 Okay, so first I'd like to thank the organizers for putting this meeting together and for inviting me And so today's talk is going to be some sort of a follow-up to Alexandra's talk yesterday And I will exclusively be talk about analysis of data that was collected by the group in Rome That and the data that Stefania explained how to obtain beautifully in the in the previous talk and This was what I'm going to talk about was also of course collaboration with Alexander Valchak And also with Francesco Ginelli at Abadian University So but first before going into Into my talk, I just wanted to to give a first slide about Maybe emphasizing the degenerality of collective behavior and especially of emergent collective behavior Not just in animal groups, but at any scale in biology. I think that's a useful thing to remember So there's you know, there's collective behavior even at the molecular scale if you think of allosteri for instance Corporating banning and then you know at the cellular scale like a neuroscience is an example of collective behavior of many cells together Or multi-cellularity problem and of course animal groups, which is what we most of us are talking about during this meeting So the the system again to talk about is a flock of stalling So this is the movie you already saw yesterday So maybe you'll have to stare at it so so much and this is the same slide you saw yesterday So this is the data that was collected by in a group of Irani Jardini and Andrea Cavagna in the past few years and They managed as you saw in the Stefania's talk to reconstruct the 3d position and velocities of up to you know thousands of birds in the same flock and And the amazing thing is the amount of coordination, but not just the amount of coordination, but also the wide length scales over which individuals are correlated with each other and When you see this as a physicist your first reaction is wow, you know this this reminds me of something and it reminds me of interacting models in in statistical mechanics the prime example of which is The ising model so you know it's this is a very old idea is the idea that you have spins that want to point in the same direction and at equilibrium Even if you're you have just have local interactions with the nearest neighbors You still get spontaneous mechanization of the entire system So you get a breaking of the symmetry and so the you know The idea that physicists have when applying this to biological systems, and I want to emphasize again that Physicists have applied this idea to many of the systems than flocking and and the collective behavior Is that maybe you can think of this as a paradigm for thinking about many interacting agents of any sort Put them on the network of interactions and maybe you'll get collective behavior in the same way you get spontaneous Mechanization so you can put for instance flocks of starlings on this and you know this idea sounds a bit maybe a naive You know say okay. Yeah, we're physicists. We know better than everybody else. So we just apply our tools But in fact, you know, you can do this construction explicitly and this is what this is just a summary of x and y stocks yesterday By using the principle of maximum entropy where you just try to have a probability distribution over the orientations of velocity of many birds in the flock and All you do is constrain the correlation functions between pairs of birds and what you end up with is precisely this kind of models of interacting spins that we familiar with in physics in this case is called the Heisenberg model and With this kind of thing you can actually produce the long range correlate the long range of correlations Simply from a from a short range of interactions So we're producing this idea from the Ising model that you get the the spontaneous order at the large scale from just the small scale and So so now what I'm gonna do is I'm going to try and first this construct this idea and and maybe explain to you why you know, maybe The doubt that fair criticism against this kind of idea and then you know, otherwise I would be I wouldn't be talking about it this way I will rehabilitate the idea at the end So what's what's the main caveat with this kind of approach is that it's purely phenomenological It's based on probability theory and maximum entropy, but then we give this interpretation in terms of physics of having a Hamiltonian But you know like a physicist have studied talking before you know independently of the Heisenberg model and They they had a very big different picture in mind. So let me explain this to you So so far, you know when I was just interested as an introduction. We implicitly assume a equilibrium, right? but there's a big difference between birds and and spins and this was realized of course, you know in the 90s as you will see is that birds They actually exchange neighbors the network is not fixed and this is very important for two reasons I mean first of all is that because it from the physics point of view with it brings it takes the system out of equilibrium So maybe this should question the approach of trying to have an equilibrium like inference procedure But even from the from the biological point of view like so, you know the exchange neighbors What is within this also entails is that maybe you know the the the effective number of interaction partners that each bird has is not the one That we we get from Instantly so let me explain this like if your bird basically as you you know, if you integrate information from your neighbors But you know if you if you integrate over some some timescale. So remember, you know what you learn from