Chapter
3.3 The Laplace Transform*
4 The Active Isopotential Cell
4.1 The Delayed Rectifier Potassium Channel
4.3 The Hodgkin-Huxley Equations
4.4 The Transient Potassium Channel*
4.5 The Sodium-Potassium Pump*
5 The Quasi-Active Isopotential Cell
5.1 The Quasi-Active Model
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*
6.1 The Discrete Passive Cable Equation
6.2 Exact Solution via Eigenvector Expansion
6.4 The Passive Cable Equation
7 Fourier Series and Transforms
7.2 The Discrete Fourier Transform
7.3 The Fourier Transform
7.4 Reconciling the Discrete and Continuous Fourier Transforms
8 The Passive Dendritic Tree
8.1 The Discrete Passive Tree
8.2 Eigenvector Expansion
8.4 The Passive Dendrite Equation
8.5 The Equivalent Cylinder*
8.6 Branched Eigenfunctions*
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
10 Extracellular Potential
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
11 Reduced Single Neuron Models
11.1 The Leaky Integrate-and-Fire Neuron
11.3 Simplified Models of Bursting Neurons
12 Probability and Random Variables
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*
12.10 Exponential and Gamma Distributed Random Variables
12.11 The Homogeneous Poisson Process
12.12 Summary and Sources
13 Synaptic Transmission and Quantal Release
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
14 Neuronal Calcium Signaling*
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
15 Neurovascular Coupling, the BOLD Signal and MRI
15.1 The Metabolic Cost of Neural Signaling
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
16 The Singular Value Decomposition and Applications*
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
17 Quantification of Spike Train Variability
17.1 Interspike Interval Histograms and Coefficient of Variation
17.3 Spike Count Distribution and Fano Factor
17.5 Return Maps and Serial Correlation Coefficients
18.1 Definition and General Properties
18.4 The Inhomogeneous Poisson Process
19.1 Two-State Channel Model
19.2 Multi-State Channel Models
19.3 The Ornstein-Uhlenbeck Process
20 Power and Cross-Spectra
20.1 Cross-Correlation and Coherence
20.2 Estimator Bias and Variance
20.3 Numerical Estimate of the Power Spectrum*
21 Natural Light Signals and Phototransduction
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
22 Firing Rate Codes and Early Vision
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*
23 Models of Simple and Complex Cells
23.2 Non-Separable Receptive Fields
23.3 Receptive Fields of Complex Cells
23.6 Multiscale Representation of Visual Information
24 Models of Motion Detection
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
25 Stochastic Estimation Theory
25.1 Minimum Mean-Square Error Estimation
25.2 Estimation of Gaussian Signals*
25.3 Linear Non-Linear (LN) Models*
26 Reverse-Correlation and Spike Train Decoding
26.2 Stimulus Reconstruction
27 Signal Detection Theory
27.2 Ideal Decision Rules
27.4 Multi-Dimensional Gaussian Signals*
27.5 Fisher Linear Discriminant*
28 Relating Neuronal Responses and Psychophysics
28.1 Single Photon Detection
28.2 Signal Detection Theory and Psychophysics
29.1 Cartesian Coordinate Systems
29.2 Overcomplete Representations
29.5 Estimation Error and Cramer-Rao Bound*
29.6 Population Coding in the Superior Colliculus
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
30.9 Brain Maps and Self-Organizing Maps
30.10 Summary and Sources
31 Solutions to Exercises
Mathematics for Neuroscientists
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