Instant Genius - How maths can help us to understand the human brain

Episode Date: July 7, 2024

It’s often said that the human brain is the most complex structure in the known Universe. So how do we go about studying it? You may think that we should leave this to biologists or neuroscientists, but approaching the brain as a mathematical object and investigating its geometry and structure is providing researchers with more and more new insights. In this episode we catch up with mathematician Alain Goriely, professor of geometry at Gresham College, London ahead of his series of free public lectures entitled Mathematics and the Brain. He tells us how the brain’s shape, structure and size relate to intelligence, how mathematical models can help us deepen our understanding of diseases such as Alzheimer’s and how advances in scanning technology have helped us begin to uncover its many mysteries. Learn more about your ad choices. Visit podcastchoices.com/adchoices

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Every Monday and Friday, you'll hear world-leading experts and scientists talking about the most fascinating ideas in science and technology today. I'm Jason Goodyear, commissioning editor at BBC Science Focus. It's often said that the human brain is the most complex structure in the known universe.

So how do we go about studies? it. You may think we should leave this to biologists or neuroscientists, but approaching the brain as a mathematical object and investigating its geometry and structure is providing researchers with more and more new insights into its functions. In this episode, we catch up with mathematician Alan Gorelli, Professor of Geometry at Gresham College London, ahead of this new series of three public lectures entitled Mathematics and the Brain. He tells us how the brain's shape, structure and size relate to intelligence, how mathematical models can help us deepen our understanding of diseases such as Alzheimer's, and how advances in scanning technology have helped us begin to uncover

its many mysteries. So welcome to the podcast. Thanks very much for joining us. Oh, thank you very much for having me. So the first question then is, what's your background and what do you do day to day? So my background is more or less in mathematics and physics and science. I studied in the University of Brussels and then went to the United States where I was a professor at the University of Arizona for about 17 years until I moved to Oxford to take up the position of professorship of mathematical modeling in the Mathematical Institute. And I've been there since then. And recently I became appointed to the position of Gresham Professor of Geometry.

Great. So today we're talking about using maths to help us understand the brain. So how do we even approach that? Yes, so this will be the topic of my lecture, public lecture at Gresham College in London. And it's something I've been pondering for many years and working actively as part of my research. It is a fascinating topic because everybody has questions about our brain works, our own brain works, but also trying to understand what is in other people's head. And I think that most of societies organize along this line, and most of the human condition is trying to understand oneself and trying to understand the others.

So everything stands from that in a way, very self-reflective. And from a mathematical perspective, it's also very interesting, because we can look at it from a biological perspective and say, well, the brain is made of cells and the cell interact in a certain way. Or we can start quite differently, and that will be my approach. Look at the brain as an organ. Here is an object. What are you questioned?

And what were the question that people had originally? And if you're asking anybody, he says, oh, here's a brain. The brain has different feature. Or does that correlate to our cognitive ability or ability to interact, or more or intelligence or anything else. And I will start, in first lecture, talking about how the notion of relating, what we call neuronal correlates to intelligence,

has developed through time and what we can say if we look at it very precisely. And that's the whole problem. How do we define precisely things that are associated with the brain? The first idea that you might have is, okay, bigger heads have better minds. that's a natural, what people say, oh, he has a big head, he must be smart. Is that true?

That's what people thought for a long time. And we can try to be very precise about that, look at that, and try to understand if there is a relation of this type. Of course, as soon as you say anything like that, you want to do mathematics, so you have to understand the object that you work with very precisely. How do we actually go about measuring volume? Is there a good way to do that from a scientific perspective? and what are the possible measure of intelligence? And right away, that develops into a whole lot of very deep and sometimes dangerous questions. So if we're talking about maths, we're talking about things like geometry when it comes to objects, as you say.

So mathematically speaking then, what can we say about the shape and the structure of the brain? So the first part is let's look at the volume. That would be the first thing, away the brain, and people have done that. out to be disappointing because it doesn't give much information about anything. They are bigger brain and their smaller brain. And it turns out that the idea that men of genius will have, of men and women of genius, would have bigger brain turns out to be completely wrong

because their pathology associated with very small, very large brain. And people fall more or less on average around the Gaussian, around the normal distribution independently of the brain. So the next question is, well, maybe there is question related to the actual shape of the brain. And historically, that's what people started looking after realizing there was nothing so special about brain size. And that's very interesting because everybody probably has seen a picture of a brain.

It looks like a dried walnut, you know, all these valleys and fall. They call solaceae in gyri in neuroscience. Very complicated structure that's very highly folded. And right away, you have questions. Or do I characterize that? what are the geometric characterization of such object, and their natural way in geometry to do that. But then also pops up another question, he says, how does this structure come about? Or is it created through development in gestation?

