 A great time here. And it's actually my first visit, not only to Trieste and to Italy as well. So overall, a great experience. So in one or two centers, what the general long-term aim of my lab is to understand all of the steps from sensation to action. That is, from the initial detection of sensory stimuli by receptor on an animal's periphery to all of the processing steps in the brain. And finally, for the biomechanics of the interaction of the environment, of the animal with the environment. So that's what we hope to achieve in the long term. And much of my lab's work is on flies. And one obvious reason is the fact that flies have the great genetic tools in the flies. But there are a couple of other important reasons, maybe equally obvious to some. But I think I want to articulate those. The first is the fact that although flies cannot read or write, they still have a large array of very interesting behaviors that resemble behaviors of larger animals. And underlying these behaviors is a relatively simple brain. And there are two points that I want to make here. The first one is that clearly compared to, say, a human brain, the fly brain is numerically much simpler. But equally important point is that it's still a substantial brain and has about 100,000 neurons. And when I decided to work on flies, one of the ideas was that a lot of the principles that emerge from a brain of this size are likely to translate to larger organisms. So that's how we think about the transformation from sensation to action and our choice of model system. In my talk today, I will focus on just on locomotion. And what I want to emphasize, the thing I want to emphasize about locomotion that has already come up a few times in this conference is the fact that you can think about locomotion on multiple spatial and temporal scales. At the most gross or large scale level, you can think of flies as a point object, sort of the Google map view of us. And where you can see the flies having either straight paths or taking these sharp turns in red, straight paths in black. Now if you zoom in, one realizes that if you look at now the body orientation here shown by this gray, the body orientation at many times in its trajectory is actually pointed almost perpendicular to the direction of movement. And so that adds in the idea of how is the body orientation coordinated with the orientation of a track. And finally, any movement, any change in either the orientation of the body or its translation in space has a concomitant interaction between the flies' legs and the environment. And so then you can add in the legs and these yellow lines just demonstrate when one of its legs starts the stance phase. That is when it touches the ground. And so essentially, at that spatial scale, one has to understand the physics of the interaction between the leg and the environment. And so in terms of understanding locomotion, so the basic questions that we are interested in is trying to understand the behavioral algorithms at these different spatial and temporal scales, how these algorithms are executed by neurons in the fly's nervous system, and sort of the role of mechanics in the fly's behavior. So these are the sorts of questions that we are interested in. And what I'm going to do in today's talk is present sort of two projects. One is aimed at the fly at a point object, the other sort of the nuts and bolts of locomotion. So let's start with the part one. And in this part, what we have been studying for the last few years is to understand sort of the structure of locomotion in sort of a small arena and how odors affect the fly's movement in this arena. And so this is just a scanning EM of a fly's head. And you have the olfactory appendages, the antenna, and the maxillary valve. And odors are detected when the receptors present on these olfactory receptor neurons or ORNs are they bind to odors. And that results in a change in firing rate or the rate of axiom potentials in these ORNs. And that's how odors are detected. And then the changes in activity is transmitted from the ORNs to the next layer in the brain. And these principles of order encoding, what's called order encoding, is well understood in the flies. And I want to point out two or three features of this. So in this schematic, these are the olfactory receptor neurons or ORNs that I was just talking about in the antenna and the valve. And each olfactory receptor neuron expresses a single member of a large family of receptors that not only decides which order it will bind to, but where in the brain it will project to. All ORNs that have a given receptor project to a single region in the brain called a glomerulus, where it makes connection with the projection neuron, which are second order neurons. And a large class of projection neuron to project to a single glomerulus, sort of setting up a one-to-one connections, the connection between ORNs and projection neurons. Now, most odors activate sort of the population of ORNs. And therefore, affect activity-improperty population of glomeruli and a population of projection neuron. Now, different odors will activate different but overlapping subset of ORNs. And so you can sort of conceptualize the order encoding problem as each order, in this case, say apple cider vinegar, being encoded by