# Stadium wave in the nerves: a new mathematical model

Electrical signals travel like a wave through our neural pathways. The mathematical models for these movements could not yet properly describe all the biological properties of the nerves. PhD student Willem Schouten-Straatman changed this by improving the existing models. ‘I hope that one day we will be able to predict the behaviour of electrical signals in our bodies.’ Promotion on March 2.

## Do the wave!

‘The problem with the old models was that they didn't take into account the fact that our nerves have what is called a discrete structure,’ Schouten-Straatman explains. The difference lies with the medium through which the wave travels, he explains.

‘Think, for example, of waves in the water, or the vibration of a string: at every spot the medium is the same, we call that continuous. With a discrete structure, however, there is a distance between the medium through which the wave travels.’ The best-known example of this is a stadium wave. Here, supporters form a wave-movement through the audience by standing up in alternating order. The medium through which the wave travels, the spectators, do not form a whole: there is a distance between the people. ‘In this day and age even a little more than in the past,’ laughs the mathematician. But we don’t call it a “wave” for nothing, we still speak of a wave phenomenon.’

## Stadium wave during a football match between Real Madrid en Manchester United

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You can leave our website to view this video.## Biologically correct

Nerve signals resemble this stadium wave: they jump between so-called Ranvier nodes in the nerve pathway. However, this discrete structure is not reflected in the mathematical models that describe this phenomenon. Instead, the models approximate reality by stating that the nerve pathway is a continuous entity. This, according to Schouten-Straatman, is because there is much less mathematical theory available for discrete systems than for continuous ones. ‘However, that discrete structure is an essential part of the underlying biological process. Mathematicians therefore went to work on a discrete model (see box), which my colleagues and I have now improved.’

### A lesson in math history

Mathematicians have been investigating continuous-wave models since 1850. In 1960, the American Richard FitzHugh and the Japanese Jinichi Nagumo introduced a simplified model for electrical signals through the nervous system: the FitzHugh-Nagumo models. It was not until 1998 that American mathematicians James Keener and James Sneyd formulated the first discrete version of this model. Proving the model mathematically proved to be difficult. Only in 2009 did Hermen Jan Hupkes (Leiden University) and his then supervisor Bjorn Sandstede succeed in proving that wave solutions exist in the discrete FitzHugh-Nagumo model. Only in the last few years have discrete models really emerged.

Subsequently, Schouten-Straatman had to prove mathematically that electrical signals could actually run through this new model. And that was not so easy: 'A lot of situations and exceptions occur that do not occur in less complicated models. This means that many known techniques cannot be used just like that.

But Schouten-Straatman and his colleagues succeeded and were the first to demonstrate that the models were correct. They also showed in a number of cases that the electrical signals are stable. This means that the system itself can ensure that disturbed signals - for example due to imperfections in the nerve pathway – can recover themselves.

## Predictions in the body

‘I hope that one day we can use this model to predict the behaviour of electrical signals in the human body,’ says Schouten-Straatman. ‘For example, think about how a signal reacts when it encounters an imperfection in the nerve pathway, such as a lesion. You can currently measure this in specific cases in patients, but in many cases this is difficult. How nice would it be if we could then make predictions with our model?’

At the moment, the importance of the research is mainly theoretical. Even these more complicated models are not yet precise enough to make the above predictions. ‘However, the even more precise models are so complicated that it is virtually impossible to prove anything about them,’ says Schouten-Straatman. ‘But it gives a lot of hope for the future that our already quite complicated models have managed to prove a number of important properties.’