Thank you very much, Martin, for the nice introduction. Before I start talking about the brain itself,
let me indulge a few general remarks about mathematics in the spirit of the little book that Martin mentioned.
So here I have on the right mathematics, abstract mathematics, fundamental of pure mathematics.
It constitutes theorem proof structure and things like that.
And on the left, we have the real world. Not exactly sure what it is, but I'm told it exists somewhere outside the building.
And in the real world you have application, you have problems, you have science, you have a lot of things.
And the naive idea of application of mathematics or applied mathematics is that whenever you have a problem in that box,
well, you'll find something in that box and you use it to solve the problem here.
Of course, this might have been true in the 19th century or the 18th century, but it's certainly not true nowadays.
When we use mathematics, what we mostly use are essentially methods.
This is the interface with the real world. We use computation algorithm.
We have things to solve equation. We have things to analyse, data, we use statistics and so on.
We use math matrices. We use linear algebra. We are a lot of it is based on fundamental mathematics, and it's very important to have that basis.
A lot of it nowadays is not part of mathematics. No more than accounting is considered to be part of mathematics.
It has its own development. If you really want to understand the world and understand the problem, what you need on top of that is theory.
And theory is really all the fundamental principle that people have extracted over centuries about the world around us in different disciplines.
It is physics, biology, chemistry, science, engineering, economy.
This is the body of knowledge that we have on which we can draw.
Different models use the method to solve, simplify, or complicated model and go back to the real world and try to understand it better.
So where the methods give us a way to interface with the world, the understanding really comes from the theory.
So naturally you would ask me where is applied mathematics in all that?
Where applied mathematics is really everywhere around here.
Some might include that whole thing around here, but let's not let's not create a debate about that.
You'll find people working in applied mathematics, working in all these area,
some of them just doing experiments, some of them just developing theory.
Many of them very much enjoying solving equation, developing new method to solve equation numerically or particularly analytically and so on,
developing new idea, how to handle data and so on.
And a lot of this theory might even be used as an Irishman to create new mathematics,
just like in the old days, when mathematics was first established based on simple question from the real world.
So you'll find question everywhere around here. So no, to get to the brain.
Where does the brain fit in that? We're in the brain.
We have very well-established theory. Neuroscience nowadays is a very mathematics theory.
There is computational neuroscience that's essentially a type of applied
mathematics for the brain that relies on a lot of partial differential equation,
network theory, dynamical system and other ideas.
Most of the interest there is about information or the brain receive receiving inputs,
and others who treat that or that input is being propagated in the brain in a
big kind of network and trying to understand the structure of the network.
There is a yet another area where mathematics is heavily used and that is to do image analysis whenever you put your head in a scan,
whatever it is, am I right? Or CT scan or anything like that?
There is a lot of algorithm going there to extract the information so that it's useful information.
I know it is the best scan, the MRI.
Over the last five years, there's been tremendous revolution based on method that comes not only from the development of methods in computer science,
but also from fundamental mathematics. The best techniques that allow us to take very quick MRI actually come from
the work of essentially applied mathematics and other disciplines correlate.
So what is my interest in the brain? So when I look at the brain personally, I have always been fascinated.
The first fascination I have is about its structure, its convolution.
And I have my question. And in trying to understand the brain, I wanted to build systematically on that.
And over the last five, six years or so, I've been trying to ask questions and trying to find basic answer.
And to my surprise, whereas there is a lot of work at the brain level, at the functional level,
I found that a very little was known about the basic question that I had, and I try to try to understand and answer them.
So when I have a title, can mathematics understand the brain?
I still not have a definite answer. When I started, I said, Let's try and let's see if I can understand or can answer some of my question.
I saw the work that can happen to me that I would learn something in the process and maybe I won't be able to do any mathematics.
But things are such that often when you start looking at a problem and you try to dissect it with a mathematician, I you can extract from it.
Interesting problem. And this is what I'm going to try to convince you.
So my first question was, what about the shape of the brain or does it attractive shape is geometry?
What are the basic things that we know? Another question is what happens during trauma?
We always talk about we hear about traumatic brain injury and things like that.
What actually happened? What is the chain of event and what are the consequences of that?
Can I understand it from a mathematics perspective? Again, I found that very little was known on the description of mathematical description.
The modelling of these and most of it in the medical literature was unfortunately not correct.
Yet. The third aspect that I want to address today is about dementia.
And this is really fascinating. It's a process that we all know people that are affected.
Something is happening to a brain. We have a disease, progressive disease, degenerative disease that removes some of the basic faculty.
And I just wanted to understand what happens. How can we describe this process?
So these are the three things that I'm going to try very quickly to sketch you and type and in the process try to fill these different bubbles.
So here I start with my first slides.
You see it's is a painting. So when you see it's. Your brain works.