activity in a small set of ORN classes. So in this case, you have these seven ORN classes. The numbers indicate which receptor it expresses, which then labels which neurons are important for encoding, as well as which glomerulus and projection neuron. So you can set up this problem. So that's how orders are encoded. So any given order in the environment will activate some subset of receptors, which is its signature. And so when I started my lab, I think one of the problems that we were interested in is understanding the transformation between the set of ORNs activated and the resulting behavior. And to that end, we created this kind of arena where the fly is basically walking in the circular arena. This is a side sort of schematic. And the order either sort of fresh air or odorized air is coming from the top. And then you have vacuum at the bottom. The whole point here is to create a region of a precise concentration of order inside in the center of the arena. This annulus has no order at all. So creating a precise boundary between no order region and an order region. So here are the tracks of the fly in this kind of arena. This is before the order in the presence of apple cider vinegar, which is an attractive order. The fly spends much more time in the region with this order. And then afterwards, they scoot back and sort of hang out much more at the border. You can do this kind of experiment. Oh, in front of it, just to visualize the fly. So you can do this experiment on many flies, quantify how much time it spends inside this region as a measure of attraction. And so the design principles here would be clear from a lot of the things that Ring mentioned in stock. I think one of the things we wanted to do is have sort of manic control or stimulus. We know exactly where the stimulus is and how much it is. The second thing was also mentioned by Ring is sort of the ideas from Kennedy. Kennedy's work back in the 70s is that it's wrong to just focus on attraction. So we wanted to have a measure of attraction, but we also wanted to understand sort of underlying locomotion, how these orders is changing different locomotor parameters by using sort of deriving those parameters from these tracks. And so when we did the experiments, we quickly realized that orders change many different aspects of a fly's behavior. And these changes in behavior seem to be independent of each other. So one fly could run slowly, but not stop, run slowly, but not turn much, and so on. And so to capture this idea, we sort of created this, what we call multi-dimensional behavioral space. This is a very brute force analysis based on simply parameters we thought were important and asked whether orders change those parameters. So we don't have to go this in great detail, but the top row is all of the parameters that have to do with the distribution of the fly in the arena. The second row has to do with speed. The third row has to do with things that distribute how flies locomotor and distribute into runs and stops. Then final five parameters have to do with some aspect of turning. And what you see here is how two different orders change these parameters. And one of the points that this kind of plot shows is that different orders, both of them quite attractive, change different aspects of the fly's behavior. So some one orders affect these two parameters, but the other order does not, and these other parameters in sort of magenta are affected by apple cider vinegar, but not by tube uterine. So this kind of experiment and this kind of analytical framework gave us a lot of insights, such as the fact that these parameters are controlled independently, the fact that this control is dependent on the identity of the order, and therefore the identity of the ORS that are being stimulated by these orders. But there are obvious flaws or limitations here. The one is ad hoc parameterization. So we have no way of knowing that there is no 18 parameter that is all important. And then if I gave you the 17 parameters, you can't take the 17 parameters and draw a trajectory of the fly to sort of encapsulate what the fly is doing. So there's no way to get from these parameters to a behavioral algorithm. And so what I'm going to show you next is our attempts at creating a generative model of behavior to get at the algorithm underlying sort of behavior and its response to orders. So what we'll do is measure parameters like here, but then create synthetic flies based on these parameters and compare how close the synthetic flies are to real flies or to empirical flies. And because in the interim from the time that we created the arena to when we decided to make this model, there was large progress in fly genetics. One of the big ones was the fact that typically to activate fly neurons using optogenetic methods, like activating by light, people used a blue light-sensitive channel, and then people discovered the red light-sensitive channel rhodopsin, which is called crimson. And that made it possible to do behavioral experiments because the flies cannot see red light. And so it was a clean way to activate neurons. So we also decided to update our behavioral paradigm by replacing the order null with light. The advantages of using light instead of order is that the light, although we control our order