 Can you hear me? Yeah. So, yeah. Okay. So, I'd like to talk to you about a simple way of modeling normal common active oscillations. I want to mention that this project was developed here in Trieste during my PhD. And yes, so let's start. So, in nature, there are a lot of examples of systems which oscillate actively, meaning that they develop oscillating dynamics by consumption of energy and production of entropy. You can find it, for example, in circadian rhythm, heart beating, also the fact that the modulation of canary singing. And also, there are some really nice recent examples on the dynamics of sitting and standing of milkhouse and also in the cell crawling in wet environment. So, what we've learned from previous talks is that whenever you go screen these types of dynamics, you end up with system which are non-marcovian in a sense that the waiting times between two states of the system satisfy non-monotone waiting time distribution. And, okay, here I mean it's done from experiment. So, this is something that we have to keep in mind for modeling this type of oscillation. But the model that we are interested in is one of the systems that we are interested in in the case of air bundles, spontaneous oscillation of the bullfrogs air bundle. So, inside the inner ear of the air bundle, in particular inside its sacros, so I don't know how this works. Yes, sometimes when this happens you just go and press the button there. Laser, yes. Laser should work, that should work. Okay, thanks. Okay, inside the sacros of the inner ear of the bullfrog, there is an epithelium of ear cells. And each of these cells has on its top this kind of cylindrical structure which is called air bundle. This air bundle consists of microtubules which are linked by some elastic proteins which are called tiplings. And it has like a preferred symmetry plane along which the size of these microtubules increases and also in this plane the tip of the this air bundle is able to oscillate. And here you display some typical oscillating trajectory of the system. So, usually this type of, the description of this type of system relies on some microscopic mean field theory. Mean field theory, let's say, description where you have non-linear, oops, sorry, non-linear and coupled differential equations. So, very complicated stuff. What we, every month is try to mimic, let's say, the behavior of these trajectories by introducing a simpler model. So, the model that we consider is, let's say, a generalization of the Orson-Ulembeck process. So, this is just thermal noise. This is the quadratic potential from which the particle moves. Here, the center of the potential is itself a stochastic process. So, this is the quantum stochastic process which takes two value, plus and minus C0. And the time in each of the two states is drawn from some weight in time distribution which in general can be any type. So, in this case, in this sense, the system is non-marcogan. However, the fact that we have simple quadratic potential makes the dynamics linear. So, the equations are linear. So, this means that you can calculate but they are difficult basically. So, this year we show a typical trajectory of this type of dynamic. So, you have an alternate relaxation towards the two minima of the potential. So, as you see, the potential switches between the two times. There is also a change in energy associated to each switch. And the particle relaxes between these two minima. So, just to give a simple description of how one handles this type of process, let's stick to the simplest case which is the Markovian case where the weight in time distribution is just an exponential distribution. In this case, the switching between the two states happens with a constant rate r in time. And we can characterize a state of the system by simply looking at the sign of the minimum of the potential, the sigma, and the particle position x. So, the density of the two species is just the evolution of the density of the two species. So, this is described by a standard, let's say, focal-plank operator plus this exchange term where you have, let's say, a rate of exchange of particles between the two species. So, this type of equation were derived already in the context of lamentable particles in combining potential, et cetera. But if you want to generalize this type of procedure to a non-Markovian system, you have to use a renewal approach which is not, let's say, okay. So, to solve the dynamic, calculate the stationary distribution of this equation, you see that it just depends on two parameters, which are this chi, which tells you basically how much the two centers of the potential are distinguishable with respect to thermal fluctuation. And this zeta, which is just the ratio of the two timescale of the system, namely the relaxation timescale inside the trap and the time between two consecutive switches. So, this can be generalized also for non-Markovian case. Already at the Markovian level, one can distinguish two types of trajectories. One is a property by stable trajectory which comes from a bimodal stationary probability distribution and another type of trajectory which is noisy in the sense that it has just some fluctuation around the average of the two centers. And this corresponds to an immodal probability density. So, here is just displayed the phase diagram. You can immediately guess that when the two centers are far enough with respect to thermal fluctuation, you are in a bimodal state and also you have to reset much slower than the time that it takes for the particle to relax inside the potential. So, you can do this analytically for the Markovian case, but what is interesting is that the model is linear so it doesn't fully recovers all the cases of the bullfrog because the bullfrog has a part in the phase diagram which is purely non-linear and it has to be patched with this type of model. So, as I was saying before, we are interested in cases where psi is a non-decreasing function and so for this purpose we introduce a gamma distribution. So, in the case, the gamma distribution has two parameters, one theta which takes into account the exponential of the k of the distribution over time and then there is another parameter k where basically how much your system will be non-Markovian. So, in the case of k equal one you retrieve the exponential distribution and you can see that the type of oscillation are not very regular and this can be seen also at the level of the power spectral density by noticing that there is no time scale in the system so there is no peak in the power spectrum. So, this is a comparison between analytical formulas and numerics. While, if you consider instead within time distribution with a proper time