 This always works for me. OK, good morning. I'd like to thank the organizers first for having me here. Great pleasure. So today, I'll be talking about cooperative transport in ANTS. And I guess this conference is all about the collective phenomenon in biology. And we know that there's many, many examples. We've seen some wonderful ones here where we have many similar, more or less similar entities interacting to produce some whole, which does more, gives more than the parts new kind of phenomenon. So we have many examples of this in nature, but the theoretical language by which we should describe this if it even exists, it's not too clear what it is. And one field in which we have some tools for describing large systems of similar particles is physics. So we have some decent tools that were developed over the years. And the question arises is, can we take these tools or these phenomena that we know from physics and try to use them to understand these collective biological groups? And so it's interesting how far we can get, but it's also interesting perhaps when we'll stop. What is the limits of this approach? Because when we look at these systems, we have a tendency to call them cognitive. Well, these physical systems are indifferent. And the question is, where does this arise? If we could just describe all these systems by very simple physical laws. So where does this at least intuitive difference, if it even exists, arrive from? So this is the approach that I'll be trying to convey in this talk. And I study ants. And to approach this question that I just talked about, you need two necessary things. One of them is because we want to look at the relations between the individual and the group. So it's good if, experimentally, you can track each individual and all individual ants in our cases. So there's one thing. And the second thing, since we're interested in this indifferent versus a cognitive, we better give the animals some kind of a goal, a clear collective goal. And these two things are typically not that simple to achieve in many biological systems. And luckily, the ants, it's a bit simpler. So actually, all that we have to do is go to the nearest grocery shop, get a box of Cheerios, take one of them, and then go outside to the field near our building, put the Cheerio near the ant nest. And now we get all this for free or for the price of a Cheerio. Because first, we've provided the ants with a very clear goal. They want to get this Cheerio from where it is to the nest as fast as possible, well-defined. We know exactly what they're trying to do. So there's one thing. And the second thing, another thing is this is a collective behavior. So no ant can move it on its own. It's very heavy for a single ant, very big. So we get a collective goal. Another thing, everything is just confined to this small area. We can easily see everything, track everything, each ant and all ants. So this is what it looks like after it's tracked. You can see the numbers here. So this is in the field. We don't have to tag them or anything. The ants' legs are pretty spread. So it's easy to differentiate between them. And they're going towards the nest. And the bonus is cooperative transport is a very physical kind of phenomenon. There's forces. There's motion. So these are the things we're used to deal with with physical tools. So pretty easy, maybe straightforward interpretations that we can use here. So again, we're trying to take our approach. We're going to try to look at the ants as particles and see how far we can go. So first thing you need to do when you move a large object is you need some kind of consensus in the application of forces. Because if you carry something big together and each person, each ant pulls in a different direction, you're not going to get too far. You're not going to be too efficient. And here we're going to see some microscopic evidence for this. So this is a side view. You can see that this kind of slant of the object, it's tilted in this direction. These ants here are pulling. Well, those in the back are lifting. So there is some kind of consensus. The ants are not just pulling randomly everywhere, but some kind of order in the system. You can think of two possible explanations for this. One is that we have smart and independent ants. OK, I'll discuss them in the next slide. And the second is very simple ants, like particles, but coupled. So let's look at the, sorry. And before we compare the two models, which I didn't describe yet, actually, the experimental evidence, which I'm going to pit them against, is the trajectory smoothness and speed as the dependent group size. So in the lab, we can prepare these artificial objects. So these are not cheerios. So we can cheerios are very uniform in size of different sizes. And check the characteristics of the motion. So we can see that the very large object, the blue here, go in very smooth lines. Well, the smaller objects, they wander around much more. So this is one thing. And second thing, the large objects move faster than the smaller ones. So these are two observations. First observations that I want you to remember. So now let's look at our other two models. First is smart and independent ants. So the single ant rule is, you're smart. You know where the nest is, grab the object, and pull in that direction independently of others. You don't care what the others are doing. Act