Grün / Rotter Analysis of Parallel Spike Trains

1. Auflage 2010
ISBN: 978-1-4419-5675-0
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 7, 444 Seiten

Reihe: Springer Series in Computational Neuroscience

Solid and transparent data analysis is the most important basis for reliable interpretation of experiments. The technique of parallel spike train recordings using multi-electrode arrangements has been available for many decades now, but only recently gained wide popularity among electro physiologists. Many traditional analysis methods are based on firing rates obtained by trial-averaging, and some of the assumptions for such procedures to work can be ignored without serious consequences. The situation is different for correlation analysis, the result of which may be considerably distorted if certain critical assumptions are violated. The focus of this book is on concepts and methods of correlation analysis (synchrony, patterns, rate covariance), combined with a solid introduction into approaches for single spike trains, which represent the basis of correlations analysis. The book also emphasizes pitfalls and potential wrong interpretations of data due to violations of critical assumptions.

Autoren/Hrsg.

Grün, Sonja

Rotter, Stefan

Inhaltsverzeichnis

1;Foreword;4
1.1;References;7
2;Preface;8
2.1;Why Data Analysis of Parallel Spike Trains is Needed;8
2.2;Purpose of the Book;9
2.3;Intended Audience;10
2.4;Organization of the Book;10
2.5;How to Read This Book;11
2.6;Software Download;11
2.7;Acknowledgements;12
3;Contents;13
4;Contributors;15
5;Basic Spike Train Statistics: Point Process Models;18
5.1;Stochastic Models of Spike Trains;19
5.1.1;Introduction;19
5.1.2;Renewal Processes;20
5.1.2.1;Some General Properties of Renewal Processes;21
5.1.3;The Laplace Transform;22
5.1.3.1;Examples of Renewal Processes;24
5.1.3.2;Autocorrelation;25
5.1.3.3;Spike Count and Fano Factor;26
5.1.3.4;Variable Rates;28
5.1.4;Spike-Train Models with Memory;30
5.1.4.1;Some Models of Stochastic Spike Trains with Serial Correlation;30
5.1.4.2;General Description of Stochastic Point Processes with Memory;31
5.1.4.3;Properties of Stochastic Point-Processes with Memory;32
5.1.4.4;Dependence of the Fano Factor on Serial Correlations;34
5.1.4.5;Sensitivity of CV2 to Serial Correlations;35
5.1.5;References;36
5.2;Estimating the Firing Rate;37
5.2.1;Introduction;37
5.2.2;Methods for Estimating the Firing Rate;38
5.2.2.1;PSTH;39
5.2.2.2;The Kernel Density Estimation;39
5.2.3;Methods for Optimizing the Rate Estimation;41
5.2.3.1;MISE Principle;42
5.2.3.1.1;MISE Optimization of PSTH;42
5.2.3.1.2;MISE Optimization of Kernel Density Estimation;44
5.2.3.1.3;TIPS;44
5.2.3.2;Likelihood Principle;45
5.2.3.2.1;Empirical Bayes Method of Rate Estimation;45
5.2.3.2.2;TIPS;47
5.2.4;Discussion;48
5.2.5;References;49
5.3;Analysis and Interpretation of Interval and Count Variability in Neural Spike Trains;52
5.3.1;Introduction;52
5.3.2;The Analysis of Inter-Spike Interval Variability;53
5.3.2.1;The Coefficient of Variation and Bias of Estimation;53
5.3.2.2;Analysis of Rate-Modulated Spike Trains;55
5.3.2.2.1;Step 1. Estimation of the Rate Function;56
5.3.2.2.2;Step 2. Demodulation and Analysis in Operational Time;57
5.3.2.2.3;Time-Resolved Analysis of the CV;57
5.3.2.2.4;Alternative Methods;58
5.3.3;The Combined Analysis of Interval and Count Variability;59
5.3.3.1;Fano Factor and Bias of Estimation;59
5.3.3.2;Fano Factor vs. Squared Coefficient of Variation;60