But also, all this question relates is that, if we understand something about human, are we special in any way? That was always the goal in the old days, most of the 19 and part of the 20th century, was to try to identify if there was something special about the human brain. So is the geometry special for the human brain? Is there something that is different? Is it more convoluted? Is it more complex? And that's also a question that we can explore as soon as we develop the right geometric to characterize these shapes. So what is that tool then? Can you talk us through that? Yes. So in geometry, there are ways to compute what is called curvature, which tells you it's a local notion that tells you of

curve surface is if you take a sheet of paper and make a little ball of it, you'll have an extremely curved surfaces. So you can define, once you have representation of this surface, you can define locally this curvature. That would be one way. Another way is to look at how much it is pack. So if you unfold it, what would be the total area of the brain? It's about 2,500 square centimeter, which correspond more or less at the size of an unfolded newspaper. And this is all very crumpled into our little skulls. So that also is a measure. What is the volume compared to the area?

We can obtain another measure of how tightly packed we are. And then we can go around and try to compare that to our cousins around the animal kingdom and see how well we fare. Is there something special about the brain folding of humans? So how do we compare then to other animals? Well, for each species, we can go about and characterise by the same quantity, how much packing do you have? And then you have different arguments.

You can say, well, we really should normalize compare that with respect to the volume or the total mass of the animal. And then what you realize is that humans are not particularly doing so well because dolphins, for instance, have much more convoluted brain than we have. Hence, you have to start looking about other feature of the brain, what characterizes the human brain that is really different. It's not something that we have a definite answer, but we understand more or less that geometry itself is probably not enough, even though very recent work have shown that the

way the brain works is actually well represented by its geometry, that they are way to decompose the type of dynamical waves that's found on the brain based on the geometry of the brain. So there are a lot of work now trying to relate directly brain geometry to cognitive function. So you mentioned there the surface of the brain. It's very characteristic. Like if somebody were to picture a brain in their head, they'd see all of these ridges and like you say valleys and crenellations. So what do we know about the purpose of those and what effect do they have on cognition? Yes. Purpose is more or less clear is that we know that the seat of higher cognitive function reside mostly in the cortic aleria, which is that thin layer, maybe two to four

millimeter of tissue that's around the brain, and that's the one that's highly folded. So if you want more processing units, well, you have physiological constraint related to birth and gestation and all that. So you want to be able to have as bigger cortex as possible, hence you have to fold it. That's probably a simple but mostly correct explanation. Now, it has another interesting function, which is that it creates, to the folding, little different region, modules or lobes and things like that, that function as little entities by themselves, which are functionally independent and can interact with other ones. So that goes now that we have this type of structure, it supports underneath architecture of a network of information transfer within the process. and that's also the function, or believe to be the function of these intricate shapes. So let's have a look at the communication between these lobes then.

What do we know about that? Yeah, so that probably was one of the greatest revolution, scientific revolution, acquired one over the last, I would say, 20 to 40 years. Around the 1980s, people started developing the MRI machine, the magnetic resonance imaging system. People know, people have gone to the hospital and all that. So right away, people realize they could use that for the brain because it has the ability to image soft tissue

and get that structure of the brain as it works. And doing that, with progress, people realize that they could track the way different regions of the brain are connected through axons. So most of the computing in the brain is done by neurons. And neurons, they are somewhere like 60 to 90. billion neurons in the human brain. And these neurons are connected to each other through a very long process with respect to their size, which are called axons. So neurons are connected to neurons, and that's how information is propagated through electrical mechanism by changing the action

potential of the membrane, a little bit like a battery would work. It sent a signal from one neuron to the next one, and that signal is amplified. Now, we have the abysmal. Now, we have the ability to image T's accent through MRI techniques called diffusion-tensure imaging, and we have the ability to reconstitute more coarse-grained the network of the brain. And it's becoming so important that it even has a name. It's called the connectome. The connectome is the brain network. And more or less, if you define different region of the brain, you can think of, okay, here and maybe in my frontal cortex, I'm going to decide of a very typical region, you know, with fancy scientific names and all that, then I can try to see

how these regions are connected to each other by counting the number of neurons, so the number of accents going from one to the next one, a bundle of accents. And right away, I can extract out of that what we call a network or a graph in mathematics. So we've gone from the brain, from its geometry, very complicated, from the shape and all that, through imaging, very completely technology, now we have extracted the graph of the brain. So imagine that you only, let's say, interested in four different regions of the brain, you know, the frontal cortex, the visual cortex, and other ones. Then you can draw, you can write, put on a piece of paper, four big circle, and if one region is connected to another one, you draw a line. And then you draw a line every time

one region is connected to another one. That's called a graph, just like you can think of the graph of the internet or the graph of the connection of airlines. You see, if you have two cities, an airplane connected to two, if there is a commercial airline, you would draw on the map a line connecting the two. You can do the same for the brain, and you can tell if one region talks to the other one, to what extent? How many airplanes do we have, which corresponds to how many connections you'd have, how much traffic there is. And that, again, we know from graph theory, we can make a matrix sort of that. So our array of numbers, just like we learned in school, our array of number. And now the brain is encoded. The network of the brain is encoded in our array of numbers.