as if you were alone. And then that's what all the ants are doing. And then we can calculate the forces, et cetera. And the collective motion is just overdammed motion. Of course, the ants are applying force, but it doesn't mean they're accelerating forever. So actually, the speed is proportional to the total force of all ants. And the consequence of this is wisdom of the crowds. So what is the total force that we get on the objects? It's just the sum of all forces. So it's just like averaging many, many forces. And the more ants we have, if each of them has some notion of where home is, but there is some noise in this, if each of them is making a mistake, then the more ants we have, we get a better and better estimate. Or the total force is a better and better estimate of the correct direction to the nest. So this is like the wisdom of the crowds, like the large numbers, if you want. The more you average, the more accurate you get. But you get the same kind of mean. So the mean stays the same. You get smaller and smaller fluctuations, but the mean stays the same. So this means that if we look at larger items, where more ants are needed to carry it, we expect to have same mean speed and smaller fluctuations. So this fits with one of our observations. We did get smaller fluctuations for larger objects, but they were also faster. We didn't get the same mean speed. Okay, so now let's look at the second model. So here the ants are simpler. They don't really know where home is. So the single ant rule is you're lost. You don't know where to go. Just feel the current direction. Feel the force at your point of attachment. And just go with the flow. So if you're in the front and it's coming towards you, so you pull also, you help the group. If you're in the back, lift. Okay, so we can capture it by this transition rate rules that ants measure the force once every few seconds and decide if they want to pull or lift, switch between pulling and lifting. Okay, so again, they tend to be lift pullers on the leading edge and lifters on the back edge, on the trailing edge. And the collective motion is again over-damped. So what do we expect to get here? So, okay, one more detail here. Sorry, I forgot to say. So what they actually do in our model is they take the force that they feel and if this force is over, okay, so this is the total force that they feel at the point of attachment, if this force is over some force that they have measured the scale that they have in their head, F independence, F end here. So if the force is over, stronger than this, then they will tend to align. If the force is very weak, compared to the scale that they have in their head, they will just act randomly and they can pull in the front, lift in the back, et cetera. Okay, so the consequence of this is that if the forces that an ant feel is much or independence, it's much smaller than total force, then all the ants will tend to align. Okay, so there's this force which is much stronger than independence. All the ants will tend to align and we get high speed and smooth trajectories. On the other hand, if the independence of the ant is very large, so they have a very, very high independence parameter in their head, it's very difficult to convince them, then each ant will pull in different direction, we will get low speed and irregular trajectories. Okay, so actually we can achieve these two inequalities either by playing with the independence level, the scale that each ant, that the ants have in their head, which is very difficult to do, but we can also play with the total force just by increasing the object's size. So it's like playing with the social pressure. So very large object, there'll be a large social pressure, all the ants will tend to align and here there's a small social pressure because the total force doesn't pass this scale. Okay, so we expect to get high speed, smooth trajectories for the large load and low speed irregular trajectories for the small load. Okay, so this fits better with this observations, first observations that I showed you. Okay, so just a quick summary here. So it seems that our observations fit with ants that act like stupid particles, they don't really know where home is, they just go with the flow and they're trapped in some collective state with a broken symmetry. Okay, so this is actually the same species of ants in different behavior, the ant entrance, we just put like a toilet paper roll around it, they can go, they can escape easily, they can go back, do whatever they want, but they're trapped in this collective state where they go around and around and this maintained itself for 10 days, I think, during which no ant died, because if an ant got tired, she just went back to the nest, another replaced her, but it's really a collective state. Okay, so now that we saw that we can describe, okay, we can hope to describe the ants as simple particles, let's go to this next stage where we actually show that ants are important as individuals. So we have two puzzles that are actually left by this model that I just described. First, it doesn't count at all for directionality, so it just says that wherever the object is moving, the ants will try to keep moving in the same direction, go with the flow, but this will not really bring them anywhere or to the nest. And second, the path that we measure are quite convoluted and this was very strange for