 Hi everyone, I want to thank the organizers for inviting me to present my work. I'm going to be talking about the regulation of harvester and forging as a closed loop excitable system. This is a joint project with my supervisor Naomi Leonard at Princeton University and Deborah Gordon from Stanford University who has been studying these harvester ants for over 30 years. She's actually given a talk on Wednesday morning and I think it will be great to see more detail about this system. Give me a second. So we're interested in social insects because the colonies exhibit remarkable collective social behaviors that maintain coherence and plasticity even when conditions are changing and in the face of perturbations and when the individuals of these groups have limited information on the state of the group and the state of the environment. Let harvester ants live in the hot and dry Arizona desert where they collect for seeds that are dispersed by the wind and this process works in the following way. Foragers leave the nest through the nest entrance and they go to a forging area where they disperse and they search for seeds and once they find a seed they come back through the same foraging trail and they get into the entrance chamber and when they get into the entrance chamber they touch antennae with other foragers that are waiting inside the nest and what this does is that this excites the available foragers inside the entrance chamber in the sense that Deborah has found that the rate at which interactions happen is proportional to the likelihood of these foragers going out to forage themselves. And one of the really interesting things that Deborah has found is that how they regulate this behavior in terms of what the environment is like outside feels very hot or feels more humid actually dictates the reproductive success of the colony she's found that when conditions outside are very nice there is not much variation in the foraging activity between colonies but when the conditions are hotter outside for colonies that tend not to forage actually do better in the reproductive sense in the long term. So some of the key questions is how does this little interaction process inside the entrance chamber regulates the behavior across minute two hours long time scales and gives rise to flexible and robust behaviors that manage uncertainty in what information the foragers have and the state of the environment and how does this sensitivity to environmental parameters affects the group resilience and moreover whether there are any key parameters that we can use in order to distinguish the collective behaviors of one colony from another. We work with data from the foraging rates at which the ants live and enter the nest so we collect videos and manual data of the rate of which foragers come in and out of the nest and if you look at some of the time series here I'm pulling two examples for two different colonies on the same day we see very interesting behavior which very early in the morning when they start going out both the incoming foraging rate which is in blue and the outgoing foraging rate which is in red start going up and we see that after a couple hours they seem to settle in this little phenomenon that would like to call a quasi steady state in which the incoming and outgoing foraging rates equalize and when it gets really hot later at around midday then they start coming back to the nest and they stop foraging and what's very interesting is that this quasi steady state varies a lot between colonies and also within the same colony between days moreover we also see a lot of variation in transient behaviors and here I'm giving you three examples so here I'm also plotting on the left panels I'm pulling the time series which again the red line is the outgoing foraging rate the blue line is the incoming foraging rate and I've added a green line which is the integral of the difference between these two that is this keeps track of how many foragers are actively foraging outside the nest and we see a wide range of foraging behaviors and some of them when you plot them in this way in which you plot the incoming foraging rate on the x-axis and the outgoing foraging rate on the y-axis you see these beautiful dynamics in which if the trajectories are above this diagonal line where the incoming and outgoing foraging rate are equal if the trajectory is above that then that means there are more outgoing foragers and incoming foragers and the number of active foragers outside the nest is increasing and similarly if the line if the trajectory is below like in this case that means that they're coming back into the nest because the foraging rates are decreasing and we see the quasi steady state is represented by a little region that of the trajectory that spends time along this diagonal line and more importantly you can see that these two represent the same colony on different days where we see different behaviors they both seem to achieve some some more quasi steady state but they do it at very different rates and this corresponds to the intuition that on this hotter day the colony would like to forage less in order to conserve water because one of the biggest problems for the colony is that they need to regulate their foraging behavior because they get water by metabolizing the fats from the seeds so they need to manage the tradeoff between how much they forage for seeds and how much water they lose through forager desiccation and very interesting we also see these type of behaviors in which sometimes they start going out and all the sudden it seems as if they change their mind and they just come back in and in these input output plots that looks like a little loop so our goal here is to create a model that can capture this wide range of behaviors we will want to do it in a way that is simply parameterized and analytically tractable so we also want to see this as an iterative procedure in which we develop a model that can then suggest testable hypothesis which we can then use to perform field experiments and then we can use the resource of these field experiments to refine our model so our model consists of three main blocks the first block is the interactions between the incoming foragers and available foragers inside an S and this is what stimulates the foragers inside an S entering chambers to go out and force themselves then we have a second block that models the response of available foragers to these stimulus that is this little block here