1 00:00:11,920 --> 00:00:19,209 Thank you very much, Martin, for the nice introduction. Before I start talking about the brain itself, 2 00:00:19,210 --> 00:00:25,810 let me indulge a few general remarks about mathematics in the spirit of the little book that Martin mentioned. 3 00:00:27,620 --> 00:00:34,280 So here I have on the right mathematics, abstract mathematics, fundamental of pure mathematics. 4 00:00:34,370 --> 00:00:37,850 It constitutes theorem proof structure and things like that. 5 00:00:38,480 --> 00:00:45,580 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. 6 00:00:45,950 --> 00:00:50,390 And in the real world you have application, you have problems, you have science, you have a lot of things. 7 00:00:50,990 --> 00:00:57,470 And the naive idea of application of mathematics or applied mathematics is that whenever you have a problem in that box, 8 00:00:57,740 --> 00:01:01,370 well, you'll find something in that box and you use it to solve the problem here. 9 00:01:02,060 --> 00:01:08,450 Of course, this might have been true in the 19th century or the 18th century, but it's certainly not true nowadays. 10 00:01:09,020 --> 00:01:14,300 When we use mathematics, what we mostly use are essentially methods. 11 00:01:15,110 --> 00:01:19,850 This is the interface with the real world. We use computation algorithm. 12 00:01:19,970 --> 00:01:24,530 We have things to solve equation. We have things to analyse, data, we use statistics and so on. 13 00:01:24,950 --> 00:01:33,020 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. 14 00:01:33,320 --> 00:01:39,740 A lot of it nowadays is not part of mathematics. No more than accounting is considered to be part of mathematics. 15 00:01:40,040 --> 00:01:49,230 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. 16 00:01:49,740 --> 00:01:58,750 And theory is really all the fundamental principle that people have extracted over centuries about the world around us in different disciplines. 17 00:01:59,130 --> 00:02:02,850 It is physics, biology, chemistry, science, engineering, economy. 18 00:02:03,120 --> 00:02:07,020 This is the body of knowledge that we have on which we can draw. 19 00:02:07,740 --> 00:02:17,970 Different models use the method to solve, simplify, or complicated model and go back to the real world and try to understand it better. 20 00:02:18,690 --> 00:02:25,650 So where the methods give us a way to interface with the world, the understanding really comes from the theory. 21 00:02:25,950 --> 00:02:29,670 So naturally you would ask me where is applied mathematics in all that? 22 00:02:30,000 --> 00:02:34,470 Where applied mathematics is really everywhere around here. 23 00:02:35,460 --> 00:02:40,590 Some might include that whole thing around here, but let's not let's not create a debate about that. 24 00:02:41,700 --> 00:02:45,350 You'll find people working in applied mathematics, working in all these area, 25 00:02:45,480 --> 00:02:48,990 some of them just doing experiments, some of them just developing theory. 26 00:02:49,200 --> 00:02:57,450 Many of them very much enjoying solving equation, developing new method to solve equation numerically or particularly analytically and so on, 27 00:02:57,810 --> 00:03:01,260 developing new idea, how to handle data and so on. 28 00:03:01,500 --> 00:03:07,680 And a lot of this theory might even be used as an Irishman to create new mathematics, 29 00:03:07,680 --> 00:03:13,770 just like in the old days, when mathematics was first established based on simple question from the real world. 30 00:03:14,460 --> 00:03:18,930 So you'll find question everywhere around here. So no, to get to the brain. 31 00:03:18,930 --> 00:03:22,290 Where does the brain fit in that? We're in the brain. 32 00:03:22,290 --> 00:03:28,889 We have very well-established theory. Neuroscience nowadays is a very mathematics theory. 33 00:03:28,890 --> 00:03:33,209 There is computational neuroscience that's essentially a type of applied 34 00:03:33,210 --> 00:03:37,710 mathematics for the brain that relies on a lot of partial differential equation, 35 00:03:37,920 --> 00:03:41,610 network theory, dynamical system and other ideas. 36 00:03:42,120 --> 00:03:47,189 Most of the interest there is about information or the brain receive receiving inputs, 37 00:03:47,190 --> 00:03:51,870 and others who treat that or that input is being propagated in the brain in a 38 00:03:51,870 --> 00:03:56,400 big kind of network and trying to understand the structure of the network. 39 00:03:57,720 --> 00:04:06,420 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, 40 00:04:06,420 --> 00:04:10,020 whatever it is, am I right? Or CT scan or anything like that? 41 00:04:10,260 --> 00:04:15,030 There is a lot of algorithm going there to extract the information so that it's useful information. 42 00:04:16,060 --> 00:04:18,610 I know it is the best scan, the MRI. 43 00:04:18,610 --> 00:04:26,320 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, 44 00:04:26,500 --> 00:04:33,909 but also from fundamental mathematics. The best techniques that allow us to take very quick MRI actually come from 45 00:04:33,910 --> 00:04:38,170 the work of essentially applied mathematics and other disciplines correlate. 46 00:04:39,250 --> 00:04:46,190 So what is my interest in the brain? So when I look at the brain personally, I have always been fascinated. 47 00:04:46,210 --> 00:04:49,540 The first fascination I have is about its structure, its convolution. 48 00:04:50,260 --> 00:04:56,310 And I have my question. And in trying to understand the brain, I wanted to build systematically on that. 49 00:04:56,320 --> 00:05:02,650 And over the last five, six years or so, I've been trying to ask questions and trying to find basic answer. 50 00:05:03,010 --> 00:05:08,710 And to my surprise, whereas there is a lot of work at the brain level, at the functional level, 51 00:05:09,010 --> 00:05:17,320 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. 52 00:05:17,560 --> 00:05:20,650 So when I have a title, can mathematics understand the brain? 53 00:05:20,860 --> 00:05:29,020 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. 54 00:05:29,440 --> 00:05:35,920 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. 55 00:05:36,250 --> 00:05:45,459 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. 56 00:05:45,460 --> 00:05:49,150 Interesting problem. And this is what I'm going to try to convince you. 57 00:05:50,470 --> 00:05:55,520 So my first question was, what about the shape of the brain or does it attractive shape is geometry? 58 00:05:55,540 --> 00:05:59,800 What are the basic things that we know? Another question is what happens during trauma? 59 00:06:00,040 --> 00:06:03,760 We always talk about we hear about traumatic brain injury and things like that. 60 00:06:03,970 --> 00:06:08,470 What actually happened? What is the chain of event and what are the consequences of that? 61 00:06:08,500 --> 00:06:16,930 Can I understand it from a mathematics perspective? Again, I found that very little was known on the description of mathematical description. 62 00:06:16,930 --> 00:06:22,270 The modelling of these and most of it in the medical literature was unfortunately not correct. 63 00:06:24,150 --> 00:06:27,630 Yet. The third aspect that I want to address today is about dementia. 64 00:06:28,860 --> 00:06:34,260 And this is really fascinating. It's a process that we all know people that are affected. 65 00:06:34,770 --> 00:06:44,990 Something is happening to a brain. We have a disease, progressive disease, degenerative disease that removes some of the basic faculty. 66 00:06:45,320 --> 00:06:49,220 And I just wanted to understand what happens. How can we describe this process? 67 00:06:49,670 --> 00:06:57,560 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. 