Analysis of Parallel Spike Trains (eBook)

Foreword 4
References 7
Preface 8
Why Data Analysis of Parallel Spike Trains is Needed 8
Purpose of the Book 9
Intended Audience 10
Organization of the Book 10
How to Read This Book 11
Software Download 11
Acknowledgements 12
Contents 13
Contributors 15
Basic Spike Train Statistics: Point Process Models 18
Stochastic Models of Spike Trains 19
Introduction 19
Renewal Processes 20
Some General Properties of Renewal Processes 21
The Laplace Transform 22
Examples of Renewal Processes 24
Autocorrelation 25
Spike Count and Fano Factor 26
Variable Rates 28
Spike-Train Models with Memory 30
Some Models of Stochastic Spike Trains with Serial Correlation 30
General Description of Stochastic Point Processes with Memory 31
Properties of Stochastic Point-Processes with Memory 32
Dependence of the Fano Factor on Serial Correlations 34
Sensitivity of CV2 to Serial Correlations 35
References 36
Estimating the Firing Rate 37
Introduction 37
Methods for Estimating the Firing Rate 38
PSTH 39
The Kernel Density Estimation 39
Methods for Optimizing the Rate Estimation 41
MISE Principle 42
MISE Optimization of PSTH 42
MISE Optimization of Kernel Density Estimation 44
TIPS 44
Likelihood Principle 45
Empirical Bayes Method of Rate Estimation 45
TIPS 47
Discussion 48
References 49
Analysis and Interpretation of Interval and Count Variability in Neural Spike Trains 52
Introduction 52
The Analysis of Inter-Spike Interval Variability 53
The Coefficient of Variation and Bias of Estimation 53
Analysis of Rate-Modulated Spike Trains 55
Step 1. Estimation of the Rate Function 56
Step 2. Demodulation and Analysis in Operational Time 57
Time-Resolved Analysis of the CV 57
Alternative Methods 58
The Combined Analysis of Interval and Count Variability 59
Fano Factor and Bias of Estimation 59
Fano Factor vs. Squared Coefficient of Variation 60
The Effect of Serial Interval Correlation 61
The Effect of Nonstationarity 62
Slow Activity Fluctuations Introduce Across-Trial Nonstationarity 63
Task-Related Variability Dynamics 65
Interpretations 66
Components of Single Neuron Variability 66
Serial Interval Correlations 67
Nonstationary Conditions in the Living Brain 68
Appendix 70
Matlab Tools for Simulation and Analysis 70
Point Process Models 70
References 71
Processing of Phase-Locked Spikes and Periodic Signals 74
Introduction 74
Analysis of Angular Data 75
Calculating Vector Strength 77
Sources of Variation in Phase-Locking, Frequency Dependence and Temporal Dispersion 79
Rayleigh Test and Statistical Tests of Vector Strength 82
Relationship to Fourier Analysis 82
Additional Measures and Some Technical Issues 83
Modeling Phase-Locked Spike Trains 84
References 88
Pairwise Comparison of Spike Trains 90
Pair-Correlation in the Time and Frequency Domain 91
Dual Worlds: Time and Frequency Domain Representations are Related by Fourier Transforms 91
Cross-Correlation and Cross-Spectrum: FFT Pairs 92
Practical Issues in Estimating Correlations and Spectra 93
Impulse Response and Cross-Correlation 93
Correlation and Coherence: Symmetry Breaking Through Normalization 94
Practical Pair-Correlation Estimation for Neural Spike Trains Under Spontaneous Firing Conditions 95
Representation of the Cross-Correlogram in Terms of Coincidences 95
Representations of the Cross-Correlogram in Terms of Firing Rate 96
Testing for Stationarity of Individual Spike Trains 96
Time-Dependent Cross-Correlation Functions 96
Stationarity Tests for Spike Trains 98
Pair-Correlation Under Stimulus Conditions 98
Practical Estimation of Stimulus Correlation 101
Rate Correlation and Event Correlation 102
Stimulus-Dependent Neural Correlation 103
Pair Coherence for Neural Spike Trains 105
Correcting for Common Input Firing Periodicities and Firing Rates 105
Oscillatory Cross-Correlograms 107
Stimulus Corrections in the Frequency Domain 107
Time-Dependent Coherence 111
Correlation and Connectivity 111
Effects of Spike Sorting on Pair-Correlations 113
Correlations and the Brain 113
References 114
Dependence of Spike-Count Correlations on Spike-Train Statistics and Observation Time Scale 117
Introduction 117
Shot-Noise Correlations 118
Shot-Noise 119
Correlation Functions 119
Variance, Covariance and Correlation Coefficient 121
Spectra and Coherence 122
Spike-Count Correlations in a Simple Common-Input Model 123
Model Definition 124
Correlation Functions 124
Gamma Source 125
Inhomogeneous Poisson Source 126