your neighbors before Then if you change neighbors over time Then you accumulate some information from from neighbors over a long time and therefore Over many more neighbors than you integrate at any given time, right? So it's basically a combination of evidence. You have a larger effective number of neighbors and from even from a physics point of view is very important because this means that you can actually Carry information physically So if you think about the alignment of spins for instance, basically the alignment is just done locally from neighbor to neighbor But all the spins are not you know, none of them are moving But here what happens is that not only you can transmit information by communication to the next individual But you also physically move around the network and so you can also Propagate that information physically and this from you know, in fact, this completely changes the phenomenology I mean the the behavior of this kind of system So this was realized in the 90s, especially by two and Turner in 95 and What they showed is for instance if you look at continuous spins in two dimensions, which is The sort of model one has in mind a continuous spin is just a continuous direction of motion But if you're in two-dimension There's a theorem that tells you that you cannot get all the in two dimensions You cannot get a spontaneous order of the kind I was showing for the for the ising model But if you include this this active Element the fact that the neighbors change then you you you know You basically rescue the order and you can actually get it into them and also changes the critical response and everything and you know these studies were basically Based on the study of the of the Vechek model Which is this one where basically each bird as a function of time is taking some sort of a special average of its neighbors To determine its next direction of motion So you see it's quite different model than the Heisenberg model so We thought about this and you know you see that here We we have what looks like two contradictory descriptions once one is an equilibrium description From which you can actually learn useful biology and phenomenology, but then you have this dynamical Picture which seems to predict different things even on the on the theoretical level So to to address this question. We decided to to go back to similar data and actually quite different data but Where now we don't just focus on the on the equilibrium properties which you may also view as static properties It's just single snapshots of the velocities of the bird and now examine the dynamics and try to learn the dynamics, right? So essentially and try can we learn something like the equations of motion of The flock or something of the sort as the Vechek model So of course to do this the first step is to actually record the dynamics and so this is what Defining I talked about So this is we analyzed You know the the flocks that she she or she and others are recorded So I'm not going to go too much into details there. This is this was already covered. So They managed to to get the identity in In positions or in three dimensions of all these flocks or all these birds in the flock and that way can reconstruct the collective trajectory of Foldables So this is a starting point. We we take these traces and try to analyze them to learn what are the equations of motion from the traces and So in doing this we we want to to stay a bit agnostic and follow the same kind of philosophy that we did for just looking at single snapshots and by this I mean that we We don't want to maybe assume the dynamics from the beginning But we want to learn it from the data from from first principles and And and to this we you know, we basically generalize the the idea that we had before But now instead of thinking of probability distribution over all the velocities of individuals in the flock we think of The statistics of all possible trajectories. So the collective trajectories of all the birds in the flock at all times It can you can do it this way. It's like it's like a network of interacting Individuals, but they also interact with themselves at different time slices, right? and So we describe, you know by probability distribution over the states of the velocity of the orientations at all times and In physics we call this in actions is where I'm writing this way It's just a fancy word for the logarithm essentially of of this probability distribution So we'd like to learn basically what this action is from the data, right? So again, you know, how do we do this? We need to make some assumptions and what we will use the same exact same trick as before the trick of maximum entropy so but now So we want to maximize the entropy of the distribution over trajectories But now we want to put constraints Not just on the correlation functions of pairs of individuals at the same time But also we'll go cross correlations. So the correlation function of one individual at empty with another individual at empty plus one and if we do this the the kind of Action that we get looks like this. It's just using the same trick of maximum entropy So you get an exponential form and here you get these Lagrange multipliers. They essentially enforce this constraint, right? So called J1 and J2. It looks like a complicated Formula, but it's essentially the same thing as before except that we added this link this links between neighbors across two different time points, right? Can view it as a we unfold the network in the time direction and By the way, this is also called maximum caliber when applied to this time direction. So Now we that we've done