Lecture 26

Lectures: 1 | 2 | 3 | 4-5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 15 | 16 | 17 | 19 | 20 | 21 | 22 | 23 | 26

Dynamic Behaviors

1. Stability of Fixed Points:
   • Linear Stability Matrix; Eigenvectors and Eigenvalues (Routh-Hurwitz Theorem)
   • Loss of Stability from a Real Eigenvalue; Stability Exchange, Bifurcation (Page 1 (GIF))
   • Loss of Stability from Imaginary Eigenvalues; Poincare Cycles
   • Simple (Harmonic) Oscillator, and Complex Representation (Page 2 (GIF))
2. Biochemical Clocks:
   • The Belousov-Zhabotinsky Reaction
   • The Cogs of a Biological Clock
   • Animations of Biological Clocks
   • The Repressilator (PDF) [Elowitz and Liebler. Nature 403, no. 305 (2000)]
3. Synchronization:
   • Examples: Heart Pacemaker Cells, Fireflies, Cycada.
   • The Kuramoto Model (Java® Applet by Albert Diaz-Guilera)
   • Collective Synchronization (Page 3 (GIF) and Page 4 (GIF) of Lecture Notes)
4. Biological Patterns:
   • Morphogenesis is the Process Whereby a Living Organism Develops Form and Structure, e.g.
     • Animal Coats and Pigmentation Patterns
   • Where do Spots Come From? Turing's Answer
     • Reaction-Diffusion Equations (Page 5 (GIF) of Lecture Notes)
     • Constraints for a Finite Wave-length Instability (Page 6 (GIF) of Lecture Notes)
     • Long-range Inhibition and Short-range Excitation (Page 7 (GIF) of Lecture Notes)

Neural Networks

• Computation in the Brain
• Associative Memories in the Hopfield Model
• An Applet for Demonstrating Associative Memory in the Hopfield Model

Some Related Links (Biological Clocks)

• Poicare-Bendixon Theorem
• Biotiming Tutorial
• Clockwork Genes (2000 Holiday Lectures)
• How Biological Clocks Work (NIMH)
• The Repressilator Figures
• The Belousov-Zhabotinsky Reaction
• Firefly (Flash Synchrony)

Some Related Links (Morphogenesis)

• Visual Models of Morphogenesis
• Reaction Diffusion Equations and Animal Coat Patterns
• Morphological Processes and Reaction-diffusion Systems
• Uncovering the Secret of Life's Patterns (PDF)
• Theoretical Aspects of Pattern Formation and Neuronal Development (MPI Tübingen)
• Primary Pattern Formation, Organizing Regions and Regeneration (MPI Tübingen)
• Modelling Pigmentation Patterns
• The Alan Turing Internet Scrapbook Growth, Form and Crisis
• 1951-2: Belousov, Turing, Hodgkin and Huxley