very well, light gives you just a little bit of better control and reliability from day to day. The other importance and a bigger one is the fact that now you can essentially express the red light-activated channels in designer combinations of olfactory neurons. And therefore, you have much more control over which olfactory neurons that you're activating that you would not have the same extent with order. All right, so that's the arena. And this plot just shows that, again, the red light is limited here and then drops dramatically outside the center of the arena. So here is the behavior of a fly showing that there is a large change in behavior when you have this red light-activated channel expressed in a large set of ORNs. So this Orco-Galphor is driving US crimson where it's essentially driving this red light-activated channel in a large number of ORNs. So the idea is that when the fly essentially is outside sees no light and is inactivated or not activated and then it goes inside, it gets activated. And it lights that activation in this case and so sticks around in the center of the arena. Now, so what we can also do is essentially because we know the intensity of light as a function of space and we know the tracks of the fly, we can create the virtual stimulus that this particular fly is experiencing. We can then replicate the same stimulus in an electrophysiological rig and then record from the ORNs in a different fly in an electrophysiological rig and you get the ORN responses that corresponds to this kind of track. And so what we have in essence is both the moment by moment account of behavior as well as its neural responses. And so this is just an example of how this all looks. So what you have here is, again, the arena. The solid line is where the intensity of light is constant and then gray line is where it goes to 10% of the intensity inside. And here you are seeing the ORN recording, recording from the ORN. And these are the sort of sorted out spikes. Each sensilla, so these are sensillum recordings. Each sensilla has two neurons. So you're going to see two kinds of spikes. And in this, I think you should be able to hear some chirps. So those are, again, spikes. All right, so that's basically the data set. You have behaviors from a lot of animals. You have the moment by moment spike rates. And so now we will try to make sense of all of this. And I'll start by sort of talking about the generative model. So first we started with just modeling the fly's behavior before the light turns on. And that's shown here. It's just a simple model. Let's start with sort of flies walking around. And it's quite noticeable that even when the fly is kind of walking straight, there is a curvature to these parts. And that curvature is very important. And therefore we model that as a curved walk. And from the curved walk, either it can then do a sharp turn, or it can stop and turn, or it can reach the boundary. And at the boundary, we apply a boundary condition. Because behavior at the boundary is quite different from the behavior at the center. And each of these states can be modeled by two or three parameters. And essentially what you can show, so that's the actual and that's the model of this track. And you can see that the model does a reasonable job of encapsulating the behavior of the fly. And you can see that this is the RMS error is of the order of the fly length of the fly length. And so the bottom line is that this model, which is purely kinematic model, which assumes that all the fly is doing is choosing a speed and curvature and walking with it or turning with it until it reaches the boundary. And also its behavior is defined by speed, essentially, can recapitulate the behavior of the fly in the absence of the order. Now the next question we ask, will this kinematic model work to describe the behavior of the fly? How does the model get every? I think that all the model is doing is dividing the tracks into stops, turns, and the curve walks. And there's no trick to it. I think I can tell you what basic insight it gives you. But that's all we are doing. And then we are saying that within each segment, what matters is average speed and average curvature. And that gets you that close. And what it means is that the fly is going at the same speed and curvature over that track. And that's why it gives you a fairly good representation of the data. Yes, for each track, not for each point. So at each point is choosing the duration and the speed and curvature. So just so I understand, whenever there is a very short turn, you reset your model and say, OK, you've turned so much, now that's a new track. Let's see if you can hold that. Yes, that's right. Exactly. And so, all right. So that's, yeah, we can continue to discuss it later. So the next we ask that the kinematic model can explain a fly's attraction. Now, what that means is that from the 17-parameter study, we already know that orders change a lot of the kinematic parameters, such as the turn speed, run duration, like almost everything is changed by order. And what we are basically going to ask is, are these changes sufficient to bring about the changes in distribution of the fly, like spatial distribution of the fly? So what we do is, so essentially, you have your empirical flies. And we