5.3.3.3;The Effect of Serial Interval Correlation;61
5.3.3.4;The Effect of Nonstationarity;62
5.3.3.4.1;Slow Activity Fluctuations Introduce Across-Trial Nonstationarity;63
5.3.3.4.2;Task-Related Variability Dynamics;65
5.3.4;Interpretations;66
5.3.4.1;Components of Single Neuron Variability;66
5.3.4.2;Serial Interval Correlations;67
5.3.4.3;Nonstationary Conditions in the Living Brain;68
5.3.5;Appendix;70
5.3.5.1;Matlab Tools for Simulation and Analysis;70
5.3.5.2;Point Process Models;70
5.3.6;References;71
5.4;Processing of Phase-Locked Spikes and Periodic Signals;74
5.4.1;Introduction;74
5.4.2;Analysis of Angular Data;75
5.4.3;Calculating Vector Strength;77
5.4.4;Sources of Variation in Phase-Locking, Frequency Dependence and Temporal Dispersion;79
5.4.5;Rayleigh Test and Statistical Tests of Vector Strength;82
5.4.6;Relationship to Fourier Analysis;82
5.4.7;Additional Measures and Some Technical Issues;83
5.4.8;Modeling Phase-Locked Spike Trains;84
5.4.9;References;88
6;Pairwise Comparison of Spike Trains;90
6.1;Pair-Correlation in the Time and Frequency Domain;91
6.1.1;Dual Worlds: Time and Frequency Domain Representations are Related by Fourier Transforms;91
6.1.1.1;Cross-Correlation and Cross-Spectrum: FFT Pairs;92
6.1.1.2;Practical Issues in Estimating Correlations and Spectra;93
6.1.1.3;Impulse Response and Cross-Correlation;93
6.1.1.4;Correlation and Coherence: Symmetry Breaking Through Normalization;94
6.1.2;Practical Pair-Correlation Estimation for Neural Spike Trains Under Spontaneous Firing Conditions;95
6.1.2.1;Representation of the Cross-Correlogram in Terms of Coincidences;95
6.1.2.2;Representations of the Cross-Correlogram in Terms of Firing Rate;96
6.1.3;Testing for Stationarity of Individual Spike Trains;96
6.1.3.1;Time-Dependent Cross-Correlation Functions;96
6.1.3.2;Stationarity Tests for Spike Trains;98
6.1.4;Pair-Correlation Under Stimulus Conditions;98
6.1.4.1;Practical Estimation of Stimulus Correlation;101
6.1.4.2;Rate Correlation and Event Correlation;102
6.1.4.3;Stimulus-Dependent Neural Correlation;103
6.1.5;Pair Coherence for Neural Spike Trains;105
6.1.5.1;Correcting for Common Input Firing Periodicities and Firing Rates;105
6.1.5.2;Oscillatory Cross-Correlograms;107
6.1.5.3;Stimulus Corrections in the Frequency Domain;107
6.1.5.4;Time-Dependent Coherence;111
6.1.6;Correlation and Connectivity;111
6.1.7;Effects of Spike Sorting on Pair-Correlations;113
6.1.8;Correlations and the Brain;113
6.1.9;References;114
6.2;Dependence of Spike-Count Correlations on Spike-Train Statistics and Observation Time Scale;117
6.2.1;Introduction;117
6.2.2;Shot-Noise Correlations;118
6.2.2.1;Shot-Noise;119
6.2.2.2;Correlation Functions;119
6.2.2.3;Variance, Covariance and Correlation Coefficient;121
6.2.2.4;Spectra and Coherence;122
6.2.3;Spike-Count Correlations in a Simple Common-Input Model;123
6.2.3.1;Model Definition;124
6.2.3.2;Correlation Functions;124
6.2.3.2.1;Gamma Source;125
6.2.3.2.2;Inhomogeneous Poisson Source;126
6.2.3.3;Bin-Size and Autocorrelation Dependence of Spike-Count Correlations;129
6.2.3.4;Coherence;130
6.2.3.5;Jittered Correlations;133
6.2.4;Summary;135
6.2.5;Appendix;136
6.2.5.1;Estimation of the Common-Input Strength for Correlated Spike Trains;136
6.2.5.2;Count Covariances for Jittered Correlations;137
6.2.5.2.1;Rectangular Cross-Correlations;137
6.2.5.2.2;Gaussian Cross-Correlations;138
6.2.5.3;Notation;138
6.2.6;References;140
6.3;Spike Metrics;142
