The brain as a complex system

We know the brain is a complex system. It's full of neurons and these neurons are all connected together. And when they fire together and work together, they produce not only sensible behaviour, or hopefully sensible behaviour from us humans, but also in animals as well. How does that actually happen? Well, to understand a little bit more about the brain, we are joined at St. Anne's show by Christopher Lynn, assistant professor of physics at Yale University. Christopher's going to talk about where we are in terms of physically understanding how brains are connected together. We're going to talk about the human brain, we're going to talk about the fruit fly brain, and we're going to talk about a worm's brain. Then you're going to talk about network science and how we can take the concepts of network science, which you've heard about on the show before, and how we can apply them to better understanding what a brain structure actually looks like. And what it actually means. This show is really about trying to attempt to understand how a collection of neurons fitted together in a system can produce the glorious thing that we call consciousness. This is simplifying complexity. A podcast where we explore the underlying principles of complex systems. Systems that seem to defy our rational view of the world, like economies, ecologies, or even you or me. I'm forensic engineer, Sean Brady, and I'll be your host. Christopher, welcome to the show. Thanks for having me. So today we're going to talk about the brain and network science applied to the brain. How did you get into this sort of stuff? So I started off, I've always been in physics departments. So going into my PhD, I actually wanted to do quantum mechanics. And then that didn't work out. The person I wanted to work for stopped taking students. So through a random walk process, I ended up in the lab of Danny Bassett, who I noticed was a previous guest on your show. And they do network neuroscience. And what intrigued me about the things I was studying in quantum mechanics were the fact that you could get very complicated phenomena that you might not have predicted if all you knew were the microscopic rules. And a useful way of studying these emergent phenomena was through complex systems or network science. And then so these sort of more general principles of emergence and complexity were what I realized I was actually interested in. And the brain turned out to be an awesome place to study these sorts of questions. And from then on, I realized that network science was a very useful way to be thinking about these very complicated systems and trying to make sense of something that seems just like a tangled mess. So tell me about the brain in terms of how do you see it? Yeah, so the brain, I think of as this vast web of connections-- so you have these individual units, which are nerve cells that we call neurons. And then they link together in this intricate web of wires. So we know that those things are true. And then at a much larger scale, we know that somehow this vast web of wires is able to perform quite impressive feats of cognition, consciousness, and automated image processing, and linking these small scale rules. So basically, the really interesting question about the brain is how do you take these tiny little neurons and then with relatively simple rules that each neuron abides by. And then once you link them all together into some very, very large network, you end up with these phenomena that we're all familiar with and perhaps take for granted. But in our no ways, obvious given the individual cells that we started with. So how many neurons have we got in our brains, roughly? So in a human brain, you have roughly 100 billion neurons. For a sense of scale, once you start talking about numbers with a lot of zeros, I like to think about the population of the earth. So the population of the earth is about 7 or 8 billion. And so inside your head, you have about two orders of magnitude. So 100 times more neurons in your head than there are people on Earth. I think there's actually useful analogy to be made there where if you think of psychology as the study of an individual person, then sociology or political science might be the study of the large groups of people all together. Neuroscience is the study of the brain. But really, there's many, many different scales you could study it as. There's neuroscientists that spend their whole career thinking about individual neurons, which would be like psychology in our analogy. More, you could study the entire system itself, which people call systems neuroscience. If you want to study the whole brain, or you can study anywhere in between. So you could study circuits of like 10 or 100 or 1,000 neurons. And so there's a whole range of scales at which to ask these questions about emergence. It might be useful to also think about other animals that people are used to studying in the laboratory. So you'll have to check me on this. But I believe a mouse has around 100 million neurons. So I think roughly 1,000th, the number that a human has. And then the fruit fly brain, which recently famously was fully mapped, has about 100,000 or 150,000 neurons. So it's another, again, another from the mouse, another step of 1,000 times down from there. So we have about a million times more neurons than a fruit fly. And so those are the scales that we're talking about. So you're saying we have all these neurons. And they have very simple rules for interacting with each other. Just have simple out of those rules. So again, there's varying levels of realism that you can talk about. There's a famous quote that every model is wrong, but some are useful. But the simplest picture to have in your head, which gets you quite far, is that individual neurons receive signals from their neighbors in the network. So they feel impulses from their neighbors. So their neighbors will fire. And they feel these impulses with some weight. So they don't feel all impulses equally. So if I'm one neuron and you're upstream from me and you fire, and I'm strongly connected to you, then that will likely lead me to fire. But if you were weakly connected, then you firing would only give me a small impulse. And that probably wouldn't be enough for me to fire. I'd probably need other neurons firing at the same time to induce me to fire. So the simplest picture in mathematical terms is you have some neurons that this neuron is connected to. And then they fire or they don't fire. And then you have this linear sum of these inputs, meaning that there's a weight on each input. You add them all together. And then you just have this threshold that says, does that combined weight of all the neurons