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Mathematics is as sharp as a scalpel and cuts brain malfunctioning into pieces.
The first part of my title is copied from an awareness campaign of the Society for Industrial and Applied Mathematics, better known as SIAM.
In this campaign, mathematics behind everyday life is explained on 16 posters, such as “The Math behind Stopping and Preventing Fires” or “The Math behind Cardiology and Heart Attacks”.
In some sense my blog is surely also an awareness campaign: The Math behind Neurology.
In which way do I think math matters in neurology?
Mathematics certainly matters in many ways in neurology, but most are, in fact, not specific to neurology. Take statistics or other mathematical techniques to analyse data. Tomographic reconstruction is a good example. It is the basis of non-invasive medical imaging like CT or MRI. Without CT and MRI neurology would be quite different today. But so would cardiology.
So, which mathematical concepts are more specific to neurology? To get to this point, we should take a closer look at the brain and its functions, or rather by which means these functions are accomplished. The brain and its functions make us what we are, at least more than any other of our organs and their functions.
The brain consists of neurons that form networks, in which the neurons can synchronize their activity in infinite many ways. This neural activity in the brain forms complex spatial and temporal patterns. These pattern formation processes are to some extent amenable to mathematical analysis. Similarly, if brain functions fail to work normally, there are specific neural activity patterns that have caused this malfunctioning, e.g., neurological diseases.
I don’t think we have mathematical concepts for what finally results from normal brain functioning, that is, for the mind. And my guess is we will never have. But we have mathematical concepts for the dynamical behavior of single cells and their activity in small networks, like reflex loops or circuits that can perform edge or motion detection tasks in our visual cortex. And there are many more such examples.
We also have mathematical concepts describing the onset of neural activity in neurological diseases, like epilepsy, migraine, essential tremor, and others. For example, how a healthy state of the brain becomes unstable and bifurcates into a diseased state.
Bifurcation theory is the name of this mathematical field. It provides precise definitions for transitions into diseased brain states. We can apply these mathematical concepts because these transitions, or rather bifurcations in the language of mathematics, occur in rather simple networks with simple dynamics. Such a bifurcation can be, for example, the emergence of a synchronization in the firring pattern of cells in a neural population.
Why did I mention only neurology, what about psychiatric diseases? There are also mathematical concepts of neural activity related to psychiatric diseases. But, in general, psychiatric diseases are more closely related to the mind. There is almost never a simple network, as far as I know, that is exclusively affected. Mental states, diseased or not, are said to have neural correlates but they never have a clear one to one association with neural network activity, such as focal epileptiform activity causing seizures.
I am deliberately provocative at the following point. I would go as far as suggesting that if we understand a diseased state of the brain in terms of bifurcation theory or, at least, we see a chance to achieve this, it should be considered neurological by definition and otherwise psychiatric. Thus, math can be used to define a non-stigmatizing wall between neurology and psychiatry. I shall reserve that as another post to elaborate further.
Let me give an example and consider visual hallucinations. They can be classified as simple or complex. An example of a simple visual hallucination is seeing moving zigzag lines. An example of a complex visual hallucination is seeing your high school math teacher talking to you. In the former case, you should consult a neurologist, in the latter a psychiatrist (unless, of course, your math teacher is present, than you should pay attention).
As mentioned above, there are mathematical models of the neural circuits that can perfom edge detection tasks. These circuits can be directly driven by endogenous brain activity under certain pathological conditions and thus causing visual zigzag hallucinations. A zigzag pattern is nothing but a combination of edges. Therefore, a few edge detectors, that is, certain neural circuits going mad is a rather simple explanation. Zigzag hallucinations typically occur in migraine.
I have never experienced these zigzag hallucination myself, but I built about ten years ago a mathematical model of it. That clearly was more fun than having migraine. It was a small piece of math applied to a problem in neurology. The result is seen in this video.1
So applying math can help in neurology in many ways, in this example, it helped patients to recognize and communicate their neurological symptoms during migraine.
I will report in this blog mostly about my own work. And, like in this post, about the context of my research and why it matters.
I included a new video after trantioning the blogpost from SciLogs.eu to Altamirage↩︎
