Mathematics for Neuroscientists.
Title:
Mathematics for Neuroscientists.
Author:
Gabbiani, Fabrizio.
ISBN:
9780080890494
Personal Author:
Physical Description:
1 online resource (505 pages)
Contents:
Front cover -- Mathematics for Neuroscientists -- Copyright page -- 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.
Abstract:
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. All code will be available via a companion website, which will be continuously updated with additional code and updates necessitated by software releases. 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 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.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
Genre:
Added Author:
Electronic Access:
Click to View