Multiscale Analysis and Nonlinear Dynamics: From Genes to the Brain -- Contents -- List of Contributors -- Preface -- 1 Introduction: Multiscale Analysis - Modeling, Data, Networks, and Nonlinear Dynamics -- 1.1 Multiscale Modeling -- 1.1.1 Domain-Specific Modeling -- 1.1.2 Analysis -- 1.1.3 Model Interpretation and Verification: Experimental/Simulation Data -- 1.2 Multiresolution Analysis and Processing of High-Dimensional Information/Data -- 1.3 Multiscale Analysis, Networks, and Nonlinear Dynamics -- 1.4 Conclusions -- References -- Part One: Multiscale Analysis -- 2 Modeling Across Scales: Discrete Geometric Structures in Homogenization and Inverse Homogenization -- 2.1 Introduction -- 2.2 Homogenization of Conductivity Space -- 2.2.1 Homogenization as a Nonlinear Operator -- 2.2.2 Parameterization of the Conductivity Space -- 2.3 Discrete Geometric Homogenization -- 2.3.1 Homogenization by Volume Averaging -- 2.3.2 Homogenization by Linear Interpolation -- 2.3.3 Semigroup Properties in Geometric Homogenization -- 2.4 Optimal Weighted Delaunay Triangulations -- 2.4.1 Construction of Positive Dirichlet Weights -- 2.4.2 Weighted Delaunay and Q-Adapted Triangulations -- 2.4.3 Computing Optimal Weighted Delaunay Meshes -- 2.5 Relationship to Inverse Homogenization -- 2.6 Electrical Impedance Tomography -- 2.6.1 Numerical Tests -- 2.6.1.1 Harmonic Coordinate Iteration -- 2.6.1.2 Divergence-Free Parameterization Recovery -- References -- 3 Multiresolution Analysis on Compact Riemannian Manifolds -- 3.1 Introduction -- 3.2 Compact Manifolds and Operators -- 3.2.1 Manifolds without Boundary -- 3.2.2 Compact Homogeneous Manifolds -- 3.2.3 Bounded Domains with Smooth Boundaries -- 3.3 Hilbert Frames -- 3.4 Multiresolution and Sampling -- 3.5 Shannon Sampling of Band-Limited Functions on Manifolds -- 3.6 Localized Frames on Compact Manifolds.

3.7 Parseval Frames on Homogeneous Manifolds -- 3.8 Variational Splines on Manifolds -- 3.9 Conclusions -- References -- Part Two: Nonlinear Dynamics: Genelets and Synthetic Biochemical Circuits -- 4 Transcriptional Oscillators -- 4.1 Introduction -- 4.2 Synthetic Transcriptional Modules -- 4.2.1 Elementary Activation and Inhibition Pathways, and Simple Loops -- 4.2.2 Experimental Implementation -- 4.3 Molecular Clocks -- 4.3.1 A Two-Node Molecular Oscillator -- 4.3.2 Analysis of the Oscillatory Regime -- 4.3.3 Experimental Implementation and Data -- 4.4 Scaling Up Molecular Circuits: Synchronization of Molecular Processes -- 4.4.1 Analysis of the Load Dynamics -- 4.4.1.1 Quasisteady State Approximation of the Load Dynamics -- 4.4.1.2 Efficiency of Signal Transmission -- 4.4.2 Perturbation of the Oscillator Caused by the Load -- 4.4.2.1 Consumptive Coupling -- 4.4.2.2 Nonconsumptive Coupling and Retroactivity -- 4.4.3 Insulation -- 4.4.3.1 Reduction of Perturbations on the Oscillator Dynamics -- 4.4.3.2 Signal Transmission to the Insulated Load -- 4.5 Oscillator Driving a Load: Experimental Implementation and Data -- 4.6 Deterministic Predictive Models for Complex Reaction Networks -- 4.7 Stochastic Effects -- 4.8 Conclusions -- References -- 5 Synthetic Biochemical Dynamic Circuits -- 5.1 Introduction -- 5.2 Out-of-Equilibrium Chemical Systems -- 5.2.1 A Short Historical Overview -- 5.2.1.1 Discovery of Nonlinear Chemical Systems -- 5.2.1.2 Unexpected Oscillating Chemical Systems -- 5.2.2 Building Nonequilibrium Systems -- 5.2.2.1 Energetic Requirements -- 5.2.2.2 System Closure -- 5.2.2.3 Instabilities and Dynamic Stability -- 5.2.3 Design Principles -- 5.2.3.1 Dynamism -- 5.2.3.2 Interacting Feedbacks Processes -- 5.2.3.3 Modularity -- 5.3 Biological Circuits -- 5.3.1 Biological Networks Modeled by Chemistry.