this Okay, so here I have the convention of TT plus one and then I'll move to you know T plus delta T, but you can get it's the same idea so we have this maximum entropy principle with the constraints on these on these two kinds of observables and The first thing is that one can show In a spin spin web approximation So the spin web approximation was already explained by Sandra yesterday But let me remind you what this means. It just means that the the the direction of flights the SI which are normalized vectors Are all very close to the common direction of flights of the entire flock. So we can basically You know summarize There's viable by how much they differ they deviate from from this main direction. It is this and this new Variable which is equivalent to s would just be called pi. Okay. It's just a deviation from the main direction and So the spin web approximation just means that these Fluctuations are small which is equivalent to saying that the flock is very polarized Which is indeed the case in the data that we've been analyzing So in this approximation basically and this is an approximate equation You can show that the maximum entropy distribution. I was showing before is equivalent to Stochastic differential equation or stochastic Recursion equation where each bird is basically taking some sort of weighted average over You know all the other birds with some weights mij given by this matrix plus some noise and where the noise can be is delta correlated in time, but can have some correlation structure and both m and C I just transformations of these two Matrices J1 and J2 which were the lack of multipliers Enforcing the constraints Okay, so here already you can see that this looks very much like a social force model I will try to each bird is trying to make some sort of an average of its neighbors So on the technical side, I mean the way we show this is because you know the whole process is a mark of chain and In the spinner approximation everything can be considered Gaussian because we just expand to second order And so it's just a mark of God, you know, it's called Gaussian process in some cases with collective Gaussian process. So we can just write explicitly the the form of the conditional Probability of going from one state to the next and this is what gives us this equation It's okay, so so we we've already shown and this would be turned out to be useful to have this kind of formula for inferring the dynamics explicitly But before we do that, we you know here we have a delta T which is somewhat arbitrary Parameter and you know, we will only know what to do with it So what we do with it is that we just send it to zero just take the continuous limit and And it kind of you know, it's a limit that sort of makes sense It's also it means that on top of the constraint on the same time coefficient function Now we replace this cross-collection function by the correlation function between The direction of light and how it changes in time It's a correlation between these two things and this will be our new constraints and the continuous limit And if you do this then the social force model I just showed Now actually looks like a stochastic differential equation Which has the same interpretation as before Where each bird is trying to adjust its its direction of flights according as a weighted average of its neighbors And if you look at this and you start it you realize it's exactly the same thing as the Vichyck model Only you can have arbitrary weights you can have arbitrary Correlated noise, but it's the same kind of structure so Okay, once we have this we still have a many things to parametrize in principle So we have this for instance the correlation of the noise Well, you know to simplify we'll assume that the noise is actually not correlated between different birds, right? That's a simplification This allows us to so we just say has some some delta. I forgot delta ij So it's basically determined entirely by this parameter t d here's just a dimension of the problem So you can forget about it and t is can be literally Interpreted as a temperature gives you the amount of jitter that there is when each bird is making this its decision another nice thing about this dynamics is that if you assume that the That the network of neighbors is constant and also later has its symmetric Then you can actually show that this equation has a steady state and It's in fact an equilibrium steady states, you know technically from the physics point of view and it's given by exactly this Which is exactly that has a model I was I was presenting when As a consequence of the maximum entropy principle on single snapshots, right? But here you see here. There's a big this is there's a somewhat important difference Which is that now we have explicitly this temperature here, right? So when we're doing the static in France or the equilibrium in France, we did not see this t because it was basically Folded into the J matrix was here in principle from from this description. You can distinguish the two things We'll see that we can actually do this we can actually do this and for this this thing Okay, so I already said how to parameterize the noise now We're also going to make an assumption about the nature of the interaction of the interaction matrix so this interaction matrix Jij and To do this we will make an assumption that's that's been somewhat That's been motivated by by previous work We'll assume that the interaction matrix is such as a decaying function of the order of the neighbors that you're interacting with So you interact with your first neighbor with the strength you interact with the