You received a flat on the back of your retina.
That's that information is being sent to the by the optic nerve all the way to the occipital lobe where it's treated.
It's string part of the memory.
There is a very sophisticated algorithm that people have discovered that allows you to recognise that it's a face, eye and so on.
And based on that, you trigger some memory and say, ah ha, yes, on top of that.
And you've all done that within a segment. This is Ludwig van Beethoven.
A lot of functional process happened there. Right.
And so you can see or maybe part of you memories being triggered and some music comes off,
maybe wondering, why is he showing us a picture of Beethoven in a mass talk in the first place?
So this is all the function of your brain and what it works.
There is physics behind it. Blood is being delivered. A different part of the brain signal is being processed.
So there is a lot to understand there. But again, I'm showing this picture of Beethoven because I want to talk to you about his brain.
So what do we know about his brain? Well, let's hear from Beethoven itself.
When Pontus brothers sent him a letter and signed it from your brother, Your Honour, Beethoven replied.
From your brother, Ludwig, brain owner. So we know you had a brain.
Or at least he thought he had a brain by the time of his death in 1827.
In middle or early in the 19th century, scientists were obsessed about brain and the morphology of the brain and how it
was related to the criminality or to the insane or to the idiot and all that,
but also to the genius. What makes somebody so remarkable?
And indeed, the question is the following. As Ludwig says, there are many princes and they will continue to be thousand more.
But there is only one Beethoven, and I don't think he meant his brother when you talk about that.
So what was remarkable about Beethoven's brain was good thing we have a is
autopsy report makes very light and enjoyable reading so we can look into that.
So the first thing you think that at the time it was an idea for the whole 19th century and the idea was bigger, brains are better.
It's something that we still always the big brain and things like that.
So my first parts, first part of the talk, the first Mothman would be or because your brain would fit.
So when you look at a human brain like in this drawing, I want you to have an idea of the physicality of your brain.
So how big is your brain? Well, so you take if you ask people to say, oh, that's my my brain, but that's actually your skull.
So if you put your two hands together, you're welcome to do it. But you and you put them like that.
This is about the size of your brain. Okay. So to be precise, let me take a brain here to see this is the size of your brain, right?
It's about 1.2 to 1.4 kg.
About right. It's about the volume of this little ball, which is 70 centimetre radius.
Seven centimetre radius. That's right. So this is your brain.
1.3 kg, let's say on average and so on. So.
If bigger brain are better, these brains should be even better.
This is Africa, NELSON And it's three times as big as our brain.
I said, Well, we know that African elephants are probably very wise, but they don't do crosswords.
They don't tweet about Kim Kardashian and all that.
I know it's not as is is not a measure of intelligence, but even to do these kind of things, you do need higher cognitive ability.
You'd have to agree with that. So you've got to say, aha, but wait a second.
This is only three times bigger. An elephant are much bigger, maybe 50 to 100 times an elephant.
African elephant, 6000 kilos. So you should really scale.
It should really brain should be the weight of the brain per the mass of the animal.
And and indeed, if you scale it up, this is what it would look like.
He's a random individual next to an African elephant.
And if you scale it, indeed, the brain does not grow as fast as the rest of the body.
As you go and look at different animal. And this is a law that was discovered by the Swiss physiologist Von Heller in 1762.
It does mention the African elephant. I'm not sure he mentioned any royalty in his description.
And the idea that is, as animal gets much bigger, the brain doesn't grow as fast.
So how would we quantify that kind of relationship? Well, let's assume we get a lot of data about brain weight and body weight.
And the first thing we want to do, we could plot them. Let's say X is brain weight and an X is body weight and Y is brain weight.
Now, if it falls more or less on that line, like that's what you want is put a line and then you can find the best fit line
and you'd find a correlation or you'd find the coefficient and B in the process.
Very traditional. It's very easy to do.
But what we want is not a linear low that we expect from very low, that it doesn't grow in the same proportion.
It grows slower than the brain.
What we express is a non-denial, so the data might look something like that where if you try to fit a line, you see the line is really not good there.
So that would be a terrible fit for low. So what you really want is a low that looks like like this one, right?
What you want is to find this curve. And if it's bigger, that one, in such case, it goes faster than the body weight.
And if it's less than one, then it goes slower. Like what we would expect from one other slow.
So you can try with different points. Oh, it looks like a parabola.
So I can fit a parabola. But what you really want is to extract this coefficient.
So the smart trick that you can do is you take the logarithm of that relationship.
And if you remember that the logarithm of a product is the sum of the log written, it transform into this low log y equals eight x plus log b.
So if it follows a power law like this in logarithmic coordinates,
if I plot log knock plot it would look like a line and it will be the gradient of this line.
And if it's less than one, we have something like one plus line is bigger than and what we expect we want meant to be superior.
