Multiscale Analysis and Nonlinear Dynamics: From Genes to the Brain

Since modeling multiscale phenomena in systems biology and neuroscience is a highly interdisciplinary task, the editor of the book invited experts in bio–engineering, chemistry, cardiology, neuroscience, computer science, and applied mathematics, to provide their perspectives. Each chapter is a window into the current state of the art in the areas of research discussed and the book is intended for advanced researchers interested in recent developments in these fi elds. While multiscale analysis is the major integrating theme of the book, its subtitle does not call for bridging the scales from genes to behavior, but rather stresses the unifying perspective offered by the concepts referred to in the title. It is believed that the interdisciplinary approach adopted here will be beneficial for all the above mentioned fi elds. Indeed, the roads between different sciences, “while often the quickest shortcut to another part of our own science, are not visible from the viewpoint of one science alone” (P.W. Anderson, More is Different). From the contents: MULTIRESOLUTION ANALYSIS — Discrete Geometric Structures in Homogenization and Inverse Homogenization — Multiresolution Analysis on Compact Riemannian Manifolds NONLINEAR DYNAMICS IN SYNTHETIC BIOCHEMICAL CIRCUITS — Dynamics of Synthetic Transcription Networks — Synthetic Biochemical Dynamic Circuits NONLINEAR DYNAMICS: THE HEART AND THE BRAIN — Theoretical and Experimental Electrophysiology in Human Neocortex:    Multiscale Dynamic Correlates of Conscious Experience — Multiscale Network Organization in the Human Brain — Neuronal Oscillations Scale up and Scale Down the Brain Dynamics — Linking Nonlinear Neural Dynamics to Single–trial Human Behavior — Brain Dynamics at Rest: How Structure Shapes Dynamics — Adaptive Multiscale Encoding – a Computational Function of Neuronal Synchronization — Multiscale Nonlinear Dynamics in Cardiac Electrophysiology: From Sparks to Sudden Death — Measures of Spike Train Synchrony: From Single Neurons to Populations  