second neighbor with this You know a bit a bit less with your third neighbor a bit less So you number your order your rank order your neighbors like this and then you you determine The strength as a function just of that rank and not of the distance or anything else just a rank Right and was shown in the context of static or equilibrium in France That's this is actually what you you know you get an actual exponential decay. You can see it here on the log scale Very nicely. So that's what we'll assume here We assume this form but of course once we have this we We also know we left with two parameters with J So J can be interpreted as the interaction strength and NC Would be interpreted as the interaction range. So it tells you To to what you know up to what neighbor you really effectively interact So that's it. I mean that the model is not parametrized So we just have three parameters the interaction range interaction strength and the temperature that was considering the noise that I was talking about before and Now that we have that we we just use maximum likelihood fit which can be shown to be equivalent to solving the maximum entropy model you know using some Some integration methods that I will just briefly go over now. So Let me explain why we need this method in the first place So if you look at trajectories of this of these birds in the flocks You notice that they often have this kind of oscillations, right? And that's because you look at the center of mass or the barycenter as I was saying and If you if you do this because the the birds flap the barycenter goes up and down, right? So you get this oscillation and and because of that It's somewhat ultimately limits the effective sampling rate with which you're taking different snapshots and limits precisely to about Sorry, it's ten hertz. It's point one seconds. Okay, there's a mistake here so the problem is that With such a long time the the usual integration methods that we use Which is all those methods is is likely to be in precise. So in instead what we do is that we integrate We linearize the equation And we integrate it explicitly as a function of time between t and t plus delta t Okay, so this is done here and this is only valid if you have a fixed network But it's it's it's exact if you assume a fixed network and you if you're in the spin web approximation So just more technical details when once you have that you have an explicit expression of the likelihood and Then this is exactly this is this thing that we maximize with respect to our three parameters and see giant, right? which are all hidden here in these in these formulas t is here j is here and C is here So we do this maximization and we get an answer so but first before Actually talking about the application of this to the data Well, we first wanted to test it on synthetic data to make sure that the method actually works so You know, we just simulated The process we want to infer right and then we try to infer back So here we put the birds in the in the box to to avoid any any problems with the periodic boundary conditions And then we we take the data as if it were Experimental data and we try to infer back the parameters of the model And so, you know, this this is supposed to show you that it works very well but also it's it's it's the important point here is that You know, we do with different value of the true interaction range and see where we get back So it works well and we do it here for two different sampling rates So delta t here is the sampling rates with which we actually know It's not the rate with which we we do the simulation, but the simulation is done in continuous time But then when we do the inference we only allow ourselves To have one snapshot every second here Right So if we use Euler's methods or his method is just integration, you know, simple integration rule You can see that when this delta t the some the sampling time is large You you're gonna get you're gonna make a big error on the inferred On the inferred range, but if you use this exact integration method I was just telling you about then that case you get the right answer or most no matter what delta t is Okay, so this is to show that you know, it's not not it works and it's a useful step to go beyond Euler's approximation then Still on synthetic data we can ask Okay, so we're gonna get an answer for NC for J and for T and Can we make a comparison with what we would get from just so this inference here was done on the as I said on the On the dynamics so trying to learn directly the equations of motion but can we compare to what we would get if we did the so-called static inference or the equilibrium inference and this is what this gives and You can see that the the answer is actually depends on on the parameters of model so When the interaction strength is fairly large You see that the static and equilibrium inference give pretty much the same answer So equilibrium here is it's just doing inference on on the single snapshots Whereas the exact inferences is actually doing it on successive snapshots So in that case you get The right answer, but now imagine that the interaction Strengths is a bit lower in that case You can see that when you take the equilibrium inference you completely overestimate the interaction range And this is exactly what I was telling you about before which is that when you do this equilibrium inference You know, it's actually what you capturing is the effective number of neighbors and Here the effective number of neighbors is actually much larger than the instantaneous one. It's one you get this discrepancy so Why do you get the right answer in