68 00:06:58,640 --> 00:07:02,060 So here I start with my first slides. 69 00:07:02,060 --> 00:07:06,740 You see it's is a painting. So when you see it's. Your brain works. 70 00:07:07,130 --> 00:07:09,260 You received a flat on the back of your retina. 71 00:07:09,280 --> 00:07:16,310 That's that information is being sent to the by the optic nerve all the way to the occipital lobe where it's treated. 72 00:07:16,520 --> 00:07:17,929 It's string part of the memory. 73 00:07:17,930 --> 00:07:26,030 There is a very sophisticated algorithm that people have discovered that allows you to recognise that it's a face, eye and so on. 74 00:07:26,840 --> 00:07:32,180 And based on that, you trigger some memory and say, ah ha, yes, on top of that. 75 00:07:32,180 --> 00:07:35,810 And you've all done that within a segment. This is Ludwig van Beethoven. 76 00:07:36,080 --> 00:07:39,500 A lot of functional process happened there. Right. 77 00:07:40,160 --> 00:07:45,080 And so you can see or maybe part of you memories being triggered and some music comes off, 78 00:07:45,340 --> 00:07:50,060 maybe wondering, why is he showing us a picture of Beethoven in a mass talk in the first place? 79 00:07:51,820 --> 00:07:55,560 So this is all the function of your brain and what it works. 80 00:07:55,570 --> 00:08:00,730 There is physics behind it. Blood is being delivered. A different part of the brain signal is being processed. 81 00:08:01,060 --> 00:08:07,930 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. 82 00:08:08,620 --> 00:08:12,190 So what do we know about his brain? Well, let's hear from Beethoven itself. 83 00:08:12,880 --> 00:08:20,560 When Pontus brothers sent him a letter and signed it from your brother, Your Honour, Beethoven replied. 84 00:08:20,560 --> 00:08:25,240 From your brother, Ludwig, brain owner. So we know you had a brain. 85 00:08:25,240 --> 00:08:30,430 Or at least he thought he had a brain by the time of his death in 1827. 86 00:08:30,730 --> 00:08:38,350 In middle or early in the 19th century, scientists were obsessed about brain and the morphology of the brain and how it 87 00:08:38,350 --> 00:08:43,840 was related to the criminality or to the insane or to the idiot and all that, 88 00:08:44,200 --> 00:08:48,250 but also to the genius. What makes somebody so remarkable? 89 00:08:48,730 --> 00:08:55,470 And indeed, the question is the following. As Ludwig says, there are many princes and they will continue to be thousand more. 90 00:08:55,630 --> 00:09:00,400 But there is only one Beethoven, and I don't think he meant his brother when you talk about that. 91 00:09:02,020 --> 00:09:06,879 So what was remarkable about Beethoven's brain was good thing we have a is 92 00:09:06,880 --> 00:09:12,070 autopsy report makes very light and enjoyable reading so we can look into that. 93 00:09:12,250 --> 00:09:20,980 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. 94 00:09:21,040 --> 00:09:24,010 It's something that we still always the big brain and things like that. 95 00:09:24,310 --> 00:09:30,610 So my first parts, first part of the talk, the first Mothman would be or because your brain would fit. 96 00:09:31,120 --> 00:09:38,110 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. 97 00:09:38,320 --> 00:09:44,950 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. 98 00:09:45,160 --> 00:09:49,450 So if you put your two hands together, you're welcome to do it. But you and you put them like that. 99 00:09:49,650 --> 00:09:58,060 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? 100 00:09:58,270 --> 00:10:01,839 It's about 1.2 to 1.4 kg. 101 00:10:01,840 --> 00:10:08,140 About right. It's about the volume of this little ball, which is 70 centimetre radius. 102 00:10:08,680 --> 00:10:12,430 Seven centimetre radius. That's right. So this is your brain. 103 00:10:12,580 --> 00:10:15,970 1.3 kg, let's say on average and so on. So. 104 00:10:17,060 --> 00:10:21,050 If bigger brain are better, these brains should be even better. 105 00:10:21,800 --> 00:10:25,700 This is Africa, NELSON And it's three times as big as our brain. 106 00:10:26,210 --> 00:10:32,360 I said, Well, we know that African elephants are probably very wise, but they don't do crosswords. 107 00:10:32,630 --> 00:10:35,300 They don't tweet about Kim Kardashian and all that. 108 00:10:36,830 --> 00:10:44,880 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. 109 00:10:44,900 --> 00:10:49,460 You'd have to agree with that. So you've got to say, aha, but wait a second. 110 00:10:50,030 --> 00:10:55,669 This is only three times bigger. An elephant are much bigger, maybe 50 to 100 times an elephant. 111 00:10:55,670 --> 00:10:59,090 African elephant, 6000 kilos. So you should really scale. 112 00:10:59,240 --> 00:11:03,620 It should really brain should be the weight of the brain per the mass of the animal. 113 00:11:04,010 --> 00:11:09,070 And and indeed, if you scale it up, this is what it would look like. 114 00:11:09,080 --> 00:11:13,070 He's a random individual next to an African elephant. 115 00:11:13,670 --> 00:11:19,790 And if you scale it, indeed, the brain does not grow as fast as the rest of the body. 116 00:11:19,970 --> 00:11:27,110 As you go and look at different animal. And this is a law that was discovered by the Swiss physiologist Von Heller in 1762. 117 00:11:27,360 --> 00:11:32,750 It does mention the African elephant. I'm not sure he mentioned any royalty in his description. 118 00:11:33,920 --> 00:11:38,900 And the idea that is, as animal gets much bigger, the brain doesn't grow as fast. 119 00:11:39,050 --> 00:11:46,250 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. 120 00:11:46,790 --> 00:11:54,290 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. 121 00:11:54,590 --> 00:12:02,450 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 122 00:12:02,600 --> 00:12:07,310 and you'd find a correlation or you'd find the coefficient and B in the process. 123 00:12:07,970 --> 00:12:09,890 Very traditional. It's very easy to do. 124 00:12:10,370 --> 00:12:17,740 But what we want is not a linear low that we expect from very low, that it doesn't grow in the same proportion. 125 00:12:17,750 --> 00:12:20,330 It grows slower than the brain. 126 00:12:20,510 --> 00:12:28,640 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. 127 00:12:29,000 --> 00:12:35,870 So that would be a terrible fit for low. So what you really want is a low that looks like like this one, right? 128 00:12:35,900 --> 00:12:42,590 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. 129 00:12:42,980 --> 00:12:48,170 And if it's less than one, then it goes slower. Like what we would expect from one other slow. 130 00:12:49,310 --> 00:12:53,209 So you can try with different points. Oh, it looks like a parabola. 131 00:12:53,210 --> 00:12:57,170 So I can fit a parabola. But what you really want is to extract this coefficient. 132 00:12:57,770 --> 00:13:03,229 So the smart trick that you can do is you take the logarithm of that relationship. 133 00:13:03,230 --> 00:13:12,710 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. 134 00:13:13,010 --> 00:13:17,540 So if it follows a power law like this in logarithmic coordinates, 135 00:13:17,540 --> 00:13:23,270 if I plot log knock plot it would look like a line and it will be the gradient of this line. 136 00:13:23,630 --> 00:13:31,190 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. 137 00:13:31,370 --> 00:13:34,820 So we hope that somewhere will be so. 138 00:13:34,940 --> 00:13:44,149 We'll be right. Here. And we can see animals are all over here that follow natural evolution and we are very special 139 00:13:44,150 --> 00:13:49,280 because we do not follow that this scaling that does not apply to us and man is superior. 140 00:13:50,840 --> 00:13:59,479 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. 