measure the kinematic parameters in absence of light, feed it to this generative model to create synthetic flies. And then we have one activation when you activate the lights. We again measure the kinematic parameters and again feed it to the generative model. This time we have tracks in the presence of order. And these are the red tracks. And what we are going to ask is, are the red tracks distribution in space different from those of the green tracks? And it turns out, so this is just examples. And you can see that one thing also to point out is that the reason we are showing two examples is to show that the empirical data is variable. The behavior of the fly, even when you have highly controlled stimulation, is variable. And there's some variability in the kinematic data as well. So these are just samples from the empirical data and kinematic model. And here is one measure of the distribution of the fly. And you can see by attraction index or the time that is spent inside the light zone. And you can see that the empirical flies are much more attracted than what the kinematic model flies. It's also important to note that changes in kinematics that we observe can also produce significant attraction. But that attraction is significantly less than that of the empirical flies. And this is another way to sort of look at this data. And what you can see is that the kinematic flies does not recapitulate the distribution of the empirical flies, implying that we are clearly missing something. Although the kinematic model is a good model of the flies behavior before the orders, it's not capturing some aspect of its behavior. And so what is missing is quite apparent from this kind of tracks. And what you see is that the flies, when they exit the light zone, they have a tendency to turn sharply right after they exit the light zone. And this can be quantified by plotting what we call decision density, which is the extra number of sharp turns that the flies take right at the order border. And you can see that there's a large increase in those turns that matches the gradient of the light. So this kind of behavior is not being captured by the kinematic model. Moreover, it turns out that this kind of behavior we will now capture with a parameter that we call border choice, where we increase the chance of the turn to match sort of this distribution. Now, there is another phenomenon that occurs that it's not easy to see here. But these turns are also biased. That is, a larger number of these turns tend to bring the fly inside the horizon. And we capture this by creating a term called turn bias, which essentially compares the angle between the radial vector and the initial direction and the radial vector and the final direction after the turn. And if theta 2 is less than theta 1 or final direction is less than theta 1, that means the fly going in an inward direction. And that gives you the turn bias. So we will next model this idea by using these two parameters, border choice and turn bias, which we refer to as decisional parameters. Not crazy about the name either. But that's what we are going to call it. It turns out that the interesting feature in the empirical data is that the first two turns after the fly exits the arena or enters the arena is what turns out to be important. And that's shown here. So first for the sort of the decision for the border choice, if you look at the decision density for the first two turns, you see this peakedness. And particularly outside, after the first two turns, basically it returns back to baseline. And the same thing we observe in turn bias. The first two turns outside have a very large turn bias of almost 100%, close to 90%, whereas the subsequent turns are at random. So that's the feature of the empirical data of the border choice and turn bias that we are now going to put into the model. Obviously, the before period remains the same. There's no change. And then after one activation, we are saying that there are not only changes in kinematic parameter, but also decisional parameters. And now see whether we can recapitulate the attraction. And the answer is that yes, we can. That if you look at both sort of the attraction index, as well as the radial occupancy of the fly, the synthetic flies now recapitulate the behavior of the empirical flies, implying that this seems to be at least a sufficient set of parameters to capture the behavior of the fly. Now, we look further into sort of the mechanism of the decisional parameters to ask what's causing the flies to turn right after it exits the light zone. And the first experiment we thought of is taking out one of the antenna because of work in many insects and in flies, as well as physiological work in the fly, showing that flies should be able to make inter-antanormal comparisons. We decided to take out one antenna. And that's the experiment. This is one example fly. And this is sort of the distribution of the fly when it has only one antenna compared to when it has both antennas. And at least at this level, it does not seem to be that the inter-antanormal comparison is a dominant mechanism. So the next thing we wanted to ask is sort of more empirically why is what is it that happens when the flies, to increase the rate of turns right outside the light zone? Is it are