one case and not in another case? I leave that to a further slide, but it's just something for you to to think about Now, you know the the real question for us is if we look at the data and what in which of these two situations are we and The answer is that we in this situation, which is the good one in which the two things agree so here this is application to the actual flux that was just showing before and What I want you to really You know, what I really want to stress is that the two answers here are obtained by completely different types of information This is just obtained by single snapshot. They're just looking at the state In the statistics of things you see in a single snapshot Where's here? This is obtained by learning what happens when you look at one snapshot and then the next and It's really a non-trivial Prediction that the you know the two things should agree because it really means that a steady-state distribution here Can be can be predicted as a consequence of the equations of motion and again, you know here you know Here we I'm showing just interaction range the fact that we get it right So these are 14 flocking events and each of them has a different interaction range And there's something else that we can infer which is the interaction strength and temperature and that case where we need to compare Of course is j over t because I said it's folded into It's folded into the the j in the in the equilibrium inference and For that also we get a fairly good women's So why is it that after everything I told you that? The other equilibrium aspect the fact that the neighbors change that the effective number of neighbors should be higher How come that this actually still works on the actual data that the fact that this equilibrium inference gives the right answer so to speak Well to understand this I need to maybe introduce the time scales that are in the problem So you you you can think of I think that two main time scales One of them is how do birds? How fast do birds align with the neighbors? So imagine you have a bird here, which is misaligned with all its neighbors and Because it tries to it's it's it's driven by this social force. They would try to realign with the neighbors and Like this and it takes some time and that's what we call towel relax. It's a relaxation time of the aligners So that's one time scale and then there's a second time scale, which is How fast it takes for a bird to change its neighbors because it's Yeah, sorry the first one this recession time time scale as I argue in the next slide and a couple of sites It's essentially given by by J times NC. You can see it scales like one of the time So tar relax is given by the inverse of this But so the second one is how fast each bird Renews its neighborhoods So how long it takes to completely to have a complete turnover in the set of neighbors, right? and that's what we call town network and That we can directly read off from the data I'm not going to show how but just looking at the auto correlation function of the of the net of the network neighborhoods so What happens in this system in the actual data is that if we recall these two guys you see that the The timescale that governs the network rearrangements is much slower than the ones of relaxation So what this means that what I was saying about the birds keeping memory of the alignment and carrying it to the neighbors and everything That's not actually really true because they keep very little memory of Things over time. This is what this time relaxation timescale means. So it's but it's less than a second, right? So after, you know, less than a second they completely forgotten information from previous neighbors And since they changed neighbors of the timescale of 10 seconds by the time they've renewed the neighborhood the neighborhood They will have forgotten the information about the previous neighborhood So in other words, they always, you know, really aligned with the local neighbors that they have now and This is essentially this is really the the the fundamental reason why The the equilibrium inference works and now we can come back to this thing I already showed So this was the case where it didn't work and the reason why it didn't work is because in that case the tau relax So which is given by the inverse of this was of the same order of magnitude as the network relaxation The network rearrangements and timescale you can see the ratio of the two in this in this plot here was about a hundred Sorry in this plot was about a hundred to get a clear separation of timescales Why is in this case where you get the overestimation? It's when it was of the order of one Okay, so now you might might be worried that You know, does it mean that you know have to throw away all the nice theory of turner into that was introducing the talk with And answer answer, of course, is no we shouldn't do that Right and the reason is because you know the this relaxation timescale is not the only Timescale that that matters in fact to so to do this you you It's useful to to take the check like model and Put it on the square lattice, right? So forget about the name the neighbor exchanges because I already said that it's slow compared to the other timescales And let's assume that all our birds on the on the regular lattice in that case You can take some continuous limits in which this this pie, which is basically remember the fluctuation With respect to the main direction is is is driven by this equation and You know we can go in four-year space and write the so-called Green's function that describes fluctuations of this and from that you can actually write an