So we hope that somewhere will be so.
We'll be right. Here. And we can see animals are all over here that follow natural evolution and we are very special
because we do not follow that this scaling that does not apply to us and man is superior.
Well, if you look at the data, it's not quite as clear this is what the data looks like and man is still in red is right in the middle of the pack.
Really. You see the pesky, the dusky dolphin right here is as good as we are really.
And brain per pound while the mouse is actually the same ratio.
So really that's racial argument really does not apply.
And indeed the exponent here it is a slow that is between two third and three quarter.
So if you do like scaling laws and there is a good talk coming in April seven public lecture here by Geoffrey West who spent his life doing
that you say haha point 66 that two thirds that I know what is coming from something like surface area in volume or people would say aha,
it's three quarter, I know where it's coming from. This is the metabolism blow and you can make a whole theory about that.
But that's not what I want to do today. I don't want to speculate. I want to understand the mechanism about that.
And you see that this idea of brain wage really doesn't doesn't work.
And by the end of the 19th century, Darwin was really aware of this.
When you wrote in The Descent of Meant,
no one supposes that the intellect of any two men can be accurately gauged by the cubic contents of the skulls.
And really that was the end of that, that line of reasoning.
So going back to going back to Beethoven, if his brain was when his brain was looked at during the autopsy,
unfortunately it was not remove that distinction came to LE Plus actually the mathematician and Laplace
sat in a jar of alcohol as a trophy for many years on the desk of the pathologist that took it out.
So Beethoven's brain was not removed, but it was described as being very small in atrophy.
And we'll go back to that in a second, just remembered. But probably the doctors were desperate to find something special about Beethoven's brain.
And in the description they say the convolutions appear twice as numerous as in ordinary brain.
Hmm. That's interesting. So maybe it's not the weight.
That is important for being a remarkable genius.
Maybe it's the convolution. Ah ha. So maybe we should look at that.
So our next chapter or next movement is all convoluted.
Ah, you look, Rick and I show you as a proof,
as other evidence of t of you statement that come related brain that better this I offer you to his brain.
And I'm sure mathematician would directly recognise anything coming from his great brain because this is Carl Friedrich Gauss,
the the greatest arguably one of the greatest mathematician who was described also by a doctor
named Wagner as the cerebrum is remarkable for the great complexity of the convolutions.
Goes up a little later is brain was not either remarkable in volume, but maybe there was something special about it.
So what do we mean about convolution? Let's think about it a little bit deeper here.
So if you take a brain and you take a slice, so this is called a coronal slice.
You slice it like that, what you'll see is this convolution and you see two type of tissue.
The outer tissue is called the grey matter because it appears grey when you staining that you remove it.
It's mostly made of cell body, really highly packed.
It's where a lot of mental activity I cognitive activities take place and the white matter inside.
It's a more fatty tissue. That's why it looks a little white.
And essentially you have axon bundles carrying information between different parts of the brain.
You can think of that. Here is where the processing takes place and the highway of information.
The cables in between essentially vary. But what you see is the convolution where you have this thrust and it is richest,
this valley and Montell that I call sulcus and gyrus or circadian gyri, if you have more than one, and this is what we're going to look at.
And in the brain, it it it takes it first appears around seven months here.
The in gestation pregnancy, the breast is smooth and then at some point it becomes wrinkled and is wrinkled,
further develop and then gets convoluted to a normal brain.
And let's look a little bit deeper. We have good scanner for this kind of thing.
We have amazing scan, actually. It's a whole field that generates a lot of data.
In the scan, you see right at the 27, 28 week of gestation, you go from the smooth brain to it.
You see here the wrinkle really developing here to be very convoluted.
And indeed, if you look and the idea is that at that time the area increased much faster than the volume, right.
So if you look indeed at the cerebral area at that time, it increased very quickly compared to the volume.
So if you had a brain that just expand like a sphere, the area would go much slower than the volume.
It goes like a square and the volume grows cube.
So but if the area grows quicker, it gets folded inside.
And what you will measure is the area and for a smaller volume.
So the folding comes inside and that's why you have this different.
So how do we quantify that? A nice way to quantify as it looks is to look at what's called the diversification index,
which is the ratio of the curve where you follow the contour by the curve that you obtain around the control,
the outer contour here you see it well is the blue, the length of the blue curve divided by the red curve,
the one that you put if you put like an elastic band around the whole thing?
In this case, for instance, the gentrification index is 1.47.
Okay, so now we have a way to measure things and we can go and look at all animals and all that.
Could brains and be happy. Aha! Hooray!
Here we are again. The red dot sitting on top of all the animals.
Way ahead of the competition. So finally here, we found something that distinguish us from all these animals.
And indeed, if you look closely and you compare us to a primate cousin, we do much better in verification.
Of course, the only problem is when you include the marine mammals.