that fire did it reach the threshold? And if so, then I fire, if not, then I don't fire. And this very simple picture is known as the perceptron. It's perhaps one of the earliest computational models we have of how neurons work. And it actually gets a lot of the qualitative details about right. So this simple picture is very nice for people interested in computation and interested in deriving mathematical rules. But there are other biological details that give you an even richer picture than just this, that people are still exploring. So an assumption we're kind of making here is that this combined influence I feel is all that really matters for determining whether or not I'm going to fire if I'm a neuron. But it turns out that in real neurons, there's increasing evidence that sometimes-- not always, but sometimes-- it's actually a bit more complicated. The more complicated combination of which neurons are firing upstream from me actually plays an important role, not just the combined influence, but actually which specific combinations of neurons are firing, it turns out seems to matter in some circumstances for determining if I'm going to fire or not. It's a bit of a complex distinction to make. But it turns out that you can't get these more complicated combinations of dependencies just from the simple picture I described before. I used to say that it's not only how other neurons are firing that makes you fire. It's how other groups of neurons are firing. Is it more collective? Is that what we're getting at? Yeah, I always like to go back to social networks as an analogy in my head. So you could imagine that oftentimes voting is a useful thing because it's binary. It's yes or no. And neurons are similarly binary. They fire. They don't fire. So you could imagine that you deciding to vote yes or no on something is due to some combination of influence of all the people in your social network. And typically, if you have a bunch of friends that vote yes on something, then you'll be influenced by voting yes on something. And so the simple picture I would talk about before basically says that you combine the full sum of the influence from your neighbors and the network, and then you decide to fire. So for example, suppose you have the simplest pictures, you just have two friends. And suppose you're positively interacting with both. Then if one votes yes or the other votes yes, then you're likely to vote yes. But I can give you an example of a type of interaction that isn't captured by this picture. So suppose if one of your friends votes yes or the other one votes yes, then you're likely to vote yes. But actually, if both of them vote yes, then all of a sudden you're turned off by the idea and you want to vote no. That circumstance is an inherently what's known as a combinatorial interaction. And it inherently involves understanding this more complicated, not just the sum influence of the two, but actually, you need to know the combination of who voted and who didn't vote yes. And it turns out that recently they're starting to see evidence for these sorts of interactions and individual in their own. That sounds like whether you vote or not is dependent on the result of the interaction of the other voters. So it's not just the influence of individually on you. It's when they interact, which are they produce an emerging property. And that emerging property essentially affects you. Wow. Exactly. And yet for people that are interested in this, what I just described is a classic computation known [BLANK_AUDIO] an exclusive or function, or meaning one or the other of your friends voted yes or you want to vote yes, but it's exclusive. So you don't want both to vote yes or then you're no. This is a classic kind of function that people previously thought would require multiple neurons to perform this function. You would need to wire multiple neurons together and you could do something like this, but they're finding that individual neurons might be complex enough to do something of this sort. So individual neurons might be a bit more complex than we think from the simple picture I described. So if we go back to the simple picture, so we've got which is quite extraordinary. You've got a brain met up of all these neurons connected in certain ways and out of whether they decide to fire or not, we get consciousness and cognition and computation and all those sorts of things. So then talk to me about network science, probably generally with network science and then we go and talk about why it fits so well with the brain. I mean, obviously it would because it's a network, but let's just step through it. The way I view network science is it's a very useful lens through which to look or tool to use to study complex systems because at the end of the day, we don't understand something unless it's nice and ordered. So we need to take this very complicated thing and boil it down to some simpler principles and network science is really good at doing that. So the way I view network science, and this is an universal, the way I view the sort of modern history of network science is around the late 90s or better on 2000. So you start to have these big data sets and people started being able to realize that there were some sort of shared patterns between different networks that we were seeing in the real world. So there's different sort of features that pop up again and again in these different networks and then people focused on those features and tried to study them and explain them and understand where they can come from. So you have scale-free structure, which was famously studied by Barbassian and Ricca Albert. And there you're interested in studying what are known as the degrees of nodes. A node is just a single unit within the network. So in every network, there's two important things to understand. There's the nodes, which are the individual things. So in the brain, these would be neurons, in a social network, these would be individual people. And then you have the edges or the connections between things. So in the brain, these would be synaptic connections between neurons. In a social network, these would be friendships on Facebook or follower relationships on Twitter or Instagram. And so these are the important things. You have nodes and you have connections or edges. And then the number of connections that a node has is its degree. And something that you commonly find across a lot of real-world networks is that these degrees are not evenly distributed. Very often in these networks, there's some nodes with many, many, many connections. So these on social networks, these would be like influencers or something or people with a million followers, whereas the average person has maybe a hundred