using window so maybe you can should we have from the so yeah thank you so uh the title of my talk is cooperative operative maximum likelihood estimation for food flow dynamics you know by a sensor array and uh maybe somewhat unlike the previous talks in the section this is actually a real system with built so so we actually trying to see how we can model the very complicated system uh so for so to actually describe you what this real system is and it's kind of quite and uh interesting thing in it's own right so uh let's focus first on the diagram of the right hand side of the slide and oh what what happened was that a colleague of whose name is bruce cornell uh in in in the nineteen nineties an actually he's continue to work on that he's build a a remarkable now don't machine which can actually be uses as a by sensor and the goal of our work is to try and mortal that how does that work and how can be used that to do useful things so it's first important to to understand how this thing works because then we can model the dynamics of that uh so this by sensors actually build our of a synthetic cell number eight i mean all of the know what a cell membrane as it's on to sell and it can of two lose of fact basic bits score than that that by layer uh mathematically the tool is a fact walk around move around according to a random walk that's that's typically what people to so what what we do next is they insert in this slip by there protein now don't two these two are very easy to synthesise an approach you make that oh what you can imagine now is you have two layers moving around according to a random walk when two two combine they form of conducting or and a car can go through when to to to do not combine there is a car going to of course are several thousands of such do and so the probability of a couple of dupes combining is quite large and what you see some sort of car which goes up and down so that's really what's shown there now that's fairly easy to synthesise that the main idea behind this of course as the next stage where you attached to the top layer pacific and bodies which can detect molecules you wanted to do so suppose to interesting in to a interested in detecting H I V or H one and one or or or expose a molecule you can build an approach you makes flat specific antibodies which latch on to this talk talk Q so what happens is when you target molecule comes these antibodies bodies score to point to them and this stock so what happens is a top player cannot move anymore they graph so this changes the dynamics of the system previously we or on hindered random what what things were moving around you are on and off basically card now all the top is stuff and so the current dramatically decreases one other words that impedance increases substantially to this was the by sensor they build he publishes paper and ninety ninety seven in nature and uh it's it's quite a remarkable sense so because this can detect a low concentrations we can detect up a fan till more lower concentrations and if you think about that that's that's pretty surprising because oh one more are as as you know from high school chemistry is one have a cat was number roughly ten to the twenty three a ten to the three molecules in one or water once stand to model lower multiply by the by ten to the minus fifty so what can to the eight molecules molecule lead or more and that is extremely low concentration and that these things work remarkably we for that okay so that this is a system that field oh goal has been to try to see how can be more the system how can we predict how before was if we can do that we could possibly fine the system and make it work better and we can also maybe extend the system to work in the scenarios so we actually done a lot of work in this in the past and we have a couple of people which came out of the transaction that of technology just last year which dealt with more in the specific system what wanna talk about today is just ongoing work is suppose that you take this by sensor when you build a a of such five sensors how can you model that and it turns are be highly nontrivial problem tell you why know okay so that's for start with the individual electrodes to this is a signal by sensor what happens is you of the fluid which is still a word to this by a the you know food would delivery system so you basically have a some liquid such as sodium chloride containing the molecules tools you wish to detect and that is flowing how this by sense so this equation is the part of the to equation of of fluid flow it's the parabolic pde with the diffusion constant and so on now what happens is where the molecules in the fluid and come to this and electrode the trade off a chemical reaction which is a bunch of non than your or be different role questions so i here is the concentration both the stuff stuff you want estimate such as a H be you whatever these are the chemical reactions to trade off and what you measure eventually a some noisy version of a specific chemical in this chemical reaction plus but so these are the dynamics of the system a there pretty dirty for several reasons you see this guy and it's all is pretty nice this is just a straightforward forward flow P D to still be Q the back part lot of things we conditions you see that select lies at the bottom of the food chamber and this is where the stuff happens this is where the molecules which which we should detect re yeah with this by sensor to give you your measurement which is a increase in impedance this is a a boundary condition only and one location so it's not smooth it's not a it it's just a that location uh so when you have these complicated do rate it is that your boundary should these of uh one alignment boundary conditions that and the fairly difficult to deal with uh in addition you have noise you okay so this this system is is quite sophisticated it are so that you can construct a very nice models for this and and we've done that of the part now let's look at what we're trying to do here so now we have a a rate of such things see what is really nonstandard standard in this is but you have an array of such a electrode and particular if your concentration a is very small you of fluid goes pasta for select role the electrode graph some some of the molecules to react with that means when you measure the system you actually changing the system and that's likely non standard and signal processing in most signal processing we do typically menu measure the system you don't change so as to four flows but here the first electrode brat the molecules so the second electrode has fewer Q to detect so if you placed the second electrode very close to the first select road there's a depletion layer and no you don't a measure and fig