We are here and purpose is here in very far, even for a long way.
And there are many more Jerry ification here. So, again, as a measure of anything remarkable about the brain, this might not be its right.
So my question is not so much why are we superior?
Why is the brain?
But again, I look at this slice and I said, this is striking in beauty but also in patterns and applying mathematician really like pattern formation.
We like to understand a pattern that formed from simple physical problems.
So can we do it on this case where people, of course, have been thinking about that for a long time?
And this is there are many different hypotheses for that and we can use mathematical modelling to test them.
One of the earlier hypotheses was going back to about a century is that the brain grows.
It hits the skull. And since the skull is very hard, it's forced to fall back.
And that seems a reasonable idea. But in the fifties, Barron at Yale University did a series of experiments.
And when I say a series of experiments, this is ship number 409.
And this is really cheap, number 409. Right. So you would go and manipulate the embryo of the land during gestation and see what happens.
Very, very nice paper, a lot of data. But I mean, it must be quite a challenge if you think about it.
I call it The Silence of the Lambs experiment. So but one remarkable experiment to date is the following the wandering gestation and remove the skull.
And when you remove the skull, it still saw all the convolution. So the convolution are not created by the interaction with the skull.
Another hypothesis and the leading way of thinking about this,
this day is that the grey matter undergoes very rapid tangential expansion on top of the white matter and that force it to fall on itself.
So let's see if we can understand that in a simple case, let's be laterally model for that.
We going to take simply a slide and we're going to try to understand all of the convolution that created in a two dimensional setting.
Let's simplify the problem. So if I rectify that, it looks like this I have the white matter in blue, the grey matter in yellow, where the cortex is.
So since I'm only interested in the relative growth,
I'm going to assume that the white matter grow quickly and that there is no growth in the the white matter.
So the grey matter growth here.
So if a beam of a thin film growing very quickly and attach bonded to a substrate, an elastic substrate that does not grow.
And my question is, is that critical value of the growth so gamma equals zero mean there is no growth and gamma equals one like
twice as long as the original one is that critical value where instead of having something like that,
I'll have something like this, I'm going to start having wrinkled appearing.
And what I want to know is the number of oscillations per unit length.
And that would tell me something about the brain maybe. So why do I call this a toy model?
Where here is an analogy for you. Suppose that your scientist was mostly interested in monster trucks and this is your favourite one.
As a scientist, you've been looking at it your whole life and you know all the parts and all the details, you know, the pistons and things like that.
But when you come to the subject and you say, Well, what I really want to know is how this thing move.
What is the principle of the locomotion to start with? All this is one thing works, for instance, right?
So maybe you're not so much interested in the the wall sticker that is right here that
the scientists think is very important because he'd seen it in all the monster trucks.
But maybe we can start with the wheel. So what you want to do is you want to start with a toy model, maybe something like that,
or if you're really lucky, you thought model is really going to look very close to the original ones.
And this one comes with crushable costs. So it's really cool.
So this is my top model for you. We're going to play a ways.
And my mathematical toy model is this one.
This is the simplest possible one that's still containing a physical information so that we can learn something.
It is a relatively simple equation.
It's a fourth order equation that describe the mechanics of a beam attached to a substrate in very, very simple time.
So w is that wrinkle. What we want is w as a function of s.
So for instance, this looks like a sign of something sine x, right?
So we want to find that h here's the thickness.
Gamma is the growth and size, the ratio of the stiffness of the two layers.
Essentially a material parameter that tell you are stiff. The stiff you layer is on top of the substrate.
So it's relatively simple and I'm sure you want to see the real thing.
You want to go and see under the hood of the the real machine so I can let you peek for a little time.
This is the real system in two dimension that has all the exact mechanics.
It is a system of two rather nonlinear partial differential equation in two variable in mixed coordinates formulation takes a while and all that.
This is the real deal. This is the real thing until we on which we work.
And you have to add that boundary condition and this is work with IMSA.
I don't know if I'm XYZ around and I know you feel all very sorry for him having to deal with that.
But actually we do have techniques like every, every, every technical discipline we have way to deal.
That is not like we go by hand and and change the power or something like that.
We have plenty of way to deal, but morally, the physics of that,
a lot of the physics of that is also contained in my toy model and we all want to go back and play with it.
So let's go back to the toy model. So now you find this one is not too difficult.
So it's a one with constant coefficient. And maybe you've already seen ordinary differential equation.
It's part of some of the further math. So you you will see to some point.
So we can start playing and removing wheels, for instance, and see how it works.
So let's see what happens if we remove system. So let's remove system.
That means we don't want any substrate here. So we remove the substrate and what you have, the layer grows.
Push against the wall and buckles. Right. This is Euler buckling 1757.
As the layer grows, it push against the wall and does something like that very well.