us the first time, we saw it because we used to see ant trails that are so straight and nice and so we know that ants as a species and also these species have the propensity to make very straighter eyes, but here the trail has a lot of sinuosity and this was very strange for us. Why don't they take the straight line if a single ant can do so? And I think there is some explanation that kind of what I'll describe now kind of provides the same answer for these two questions. Okay, so how do the ants find their way home? So here we'll show a short movie, so the ants are carrying the stereo, they're lost and they're all trying to just go the flow, pull in the same direction, but they're going up instead of to the right. So because the path is convoluted, it happens a lot. So they have to somehow be corrected and let's see how it happens. So now they're going up, the nest is to the left and soon you see the rescuer, she's coming from the outside, well oriented, ants like any other insects navigate very well when they're doing free motion. She looks for a place to attach, attaches, and when she attaches you see a very sharp turn to the left. So now they're going very much to the left almost 90 degree turn, which is very strange because it's one ant pulling to the left and the rest are cooperating and going up, you would expect maybe a very, very wide turn. How could she turn them so sharply? Okay, so to try and understand what is happening here, how we got this sharp turn, we went back to the model I described before and we simulated it, okay? So we took these ants on a circular object and they attached all around and they follow these rules that I told you before over deciding whether to pull in the back, lift in the front, measuring forces, et cetera. So it's a very detailed model, but finally you actually have, it includes four free parameters, okay? So most parameters you can measure, like the friction or the size of the Cheerio, the mass of the Cheerio, et cetera, you can measure, you're left with four free parameters and because we have so many trajectories, we have many, by today many kilometers of such trajectories, you can get four different features and many more actually from the trajectories, okay? Like the speed as a function of the number of ants, the correlation distance of the trajectory, et cetera. So there's more than four features, so we can fit it pretty reliably. And what do we see when we fit the parameters to the natural motion, okay? So we fit them for a single size object on which we did most of the experiments and what we saw is that the fitted parameters actually places naturally sized loads, okay, which are about one, two centimeters, that's what they carry in nature, large insects, at the transition between a random walk and persistent motion. So very large loads go in very smooth trajectories. Okay, what we see here is actually the projection of the speed on the X axis, for example. So very large load, they have a persistent motion, ballistic, which you can see by these two peaks here, small objects have just something like a random walk, because none of the ants are cooperating, and natural sized objects are in the middle, okay? So they're not deeply in this area and not deeply in this area, somehow between two different modes of motion. Okay, we can compare this to the experiment, so we can see here that the curvature of the trails in the experiment for different load sizes and in this simulation, okay? So it fits, the parameters that we fit for one single size, they actually fit all sizes, and they place us at this transition somewhere between a random walk and ballistic motion. Okay, so what does this tell us? So transitions, we know them from physics, and we try to look in this direction, and to get this into a more physically understandable model, instead of looking at a circular load, we look just at this kind of rectangular load where the ant can attach just in the front and just in the back, to make things simpler so we don't have all the angles, and the ants can then choose if to be a puller or a lifter, according to the exact same rules as I told you before. Okay, so just a simpler model, and here they just move along a single dimension, so they can just move front or back. Okay, so just a simplified version of the previous model, and actually, even though this thing is moving, since there's translational invariance, it's actually mapped to a fully connected, Ising model at a thermal equilibrium, we can even write a Hamiltonian for it, which is something a physicist likes to do, like to do, and as we know for Ising models, you get this transition between order and disorder. So to get really the Ising model, what you have to do is take the number of particles to infinity, which is the kind of systems we deal with in physics, very, very large systems, and we can see that we take the number of particles to infinity of ants, and then we also have to raise this independence parameter at the same time in their heads, so we take them both to infinity together, and then we get an exact mapping to this mean field solution of the Ising model, where you get ordered states here, where the ants are not so independent, and they go to the group, and you get a disordered state here, and no speed, okay, so the order parameter is the speed here that you get. So this is for very large system sizes, it's clearly not what