Observed Brain Dynamics

The biomedical sciences have recently undergone revolutionary change, due to the ability to digitize and store large data sets. In neuroscience, the data sources include measurements of neural activity measured using electrode arrays, EEG and MEG, brain imaging data from PET, fMRI, and optical imaging methods. Analysis, visualization, and management of these time series data sets is a growing field of research that has become increasingly important both for experimentalists and theorists interested in brain function. Written by investigators who have played an important role in developing the subject and in its pedagogical exposition, the current volume addresses the need for a textbook in this interdisciplinary area. The book is written for a broad spectrum of readers ranging from physical scientists, mathematicians, and statisticians wishing to educate themselves about neuroscience, to biologists who would like to learn time series analysis methods in particular and refresh their mathematical and statistical knowledge in general, through self-pedagogy. It may also be used as a supplement for a quantitative course in neurobiology or as a textbook for instruction on neural signal processing. The first part of the book contains a set of essays meant to provide conceptual background which are not technical and shall be generally accessible. Salient features include the adoption of an active perspective of the nervous system, an emphasis on function, and a brief survey of different theoretical accounts in neuroscience. The second part is the longest in the book, and contains a refresher course in mathematics and statistics leading up to time series analysis techniques. The third part contains applications of data analysis techniques to the range of data sources indicated above (also available as part of the Chronux data analysis platform from http://chronux.org), and the fourth part contains special topics.

Conceptual Background1. Why Study Brain Dynamics?1.1 Why Dynamics? An Active Perspective1.2 Quantifying Dynamics: Shared Theoretical Instruments1.3 'Newtonian and Bergsonian Time'1.3.1 Reversible and Irreversible Dynamics; Entropy1.3.2 Deterministic versus Random Motion1.3.3 Biological Arrows of Time?2. Theoretical Accounts of the Nervous System2.1 Three axes in the space of theories2.1.1 Level of Organization2.1.2 Direction of Causal Explanations2.1.3 Instrumental Approach2.1.4 Conclusion3. Engineering Theories and Nervous System Function3.1 What Do Brains Do?3.2 Engineering Theories3.2.1 Control Theory3.2.2 Communication Theory3.2.3 Computation4. Methodological Considerations4.1 Conceptual Clarity and Valid Reasoning4.2 Nature of Scientific Method4.2.1 Causation and CorrelationII Tutorials5. Mathematical Preliminaries5.1 Scalars: Real and Complex Variables; Elementary Functions5.1.1 Exponential functions5.1.2 Miscellaneous Remarks5.2 Vectors and Matrices: Linear Algebra5.2.1 Vectors as Points in a High Dimensional Space5.2.2 Angles, Distances, Volumes5.2.3 Linear Independence and Basis Sets5.2.4 Subspaces and Projections5.2.5 Matrices: Linear Transformations of Vectors5.2.6 Some Classes of Matrices5.2.7 Functions of Matrices: Determinants, Traces and Exponentials5.2.8 Classical Matrix Factorization Techniques5.2.9 Pseudospectra5.3 Fourier Analysis5.3.1 Function Spaces and Basis Expansions5.3.2 Fourier Series5.3.3 Convergence of Fourier Expansions on the Interval5.3.4 Fourier Transforms5.3.5 Bandlimited Functions, the Sampling Theorem and Aliasing5.3.6 Discrete Fourier Transforms and Fast Fourier Transforms5.4 Time Frequency Analysis5.4.1 Broadband Bias and Narrowband Bias5.4.2 The Spectral Concentration Problem5.5 Probability Theory5.5.1 