8.592 - HST.452J (Fall 2025)

Statistical Physics in Biology

Lecture 21

1. Dynamics on networks (motifs):
• Node variables evolving according to inputs from connected nodes
• Examples- Chemical flux balance analysis, artificial neural networks
• Possible collective outcomes: fixed points, temporal oscillations, spatial patterns, ...
2. Time indedpendent steady-states:
   • One dimensional systems, fixed points, stochasticity and noise
   • Gradient descent in higher dimensional systems
   • Hopfield's model of neurons with graded response, PNAS 81, 3088 (1984).
   • Hopfield network (learning applet)  applet1
   • 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 applications of bifurcation theory
   • Biological switches (Genetic toggle switch by James Collins)
   • Loss of stability from imaginary eigenvalues; Poincaré–Bendixson theorem and periodic orbits
   • Simple (harmonic) oscillator, and complex representation
4. Predator-Prey equation
5. Diversity paradox and random matrix
6. Biochemical clocks:
   • The Belousov-Zhabotinsky Reaction (video, video) (Explanation)
   • The cogs of a biological clock
   • Animations of biological clocks (offline}
   • The Repressilator [Elowitz and Liebler, Nature 403, 305 (2000)]

(Lecture Notes)

• Links to Online Demos from UBC
• Computation in the brain
• Introduction to Neural Networks [8.594J by Prof. Sebastian Seung in 2005]