followers or something like that. How you would describe this very large spread in degrees is by saying that there's a heavy-tailed distribution of degrees. So there's some rare nodes with very many connections. Many, many nodes have very few connections. And in fact, if you plot these on a log log scale, they look like a line often, which is what we call scale-free. It's a power-lawed distribution of degrees. Which are listeners of a very familiar way from Jeff West's episodes and many episodes in this series. Exactly. So that's one sort of classic type of network structure. And then you could go try to explore. And I think we'll probably talk about that a bit later. Then there's others. So for example, you have Watson Stroghats right around the same time discussed small-world networks, which is this feature of networks where you're able to hop between any two nodes in only a small number of hops. And this is a feature, again, you see in social networks where even though you might feel very disconnected from like a sheep farmer in Mongolia, it turns out that very likely, if you follow connections on Facebook or something in relatively few hops, you'll actually be able to get to any other person on the whole network. So within, I think about six or seven hops on average, you can get between any two people on something like Facebook. So this is small-world structure, which is another type of structure that people see in these networks. The third sort of classic, what I think of as sort of the pillars of network science that gave rise to this big boom in network science in around 2000 was, I would say community structure and there's some classic papers by Mark Newman and others. And there you're looking at how nodes like sort of clump together into communities, into clusters. And you see this again in social networks, so you see that this idea that if you have two friends, then they themselves are more likely to be friends with each other simply due to the fact that they're friends with you. So they might have showed up to one of your dinner parties at some point and then they become friends. And so through this clustering effect, you end up with communities of nodes that are much more tightly knit than you would expect just from a random network. And so these are just some examples. Then there's many other types of network structures that you could go on and talk about. And I presume the key here is that because whether it's a fat tailed distributor or a parallel distribution or you got small world going on while the community based one, you can go and do the mats for these networks and understand how they work and break them down to that relatively simple understanding of how the network works, even though it's a mess on the surface. But by applying these principles that people have developed of understanding of these networks work. So that means you take those mathematical tools that are developing developed and then you can apply them in different systems. And for you, it's the brain. Yeah, exactly. So the key idea here is that these systems are so complicated that you can't just look at a picture and say, oh, this one looks like that one. At the end of the day, you need to be able to be quantitative about this. You need to come up with a quantity that does a good job of getting at intuitively whatever we're trying to measure. So how heavy tailed is this degree distribution? You could look at the variance of degrees or something like that. Or there's ways of trying to quantify the strength of community structure, which is still I think an ongoing area of actual research because there isn't actually a well agreed upon quantification of community structure because it's a bit more amorphous. But in small world structure, there is a well-defined notion where you can measure the average shortest path between people. So that's what I was referencing. And I was saying, about six or seven hops, you might be able to get between friendships. You might be able to get between any two people on Earth. That is referencing this average shortest path. One of the things that's curious about this is that the application of, let's say we go to a network and we apply small world techniques to it. And we discover that in this particular network, a real world network, whatever it is, we've got seven hops on average. And then we get another system that might have a hundred hops. I presume that's a quantity and we can measure each of those. What does that mean in practical terms in terms of understanding that network? If we go from like 10 to 7 to 100, what does it mean? That's where the hard work comes in. It'll depend on the context. It'll depend on the specific system that you're studying. Are you studying a social network? Are you studying the brain? Are you studying the internet or a transportation network? And in each of those contexts, what a hub or small world structure, community structure represents will mean different things. And then whether or not that's a useful thing to investigate further, will depend on one network or another. So, in the brain, having a hub node will likely mean something different than in a social network. But the useful thing is that we have this common framework so we can start to draw analogies between the two. So if we have an intuition that holds in one instance, then we might be able to port that intuition over to another context and start to try to build a similar knowledge. Oh, so you're saying that it's not really that important if we're going from one network and say, "The 7 here and the 100 here and what does that mean?" It's much more interesting to say, "Well, this network here is clearly a small world network with 7 hops." And this one here is also a small world network with 7 hops, but they could be completely different in terms of practical systems. But the value is identifying the similarity between them and then presumably the possibility of the lessons between the two of them. Is that fair? Exactly. That's what I find really exciting is if you see two networks in totally different contexts that have some sort of similarities in their structure, let's say one of them we understand already, then that already gives you hints at what might be going on in the other network. So you can try to take the lessons we learned in the first network and try to see if they apply in an analogous way in the second network. And this is actually something that I find one of the most useful things in network science is being able to translate intuitions from one context into another context and see if it helps us understand what on the surface might seem like two totally different types of systems, like a brain in a social network or the internet or