Mathematics for Neuroscientists

Mathematics for Neuroscientists, Second Edition, presents a comprehensive introduction to mathematical and computational methods used in neuroscience to describe and model neural components of the brain from ion channels to single neurons, neural networks and their relation to behavior. The book contains more than 200 figures generated using Matlab code available to the student and scholar. Mathematical concepts are introduced hand in hand with neuroscience, emphasizing the connection between experimental results and theory.

• Fully revised material and corrected text
• Additional chapters on extracellular potentials, motion detection and neurovascular coupling
• Revised selection of exercises with solutions
• More than 200 Matlab scripts reproducing the figures as well as a selection of equivalent Python scripts

Neuroscientists, experimental neuroscientists, computational neuroscientists, mathematicians

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