the flies turning more per second? And that's shown here. And incredibly, it turns out that that's not the answer. If anything, the sort of time that it turns after it leaves the light zone is, if anything, sort of larger in the flies with retina. So I did explain this before. The control here is for these light-activated channels to be active, you need to feed the fly with retina. So the control flies are basically perfect. They have the same sort of generating background. But they don't have the retina, so they are not active. And you can see that the increased turning is not because of increased rate in turning. And it turns out that this is just sort of an illusion of the fact that the flies dramatically decrease their speed as they are exiting the light zone. So essentially, as they're exiting the light zone, they're walking almost 50% slowly, if not more, just because of that, they have a higher density of turn right at the border. And so that's the biggest reason that they have this increase in turn density. Now, with respect to the turn bias, so next we look at what's happening with the turn bias. And so again, through a lot of other studies, what people have found is that the flies can have some idea of the environment. And so one of the ways to demonstrate that is as the flies exit the arena, unless they're going straight normally out of the arena, there is a preferred direction of turn and a non-optimal direction of turn and non-optimal direction of turn. And in other studies, people have found that flies can take the optimal direction of turn, showing that they have some idea of the shape of the arena. What we found, however, is that it's by chance. The direction that they're turning is by chance. And so here the reason that we see such a pronounced turn bias turns out to be the fact that there's a large increase in the angle of sharp turn. And so essentially this is the turn before and this is the turn of the first two turns that demonstrate this massive bias. And the later turns, although they are still higher, but they are much more closer to the baseline. And so essentially to conclude this part, I think that the first point is a kinematic, sort of kinematic in a uniform arena where there's no stimulus. A kinematic mechanism does just fine to describe a fly's behavior. Now in the presence of light, you have kinematic changes and you have decisional changes. And they together can describe the behavior of the fly. Furthermore, decisional changes result from the simplest sort of mechanism, I would say one can think of. In fact, we didn't think of this mechanism. We thought of all of the mechanisms that would be more complex, like osmotropotaxis or path integration kind of mechanism where you need to have some view of the environment. But really the mechanism is just slow down and turn hard. There's no, and it seems to be a very, very simple computation that the fly are doing here. Now, so we have done a lot of work at the neural end of this project that I'm not going to talk about today. And I'm very happy to talk about after the break. And at this point, I'm going to turn to the second project, which looks at the nuts and bolts of locomotion. And here what I'm going to talk about is our attempts to understand the physics of the interaction of the fly with the substrate when the fly is walking. And here our ideas were influenced by sort of this long history of work that has shown that despite the complexity in the movement of individual limbs or individual joints within those limbs, the movement of the animal itself is relatively simple and can be captured by simple mechanical model. And here is sort of the sort of instantiation of that idea using fly. So what I'm showing here is the joint position of the fly as it's walking from right to left. And you can see that they are fairly complex. If you look at the corresponding movement of a point on its body or on its thorax, which we use as a proxy of the center of mass, that motion is relatively simple. And so what I'm going to talk for the rest of my talk is our attempts to understand this, to get some simple mechanical model that will explain the fly's kinematics during forward locomotion. And so this approach can sort of divide into four parts. The first is to obtain sort of a high resolution account of the center of mass kinematics. So because the fly itself is very small, any changes in its center of mass mechanics, particularly that in the z-dimension is quite small. And we need to be able to resolve those changes to have any attempt, any hope at getting a good mechanical model. So first thing we did was to measure center of mass kinematics at high spatial and temporal resolution. And then the next step is to how do you take this sort of continuous stream of center of mass kinematics and discretize into steps? Now because the flies have six legs and those legs are not synced up all the time, so that poses some challenges. And so the next step of what we did was to find some rational way of dividing this center of mass kinematics into tracks. And that question is sort of intimately related to question of interlegged coordination, i.e. great. So that's step number two. And then we came up with a simple mechanical model that we are going to talk more about later that we