# Information Dynamics Foundations and Applications

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1 Gustavo Deco Bernd Schürmann Information Dynamics Foundations and Applications With 89 Illustrations Springer

2 PREFACE vii CHAPTER 1 Introduction 1 CHAPTER 2 Dynamical Systems: An Overview Deterministic Dynamical Systems Fundamental Concepts Attractors Strange Attractors: Chaotic Dynamics Quantitative Description of Chaos Chaotic Dynamical Systems Stochastic Dynamical Systems Gaussian White Noise Markov Processes Linear and Nonlinear Stochastic Dynamics Statistical Time-Series Analysis Nonstationarity: Slicing Windows 34

3 2.3.2 Linear Statistical Inference: Correlations and Power Spectrum Linear Filter 36 CHAPTER 3 Statistical Structure Extraction in Dynamical Systems: Parametric Formulation Basic Concepts of Information Theory Parametric Estimation: Maximum-Likelihood Principle Bayesian Estimation Maximum Likelihood Maximum-Entropy Principle Minimum Kullback-Leibler Entropy Linear Models Nonlinear Models Feedforward Neural Networks Recurrent Neural Networks Density Estimation Information-Theoretic Approach to Time-Series Modeling: Redundancy Extraction Generalities Unsupervised Learning: Independent Component Analysis for Univariate Time Series Unsupervised Learning: Independent Component Analysis for Multivariate Time Series Supervised Learning: Maximum-Likelihood 68 CHAPTER 4 Applications: Parametric Characterization of Time Series 4.1 Feedforward Learning: Chaotic Dynamics 4.2 Recurrent Learning: Chaotic Dynamics

4 xii 4.3 Dynamical Overtraining and Lyapunov Penalty Term Feedforward and Recurrent Learning of Biomedical Data Unsupervised Redundancy-Extraction-Based Modeling: Chaotic Dynamics Univariate Time Series: Mackey-Glass Multivariate Time Series: Taylor-Couette Unsupervised Redundancy Extraction Modeling: Biomedical Data 89 CHAPTER 5 Statistical Structure Extraction in Dynamical Systems: Nonparametric Formulation 5.1 Nonparametric Detection of Statistical Dependencies in Time Series Introduction and Historical Perspective Statistical Independence Measure Statistical Test: The Surrogates Method Nonstationarity A Qualitative Test of Nonlinearity 5.2 Nonparametric Characterization of Dynamics: The Information Flow Concept Introduction and Historical Perspective Information Flow for Finite Partitions Information Flow for Infinitesimal Partition 5.3 Information Flow and Coarse Graining Generalized Correlation Functions Distinguishing Different Dynamics CHAPTER 6 Applications: Nonparametric Characterization of Time Series Detecting Nonlinear Correlations in Time Series Test of Nonlinearity 128

5 XIV Contents Testing Predictability: Artificial Time Series Testing Predictability: Real-World Time Series DataSelection Sensitivity Analysis Nonparametric Analysis of Time Series: Optimal Delay Selection Nonchaotic Deterministic Linear Stochastic Chaotic Deterministic Determining the Information Flow of Dynamical Systems from Continuous Probability Distributions Dynamical Characterization of Time Signals: The Integrated Information Flow Information Flow and Coarse Graining: Numerical Experiments The Logistic Map White and Colored Noise EEG Signals 162 CHAPTER 7 Statistical Structure Extraction in Dynamical Systems: Semiparametric Formulation Markovian Characterization of Univariate Time Series Measures of Independence Markovian Dynamics and Information Flow Markovian Characterization of Multivariate Time Series Multidimensional Cumulant-Based Measure of Information Flow Nonlinear AT-dimensional Markov Models as Approximations of the Original Time Series 176

6 xv CHAPTER 8 Applications: Semiparametnc Charactenzation oftime Series Univariate Time Series: Artificial Data Autoregressive Models: Linear Correlations Nonlinear Dependencies: Non-Chaos, Chaos, and Noisy Chaos Univariate Time Series: Real-World Data Monthly Sunspot Numbers The Hidden Dynamics of the Heart Rate Variability Multivariate Time Series: Artificial Data Autoregressive Time Series Nonlinear Time Series Chaotic Time Series: The Henon Map Multivariate Time Series: Tumor Detection in EEG Time Series 199 CHAPTER 9 Information Processing and Coding in Spatiotemporal Dynamical Systems: Spiking Networks Spiking Neurons Theoretical Models Rate Coding versus Temporal Coding Information Processing and Coding in Single Spiking Neurons Information Processing and Coding in Networks of Spiking Neurons The Processing and Coding of Dynamical Systems 225

7 CHAPTER 10 Applications: Information Processing and Coding in Spatiotemporal Dynamical Systems The Binding Problem Discrimination of Stimulus by Spiking Neural Networks The Task: Visual Stimulus Discrimination The Neural Network: Cortical Architecture Numerical Experiments 232 EPILOGUE 239 APPENDIX A Chain Rules, Inequalities and Other Useful Theorems in Information Theory 241 A.l Chain Rules 241 A.2 Fundamental Inequalities of Information Theory 245 APPENDIX B Univariate and Multivariate Cumulants 251 APPENDIX C Information Flow of Chaotic Systems: Thermodynamical Formulation 255 APPENDIX D Generalized Discriminability by the Spike Response Model of a Single Spiking Neuron: Analytical Results 259 REFERENCES 263 INDEX 275

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