CONTENTS -- Preface -- Multiple Rhythms and Switches in the Nervous System N. Kopell, D. Pervouchine, H. G. Rotstein, T. Netoff, M. Whittington and T. Gloveli -- 1. Rhythmic dynamics in the nervous system -- 2. Gamma, Theta and in vitro preparations -- 3. The Mathware: Voltage-gated conductance equations -- 4. Some mechanisms for the gamma rhythm -- 5. A theta-rhythmic cell -- 6. Structure of Hodgkin-Huxley equations and low-dimensional maps -- 7. Where does the theta come from? -- 8. All together now: a larger network -- 9. Multiple rhythms, switches and bursting -- 10. Discussion -- References -- Some Ideas on Action Potentials D. Y. Hsieh -- 1. INTRODUCTION -- 2. THE TELEGRAPH EQUATION -- 3. THE ION-ACOUSTIC WAVES -- 4. THE SENSORY CODING -- 5. DISCUSSIONS -- Appendix A. Ion-Acoustic Waves in Plasm -- Appendix B. Non-linear Oscillators -- References -- Negative Feedback in Morphogen Gradients M. Khong and F. Y. M. Wan -- 1. Introduction -- 2. An Extracellular Formulation with Receptor Synthesis -- 3. Time Independent Steady State Behavior -- 3.1. Reduction of the Steady State Equations -- 3.2. Existence of a Unique Set of Steady State Gradients -- 3.3. Monotonicity -- 4. Linear Stability -- 4.1. A Nonlinear Eigenvalue Problem -- 4.2. Positive Eigenvalues and Asymptotic Stability -- 4.3. A Rayleigh Quotient -- 5. Some Steady State Gradients -- 5.1. Simplification for n = 1 -- 5.2. Exact Solution -- 5.3. A Narrow Ligand Source Width -- 6. Effects of the Negative Feedback -- 7. Conclusion -- Acknowledgments -- References -- CSAW: Stochastic Approach to Protein Folding K. Huang -- 1. Introduction -- 2. Protein folding -- 3. Brownian motion -- 4. Monte-Carlo -- 5 . SAW -- 6. CSAW -- 7. Hydrophobic model -- 8. Hydrogen bonding -- 9. Molten globule -- 10. Folding pathways of chignolin -- 11. Correlation functions -- 12. Outlook -- Acknowledgments.

References -- "Collapse and Search" Dynamics of Protein Folding Detected by Time-Resolved Small- Angle X-Ray Scattering S. Takahashi and T. Fzljisawa -- 1 Introduction -- 2 Stepwise Folding of Cytochrome c -- 3 Collapse and Search Dynamics of Apomyoglobin Folding -- 4 A Significant Collapse in the Folding of Single Chain Monellin -- 5 Coil-Globule Transition of the Initial Collapse -- 6 Summary and Perspective -- Acknowledgments -- References -- Structural Transitions in Biopolymers: From DNA to Protein to Spider Silk H. Zhou -- 1. Introduction -- 2. Simple polymer elasticity models -- 2.1. The Gaussian chain model -- 2.2. The freely jointed chain (FJC) model -- 2.3. The worm-like chain (WLC) model -- 2.4. Comparison of the three polymer models -- 3. The over-stretching transition of double-stranded DNA -- 4. Denaturation of RNA secondary structures -- 5. The -sheet secondary structure of proteins -- 5.1. The model and the major analytical procedure -- 5.2. Analytical and simulational results -- 6. Elasticity of spider capture silk -- 7. Conclusion -- Acknowledgments -- References -- The Structure, Evolution and Instability of a Self-Gravitating Gaseous Disk under the Influence of Periodic Forcings C. Yuan -- 1. Introduction -- 2. Rotation Curve and Bar Potential -- 3. Spiral Waves excited by a Fast Rotating Bar -- 4. Spiral Waves Excited by a Slowly Rotating Bar -- 5. Doubly-Periodic Shock Solutions -- 5.1 Background -- 5.2. Numerical One-dimension Asymptotic Theory -- 5.3. Two-Dimensional Numerical Results -- 5.4. Instability and Chaotic Sub-structure -- 5.5. Effect of self-gravity -- 6 . Concluding Remarks -- Acknowledgments -- References -- Dynamics of Spiral Galaxies G. Bertin -- 1 Introduction -- 2 Morphology of spiral galaxies -- 3 Density Wave Theory of spiral structure in galaxies -- 3.1. Quasi-stationary spiral structure?.