141 00:13:59,480 --> 00:14:06,170 Really. You see the pesky, the dusky dolphin right here is as good as we are really. 142 00:14:06,350 --> 00:14:11,060 And brain per pound while the mouse is actually the same ratio. 143 00:14:11,570 --> 00:14:15,650 So really that's racial argument really does not apply. 144 00:14:15,830 --> 00:14:21,680 And indeed the exponent here it is a slow that is between two third and three quarter. 145 00:14:22,100 --> 00:14:30,259 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 146 00:14:30,260 --> 00:14:39,200 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, 147 00:14:39,200 --> 00:14:44,510 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. 148 00:14:44,870 --> 00:14:49,820 But that's not what I want to do today. I don't want to speculate. I want to understand the mechanism about that. 149 00:14:50,150 --> 00:14:53,390 And you see that this idea of brain wage really doesn't doesn't work. 150 00:14:53,570 --> 00:14:57,320 And by the end of the 19th century, Darwin was really aware of this. 151 00:14:57,530 --> 00:14:59,360 When you wrote in The Descent of Meant, 152 00:14:59,870 --> 00:15:05,360 no one supposes that the intellect of any two men can be accurately gauged by the cubic contents of the skulls. 153 00:15:05,870 --> 00:15:08,630 And really that was the end of that, that line of reasoning. 154 00:15:09,320 --> 00:15:19,129 So going back to going back to Beethoven, if his brain was when his brain was looked at during the autopsy, 155 00:15:19,130 --> 00:15:26,450 unfortunately it was not remove that distinction came to LE Plus actually the mathematician and Laplace 156 00:15:26,450 --> 00:15:32,150 sat in a jar of alcohol as a trophy for many years on the desk of the pathologist that took it out. 157 00:15:32,480 --> 00:15:38,510 So Beethoven's brain was not removed, but it was described as being very small in atrophy. 158 00:15:38,750 --> 00:15:46,310 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. 159 00:15:46,590 --> 00:15:52,900 And in the description they say the convolutions appear twice as numerous as in ordinary brain. 160 00:15:53,810 --> 00:15:57,440 Hmm. That's interesting. So maybe it's not the weight. 161 00:15:58,520 --> 00:16:02,030 That is important for being a remarkable genius. 162 00:16:02,210 --> 00:16:06,380 Maybe it's the convolution. Ah ha. So maybe we should look at that. 163 00:16:06,740 --> 00:16:11,170 So our next chapter or next movement is all convoluted. 164 00:16:11,180 --> 00:16:14,210 Ah, you look, Rick and I show you as a proof, 165 00:16:14,240 --> 00:16:21,080 as other evidence of t of you statement that come related brain that better this I offer you to his brain. 166 00:16:21,920 --> 00:16:28,370 And I'm sure mathematician would directly recognise anything coming from his great brain because this is Carl Friedrich Gauss, 167 00:16:29,060 --> 00:16:36,590 the the greatest arguably one of the greatest mathematician who was described also by a doctor 168 00:16:36,590 --> 00:16:41,780 named Wagner as the cerebrum is remarkable for the great complexity of the convolutions. 169 00:16:44,080 --> 00:16:52,270 Goes up a little later is brain was not either remarkable in volume, but maybe there was something special about it. 170 00:16:53,080 --> 00:16:57,100 So what do we mean about convolution? Let's think about it a little bit deeper here. 171 00:16:57,580 --> 00:17:01,970 So if you take a brain and you take a slice, so this is called a coronal slice. 172 00:17:02,060 --> 00:17:07,270 You slice it like that, what you'll see is this convolution and you see two type of tissue. 173 00:17:07,570 --> 00:17:13,020 The outer tissue is called the grey matter because it appears grey when you staining that you remove it. 174 00:17:13,240 --> 00:17:15,880 It's mostly made of cell body, really highly packed. 175 00:17:16,180 --> 00:17:24,670 It's where a lot of mental activity I cognitive activities take place and the white matter inside. 176 00:17:25,030 --> 00:17:28,030 It's a more fatty tissue. That's why it looks a little white. 177 00:17:28,330 --> 00:17:33,010 And essentially you have axon bundles carrying information between different parts of the brain. 178 00:17:33,130 --> 00:17:38,230 You can think of that. Here is where the processing takes place and the highway of information. 179 00:17:38,230 --> 00:17:47,200 The cables in between essentially vary. But what you see is the convolution where you have this thrust and it is richest, 180 00:17:47,200 --> 00:17:55,330 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. 181 00:17:55,800 --> 00:18:02,500 And in the brain, it it it takes it first appears around seven months here. 182 00:18:02,500 --> 00:18:09,100 The in gestation pregnancy, the breast is smooth and then at some point it becomes wrinkled and is wrinkled, 183 00:18:09,100 --> 00:18:13,450 further develop and then gets convoluted to a normal brain. 184 00:18:13,870 --> 00:18:17,260 And let's look a little bit deeper. We have good scanner for this kind of thing. 185 00:18:17,740 --> 00:18:21,700 We have amazing scan, actually. It's a whole field that generates a lot of data. 186 00:18:22,210 --> 00:18:30,510 In the scan, you see right at the 27, 28 week of gestation, you go from the smooth brain to it. 187 00:18:30,550 --> 00:18:35,050 You see here the wrinkle really developing here to be very convoluted. 188 00:18:35,980 --> 00:18:45,139 And indeed, if you look and the idea is that at that time the area increased much faster than the volume, right. 189 00:18:45,140 --> 00:18:50,890 So if you look indeed at the cerebral area at that time, it increased very quickly compared to the volume. 190 00:18:51,370 --> 00:18:58,179 So if you had a brain that just expand like a sphere, the area would go much slower than the volume. 191 00:18:58,180 --> 00:19:01,240 It goes like a square and the volume grows cube. 192 00:19:01,600 --> 00:19:05,380 So but if the area grows quicker, it gets folded inside. 193 00:19:05,620 --> 00:19:09,910 And what you will measure is the area and for a smaller volume. 194 00:19:10,180 --> 00:19:14,320 So the folding comes inside and that's why you have this different. 195 00:19:14,620 --> 00:19:21,850 So how do we quantify that? A nice way to quantify as it looks is to look at what's called the diversification index, 196 00:19:22,180 --> 00:19:28,089 which is the ratio of the curve where you follow the contour by the curve that you obtain around the control, 197 00:19:28,090 --> 00:19:34,810 the outer contour here you see it well is the blue, the length of the blue curve divided by the red curve, 198 00:19:34,840 --> 00:19:37,930 the one that you put if you put like an elastic band around the whole thing? 199 00:19:39,590 --> 00:19:43,520 In this case, for instance, the gentrification index is 1.47. 200 00:19:44,520 --> 00:19:49,759 Okay, so now we have a way to measure things and we can go and look at all animals and all that. 201 00:19:49,760 --> 00:19:53,530 Could brains and be happy. Aha! Hooray! 202 00:19:54,350 --> 00:19:58,790 Here we are again. The red dot sitting on top of all the animals. 203 00:19:58,790 --> 00:20:06,060 Way ahead of the competition. So finally here, we found something that distinguish us from all these animals. 204 00:20:06,890 --> 00:20:14,090 And indeed, if you look closely and you compare us to a primate cousin, we do much better in verification. 205 00:20:15,490 --> 00:20:19,990 Of course, the only problem is when you include the marine mammals. 206 00:20:20,830 --> 00:20:25,960 We are here and purpose is here in very far, even for a long way. 207 00:20:26,260 --> 00:20:37,630 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. 208 00:20:38,920 --> 00:20:42,129 So my question is not so much why are we superior? 209 00:20:42,130 --> 00:20:42,880 Why is the brain? 