User:Johnsy/Advanced Modelling in Biology

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Advanced Modelling in Biology

Spring 2008 Session

Lecturer: Dr. Mauricio Barahona

Topics

  • Optimization
    • Introduction to optimization: definitions and concepts, standard formulation. Convexity. Combinatorial explosion and computationally hard problems.
    • Least squares solution: pseudo-inverse; multivariable case. Applications: data fitting.
    • Constrained optimization:
      • Linear equality constraints: Lagrange multipliers
      • Linear inequality constraints: Linear programming. Simplex algorithm. Applications.
    • Gradient methods: steepest descent; dissipative gradient dynamics; improved gradient methods.
    • Heuristic methods:
      • Simulated annealing: Continuous version; relation to stochastic differential equations.
      • Neural networks: General architectures; nonlinear units; back-propagation; applications and relation to least squares.
    • Combinatorial optimization: ‘hard’ problems, enumeration, combinatorial explosion. Examples and formulation.
    • Heuristic algorithms: simulated annealing (discrete version); evolutionary (genetic) algorithms. Applications.
  • Discrete Systems
    • Linear difference equations: general solution; auto-regressive models; relation to z-transform and Fourier analysis.
    • Nonlinear maps: fixed points; stability; bifurcations. Poincaré section. Cobweb analysis. Examples: logistic map in population dynamics (period-doubling bifurcation and chaos); genetic populations.
    • Control and optimization in maps. Applications: management of fisheries.
  • Advanced Topics (Networks & Chaos)
    • Networks in biology: graph theoretical concepts and properties; random graphs; deterministic, constructive graphs; small-worlds; scale-free graphs. Applications in biology, economics, sociology, engineering.
    • Nonlinear control in biology: recurrence plots and embeddings; projection onto the stable manifolds; stabilization of unstable periodic orbits and anti-control. Applications to physiological monitoring.

Primer/Notes

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