if you pay a second were very far away from the first electrode it takes a while for the food to reach their and it means that you're detection times very low so so actually designing where you place you electrodes is is it itself also an interesting problem so anyway a goal at the stage is per some to be able to model this and come up with some tractable approximations as to how how this stay and i one wanna describe some of those two so you can be this is a problem of saying that given this fairly complex the would multiple measuring devices how do like estimate the concentration at the initial concentration that's that's what we wish to to me so i let's go a little bit of it you should before we start so P D's activity i mean so the first step you can think of is can be construct some sort of time scale approximation to replace this by or we differ to question in that sense we would simply we have a nonlinear D ease which which is a more tractable problem then you can deal with the non linear regression and and and a signal processing like that it for not not not reveal but least it's it's manageable okay so the way one can do it is you can use actually multi time scale dynamics it turns so that as flow goes by you see stuff at the top of the chamber which is very far away for the electrode uh it very quickly a G station stationarity in other words stuff at the top at of T reaches at infinity very quick oh it turns out you can really segment spatially this in to several compartment regions which uh far away from the electrode are are basically constant you don't need the speedy that you just have to sort regions which are very close of course you have to speed D but you make for the approximation to as described in a minute so the idea you you to use something called averaging two you with some of you may be familiar with but when you D with the filters that one at time scale where things happen slow and fast on the slow time scale you can replace the fast i i'd average which in this case is a constant away from the true and a man the stuff the electrode you have to be but a bit more careful to to see what to be done oh case that's roughly didn't you should another vehicle want to how you construct such more so uh this is actually the as that mentioned before you have this fluid would flow and then these are the equations of the chemical reactions to the sensors you don't so actually these chemical reactions of cells at two time scales and so you can for the simplified things you can actually average job the fast times deal with a so that's good and as i set a goal is to estimate the concentration at in the in light of this through chamber so this is what we don't to do is we are place this distributed parameter system or P D if you like by multiple compartment and that becomes a bunch of more than your own to use it and then you can do of a question questions so uh the way you do this this this idea of using multiple compartment model that is widely studied by people who to be on with fluid flow chemistry and and they they analyse some fairly complex devices for that sort of thing the main idea is this so if you could can our say a single electrode what you can do is i and this is that in and you can actually make this quite rigorous mathematical you uh so if you know that you P you could view heuristic lead a P well with the spatial dimension as an infinite system of all the east but you want you space okay so you to think of a great over space where at each grade you have a or T E and of course all these all these are interacting you of infinite system a to use based what what you think that show is that the only D which is very close to here is really the boundary condition so that's a chemical reaction the only which is a way from this is essentially the one which is a power with this chamber here with things happen very fast and that there's no P D it's just a that because it's a it's fluid flow approximation of right so it's you have to boundary conditions and this chamber just a bunch of than you these that's that's really that you so what you and it up with is a two compartment model here where here you have a constant here you have a nonlinear only and you have the boundary condition which is of the all that would be so you now down to a finite dimensional system which is which is tractable now as a said this was a heuristic thing because to be really uh is you have to kind of quantized of course this up sub step i learn and then you have to proof that the errors are bounded in so this is a deterministic system so it's not that difficult to do things i mean it's of a stochastic could be slightly harder so actually it it's it's not not that difficult if if you assumes so the regularity the system then then you can do one okay now so that's the to compartment model which is easy to do now you can imagine conceptually uh extended this robotic a model and it's roughly the same thing so in the first compartment is identical to to what you have to you for the first like true i'm and stuff was fast post to to the second electronic can you have a to keep up with the and and of course you have to keep in mind that because the first electrode is grad some of the and a like some of the ball was to try to better the second electrode has different boundary conditions which are vertical boundary conditions you but it's it's quite easy to take care of conceptual so that you have it to basically have a whole system of norm that you're or ease which is conceptually at least easier than dealing with a much more complex as to so once you now as i mentioned that i'm not gonna give you the details of these all these it doesn't really serve any purpose but just to kind of go back and you roughly what these ordinary we differential equations do uh you see they really model the chemical reactions you the model the movement of these now i tubes in the a number eight they model how these chemical reactions happen the chemical bonds and how when things couple they don't move so really there about seven chemical species it you know which you you model by of order to control question these are quite tractable in in fact the reaction rate constants and the one by by a of so the the all the parameters of those equations are actually or we so once to end up with this as a said you know all the reaction rate parameters you only unknown quantity is your input and like the creation which which is what we wish to be so that was really the main idea uh and just a can give you now some it you should be how this would work so if you think about even if this was a of now a bunch of ordinary differential equations questions or if you like discrete as a what time it's a bunch of difference equations a you