Understood. And if you work a little bit on this equation, you can you can find a solution in this.
Not very difficult. But no, we want to add the substrate back.
You still have a solution like that, but this one takes a lot of energy.
You have to raise the whole substrate. So this is not the preferred solution.
The one that is preferred is one with a lot of oscillation.
And it's a it's a compromise between having to bend so between system and having to lift the substrate.
Now you see that we can try a solution of the for cosine because is constant coefficient of force derivative
of cosine is the cosine the second derivative of cosine the cosine and the cosine is the cosine.
So it's the solution of the equation for some value of n and gamma.
So you can put that back in studied.
And what you find is that the critical mode number or any oscillation you have depends on gamma as the square root of gamma divided by the thickness.
So here is the thickness and I can run the whole thing. I can do a simulation of the whole thing and this is what we would like.
And we have a critical value and we have the number of peaks per unit length or something
like that is this is going to be given and you recover that in the full system.
So the simplified system contains already the physics there.
What is more important is what we can deduce from this formula.
Why? That's three things we can look at. We can see that we will have more wrinkled, larger N if you have more growth.
So gamma increase, you'd have more wrinkled with thinner cortex.
If H decrease makes and bigger and you have less wrinkle with thicker cortex.
So let's see what's basic trends compared to what we know about the brain.
Go back to the brain and in the brain you have that.
In pathology, you have interesting case. So for instance, in the case of schizophrenia,
you have excess growth and the excess growth is related to a lot of wrinkle and the higher diversification index.
So indeed, an increase with gamma.
You can look at all the pathological case where you have a thinner cortex, and this is probably called probably micro diarrhoea.
And when you have thinner cortex, you have more and more oscillation.
And when you have a thicker cortex, this is called listen carefully and you can have an almost completely smooth brain.
So this is all consistent. So let's reflect back before we move on to the next topic to what we've learned.
Skating knows a good way to organise data and try to look at some laws with it.
Give us good suggestion of where to look. But if there is any conclusion, is that based on scaling?
No, there is really nothing special about the human brain, at least not the one we know.
But for brain morphogenesis. Now we have models that allows us to identify the correct mechanism that leads to the convolution we
can properties model to both the genetic and the cellular level and try to answer a final question.
We can try. We can generalise that.
We can look at different species, for instance, and different geometry, very the parameter and all that, and recover other trends that observe.
And since convolution are an important biomarker for a lot of neurological disorders like autism, it is a freena.
It is crucial to understand the origin and or the form during morphogenesis of doing brain development.
And in the process we see that we had to develop theories to understand our.
Continue material grow and we had to formulate a new theory for that.
That's what we've been doing with colleagues for the last ten or 15 years or so.
We had to develop new methods in order to deal with differential equation and that nice aspect
of pure mathematics that I have no time to discuss that you can also generate in the process.
So let me go back now to Beethoven.
Oh, what trauma? So, as you know, Beethoven was dead for a large part of his life.
He says that you didn't hear the Ninth Symphony being played for the first time.
And the reason there are multiple hypotheses of why. Why wasn't there?
What was the cause of his deafness? Beethoven himself attributed that to a fall which says that the E heard is brain and that caused swelling.
Other people believe that it was the abuse of repeated abuse of his father.
And the idea is that that swelling in the brain caused compression on the nerve that damages hearing.
Now, the hypothesis that this was discussing, but for us, this is an interesting point.
Swelling in the brain, swelling is really bad in the brain.
It can occurs in different case in an accident,
but it can also be due to rather stupid human activities like punching each other in the face or using the head as a weapon.
It's very strange when you think from the evolutionary perspective,
when evolution has been something over millions of years to try to protect the brain.
Oh, yes, but let's put a helmet around it. That's really going to help.
Or it can also be due to a stroke or a brain tumour, also creating inflammation that produces brain swelling.
High altitudes can also talk brain swelling. Medical is called cerebral.
Oedema is just mean brain swelling. Oedema is the accumulation of fluid in the interstitial.
At the microscopic level, it is due to the fact that fruit from the blood vessel should think of these as tissue is being drawn inside the tissue.
And this is due to the fact when cell die, the release a lot of small iron, positive and negative charge, a neutral subject into the brain tissue.
And that creates a huge osmotic pressure difference between the tissue and the blood vessel that drains the fluid from the blood into the tissue,
a very complicated mechanism that has electrochemistry, fluid dynamics, solid mechanics and so on that you have to incorporate.
And we've done, we've done some modelling that then. So some of the mechanics question about that, but that takes a little to.
The interesting thing is that when it starts happening, it's really bad.
Why? Because when it starts happening, the brains swell.
But there's no places to go. So it increases the pressure and it increases the intracranial pressure.
Blood vessels are occluded and cannot deliver oxygen to other parts of the brain.
