Observed Brain Dynamics

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