we have with the ants, we don't have a number of ants, but we can, using simulations, we can see also with the ants going as low as 10 ants, I can't even read it from here, you can see that we get the same behavior even for very, very low number of ants. So we get this transition between ordered state, where you get a collective speed, collective motion, and a disordered state where the thing moves very, very slowly, okay, and as we said, the ants are positioned, the real system is positioned somewhere in the middle, okay, not deeply in the ordered state, and not deeply in the disordered state, okay, so what could be the advantage of being there? So this kind of a criticality was observed in many other systems, in bird flocks and fish, and the idea here is that the system may have very strong responsiveness, okay, so we know that in physical systems, near the transition here, you have strong susceptibility to external fields, so if you apply a small external field on the Ising magnet, it will respond very strongly when you are here, okay, so what is the external field, how can it be here? External field is just a constant force that you add to the system, okay, so this constant force, how can you get it? You can get it by having a single ant, which is not following the rules, she's an informed ant, she's just pulling in a constant direction, she knows this to be the direction to the nest, she just pulls there, she acts as an external field, and we expect to have a maximal response to her since the system is sitting here, okay, so this ant doesn't have to communicate anything to the group to be able to get control like this, she might get the control just by the group being placed at this critical position and near this critical position, okay, so we add informed ants to the model and informed ants are ants that just attached to the load, so they came from the outside, they're well oriented, just attached to the load, and the rule is you know best, others don't know, so ignore them, just attach and pull to the nest, okay, and after about 10 seconds, you consider yourself lost also, okay, so this comes from experimental measurements, and then you become a regular carrier and go on to follow the rules before, okay, so you have the ant had its time of it's a five minutes of fame, 10 seconds of fame, and then she becomes a regular carrier, okay, so what does this give us? So in the simplified model, this is just an Ising model, if you take an end to infinity, you really get a divergence of the response, it's a red line here for small numbers or for a finite times, you still get a peak here, and the full model, we get the same kind of peak, and this is what we think we see here, so this ant is actually not interacting with these ants there, she's just looking for a place to attach, and once she attaches, because the other ants are placed at this critical position, she can get control of the whole system, okay, so experimental evidence that there is an optimal group size, okay, so if we have a very small load, the social pressure or the total force is not enough to impress the ants, and one ant actually does much better when the second ant attaches, they start to fight each other, they go right, left, right, left, and they can't seem to coordinate, if you have a large load, as we saw before, it goes in very straight lines, so very smooth, this will be the best, but the problem, if we confront it with an obstacle, so similar to the obstacles that Odre just showed, except this one's a bit more confusing because there is a small hole in here in the center, it will stay here for 20, 30 minutes, it cannot get the new information, okay, so the ants are very, very well coordinated with each other, all the carriers, but they're not open to external information that will tell them how to pass, and this natural size load, size of an insect, it walks in lines that are kind of convoluted, okay, so it doesn't walk in the straightest line possible, but when it hits the same obstacle, quickly gets information of how to pass it, we'll talk about this later, and they pass, okay, so I think this is a nice result here, actually what we see is transient leadership, so the object is moving towards home, each time a new ant comes, she has influence, she leads for about 10 seconds, but then she stops to lead, the object gets lost again, and then another ant picks up, and then another each color here is a new leader, and slowly the cheerio goes towards home, okay, so when you look at an ant system and you ask yourself, okay, is it a distributed system, is it hierarchical, is somebody in control, it depends on the time scale here, so if you look at a single, five second period, there will be somebody that is leading the entire group, but there's somebody, there's nothing too special about it, if you look at this whole period, there are many leaders, okay, so the leaders are democratically changing between them, and at a longer time period, it actually looks much more distributed in the non-hierarchical system, okay, so now we saw something kind of funny, that first I described the ants as particles, very simple particles that follow simple rules, but actually what this did is transfer the control to a single ant that was very knowledgeable and she knows how to get home, so actually by describing the ants as a physical system, we move the control to, okay, like particles