 Hi, so I changed the title because I think it's fitting more about what I will speak today. And I hope that you will enjoy my French accent all along this presentation. So I will first present what is trail formation. And then I will explain, I will present some experimental results and show you what I mean by marginality. So trail formation occurs when you have a non-explore area, and you have a first individual going through and leaving some attractive trail behind it. And then when the second individual come, it has a higher chance to follow the same trails and leaving his own attractive trails and so on and so on. And you have path formation. The master in path formation ends. They are using pheromones, which is attractive substances, attractive for the nest mates. And they use it to deploy it to transport network and run their nests, which help them to efficiently explore their environment and to optimally exploit it. For instance, here, they are finding the shortest path between the nest entry and the thought source. To understand this phenomenon, we need to really understand what are the individual rules here. And there are a lot of studies carrying on here. But there are two problems in that phenomenon, which make it hard to study. The first one is that pheromones are really hard to detect, especially if you have a large group in a large arena. And what we usually do is that we use the previous passages as a proxy to estimate the pheromone concentration. The second problem is that when you study such animals in large groups, you have tracking issue. You lose the identity of your animals. And also, you have a high frequency of direct interaction. So when you observe a behavior, you don't know if it gets influenced by direct contact information or by pheromones. So to overcome these two problems, I have done several experiments based on Tingen end. So you have only one end in the experiment. And I focus on today, I will speak mostly about characterization of the trajectory of an isolated end. And also, I will answer to a very simple question, but which has been never addressed, is a single end influenced by its own pheromones. So here is my experimental setup. You take an end. You put it in an Argentine end. You put it in a triangle-shaped maze. And you just record its trajectory for two hours without food. This experiment is easy to replicate and to track. So I have about 500 of precisely tracked trajectory. So it's a behavior of an isolated end, like I said. And what is interesting is that the trajectory is confined. So the end is visiting repeatedly the same areas. And also is facing frequently binary choice when it is leaving the chamber. So all of that will help to address the question I have asked previously. And so I represent two experimental results. And these results provide marginality. The first one is on the end activity. So the hand has active period when it's walking and unactive period when it's stopping. Just to let you know, I have found that while walking, the end have a constant probability to stop. So nothing new. What is interesting is the stopping time distribution. It's a parallel of parameter 1. So just for the intuition, a parallel means that the more the end stops, the less likely it will walk again. So there is a self-implification in the stop process. There are a lot of really interesting features in the parallel for the end. But today, I will stress on the marginality of the value 1. Because if the stopping time we're following a parallel of parameter greater than 1, the stopping time will have a finite mean. So it means that you can predict in average how long the end will stop. And if you had a look to the evolution of the proportion of active time of your end, it will stay constant. But if the parameter is less than 1, stopping time will have a non-finite mean. So it's not really easy to get it intuitively. For the end, that means that you cannot predict how long the end will stop. And you can observe very long, long, long stopping time. And also, if you had a look to the evolution of the proportion of active time, it will decrease over time. So activity will decrease. So that is my first experimental results. The second one is marginality, again, in the binary choice. Like I told you, when the end is exiting the room, it's facing a binary choice, right or left. And we have a look to that to see if there is a reinforcement in the barrier of the end. And so we will see if we can observe path selection. This means that after a while, the end when it's exiting the chamber will only use exiting by the same corridor or observe the uniform traffic. It will use both the same way. Or do we have something in between? So before to show my experimental results, I will present to you the model I have choose to model with binary choices. So you consider that the end is leaving a quantity b at its choice between the connection of the chamber and the chosen corridor. And then the probability for the end to choose, let's say, the right branch, the right corridor, will depend, of course, of the weight. And this weight is expressed with the total ferroman quantity left here. So it's b times the number of passages. And also I add a coefficient alpha, which is fair to model the end sensitivity to the ferroman. So the role of the parameter alpha b in the model. So to understand that, we just follow the evolution of the proportion of passages of the end through one of the branches. So here it's right. One of the corridors here is right. And here are sketches of this distribution. The two parameters can have two effects on the model. The first one is path selection. When they are high, if you have a high quantity of ferroman left or a high sensitivity, so you will observe path selection. They will contribute to a path selection. And the contrary, if they are low, you will observe a uniform traffic. But they don't affect at the same time in the convergence of the proportion. B will have a short-term influence. It