think describes the center of mass kinematics quite well. And finally, we'll show that if you think of a walking fly as a point mass supported by three legs of its tripod, or three legs that are on its ground at any time, that springy tripod reduces to this AR slip model. So let's start with the experimental data. It's a simple sort of small chamber where the fly is walking around. And we have a camera that's looking both as side view as well as the reflected view through a mirror. And so that's sort of how the data looks. We can track both the movement of a point on its thorax as well as the stance legs as the location of the stance legs in each frame. And the important feature to notice here is these sort of nice rhythmic bumps in the center of mass Z dimension showing that, and these are sort of step-to-step changes in the center of mass height, showing that we can resolve the kinematics reasonably well to be able to fit a mechanical model. So that's the data. And the next step is to look at the gate. In flies, in particular, and other insect as well, what people say, sort of current view, is that tripod is the dominant gate at high speeds. And tripod is basically where you have these three legs, two legs of one side and the middle leg of the other side. They move synchronously. And in opposition to the other three legs. So that's tripod. And then you have the straight-to-port gate where the legs marked with these lines are the ones that move together. And they are sort of two versions of the straight-to-port. And so to try to sort of tease out the gate that we see, the first thing we did was just plot the stance start times. That is, the time at which each leg hits the ground on each step aligned to the first right leg hitting the ground. And that's shown here. And so that's the first right leg. So that's what all of the rest of the data is aligned to. And what you can see is that the legs of one of the tripods generally come together, whereas the other tripod legs also come together, implying. And notice that this is organized according to increasing speed. And you don't see sort of wild changes as the speed increases, implying strongly that a tripod gate is used across all speeds. Now, we did a lot of different analysis to sort of form up this conclusion. And one of the analysis was to come up with a new description of gate, what we call as the gate delay index. And what the gate delay index is, if you think of each leg cycling through stance and swing, then a gate is basically determined by the phase differences between these cycles. And so if I show, so there are basically five delays that determine the gate. And just for convenience, let me show you with these three delays, you can create a space based on this phase difference or delay vectors, delay. And then within the space, a given gate, like an ideal tripod, would be a vector. And then you can compute your experimental vector on a step-by-step basis. And you can ask how close to an ideal tripod this experimental vector is. And here is the result of that kind of analysis. What you can see is, and so what I'm plotting here is cost of that angle. So one meaning it's ideal tripod. And you can see that most of the steps are quite close to the tripod. And there are no sort of steps that are hanging around the two tetrapod lines. In fact, the real gate is what we call as either metatonal tripod, or you can just think of as modified tripod, where the three legs of the tripod are not going into stance synchronously, but with delay between the front leg. So front leg goes first, the middle leg goes next, and the hind leg goes last with the small phase delays. And so that's what we see as a gate. And then those delays sort of become shorter and shorter as the flies become faster. Therefore, the gate starts to become much more like pure tripod. So essentially, the conclusion is that the flies predominantly use a modified tripod gate. And that gives us a rationale for how to chunk up the data into steps. And this is shown here. And this is what is referred to as gate map. The black areas are where the legs are in the stance phase. And what we decided to define as the start of a step is the midpoint between when the last leg of the previous tripod leaves the ground and when the last leg of this tripod hits the ground. And that's what we decided to put as sort of the start of a step. Here are the speed and height now aligned to these steps. And you can clearly now see the pattern that there is an increase in speed around mid-stands in most of the steps. There's also an increase in height. But that's less consistent. Here is the data across all the trials. I don't know why the error bars are not showing up. So there is an error bar here that's not showing up. It's a different state of gate, I guess. But the conclusion is that the speed changes. So there's a large increase in speed at mid-stands. There's also a increase in height, significant increase in height. But that increase is much, much smaller than what you see for changes in speed. Now at this point, it's important to mention that this is very different from what the well-known model for biomechanics and insect the cockroach does. That is, cockroaches do just the opposite where the speed and height both decrease to a minimum at