And that's where we start doing mathematics. We have a very simple object that has deep significance. And it was realized now, it's only started 20 years ago, that the use of mathematical ideas from graph theory and from its equivalent, which is used by physicists network theory, could be directly applied to understand how the brain is organized. And so that started a whole big field of brain topology, if you want, trying to understand the organization of the brain through mathematics. I know it relates to certain type of cognition,

certain type of behavior or even learning, for instance. That's fascinating. So what sort of concrete findings have we found from that idea, from that concept? So, for instance, we know that there are region in the brains that hacked as hubs, you know, just like plane hubs. If you look at the map of the US in particular, big airlines, FTs, big hubs where all their planes come and then ascend to other place. The brain is organized like that.

Their region that are highly connected to other regions. Another finding is that it has the so-called strong. of small world network. It is an idea that came out from the work of Watts and my friend Steve Strogett, in the US about more than 25 years ago, that if you take such a network and you have a long-range connection

between different nodes, then the number of steps you have to go from one node to the other is actually very short. And this is typically in popular science is a friend of a friend of six degrees of separation. You know, I know people. people in Oxford, I know my friend, but I also know somebody in New York, and the people in Oxford, through me, are connected to all my friends in New York. It turns out that the same

structure is also found in the brain, and that was a big surprise and really started the whole field of network science. So what happens then if one of these areas or these nodes becomes damaged or compromised? Is the brain able to reorganize itself? Yeah, so now you come to another feel which is all related to brain damage, which is also fascinating or trauma. We know that there is a certain what's called plasticity in the brain that's not related to plasticity of a plastic sheet or anything like that, but it's the ability of the brain to rebuild this connection or rewrote his connection up to a point. There are certain things you can relearn if you have a stroke, but there are other things that you might never be able to regain. There are certain

abilities that you develop in very young, but that you cannot acquire later on for the same reason. Now, what we've been also interested, and that will be also a topic of one of the last lecture, is how very serious neurological condition, neurodegenerative disease, dementia, Alzheimer, Parkinson, affect cognition, but also evolve to the brain in mathematical terms. Oh, that's really interesting. So can you dig into that a little bit more, please? It is a fascinating topic, very sad in many ways, because you see exactly how it affects people and families and the social burden it creates. But as a science problem, it's also very fascinating. So let's take Alzheimer, for instance, Alzheimer's disease, which is probably cover 60 to 70% of all known dementia.

Alzheimer is believed to be associated with the progression of proteins within the brain. So the same network that allow you to function so well by connecting. different parts of the brain, you know, memory and action and motor and all that, the same network is used by these toxic protein to systematically invade the brains. And that progression is well known. It has been the different stage and known, and which each stage of progression comes further and further damage within the brain, starting with amnestic syndrome, symptoms, you know, typical memory problem, and followed by systematic cognitive decay. and inability to perform a physical function, all the way to very, very late stage of apathy

and violent oboebus and anxiety. Very sad. But that progression is very systematic. So to the eye of a mathematician, it says, well, there must be a rather simple basic process underlying this systematic progression. And what we've been able to establish with other colleagues around the world is that you can model this very precisely, mathematically, as a process where you have expansion of these toxic protein in a way that we understand that they can accumulate very much like you would think of

a pandemic. You know, one toxic protein can turn a healthy one into a toxic one and further expand that population and be transported. Again, you can think of COVID being transported in the network of airplanes from one city to the novel. And jumping on these, network, the connecto, these proteins systematically invade the brain. Just based on these ideas, with very little minor tweak, you can actually have a full progression within the brain that perfectly match what is known, and that can even be predictive if you know enough about the basic initial point. So you can tell how the disease evolve. That doesn't necessarily help you cure it, but it goes a long way into a basic understanding

of this mechanism, which is very important if you want to develop any therapeutic and all that. So the next point is, okay, now that I know that it evolves through the brain, how does it affect the brain? How does it change locally the brain? And we know the effect of these toxic proteins on the cell, on the local cell and the ability to kill cell and result in a brain atrophies. One of the symptoms that can be also measured as a biomarker for the disease is the reduction of the brain size. damage now can also be recorded by looking at the brain dynamics, these waves going in the brain that you can record, for instance, with electroencephalograms.