very simple, we move the control to this single ant, which we can't really describe by physics, how she navigates and what she remembers, okay, this is a bit less like physical laws, but still maybe we still have a chance, because all these ants that are coming and in most of the examples we saw, they just want to pull to the nest and the nest is always to the west, to the east, maybe it's still simple, okay, so if you just model them as a simple particles that have some attraction to the east, maybe that'll be enough, so let's see how far we can still go and use this physics language, so what we did is use the invisible constraint and the invisible constraint, it's different than the constraint I showed before, which were walls, this is a piece of hair which is tethered here and tied to the load, okay, the movie was not supposed to begin, and what happens is that the ants go towards the nest and then when the hair becomes taut, they start this pendulum-like motion, okay, so we're going to look at the system, it's a different system than we saw before, it will help us test our model as well, okay, so we want to test our model on a different system, and we're going to use this one as a test to the model we developed on moving in open space, okay, so and the reason that we use this invisible constraint is we don't want the ants to have any other information than the nest is in this direction, okay, so we want the leader ants to be as simple as possible just to pull towards the south, okay, and this is what we get, okay, so first let's try to see if this really supports our model, okay, so this is, we have these two different alternatives, the smart independent with the averaging and the coupled particles, so in free motion that the puller ants in the front could be there for two reasons, either they're going with the direction because or they're going towards the nest, typically these two are more or less aligned, okay, so the trajectory is wavy but more or less goes towards home, so these are aligned, so it's really different, difficult to tell between these two, but when we look at this constraint motion, the direction of transport is different from the direction of the nest, right, they're doing this pendulum, the nest could be there and they're moving to the right, so here we could see are the pulling ants in the direction of the nest because they're all nowhere it is or are in the direction of the leading edge, okay, what we can see actually is the slant is towards the direction of transport and not towards the direction of the nest, okay, so again this is microscopic support for this coupled particles model, that's the answer, just going with the flow, okay, second kind of support is just to take our model as is, okay, so this is what we measured experimentally, these kind of shark fin oscillations, so we took take our model as is and we just add this piece of hair as a Lagrange multiplier that restricts it to go on the circle, so what happens when we do this for our coupled model we get very similar kind of oscillations and for the uncoupled model all the ants are trying to pull to the nest, so it actually stays near the center, okay, so they're just trying to pull there just so the more they are, the more they'll stay near the center, okay, and actually doing this we can get very, very nice fit, so this is the experimental data in blue and this is this simulation in red, very similar and also this is an analytical kind of a model that I will show you next, also fits very, very well, okay, which is another piece of evidence that our model seems to be correct and these oscillations are happening just do, not because any ants is realizing what is happening, okay, the ants are just following the same rules that they did in open space, okay, this is just a collective phenomenon, okay, or emergent phenomenon that gives rise to these oscillations, okay, so let's just try to give us very small physical intuition for the oscillations, so here again we take the simpler model, we have an object that has two sides, front and the back and it moves along this circle, so we can vary theta and we can write a very simple equation for this, so we have the informed ants, they just come in as a force towards the nest, so we g here like gravity and with the uninformed ants kind of give the object its momentum, so they keep on going left and right, so we can capture it by these equations, I won't go into it of course, and we can linearize it, so if we linearize it, we get an equation of the speed change according to the speed and we can see that if here, if this thing here is negative, so the speed that change varies negatively with the speed and this can give us, okay, so this happens with a, for a small number of ants, the number of ants here is n, for a small number of ants we can get a fixed point at zero zero, okay, so this is negative, if you get a speed in the right direction, it will pull you back to the left, so for a small number of ants, we expect it to be stuck at zero, but what happens if the number of ants becomes larger and then the ant starts actually to fill each other, the social pressure they start to cooperate, so what happens when the number of ants goes over this, so when the number of ants becomes higher, we get these, this is what we think, we get these oscillations in the experiment, so you can see in the oscillations, the