User:Johnsy/Advanced Modelling in Biology

=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

 * AMB Primer (.pdf)