6.3.1;Introduction;142
6.3.1.1;Mathematics and Laboratory Data;142
6.3.1.2;Representing Spike Trains as Samples of Point Processes;143
6.3.1.3;Analyzing Point Processes: The Rationale for a Metric-Space Approach;143
6.3.1.4;Plan for this Chapter;145
6.3.2;Spike Train Metrics;145
6.3.2.1;Notation and Preliminaries;146
6.3.2.2;Cost-Based (Edit Length) Metrics;146
6.3.2.2.1;General Definition;146
6.3.2.2.2;Spike Time Metrics;147
6.3.2.2.3;Spike Interval Metrics;148
6.3.2.2.4;Multineuronal Cost-Based Metrics;149
6.3.2.2.5;Other Cost-Based Metrics;149
6.3.2.2.5.1;More Flexible Assignments of Costs to the Elementary Steps;149
6.3.2.2.5.2;Other Kinds of Elementary Steps;150
6.3.2.2.5.3;Further Generalizations;151
6.3.2.2.6;Algorithms;151
6.3.2.2.6.1;The Basic Dynamic Programming Algorithm;152
6.3.2.2.6.2;Extensions of the Dynamic Programming Algorithm;153
6.3.2.3;Spike-Train Metrics Based on Vector-Space Embeddings;154
6.3.2.3.1;Single-Neuron Metrics Based on Vector Space Embeddings;155
6.3.2.3.2;Multineuronal Metrics Based on Vector-Space Embeddings;156
6.3.2.3.3;Computational Considerations;158
6.3.2.4;Applications;158
6.3.2.4.1;Overview;158
6.3.2.4.2;Assessment of Variability;159
6.3.2.4.3;Construction of Response Spaces;159
6.3.2.4.3.1;Standard Multidimensional Scaling;160
6.3.2.4.3.2;Examples;160
6.3.2.4.3.3;Implications of Non-Euclidean Nature of Spike Metrics;161
6.3.2.4.3.4;Relationship to the "Kernel Trick" and van Rossum-Type Metrics;161
6.3.2.4.3.5;Nonlinear Scaling;162
6.3.2.4.4;Applications to Information-Theoretic Analysis;162
6.3.2.4.4.1;Examples;165
6.3.3;Conclusion;166
6.3.4;References;167
6.4;Gravitational Clustering;170
6.4.1;Introduction;170
6.4.2;The Basic Gravity Representation;171
6.4.3;Visualization of the Gravitational Analysis;174
6.4.4;Significance Testing for Distance Trajectories;178
6.4.5;Variations on the Basic Gravity Computation;178
6.4.5.1;Forward and Backward Charges;178
6.4.5.2;Nonlinear Functions of Charge Product;179
6.4.5.3;Temporal Clustering of Pair Interactions;180
6.4.6;Tuned Gravity;181
6.4.7;Repeating Synchrony Patterns and Time Markers;181
6.4.8;Other Visualizations;182
6.4.9;Conclusion;184
6.4.10;References;184
7;Multiple-Neuron Spike Patterns;186
7.1;Spatio-Temporal Patterns;187
7.1.1;Introduction;187
7.1.2;Types of Precise Spatio-Temporal Patterns;188
7.1.2.1;Statistical Significance;191
7.1.2.2;Significance of a Particular PFS;193
7.1.2.3;Multiple Comparisons;195
7.1.2.3.1;PDF of Probabilities;196
7.1.2.3.2;A Summarizing Statistic;197
7.1.2.4;Further Work;200
7.1.3;References;201
7.2;Unitary Event Analysis;202
7.2.1;Introduction;202
7.2.2;Basic Elements of the UE approach;203
7.2.2.1;Detection of Joint-Spike Events;204
7.2.2.2;Null Hypothesis;204
7.2.2.3;Significance of Joint-Spike Events;205
7.2.2.4;Capturing Dynamics of Correlation;206
7.2.3;Parameter Dependencies;208
7.2.3.1;Analysis Window Width;208
7.2.3.2;Firing Rate;211
7.2.3.3;Temporal Precision of Joint-Spike Events;212
7.2.4;Impact of Nonstationarities and Other Violations of Assumptions;215
7.2.4.1;Nonstationary Rates;215
7.2.4.2;Cross-Trial Nonstationarity;217
7.2.4.2.1;Intermediate Summary on Nonstationarities;220
7.2.4.3;Non-Poisson Processes;221
7.2.5;Discussion;222
7.2.5.1;Surrogates;223
7.2.5.2;Population Measures;225
7.2.5.3;Relation to Other Analysis Methods;226
7.2.5.4;Conclusion;228
7.2.6;References;229
7.3;Information Geometry of Multiple Spike Trains;232
7.3.1;Introduction;232