You know, we naturally have brain states that we have an electrical oscillation in the brain, and depending on whether or not you are alert or meditative or fully relaxed or sleeping, you'll have different frequency in the brain that activate it. And we know that the effects of the disease is to change these frequency and to change the amplitude of these frequencies. Ambition comes in all shapes and sizes. At First Citizens Bank, we roll with your goals because we're built for what you're building. Fit for your ambition for Citizens Bank.

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I'm thinking of senses and our perception. Oh yeah, that's another fascinating. topic, and it would be a wonderful series of lecture by itself. I'll probably just give one, but I've recently become extremely interested in the visual system. The visual system is by far the most study of any sensory system. It plays a very big part in a world, in all our cognitive world. We talk about seeing something when we understand a problem, and so we really rely on the visual cortex. It's also a huge part of a brain at the back of a brain, so we don't do a lot of processing. Now, there are a lot of different effect, but I've been interested in the edge effect. By that, I mean things are not exactly like we think they are, and that can be hallucination,

where you see things that are not there, or illusion, visual illusion, which are the deformation of physical reality. You are presented with lines, let's say, that are parallel, and they have a background, and they appear to you as being bored or not quite right. All cases. All cases, have played with this visual illusion, they're great wonder. But there are also a window into understanding how the brain sense the world. Because we are presented with an object, let's say, a circle, a line, or something like that on a piece of paper. This line is sent to the back of our retina. That's the information. And then the information is sent to the brain. But in order for the brain to process this information, it has to do a series of transformation. And this

series of transformation is not perfect because we cannot have all that would be a waste. And so it has to transform the primary image into pieces of information that allows it to do a lot of very good trick, like detecting an edge and so on and detecting a motion. But there are cases where that doesn't work exactly because you have trade-offs. And these trade-offs turns out to be the visual illusion that we have. And if you understand the different step, the processing step that occurs between the image that's in the back of the retina all the way to back of the brain in the visual cortex area, then you can start seeing where it can go wrong. Either the cell in the back of the brain being activated when there is no signal, in that case it would be hallucination. And that can happen, for instance, if you take drugs or something like that, the special drugs like LSD that would create the illusion.

of an object without the object, so an hallucination, or transform a real object that's there into another one, and these are the source of many of the visual illusion. So by looking at this, you really understand how the brain process information

and or it can be twisted and or it can deceive you. Perception is misconception, who said Joseph Brentano, the psychologist. And so that's really fascinating. And another aspect of that is that artists have also been fully aware of that, you know, painters, the whole movement of abstraction in the 20th century

was a search to simplify object and understand the interaction of different elements or visual elements percepts with each other. And you can see that they essentially had the same idea that you can now understand in both physiological and mathematical terms. So we've covered quite a lot of different areas there. It's a really fascinating subject. So if we go right up to date, what are the current sort of big questions that we're trying to answer using this method? Oh, well, there are multiple methods, and the field of neuroscience is really blooming. Of course, I mean, a lot of people know are talking about artificial intelligence. It's in the mind of everyone these days with all the progress.

But artificial intelligence is very interesting because you have to define first what you mean by intelligent, artificial general intelligence. So the whole field of intelligence is at this time trying to be more precise to see if we can measure, you know, abilities both human and machine at the same time. It also naturally rises another question, is that this system, network, originally they were inspired by a brain network. But it seems like from the technology of artificial network that the best result do not use the same techniques that the brain use in terms of learning. Which is perfectly fine. I mean, we have planes, and we started making planes by thinking of birds, but there is no big planes that look like a bird. Right?

We develop technology with a certain inspiration, and we move to it. But my point was the following. It is a very big and complex network. And what we've observed, just like in the human in the sense, is that when they reach a certain level of complexity in terms of hierarchy and all that, all of a sudden you have emergence of ability, that are really surprising at a much higher level than what you would anticipate. And I think that's also the key to understand the ability of the human brain,

this hierarchical aspect of the network that becomes sufficiently complex, to all of a sudden be much better at cognitive tasks than almost all animals. Thank you very much for listening to this episode of Instant Genius, brought to you from the team behind BBC Science Focus. If you'd like to hear more about the topics we've just discussed, you can attend Professor Gorilli's series of free lectures at Gresham College London this autumn. For more information, visit their website at gresham.ac.uk. If you liked what you just heard,

then please do consider subscribing to Instant Genius on your preferred podcast platform. The current issue of BBC Science Focus magazine is out now. Pick up a copy wherever you buy your favourite magazines, or download us on your app store of choice. also find us online at sciencefocus.com. This podcast is sponsored by Name, Audio and Focal. The texture and emotional depth of music can be lost through digital sources or poor signal. Name Audio believes you can have digital precision with analogue warmth.

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