something like this. They seem totally unrelated, but by realizing that there might be similarities in their structure you can start to guess maybe it's similar underlying mechanisms at what might be going on. So what happens when you take out this and apply it to the brain? So I think the brain is a classic example of a complex network, in my opinion. I mean, you have these neurons. If I guess for another sense of scale, a useful thing to know is that at least in the human brain a typical neuron has about a thousand or ten thousand connections to other neurons. So again, you might be useful to think about this on the scale of what might happen in a social network where everyone has a Facebook account or an Instagram account and they might have around a hundred, maybe tops a thousand connections or something like that. Well, I mean, on Instagram you might have like followers, but on Facebook you probably don't have many more than a thousand actual friends in the real world or something like that. Whereas a neuron has like a thousand or ten thousand connections. And so I think I think of the brain as a system that is classically amenable to these kinds of network type approaches. The important thing is that network science to make any progress you need actual data. So we can sit here and theorize and philosophize all day about how the brain might work as a network. But until you actually have data about which neurons are connected to which other neurons or which neurons are correlated with which other neurons, then we can't actually make quantitative progress about trying to build these connections that we were talking about and use these tools that network science has built up. So this has been quite successful for a while, I would say, in what's known as systems neuroscience, which is the study of the whole brain. And in humans, this is done usually to give you a sense of where this data usually comes from. You have, for example, MRI machines, if anyone's ever hit their head or something like that, you might have had an MRI scan. MRI scans look at the structure of your brain and then there's what's called FMI, which is functional magnetic resonance imaging, which allows you to sort of measure over time which brain regions are becoming more active or less active over time. And then from that, what you can do is you can ask, well, which pairs of brain regions tend to be correlated over time. And if we think two brain regions are highly correlated, correlated meaning if one brain region is tends to be active and the other one tends to be active. And if we see them always active together and inactive together, then they're going to be highly correlated. And then you might think that they're connected in some way, either they're actually physically connected and interacting with one another or maybe they play a similar role in information processing or something. So with sort of the proliferation of FMI, people started looking at these functional networks in the human brain. And there's been a lot of research in systems neuroscience looking at the structure of these functional networks, these networks of correlations. And for example, how do these functional networks change within one person? How does it change from one task to another? So when you're just lying in the scanner, maybe the network will look different than if you're performing some tasks where you really have to concentrate. And it might change as you age, it might change from healthy subjects to subjects with some disease. Then you can start to use the structure of these networks to try to think about the actual function of the brain by doing these sorts of experiments. So going back to the idea that a lot of science is driven by really experimental advances, particularly in network science. We need the data to be able to say anything about the network. More recently, we now have actual maps of the synaptic connections between neurons. So at a very, very small scale, we see the individual little connections between pairs of neurons at much larger scales where we can start to think about these as networks. And I should say that that data doesn't exist in humans. So usually you talk about whole brain imaging in humans and you can think about networks of brain regions. Or increasingly you can start to think about the physical connections between individual neurons. But this is not in humans because it requires killing the animal and freezing its brain and slicing the brain up into a thousands of slices. So you can't do that to a human. Is this probably a good time to go to the fruit fly? So they've mapped only recently the whole fruit fly brain. Talk about that. Yeah. So what I'm describing here is this where we're mapping out the physical connections between neurons. This is broadly classified under the name connectomics. So if you ever hear the word connectome, that's used to refer to the network of physical connections between neurons. And when I say physical connections, I mean, so individual neurons touch one another at points that are called synapses. And then these synapses are where the sort of impulses are transferred from one neuron to the next. And so you have synapse level connectomes now. And the history of these connectomes is quite interesting. So the first one was published in the mid 80s actually. So quite a while ago coming on 40 years now in 1986, I think they published a full connectome of this tiny little worm called C elegans. C elegans is a it just pops up all over sort of science and medical science just because it's one of the simplest organisms that does some complex things. C elegans is this tiny little worm. It has exactly 300 and two neurons. And it's such a small network that we have guesses at least of what each individual neuron is meant to be doing. In the mid 80s, they were able to map out the full connectivity map, the connectome of one worm. And I don't think they did the whole worm. I think there it was like the brain of the worm, which is somewhere between 100 and 200 neurons. And since then, the people have mapped out, I think roughly the connectome of the whole worm. So that was the 80s. And then there was a long gap with some data sets coming out. And then much more recently over the last 10 or 15 years, there's been quite a big explosion in experiments able to map larger and larger networks of these connectomes. So you have networks of connections between neurons in the mouse retina. You have other invertebrates, other little worms, then around, I want to say like five or 10 years ago, scientists published a quarter of the fruit fly brain, which is around 20,000 neurons, 25,000 neurons. And yeah, very recently, which is very exciting, a large consortium of