Bin-Size and Autocorrelation Dependence of Spike-Count Correlations 129
Coherence 130
Jittered Correlations 133
Summary 135
Appendix 136
Estimation of the Common-Input Strength for Correlated Spike Trains 136
Count Covariances for Jittered Correlations 137
Rectangular Cross-Correlations 137
Gaussian Cross-Correlations 138
Notation 138
References 140
Spike Metrics 142
Introduction 142
Mathematics and Laboratory Data 142
Representing Spike Trains as Samples of Point Processes 143
Analyzing Point Processes: The Rationale for a Metric-Space Approach 143
Plan for this Chapter 145
Spike Train Metrics 145
Notation and Preliminaries 146
Cost-Based (Edit Length) Metrics 146
General Definition 146
Spike Time Metrics 147
Spike Interval Metrics 148
Multineuronal Cost-Based Metrics 149
Other Cost-Based Metrics 149
More Flexible Assignments of Costs to the Elementary Steps 149
Other Kinds of Elementary Steps 150
Further Generalizations 151
Algorithms 151
The Basic Dynamic Programming Algorithm 152
Extensions of the Dynamic Programming Algorithm 153
Spike-Train Metrics Based on Vector-Space Embeddings 154
Single-Neuron Metrics Based on Vector Space Embeddings 155
Multineuronal Metrics Based on Vector-Space Embeddings 156
Computational Considerations 158
Applications 158
Overview 158
Assessment of Variability 159
Construction of Response Spaces 159
Standard Multidimensional Scaling 160
Examples 160
Implications of Non-Euclidean Nature of Spike Metrics 161
Relationship to the "Kernel Trick" and van Rossum-Type Metrics 161
Nonlinear Scaling 162
Applications to Information-Theoretic Analysis 162
Examples 165
Conclusion 166
References 167
Gravitational Clustering 170
Introduction 170
The Basic Gravity Representation 171
Visualization of the Gravitational Analysis 174
Significance Testing for Distance Trajectories 178
Variations on the Basic Gravity Computation 178
Forward and Backward Charges 178
Nonlinear Functions of Charge Product 179
Temporal Clustering of Pair Interactions 180
Tuned Gravity 181
Repeating Synchrony Patterns and Time Markers 181
Other Visualizations 182
Conclusion 184
References 184
Multiple-Neuron Spike Patterns 186
Spatio-Temporal Patterns 187
Introduction 187
Types of Precise Spatio-Temporal Patterns 188
Statistical Significance 191
Significance of a Particular PFS 193
Multiple Comparisons 195
PDF of Probabilities 196
A Summarizing Statistic 197
Further Work 200
References 201
Unitary Event Analysis 202
Introduction 202
Basic Elements of the UE approach 203
Detection of Joint-Spike Events 204
Null Hypothesis 204
Significance of Joint-Spike Events 205
Capturing Dynamics of Correlation 206
Parameter Dependencies 208
Analysis Window Width 208
Firing Rate 211
Temporal Precision of Joint-Spike Events 212
Impact of Nonstationarities and Other Violations of Assumptions 215
Nonstationary Rates 215
Cross-Trial Nonstationarity 217
Intermediate Summary on Nonstationarities 220
Non-Poisson Processes 221
Discussion 222
Surrogates 223
Population Measures 225
Relation to Other Analysis Methods 226
Conclusion 228
References 229
Information Geometry of Multiple Spike Trains 232
Introduction 232
Joint Probability Distributions of Neural Firing 234
Joint Firing Coordinates of Full Statistical Model 234
Log Interaction Coordinates 235
Coordinate Transformation Between theta and eta 236
Fisher Information 237
Geometry of Sn and Orthogonal Parameters 239
Separation of Correlations from Firing Rates 240
Orthogonal Measure of Correlation 240
Kullback-Leibler Divergence 242
Higher-Order Correlations 244
Tractable Models of Probability Distributions 246
Homogeneous Model 246
Boltzmann Machine 247
Marginal Models 248
Higher-Order Correlations Generated from Common Inputs 250
Mixture Model and Its Dynamics 251
Temporal Correlations of Spikes in a Single Neuron 255
Full Model of Train of Spikes and Stationary Model 255
Markov Chain 255
Estimation of Shape Parameter in Renewal Process 258
Discrimination of ISI Distributions 261
Conclusions 261
References 262
Higher-Order Correlations and Cumulants 264
Introduction 264
Correlation 266
Pairwise Correlation 266
Higher-Order Correlations (N> 2)
The Additive Common Component Model 270
Correlated Poisson Processes 272
Firing Rate and Pairwise Correlations 274
Examples 275
SIP-Like Models 276
MIP-Like Models 277
Data Analysis 277
CuBIC 278
De-Poissonization 279
Cumulants vs. Exponential Interactions 280
Conclusions 283
Appendix A: Cumulants 283
