# 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.