equation for the relaxation autocorrelation function Which takes this form in the end right and so so forget about the details may be here But this tells you basically how much power there is in the fluctuations at each At each k where k basically gives you another man into the one of the distance, right? What they said when if you look at this equation You see that you have a one of a k squared and this thing actually diverges when k goes to zero So it means that you get so-called infinite power of fluctuations for very very long timescales right and When you when you end that situation Also, if you look at the relaxation when k goes to zero the relaxation here is infinitely slow so what happens in the you know in in a system like this is that Things the you know any perturbations will propagate to the entire flock, but it will take infinitely long to do so So if you think of this relaxation here the global relaxation time it would be given by the By the by the cutoff here in k, which is 1 over L And is given by this thing that diverges with the system size L is the system size in this case So the global relaxation timescale is extremely large if you have an infinite flock However, that's not what concerns us here because what we really care about, you know, I'm sorry Say it's is is really the relaxation timescale Locally right the relaxation timescale respect to just your neighbors and That's given by you know the same equation here But instead of putting cutoff at one of L just the size of the entire network It's given by one of RC where RC is the is the range in the interaction range expressed in Expected expressed in an actual length right in that case you get this star relax Which is pretty much what I was assuming before with this corrective factor, which is of order one What's really important here is that the relaxation timescale of the wish the inference is done, which is a local timescale Is fast enough right it doesn't mean that the scale of the entire system is fast and Okay, so So that's it. I mean we we you know at this point. We're fairly happy because we we've done a Dynamical inference we've done a static inference the two things agree. We understand why But there's a but there's kind of a problem Which is that we've been assuming some dynamics that's inconsistent with some other features of the data and let me just Tell you briefly before before wrapping up. Well, that's the problem So in the previous paper The wrong collaboration Showed that there was a ballistic transfer transport of information in the in the flock meaning so they saw that during turns Basically what this means is that when some bird is changing direction the way this information propagates Through the rest of the flock does it in the in a linear way in time like a wave, right? but if you think about the v-check model or the kind of model that we have this kind of model and you know the What this model would with instead predict is that when you have a turn the information Would basically decay as a function of distance and the reason for that is because if you take this equation and you Do the same trick of taking the continuous limit. I just explained before if you look at the structure of this equation It's essentially a diffusion equation. So the diffusion equation that you get you get a but you get a perturbation somewhere and then basically Diffuses very slowly like the square root of time and also decays So not exactly what we will expect from from from from these data And so to to rescue this you you need basically that's why we They showed in the first paper and then we we proposed a more detailed model in the second paper here is that you actually need to Go to a second-order dynamics and basically what so this is the equation But basically the basic idea is that it's like adding a mass or adding momentum to your system. So If you think of a of a of a dental see later We add this the second-order term which corresponds to this inertial term and that's what will allow the The you know to waste to propagate because if you linearize this equation the same way I did before then you get an equation of that form and So this is the damping as before but here now you see this looks exactly like a wave equation Right, so you get a second-order derivative in time and then it's known, you know from since the 18th century that this kind of equation can support wave solution and therefore Can explain why information propagates linearly and we know we showed this Explicitly by simulating this model and by showing that a turn can actually Propagate so that the entire flock can turn together instead of having just a few leaders and Do the turn and then the flock splitting So Okay, so says, you know that that's another piece of information for you and basically You know what we need to do at some point is is reconcile these two things. So these are my conclusions I mean the domain the main really message I want you to to remember is that you know in the real flux there's a separation of time scales between the the rearrangements of the neighbors and the relaxation timescale and this is why The this this equilibrium inference and dynamic inferences give the same answer and Then you know the caveat is that you know really to explain what happens in these flocks We would need a second order dynamics with this behavior of mass to explain inner propagation So what we need to do next is to do this kind of inference both in birds But also in swarms because as Andrea showed yesterday There's also evidence for this kind of second order dynamics in swarms and with that I thank you for your attention