List of Contributors XIII Preface XVII 1 Introduction: Multiscale Analysis – Modeling, Data, Networks, and Nonlinear Dynamics 1 Misha (Meyer) Z. Pesenson 1.1 Multiscale Modeling 5 1.2 Multiresolution Analysis and Processing of High–Dimensional Information/Data 10 1.3 Multiscale Analysis, Networks, and Nonlinear Dynamics 11 1.4 Conclusions 14 References 14 Part One Multiscale Analysis 19 2 Modeling Across Scales: Discrete Geometric Structures in Homogenization and Inverse Homogenization 21 Mathieu Desbrun, Roger D. Donaldson, and Houman Owhadi 2.1 Introduction 21 2.2 Homogenization of Conductivity Space 23 2.3 Discrete Geometric Homogenization 31 2.4 Optimal Weighted Delaunay Triangulations 39 2.5 Relationship to Inverse Homogenization 47 2.6 Electrical Impedance Tomography 49 References 61 3 Multiresolution Analysis on Compact Riemannian Manifolds 65 Isaac Z. Pesenson 3.1 Introduction 65 3.2 Compact Manifolds and Operators 66 3.3 Hilbert Frames 69 3.4 Multiresolution and Sampling 70 3.5 Shannon Sampling of Band–Limited Functions on Manifolds 72 3.6 Localized Frames on Compact Manifolds 73 3.7 Parseval Frames on Homogeneous Manifolds 76 3.8 Variational Splines on Manifolds 79 3.9 Conclusions 81 References 81 Part Two Nonlinear Dynamics: Genelets and Synthetic Biochemical Circuits 83 4 Transcriptional Oscillators 85 Elisa Franco, Jongmin Kim, and Friedrich C. Simmel 4.1 Introduction 85 4.2 Synthetic Transcriptional Modules 86 4.3 Molecular Clocks 89 4.4 Scaling Up Molecular Circuits: Synchronization of Molecular Processes 96 4.5 Oscillator Driving a Load: Experimental Implementation and Data 105 4.6 Deterministic Predictive Models for Complex Reaction Networks 105 4.7 Stochastic Effects 107 4.8 Conclusions 110 References 110 5 Synthetic Biochemical Dynamic Circuits 113 Rapha€el Plasson and Yannick Rondelez 5.1 Introduction 113 5.2 Out–of–Equilibrium Chemical Systems 114 5.3 Biological Circuits 123 5.4 Programmable In Vitro Dynamics 130 5.5 Perspectives 139 References 142 Part Three Nonlinear Dynamics: the Brain and the Heart 147 6 Theoretical and Experimental Electrophysiology in Human Neocortex: Multiscale Dynamic Correlates of Conscious Experience 149 Paul L. Nunez, Ramesh Srinivasan, and Lester Ingber 6.1 Introduction to Brain Complexity 149 6.2 Brief Overview of Neocortical Anatomy and Physiology 154 6.3 Multiscale Theory in Electrophysiology 160 6.4 Statistical Mechanics of Neocortical Interactions 166 6.5 Concluding Remarks 173 References 174 7 Multiscale Network Organization in the Human Brain 179 Danielle S. Bassett and Felix Siebenh€uhner 7.1 Introduction 179 7.2 Mathematical Concepts 181 7.3 Structural Multiscale Organization 182 7.4 Functional Multiscale Organization 187 7.5 Discussion 191 References 195 8 Neuronal Oscillations Scale Up and Scale Down Brain Dynamics 205 Michel Le Van Quyen, Vicente Botella–Soler, and Mario Valderrama 8.1 Introduction 205 8.2 The Brain Web of Cross–Scale Interactions 206 8.3 Multiscale Recordings of the Human Brain 208 8.4 Physiological Correlates of Cross–Level Interactions 210 8.5 Level Entanglement and Cross–Scale Coupling of Neuronal Oscillations 212 8.6 Conclusions 213 References 214 9 Linking Nonlinear Neural Dynamics to Single–Trial Human Behavior 217 Michael X Cohen and Bradley Voytek 9.1 Neural Dynamics Are Complex 217 9.2 Data Analysis Techniques and Possibilities Are Expanding Rapidly 218 9.3 The Importance of Linking Neural Dynamics to Behavior Dynamics 219 9.4 Linear Approaches of Linking Neural and Behavior Dynamics 220 9.5 Nonlinear Dynamics and Behavior: Phase Modulations 221 9.6 Cross–Frequency Coupling 224 9.7 Linking Cross–Frequency Coupling to Behavior 226 9.8 Testing for Causal Involvement of Nonlinear Dynamics in Cognition and Behavior 228 9.9 Conclusions 229 References 229 10 Brain Dynamics at Rest: How Structure Shapes Dynamics 233 Etienne Hugues, Juan R. Vidal, Jean–Philippe Lachaux, and Gustavo Deco 10.1 Introduction 233 10.2 Model 234 10.3 Results 236 10.4 Comparison with Experimental Data 239 10.5 Discussion 240 References 242 11 Adaptive Multiscale Encoding: A Computational Function of Neuronal Synchronization 245 Misha (Meyer) Z. Pesenson 11.1 Introduction 245 11.2 Some Basic Mathematical Concepts 247 11.3 Neural Synchronization 247 11.4 Concluding Remarks 253 References 254 12 Multiscale Nonlinear Dynamics in Cardiac Electrophysiology: From Sparks to Sudden Death 257 Zhilin Qu and Michael Nivala 12.1 Introduction 257 12.2 Subcellular Scale: Criticality in the Transition from Ca Sparks to Ca Waves 258 12.3 Cellular Scale: Action Potential and Ca Cycling Dynamics 260 12.4 Excitation Dynamics on the Tissue and Organ Scales 266 12.5 Conclusions 271 References 271 13 Measures of Spike Train Synchrony: From Single Neurons to Populations 277 Conor Houghton and Thomas Kreuz 13.1 Introduction 277 13.2 Measures of Spike Train Distance 278 13.3 Comparisons 286 13.4 Measuring the Dissimilarity within a Population 290 13.5 Measuring the Dissimilarity between Populations 292 13.6 Discussion 294 References 296 Index 299

Dr. Misha (Meyer) Z. Pesenson has held positions in academia, including UCLA and Caltech, since 1990. He is currently Research Scientist at the Computing and Mathematical Sciences Dept., California Institute of Technology. Dr. Pesenson′s research focuses on multiscale modeling, nonlinear dynamics, neural networks, and complex information processing. The Series Editor Heinz Georg Schuster is Professor of Theoretical Physics at the University of Kiel in Germany. He was a visiting professor at the Weizmann–Institute of Science in Israel and at the California Institute of Technology in Pasadena, USA. He authored and edited many books on nonlinear phenomena and chaos control.