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.
Author(s): Partha Mitra
Edition: Hardcover
Publisher: Oxford University Press, USA
Year: 2007
Language: English
Pages: 408
PART I Conceptual Background/1
1 Why Study Brain Dynamics?/3
1.1 Why Dynamics? An Active Perspective/3
1.2 Quantifying Dynamics: Shared Theoretical Instruments/6
1.3 ‘‘Newtonian and Bergsonian Time’’/7
1.3.1 Reversible and Irreversible Dynamics; Entropy/9
1.3.2 Deterministic Versus Random Motion/12
1.3.3 Biological Arrows of Time?/12
2 Theoretical Accounts of the Nervous System/14
2.1 Three Axes in the Space of Theories/15
2.1.1 Level of Organization/17
2.1.2 Direction of Causal Explanations/22
2.1.3 Instrumental Approach/24
2.1.4 Conclusion/25
3 Engineering Theories and Nervous System Function/27
3.1 What Do Brains Do?/27
3.2 Engineering Theories/29
3.2.1 Control Theory/31
3.2.2 Communication Theory/32
3.2.3 Computation/36
4 Methodological Considerations/40
4.1 Conceptual Clarity and Valid Reasoning/41
4.1.1 Syntax: Well-Formed Statements/41
4.1.2 Logic: Consequence/42
4.2 Nature of Scientific Method/42
4.2.1 Empirical and Controlled Experimental Methods/43
4.2.2 Deductive and Inductive Methods/44
4.2.3 Causation and Correlation/46
PART II Tutorials/49
5 Mathematical Preliminaries/51
5.1 Scalars: Real and Complex Variables; Elementary Functions/52
5.1.1 Exponential Functions/54
5.1.2 Miscellaneous Remarks/56
5.2 Vectors and Matrices: Linear Algebra/56
5.2.1 Vectors as Points in a High-Dimensional Space/57
5.2.2 Angles, Distances, and Volumes/58
5.2.3 Linear Independence and Basis Sets/61
5.2.4 Subspaces and Projections/62
5.2.5 Matrices: Linear Transformations of Vectors/63
5.2.6 Some Classes of Matrices/64
5.2.7 Functions of Matrices: Determinants, Traces, and Exponentials/66
5.2.8 Classical Matrix Factorization Techniques/67
5.2.9 Pseudospectra/70
5.3 Fourier Analysis/72
5.3.1 Function Spaces and Basis Expansions/74
5.3.2 Fourier Series/77
5.3.3 Convergence of Fourier Expansions on the Interval/81
5.3.4 Fourier Transforms/83
5.3.5 Bandlimited Functions, the Sampling Theorem, and Aliasing/84
5.3.6 Discrete Fourier Transforms and Fast Fourier Transforms/86
5.4 Time Frequency Analysis/89
5.4.1 Broadband Bias and Narrowband Bias/90
5.4.2 The Spectral Concentration Problem/94
5.5 Probability Theory/98
5.5.1 Sample Space, Events, and Probability Axioms/100
5.5.2 Random Variables and Characteristic Function/102
5.5.3 Some Common Probability Measures/105
5.5.4 Law of Large Numbers/111
5.5.5 Central Limit Theorems/112
5.6 Stochastic Processes/113
5.6.1 Defining Stochastic Processes/114
5.6.2 Time Translational Invariance/116
5.6.3 Ergodicity/117
5.6.4 Time Translation Invariance and Spectral Analysis/118
5.6.5 Gaussian Processes/118
5.6.6 Non-Gaussian Processes/123
5.6.7 Point Processes/124
6 Statistical Protocols/148
6.1 Data Analysis Goals/149
6.2 An Example of a Protocol: Method of Least Squares/150
6.3 Classical and Modern Approaches/151
6.3.1 Data Visualization/152
6.4 Classical Approaches: Estimation and Inference/153
6.4.1 Point Estimation/154
6.4.2 Method of Least Squares: The Linear Model/161
6.4.3 Generalized Linear Models/167
6.4.4 Interval Estimation/171
6.4.5 Hypothesis Testing/172
6.4.6 Nonparametric Tests/178
6.4.7 Bayesian Estimation and Inference/181
7 Time Series Analysis/184
7.1 Method of Moments/185
7.2 Evoked Potentials and Peristimulus Time Histogram/187
7.3 Univariate Spectral Analysis/189
7.3.1 Periodogram Estimate: Problems of Bias and Variance/190
