1. Dynamics on networks:
   • Node variables evolving according to inputs from connected nodes
   • Examples- Chemical flux balance analysis (FBA), Neural networks
   • Possible collective outcomes: fixed points, temporal oscillations, spatial patterns, ...
2. Time indedpendent steady-states:
   • One dimensional systems, fixed points, Langevin noise
   • Gradient descent in higher dimensional systems
   • Hopfiel's model of neurons with graded response, PNAS 81, 3088 (1984).
   • An applet for demonstrating Associative Memory in the Hopfield Model
3. Stability of Fixed Points:
   • Linear stability matrix; eigenvectors and eigenvalues
   • Loss of stability from a real eigenvalue; stability exchange, bifurcation
   • Biological switches
   • Loss of stability from imaginary eigenvalues; Poincaré–Bendixson theorem and periodic orbits
   • Simple (harmonic) oscillator, and complex representation

1. Biochemical clocks:
   • The Belousov-Zhabotinsky Reaction (Video) (Explanation)
   • The cogs of a biological clock
   • Animations of biological clocks
   • The Repressilator [Elowitz and Liebler, Nature 403, 305 (2000)]
2. Synchronization:
   • Examples: firefly, fireflies, heart pacemaker cells, cicada
   • The Kuramoto model: (Math Insight Applet) (Cactus Applet) (Applet by Albert Diaz-Guilera)
   • Collective synchronization

1. 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
     • "The chemical basis of morphogenesis," A.M. Turing, Phil. Trans. Roy. Soc. 237, 37 (1952) (offline)
     • Applets:  1-dimensional instability, 2-dimensional patterns:

       

       (from web-page of H. Meinhardt at MPU Tuebingen)

2. Reaction-Diffusion equations
   • Constraints for a finite wave-length instability
   • Long-range inhibition & short-range excitation

[Lecture Notes]

[Exercises]

Related links

• Firefly (flash synchrony)
• Steven Strogatz on sync
• Many Oscillators and Biological Sync
• Visual Models of Morphogenesis
• Reaction Diffusion Equations and Animal Coat Patterns
• Morphological processes and reaction-diffusion systems
• 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
• "The impact of Turing's work on pattern formation in biology," P.K. Maini, Math. Today 40, 140 (2004)