Mathematical models can be based on simple concepts and still have a complex temporal evolution. Thus they can help us to understand many complicated phenomena in the real world. One example of this are the networks formed by pendulum clocks connected in order to synchronize the oscillations of all pendulums of identical characteristics that make up the network.

Surprisingly, these networks behave in such a way that they can be separated into two incoherent groups; one group oscillates in perfect regular synchronization while the other does so incoherently and erratically. To describe this inconsistency, this phenomenon has received the name of “chimera”, thus establishing a comparison with the imaginary monster from Greek mythology whose body was composed of anatomical parts of different animals.

In this context, the existence of chimera states is certainly intriguing and has drawn the attention of the scientific community. The disconcertion is even greater, given the observation that, at an unpredictable moment of time, these states will collapse, and thus cause the synchronous oscillation of all pendulums.

Explanation for this sudden collapse

A study published on 9 March in the journal Scientific Reports by Ralph G. Andrzejak, coordinator of the Nonlinear Time Series Analysis Group (NTSA), of the Department of Information and Communication Technologies (DTIC), in collaboration with physicists and neurologists from research centres in Switzerland and Germany, has found a surprising explanation for this sudden collapse.

Although intuition might suggest that this collapse comes about by critically high synchronization, Andrzejak and colleagues show that the opposite happens; collapse is driven by a decrease in synchronization. This surprising discovery that comes from extensive numerical analysis of mathematical models whose execution has been made possible thanks to the help of the  High-Performance Cluster of the DTIC, to which it has access at Pompeu Fabra University.

Analogy between the collapse of chimera states and epileptic seizures

The most important aspect of Ralph G. Andrzejak’s work, however, is the evidence that a similar phenomenon occurs in nature, specifically in the field of neurology. In the study published in Scientific Reports, the signals of electroencephalograms (EEG) recorded during the attacks of patients with epilepsy have been analysed. Epilepsy had often been considered as being due to an excess of synchronization, whereas, in this study, in line with other recent studies, a pronounced decrease in synchronization has been observed at the start of the epileptic episode. Therefore, the authors establish a close analogy between the sudden collapse of chimera states in mathematical models of connected pendulums and seizures.

The EEG signals that have been used are sourced from the Neurology Department of Bern University Hospital (Switzerland). At these neurological centres, EEGs are recorded solely to optimize patients’ diagnosis and treatment. Thanks to the cooperation of neurologists like Kaspar Schindler and Florian Mormann, both co-authors of this study, the EEG records were made available for retrospective scientific analysis.

The control of global synchronization opens the way for new applications

Returning to the mathematical model of systems, Andrzejak and colleagues, applying their findings to a control paradigm, managed to both cause and prevent the outbreak of global synchronization. Therefore, not only are these results a step forward in the understanding of networks of coupled dynamics, but it can also open up new ways for their control, and provide a wide range of potential new applications.

Ralph G. Andrzejak stated: “Our work is a piece of basic research. When it comes to neurological treatments it is very important to avoid overly high expectations. In the future, further research will be required in this line to apply our findings to the real world. Although our findings cannot be applied immediately to improve the diagnosis or treatment of patients with epilepsy, we should have a positive attitude because our findings can contribute to better understanding epilepsy, that is to say, the fight against a disease that affects more than 50 million people around the world”.

And he adds: “in the long term, an application of our control scheme could be used in the treatment of various neurological diseases in which the balance between synchrony and asynchrony is altered. In addition to epilepsy, this group includes such diseases as schizophrenia and Parkinson’s disease.

Andrzejak conducted this study with the support of project, COSMOS, the financing of the Spanish Ministry of Economy and Competitiveness (GrantFIS2014-54177-R) and the German Volkswagen Foundation.

Reference work:

Ralph G. Andrzejak, Christian Rummel, Florian Mormann, Kaspar Schindler (2016), “All  together now: analogies between chimera state collapses and Epileptic seizures”, Scientific Reports 6, doi: 10.1038 / srep23000.