210 00:20:42,880 --> 00:20:51,580 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. 211 00:20:51,760 --> 00:20:55,660 We like to understand a pattern that formed from simple physical problems. 212 00:20:55,960 --> 00:21:00,880 So can we do it on this case where people, of course, have been thinking about that for a long time? 213 00:21:01,180 --> 00:21:06,910 And this is there are many different hypotheses for that and we can use mathematical modelling to test them. 214 00:21:07,330 --> 00:21:12,500 One of the earlier hypotheses was going back to about a century is that the brain grows. 215 00:21:12,730 --> 00:21:16,870 It hits the skull. And since the skull is very hard, it's forced to fall back. 216 00:21:17,470 --> 00:21:24,400 And that seems a reasonable idea. But in the fifties, Barron at Yale University did a series of experiments. 217 00:21:24,730 --> 00:21:29,230 And when I say a series of experiments, this is ship number 409. 218 00:21:30,550 --> 00:21:40,570 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. 219 00:21:40,940 --> 00:21:47,350 Very, very nice paper, a lot of data. But I mean, it must be quite a challenge if you think about it. 220 00:21:47,980 --> 00:21:57,700 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. 221 00:21:58,300 --> 00:22:05,860 And when you remove the skull, it still saw all the convolution. So the convolution are not created by the interaction with the skull. 222 00:22:06,880 --> 00:22:10,060 Another hypothesis and the leading way of thinking about this, 223 00:22:10,090 --> 00:22:21,070 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. 224 00:22:22,410 --> 00:22:27,150 So let's see if we can understand that in a simple case, let's be laterally model for that. 225 00:22:28,320 --> 00:22:36,719 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. 226 00:22:36,720 --> 00:22:44,910 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. 227 00:22:45,270 --> 00:22:47,700 So since I'm only interested in the relative growth, 228 00:22:47,910 --> 00:22:55,799 I'm going to assume that the white matter grow quickly and that there is no growth in the the white matter. 229 00:22:55,800 --> 00:22:57,630 So the grey matter growth here. 230 00:22:57,900 --> 00:23:08,040 So if a beam of a thin film growing very quickly and attach bonded to a substrate, an elastic substrate that does not grow. 231 00:23:08,580 --> 00:23:16,230 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 232 00:23:16,260 --> 00:23:21,630 twice as long as the original one is that critical value where instead of having something like that, 233 00:23:21,990 --> 00:23:26,070 I'll have something like this, I'm going to start having wrinkled appearing. 234 00:23:26,370 --> 00:23:29,400 And what I want to know is the number of oscillations per unit length. 235 00:23:29,580 --> 00:23:35,130 And that would tell me something about the brain maybe. So why do I call this a toy model? 236 00:23:36,150 --> 00:23:44,220 Where here is an analogy for you. Suppose that your scientist was mostly interested in monster trucks and this is your favourite one. 237 00:23:46,320 --> 00:23:53,370 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. 238 00:23:53,910 --> 00:23:59,280 But when you come to the subject and you say, Well, what I really want to know is how this thing move. 239 00:24:00,060 --> 00:24:06,030 What is the principle of the locomotion to start with? All this is one thing works, for instance, right? 240 00:24:06,150 --> 00:24:10,620 So maybe you're not so much interested in the the wall sticker that is right here that 241 00:24:10,620 --> 00:24:14,250 the scientists think is very important because he'd seen it in all the monster trucks. 242 00:24:14,490 --> 00:24:22,140 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, 243 00:24:22,500 --> 00:24:26,970 or if you're really lucky, you thought model is really going to look very close to the original ones. 244 00:24:27,660 --> 00:24:30,710 And this one comes with crushable costs. So it's really cool. 245 00:24:33,320 --> 00:24:36,530 So this is my top model for you. We're going to play a ways. 246 00:24:36,950 --> 00:24:39,380 And my mathematical toy model is this one. 247 00:24:39,410 --> 00:24:45,170 This is the simplest possible one that's still containing a physical information so that we can learn something. 248 00:24:46,160 --> 00:24:48,620 It is a relatively simple equation. 249 00:24:48,800 --> 00:24:55,730 It's a fourth order equation that describe the mechanics of a beam attached to a substrate in very, very simple time. 250 00:24:56,300 --> 00:25:00,320 So w is that wrinkle. What we want is w as a function of s. 251 00:25:00,590 --> 00:25:03,950 So for instance, this looks like a sign of something sine x, right? 252 00:25:04,190 --> 00:25:08,840 So we want to find that h here's the thickness. 253 00:25:09,350 --> 00:25:13,850 Gamma is the growth and size, the ratio of the stiffness of the two layers. 254 00:25:14,180 --> 00:25:20,060 Essentially a material parameter that tell you are stiff. The stiff you layer is on top of the substrate. 255 00:25:21,980 --> 00:25:26,299 So it's relatively simple and I'm sure you want to see the real thing. 256 00:25:26,300 --> 00:25:32,660 You want to go and see under the hood of the the real machine so I can let you peek for a little time. 257 00:25:33,050 --> 00:25:37,460 This is the real system in two dimension that has all the exact mechanics. 258 00:25:37,820 --> 00:25:48,740 It is a system of two rather nonlinear partial differential equation in two variable in mixed coordinates formulation takes a while and all that. 259 00:25:48,950 --> 00:25:52,970 This is the real deal. This is the real thing until we on which we work. 260 00:25:53,120 --> 00:25:57,230 And you have to add that boundary condition and this is work with IMSA. 261 00:25:57,290 --> 00:26:02,000 I don't know if I'm XYZ around and I know you feel all very sorry for him having to deal with that. 262 00:26:02,240 --> 00:26:08,209 But actually we do have techniques like every, every, every technical discipline we have way to deal. 263 00:26:08,210 --> 00:26:11,930 That is not like we go by hand and and change the power or something like that. 264 00:26:11,930 --> 00:26:16,459 We have plenty of way to deal, but morally, the physics of that, 265 00:26:16,460 --> 00:26:21,740 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. 266 00:26:22,880 --> 00:26:28,060 So let's go back to the toy model. So now you find this one is not too difficult. 267 00:26:28,600 --> 00:26:34,990 So it's a one with constant coefficient. And maybe you've already seen ordinary differential equation. 268 00:26:35,200 --> 00:26:39,130 It's part of some of the further math. So you you will see to some point. 269 00:26:40,360 --> 00:26:44,049 So we can start playing and removing wheels, for instance, and see how it works. 270 00:26:44,050 --> 00:26:48,370 So let's see what happens if we remove system. So let's remove system. 271 00:26:48,520 --> 00:26:55,600 That means we don't want any substrate here. So we remove the substrate and what you have, the layer grows. 272 00:26:56,020 --> 00:27:00,910 Push against the wall and buckles. Right. This is Euler buckling 1757. 273 00:27:01,210 --> 00:27:05,560 As the layer grows, it push against the wall and does something like that very well. 274 00:27:05,560 --> 00:27:10,960 Understood. And if you work a little bit on this equation, you can you can find a solution in this. 275 00:27:11,530 --> 00:27:14,680 Not very difficult. But no, we want to add the substrate back. 276 00:27:15,400 --> 00:27:18,639 You still have a solution like that, but this one takes a lot of energy. 277 00:27:18,640 --> 00:27:22,390 You have to raise the whole substrate. So this is not the preferred solution. 278 00:27:22,720 --> 00:27:25,330 The one that is preferred is one with a lot of oscillation. 