5.3.2 Biosystems: A Multilevel Complexity -- 5.3.3 A First Example of a Biological Reaction Circuit -- 5.3.4 Biological Networking Strategy -- 5.3.4.1 GRNs Are Templated Networks -- 5.3.4.2 Regulation and Feedback Loops -- 5.3.4.3 Nonlinearities in Genetic Regulation -- 5.3.4.4 Delays -- 5.3.4.5 Titration Effects -- 5.3.5 Higher Level Motifs and Modularity of Biochemical Networks -- 5.4 Programmable In Vitro Dynamics -- 5.4.1 Enzymatic Systems -- 5.4.1.1 DNA-RNA Sequence Amplification -- 5.4.1.2 The Genelet System -- 5.4.1.3 The PEN Toolbox -- 5.4.2 Nonenzymatic Networks: Strand Displacement Systems -- 5.4.3 Numerical Modeling -- 5.4.3.1 Mathematical Descriptions -- 5.4.3.2 LSA and Bifurcation Analysis for Design -- 5.4.3.3 Time Evolutions -- 5.4.3.4 Robustness Analysis and In Silico Evolutions -- 5.5 Perspectives -- 5.5.1 DNA Computing -- 5.5.2 Self Organizing Spatial Patterns -- 5.5.3 Models of Biological Networks -- 5.5.4 Origin of Life -- References -- Part Three: Nonlinear Dynamics: the Brain and the Heart -- 6 Theoretical and Experimental Electrophysiology in Human Neocortex: Multiscale Dynamic Correlates of Conscious Experience -- 6.1 Introduction to Brain Complexity -- 6.1.1 Human Brains and Other Complex Adaptive Systems -- 6.1.2 Is "Consciousness" a Four-Letter Word? -- 6.1.3 Motivations and Target Audiences for this Chapter -- 6.1.4 Brain Imaging at Multiple Spatial and Temporal Scales -- 6.1.5 Multiple Scales of Brain Dynamics in Consciousness -- 6.2 Brief Overview of Neocortical Anatomy and Physiology -- 6.2.1 The Human Brain at Large Scales -- 6.2.2 Chemical Control of Brain and Behavior -- 6.2.3 Electrical Transmission -- 6.2.4 Neocortex -- 6.2.5 The Nested Hierarchy of Neocortex: Multiple Scales of Brain Tissue -- 6.2.6 Corticocortical Connections Are Nonlocal and "Small World" -- 6.3 Multiscale Theory in Electrophysiology.

