"This cannot happen"

The statement made in the title often marks a major step in our understanding of something that happened but should not have. In a scientific context, this can result in a shift of paradigm.

This cannot happen ... unless, of course, my assumptions were wrong.

You may ask What cannot happen? I have indeed something in mind. But let me just for a little longer stay in the abstract and call it it.

It cannot happen. Three words that say so much more. Who ever said or wrote them obviously thought about the possibility of it, or was forced by someone else to think about it, but than asserts that it cannot happen.

So what is my actual topic? What is it?

 © Copyright Penny Mayes and licensed for reuse under this Creative Commons Licence.

Well, if you have read earlier posts, you know what will come: more about migraine research from the perspective of a physicist.

But apart from this, if we abstract the specifically migraine research content, I illustrate with examples a specific case of general interest. Namely observing something that should not have happened because it is unstable behavior (or rather an unstable solution in mathematical terms). Just look at the image above, and you will know what I mean.

Observing unstable behavior does not really contradict the theory you have, yet there clearly is a problem. Or would you ever pitch your tent near this rock?

Artefact or scientific revolution

Let me start with a simple example.

What if your are a scientist and you just have observed something that should not have happened according to current scientific believe? You are excited of course. This could manifest a shift of scientific paradigm and your are right in front of it.

To be more concrete, imagine you are interested in the theory of gravity. On a sunny day you hike through North Yorkshire, England and you see the rock shown in the image above. Intuitively, you may think:

This cannot happen. Unless, of course, the theory of gravity is wrong. 

In view of the bizarre rock formation you start to ponder:

"... Maybe there is a difference between inertial mass and gravitational mass? No, that would not really explain this bizarre rock formation. Wait a minute, there may be actually nothing wrong with the theory of gravity. This rock formation is a perfectly valid solution of how rocks could behave within this theory. The only problem is that this solution is unstable! ..."

To me, unstable solutions create very interesting situations. In a different dress, such a problem puzzled me many years. 

Unstable migraine waves

I was puzzling over waves in the brain that cause migraine. In particular, I investigated how are the chances that such waves can occur. You may ask How does that relate to the rock example? Well to see this, you need to replace the landscape of North Yorkshire with the landscape created by the phase space of waves described by a mathematical model.

Don't worry, if the last sentence did not make any sense.

You don't need to understand the details. Just try to imagine that we can identify within an abstract space valleys and hills, that is, stable and unstable states, respectively. This space results from a problem like the occurrence of waves in the brain. Once we have a mathematical model for such problems, that is, the migraine waves, we have also a landscape, much like in the picture below. 

 

  © Copyright Markus A. Dahlem

So the problem was that any of the observed waves in human brain during a migraine attack are more like the one shown in the middle part of the figure. Such spatially confined waves are unstable solutions and we should not observe them (unless, of course, the current model is too simple!).

Unstable solutions are not good solution to real life problems for the same reason why you would never pitch your tent near the bizarre rock. They collapse usually in no time. So I needed an extended theory like in the case of the weird rocks to explain spatially confined migraine waves.

The rocks are eroded by water and wind. This is obviously a delicate process. There is no direct analogy of the wind, water and erosion in the case of migraine waves. So I leave it at that and continue with the migraine waves.

Simply speaking, I believe the brain found a method to merge the stability of the wave shown in the left side, that is, a wave that engulfs all of one hemisphere and the spatially confined wave shown in the middle. This is a phenomenon, termed saddle-node bifurcation, that shows very universal behavior and is found in many dynamical systems. In other words, the brain found a clever way to avoid the global wave, that is, more damaging waves, and yet allow for a highly excitable medium, like the brain clearly is.

The main purpose of this blog is to explain in some loose way how mathematics and physics enters neurology. It is about creating models of the brain and investigate their descriptive and predictive power. Probably everyone is awestruck in view of the famous Brimham rocks. Unfortunately, it needs quite a bit of mathematical training to see what in a dynamical systems like our brain can and what cannot happen, or how the dynamics must be modified so that it can.

The goal is to make predictions based on such mathematical models. For instance,  how to avoid unwanted behavior like migraine waves. If we can proof that the migraine waves are really created by such a saddle-node bifurcation, than we also can exploit the universal dynamics near such a phenomenon in migraine therapy. 

EPILOG

If an assertion fails

It was in the early 90ties when I first read 

This cannot happen.

I was, at most, only partly aware of the irony. The sentence was the message I read as a computer program crashed.

I only later learned what it meant. First, I thought this is just funny. But as I started to develop my own computer programs, I came in the situation to write programs that would crash more often than I wished.  So I read books about programming, like Code Complete. One thing I learned was that this sentence was an assertion. And that such asserstions can help me to develop better programs.

In computer programming, an assertion is a programmed condition that the developer thinks is clearly true. It is good style to place assertions at routines to document what your assumtions are. Not as comments, which do not get executed, but as actual code that stops the program if an assertion fails. This should not be mistaken as error handling. Assertions check for things that should never happen, while error handling deals with, for example, the situation that the file you tried to open is defect. So, erros can happen.

To cut a long story short, if an assertion fails, something is very wrong. In a computer program, it means you should not trust the program at all. From this point on, the behavior is unpredictable, as the programmer obvioulsly did not thought about this to happen. That is why the program must stop.

The actual message in this case could be more diagnostic, i.e., describing the condition that should have been true but was not, but this is not the point here. This epilog emphasizes what assertions are good for and that it helps us to make our assertions explicit, not only in programming.