279 00:27:25,630 --> 00:27:32,950 And it's a it's a compromise between having to bend so between system and having to lift the substrate. 280 00:27:33,900 --> 00:27:40,740 Now you see that we can try a solution of the for cosine because is constant coefficient of force derivative 281 00:27:40,740 --> 00:27:45,330 of cosine is the cosine the second derivative of cosine the cosine and the cosine is the cosine. 282 00:27:45,570 --> 00:27:49,080 So it's the solution of the equation for some value of n and gamma. 283 00:27:49,800 --> 00:27:51,710 So you can put that back in studied. 284 00:27:51,720 --> 00:28:00,690 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. 285 00:28:01,170 --> 00:28:07,739 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. 286 00:28:07,740 --> 00:28:12,809 And we have a critical value and we have the number of peaks per unit length or something 287 00:28:12,810 --> 00:28:17,250 like that is this is going to be given and you recover that in the full system. 288 00:28:17,550 --> 00:28:21,310 So the simplified system contains already the physics there. 289 00:28:21,660 --> 00:28:27,330 What is more important is what we can deduce from this formula. 290 00:28:28,020 --> 00:28:34,500 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. 291 00:28:34,710 --> 00:28:38,160 So gamma increase, you'd have more wrinkled with thinner cortex. 292 00:28:38,550 --> 00:28:43,620 If H decrease makes and bigger and you have less wrinkle with thicker cortex. 293 00:28:44,220 --> 00:28:47,879 So let's see what's basic trends compared to what we know about the brain. 294 00:28:47,880 --> 00:28:52,290 Go back to the brain and in the brain you have that. 295 00:28:52,290 --> 00:28:58,319 In pathology, you have interesting case. So for instance, in the case of schizophrenia, 296 00:28:58,320 --> 00:29:04,350 you have excess growth and the excess growth is related to a lot of wrinkle and the higher diversification index. 297 00:29:04,770 --> 00:29:06,960 So indeed, an increase with gamma. 298 00:29:08,400 --> 00:29:16,050 You can look at all the pathological case where you have a thinner cortex, and this is probably called probably micro diarrhoea. 299 00:29:16,410 --> 00:29:20,040 And when you have thinner cortex, you have more and more oscillation. 300 00:29:20,490 --> 00:29:26,370 And when you have a thicker cortex, this is called listen carefully and you can have an almost completely smooth brain. 301 00:29:27,090 --> 00:29:32,610 So this is all consistent. So let's reflect back before we move on to the next topic to what we've learned. 302 00:29:33,210 --> 00:29:39,450 Skating knows a good way to organise data and try to look at some laws with it. 303 00:29:39,450 --> 00:29:44,790 Give us good suggestion of where to look. But if there is any conclusion, is that based on scaling? 304 00:29:44,790 --> 00:29:49,140 No, there is really nothing special about the human brain, at least not the one we know. 305 00:29:50,050 --> 00:29:58,360 But for brain morphogenesis. Now we have models that allows us to identify the correct mechanism that leads to the convolution we 306 00:29:58,360 --> 00:30:04,420 can properties model to both the genetic and the cellular level and try to answer a final question. 307 00:30:04,660 --> 00:30:07,030 We can try. We can generalise that. 308 00:30:07,030 --> 00:30:14,140 We can look at different species, for instance, and different geometry, very the parameter and all that, and recover other trends that observe. 309 00:30:14,590 --> 00:30:21,460 And since convolution are an important biomarker for a lot of neurological disorders like autism, it is a freena. 310 00:30:21,820 --> 00:30:27,730 It is crucial to understand the origin and or the form during morphogenesis of doing brain development. 311 00:30:28,270 --> 00:30:33,280 And in the process we see that we had to develop theories to understand our. 312 00:30:34,280 --> 00:30:37,880 Continue material grow and we had to formulate a new theory for that. 313 00:30:37,890 --> 00:30:41,660 That's what we've been doing with colleagues for the last ten or 15 years or so. 314 00:30:41,930 --> 00:30:46,969 We had to develop new methods in order to deal with differential equation and that nice aspect 315 00:30:46,970 --> 00:30:52,430 of pure mathematics that I have no time to discuss that you can also generate in the process. 316 00:30:55,570 --> 00:30:59,470 So let me go back now to Beethoven. 317 00:31:00,380 --> 00:31:05,100 Oh, what trauma? So, as you know, Beethoven was dead for a large part of his life. 318 00:31:05,110 --> 00:31:08,950 He says that you didn't hear the Ninth Symphony being played for the first time. 319 00:31:10,000 --> 00:31:13,870 And the reason there are multiple hypotheses of why. Why wasn't there? 320 00:31:13,960 --> 00:31:22,840 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. 321 00:31:23,080 --> 00:31:27,250 Other people believe that it was the abuse of repeated abuse of his father. 322 00:31:27,760 --> 00:31:33,340 And the idea is that that swelling in the brain caused compression on the nerve that damages hearing. 323 00:31:33,670 --> 00:31:40,300 Now, the hypothesis that this was discussing, but for us, this is an interesting point. 324 00:31:40,510 --> 00:31:45,069 Swelling in the brain, swelling is really bad in the brain. 325 00:31:45,070 --> 00:31:48,010 It can occurs in different case in an accident, 326 00:31:48,310 --> 00:31:57,340 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. 327 00:31:57,460 --> 00:32:00,250 It's very strange when you think from the evolutionary perspective, 328 00:32:00,610 --> 00:32:04,810 when evolution has been something over millions of years to try to protect the brain. 329 00:32:05,590 --> 00:32:08,500 Oh, yes, but let's put a helmet around it. That's really going to help. 330 00:32:11,570 --> 00:32:18,620 Or it can also be due to a stroke or a brain tumour, also creating inflammation that produces brain swelling. 331 00:32:18,920 --> 00:32:23,090 High altitudes can also talk brain swelling. Medical is called cerebral. 332 00:32:23,090 --> 00:32:27,980 Oedema is just mean brain swelling. Oedema is the accumulation of fluid in the interstitial. 333 00:32:29,050 --> 00:32:38,080 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. 334 00:32:38,350 --> 00:32:46,570 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. 335 00:32:46,750 --> 00:32:54,730 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, 336 00:32:55,030 --> 00:33:01,360 a very complicated mechanism that has electrochemistry, fluid dynamics, solid mechanics and so on that you have to incorporate. 337 00:33:01,540 --> 00:33:07,300 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. 338 00:33:08,450 --> 00:33:12,560 The interesting thing is that when it starts happening, it's really bad. 339 00:33:13,010 --> 00:33:17,659 Why? Because when it starts happening, the brains swell. 340 00:33:17,660 --> 00:33:22,760 But there's no places to go. So it increases the pressure and it increases the intracranial pressure. 341 00:33:23,360 --> 00:33:28,830 Blood vessels are occluded and cannot deliver oxygen to other parts of the brain. 342 00:33:28,850 --> 00:33:35,450 Other cells start dying. They die. The release, the solute, the release, the ions and the brain swell even more. 343 00:33:35,840 --> 00:33:42,229 So you have catastrophic event, a cascade that takes place in the brain, and that's in very tissues. 344 00:33:42,230 --> 00:33:46,880 The part that is necrotic, that's invade the whole brain within a few days. 345 00:33:47,900 --> 00:33:56,180 So when doctor sees an elevated intracranial pressure of 20 millimetre for more than 20 minutes, nowadays they have to intervene. 346 00:33:56,810 --> 00:34:02,570 And the one thing they can do is rather drastic. But it's one of the few things they can do. 347 00:34:02,840 --> 00:34:05,900 Is this removing a big part of the skull? 