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Mathematics for Neuroscientists,

Edition 2

Editors: By Fabrizio Gabbiani and Steven James Cox

Publication Date: 23 Feb 2017

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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.

Key Features

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

About the author

By Fabrizio Gabbiani, Baylor College of Medicine, Houston, TX, USA and Steven James Cox, Computational and Applied Mathematics, Rice University, Houston, TX, USA

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

Book Reviews

"This is a big book in more than one sense. It has a large page format measuring about 20cm x 27cm making it easy to open up and take in large swathes of text, equations, and figures. More importantly, it covers a very wide range of mathematical methodologies relevant to neuroscience. ...I would highly recommend this book to those with an interest in computational neuroscience who wish to delve more deeply into the biophysics underlying cell-based dynamics and computations, especially if they are interested in flexing their mathematical muscles."—MathSciNet

Amazon Editorial Reviews for first edition: "I really think this book is very, very important. This is precisely what has been missing from the field and is badly needed. "—Dr. Kevin Franks, research fellow, Richard Axel's laboratory Columbia University, NYC

"The idea of presenting sufficient maths to understand the theoretical neuroscience, alongside the neuroscience itself, is appealing. The inclusion of Matlab code for all examples and computational figures is an excellent idea. "—David Corney, research fellow, Institute of Ophthalmology, University College London

ISBN: 9780128018958

Page Count: 628

Retail Price :

Van Drongelen, Signal Processing for Neuroscientists, 320pp, 2006, 9780123708670, $104.00
Wallisch, MATLAB for Neuroscientists 2e, 570pp, Nov 2013, 9780123838360, $89.95
Brzychczy, Mathematical Neuroscience, 208pp, Sep 2013, 9780124114685, $129.95

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