scientists. It's a huge collaboration involving even non-scientists that hiring people outsourcing the process of labeling all the neurons and all this inapses. So there's very, very large collection of scientists and non-science people getting together to map these connections resulted in mapping the whole fruit fly brain in one fruit fly, I should say. So there's this is one fruit fly. It took about 10 years and we mapped all the connections between roughly 150,000 neurons and this was published about a month ago in nature. They published like nine papers or something. And is this both just working at what's connected to wash, but also whether or not one five makes the other fire? Is that part of it as well? Yeah, ideally what you would really like to understand is how does the structure of the brain relate or guide the function of the brain, which is arguably, well, it's one of the holy grails of neuroscience. So it's like a computer. If you show me a picture of all the things being wired to get, where all the wires go in the computer, that isn't quite the same as telling me how to use word or telling me how to get on a zoom call. They're quite different things. The function of the computer might not be obvious just by looking at the wiring diagram. Have we only got the wiring diagram now of the fruit fly? So you have both in different ways, but not in the same fly yet. Okay, go ahead, explain that. So in one fruit fly, you have the full wiring diagram. And then there are many, many people that record the activity of neurons in the fruit fly. So they'll stick probes in a fruit fly or they'll cut open its head and attach a microscope and watch the neurons light up, which I could describe how you record the activity of different neurons. There's many people also simultaneously looking at which neurons are actually firing in the fruit fly. So now we can start to compare the two. Now that we have the wiring diagram, we can start to compare it to what patterns of activity are we actually seeing in the brain. So if two neurons are really correlated in their activity in the fruit fly brain, does that correspond to a strong physical connection between those neurons in this wiring diagram? The complication or one big caveat here is that we don't have recordings of the activity and the wiring diagram in the same brain. Because as I mentioned before, to map out these physical connections, you have to kill the animal and freeze its brain and slice up the brain and map it all back together. But there are a couple labs trying to do precisely this, record the activity in an animal, either a mouse or a fruit fly or a worm, and then do the connectomics and try to map the connections. So then that would give the first glimpses of activity or what's sometimes referred to as function and the structure of the connections in the same brain. And that would be a massive step forward to presume in understanding how any animal works or how any of us, what neurons work. So where are we with respect to applying network theory to these? So presumably you've got the structure now in some cases of something and you've got the responses. And are you using the network theory tied to two of them together? Is that ultimately we're trying to do? Or are you just purely trying to say, I know what the connections look like in the brain from a network perspective? Explain that. Is it just purely that piece? I think it's both. It's on any and all. So basically we're still very long way away from understanding how the brain works. So we are basically doing everything at the wall and seeing what sticks. And there's many sub-goals to the large goal of solving the brain in some sense, understanding how it all works. So one clear set of questions is like what you're saying, how can we connect the correlations and activity that we see to the actual structure of connections now that we have them in the fruit fly? That's very much an ongoing thing that people are thinking about. And this is ongoing not only in the fruit fly, but also in the worm sea elegance that I was talking about. Even though we map the connections in the 1980s, we're still trying to understand how that connects to functionally which neurons are interacting with which other neurons and why. So we have the network of which neurons are actually connected to which other ones in the worm. There's plenty of experiments where you can record the activities of these neurons. And so you might naively say, well, if two neurons are strongly connected, then they should fire together a lot. I mean, if one fire is it probably will lead the other one to fire like from our simple picture that we talked about before. And so this is a very clear hypothesis that we should have. And you can go test it. And what you find is that indeed those two things are related. If two neurons are strongly connected in their physical connection, they are likely to be more correlated, but it's not always the case. And it's not this nice one-to-one mapping. And you have there's a lot of noise. So it's still very much so understanding what else is giving rise to correlations. What that means is that it's not just the physical wiring that's giving rise to these correlations between neurons. There must be other things we're not accounting for. And even Even in this simple worm, where we only have 300 neurons, it's quite a complicated test. So for example, well, the worm is a funny case because the worm is so small that neurons can actually communicate not just through the wires, but just by basically releasing some hormones that other neurons sense. So the worm is so small that it can just release some chemicals. And then the other neurons are close enough that it can kind of tell that that neuron fired just because it released some chemicals, even though maybe they're not physically wired with a wire. The worm brain is a little bit of a special case where that is true. In the fruit flyer humans, that's not the case. They only communicate via the wires or so we think. But another big confound or complicating fact about this, and this is true of all animals, is that two neurons can become strongly correlated, even if they're not directly connected, because maybe they're receiving common inputs. So you could imagine a neuron in the retina on your right eye and a neuron in the retina of your left eye might be focused on the same object in your visual field. And then they're going to fire at the same time because they're seeing the same thing, even though they're not connected at all. But they become correlated in their activity because they're doing the same thing. They're performing the same computation in some sense. So there are reasons why neurons might be correlated in their activity