348 00:34:06,410 --> 00:34:14,510 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. 349 00:34:16,820 --> 00:34:21,590 So that's I thought was very interesting is a process called images, too. 350 00:34:22,100 --> 00:34:26,360 So I decided to look a little bit at that. And again, we had a simple question. 351 00:34:26,900 --> 00:34:31,730 And if you look at that, my first question is, do I understand this shape or this? 352 00:34:32,360 --> 00:34:37,680 What is the shape of that bulging brain? But also, what are the forces generated during that process? 353 00:34:37,700 --> 00:34:43,460 If I have swelling here and what are the possible damage that occurs during that swelling process? 354 00:34:44,030 --> 00:34:45,430 So started with a toy model. 355 00:34:45,530 --> 00:34:53,179 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. 356 00:34:53,180 --> 00:34:58,200 Essentially, it's about the same mechanically as pressing on it. And you probably can see it and it works. 357 00:34:58,220 --> 00:35:01,610 But here is the movie. So this is swelling to an opening. 358 00:35:01,910 --> 00:35:06,860 Imagine that is the brain and that is the opening here. And what you obtain is a bunch. 359 00:35:08,100 --> 00:35:15,929 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, 360 00:35:15,930 --> 00:35:19,200 even though a lot of problem in contact mechanics had been studied. 361 00:35:19,680 --> 00:35:25,050 So the bulging problem, you assume you have a big that's a big cylinder and you fix. 362 00:35:26,030 --> 00:35:29,870 You fix the displacement around here and you let the brain bulge out like this. 363 00:35:30,260 --> 00:35:35,410 So just like this box, you have a small opening and you squeeze it and what you want is what happens there. 364 00:35:36,800 --> 00:35:43,970 This is a finite element simulation. And what you see here, shear forces, sheer stress that presents a singularity here. 365 00:35:44,240 --> 00:35:48,400 The stress goes to infinity around this point. Okay. 366 00:35:49,270 --> 00:35:52,390 And the shallow singularity is always a clue. 367 00:35:52,600 --> 00:35:59,440 This is always a good point to look at so you can formulate this problem and do some approximation in linear elasticity. 368 00:35:59,920 --> 00:36:07,060 And when you do that, you can further simplify it and try to find the pressure at all point that maintains its shape. 369 00:36:07,480 --> 00:36:12,610 And when you do that after a long computation, you actually obtain a very simple solution. 370 00:36:13,000 --> 00:36:18,430 This is the exact solution for this problem. It says it tells you the shape of this bulge. 371 00:36:18,760 --> 00:36:24,910 And if you look, it's actually almost like a sphere that is squashed by a factor here in front of it. 372 00:36:25,180 --> 00:36:31,090 So this is the exact solution. It's a project. It's a half a sphere for the for the problem. 373 00:36:32,220 --> 00:36:35,910 And once you have that, if you have that information, you actually have everything. 374 00:36:36,210 --> 00:36:40,110 You can start looking at the force generated inside the material. 375 00:36:40,380 --> 00:36:43,410 And you can look at this point where you have a singularity, 376 00:36:43,680 --> 00:36:49,170 where the stress are very large because you suspect that to the point where you would have problem if you have a tissue, 377 00:36:49,560 --> 00:36:53,490 if you have too much compression of shear, this is where the cells are going to die. 378 00:36:55,000 --> 00:36:58,540 And what you see here are what we call damage drops. 379 00:36:58,870 --> 00:37:00,579 And you see they always have this shape, 380 00:37:00,580 --> 00:37:07,720 very nice regular shape when you look closer and you can actually find exactly the angle given by that nice formula, 381 00:37:07,990 --> 00:37:17,470 the attention of a, of a nice, uh, root number which amount to 111 and 32 degree where you have these two directions. 382 00:37:18,810 --> 00:37:22,940 Of course you're going to say, well, this is you top model or relevant is it for the real thing? 383 00:37:24,010 --> 00:37:28,830 So we should do the real thing. For that. 384 00:37:29,430 --> 00:37:32,790 You need a brain. If only I had a brain. 385 00:37:34,020 --> 00:37:43,220 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. 386 00:37:44,250 --> 00:37:52,380 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. 387 00:37:52,800 --> 00:37:56,900 And once you have this slice, you can do what's called a segmentation. 388 00:37:56,910 --> 00:38:00,570 You can tell and say what tissue is is at what place in the slide. 389 00:38:01,080 --> 00:38:07,680 And you can do that for all the slices. And then you have a description of the tissue at all points into different slices. 390 00:38:08,250 --> 00:38:12,180 And then you reconstruct the full 3D model. 391 00:38:12,480 --> 00:38:15,540 You put all this slice together and this is what you obtain. 392 00:38:21,260 --> 00:38:29,240 So what you have is a match. And that message has 1.3 million tetrahedral elements that describe both the skull, 393 00:38:29,660 --> 00:38:34,340 but also the different type of tissue and tissue is the 3D print from my office. 394 00:38:34,640 --> 00:38:39,470 This is actually the same brain as this one that I show you, and that is the skull that goes with it. 395 00:38:41,440 --> 00:38:45,040 So here you see the white matter. Here you see the cerebellum. 396 00:38:45,250 --> 00:38:48,640 And now we have everything we can. Let it swell with the skull. 397 00:38:49,000 --> 00:38:52,240 Right. So we put. We put back the skull. 398 00:38:54,040 --> 00:38:57,940 I know what I want to do is I want to open it. 399 00:38:58,060 --> 00:39:02,770 Here is the opening. I do the compressive craniotomy. This is glow in the dark. 400 00:39:02,770 --> 00:39:08,350 But you can't really appreciate that. So you do the compressive craniotomy and united swell. 401 00:39:09,070 --> 00:39:18,230 And this is what you have. And once since you do the full computation, you not only have the displacement of geometry, 402 00:39:18,440 --> 00:39:23,330 but you have all the forces that are developing that very complex material and very complex geometry. 403 00:39:23,540 --> 00:39:26,210 You have the displacement, the sheer force, the strain, and so on. 404 00:39:26,720 --> 00:39:34,880 So you can go back and compare comparison is not that great, is not the best the best cut here. 405 00:39:35,090 --> 00:39:41,540 But you can compare the damage drop that occurs here. You find them again with the same angle on the to dislike. 406 00:39:43,030 --> 00:39:49,000 Okay. So the same phenomena that you see on the very simple model is actually very generic in all these problems. 407 00:39:49,750 --> 00:39:56,220 And if you had us before, to the neurosurgeon, where are the possible damage that occurs with these that would say, 408 00:39:56,230 --> 00:40:00,250 of course, it occurs right at the boundary that they know because there is a lot of pressure. 409 00:40:00,670 --> 00:40:04,330 But there is no there was no understanding. Await occurs inside the material. 410 00:40:04,690 --> 00:40:06,310 That's only one type of damage. 411 00:40:07,760 --> 00:40:14,960 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, 412 00:40:15,260 --> 00:40:17,780 you actually going to stretch the axon. 413 00:40:18,260 --> 00:40:26,690 And we know from external studies that stretch that even like less than 20% in stretching the axons can lead to permanent damage. 414 00:40:27,500 --> 00:40:30,889 And when you look at this map, you'll realise that for this small open, 415 00:40:30,890 --> 00:40:36,770 you actually have extremely large 50% damage, but also very deep inside the tissue. 