7.3.2;Joint Probability Distributions of Neural Firing;234
7.3.2.1;Joint Firing Coordinates of Full Statistical Model;234
7.3.2.2;Log Interaction Coordinates;235
7.3.2.3;Coordinate Transformation Between theta and eta;236
7.3.2.4;Fisher Information;237
7.3.2.5;Geometry of Sn and Orthogonal Parameters;239
7.3.3;Separation of Correlations from Firing Rates;240
7.3.3.1;Orthogonal Measure of Correlation;240
7.3.3.2;Kullback-Leibler Divergence;242
7.3.3.3;Higher-Order Correlations;244
7.3.4;Tractable Models of Probability Distributions;246
7.3.4.1;Homogeneous Model;246
7.3.4.2;Boltzmann Machine;247
7.3.4.3;Marginal Models;248
7.3.4.4;Higher-Order Correlations Generated from Common Inputs;250
7.3.5;Mixture Model and Its Dynamics;251
7.3.6;Temporal Correlations of Spikes in a Single Neuron;255
7.3.6.1;Full Model of Train of Spikes and Stationary Model;255
7.3.6.2;Markov Chain;255
7.3.6.3;Estimation of Shape Parameter in Renewal Process;258
7.3.6.4;Discrimination of ISI Distributions;261
7.3.7;Conclusions;261
7.3.8;References;262
7.4;Higher-Order Correlations and Cumulants;264
7.4.1;Introduction;264
7.4.2;Correlation;266
7.4.2.1;Pairwise Correlation;266
7.4.2.2;Higher-Order Correlations (N>2);267
7.4.2.3;The Additive Common Component Model;270
7.4.3;Correlated Poisson Processes;272
7.4.3.1;Firing Rate and Pairwise Correlations;274
7.4.3.2;Examples;275
7.4.3.2.1;SIP-Like Models;276
7.4.3.2.2;MIP-Like Models;277
7.4.4;Data Analysis;277
7.4.4.1;CuBIC;278
7.4.4.2;De-Poissonization;279
7.4.5;Cumulants vs. Exponential Interactions;280
7.4.6;Conclusions;283
7.4.7;Appendix A: Cumulants;283
7.4.7.1;Univariate Random Variables;283
7.4.7.1.1;Moments;283
7.4.7.1.2;Cumulants;284
7.4.7.2;Multivariate Random Variables and Correlation;285
7.4.7.3;Proof of Theorem 1;286
7.4.7.4;Cumulants of the Population Spike Count in the Additive Poisson Model;287
7.4.8;Appendix B: Computing Correlation Parameters in Practice;287
7.4.9;References;289
8;Population-Based Approaches;292
8.1;Information Theory and Systems Neuroscience;293
8.1.1;Introduction;293
8.1.2;The Encoder;295
8.1.2.1;Entropy;295
8.1.2.2;Entropy and Neuroscience;296
8.1.3;The Channel;298
8.1.3.1;Mutual Information;299
8.1.3.2;Capacity and Reliable Communication;300
8.1.3.3;Capacity and Neuroscience;301
8.1.4;Entire System Analysis;304
8.1.4.1;Rate-Distortion Theory;304
8.1.4.2;Rate-Distortion Theory and Neuroscience;306
8.1.5;Post-Shannon Information Theory;307
8.1.5.1;Kullback-Leibler Distance;307
8.1.5.2;Data Processing Theorem;308
8.1.6;Measuring Information Theoretic Quantities;308
8.1.6.1;Entropy and Mutual Information;309
8.1.6.2;Kullback-Leibler Distance;309
8.1.7;Conclusions;310
8.1.8;References;310
8.2;Population Coding;312
8.2.1;Introduction;312
8.2.2;Definitions of the Experimental Quantities;313
8.2.3;Quantifying the Role of Correlated Firing in Population Coding;314
8.2.3.1;Signal vs Noise Correlations;314
8.2.3.2;The Information Breakdown;317
8.2.3.2.1;The Linear Term;318
8.2.3.2.2;The Signal Similarity Term;319
8.2.3.2.3;The Terms Quantifying the Impact of Noise Correlation;319
8.2.3.3;Examples of Application of the Information Breakdown to Neural Data;321
8.2.3.3.1;Role of Correlated Firing in Coding Whisker Stimuli;321
8.2.3.3.2;Role of Correlations in Encoding Stimulus Contrast in Monkey Visual Cortex;322
8.2.4;Studying the Information Content Through Decoding Algorithms;323
8.2.5;Pooling as a Strategy for Population Coding;324
8.2.6;Software Implementation;326