without ever actually communicating with one another. And this is disentangling these two things is still an act, yeah, very active, very research, even in the simple worm. And gone back to the idea of the power law connection where we see some neurons reloads connections, some would not, or the small world. What are we seeing in terms of applying those techniques to brains of either the fruit fry or anything else? I guess I didn't finish the full answer to your previous question, which was then, so there's one thing that you can do, which is, yeah, try to look at the connection between the structure of these connections and then the correlations in activity. Another thing you can do is just look at these wiring diagrams in their own right and try to understand what are the structures we're seeing in these very interesting big new networks. Are there features in these networks that look anything like what we're used to seeing in other networks, like social networks or the internet or something like this? So for example, all the structures that we talked about before are on the table. So when we go back into these big new connectomes, do we see any of these structures? So this data so it was only published a month ago. So people are scrambling trying to see if you see these structures or not. For my own personal tinkering around with the data, it doesn't quite look like you see scale-free structure in the famous way or in the traditional way that we're used to talking about it. So before when I describe scale-free structure, what we're talking about specifically is scale-free distributions of degrees, which is again the number of connections that a node has. And when you look in the brain or at least in the brains that we have access to right now, it doesn't look quite like that's the case. So I mentioned before that a typical neuron, at least in the human brain, has around a thousand or ten thousand connections. And it doesn't vary wildly from that. If we did truly have scale-free structure, it would vary wildly from one neuron to the next. So there would be what that would imply is that there are some neurons with many, many connections, like millions of connections, whereas the vast majority of neurons have many fewer, like a thousand or a hundred. But actually, we see that neurons do typically have not exactly a thousand or ten thousand, but roughly around there and it doesn't vary wildly in the scale-free way from there. And just to sort of tease out scale-free there, if we sort of zoomed in close to a bunch in neurons and then we zoomed out in a scale-free world, that would look like a version of itself. It would look like pulling back from a river network. We just see if it starts to look similar and that follows the power laws. And we're not seeing that. So if you're seeing we zoom in on a power of a brain, we're looking at the neurons and then we zoom out a bit, we will see a different structure. Yeah, that's a great way of putting it. So if you zoom in versus zoom out, you will still see that the typical neuron has sort of a typical number of connections, like around a thousand or ten thousand, whereas in a social network or something like that, that might have scale-free structure. If you zoom in, you'll see that there's always some hub nodes and some very weekly connected people. So you could zoom in on your own social network. There might perhaps you're the hub of your own social network where you're really connected to people. But you have a hundred connections and everyone else might have one or two connections to each other. And then you zoom out and you realize, oh, you're just a medium sized hub in a bigger sea where really there's these, in the brain, really neurons do typically tend to have a sort of a typical number of connections. So it's not scale-free in the same way where you zoom out and all of a sudden you see this phrase that there's always a bigger fish in the sea doesn't quite apply to the brain. Going back to the concept now that we can, when we understand that about the network, and we can go and say, well, are there any other networks that look like that? Is there any other obvious networks that scale like this as opposed to word power laws and scale-free? Yeah, that's a good question. So for example, if you were to, I think a useful thing that for people to have in their heads is just a random network. So anytime I'm trying to understand, when I look at a network and I plot something or I measure something, I'm always asking myself, it doesn't make sense unless you compare it to something. Or it's hard to interpret anything unless you compare it to something else. Just like any number, like you could, you stand on the scale and you see a number, is this heavier or not? Well, it depends what you weighed maybe five years ago or something. In just the same way, it's always used to that have in your head, what would a random network of connections between things look like? And there, a random network does have sort of a typical number of connections. So you do not see scale-free structure in this way. So in a random network, if you have N nodes and E edges, big E edges, the average number of connections per node is just two times E over N. So this would be the average degree. And things don't really vary too far from that. So if we have 100 nodes and 100 connections, the never-e-node on average will be connected to two things. And you're not going to see a vast difference from two. You might see five, you might even see ten, but you're not going to see a node connected to everything else. That will never happen. Whereas, roughly something akin to that happens in scale-free networks. So a random network is not scale-free. Another good example to have in your head is perhaps something like a lattice, which in physics people are really used to thinking about. So if you imagine just a grid in two-dimensional space, like a sheet, each node is connected to the thing to the front, the back, the left, and the right, that's a clear example of something that doesn't have scale-free structure because every node has exactly four connections. So there, by definition, there's no hub nodes to be had. There's no node with many, many connections. In fact, every node has exactly the same number of connections for. So this is another example of a type of network. And we'd say a spider web is like that as well, isn't it? Each intersection has got four or however many coming out from it. Yes, spider webs are very lattice-like. What does that mean? So there's many axes in which you could compare the structure. So we talked about all these different structures that networks could have. So along this one axis where we're looking at scale-free structure