Principles of Neurobiological Signal Analysis

Preface

Chapter 1: Some Properties of Biological Signals


1.1 Introduction


1.2 Continuous Signals and Their Discrete Counterparts


1.3 Repetitive and Periodic Signals


1.4 Sampled Representation of a Signal


1.5 Fourier Series Representation of a Signal


1.6 Bandwidth Limited Signals


1.7 Autocovariance Functions and Power Spectra of Periodic Signals


1.8 Aperiodic Signals


1.9 Autocovariance Functions and Power Spectra of Aperiodic Signals


1.10 Cross Covariance Functions and Cross Spectra for a Pair of Periodic Signals


1.11 A Summary of Properties of Covariance Functions and Spectra


1.12 Random or Probabilistic Signals


1.13 Some Important Probability Distributions

A. Probabilistic Descriptions of Dynamic Processes

B. The Gaussian Distribution

C. The Chi-Squared Distribution


1.14 Ensemble Autocovariance Functions


1.15 Ensemble Autocovariance and Cross Covariance Functions, and Stationarity


1.16 The Relationship between Ensemble and Time Statistics


1.17 Mixtures of Signal and Nois


1.18 Response Detection and Classification - Hypothesis Testing

Chapter 2 Basics of Signal Processing


2.1 Introduction


2.2 Analog-to Digital Conversion


2.3 Quantization Noise


2.4 Multiplexing: Monitoring Data Sources Simultaneously


2.5 Data Filtering


2.6 The Digital Filter

A. Filtering of the Constant Component

B. Filtering the/mth Frequency Component


2.7 Impulse Response of a Digital Filter


2.8 Spectral Relations between Filter Input and Output-The Discrete Fourier Transform


2.9 Filtering Aperiodic Signals

A. Short Duration Signals

Β. Maintained Signals


2.10 Data Smoothing by Digital Filtering


2.11 Digital Filters with Feedback - Recursive Filters


2.12 The Linear Analog Filter


2.13 The Laplace Transform, the Filter Transfer Function, and Impulse Response


2.14 The Operational Amplifier


2.15 The Amplitude Comparator


2.16 Time-Varying and Nonlinear Filters 100

Chapter 3: Power Spectra and Covariance Functions


3.1 Introduction


3.2 Discrete Fourier Representations of Continuous Processes


3.3 Aliasing


3.4 Leakage

A. Fourier Series

B. Discrete Fourier Transforms


3.5 Trend


3.6 The Power Spectrum, General Considerations


3.7 Power Spectrum of Continuous Random Signals


3.8 The Power Spectrum of T-Discrete Signals


3.9 The Fourier Transform for T-Discrete Signals


3.10 The Periodogram


3.11 Statistical Errors of the Periodogram—Bias


3.12 Statistical Errors of the Periodogram—Variance


3.13 Averaging the Periodogram the Bartlett Estimator


3.14 Variance of the Bartlett Estimator


3.15 The Fast Fourier Transform and Power Spectrum Estimation


3.16 Smoothing of Spectral Estimates by Windowing


3.17 The Cross Spectrum


3.18 Covariance Functions

A. Some Statistical Properties of the ACVF Estimators

B. Estimation of the ACVF

C. Cross Covariance Function Estimation


3.19 Coherence Functions


3.20 Phase Estimation

Chapter 4: Evoked Potentials: Averaging and Discriminant Analysis


4.1 Introduction


4.2 Estimation of Variability


4.3 Confidence Intervals


4.4 Comments on Assumptions


4.5 Alternative Measures of Variability

A. Split Sweep Assessment of Significance

B. Plus-Minus Reference Method


4.6 Correlated Noise, Overlap and Stimulus Spacing

A. Aperiodic Stimuli

B. Narrow Band Noise and Aperiodic Stimuli


4.7 The Median Evoked Response


4.8 Nonhomogeneous Sets of Evoked Potentials


4.9 Correlation Estimation of a Constant Waveform with Varying Latency


4.10 Homogeneous Subsets


4.11 The Cumsum Procedure

A. The Precum Method

B. Distribution of Precums for Homogeneous Responses

C. An Algorithm for Computing Precum


4.12 The Sort Method


4.13 Discriminant Analysis


4.14 Stepwise Discriminant Analysis


4.15 Appendix: Histograms, Smoothing, N Mode

Chapter 5: Evoked Potentials: Principal Components and Varimax Analysis


5.1 Introduction


5.2 Linear Representation of Waveforms


5.3 The Cross Correlation Coefficient