Other cells start dying. They die. The release, the solute, the release, the ions and the brain swell even more.
So you have catastrophic event, a cascade that takes place in the brain, and that's in very tissues.
The part that is necrotic, that's invade the whole brain within a few days.
So when doctor sees an elevated intracranial pressure of 20 millimetre for more than 20 minutes, nowadays they have to intervene.
And the one thing they can do is rather drastic. But it's one of the few things they can do.
Is this removing a big part of the skull?
This is called the compressive connectome. You take a big part of the skull out to let the brain swell and release the pressure so cells don't die.
So that's I thought was very interesting is a process called images, too.
So I decided to look a little bit at that. And again, we had a simple question.
And if you look at that, my first question is, do I understand this shape or this?
What is the shape of that bulging brain? But also, what are the forces generated during that process?
If I have swelling here and what are the possible damage that occurs during that swelling process?
So started with a toy model.
Here's a little experiment experiment that that Finn has been doing here in the lab where you put a judge and you let it swirl.
Essentially, it's about the same mechanically as pressing on it. And you probably can see it and it works.
But here is the movie. So this is swelling to an opening.
Imagine that is the brain and that is the opening here. And what you obtain is a bunch.
So a problem was, okay, let's study the purging problem. And to my surprise, again, the bulging problem has not actually been studied at all,
even though a lot of problem in contact mechanics had been studied.
So the bulging problem, you assume you have a big that's a big cylinder and you fix.
You fix the displacement around here and you let the brain bulge out like this.
So just like this box, you have a small opening and you squeeze it and what you want is what happens there.
This is a finite element simulation. And what you see here, shear forces, sheer stress that presents a singularity here.
The stress goes to infinity around this point. Okay.
And the shallow singularity is always a clue.
This is always a good point to look at so you can formulate this problem and do some approximation in linear elasticity.
And when you do that, you can further simplify it and try to find the pressure at all point that maintains its shape.
And when you do that after a long computation, you actually obtain a very simple solution.
This is the exact solution for this problem. It says it tells you the shape of this bulge.
And if you look, it's actually almost like a sphere that is squashed by a factor here in front of it.
So this is the exact solution. It's a project. It's a half a sphere for the for the problem.
And once you have that, if you have that information, you actually have everything.
You can start looking at the force generated inside the material.
And you can look at this point where you have a singularity,
where the stress are very large because you suspect that to the point where you would have problem if you have a tissue,
if you have too much compression of shear, this is where the cells are going to die.
And what you see here are what we call damage drops.
And you see they always have this shape,
very nice regular shape when you look closer and you can actually find exactly the angle given by that nice formula,
the attention of a, of a nice, uh, root number which amount to 111 and 32 degree where you have these two directions.
Of course you're going to say, well, this is you top model or relevant is it for the real thing?
So we should do the real thing. For that.
You need a brain. If only I had a brain.
So. So in order to get a brain, here is Allen KUHN, my collaborator from Stanford, which I've done with whom I've done most of these studies with.
And here is Bennett. I putting Alan in the MRI machine, and you scan it very, very quick scan and you get all this slice.
And once you have this slice, you can do what's called a segmentation.
You can tell and say what tissue is is at what place in the slide.
And you can do that for all the slices. And then you have a description of the tissue at all points into different slices.
And then you reconstruct the full 3D model.
You put all this slice together and this is what you obtain.
So what you have is a match. And that message has 1.3 million tetrahedral elements that describe both the skull,
but also the different type of tissue and tissue is the 3D print from my office.
This is actually the same brain as this one that I show you, and that is the skull that goes with it.
So here you see the white matter. Here you see the cerebellum.
And now we have everything we can. Let it swell with the skull.
Right. So we put. We put back the skull.
I know what I want to do is I want to open it.
Here is the opening. I do the compressive craniotomy. This is glow in the dark.
But you can't really appreciate that. So you do the compressive craniotomy and united swell.
And this is what you have. And once since you do the full computation, you not only have the displacement of geometry,
but you have all the forces that are developing that very complex material and very complex geometry.
You have the displacement, the sheer force, the strain, and so on.
So you can go back and compare comparison is not that great, is not the best the best cut here.
But you can compare the damage drop that occurs here. You find them again with the same angle on the to dislike.
Okay. So the same phenomena that you see on the very simple model is actually very generic in all these problems.
And if you had us before, to the neurosurgeon, where are the possible damage that occurs with these that would say,
of course, it occurs right at the boundary that they know because there is a lot of pressure.
But there is no there was no understanding. Await occurs inside the material.
That's only one type of damage.
And you can deal with that if the opening is large enough or whether you have another type of damage is that when you open the brain,
you actually going to stretch the axon.
And we know from external studies that stretch that even like less than 20% in stretching the axons can lead to permanent damage.