416 00:40:37,370 --> 00:40:45,020 So you do have damage and the damage is very deep inside, inside the tissue, which was also a bit of a surprise. 417 00:40:45,560 --> 00:40:50,959 So no, what we have is really computationally and a theoretical framework on which you can 418 00:40:50,960 --> 00:40:57,260 test hypotheses about different option and also gain understanding about this process. 419 00:40:58,320 --> 00:41:03,360 We understand piece about swelling. We understand about trauma and craniotomy and about damage. 420 00:41:04,200 --> 00:41:07,290 It's very early on and we still have a lot to learn, of course. 421 00:41:07,590 --> 00:41:12,660 And maybe we are the Mad Men treating the Mad Men just like in this painting of Bush. 422 00:41:14,740 --> 00:41:19,690 So as I told you, Beethoven's brain was rather small, unfortunately. 423 00:41:20,320 --> 00:41:24,370 And one of the reasons why it was so small is that probably. 424 00:41:27,180 --> 00:41:35,460 Beethoven like to drink a little bit too much and actually die of cirrhosis infection of the liver due to excess alcohol. 425 00:41:35,790 --> 00:41:39,000 Not helped by the fact that the cure at the time was to drink more alcohol. 426 00:41:40,020 --> 00:41:43,700 Okay. So understanding is good for teens. 427 00:41:45,210 --> 00:41:52,230 So the problem is that if you have a normal man, here's is a normal 43 year old compared to alcoholic. 428 00:41:52,470 --> 00:42:00,230 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. 429 00:42:00,240 --> 00:42:05,640 You do lose a lot of sense due to excess alcohol. And here is an extreme case. 430 00:42:06,960 --> 00:42:10,770 You probably recognise too many duff beer, I think. 431 00:42:11,310 --> 00:42:16,440 So in the normal process we want to talk about ageing, atrophy and apathy, dementia. 432 00:42:16,440 --> 00:42:20,190 Eventually, in the normal process we know that we lose brain mass. 433 00:42:20,490 --> 00:42:26,970 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. 434 00:42:27,210 --> 00:42:34,410 And then we start declining in brain mass. So you can all place yourself somewhere around here and you know where you stand. 435 00:42:37,620 --> 00:42:43,140 But that process can be accelerated tremendously due to some type of disease. 436 00:42:43,350 --> 00:42:47,460 For instance, in Alzheimer disease, it's well known that you have very, 437 00:42:48,150 --> 00:42:54,270 very strong atrophy due to the propagation of some type of toxic protein in the material. 438 00:42:54,840 --> 00:43:01,950 So the way it happens is that there are certain type of molecule, there are different hypothesis which one are important and so on. 439 00:43:02,250 --> 00:43:05,520 Of course, a lot of complexity, a certain type of molecule, 440 00:43:05,520 --> 00:43:12,690 very small molecule protein in inside the cell that are toxic, that are turning to toxic proteins. 441 00:43:12,990 --> 00:43:17,430 And when they initially see that, they propagate through the brain and propagate, 442 00:43:17,550 --> 00:43:23,910 and when the propagation, the concentration reach a certain level, it fully damage the brain. 443 00:43:23,910 --> 00:43:32,360 And that part of the tissue is removed. So diffusion process or transport process are very interesting. 444 00:43:32,660 --> 00:43:40,790 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. 445 00:43:41,480 --> 00:43:46,190 And the idea is the following here is a structural map of Axum. 446 00:43:46,520 --> 00:43:51,830 So this is taken from image analysis. You can find Axon, the axon bundle I told you about. 447 00:43:52,780 --> 00:43:56,800 It's beautiful. The object you see, what you can extract from that is a graph. 448 00:43:57,220 --> 00:44:02,110 You can identify a region of interest and see all the connected to the region. 449 00:44:02,440 --> 00:44:09,460 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. 450 00:44:10,030 --> 00:44:13,510 But this is nice because that's a very neat mathematical object. 451 00:44:13,720 --> 00:44:17,750 This is a graph and a graph. We can start labelling notes. 452 00:44:17,770 --> 00:44:20,800 Let's call that one and two and three and four. 453 00:44:21,190 --> 00:44:23,680 A lot of times you go on like, let's say 100. 454 00:44:24,280 --> 00:44:32,049 And out of that, you can create matrices that allows you to study this object and the matrix you can create. 455 00:44:32,050 --> 00:44:36,580 For instance, let's take a very simple one with six nodes. One, two, three, four, five, six. 456 00:44:36,820 --> 00:44:39,940 I'm going to create a matrix. And I'll tell you, I create it. 457 00:44:40,270 --> 00:44:44,610 Let's call that L and matrix on the diagonal will be the number of neighbour. 458 00:44:44,710 --> 00:44:50,590 This one has two neighbour, five and two. So I put two to a three neighbours, so I put three in and so on. 459 00:44:50,860 --> 00:44:53,230 Very easy. And I put minus one. 460 00:44:53,500 --> 00:45:01,420 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. 461 00:45:01,660 --> 00:45:04,900 So minus one. Minus one. And I create a symmetric matrix. 462 00:45:05,720 --> 00:45:08,440 Okay. We all we all agree with the rule. Very simple. 463 00:45:09,500 --> 00:45:15,469 This matrix is very important in graph theory and in all network problems because it is called the graph 464 00:45:15,470 --> 00:45:23,750 la pression and it is the matrix that contains information about our process diffuses on a graph. 465 00:45:24,080 --> 00:45:32,480 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. 466 00:45:32,780 --> 00:45:35,980 All that information is contained into that matrix, 467 00:45:36,000 --> 00:45:43,850 the graph la pression and you can extract it by taking things that are called eigenvalues that you have ever either seen. 468 00:45:43,850 --> 00:45:48,259 And you know what I'm talking about or you have not yet seen. And when you say them, you'll go. 469 00:45:48,260 --> 00:45:51,920 I remember it was important for the brain. So trust me on that. 470 00:45:52,160 --> 00:45:59,780 You can extract that knowledge. And when you extract that knowledge, here is the second eigenvector of the network diffusion. 471 00:46:00,440 --> 00:46:06,840 So the of the graph la pression. For now I make a big one and the colour represent the intensity of this eigenvector. 472 00:46:06,900 --> 00:46:09,350 This is a nice work from 2012. 473 00:46:10,360 --> 00:46:19,000 And the surprising and remarkable thing is that if you compare that to atrophy patterns due to Alzheimer, you have extremely high correlation. 474 00:46:19,480 --> 00:46:30,820 So the second eigenvector of the graph pression of the brain connector is somehow correlated to the atrophy pattern of Alzheimer disease. 475 00:46:31,910 --> 00:46:36,360 Amazing result. But I wanted to know more. 476 00:46:36,670 --> 00:46:39,960 I said, This is nice, but I really again, I want to go back to the mechanic. 477 00:46:40,020 --> 00:46:43,770 This really doesn't tell me how things flow. What is the evolution process? 478 00:46:44,850 --> 00:46:48,180 So first we have to understand what happened in the brain. 479 00:46:48,390 --> 00:46:54,810 I told you about toxic proteins. What are they? Well, you have a lot of protein that does normal cellular work inside the brain. 480 00:46:55,440 --> 00:47:02,430 And these letters are in blue here. And sometimes the misfolded, instead of the normal configuration, they take another shape. 481 00:47:02,790 --> 00:47:09,420 And some of these fold are toxic. Think of them a little bit like a zombie apocalypse in a city. 482 00:47:10,020 --> 00:47:13,800 Right. So you have a normal guy that become, for some reason a zombie. 483 00:47:14,370 --> 00:47:19,290 And that zombie, whenever it meets another normal guy, transform them into a toxic guy. 484 00:47:19,590 --> 00:47:24,870 Now you have two zombies in the in the city and the zombies that bunch together. 