will influence only the first choices, which is totally opposite to alpha, which has an asymptotic effect. And alpha here will be the only one to decide which behavior we will observe eventually. If alpha is greater than 1, we will observe a probability. One path selection. And if alpha is less than 1, we will observe uniform traffic. So here it's important to retain that choosing the value of alpha different than 1 fix what you will observe at the end. So it's deterministic. What is interesting here is the marginality. So when alpha is equal to 1, it's B equal to 1. Here, alpha kind of disappears. And B is the only one to decide what will happen eventually. And it's no more deterministic. You will have a certain probability to observe path selection and a certain probability to observe uniform traffic. And tricking the value of B will change this probability. If B is less than 1, you will have a higher probability to observe uniform traffic. And if B is bigger than 1, you will observe the contrary. And when B is equal to 1, the marginal value, you will have the same probability to observe path selection or uniform traffic. OK, now let's go back to my experiments. And of course, the question is what are the good value alpha B to reproduce what we observe. And I claim that we should take alpha and B close to 1, 1. Why? I will present two arguments based on the data and some modeling. The first one is that the binary choice probability is following the verbals law. So I will not explain what you see here, because we already spent time yesterday about that. So what it means here is just that the probability to choose the right branch depend linearly to the binary choice probability. So here it's not enough to decide that alpha and B is close to 1, because we could maybe find over parameter value who provide such a linear relation. So to explore the parameter space, I had a look to the diffusion. So what I mean by diffusion? You take the triangle divided in six zones. And you don't take into consideration what is happening in each of the zones. You don't just consider that the end is jumping from one zone to another one after some waiting time. And then you unfold the space so that if the end is always jumping in the same direction, you consider that it's converging to infinity. And whereas if it's jumping between two zones, you consider that it is staying close to 0. So what is the experimental diffusion? It's that. So here you have the distribution of the end on the unfold space for several times. And here you have the variance over time. So it's a diffusion because the variance depends linearly on the time, so that is not new. But what is really interesting is that if you put a log on the y-axis, you see that the distributions are exponential. So that means that the diffusion is not Gaussian. So that it's a clear evidence that we have a reinforcement. Because if the process was memorialized, it would be a Gaussian. And it's not what you observe. We observe exponential diffusion. So now the question is, which value of alpha B I have to choose to reproduce this specific diffusion? And so for that, I have done simulation by fixing different value of alpha and B. Most of the time for value of alpha B, I observe diffusion. So the variance depends linearly over time. So good. And here is a diagram of B and alpha. And I plot here the result of the simulation for a different value of alpha and B. Here you have alpha and B equal 1. Here alpha and B less than 1, and et cetera. So you can observe that we have a diffusion which are not exponential. This one on here also, which is Gaussian. But we also observe exponential diffusion on that for several values of alpha and B. So to efficiently explore the parameter space, what I have do is that I have, for several values of parameter value, I have simulated two hours of experiments. And then I have fitted the resulting distribution of the ends with an exponential function. And I have reported the quality of this fit on this heat map. The darker it is, the better the fit is. And you can see that we have a whole region of good parameter value, which will produce an exponential diffusion. What does that mean for the end? So this region relies when the parameter alpha and B have an antagonist effect. For instance, this extreme case is when the sensitivity is high, alpha is bigger than 1, and the quantity of element is low, B less than 1. And the same here is the contrary here in that region. So if you think about the end, it's a waste of energy to take alpha and B in that region. Because it means that if you have ferriments, which if you lay a lot of ferriments at each passage, this means that you have to be really poorly sensitive to this ferriments. And the contrary, if you leave a very weak quantity of ferriments, you have to be really sensitive to this ferriments. So that is not smart for the end. And like scale down the flexibility of the Bayer view. And of course, the marginal value 1, 1 is inside this region. And I think of this marginal value, it's a suicidal compromise between the two effects. And so that is that argument to take alpha and B close to 1, 1. So to conclude, the answer is yes, ends are influenced by their own ferriments, but by a marginal way. And we have found marginality, like I told you, in the end activity between path selection and uniform traffic, we have, oh, there is a problem. And I have also found some marginality between constant activity and decreasing activity. And I didn't have time today to explain another sign of marginality I have found in the still magic Bayer view between nonlinear reinforcement and no reinforcement at all. And yeah, so thank you to have listening. And I want to thanks my PIs and collaborators who have helped me in this project, and notably Gitterolas and Yuxiate. And also I want to thank Offer Feynman, who welcomed me in his team since one year. And I hope that the next time that we meet, I will present to you my work with Offer. Thank you.