7.3.2 Nonparametric Quadratic Estimates/191
7.3.3 Autoregressive Parametric Estimates/197
7.3.4 Harmonic Analysis and Mixed Spectral Estimation/200
7.3.5 Dynamic Spectra/202
7.4 Bivariate Spectral Analysis/207
7.4.1 Cross-Coherence/208
7.5 Multivariate Spectral Analysis/209
7.5.1 Singular Value Decomposition of Cross-Spectral Matrix/209
7.6 Prediction/211
7.6.1 Linear Prediction Using Autoregressive Models/212
7.7 Point Process Spectral Estimation/213
7.7.1 Degrees of Freedom/214
7.7.2 Hybrid Multivariate Processes/214
7.8 Higher Order Correlations/215
7.8.1 Correlations Between Spectral Power at Different Frequencies/216
PART III Applications/217
8 Electrophysiology: Microelectrode Recordings/219
8.1 Introduction/219
8.2 Experimental Approaches/220
8.3 Biophysics of Neurons/221
8.3.1 Transmembrane Resting Potential/221
8.3.2 Action Potentials and Synaptic Potentials/221
8.3.3 Extracellular Potentials/223
8.4 Measurement Techniques/224
8.4.1 Intracellular Measurements/224
8.4.2 Extracellular Measurements/224
8.4.3 Noise Sources/225
8.5 Analysis Protocol/225
8.5.1 Data Conditioning/225
8.5.2 Analysis of Spike Trains/228
8.5.3 Local Field Potentials/239
8.5.4 Measures of Association/242
8.5.5 Periodic Stimulation/246
8.6 Parametric Methods/248
8.6.1 Goodness of Fit/250
8.6.2 Example/250
8.7 Predicting Behavior From Neural Activity/251
8.7.1 Selecting Feature Vectors/253
8.7.2 Discrete Categories/254
8.7.3 Continuous Movements/255
9 Spike Sorting/257
9.1 Introduction/257
9.2 General Framework/258
9.2.1 Manual Sorting/258
9.3 Data Acquisition/259
9.3.1 Multiple Electrodes/259
9.3.2 Sampling/259
9.3.3 Data Windows/260
9.4 Spike Detection/262
9.4.1 Alignment/263
9.4.2 Outlier Removal/265
9.4.3 Data Visualization/265
9.5 Clustering/266
9.6 Quality Metrics/268
9.6.1 Manual Review/268
10 Electro- and Magnetoencephalography/271
10.1 Introduction/271
10.2 Analysis of Electroencephalographic Signals: Early Work/271
10.3 Physics of Encephalographic Signals/273
10.4 Measurement Techniques/273
10.4.1 Noise/275
10.5 Analysis/275
10.5.1 Denoising and Dimensionality Reduction/275
10.5.2 Confirmatory Analysis/283
11 PET and fMRI/294
11.1 Introduction/294
11.2 Biophysics of PET and fMRI/295
11.2.1 PET/295
11.2.2 fMRI/295
11.2.3 Noise Sources/296
11.3 Experimental Overview/297
11.3.1 Experimental Protocols/298
11.4 Analysis/299
11.4.1 Data Conditioning/299
11.4.2 Harmonic Analysis/305
11.4.3 Statistical Parametric Mapping/306
11.4.4 Multiple Hypothesis Tests/310
11.4.5 Anatomical Considerations/311
12 Optical Imaging/313
12.1 Introduction/313
12.2 Biophysical Considerations/313
12.2.1 Noise Sources/314
12.3 Analysis/314
12.3.1 Difference and Ratio Maps/315
12.3.2 Multivariate Methods/315
PART IV Special Topics/321
13 Local Regression and Likelihood/323
13.1 Local Regression/323
13.2 Local Likelihood/326
13.2.1 Local Logistic Regression/327
13.2.2 Local Poisson Regression/327
13.3 Density Estimation/328
13.4 Model Assessment and Selection/328
13.4.1 Degrees of Freedom/328
13.4.2 Selection of the Bandwidth and Polynomial Degree/329
13.4.3 Residuals/331
13.4.4 Confidence Intervals/332
14 Entropy and Mutual Information/333
14.1 Entropy and Mutual Information for Discrete Random Variables/334
14.2 Continuous Random Variables/336
14.3 Discrete-Valued Discrete-Time Stochastic Processes/337
14.4 Continuous-Valued Discrete-Time Stochastic Processes/338
14.5 Point Processes/339
14.6 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
