Computational Biomedical Engineering Master

by Sergi Valverde and Josep Sardanyés

Course 2017-2018

Course Description

The objective of this course is to introduce the basic methods for the modeling and analysis of complex systems. Special emphasis will be made on the nonlinear processes underlying complexity and collective dynamics in biological and artificial systems at all levels, ranging from the behavior of individual cells to populations of organisms and their associated social behavior. At the end of the course, the student will be able to interpret nature in terms of emergence and self-organization.

The course consists of 7 theoretical lectures (2 hours each) and computer lab activities (11 hours) in the first trimester. 


The final grade will be based on the following: 

1. Final Examination. 

2. Personal Project: Early in the course, students will pick a topic of their interest that will be developed in the course (or personal project). By the end of the course, students will present both a written report and an oral defence describing their study. 

Please check the reference list below to decide the topic of your project. Recall that personal projects are individual.


1. Dynamical Systems: Mean Field Models, Ordinary Differential Equations (ODEs), Equilibria and Stability, Local Bifurcations and Normal Forms, Universality and Chaos. Numerical methods for solving ODEs.

2. Fractals: Self-similarity, Geographic scaling, von Koch curve, Fractal dimension, Box-counting algorithm, Chaos game. 

3. Evolutionary algorithms: Biomimicry, Genetic Algorithms, Fitness Landscape, Quasispecies Equation, Error Threshold, Learning, Ant Colony Optimization.  

4. Spatially-extended systems: Cellular automata. 

5. Complex Networks I: Network Properties, Random Graph, Percolation Transition, Hub, Connectors and Paths. 

6. Complex Networks II: Small-Worlds, Scale-Free Networks, Modularity. 

7. Scaling: Universality and power-laws. Optimization and Transport Networks. Phase transitions. 


1. Network Attacks: Learn about network fragmentation and the concept of percolation with this interactive website. How many nodes would you have to remove to break up an urban network? 

2. Path Length
: An interactive visualization of shortest paths. Explore how the properties of the shortest path depend on the rate of node failures.   

3. Random Graph: Random graphs are used as null models when assessing the significance of real-world networks. Used the Erdos-Renyi model to create random graphs with a given number of nodes and a fixed link probability. 

4. Small Worlds
: Add 10 shortcuts in order to minimise the average path length in the urban network. 

5. Preferential Attachment
: Generates a scale-free graph by a process of "preferential attachment", in which new nodes prefer to make a connection to the more popular existing nodes. 

6. Network Modularity
: Generates a random modular network and computes its modularity values. 

7. Vaccination game: Fight against the spreading of infectious diseases in different types of urban networks. After distributing a number of limited vaccines, the infectious outbreak spreads and the players quarantines individuals at the risk of becoming infected. 

The above models have been all developed by Sergi Valverde. They require a modern, WebGL-compliant, web browser.


Melanie Mitchell (2009) Complexity: a guided tour, Oxford University Press.

Steven H. Strogatz (2001) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering  (studies in nonlinearity), Westview Press. 

Ricard V. Solé and Susanna C. Manrubia (1996) Orden y caos en sistemas complejos, Politex UPC.

Ricard V. Solé and Brian Goodwin (2001) Signs of life: how complexity pervades biology, Basic Books.

Benoit B. Mandelbrot, The fractal geometry of nature, W. H. Freeman and Company. 

Przemylslaw Prusinkiewicz and Aristid Lindenmayer (2004) The algorithmic beauty of plants, Springer-Verlag.

Michael Barnsley (1988) Fractals Everywhere, Academic Press. (chapter IX is online:

James Gleick (1987) Chaos: making a new science, Viking Books.

Stuart Kauffman (2000), Investigations, Oxford University Press. 

Stuart Kauffman, Antichaos and adaptation.

Goldberger and B. West (1973), Fractals in physiology and medicine, J. Biol. Med.

John Koza et al (2003), Evolving inventions, Scientific American.

Martin Gardner (1970), Mathematical games - the fantastic combinations of John Conway’s new solitaire game “life”, Scientific American 223: 120-123.

Stephen Wolfram (1984) Cellular automata as models of complexity, Nature 311(5985), 419-424.

Stephen Wolfram (2002) A new kind of science, Wolfram research.

Ricard V. Solé (2009) Redes complejas: del genoma a internet, Tusquets (Metatemas). 

Ricard V. Solé (2011) Phase transitions, Princeton University Press. 

Duncan J. Watts and Steven H. Strogatz (1998) Collective dynamics of ‘small-world’ networks, Nature 393 (6684).

Albert-Laszlo Barabasi and Reka Albert (1999) Emergence of scaling in random networks, Science 286 (5439).

Mark Newman (2010) Networks: an introduction, Oxford University Press. 

Geoffrey West (1999) The origin of universal scaling laws in biology, Physica A.

Sergi Valverde and Jordi Garcia-Ojalvo (2016) Hacia una teoría unificada de la criticalidad biológica. Investigación y Ciencia, Marzo 2016 (N. 474).

Sergi Valverde and Ricard V. Solé (2013) Networks and City, 83(4): 112-119. (link).



During the course, the Netlogo environment will be used. Students are encouraged to use their favourite programming environment to develop the models proposed in the course and also to explore other models of their interest. Netlogo is freely available here: 


Check the course schedule here:

Where we are


Biomedical Research Park Barcelona (PRBB)

Complex Systems Lab  (UPF-IBE)

Planta 4, D-491

Dr. Aiguader, 88 - 08003 Barcelona


Mar campus (UPF)  - Dr. Aiguader Building

Dr. Aiguader, 80 - 08003 Barcelona 

Metro line 4 Ciutadella - Vila Olympic

BUS V27, V21, D20, 92, 59, 45, 36

Edifici Dr. Aiguader