3.2. Density waves -- 3.3. The importance of gas, self-regulation, and feedback -- 3.4. Overreflection at corotation -- 3.5. Discrete spectrum of self-excited global modes -- 3.6. A unified theory of normal and barred spiral modes -- 3.7. Role of Inner Lindblad Resonance and of some non-linear processes -- 3.8. A unified framework for the Hubble morphological classification -- 3.9. The decisive "proop" of near-infrared observations -- 3.10. Concluding remarks -- 4 Other interesting problems in the dynamics of spiral galaxies and future prospects -- 4.1. Selfgravitating accretion disks -- 4.2. Extraplanar gas -- 4.3. Hi disks in elliptical galaxies -- 4.4. Prospects for new advances in the Density Wave Theory -- Acknowledgments -- References -- Dark Matter Dynamics in Galaxies C.-P. Ma -- 1. The New Cosmology -- 2. Pros and Cons of Numerical Simulations -- 3. Cosmological Kinetic Theory -- 3.1. A sketch of the derivation -- 3.2. Subtleties -- 3.3. Extension into the nonlinear regime -- 4. Acknowledgments -- References -- Asymptotics and Star Formation F. H. Shu -- 1 Introduction -- 1.1. Efficacy of Asymptotic in Astrophysics -- 1.2. Four Phases of Star Formation Whether Isolated or Clustered -- 1.3. Equations of Non-ideal Magnetohydrodynamics -- 2 Molecular Cloud Cores -- 2.1. Gravitational Condensation by Ambipolar Diffusion -- 2.2. Pivotal State -- 3 Gravitational Collapse of Self-similar Pivotal States -- 3.1. Catastrophic Magnetic Breaking if Field Freezing Applies -- 3.2. Field Freezing Produces a Split Monopole at the Center -- 4 Effects of Non-ideal Magnetohydrodynamics -- 4.1. Breakdown of Ideal MHD at Small Scales -- 4.2. Ohmic Dissipation Yields Central Region with Low Uniform Field -- 5 X-winds and Protostellar Jets -- 5.1. Interaction of an Accretion Disk and a Strongly Magnetized Central Star.

5.2. Asymptotic Behavior of X-wind at Large Distances from the Source -- 5.3. Optical Illusion of Jet when Outflow is Lit by Turbulent Shockwaves -- 6 Summary -- Acknowledgments -- References -- Solitary Waves from Optics to Fluid Dynamics M. J. Ablowitz and A. Docherty -- 1. Introduction -- 2. Dispersion-managed systems -- 3. NLSM systems -- 4. Spectral renormalization -- 5. A Nonlocal Formulation of Water Waves in Three Dimensions -- 5.1. The Dynamic Boundary Condition - Bernoulli's Equation -- 5.2. The Analysis of Equations (5.1) - (5.4) -- 5.3. The Linear Limit -- 5.4. The Nondimensional Form -- 5.5. Two dimensional Benney-Luke and KP equations -- 5.6. Computational Studies - 2+1 dimensional lumps -- Acknowledgments -- References -- Resonance Problems in Photonics M. I. Weinstein -- 1. Introduction and outline -- 2. Nonlinear waves at Bragg resonance - envelope equations -- 3. Coherent structures -- 4. Controlling nonlinear pulses -- 4.1. Nonlinear scattering -- 4.2. Nonlinear defect modes -- 5. Large time evolution of a captured soliton - ground state selection for NLS / G-P -- References -- Some Mathematical Properties of Long Waves D. J. Benney -- 1. Introduction -- 2. Two-dimensional analysis -- 3. An analogous mathematical system -- 4. Three-dimensional analysis -- References -- New Solitary Wave Structures in TweDimensional Periodic Media Z. Shi and J. Yang -- 1. Introduction -- 2. The Mathematical Model -- 3. Bloch bands and band gaps -- 4. Asymptotic derivation of coupled envelope equations -- 5 . Solutions of the coupled envelope equations -- 6. Numerical solutions of solitary waves at arbitrary amplitudes -- 7. Conclusion -- Acknowledgements -- References.