scale then you see that oscillation are more regular and also the power spectrum develops a peak and so the system is properly oscillating. This is also, once again, a comparison between analytics and numerics. I would like to give some formulas. The power spectrum with power spectral density as was also introduced before can be calculated by taking the free transform of the autocorrelation function and we see that in this linear system the full power spectrum is connected to the power spectrum of the process C so the center of the potential and with this we can calculate explicitly in the Laplace transform. As you can see this depends only on the Laplace transforms of the within time distribution and on the average period of oscillations and if you plug this inside here you get the full power spectrum and this is general for any within time distribution. Okay. So, we want now to fit data from the oscillation of our friend Alfred. We have we have done this for three species of bullfrogs. So, first we fit as you can see the weight in time distribution for example and then we see how we can basically retrieve the the physics of this oscillation. So, here is the fit of the autocorrelation function, the power spectrum and here is just the stationary density. So, we do this with the other two species Manfred and Lothar and okay also here we find a good agreement. So, we fit this with a symmetric model but of course in general these are not the symmetric but still we get a really good agreement at this level of say approximation. So, finally we want to characterize if you want the energetics of this type of system we are adding the statistics of the work. Here the work is just calculated as the variation of the energy with respect to our external parameter which is this C the center of the potential and the contribution to the work come only when this variation of the process is non-zero which means that this happens only in correspondence of our switch. So, this allows to simplify the calculation and once again we can calculate the average work. As you can see this average work scales linearly in time because of large division principle basically and we can extract the average stationary power. Once again this depends only on the weight in time distribution the process form of the weight in time distribution calculated as some typical time scales and also once again with respect to the period of oscillation. Finally we use our inferred parameters from the trajectory to calculate the the the dissipated power per cycle and we find that so the average energy dissipated per cycle and we see that it is of the order of 10 ATP which is something that we expect from experiments and also we had some bounds which were calculated at the level of the macroscopic model for the system. So, with this I thank you for your attention and I just conclude by seeing that we've just introduced this simple model, the minimal model to describe active oscillation and we see how even though it's a normal curve and it's still solvable. Questions? Yes. So, what is the requirement in the weight in time distribution to get regular oscillation? Sorry, I've not heard. Is there a necessary condition on the weight in time distribution to get regular oscillation? You mean a relevant application, sir? Like regular oscillation. Ah, okay. Yeah, it just to have a peak basically. So, you have to have a time scale which tells you let's say on average after how much time do you have to do it and I have a second question. When you fit your data do you get like similar parameters for one species or like good variations between an individual in the same species? Well, I mean also I mean probably they're not even made for the same type of you know, air bundle because for each type for different air bundles you have different characteristic frequencies so I mean it's not the same frequency, they're different here the frequency are different so the peaks of the spectrum are different because you take also of course different species but also let's say different air bundles. Thank you. Yeah, exactly. Yeah, exactly. Yes, exactly. For me species in the sense different animals. Sorry. So maybe I missed something related question is that you had this exponent k sigma in the waiting time distribution. Yes. So how much does it vary from Alfred to Manfred to Lothar? Okay. Okay, yes. So usually they vary of the order of maximum one order of market. So for example. It's not universal. No. Biologically relevant question. Is there a correlation between the fitness and shape of the frog and the dissipation? So do the frogs that dissipate more are they thinner, more sporty? I don't know. Maybe if they don't do sports they're like less healthy and they heal. I don't know. You had this three history. That was the question. That's better because it has to stay there. I don't know. Are there other questions? Yes. So to what extent do you think like picturing this as a like refractory time is a good so is this like a refractory time that prevents like switches from happening immediately after you refractory time in the sense that you have to wait at least some time before this. Is there a reason why this wouldn't switch immediately because of some No. We don't consider that. Okay. And my follow up question is I mean you were saying this is effectively has the same sort of mathematical structure of run and tumble in a harmonic potential. Do you know if similar analysis with non-Markovian different waiting time distributions I don't know. I know just about exponential. Okay. So I think this is the first time. I don't know. I looked for it but I've not found it. Well there are some similar things in resetting but not in this context. Thank you. So you know that for this system there was this model by Julia and co-workers. Is there something really wrong with the model that motivated you to propose another one? It's just that this is simpler. It's as simple as that. We need something that is like and then this is also more general in this simplicity because it just keeps minimal elements on these deep types of oscillations and so you can in principle adapt it to other models. But there is no interpretation microscopic interpretation for this model. Have you considered what happens to your process if you present an external stimulus as it would be the case in the... So how it responds to noise? Yeah. It should be doable. It should be doable. Yeah. Are there any questions in the chat box? There is a month. No. Okay, so in that case let's thank all these speakers of the session and we go for lunch. I think we have the group picture, right? So we have to go where? Upstairs? So comb, dress properly, you know, these things.