485 00:47:25,210 --> 00:47:31,650 They're quite slow in the progression, but they always go on and and neurodegenerative diseases take a long time. 486 00:47:31,950 --> 00:47:34,710 But it goes on and on and we have very few ways to stop it. 487 00:47:35,370 --> 00:47:43,480 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. 488 00:47:43,980 --> 00:47:45,450 And when they become large enough, 489 00:47:46,080 --> 00:47:52,470 then the whole neighbourhood starts functioning and it's dead essentially right to stop the normal process from working. 490 00:47:52,680 --> 00:47:57,260 And that's the end of. So we can build a model out of that. 491 00:47:57,800 --> 00:48:01,250 And the model is to follow both toxic and non-toxic protein. 492 00:48:01,910 --> 00:48:05,810 You follow the concentration of the two, both in space and time within the brain. 493 00:48:06,820 --> 00:48:12,400 And what you do. You write rate equation for both aggregation and fragmentation process here. 494 00:48:12,700 --> 00:48:18,690 And you can borrow ideas from sync from other or the field of applied mathematics. 495 00:48:18,700 --> 00:48:20,200 What is processes take place? 496 00:48:22,200 --> 00:48:28,990 And then you realise that what happens is that you have very fast transport along the axon and that's why the graph works. 497 00:48:29,310 --> 00:48:37,740 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. 498 00:48:38,520 --> 00:48:46,080 And when they reach a certain concentration level, the axon dies because there's too much concentration and is removed from the material. 499 00:48:47,010 --> 00:48:48,210 Okay. So you can write. 500 00:48:48,270 --> 00:48:57,150 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. 501 00:48:57,780 --> 00:49:08,490 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. 502 00:49:08,790 --> 00:49:16,260 This is the sum up of the knowledge for maybe 20, 30 years of how neurodegenerative progress in the brain. 503 00:49:17,190 --> 00:49:23,790 Now what you can do is you can see that the same place here and you let the process run. 504 00:49:24,680 --> 00:49:35,540 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. 505 00:49:35,810 --> 00:49:39,980 The same in the back, in the temporal lobe and so on. So you can match a lot of things. 506 00:49:39,980 --> 00:49:46,570 You just let the process run. Then you can say, well, if this works for this particular system, what about the other ones? 507 00:49:47,440 --> 00:49:51,130 So you can go back and you can look at different molecules, 508 00:49:51,140 --> 00:49:57,130 different seeding and different diseases, including Parkinson's disease and lateral sclerosis. 509 00:49:57,370 --> 00:50:02,740 What's remarkable about this, I find, is that it is very systematic and people have been able to follow that. 510 00:50:03,070 --> 00:50:06,700 It always more or less takes place in the same characteristic way. 511 00:50:06,880 --> 00:50:15,370 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. 512 00:50:16,250 --> 00:50:20,330 But of course you say, well, that's quite not enough right now, if I know that, can I do more? 513 00:50:21,220 --> 00:50:24,700 People always want more. So what about atrophy? 514 00:50:25,030 --> 00:50:29,360 Here is an Alzheimer patient. Eight years zero. When? The first year when the scan was. 515 00:50:29,470 --> 00:50:37,600 Was made. Zero. So you extract out of that you segment and you have you have the full tissue white, white and grey matter. 516 00:50:37,930 --> 00:50:41,590 And you let you see this here at the base in two dimension. 517 00:50:41,920 --> 00:50:52,570 It sees you and you let the process run and this one, CO2 and you three and you can actually superimpose that almost identically. 518 00:50:53,260 --> 00:51:01,810 So the atrophy patterns that this process created by the evolution of this toxic protein seems to be extremely consistent, 519 00:51:01,960 --> 00:51:05,680 certainly correlated with what is observed during atrophy. 520 00:51:07,360 --> 00:51:14,950 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. 521 00:51:15,880 --> 00:51:18,490 You're not supposed to talk about things that are not published yet. 522 00:51:19,360 --> 00:51:27,310 But between us and my 20,000 Facebook friends, I thought that it would remain in the room, so to speak. 523 00:51:28,240 --> 00:51:34,610 So I thought that was. So interesting, it's so compelling that there is something there to study. 524 00:51:35,120 --> 00:51:40,820 So let me conclude. What did we learn from this based on atrophy and amnesia? 525 00:51:41,450 --> 00:51:47,089 Well, the interesting part that I found from a modelling perspective is that these phenomena are 526 00:51:47,090 --> 00:51:52,910 both multiscale and multi physics that include process that takes place at very short time. 527 00:51:53,000 --> 00:51:57,620 Folding of protein or shock in the brain, milliseconds, all the ways to year. 528 00:51:57,630 --> 00:52:04,680 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. 529 00:52:05,400 --> 00:52:12,450 It also includes different type of physics, different process transport, but also solid fluid mechanics, protein. 530 00:52:12,870 --> 00:52:14,790 Understanding what happened to the protein and all that. 531 00:52:14,970 --> 00:52:21,990 So you have to combine that and that is natural ground for applied mathematics, combining different field, different expertise, 532 00:52:22,230 --> 00:52:30,380 and looking at complicated equation and trying to extract from it simplicity against this tremendous complexity. 533 00:52:30,390 --> 00:52:33,810 I don't want to say that these diseases are easy to understand. 534 00:52:34,410 --> 00:52:39,360 30 years or more, 50 years of research have shown that sees extremely complex problems. 535 00:52:39,750 --> 00:52:47,940 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. 536 00:52:48,780 --> 00:52:51,930 And the last part, which is really where we're going now, 537 00:52:52,320 --> 00:52:58,680 is that once you understand what this process and of dementia takes place, that when atrophy takes place, 538 00:52:58,920 --> 00:53:06,149 you cannot go back and couple the physical atrophy with the functional connectome and try to 539 00:53:06,150 --> 00:53:12,840 understand how the network of brain function start degrading based on the underlying physics. 540 00:53:13,590 --> 00:53:19,049 So you couple the structural part, which is the physics part to the functional part, 541 00:53:19,050 --> 00:53:24,510 which is all the brain process information and it loses its ability to process information. 542 00:53:24,510 --> 00:53:33,570 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. 543 00:53:33,750 --> 00:53:39,480 We try to create a microcosm trying and inside the head you have a microcosm the whole world. 544 00:53:39,660 --> 00:53:43,890 We're trying to do the same. We're trying to model things and include different parts. 545 00:53:44,280 --> 00:53:51,389 We try to do physics and geometry and and genetics and and all different bubble, all different onion. 546 00:53:51,390 --> 00:53:55,350 And every time you peel one, you come to other question and you try to answer them. 547 00:53:55,740 --> 00:53:57,180 And it's really mathematics. 548 00:53:57,720 --> 00:54:04,740 And the way we applied that give us a chance to understand all these different world within our heads are really connected. 549 00:54:05,040 --> 00:54:11,490 And here is we started feeling a bubble. And again, I think applied mathematics is all the way there. 550 00:54:12,210 --> 00:54:20,160 So, of course, I want to give the last words to our friend Ludovic, who said he had a good sense of humour. 551 00:54:21,030 --> 00:54:23,900 Friends applaud. The comedy's over. Thank you very much.