6.3.1 Characteristic EEG and Physiological Time Scales -- 6.3.2 Local versus Global Brain Models and Spatial Scale -- 6.3.3 A Large-Scale Model of EEG Standing Waves -- 6.3.4 Relationships between Small, Intermediate, and Large Scales: A Simple Mechanical Analog -- 6.4 Statistical Mechanics of Neocortical Interactions -- 6.4.1 SMNI on Short-Term Memory and EEG -- 6.4.1.1 SMNI STM -- 6.4.1.2 SMNI EEG -- 6.4.2 Euler-Lagrange Equations -- 6.4.2.1 Columnar EL -- 6.4.2.2 Strings EL -- 6.4.2.3 Springs EL -- 6.4.3 Smoking Gun -- 6.4.3.1 Neocortical Magnetic Fields -- 6.4.3.2 SMNI Vector Potential -- 6.5 Concluding Remarks -- References -- 7 Multiscale Network Organization in the Human Brain -- 7.1 Introduction -- 7.2 Mathematical Concepts -- 7.3 Structural Multiscale Organization -- 7.4 Functional Multiscale Organization -- 7.5 Discussion -- 7.5.1 Structure and Function -- 7.5.2 Hierarchical Modularity -- 7.5.3 Power-Law Scaling -- 7.5.4 Network Models of Multiscale Structure -- References -- 8 Neuronal Oscillations Scale Up and Scale Down Brain Dynamics -- 8.1 Introduction -- 8.2 The Brain Web of Cross-Scale Interactions -- 8.3 Multiscale Recordings of the Human Brain -- 8.4 Physiological Correlates of Cross-Level Interactions -- 8.5 Level Entanglement and Cross-Scale Coupling of Neuronal Oscillations -- 8.6 Conclusions -- References -- 9 Linking Nonlinear Neural Dynamics to Single-Trial Human Behavior -- 9.1 Neural Dynamics Are Complex -- 9.2 Data Analysis Techniques and Possibilities Are Expanding Rapidly -- 9.3 The Importance of Linking Neural Dynamics to Behavior Dynamics -- 9.4 Linear Approaches of Linking Neural and Behavior Dynamics -- 9.5 Nonlinear Dynamics and Behavior: Phase Modulations -- 9.6 Cross-Frequency Coupling -- 9.7 Linking Cross-Frequency Coupling to Behavior.

9.8 Testing for Causal Involvement of Nonlinear Dynamics in Cognition and Behavior -- 9.9 Conclusions -- References -- 10 Brain Dynamics at Rest: How Structure Shapes Dynamics -- 10.1 Introduction -- 10.2 Model -- 10.3 Results -- 10.3.1 Neural Dynamics -- 10.3.1.1 Case of Infinite Conduction Velocity -- 10.3.1.2 Case of Finite Conduction Velocity -- 10.3.2 BOLD Dynamics -- 10.4 Comparison with Experimental Data -- 10.5 Discussion -- References -- 11 Adaptive Multiscale Encoding: A Computational Function of Neuronal Synchronization -- 11.1 Introduction -- 11.2 Some Basic Mathematical Concepts -- 11.3 Neural Synchronization -- 11.3.1 Connections with Some Existing Approaches to MRA -- 11.4 Concluding Remarks -- References -- 12 Multiscale Nonlinear Dynamics in Cardiac Electrophysiology: From Sparks to Sudden Death -- 12.1 Introduction -- 12.2 Subcellular Scale: Criticality in the Transition from Ca Sparks to Ca Waves -- 12.3 Cellular Scale: Action Potential and Ca Cycling Dynamics -- 12.3.1 Intracellular Ca Alternans -- 12.3.2 Fast Pacing-Induced Complex APD Dynamics -- 12.3.3 EAD-Mediated Nonlinear Dynamics at Slow Heart Rates -- 12.4 Excitation Dynamics on the Tissue and Organ Scales -- 12.4.1 Spatially Discordant APD Alternans -- 12.4.2 Spiral and Scroll Wave Dynamics -- 12.4.3 Chaos Synchronization -- 12.5 Conclusions -- References -- 13 Measures of Spike Train Synchrony: From Single Neurons to Populations -- 13.1 Introduction -- 13.2 Measures of Spike Train Distance -- 13.2.1 Notation -- 13.2.2 The Victor-Purpura Metric -- 13.2.3 The van Rossum Metric -- 13.2.4 The ISI- and the SPIKE-Distance -- 13.2.4.1 The ISI-Distance -- 13.2.4.2 The SPIKE-Distance -- 13.2.5 Entropy-Based Measure -- 13.3 Comparisons -- 13.3.1 The ISI- and the SPIKE-Distance -- 13.3.2 The ISI-Distance and the van Rossum Metric.

13.3.3 The SPIKE-Distance and the Victor-Purpura Metric.