Sample Space, Events, Probability Axioms5.5.2 Random Variables, Characteristic Function5.5.3 Some Common Probability Measures5.5.4 Law of Large Numbers5.5.5 Central Limit Theorems5.6 Stochastic Processes5.6.1 Defining Stochastic Processes5.6.2 Time Translational Invariance5.6.3 Ergodicity5.6.4 Time Translation Invariance and Spectral Analysis5.6.5 Gaussian Processes5.6.6 Non-Gaussian Processes5.6.7 Point Processes6. Statistical Protocols6.1 Data Analysis Goals6.2 An Example of a Protocol: Method of Least Squares6.3 Classical and Modern Approaches6.3.1 Data Visualization6.4 Classical Approaches: Estimation and Inference6.4.1 Point estimation6.4.2 Method of Least Squares: The Linear Model6.4.3 Generalized Linear Models6.4.4 Interval Estimation6.4.5 Hypothesis Testing6.4.6 Nonparametric Tests6.4.7 Bayesian Estimation and Inference7. Time Series Analysis7.1 Method of Moments7.2 Evoked Potentials and PSTH7.3 Univariate Spectral Analysis7.3.1 Nonparametric Quadratic Estimates7.3.2 Autoregressive Parametric Estimates7.3.3 Harmonic Analysis and Mixed Spectral Estimation7.3.4 Dynamic Spectra7.4 Bivariate Spectral Analysis7.4.1 Cross-Coherence7.5 Multivariate Spectral Analysis7.5.1 SVD of Cross Spectral Matrix7.6 Prediction7.6.1 Linear Prediction Using Autoregressive Models7.7 Point Process Spectral Estimation7.8 Higher Order CorrelationsIII Applications8. Electrophysiology: Microelectrode Recordings8.1 Introduction8.2 Experimental Approaches8.3 Biophysics of Neurons8.3.1 Transmembrane Resting Potential8.3.2 Action Potentials and Synaptic Potentials8.3.3 Extracellular Potentials8.4 Measurement Techniques8.4.1 Intracellular Measurements8.4.2 Extracellular Measurements8.4.3 Noise Sources8.5 Analysis Protocol8.5.1 Data Conditioning8.5.2 Analysis of Spike Trains8.5.3 Local Field Potentials8.5.4 Measures of Association8.5.5 Periodic Stimulation8.6 Parametric Methods8.6.1 Goodness of Fit8.6.2 Example8.7 Predicting Behavior from Neural Activity8.7.1 Selecting Feature Vectors8.7.2 Discrete Categories8.7.3 Continuous Movements9. Spike Sorting9.1 Introduction9.2 General Framework9.3 Data Acquisition9.3.1 Multiple Electrodes9.3.2 Sampling9.3.3 Data Windows9.4 Spike Detection9.4.1 Alignment9.4.2 Outlier Removal9.4.3 Data Visualization9.5 Clustering9.6 Quality Metrics9.6.1 Manual Review10. Electro- and Magnetoencephalography10.1 Introduction10.2 Analysis of EEG Signals: Early Work10.3 Physics of Encephalographic Signals10.4 Measurement Techniques10.4.1 Noise10.5 Analysis10.5.1 Denoising and Dimensionality Reduction10.5.2 Confirmatory Analysis11. PET and fMRI11.1 Introduction11.2 Biophysics of PET and fMRI11.2.1 PET11.2.2 fMRI11.2.3 Noise Sources11.3 Experimental Overview11.3.1 Experimental Protocols11.4 Analysis11.4.1 Data Conditioning11.4.2 Harmonic Analysis11.4.3 Statistical Parametric Mapping11.4.4 Multiple Hypothesis Tests11.4.5 Anatomical Considerations12. Optical Imaging12.1 Introduction12.2 Biophysical Considerations12.2.1 Noise Sources12.3 Analysis12.3.1 Difference and Ratio Maps12.3.2 Multivariate MethodsIV Special Topics13. Local Regression and Likelihood 36113.1 Local Regression13.2 Local likelihood13.2.1 Local Logistic Regression13.2.2 Local Poisson Regression13.3 Density Estimation13.4 Model Assessment and Selection13.4.1 Degrees of Freedom13.4.2 Selection of the Bandwidth and Polynomial Degree13.4.3 Residuals13.4.4 Confidence Intervals14. Entropy and Mutual Information14.1 Entropy and Mutual Information for Discrete Random Variables14.2 Continuous Random Variables14.3 Discrete-Valued Discrete-Time Stochastic Processes14.4 Continuous-Valued Discrete-Time Stochastic Processes14.5 Point Processes14.6 Estimation MethodsThe Bandwagon by C. E. ShannonTwo Famous Papers by Peter Elias