speed is practically, okay, it decreases here but almost constant for the entire range of motion and then there's a brap flipped and it goes to the other side and then against almost constant, so the change in the speed is almost zero, okay, so if we write the change of the speed as a derivative of this energy function, okay, so we just took the formula we had here, change of the speed and we write this as a derivative of an energy function, so this is like a Lando energy, we can write this energy, so this is just the same formula as before but now we can plot this energy at different locations along the trajectory and this will give us intuition for the oscillations, so what happens when the object is at the center, the force of the spooler ants is just totally canceled by the string so that we can forget about them and what we have is just the lift, just the other ants, the uninformed ants just trying to cooperate with each other so they want to move together, they have two alternatives either left or right and the two alternatives are equally probable and you can really see it in this W function, in this Lando energy, you have these two valleys and you spontaneously fall into one of them, so now you got the system got a positive speed, it starts to move to the right, when it starts to move to the right, now the informed ants that are coming and pulling towards the nest, this gravity, now they have an effect, okay, because they're not totally canceled by the string and their effect is in the opposite direction, so actually what happens if you plot this function, you see that the peak here, the valley here, becomes smaller and then becomes even smaller until at some point it loses stability and the system has to fall to the other side, okay, so. So actually this will tell us the different load sizes, will give us different collective modes, the larger the load size, the larger the oscillations and this we can see in the experiment and in theory and in this modeling, this physics like modeling even predicted a third kind of, which doesn't work, doesn't work, sorry, a third kind of regime, which I had the movie here but somehow it skips it, where actually the object can do full rotations around its axis, so my time is completely over. Okay, so I'll go into, so what we saw here, okay, so unfortunately there's not the experimental evidence but it fits quite well, so what we saw here, again we can describe the system very nicely as a physical system, as they describe the ants as very simple particles, which we can track using physical tools, okay, so this was in a very simple system as well, with this invisible kind of constraint, okay, but now when we go to more natural systems, then we see that the ants actually profit from both the individual and the group, okay, so it's not one or the other, we need both the collective effects and the intelligence of the individual ants, okay, so natural environments look more like this, so this is a natural movie, which was not staged at all, so there's a leaf here and the ants are carrying this seed, so the ants easily pass from left to right but the seed can't and then they have to go and find a way where the seed can actually pass, so we kind of simulated this here, we bring the ants an obstacle, here the ants can pass easily, it's the fastest way to the nest, but the theory cannot pass, okay, and what happens, so it's a confusing kind of obstacle, they think they could go here but the real solution is from the side and actually we see that they solve this in linear time, okay, so how do the ants do this, so for this we have to add a new kind of information to the system, which is the scent marks that the ants employ, so the ants actually employ scent marks while they're carrying the loads and the scent marks you can see the ants here, she's touching the end of her gaster to the floor, she's applying a scent mark and the nice thing is you can see this behavior from the top, so it's a very stereotypical behavior, she walks a bit backwards, you can see it from the top, you don't need the side view, which means that even when you look at a single ant from the top, you can infer just from the movie where exactly she laid those scent marks, which we add here in animation, and these scent marks also appear during the co-op of the transport, so here the cheer is moving and we plot, this is a real experiment, we plot the scent marks that the ants have laid and the cheer are moving on top of them and we see that cheer lost the scent marks and a new trail formed here, cheer is moving on it, loses it again and a third trail forms here, so the trail is always deforming according to the location of the cheerio, so the trail is more finger and leading the cheerio at the same time, both leading each other. Okay, and the nice thing here, when we follow the scent marks, we learned that the individuals cannot solve these kind of problems alone, so when the individuals see these kind of obstacles, they actually mark towards the shortest route to the nest, which here is under the sleeve, where the object can't pass, so even if you take many such animals and you average all of them together, okay, even if a few animals will see, a few ants will see the way around, the average will be towards the center, okay, so doing this kind of wisdom of the crowds doesn't work here at all, the wisdom of the crowds is telling you