5.4 Signal Space


5.5 Linear Expansion Methods, Factor Analysis and Other Techniques


5.6 Factor Analysis and Principal Factors


5.7 Matrices and Signal Analysis

A. Matrix Properties, Definitions

B. Matrix Addition

C. Scalar Multiplication

D. Matrix Multiplication


5.8 Matrices and Linear Expansions of Waveforms

A. Cross Correlation


5.9 Transpose of Matrix Products


5.10 Some Special Matrices


5.11 Matrix Inverse


5.12 Orthogonal Matrices


5.13 Properties of Linear Expansions Based upon Orthonormal Basic Waveforms


5.14 Principal Components


5.15 Computation of Principal Components


5.16 Covariances and Correlation Coefficients


5.17 Dimensionality and Eigenvalues


5.18 Varimax Rotation of the Weighting Coefficients


5.19 Varimax Rotation of the Basic Waveforms


5.20 Principal Component Analysis and the Karhunen-Loeve Expansion


5.21 Principal Components—Varimax Analysis of Deviation Waveforms


5.22 Covariances, Correlation Coefficients, and Implied Baselines


5.23 Principal Component—Varimax Analysis Based upon Orthonormal Weighting Coefficients


5.24 Examples

A. Auditory Evoked Responses and Masking Effects

B. Effects of Drugs upon Evoked Responses

C. Independence of Components Recorded from Scalp of Humans


5.25 Some General Remarks on Linear Expansions

Chapter 6: Spontaneous and Driven Single Unit Activity


6.1 Introduction


6.2 Point Process—an Idealization of Neuronal Spike Activity

A. Spontaneously Active Processes

B. Driven Processes


6.3 Classification of Spontaneously Active Processes

A. Renewal Processes

B . Poisson Processes


6.4 Spike Data Acquisition


6.5 Interval Distribution, Mean and Variance


6.6 Tests for Mean Interval and Rate of a Poisson Process


6.7 Tests for a Poisson Process


6.8 Tests for the Parameters of a Gamma Renewal Process


6.9 Serial Statistics and Non-renewal Processes


6.10 Interval Shuffling as a Test for Renewal Processes


6.11 The Expectation Density and Covariance Function of Point Processes


6.12 Spectral Analysis of Spike Sequences

A. Relationship to the Expectation Density

B. Smoothed Estimates of Point Process Spectra


6.13 General Considerations in Spectral Smoothing


6.14 The Spectrum of Intervals and Its Relationship to the Serial Correlogram


6.15 Driven Single Unit Activity


6.16 Peristimulus Time Histogram Analysis of Driven Activity


6.17 Tests for Response Dependency on the Stimulus


6.18 Response Trends


6.19 Data Displays

Chapter 7: Multiple Unit Activity


7.1 Introduction


7.2 Cross Covariance Methods

A. Cross-Expectation Density Analysis

B. The Cross-Expectation Density during Spontaneous Activity or Continuous Stimulation

C. The Cross-Expectation Density during Stimulation


7.3 Inter-spike Interval Tests for Unit Dependency


7.4 Data Displays for Two Stimulated Units-the Peristimulus Time Scatter Diagram


7.5 Data Displays for Three Units—the Snowflake Diagram


7.6 Separation of Activity of Concurrently Discharging Units

A. Use of Salient Waveform Parameters: Amplitude, Width, Slope

B. Use of the Entire Spike Wave form

C. Identification Errors and the Analysis of Unit Interactions


7.7 Multi-unit Activity as an Entity

Chapter 8: Relations between Slow Wave and Unit Activity


8.1 Introduction


8.2 Some General Considerations on Covariance and Spectral Analysis


8.3 Estimation of the Linear Component of Process Interactions


8.4 Estimation of Cross Covariance between a Continuous and a Point Process


8.5 Some Dangers in Cross Covariance Estimation by an Average Response Computer


8.6 Cross-Expectation Relations between a Point Process and Local Features of a Continuous Process


8.7 Cross Spectral Analysis


8.8 Transfer Function Estimation


8.9 Conditional Probability Descriptions


8.10 Continuous Process Probabilities Conditioned by Point Process Events


8.11 Relations between Processes during Stimulation - Comparison of AEPs and PSTs


8.12 Changes of State in Point and Continuous Processes

Subject Index