And when you look at this map, you'll realise that for this small open,
you actually have extremely large 50% damage, but also very deep inside the tissue.
So you do have damage and the damage is very deep inside, inside the tissue, which was also a bit of a surprise.
So no, what we have is really computationally and a theoretical framework on which you can
test hypotheses about different option and also gain understanding about this process.
We understand piece about swelling. We understand about trauma and craniotomy and about damage.
It's very early on and we still have a lot to learn, of course.
And maybe we are the Mad Men treating the Mad Men just like in this painting of Bush.
So as I told you, Beethoven's brain was rather small, unfortunately.
And one of the reasons why it was so small is that probably.
Beethoven like to drink a little bit too much and actually die of cirrhosis infection of the liver due to excess alcohol.
Not helped by the fact that the cure at the time was to drink more alcohol.
Okay. So understanding is good for teens.
So the problem is that if you have a normal man, here's is a normal 43 year old compared to alcoholic.
You see, you do lose a lot of brain mass. So when somebody tells you that's alcohol, preserve cells when they drink, it's not true.
You do lose a lot of sense due to excess alcohol. And here is an extreme case.
You probably recognise too many duff beer, I think.
So in the normal process we want to talk about ageing, atrophy and apathy, dementia.
Eventually, in the normal process we know that we lose brain mass.
We gain brain mass up to the age of very quickly, up to the age of four, then slowly up to the age of 20.
And then we start declining in brain mass. So you can all place yourself somewhere around here and you know where you stand.
But that process can be accelerated tremendously due to some type of disease.
For instance, in Alzheimer disease, it's well known that you have very,
very strong atrophy due to the propagation of some type of toxic protein in the material.
So the way it happens is that there are certain type of molecule, there are different hypothesis which one are important and so on.
Of course, a lot of complexity, a certain type of molecule,
very small molecule protein in inside the cell that are toxic, that are turning to toxic proteins.
And when they initially see that, they propagate through the brain and propagate,
and when the propagation, the concentration reach a certain level, it fully damage the brain.
And that part of the tissue is removed. So diffusion process or transport process are very interesting.
And one of the simplest things you can do at the mathematical level is really try to extract completely and make it extremely simple model.
And the idea is the following here is a structural map of Axum.
So this is taken from image analysis. You can find Axon, the axon bundle I told you about.
It's beautiful. The object you see, what you can extract from that is a graph.
You can identify a region of interest and see all the connected to the region.
This is what's called the Connector, which is in this case, a structural connector that tells you all parts of the body are connected to each other.
But this is nice because that's a very neat mathematical object.
This is a graph and a graph. We can start labelling notes.
Let's call that one and two and three and four.
A lot of times you go on like, let's say 100.
And out of that, you can create matrices that allows you to study this object and the matrix you can create.
For instance, let's take a very simple one with six nodes. One, two, three, four, five, six.
I'm going to create a matrix. And I'll tell you, I create it.
Let's call that L and matrix on the diagonal will be the number of neighbour.
This one has two neighbour, five and two. So I put two to a three neighbours, so I put three in and so on.
Very easy. And I put minus one.
Each time I have a neighbour, two, one, four and the first row as near as I put a minus one here because two is a neighbour, five is a neighbour.
So minus one. Minus one. And I create a symmetric matrix.
Okay. We all we all agree with the rule. Very simple.
This matrix is very important in graph theory and in all network problems because it is called the graph
la pression and it is the matrix that contains information about our process diffuses on a graph.
So if you think let's hit number one, I want to know or the heat is going to flow in the graph from number one and propagate to number six.
All that information is contained into that matrix,
the graph la pression and you can extract it by taking things that are called eigenvalues that you have ever either seen.
And you know what I'm talking about or you have not yet seen. And when you say them, you'll go.
I remember it was important for the brain. So trust me on that.
You can extract that knowledge. And when you extract that knowledge, here is the second eigenvector of the network diffusion.
So the of the graph la pression. For now I make a big one and the colour represent the intensity of this eigenvector.
This is a nice work from 2012.
And the surprising and remarkable thing is that if you compare that to atrophy patterns due to Alzheimer, you have extremely high correlation.
So the second eigenvector of the graph pression of the brain connector is somehow correlated to the atrophy pattern of Alzheimer disease.
Amazing result. But I wanted to know more.
I said, This is nice, but I really again, I want to go back to the mechanic.
This really doesn't tell me how things flow. What is the evolution process?
So first we have to understand what happened in the brain.
I told you about toxic proteins. What are they? Well, you have a lot of protein that does normal cellular work inside the brain.
And these letters are in blue here. And sometimes the misfolded, instead of the normal configuration, they take another shape.
And some of these fold are toxic. Think of them a little bit like a zombie apocalypse in a city.
Right. So you have a normal guy that become, for some reason a zombie.
