Observed Brain Dynamics

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1 Observed Brain Dynamics Partha P. Mitra Hemant Bokil OXTORD UNIVERSITY PRESS 2008

2 \ PART I Conceptual Background 1 1 Why Study Brain Dynamics? Why Dynamics? An Active Perspective 3 Vi Qimnü^iQ^Dv.aamics'v Shared Theoretical Instruments "Newtonian and Bergsonian Time" Reversible and Irreversible Dynamics; Entropy Deterministic Versus Random Motion Biological Arrows oftime? 12 2 Theoretical Accounts of the Nervous System Three Axes in the Space of Theories Level of Organization Direction of Causal Explanations Instrumental Approach Conclusion 25 3 Engineering Theories and Nervous System Function What Do Brains Do? Engineering Theories Control Theory Communication Theory Computation 36 4 Methodological Considerations Conceptual Clarity and Valid Reasoning 41 XVII

3 4.1.1 Syntax: Well-Formed Statements Logic: Consequence Nature of Scientific Method Empirical and Controlled Experimental Methods Deductive and Inductive Methods Causation and Correlation 46 PART II Tutorials 49 5 Mathematical Preliminaries Sealars: Real and Complex Variables; Elementary Functions Exponential Functions Miscellaneous Remarks Vectors and Matrices: Linear Algebra Vectors as Points in a High-Dimensional Space Angles, Distances, and Volumes Linear Independence and Basis Sets Subspaces and Projections Matrices: Linear Transformations of Vectors Some Classes of Matrices Functions of Matrices: Determinants, Traces, and Exponentials Classical Matrix Factorization Techniques Pseudospectra Fourier Analysis Function Spaces and Basis Expansions Fourier Series Convergence of Fourier Expansions on the Interval Fourier Transforms Bandlimited Functions, the Sampling Theorem, and Aliasing Discrete Fourier Transforms and Fast Fourier Transforms Time Frequency Analysis Broadband Bias and Narrowband Blas The Spectral Concentration Problem Probability Theory Sample Space, Events, and Probability Axioms Random Variables and Characteristic Function Some Common Probability Measures Law of Large Numbers Central Limit Theorems 112 xviii <

4 5.6 Stochastic Processes Deflning Stochastic Processes Time Translational Invariance Ergodicity Time Translation Invariance and Spectral Analysis Gaussian Processes Non-Gaussian Processes Point Processes Statistical Protocols Data Analysis Goals An Example of a Protocol: Method of Least Squares Classical and Modern Approaches Data Visualization Classical Approaches: Estimation and Inference Point Estimation Method of Least Squares: The Linear Model Generalized Linear Models Interval Estimation Hypothesis Testing Nonparametric Tests Bayesian Estimation and Inference Time Series Analysis Method of Moments Evoked Potentials and Peristimulus Time Histogram Univariate Spectral Analysis Periodogram Estimate: Problems of Bias and Variance Nonparametric Quadratic Estimates Autoregressive Parametric Estimates Harmonie Analysis and Mixed Spectral Estimation Dynamic Spectra Bivariate Spectral Analysis Cross-Coherence Multivariate Spectral Analysis Singular Value Decomposition of Cross-Spectral Matrix Prediction Linear Prediction Using Autoregressive Models Point Process Spectral Estimation Degrees of Freedom Hybrid Multivariate Processes 214 xix

5 7.8 Higher Order Correlations Correlations Between Spectral Power at Different Frequencies 216 PART III Applications Electrophysiology: Microelectrode Recordings Introduction Experimental Approaches Biophysics of Neurons Transmembrane Resting Potential Action Potentials and Synaptic Potentials Extracellular Potentials Measurement Techniques Intracellular Measurements Extracellular Measurements Noise Sources Analysis Protocol Data Conditioning Analysis of Spike Trains Local Field Potentials Measures of Association Periodic Stimulation Parametric Methods Goodness of Fit Example Predicting Behavior From Neural Activity Selecting Feature Vectors Discrete Categories Continuous Movements Spike Sorting Introduction General Framework Manual Sorting Data Acquisition Multiple Electrodes Sampling Data Windows Spike Detection Alignment Outlier Removal 265 xx

6 9.4.3 Data Visualization Clustering Quality Metrics Manual Review Electro- and Magnetoencephalography Introduction Analysis of Electroencephalographic Signals: Early Work Physics of Encephalographic Signals Measurement Techniques Noise Analysis Denoising and Dimensionality Reduction Confirmatory Analysis PET and fmri Introduction Biophysics of PET and fmri PET fmri Noise Sources Experimental Overview Experimental Protocols Analysis Data Conditioning Harmonie Analysis Statistical Parametric Mapping Multiple Hypothesis Tests Anatomkal Considerations Optical Imaging Introduction Biophysical Considerations Noise Sources Analysis Difference and Ratio Maps Multivariate Methods 315 PART IV Special Topics Local Regression and Likelihood Local Regression 323 xxi

7 13.2 Local Likelihood Local Logistic Regression Local Poisson Regression Density Estimation Model Assessment and Selection Degrees offreedom Selection ofthe Bandwidth and Polynomial Degree Residuais Confidence Intervals Entropy and Mutual Information Entropy and Mutual Information for Discrete Random Variables Continuous Random Variables Discrete-Valued Discrete-Time Stochastic Processes Continuous-Valued Discrete-Time Stochastic Processes Point Processes Estimation Methods 340 Appendix A: The Bandwagon by C. E. Shannon 343 Appendix B: Two Famous Papers by Peter Elias 345 Photograph Credits 347 Bibliography 349 Index 363 >, xxii

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