- 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, …

- 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.

- 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

- Biochemical clocks:
- The Belousov-Zhabotinsky Reaction (Explanation)
- The cogs of a biological clock
- The Repressilator

- Synchronization:
- Examples: heart pacemaker cells, fireflies, cicada
- 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
- 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)