And that zombie, whenever it meets another normal guy, transform them into a toxic guy.
Now you have two zombies in the in the city and the zombies that bunch together.
They're quite slow in the progression, but they always go on and and neurodegenerative diseases take a long time.
But it goes on and on and we have very few ways to stop it.
So it goes on the birds together and they make aggregates and when they meet other guys, they transform them into zombies, into toxic proteins.
And when they become large enough,
then the whole neighbourhood starts functioning and it's dead essentially right to stop the normal process from working.
And that's the end of. So we can build a model out of that.
And the model is to follow both toxic and non-toxic protein.
You follow the concentration of the two, both in space and time within the brain.
And what you do. You write rate equation for both aggregation and fragmentation process here.
And you can borrow ideas from sync from other or the field of applied mathematics.
What is processes take place?
And then you realise that what happens is that you have very fast transport along the axon and that's why the graph works.
But very slow transport inside the tissue. So you zombies that go very quickly in the main street but very slowly in the alleyway if you want.
And when they reach a certain concentration level, the axon dies because there's too much concentration and is removed from the material.
Okay. So you can write.
It's a little too late to write Big Equation now, but you can write big system of equation for that and you can look at the consequence of that.
So here again, the map and this is the kind of map that you find in in a quite recent article about the progression of neurodegenerative disease.
This is the sum up of the knowledge for maybe 20, 30 years of how neurodegenerative progress in the brain.
Now what you can do is you can see that the same place here and you let the process run.
And what you obtain is almost identical process where you see that part of the brain that are not attack at all are indeed free of the toxic protein.
The same in the back, in the temporal lobe and so on. So you can match a lot of things.
You just let the process run. Then you can say, well, if this works for this particular system, what about the other ones?
So you can go back and you can look at different molecules,
different seeding and different diseases, including Parkinson's disease and lateral sclerosis.
What's remarkable about this, I find, is that it is very systematic and people have been able to follow that.
It always more or less takes place in the same characteristic way.
So there is something generic in the transport process and I think that's the way we model it, extract that simplicity out of that system.
But of course you say, well, that's quite not enough right now, if I know that, can I do more?
People always want more. So what about atrophy?
Here is an Alzheimer patient. Eight years zero. When? The first year when the scan was.
Was made. Zero. So you extract out of that you segment and you have you have the full tissue white, white and grey matter.
And you let you see this here at the base in two dimension.
It sees you and you let the process run and this one, CO2 and you three and you can actually superimpose that almost identically.
So the atrophy patterns that this process created by the evolution of this toxic protein seems to be extremely consistent,
certainly correlated with what is observed during atrophy.
Showing your taste. I've done something I'm not supposed to do. This is actually the work of the last few weeks with my collaborator in Stanford.
You're not supposed to talk about things that are not published yet.
But between us and my 20,000 Facebook friends, I thought that it would remain in the room, so to speak.
So I thought that was. So interesting, it's so compelling that there is something there to study.
So let me conclude. What did we learn from this based on atrophy and amnesia?
Well, the interesting part that I found from a modelling perspective is that these phenomena are
both multiscale and multi physics that include process that takes place at very short time.
Folding of protein or shock in the brain, milliseconds, all the ways to year.
And we have to understand how we go from one to the other one. It also goes from the protein level all the way to the head level.
It also includes different type of physics, different process transport, but also solid fluid mechanics, protein.
Understanding what happened to the protein and all that.
So you have to combine that and that is natural ground for applied mathematics, combining different field, different expertise,
and looking at complicated equation and trying to extract from it simplicity against this tremendous complexity.
I don't want to say that these diseases are easy to understand.
30 years or more, 50 years of research have shown that sees extremely complex problems.
Yet despite that, I think there are aspects of it that mathematics can unravel that should be simple of it and that can improve our understanding.
And the last part, which is really where we're going now,
is that once you understand what this process and of dementia takes place, that when atrophy takes place,
you cannot go back and couple the physical atrophy with the functional connectome and try to
understand how the network of brain function start degrading based on the underlying physics.
So you couple the structural part, which is the physics part to the functional part,
which is all the brain process information and it loses its ability to process information.
And this is where we going. So what we're really trying to start, Robert Flirts a cabal is from Oxford from the 17th century.
We try to create a microcosm trying and inside the head you have a microcosm the whole world.
We're trying to do the same. We're trying to model things and include different parts.
We try to do physics and geometry and and genetics and and all different bubble, all different onion.
And every time you peel one, you come to other question and you try to answer them.
And it's really mathematics.
And the way we applied that give us a chance to understand all these different world within our heads are really connected.
And here is we started feeling a bubble. And again, I think applied mathematics is all the way there.
So, of course, I want to give the last words to our friend Ludovic, who said he had a good sense of humour.
Friends applaud. The comedy's over. Thank you very much.