Univariate Random Variables 283
Moments 283
Cumulants 284
Multivariate Random Variables and Correlation 285
Proof of Theorem 1 286
Cumulants of the Population Spike Count in the Additive Poisson Model 287
Appendix B: Computing Correlation Parameters in Practice 287
References 289
Population-Based Approaches 292
Information Theory and Systems Neuroscience 293
Introduction 293
The Encoder 295
Entropy 295
Entropy and Neuroscience 296
The Channel 298
Mutual Information 299
Capacity and Reliable Communication 300
Capacity and Neuroscience 301
Entire System Analysis 304
Rate-Distortion Theory 304
Rate-Distortion Theory and Neuroscience 306
Post-Shannon Information Theory 307
Kullback-Leibler Distance 307
Data Processing Theorem 308
Measuring Information Theoretic Quantities 308
Entropy and Mutual Information 309
Kullback-Leibler Distance 309
Conclusions 310
References 310
Population Coding 312
Introduction 312
Definitions of the Experimental Quantities 313
Quantifying the Role of Correlated Firing in Population Coding 314
Signal vs Noise Correlations 314
The Information Breakdown 317
The Linear Term 318
The Signal Similarity Term 319
The Terms Quantifying the Impact of Noise Correlation 319
Examples of Application of the Information Breakdown to Neural Data 321
Role of Correlated Firing in Coding Whisker Stimuli 321
Role of Correlations in Encoding Stimulus Contrast in Monkey Visual Cortex 322
Studying the Information Content Through Decoding Algorithms 323
Pooling as a Strategy for Population Coding 324
Software Implementation 326
Conclusions 326
References 326
Stochastic Models for Multivariate Neural Point Processes: Collective Dynamics and Neural Decoding 329
Introduction 329
Estimation of Conditional Intensity Functions 332
Generalized Linear Models 332
Penalized Generalized Linear Models and Penalized B-Splines 334
Hierarchical Bayesian P-Spline Models 335
Nonparametric Function Approximation 336
Statistical Inference 338
Neural Ensemble Decoding 340
Neuronal Ensemble Collective Dynamics 344
Summary and Future Directions 346
References 346
Practical Issues 350
Simulation of Stochastic Point Processes with Defined Properties 351
Point Processes and Thinning 351
Poisson Processes 353
Homogeneous Poisson Process 353
Inhomogeneous Poisson Process 354
Count Distribution and Operational Time 355
Correlated Poisson Processes 356
Renewal Processes 357
Ordinary Renewal Processes 357
The Master Equation of a Point Process with Time-Dependent Hazard 359
Renewal Processes in Equilibrium 359
Nonstationary Renewal Processes and Operational Time 360
Operational Time and Real Neural Data 361
References 363
Generation and Selection of Surrogate Methods for Correlation Analysis 364
Introduction 364
Example Data Sets 365
Surrogate Generation 367
Trial Shuffling 369
Spike Time Randomization 371
Spike Train Dithering 372
Spike Time Dithering 372
Joint Interspike Interval Dithering 373
Spike Exchanging 374
Correlation Analysis 376
Cross-Correlogram Analysis 376
Significance of Spike Coincidences 378
Performance of Surrogates 379
Suitability of Surrogates for Different Data Sets 381
Discussion 384
References 386
Bootstrap Tests of Hypotheses 388
Tests of Hypotheses 388
Bootstrap Tests of Hypotheses 391
Goodness of Fit Test 394
Simulation Error and Statistical Error 395
Comparing Neuronal Responses 397
Synchrony 399
Joint Null Envelope for a Function 401
References 402
Generating Random Numbers 404
Requirements for Pseudorandom Number Generators 405
Recurrence-Based Generators 406
Linear Congruential Generators 407
Lagged Fibonacci Generators 407
Combined Multiple Recursive Generators 408
Mersenne Twister and Related Generators 409
Nonlinear Random Number Generators 410
Cryptographically Strong Random Number Generators 410
Seeding Random Number Generators 411
Parallel Streams of Random Numbers 412
Transforming Random Numbers 413
Recommendations 414
References 414
Practically Trivial Parallel Data Processing in a Neuroscience Laboratory 417
Introducing a Simple Serial Program 420
The Idea of Trivial Parallelization 422
Theoretical Considerations 422
Basic Parallelization of the Example Program 424
Starting a Parallel Job on a Single Multicore Computer 425
A Manual Solution: Screen 425
An Automated Solution: Make 427
Using a Queuing System to Distribute Jobs 428
Introducing Job Dependencies 433
Using Parallelization Libraries 435
Concluding Remarks 439
References 440
Erratum to: Population Coding 441
Index 442