mid-stands. But fly is not alone here because thick insects also show the same pattern as flies. It just turns out to be different from those of the cockroaches, which is a dominant model for biomechanics and insects. So next, we attempted to model these kinematics, starting from what other people have done for cockroaches, which is that there's tons and tons of studies that have modeled cockroach locomotion using slip or spring-loaded inverted pendulum. Now, in slip, what you're doing is taking the three legs of a tripod and replacing them with this effective leg, that is, a compliant leg and can be modeled as a linear spring. And so the basic idea is quite simple. As the animal enters the stance, some of its kinetic energy is used to compress the spring. And because it's losing kinetic energy, it's going to reach a mid-stands minima. And then in the second half, that spring energy is being released again and converted into kinetic energy, causing, again, an increase in speed. Therefore, you get a mid-stands minima. Now, it's easy to see that slip, like fundamentally, won't be able to capture the increase in speed that we see in flies or stick inside for that matter. And so our first attempt at mechanical modeling was to take slip and tweak it to get a scenario to insert a mechanism that can cause this increase in kinetic energy. And the mechanism that we came up with was adding these so-called angular spring. And essentially, as the fly now goes through its stance, so the stance begins on the left and goes like that, as the fly goes through the stance, this angular spring release energy and increase that can be converted into kinetic energy. So let's look at this idea in a little more detail here. And essentially, the way we conceptualize this is by having, when at mid-stands, the angular springs are not doing anything, they're at their natural length. And then away from the mid-stands, these springs are always producing restorative forces to get the center of mass towards the mid-stands. As a result of which, during the first half, this spring is going to accelerate the center of mass, and during the second half, it's going to decelerate the center of mass, resulting in a maximum of speed at mid-stands. So here is our attempts to test whether this model will fit the data, and it does a surprisingly good job. This is for one step, so this is the best fit of this model to our kinematics data, and this is all of our steps in the data set, and you can see from the RMSE error the model does a very good job of describing the kinematics. The absurd of all of that is that you have two spring constants, KS and KA, that can describe the fly's kinematics over a given step. And here, now, to use the language that Daniel introduced to us, this then turns out to be the template for forward locomotion. So the next question we asked is, can the flies actually produce this kind of restorative angular spring like forces? And here the answer turned out to be surprisingly elegant, because it turns out that if you replace sort of the fly's body with a point mass, and these are the three legs of the tripod, and you think of them as linear springs, and take a sagittal plane projection of this, and we are basically modeling sagittal plane kinematics. Then the sagittal plane projection of the springy tripod under the conditions that there are not massive changes in R, and the step length is limited to sort of low angle approximations, it turns into this AR slip model naturally. So the absurd of all of that is for any springy tripod. So if you give me the fly's pose, all right, the battery is failed, if you give me the pose of the fly, the position of the tripod, I can tell you the equivalent AR slip model. And so what we can do with this is we can ask the question, if the fly's leg springs are of constant, do not change much during forward location, that is their spring constant does not change much, then we can ask the extent to which the changes in tripod geometry recaptures sort of the kinematics of the fly. So the essential idea is for each step, you have the tripod shape, and you can estimate K and KS by just crunching numbers, and you optimize K and KS, and we're asking the question, is the estimated K and KS similar to the optimized K and KS, i.e. is the tripod geometry affecting the kinematics of the fly? And the answer is a very big yes, because you can see that much of the variation in the center of mass kinematics actually results from the change in tripod geometry. So all in all, we are very satisfied with this model and how it's actually simply by the geometry of the fly's tripod. And now not only this, it also turns out that if you go to different parts of the K, A, KS parameter space, you can generate both fly-like and cockroaches like kinematics, among other things. And so what we think is that this AR slip model is an appropriate place to start to look for a general model for encyclopedia motion. And with that, I will end. And with acknowledgments, these are the people whose work I talked about. Sange Jung did the initial order behaviors. Lian Yu Tao and Siddhi Oyarkar did the optogenetic work. And Chen Wuchun did all of the fly mechanics work in collaboration with our physical physics collaborator, Deepavit Vishwas. Thank you all for listening.