Observed Brain Dynamics

Partha Mitra
Hemant Bokil

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.

Observed Brain Dynamics

Partha Mitra
Hemant Bokil

Description

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.

Observed Brain Dynamics

Partha Mitra
Hemant Bokil

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
3.2.2 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 Tra 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 < 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
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 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. 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 Visu 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 S
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 Interval 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