Conceptual Background
1. Why Study Brain Dynamics?
1.1 Why Dynamics? An Active Perspective
1.2 Quantifying Dynamics: Shared Theoretical Instruments
1.3 'Newtonian and Bergsonian Time'
1.3.1 Reversible and Irreversible Dynamics; Entropy
1.3.2 Deterministic versus Random Motion
1.3.3 Biological Arrows of Time?
2. Theoretical Accounts of the Nervous System
2.1 Three axes in the space of theories
2.1.1 Level of Organization
2.1.2 Direction of Causal Explanations
2.1.3 Instrumental Approach
2.1.4 Conclusion
3. Engineering Theories and Nervous System Function
3.1 What Do Brains Do?
3.2 Engineering Theories
3.2.1 Control Theory
3.2.2 Communication Theory
3.2.3 Computation
4. Methodological Considerations
4.1 Conceptual Clarity and Valid Reasoning
4.2 Nature of Scientific Method
4.2.1 Causation and Correlation
II Tutorials
5. Mathematical Preliminaries
5.1 Scalars: Real and Complex Variables; Elementary Functions
5.1.1 Exponential functions
5.1.2 Miscellaneous Remarks
5.2 Vectors and Matrices: Linear Algebra
5.2.1 Vectors as Points in a High Dimensional Space
5.2.2 Angles, Distances, Volumes
5.2.3 Linear Independence and Basis Sets
5.2.4 Subspaces and Projections
5.2.5 Matrices: Linear Transformations of Vectors
5.2.6 Some Classes of Matrices
5.2.7 Functions of Matrices: Determinants, Traces and Exponentials
5.2.8 Classical Matrix Factorization Techniques
5.2.9 Pseudospectra
5.3 Fourier Analysis
5.3.1 Function Spaces and Basis Expansions
5.3.2 Fourier Series
5.3.3 Convergence of Fourier Expansions on the Interval
5.3.4 Fourier Transforms
5.3.5 Bandlimited Functions, the Sampling Theorem and Aliasing
5.3.6 Discrete Fourier Transforms and Fast Fourier Transforms
5.4 Time Frequency Analysis
5.4.1 Broadband Bias and Narrowband Bias
5.4.2 The Spectral Concentration Problem
5.5 Probability Theory
5.5.1 Sample Space, Events, Probability Axioms
5.5.2 Random Variables, Characteristic Function
5.5.3 Some Common Probability Measures
5.5.4 Law of Large Numbers
5.5.5 Central Limit Theorems
5.6 Stochastic Processes
5.6.1 Defining Stochastic Processes
5.6.2 Time Translational Invariance
5.6.3 Ergodicity
5.6.4 Time Translation Invariance and Spectral Analysis
5.6.5 Gaussian Processes
5.6.6 Non-Gaussian Processes
5.6.7 Point Processes
6. Statistical Protocols
6.1 Data Analysis Goals
6.2 An Example of a Protocol: Method of Least Squares
6.3 Classical and Modern Approaches
6.3.1 Data Visualization
6.4 Classical Approaches: Estimation and Inference
6.4.1 Point estimation
6.4.2 Method of Least Squares: The Linear Model
6.4.3 Generalized Linear Models
6.4.4 Interval Estimation
6.4.5 Hypothesis Testing
6.4.6 Nonparametric Tests
6.4.7 Bayesian Estimation and Inference
7. Time Series Analysis
7.1 Method of Moments
7.2 Evoked Potentials and PSTH
7.3 Univariate Spectral Analysis
7.3.1 Nonparametric Quadratic Estimates
7.3.2 Autoregressive Parametric Estimates
7.3.3 Harmonic Analysis and Mixed Spectral Estimation
7.3.4 Dynamic Spectra
7.4 Bivariate Spectral Analysis
7.4.1 Cross-Coherence
7.5 Multivariate Spectral Analysis
7.5.1 SVD of Cross Spectral Matrix
7.6 Prediction
7.6.1 Linear Prediction Using Autoregressive Models
7.7 Point Process Spectral Estimation
7.8 Higher Order Correlations
III Applications
8. Electrophysiology: Microelectrode Recordings
8.1 Introduction
8.2 Experimental Approaches
8.3 Biophysics of Neurons
8.3.1 Transmembrane Resting Potential
8.3.2 Action Potentials and Synaptic Potentials
8.3.3 Extracellular Potentials
8.4 Measurement Techniques
8.4.1 Intracellular Measurements
8.4.2 Extracellular Measurements
8.4.3 Noise Sources
8.5 Analysis Protocol
8.5.1 Data Conditioning
8.5.2 Analysis of Spike Trains
8.5.3 Local Field Potentials
8.5.4 Measures of Association
8.5.5 Periodic Stimulation
8.6 Parametric Methods
8.6.1 Goodness of Fit
8.6.2 Example
8.7 Predicting Behavior from Neural Activity
8.7.1 Selecting Feature Vectors
8.7.2 Discrete Categories
8.7.3 Continuous Movements
9. Spike Sorting
9.1 Introduction
9.2 General Framework
9.3 Data Acquisition
9.3.1 Multiple Electrodes
9.3.2 Sampling
9.3.3 Data Windows
9.4 Spike Detection
9.4.1 Alignment
9.4.2 Outlier Removal
9.4.3 Data Visualization
9.5 Clustering
9.6 Quality Metrics
9.6.1 Manual Review
10. Electro- and Magnetoencephalography
10.1 Introduction
10.2 Analysis of EEG Signals: Early Work
10.3 Physics of Encephalographic Signals
10.4 Measurement Techniques
10.4.1 Noise
10.5 Analysis
10.5.1 Denoising and Dimensionality Reduction
10.5.2 Confirmatory Analysis
11. PET and fMRI
11.1 Introduction
11.2 Biophysics of PET and fMRI
11.2.1 PET
11.2.2 fMRI
11.2.3 Noise Sources
11.3 Experimental Overview
11.3.1 Experimental Protocols
11.4 Analysis
11.4.1 Data Conditioning
11.4.2 Harmonic Analysis
11.4.3 Statistical Parametric Mapping
11.4.4 Multiple Hypothesis Tests
11.4.5 Anatomical Considerations
12. Optical Imaging
12.1 Introduction
12.2 Biophysical Considerations
12.2.1 Noise Sources
12.3 Analysis
12.3.1 Difference and Ratio Maps
12.3.2 Multivariate Methods
IV Special Topics
13. Local Regression and Likelihood 361
13.1 Local Regression
13.2 Local likelihood
13.2.1 Local Logistic Regression
13.2.2 Local Poisson Regression
13.3 Density Estimation
13.4 Model Assessment and Selection
13.4.1 Degrees of Freedom
13.4.2 Selection of the Bandwidth and Polynomial Degree
13.4.3 Residuals
13.4.4 Confidence Intervals
14. Entropy and Mutual Information
14.1 Entropy and Mutual Information for Discrete Random Variables
14.2 Continuous Random Variables
14.3 Discrete-Valued Discrete-Time Stochastic Processes
14.4 Continuous-Valued Discrete-Time Stochastic Processes
14.5 Point Processes
14.6 Estimation Methods
The Bandwagon by C. E. Shannon
Two Famous Papers by Peter Elias