have this fluid flow coming in here so as you think that this guy remote say some proportion of the molecules see alpha that means you're signal power here is reduced by but for because you grab some of the guys so you can think that the second electrode what has a little or signal to noise ratio because you less signal now of course the next electrode will be out of the squared and so becomes a geometric matrix V still it's obvious that you won't get dramatic improvement as you have multiple electrodes because it dies of as a geometric matrix series i mean that's gonna a used even at the linear case now of course is the non linear regression you have to be slightly more careful uh in any case one could actually workout out the asymptotic covariance of this okay and it's it's a fairly complex question and one could actually show that in as you have more more electrodes you your performance that's really from frame to you'd expect if you and N electrodes everything was i i D you had no interaction between things you get one of and improved sure you don't so that's that's something which so be the case uh okay so this is the actual system be but we've tested this on on so it to test this several things you can do you not to run of the real experimental system and compare it with without a of different look should model that's to kind of to a model that's the first step and those that things we've done it quick you come the pa that's that that were a second stage is how can you actually shall if the multi component model is a good approximation because that's the we we want to estimate of constant and it are actually they work extremely well as we can see these diagrams you the actually approximate the P E extremely well uh the arrow it use that to be equally between six to eight percent and even maybe go to concentrations which are very very low concentrations of almost almost well below an animal are still so these these that she work quite well uh and and therefore it means that you can apply standard a question analysis to solve for these concentrations i want make a few other comments before finish the all this is still work in progress i mean it's a very nice to come up with these approximations but you don't the sort of things we have done which would not please people are rigorous not my we haven't even shown that this system of equations has a unique solution it's really down hard to show that so i mean if you think about P D easy we bad this in a function space like a stop let's space you wanna show that you it's of a solution highly nontrivial so i mean although the system works other real axis system sure you you this is really hard it's something which we working on of the moment uh the are the issues are it would be really useful to come up with a nice approximations for this pde itself it's still of just using a multi compartment model we've done because we still ending up with a bunch of all we different role equations which we don't have a so form solution but eventually do you man we much nice if we can come up with further there were approximations which allows to get some inside as to how the system works so there are still a couple of a that's to our approach but i think this is a in the sense that given the complexity of the system we can model it it works pretty good we can approximate it and we can actually estimate using elementary nonlinear linear regression the sort of concentrations we want to estimate so that's really all of wanna say uh if you're interested in any of the stuff this was the original paper we by a colleague uh and these are a couple as we did where we dealt with the signal electrode case maybe be model it that and actually did a not than you're question on that oh can thank you very much yeah right are you know oh okay if gram so a a really interesting problem uh uh of course um is a lot of comments i guess for instance with the pay T um you've got a linear spatial up right the first to terms so there's a brings function that's an an i i you could generate a one approximations and stuff like that a you've been looking in that direction we have quite a bit what really kills us is cool so is is this i a gonna this is a horrible but we could dish of the that this a are could if should which likes to the P right and and and this this really is a a a i was not a at the takes the all of you know the concentration respect to spatial axes and sets to one of the things so then one is thing of boundary element methods because uh that's the use a to the with that of stuff yeah i numerically what we done actually is because you have a a solution are obvious so you don't have strong so should we use a finite element method which takes to functions automatically but again oh for our approach is completely and hearing in the sense that we have a solution it works but from a mathematical point to we we still have even shown any structure proper okay but good of array element method is different from finite element you i don't know if you familiar menu with a a a a type find so a any whites aims to make that might worth looking at that are thing is on not entirely clear that the regression analysis because the use is i'd time varying quantity so um uh i great so for instance if you nate to in house some regularization on the a temporal times of the use signals okay so let's a go to it so a is the concentration which changes over time yes we interested a zero which is a initial concentration wrong a a is coupled to these sceptical reactions which are you which you also over time these are pretty easy to show for example that these are always non-negative and so the they just basic chemical reactions a can i the only difficulty in this is a non linearity of that but you've got gotta a construct you of tape right well one specific um of that is a what be measure in noise okay i but when you do the regression what do us to my just i it's a so one one gram use of that you of taste he is in there must be dealt with a the why something is it so what you have is but you re braces by multi can pop and model you have a a bunch of or very different role questions here and here which are in you and the interested in one specific component of you observed in which is a concentration of the diners basically uh basically the number of couple guys you because the cover is proportional to the no i i understand all that but i guess again but in the middle somewhere in that regression you are as a function of tape must some have a being with carpet in order to waste one one this upon it one can hold diffuse for so i just one where the in a some temporal regularization or or or or the that of a us we have applied in such thing so far so are yeah my help it it back or a