Study plan


Following the research-based spirit of our master, this course is centered around some of the work done at one of the participating research groups: The Nonlinear Time Series Analysis group ( It is taught by the director of this group, Ralph G. Andrzejak.

The framework of nonlinear signal analysis is a practical spinoff from complex dynamical systems theory and chaos theory. It allows one to characterize dynamical systems in which nonlinearities give rise to a complex temporal evolution. Importantly, this concept can extract information from signals that cannot be resolved using classical linear techniques such as the power spectrum, linear correlation functions, etc. Applications of nonlinear signal analysis to recordings from the brain increasingly contribute to our understanding of brain functions and malfunctions and thereby help to advance cognitive neuroscience and neurology.

In this course we will focus on the particular case of electroencephalographic (EEG) recordings from patients with the neurological disorder epilepsy. The World Health Organization (WHO) regards neurological disorders as “one of the greatest threats to public health”, and the most common serious neurological condition is epilepsy. It affects at least 50 million people worldwide. In a typical European population of 1.000 persons, between 5 and 10 persons have active epilepsy. In Spain the number of patients with active epilepsy is around 400.000, and the total cost of epilepsy accounts for some 5% of total health budget.

There is growing evidence that the application of nonlinear signal analysis to EEG recordings can provide information which is useful for an improved diagnostics and treatment of epilepsy patients. One example, to which also the Nonlinear Time Series Analysis group is contributing, is the localization of the seizure generating brain area from EEG recordings that do not contain any seizure. We will learn about the potential, but also about the limitations of this approach.

In the theory classes we will learn about nonlinear signal analysis techniques, as well as some basics about epilepsy and EEG recordings from epilepsy patients. In the lab sessions we will apply the analysis techniques to real EEG recordings provided by a clinical partner of the Nonlinear Time Series Analysis Group. A focus will be placed on a meaningful interpretation of the derived results.


The theory of the course is designed such that it can be followed by students with any engineering bachelor degree. This includes, but is not limited to, biomedical engineers. Students from physics, mathematics, or life sciences will also be able to follow the course. All these bachelor degrees equip the students with the necessary technical and mathematical skills to take this course.

No pre-knowledge in nonlinear time series analysis is required. The students should however be familiar with fundamental concepts of linear signal analysis, such as the Fourier transform, simple frequency filters, etc. No pre-knowledge on neuroscience, neurophysiology or neurology is needed.

We will use Matlab for the data analysis done in the lab sessions. The students need therefore to be able to write and understand simple source code in this programming language.



The theory is organized in seven blocks:

  • (1) Overview
    • We will start with an overview and motivation of the course content. It will be highlighted why its content is relevant for biomedical engineers regardless of whether they aim at a career in academia or in industry.
  • (2) Epilepsy and intracranial electroencephalographic recordings.
    • Those epilepsy patients who cannot be treated satisfactorily with medication can potentially benefit from epilepsy surgery. Apart from some basic facts about epilepsy, we will learn about this type of surgery and how intracranial electroencephalographic recordings are used for the preceding diagnostics. We start with this block since from the first lab session we will inspect and study real EEG recordings from epilepsy patients.
  • (3) Deterministic dynamics; (4) Stochastic dynamics
    • Suppose that you have measured a signal from a real-world dynamical system and you want to know if the system is deterministic or stochastic. Before we analyze real-world data to address this question, we at first have to understand how our analysis techniques work when applied to data with well-known properties. We therefore introduce deterministic and stochastic model dynamics to generate some exemplary toy signals under controlled conditions. This will allow us to understand the power but also the limitations of the different analysis techniques. 
  • (5) Linear signal analysis
    • The summary above states that nonlinear signal analysis techniques allow extracting information that cannot be resolved using classical linear techniques. To truly appreciate these advantages, we at first briefly revisit some fundamental linear techniques: The power spectrum and the autocorrelation function. We will show that for nonlinear dynamics these linear techniques cannot reliably distinguish between deterministic and stochastic dynamics.
  • (6) Nonlinear signal analysis
    • We at first show how delay coordinates can be used to reconstruct an estimate of the system’s state space representation from single signals. The nonlinear prediction error is then used to quantify the degree to which similar momentary states evolve in similar ways. Based on several examples we show that this predictability is a sensitive but not specific criterion for a deterministic system.
  • (7) The concept of surrogates
    • Surrogates are a very powerful and versatile concept within the framework of nonlinear signal analysis techniques. They allow us to test a variety of null hypothesis about the system underlying some measured signal. In our context, we will see how surrogates can be combined with the nonlinear prediction error to arrive at a more specific test for determinism.

Laboratory sessions

In the first lab session the students will be provided with intracranial EEG recordings from epilepsy patients. They will work in groups of two or three students, and each group will have a recording from a different patient. The data will be in Matlab files, and the information about these recordings will deliberately be kept to a minimum. In the first few labs, the students’ task will be to write some code to visualize the data, explore the data and to find potential artefacts resulting from external disturbances of the recordings. Of course, the students will receive supervision, advice and assistance by the professor of the labs. However, this situation, where some unknown data needs to be explored with only little background information available, is a typical task for biomedical engineers, and this module is meant to train them in such a situation. Subsequently, the students will have to implement the code to calculate the nonlinear prediction error from signals. Depending on the overall progress, the code for the generation of surrogates will be either implemented by the students or provided by the professor. Finally, the students will analyze their EEG recordings with a combination of the nonlinear prediction error and surrogates.

In a further lab, a scientific article on the application of nonlinear signal analysis to the same EEG recordings that we study in our lab sessions will be discussed in the format of a journal club.

Evaluation Methods

The students have to deliver

  • Written reports on their exploration of the EEG data and the tools they wrote for this purpose.
  • A well-commented source code which they developed for the nonlinear prediction error (and surrogates, if done by the students).
  • The results from the analysis of the EEG recordings with a combination of the nonlinear prediction error and surrogates will be delivered as a written report and presented in a short oral seminar.

The grades will be derived from the quality of these deliveries and from the active participation in the journal club discussion.

Bibliography and information resources

  • Lecture notes
    • Slides of all seven theory blocks are provided via Aula Global.
  • Selected scientific articles of the Nonlinear Time Series Analysis group that are related to the course (PDFs at
    • Naro D, Rummel C, Schindler K, Andrzejak RG. 2014. Detecting determinism with improved sensitivity in time series: Rank-based nonlinear predictability score. Physical Review E. 90:032913
    • Andrzejak RG, Schindler K, Rummel C. 2012. Nonrandomness, nonlinear dependence, and nonstationarity of electroencephalographic recordings from epilepsy patients. Physical Review E. 86:046206
    • Andrzejak RG, Mormann F, Widman G, Kreuz T, Elger CE, Lehnertz K. 2006. Improved spatial characterization of the epileptic brain by focusing on nonlinearity. Epilepsy Research. 69:30-44
  • Tutorial article on nonlinear signal analysis (PDF at
    • Andrzejak RG. 2011. Nonlinear time series analysis in a nutshell. Osorio et al. (eds.) Epilepsy: The Intersection of Neurosciences, Biology, Mathematics, Engineering and Physics, 125.
  • Text books
    • H Kantz, T Schreiber, Nonlinear time series analysis, 2nd ed. Cambridge University Press, Cambridge, 2004
    • WH Press et al., Numerical Recipes 3rd Edition: The Art of Scientific Computing. Cambridge University Press, Cambridge, 2007
    • Engel J Jr., Seizures and epilepsy. 2nd ed. Oxford University Press, 2012