- 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.
- An applet for demonstrating Associative Memory in the Hopfield Model
- 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
- 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)
PDF)