 Dobro. Dobro. So, we are talking about collective behavior and the simplest example of collective behavior that you can observe in a huge number of animal species is group formation. So, some of these groups will move, some will stand in the same position, but animals of different kinds, from insects to fish to mammals, can form groups. And there have been quite many experimental studies that I won't have time to cover now, to review now, trying to infer the interaction rules in these different animal species. So, what we can conclude from these studies is that these interactions apparently are different for different species. In some animals we find that they respond to a set of topological neighbors. In other species it seems that the best model is one in which the individual picks a neighbor, follows it for some time, then switches to another neighbor and so on. In other species we have evidence for the importance of vision. In other species the communication might be based on acoustic communication or something else. So, all these species are different and the interactions are different. And this should not be surprising because all these animals have different brains. So, if you look at the mammal brain, like human brain on the left, we have, for instance, for visual perception we have the visual cortex involved, and then you have brain areas that respond to movement of objects and so on. If you move to a different taxon to fish, they do not even have a cortex. This visual processing happens in a completely different area that is a part millions of years of evolution from what you can have in mammals. So, there is no reason to find universal rules of interaction across these fish because they are different. However, there is one thing that we all have in common with fish, with birds, with other humans and so on, which is that we respond to sensory information through Weber's law. And you had an example before, if you have to tell the difference between two sets of dots, then you can do it easily when the numbers are small, like here, but when you increase the numbers this becomes increasingly more difficult. And this is something that is relevant for our decisions of everyday decisions, and because it is shared with other animals, we can imagine that it has a relevance also for their own decisions. So, if we have the same deal, one euro off, but in this case we probably go, we change our mind because of this offer, while in this case we won't change our mind. So, Weber's law is something that is shared across animal species, and that can have a relevance for behavior. Now, what has this to do with animal groups? Now I have to start again. So, we know that these groups stay cohesive for some time, typically for timescales that are longer than the timescale of movement of individual animals. Some individuals may actually leave the group and join it again and so on, but the group stability is longer than this internal movement. And we don't know a priori how they interact, but we can at least know what they would do if they did not interact. And this is the same for all animals. If they don't interact, they would just disperse like this, and this is something that we can characterize in some way, because when animals they don't interact, they just diffuse, normally like particles, like sugar when you put it in the coffee, they will just move randomly in random directions, and this is something that we can describe. We cannot describe their interactions, but we can easily describe the absence of interactions. Typically, one way of explaining this is by considering a series of boxes like this, in which you have different numbers of particles, but it can be animals, and then at every time step, particles have a certain probability of moving to the left or to the right, but also the particles in the neighboring boxes have the same probability of moving to the left or to the right. So, the movement of each individual particle under diffusionally is random, but this produces a net movement from higher concentration to lower concentration, from zones of higher concentration to zones of lower concentration. Now, this flattens out all the distributions, but this is something that does not appear in animal groups. So, if groups didn't interact, we would have exactly the same thing happening, but they do interact, and they must do something that exactly compensates for this movement. So, we know that they stay together, and what is the kind of interaction that acts at the collective level as an anti-diffusion that brings them together in the same exact way as they were before, before diffusing. So, it turns out that Weber's law-based perception does exactly this. So, if each individual animal interacts with other animals based on Weber's law, it will be, say, in a group of seven, and it will have a certain probability of transiting to a group of eight, to a larger group. So, the probability of changing for a larger group depends on the number at the destination minus the number at the source, and it is normalized by the local density. This is what they would do if they follow Weber's law. And if you have larger groups, it is the same. The same rule holds all the time. Again, the difference that you need to have is larger because the group size is larger. So, if each individual animal applies this rule they will climb the gradient and the probability for each individual of climbing the gradient is given there, but because there are C-S individuals that apply this rule, you end up with a flow that has this form. So, they go up, in this example, you had the 6 and 18, and I had this sort of coefficient for responding to the gradient of 1, 6, and they go up in a way that has the form of an anti-diffusion. So, it is exactly like diffusion, except that it has the opposite sign. And what is interesting is that we have two completely different processes. On one hand, you had completely random movement of individuals. On the other, process, we have an active decision based on this Weber's law that as a result of something very similar to diffusion, because they go up the gradient, it has the opposite sign. If they went down the gradient of concentration, down the gradient of density, it would have exactly the same sign as diffusion. And so, we can imagine that if these two processes happen at the same time, then they will balance each other. So, let's see in simulation what happens. For instance, if we have here a special distribution of animals in two groups, for example, and they move randomly because of diffusion, I can see that what happens is that the border of the group blurs, they start spreading on the sides. And if then each single individual applies this Weber's law, gradient climbing, what you obtain is that they start moving back and restore exactly the original distribution. With some noise, of course, because some information is lost during the diffusion process. But if the time scale is short enough, they will just restore the original configuration. This is a problem that is analogous to something that is known in image analysis. You can imagine that the same problem exactly in which an image is the density distribution where bright areas are high density regions and dark areas are low density regions. And then when you blur the image, what you do is that you just blur, you move around these brightness levels and you obtain a blurred image. And if then you just adopt this strategy of moving up the gradient based on Weber's law, what you end up doing is that you can restore, it doesn't work, you can restore something very similar to the original special distribution. So with this we can show that applying Weber's law has the potential of restoring a distribution that is corrupted by diffusion. Now I want to test something slightly different. So if you have animals that are distributed not homogeneously across space and it can be because they interact, it can be for any other reasons, it could be because there is some geographical feature as it is the case with this data where animals prefer to be close to the river here. So for any reason they stay in a certain position of space which is not homogeneous, it is not uniformly distributed, but they move randomly around. And this is what I do in this kind of simulation. I create maps of density and I make particles move randomly, but they need to respect the local density. They need to keep the same density. So they must stay much longer in these white areas and much less time in the dark areas because this is a density map and I want this map to be preserved over time. And this is the kind of simulation. So if we run this simulation several times, we can then calculate the apparent response of individuals to the local density, which would be something similar to in interaction rules with other animals. And we can plot this. So an individual that is in a cell with n neighbors has a certain probability of moving to another cell and j neighbors. And this probability is given by this map. If I cut this map here along the vertical lines, I can look at the slope of these lines because I have the probability as a function of the difference of a number of neighbors in j and i. And so I get this parameter a. And what I see is that this slope goes with 1 over n i. So basically, just the fact of imposing that the density distribution has to be stable makes appear a sort of interaction rule that is analogous to Weber's law. I'm not saying here that animals actually follow this rule. They might follow some other different rule, but for instance they could follow an environmental feature and not interact with each other at all, but the simple fact that the group configuration is stable will make this Weber's law interaction appear. So I'm showing two slightly different things. On one hand, I'm showing that if they use Weber's law for interactions, then they can compensate diffusion. And on the other hand, I'm saying that even if they don't, even if they use a completely different rule, as long as the group is stable, then I will observe something that is analogous to Weber's law in terms of interaction to density of other animals. So in this sense, we are making a link between something that was perceptual in the first place to a sort of collective behavior. And thank you, I'm stopping here.