of these degree distributions, the brain does not look like a scale-free network and therefore might look a bit more like a random network or something. But there's other axes in which, along which a brain looks very different from a random network. So it just depends which structure you're talking about. Give us an example of that, that's intriguing. So in the brain, you do see, for example, a small world structure. Like we see in social networks, for example. So in the brain, it seems like you can get from one part of the brain to the other part of the brain quite quickly and only a small number of hops, which in the brain you might think is useful for transmitting signals really quickly and making fast decisions or something like this. In a similar way, you also see strong community structure in the brain, which you don't see in random networks. Random networks have no community structure. There's no sense in which one group of nodes wants to be more connected than any other group. In social networks, we already talked about how you form these little cliques and little friend groups and things like this. In the brain, you see something very similar where you have regions of the brain that are very strongly connected to one another, likely because they're performing some specific function. And then the connectivity to other regions of the brain is much sparser. So you'll have, in the human brain, this goes back to all these, these very old ideas of different parts of the brain being useful for different things. And so you'll do tend to get these community structures. You'll have a community of the brain that's devoted to maybe visual processing, like processing what you're seeing in the world. And then you have another region of the brain that's really important for making movements. So this would be something like the motor cortex. And within these groups, you'll have very strong connectivity because they need to do these specific computations. But then they're more sparsely connected to other regions. So these are other structures that you do see in brains, but not in random networks. So when it comes to them, what's next for the fruit fly? What's next in terms of trying to apply this sort of thinking to know that you have this data source? You could look at all these different types of network structures. It's a great big playground for people, for network scientists to play around with right now. So one of the most striking features that I've seen, and I've been thinking about quite a bit, is you don't see a very large spread in the degrees as we were just talking about, like most neurons tend to have a sort of a typical number of connections. But if you look at individual connections, those tend to have a very, very large spread in how strong they are, which I found very surprising. So I can be a bit more specific here. So if we have two neurons, we spoke about how two neurons will communicate with one another via these little connections that are called synapses. You can think of a synapses, one little unit of connection strength. But two neurons might not share just one synapse. They might share two or three or four. And you can think of how many synapses two neurons share as a decent approximation for how strongly they're connected to one another. And so you might think that if two neurons are connected, there's some typical number of synapses they might share. So for example, so in the fruit fly brain, if you tell me that tunerons are connected, they typically share on average around five synapses, which seems like a reasonable number. It's not too big. It's not too small. Despite that fact, you see some connections just between tunerons that are sharing over 100 or over a thousand individual synapses. This is in the fruit fly. Yeah, so you'll just have two neurons that are sharing over a thousand individual synapses, even though the average is like four or five. So these are connections that are ten or a hundred times stronger than most connections. And so this is sort of reminiscent of these scale-free distributions of degrees, but rather than being distributions of degrees, which is sort of the strength of a node, these are distributions of connection strength. So how strongly are neurons connected? And here you see what we call a very heavy-tailed distribution of these connection strengths. So the analogy here is that in heavy-tailed distributions of degrees, what that means is that you get these hub nodes, like influencers on Instagram or something, in the brain of the fruit fly at least. And actually, we see this pop-up in every other brain we've looked at. You see some connections that are just much, much, much stronger than most. They're like the hubs in the node analogy, but what they really represent are like strong backbones of connectivity in the network. So these really, really strong pathways of connections. Just to be sure I understand that right. So you see them that there's the physical connectivity. And that's one thing. But then there's the, it's like each of those connections has a different strength and a difference importance essentially. So it's kind of like you got two networks layered on top of one another, where one of them is like a small world type network in terms of the physical space and a community based sort of network. But then when you look at the strength of the relationships between those, that's much more akin to a heavy-tailed distribution power law. And wow. Yeah, exactly. And I should have probably mentioned this before. So in, in network sense, it's kind of two separate things you can talk about. There's the topology of the network, which is what you're describing this sort of the structure, the skeleton of like where are the connections. Is there a connection or is there not a connection? And then do you get this small world structure or community structure? And then on each connection, there's a weight. So how strongly am I connected to this other, are two things connected? And yeah, in these connection strengths, we see something very stark, which is this, yeah, these heavy-tailed distributions of connection strength, which is a mouthful, but really it just means that there are some connections that are much, much, much stronger than you would expect in a random network. Now, I should say that whether or not these distributions are scale-free, whether or not their power law, which would hint at scale-free structure in these connections is still very much up for debate. So determining whether or not something's a power law is a tricky subject mathematically. So some people do think it's a power law. Other people think it's their log normal distributed. We don't have to get into mathematical details. But the