# Dynamics on Networks 1

1. Dynamics on networks:
   - Node variables evolving according to inputs from connected nodes
   - Examples—Chemical flux balance equations, 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
   - Hopfield, J. J. "Neurons with Graded Response have Collective Computational Properties Like those of Two-state Neurons." PNAS 81, no. 10 (1984): 3088–92.
   - 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
   - Loss of stability from imaginary eigenvalues; Poincaré–Bendixson theorem and periodic orbits
   - Simple (harmonic) oscillator, and complex representation
4. Biochemical clocks:
   - The Belousov-Zhabotinsky Reaction (Explanation)
   - The cogs of a biological clock
   - The Repressilator
5. Synchronization:
   - Examples: heart pacemaker cells, fireflies, cicada
   - The Kuramoto model (Java Applet by Albert Diaz-Guilera) (Cactus Applet)
   - Collective synchronization

## Some Related Links

### Neural Networks

- Computation in the brain
- Minicourse on computational neuroscience for physicists [8.515 by Prof. Sebastian Seung]
- Neural Networks [9.641 by Prof. Sebastian Seung]

### Biological Clocks

- Internet Resources on Chronobiology, Biological Clocks and Rhythms
- Biotiming Tutorial
- Clockwork Genes (2000 Holiday lectures)
- The Belousov-Zhabotinsky Reaction applet
- BZ reaction video from Experiments on the web (scroll down just past the "Experimental aspects" section)
- Elowitz, MB, and S. Leibler. "A synthetic oscillatory network of transcriptional regulators." Nature 403 (2000): 335-8.
- Firefly (flash synchrony)

(Detailed Lecture Notes (PDF))