models the decision-making dynamics of the group again we do that the level of the group in order to maintain a low dimensionality so that we can maintain some analytical tractability and these two blocks together then are able to give me an up what is it an open loop model that can map a sequence of incoming foragers into a sequence of outgoing foragers so these little two blocks relate the incoming rate to the outgoing rate moreover there's this intrinsic feedback in the system of the foragers that leave the nest but eventually come back themselves come back into the nest again so we model this feedback by a little foraging block so first I'm going to talk about this open loop model and what kind of responses we can get from our model and then I'm going to talk about the closed loop model and what happens when we allow foragers to come back so first I'm going to tell you about this little block of the interactions so we represent the stimulus that the group of available foragers inside the nest receives as a little leaky integrator where we model the sequence of incoming foragers as they're at delta functions so we model this the sequence of foragers as a sequence of impulses and then we pass the sequence of impulses into a leaky integrator the leaky integrator just integrates this information in the sense that every time an ant comes into the nest there's a little kick of magnitude k you can see here and then when there's nothing going on when there are no ants coming in then it just decays at a natural rate tau and what this does is that this encapsulates information about the rate at which foragers are coming into the nest and therefore about the average stimulus that foragers are receiving inside a nest for the rest of the talk I'm assuming that k is equal to 0.3 and tau is equal to 0.41 seconds then we use the second block which models the response of available foragers and we do it in the following way we use what is called a fitzian agumon model which is a low dimensional model that is used to to model the action potentials in neurons and it consists of two variables one fast variable that is the V variable that provides positive feedback and one slower variable U that is a gate parameter a gate variable that provides slower negative feedback so we use V as a variable that represents the state of activation for the group of foragers inside a nest so we do it in the following way we use the fact that if you look at a bifurcation diagram of this fitzian agumon system when the stimulus s which is the signal that I talked about in this slide so when this signal s is low enough then the fitzian agumon doesn't do anything and it remains in a quiescent state but if the stimulus signal is strong enough then through hop-back vocation the fitzian agumon starts oscillating and what we do is that we take each oscillation in the output as a forager deciding to leave the nest forage that is there was sufficient stimulus to make a forager leave the nest forage and there's also a third mode which is if the stimulus is too high you can imagine if there are too many ants coming into the nest then we the fitzian agumon goes through a saturation stage and no more ants come out we use this to represent the nonlinear dynamics that happen at high incoming rates for example you know there's a limited size of the nest which the ants can come in and out or there's also effects in which at high incoming rates not everyone we were able to interact with everybody and there'll be some foragers inside the nest that will receive a lower rate of interactions compared to the incoming rate so through these two processes what we have is that we map the sequence of incoming foragers through the interactions block which gives me a stimulus signal and then the stimulus signal drives the fitzian agumon system and very importantly we introduce a parameter C that regulates how fast these oscillations happen when we are between the two hope bifurcations that is this parameter represents the volatility of available foragers inside the nest that is how willing they are to go outside if C is very high then they're really willing to go outside and they will go out at a fast rate and if it's really low then they will go at a slower rate so then through this we obtain oscillations that we just threshold in order to obtain a sequence of outgoing ants and using the assumption that the incoming rate is a personal process with a fixed rate just to obtain analytical results we can obtain an analytical approximation for the mean outgoing rate given the mean incoming rate and the volatility C where here P is the PDF of the signal S at steady state and TLC is just an asymptotic approximation for the limit cycle of the fitzian agumo system and now that we have this analytical estimation we can compare it to simulations of the this open loop model so what we do is that we provide a fixed incoming rate to the open loop model and then we measure what the outgoing rate is and if we do that you can do it for example these are simulations where if we fix the incoming rate at one and per second then depending on the volatility that is the frequency of the fitzian agumo oscillations we get different outgoing rates so then if we do a different number of simulations we can compute this input output curves and what's very interesting is that our simple analytical approximation is able to capture very well the qualitative behavior of this input output curves and we see that there's a difference between the curves in the sense that for low values of C low values of volatility the curve lies entirely below this diagonal where this diagonal again represents the points at which the incoming and outgoing rate are equal and for higher values of C we know that we see that at the start the line lives above the diagonal but for very large values because of the non-linearity in the fitzian agumo oscillations then we have that the curve lives below the diagonal and this will be really important now that I'm going to talk about closing the loop so so far I've talked about the opening of system that is how we map a sequence of incoming foragers to outgoing foragers but now I want to talk about what happens when you let the outgoing ants come back as incoming foragers and to model the foraging process outside the nest we keep it simple and for every ongoing ant we give it a random delay that represents the time that they spent foraging outside the nest before coming back and in order to