8.2.7;Conclusions;326
8.2.8;References;326
8.3;Stochastic Models for Multivariate Neural Point Processes: Collective Dynamics and Neural Decoding;329
8.3.1;Introduction;329
8.3.2;Estimation of Conditional Intensity Functions;332
8.3.2.1;Generalized Linear Models;332
8.3.2.2;Penalized Generalized Linear Models and Penalized B-Splines;334
8.3.2.3;Hierarchical Bayesian P-Spline Models;335
8.3.2.4;Nonparametric Function Approximation;336
8.3.2.5;Statistical Inference;338
8.3.3;Neural Ensemble Decoding;340
8.3.4;Neuronal Ensemble Collective Dynamics;344
8.3.5;Summary and Future Directions;346
8.3.6;References;346
9;Practical Issues;350
9.1;Simulation of Stochastic Point Processes with Defined Properties;351
9.1.1;Point Processes and Thinning;351
9.1.2;Poisson Processes;353
9.1.2.1;Homogeneous Poisson Process;353
9.1.2.2;Inhomogeneous Poisson Process;354
9.1.2.3;Count Distribution and Operational Time;355
9.1.2.4;Correlated Poisson Processes;356
9.1.3;Renewal Processes;357
9.1.3.1;Ordinary Renewal Processes;357
9.1.3.2;The Master Equation of a Point Process with Time-Dependent Hazard;359
9.1.3.3;Renewal Processes in Equilibrium;359
9.1.3.4;Nonstationary Renewal Processes and Operational Time;360
9.1.3.5;Operational Time and Real Neural Data;361
9.1.4;References;363
9.2;Generation and Selection of Surrogate Methods for Correlation Analysis;364
9.2.1;Introduction;364
9.2.2;Example Data Sets;365
9.2.3;Surrogate Generation;367
9.2.3.1;Trial Shuffling;369
9.2.3.2;Spike Time Randomization;371
9.2.3.3;Spike Train Dithering;372
9.2.3.4;Spike Time Dithering;372
9.2.3.5;Joint Interspike Interval Dithering;373
9.2.3.6;Spike Exchanging;374
9.2.4;Correlation Analysis;376
9.2.4.1;Cross-Correlogram Analysis;376
9.2.4.2;Significance of Spike Coincidences;378
9.2.4.3;Performance of Surrogates;379
9.2.4.4;Suitability of Surrogates for Different Data Sets;381
9.2.5;Discussion;384
9.2.6;References;386
9.3;Bootstrap Tests of Hypotheses;388
9.3.1;Tests of Hypotheses;388
9.3.2;Bootstrap Tests of Hypotheses;391
9.3.3;Goodness of Fit Test ;394
9.3.4;Simulation Error and Statistical Error;395
9.3.5;Comparing Neuronal Responses;397
9.3.6;Synchrony;399
9.3.7;Joint Null Envelope for a Function;401
9.3.8;References;402
9.4;Generating Random Numbers;404
9.4.1;Requirements for Pseudorandom Number Generators;405
9.4.2;Recurrence-Based Generators;406
9.4.2.1;Linear Congruential Generators;407
9.4.2.2;Lagged Fibonacci Generators;407
9.4.2.3;Combined Multiple Recursive Generators;408
9.4.2.4;Mersenne Twister and Related Generators;409
9.4.2.5;Nonlinear Random Number Generators;410
9.4.3;Cryptographically Strong Random Number Generators;410
9.4.4;Seeding Random Number Generators;411
9.4.4.1;Parallel Streams of Random Numbers;412
9.4.5;Transforming Random Numbers;413
9.4.6;Recommendations;414
9.4.7;References;414
9.5;Practically Trivial Parallel Data Processing in a Neuroscience Laboratory;417
9.5.1;Introducing a Simple Serial Program;420
9.5.2;The Idea of Trivial Parallelization;422
9.5.2.1;Theoretical Considerations;422
9.5.2.2;Basic Parallelization of the Example Program;424
9.5.3;Starting a Parallel Job on a Single Multicore Computer;425
9.5.3.1;A Manual Solution: Screen;425
9.5.3.2;An Automated Solution: Make;427
9.5.4;Using a Queuing System to Distribute Jobs;428
9.5.5;Introducing Job Dependencies;433
9.5.6;Using Parallelization Libraries;435
9.5.7;Concluding Remarks;439
9.5.8;References;440
10;Erratum to: Population Coding;441
11;Index;442