imaging
• 15.10. Summary and Sources
• 15.11. Exercises
• Bibliography

Chapter 16: The Singular Value Decomposition and ApplicationsThe Singular Value Decomposition and Applications?

• Abstract
• 16.1. The Singular Value Decomposition
• 16.2. Principal Component Analysis and Spike Sorting
• 16.3. Synaptic Plasticity and Principal Components
• 16.4. Neuronal Model Reduction via Balanced Truncation
• 16.5. Summary and Sources
• 16.6. Exercises
• Bibliography

Chapter 17: Quantification of Spike Train Variability

• Abstract
• 17.1. Interspike Interval Histograms and Coefficient of Variation
• 17.2. Refractory Period
• 17.3. Spike Count Distribution and Fano Factor
• 17.4. Renewal Processes
• 17.5. Return Maps and Serial Correlation Coefficients
• 17.6. Summary and Sources
• 17.7. Exercises
• Bibliography

Chapter 18: Stochastic Processes

• Abstract
• 18.1. Definition and General Properties
• 18.2. Gaussian Processes
• 18.3. Point Processes
• 18.4. The Inhomogeneous Poisson Process
• 18.5. Spectral Analysis
• 18.6. Summary and Sources
• 18.7. Exercises
• Bibliography

Chapter 19: Membrane NoiseMembrane Noise*

• Abstract
• 19.1. Two-State Channel Model
• 19.2. Multi-State Channel Models
• 19.3. The Ornstein–Uhlenbeck Process
• 19.4. Synaptic Noise
• 19.5. Summary and Sources
• 19.6. Exercises
• Bibliography

Chapter 20: Power and Cross-Spectra

• Abstract
• 20.1. Cross-Correlation and Coherence
• 20.2. Estimator Bias and Variance
• 20.3. Numerical Estimate of the Power Spectrum?
• 20.4. Summary and Sources
• 20.5. Exercises
• Bibliography

Chapter 21: Natural Light Signals and Phototransduction

• Abstract
• 21.1. Wavelength and Intensity
• 21.2. Spatial Properties of Natural Light Signals
• 21.3. Temporal Properties of Natural Light Signals
• 21.4. A Model of Phototransduction
• 21.5. Summary and Sources
• 21.6. Exercises
• Bibliography

Chapter 22: Firing Rate Codes and Early Vision

• Abstract
• 22.1. Definition of Mean Instantaneous Firing Rate
• 22.2. Visual System and Visual Stimuli
• 22.3. Spatial Receptive Field of Retinal Ganglion Cells
• 22.4. Characterization of Receptive Field Structure
• 22.5. Spatio-Temporal Receptive Fields
• 22.6. Static Non-Linearities?
• 22.7. Summary and Sources
• 22.8. Exercises
• Bibliography

Chapter 23: Models of Simple and Complex Cells

• Abstract
• 23.1. Simple Cell Models
• 23.2. Non-Separable Receptive Fields
• 23.3. Receptive Fields of Complex Cells
• 23.4. Motion-Energy Model
• 23.5. Hubel–Wiesel Model
• 23.6. Multiscale Representation of Visual Information
• 23.7. Summary and Sources
• 23.8. Exercises
• Bibliography

Chapter 24: Models of Motion Detection

• Abstract
• 24.1. HRC Model of Motion Detection
• 24.2. Responses to Moving Stimuli
• 24.3. Properties of the Correlation Model
• 24.4. Equivalence with the Motion-Energy Model
• 24.5. Beyond Correlation in Motion Detection
• 24.6. Summary and Sources
• 24.7. Exercises
• Bibliography

Chapter 25: Stochastic Estimation Theory

• Abstract
• 25.1. Minimum Mean-Square Error Estimation
• 25.2. Estimation of Gaussian Signals?
• 25.3. Linear Non-Linear (LN) Models?
• 25.4. Summary and Sources
• 25.5. Exercises
• Bibliography

Chapter 26: Reverse-Correlation and Spike Train Decoding

• Abstract
• 26.1. Reverse-Correlation
• 26.2. Stimulus Reconstruction
• 26.3. Summary and Sources
• 26.4. Exercises
• Bibliography

Chapter 27: Signal Detection Theory

• Abstract
• 27.1. Testing Hypotheses
• 27.2. Ideal Decision Rules
• 27.3. ROC Curves?
• 27.4. Multi-Dimensional Gaussian Signals?
• 27.5. Fisher Linear Discriminant?
• 27.6. Summary and Sources
• 27.7. Exercises
• Bibliography

Chapter 28: Relating Neuronal Responses and Psychophysics

• Abstract
• 28.1. Single Photon Detection
• 28.2. Signal Detection Theory and Psychophysics
• 28.3. Motion Detection
• 28.4. Summary and Sources
• 28.5. Exercises
• Bibliography

Chapter 29: Population CodesPopulation Codes?

• Abstract
• 29.1. Cartesian Coordinate Systems
• 29.2. Overcomplete Representations
• 29.3. Frames
• 29.4. Maximum Likelihood
• 29.5. Estimation Error and Cramer–Rao Bound?
• 29.6. Population Coding in the Superior Colliculus
• 29.7. Summary and Sources
• 29.8. Exercises
• Bibliography

Chapter 30: Neuronal Networks

• Abstract
• 30.1. Perceptrons
• 30.2. Hopfield Networks
• 30.3. Integrate and Fire Networks
• 30.4. Integrate and Fire Networks with Plastic Synapses
• 30.5. Formation of the Grid Cell Network via STDP
• 30.6. Hodgkin–Huxley Based Networks
• 30.7. Hodgkin–Huxley Based Networks with Plastic Synapses
• 30.8. Rate Based Networks
• 30.9. Brain Maps and Self-Organizing Maps
• 30.10. Summary and Sources
• 30.11. Exercises
• Bibliography

Chapter 31: Solutions to Exercises

• Abstract
• 31.1. Chapter 2
• 31.2. Chapter 3
• 31.3. Chapter 4
• 31.4. Chapter 5
• 31.5. Chapter 6
• 31.6. Chapter 7
• 31.7. Chapter 8
• 31.8. Chapter 9
• 31.9. Chapter 10
• 31.10. Chapter 11
• 31.11. Chapter 12
• 31.12. Chapter 13
• 31.13. Chapter 14
• 31.14. Chapter 15
• 31.15. Chapter 16
• 31.16. Chapter 17
• 31.17. Chapter 18
• 31.18. Chapter 19
• 31.19. Chapter 20
• 31.20. Chapter 21
• 31.21. Chapter 22
• 31.22. Chapter 23
• 31.23. Chapter 24
• 31.24. Chapter 25
• 31.25. Chapter 26
• 31.26. Chapter 27
• 31.27. Chapter 28
• 31.28. Chapter 29
• 31.29. Chapter 30
• Bibliography