to go forward while the answer is to the side, okay, so individuals and averaging individuals doesn't work at all, they have misguided information which is on the wrong scale, okay, the group also cannot solve the problem at all, if we remove knowledgeable individuals by blocking the ways from the side, we can see that instead of solving these obstacles linearly, it takes them at the group exponential time, so the group also can't solve it alone, okay, so both of them are not suffice, so how do the ants actually manage to quickly circumvent the obstacle, so I don't know if to give the answer or not? No, please give us the answer. Okay, so to understand this, we looked at all the scent marks that the ants lay near the obstacle, and what we found that the scent marks here near this opening, they're actually like a road sign that's pointing up, so these are wrong, but the scent marks, if you go here, more to the side or here, they're all to the right, so here the scent marks that are far from this entrance are to the right, okay, so the ants are marking the ants, know their way around, and they give these possible solutions, and this stays constant over time, so we have this error here, it's constant over time, it's always an error, but it's also confined in space, okay, so we have areas, the representation is wrong, but it's quenched in time, confined in space, okay, so when you have this, I'm trying to speed up, when you have this kind of rules, or road signs that are sometimes wrong, typically correct, sometimes wrong, what do you do? How can you use them to get to your destination? And what we find is the ants use this by applying this locally-bladed trail, which is, they apply each ant when it marks, it doesn't mark all the way to the nest, it just lays a small 10 pheromone droplets, for example, 10 centimeters, and goes back, so it's like a little arrow telling you, telling the object where to go next, okay, so the trajectory is not fully planned in advance, okay, just the next step each time, so this is one part of it, and the other, the group follow these trails, but not religiously, okay, so the group tends to go along these scent marks, but often loses them as we saw in the movie before, okay, so this is a different kind of trail than the one we typically know, which is very, very well defined and constant over time mostly, okay, so now why does this algorithm work? Okay, so here we want to get from A to B, so a very simplified version, mostly we have the correct instructions, but sometimes we have the wrong instructions, okay, so there can be few possible approaches, first, you trust, you say, okay, most instructions are correct, I'll follow them, the person who put them there wants my best, but if you do this, you'll get stuck here forever, so this is infinite time, second is disillusioned, so there's sometimes wrong, I'll just forget about them all together and just do a random walk, so this will get you to B, but in quadratic time, and third thing, which is kind of what the ants do, is follow with probability, you typically follow, but not always the next arrow, and on stretches of good advice, this gives you biased random walk, which is linear, on stretches of bad advice, it gives you exponentially long, because you're doing a biased random walk against the bias, but these stretches are rare, because usually the advice is for you, so together, you can analyze it, you get linear time, okay, so this is the reason why people have this dice in cars, okay, and actually when you follow, so I won't go into it because my time is out, I guess, but when you follow what the ants are doing, this is really what they're doing, so the reason that they're not stuck here and near the center is sometimes they don't follow advice and they go elsewhere, and then they pay for it, okay, so the ants here paid for it because here, they could have followed the advice and escaped, but still here, they chose not to follow it, and they go against their interest and back, but the price you pay here is minimal when compared to the price you gain here, and you get this linear time, so what I wanted to summarize or looking at the future is to try to say that ingredients for this collective cognition that we say we have cognitive systems is you can use this collective phenomenon in physics, can get you very far, but it will not give you cognition, at least for the ants, and for this, you have to add these navigational, competent, goal-oriented individuals, and you need information to transfer between these two all the time, okay, so in physical system, usually you can forget about the microscopical details, here, no, here, yeah, keep on being important even when you go to the group level, and then we want to understand, take this for the future and try to understand, okay, what can this give us? What is the class of problems that the ants can solve using this? And we try all kinds of different measures, so here you can see the ants trying to go through this rock field, the ants can easily walk straight, but what they're carrying has to take a very different kind of path, and we're trying to understand theoretically, okay, do we expect the ants to solve this? Don't we, what kind of puzzles do we expect them to solve and what we don't? So you can follow the trajectory of what they're doing until success, okay, so thank you very much, sorry for being over time.