represent the distribution of this random delay we use results from previous studies that show that this is sort of well approximated by a chi-square distribution where we let the parameter of the distribution D represent the average time that foragers spent outside foraging so now we can do closed loop simulations so we start a system we kick start a system by giving it a sequence of ants that are already outside foraging and start coming back into the nest and we keep that initial condition very low as it happens in the early morning and we see what happens is that if the volatility is high enough then you see these trajectories the rates start growing and growing and growing until they hit the diagonal line and then they stay there for the rest of the simulation and if you remember this is very similar to that quasi steady state that we saw in the observations and we also see a very interesting behavior which is if the volatility is low enough then the trajectories stay very close to the initial condition very close to the origin and what that means is that we have this bifurcation like behavior in which depending on the volatility of available foragers inside a nest the ants will either not go out and forage or forage at a quasi steady state where the quasi steady state is proportional to the volatility of the foragers so now we can you can think of this intuitively by noticing that there is a time scale separation between the dynamics inside the nest and the dynamics outside the nest inside the nest these maps incoming foragers to outgoing foragers the dynamics are very fast that happens through the interactions but outside the nest we have the foraging dynamics that takes several minutes so they're a lot slower so you can think of this as an iterator map we can approximate this foraging behavior just as a pure delay in between the incoming rate and the outgoing rate so you can think that if we start along this purple line this is the initial condition then what will happen is that we would then at the next iteration take to this line and the next iteration we will go to this point and so on until we hit this diagonal line and this represents a fixed point in the dynamics and that explains why we see this behavior in the closed-off simulations and we suggest that this can be help this can help explain what we see why we see the quasi steady state in the data also because this fixed point is stable that gives us an insight into how the colony manages to deal with perturbations in the incoming rate at which foragers come into the nest so so far if I bring back one of the data plots that I showed you before we see that they look kind of similar in the sense that they follow these curve trajectories and then they hit the diagonal line where they reach a quasi steady state however we see a bit more richness in the data in these transient behaviors also with the current system I cannot recreate what I told you before in which the examples in which the colony starts foraging and then they come back right away that little loop that creates in these input output plots so as a first approximation because we are not sure how the ants actually do this we're based on data that shows that the temperature and humidity inside a nest stays relatively constant in comparison to the temperature and humidity outside a nest then we expand the model to have two types of available foragers inside a nest the first type will correspond to foragers who have not been outside yet so they don't know anything about what the temperature and humidity is outside a nest and the second group is going to be the informed foragers that have already been outside and these two are going to be represented by again a Fizio Nagumo model where the only difference between these two is going to be that volatility so we're going to have some uninformed volatility and some informed volatility and what that does is that it allows for the system to adapt to whatever the current environmental condition outside a nest is so we keep the environmental condition inside the nest constant and for example in truly what you can think happens is that let's say that the uninformed volatility is very high so the colony is very eager and excited to go out outside but it happens that outside is very hot and very dry so as they start going out and coming back in they start reducing their volatility and they don't want to go out that easily anymore they require more interactions so what happens is that we start maybe along this line but then we change the colony transitions into this other line with a lower volatility where we reach this fixed rate and this gives us that overshoot behavior that we saw in the data or similarly if the conditions outside are really hot and really dry then they might drop to a very low volatility that doesn't hit this diagonal line that has no fixed point other than the trivial origin which means that the colony will stop foraging so now if I compare data on this side with simulations on this side you can see that we're able to capture qualitatively all of these range of rich behaviors through very simple mechanism and through a very low number of parameters in here we're only changing four parameters the uninformed volatility the informed volatility the number of available foragers and the average foraging time outside the nest and with that I would like to conclude and just say that our approach was to create a low-dimensional tractable model in order to provide insight into the model and to and to provide ideas for experimental tests that could be done in the field and we show that to this little feedback mechanism we can explain these quasi-state-state behaviors where the rates equalize and we identify group volatility as a key parameter that could be used to examine differences between colonies furthermore this is an ongoing process in which now we're thinking about field experiments that we can perform in order to refine the model and one of the bigger questions that is left unanswered by this model is that of heterogeneity we have assumed that all the ants in the system can be represented as a group because we have assumed that they're homogeneous but this might not be the case in the real system and we have pretty good idea that is not so then how can we incorporate this heterogeneity into the current system and see what the effects are in terms of the foraging behavior and with that I'd like to conclude and thanks everyone for listening to my talk