important thing from a qualitative perspective is that these distributions are heavy-tailed, meaning that they have this heavy-tail, which represents really, really strong connections. And presumably, these very, very strong connections are important for something that the brain is doing, or else they wouldn't be there. And can you get that from cutting a brain? Or is that just simply counting the number of synapses? Is it that simple? So what we're all we're doing is taking these datasets at these experiments, experimenters have worked a decade to get, thank the God, for them. And all we're doing is asking, yeah, how many counting the synapses just between two neurons? And then you can look at a distribution and ask, what is the average number of synapses between two neurons? And then are there some that are much, much, much larger than that? And that's effectively all we're doing. It's very simple to see these distributions in these networks. Is the goal here to save? If we could build a model, if we could take these relationships and plug them into a model of a network and apply the network science, which gives us a shorthand to sort of do some of this stuff, is this sort of dream here? You could build a model of a brain that would, you'd see the patterns of firing would be consistent with what we see in experiments where people are actually watching what a brain is doing. And that way, then we know we've got us. We've built it. Might only be for a flip-fry or a, or this worm, but we've got the basic building blocks. I think that there's multiple goals. I think here, when we're looking at the structure of these networks, for me, there's kind of two central goals. One is what you described, which is now that we know the wiring of the connections between neurons. Could we, with all of our knowledge about how neurons behave? Could we build a model based on the connectivity and based on how we know the biology works, of how neurons work? Could we then simulate a network on a computer that looks a lot like combining all of our knowledge, can we build something that looks like what a real brain looks like when it actually fires? So do the patterns of activity actually look the same? And in fact, one of these nine papers that were published about the fruit fly brain did precisely this. So they asked, can we put a model on top of this big complicated network of the entire fruit fly brain? And they call it a digital twin, I should say. So then you can simulate this network on your computer. And in some sense, you have a digital twin of this fruit fly that we killed for the sake of science. And you can ask, do we see patterns of activity that are similar to the types of things we see when we actually observe activity in the fruit fly? You can go further and ask, now that you have a digital twin, you can do things that you couldn't do in a real fruit fly. So you can say, well, what if we start to rewire things? What happens? In the fruit fly, you can't do that. You just have what you have. Now that it's on your computer, you can change things in any way you want. You can tune different knobs and rewire the network and ask, what are the effects on the rest of the network? I think this is one large scale goal that I think is very important. Another goal is yeah, to understand the physical network of wires in and of itself. So to understand what are the structures that are there and how does the brain wire itself in these highly non-random ways? Because sort of an overarching theme in the study of biology and neuroscience in particular is living systems like to do the easiest thing. I think we all can agree with this. You're really a chair. There's a personal level. Yeah, yeah, the personal level. There's two options. They both get the job done and one is easier than you should probably just do the easier thing. And in the same way, if you're going to wire up the brain, you could just do it randomly, which would be the easiest thing to do. You don't have to expend a lot of energy to make sure you're getting precisely the right connection to correct. Then if you could do that, then you probably would because it's the easiest. But given that the brain isn't doing that, given that you see these very highly non-random features, like these heavy-tailed distributions of connection strengths, with these very, very strong connections are highly non-random, you can immediately ask, well, how could the brain wire itself that way? And what could these features, these very strong connections be useful for? So that's kind of a separate set of questions that you could ask. I presume the real value of that, that's actually even more of that. Because the model of the first brain just tells you you've got that model right. But this is giving you the basic, the first principles of how brains work, which then I presume you can go and apply to much bigger, more complex brains like our own. Yeah, and I guess here there becomes a matter of taste about what kinds of problems you like to study. So the my taste is studying problems where you can really understand, well, because it's hard for me to keep many things in my mind at once and I have a horrible memory, it forces me to study models and questions that are so simple I can keep the whole thing in my head at one time. So doing the thing that we talked about before, designing this digital twin is, you end up with this vastly complicated thing, we're like, no single human can understand what that model is doing. That's why we have to simulate it on a computer. And that's a very useful thing to do. And it's important for people to do that, to check to make sure that all the things we think we know are correct. And if they're not correct, then that means that there's something we're getting wrong, and we need to go in there and understand that. But there's another side of the spectrum, which is where I tend to spend my time, which is trying to come up with the simplest possible explanations for the most interesting things I can try to understand. That's what I try to optimally balance simplicity and interestingness. That's what I try to do. And on that note, Christopher, thank you very much for being on the show. Thank you very much for having me. This has been awesome. Thanks for listening to Simplifying Complexity, where we look at the key concepts of complexity science with expert minds from across the world. Concepts like emergence, self-organization, adaptation, networks, scaling, tipping points, and much more. This podcast was produced by Brady Heward and Weveland Creative. To make sure you don't miss an episode, be sure to subscribe to or follow the show in your podcast app. I'm Sean Brady, and I'll see you in our next episode. [BLANK_AUDIO]