Good evening, everybody. Hope you can hear me well. Yes.
Excellent. So I'm as I said, I'm going to talk about scaling, but actually, I'm not going to talk about that type of scaling.
I'm going to talk about dynamical scaling and how we actually use that to be able to understand population dynamics.
As I said in my background, as is in zoology, I'm actually trained as an ecologist, so I'm going to talk a lot about ecology.
But we also, as I said in my group,
we have lots of diverse interests across the breadth of the life sciences where we've been interested in thinking about applying mathematics.
And we're going to I'm going to sort of delve into those sorts of things.
And I'm also going to do a little experiment halfway through or thereabouts where some of you will get to eat chocolate.
So you have to be fast. When I need six volunteers, the people who put their hands up fastest will get fed chocolate.
So I am going to is are we going to do that? And I'm going to start at the very end. And it's another little game that we used to solve.
So I explore some ideas in mathematics and biology.
So this is what I was just saying is very, very, very common in biology.
We think about scaling in in a static way. So here, just like this.
Some sort of crab with a sort of a pincer.
And we can relate it to the size of the body, the size of the crab, to its the size of its pincer.
And often those are plotted on a log scale and they show that this is very classic al symmetric scaling.
And in this case that I just nicked from the web, we can just see there's actually a grey.
The bigger the crab, the much bigger it is.
Pincer is in relation to its body size.
So this idea of scaling in this elementary scaling in biology is very common, but it's a static thing.
We tend just to measure species.
We can go out and we can collect crabs so we can measure the length of the car pieces and we can measure the length of their pincers.
And we can look to see how that how those relationships play out.
Well, I'm a population dynamics artist, and I'm interested in thinking about how things change through time and space.
But tonight, just about moment, mostly about time. And we want to think about how we do that.
How do we scale across these different types of dynamics?
How do we take into account the fact that things may operate on different temporal or spatial time, time or space spatial scales?
So I'm going to go through about three things, three different types of ways that we approach this.
And I'm going to so explain the mouse and I'm going explain the biology behind some of these with some examples.
I'm going to explain some of the maths and at the end I'm going to show and talk about the implications where this becomes really,
really important and why we think it's important and all the stuff that we do in in extending our ideas of maths and biology to the real world.
So I'm going to talk about slow, fast dynamics.
I'm going to talk about this functional response around this point here where we get to surprise interactions.
That's where we get to eat chocolate.
This point here, I'll talk about how we can how we can use approximations, and I'll talk about some stuff that we've been doing.
And so thinking about how we can understand bipolar disorder and mood fluctuations,
I'm going to finish off with some implications and some very contemporary stuff that we do in terms of biological control.
So slow, fast dynamics. What I mean by that, well, I'm going to I'm thinking, okay.
I'm thinking about these sorts of systems. So this is a sign of our regular system.
If any of you ride horses, I know it is about horses. This plant is pretty nasty to horses.
It causes lots of things that make them sick and poorly. It's a very it's a very common plant in the UK landscape.
It's a accident. It's an either an annual short lived perennial plant or long live perennial plant.
It has these very colourful and yellow flowers, but it's stuffed full of very nasty things that may make horses and other things sick.
This is the Cinnabon moth. It's as a as a caterpillar. It feeds on this this particular plant.
It only feeds on this particular plant. And it's able to sequester, capture up those those those chemicals.
And under that, these caterpillars boast that the caterpillars are distasteful and the the moths are distasteful.
This is very pretty. It's on the way out in about the middle of May, through to the end of July.
And it's a pretty common sight, a pretty common moth. We're interested in the dynamics.
How do what sort of population dynamics, how does this moth and plant interact over time?
And one of the things that we can do is we can write down an equation to capture that sort of dynamic.
So so this might be one way that we might do that. So I have resources are changing through time.
So the change in resource through time on the left. And then I've got some of the biology that describes that, that sort of dynamic.
So Lambda is going to be some input rate of resources into the system.
And then I have some consumption by the moths which I.
Usefully called G and they're consuming this resource at right alpha.
And the other thing that will happen is if the resource doesn't get eaten, it will just decay away.
And then there's some coupling and I'll come back to that coupling later in the talk.
But we're going to couple the. So what gets eaten by the what gets eaten by them also gets converted into new, important new moth.
So this is the birth rate of the moths, if you like. And, and then the only thing that's going to happen to them is that going to die.
And we can do lots of analysis on this sort of model.
We can really think about, you know, what, when the population dynamics would be stable, when they might show cycles, I was quite involved.
And we really want to think about, well, actually, if we're thinking about scaling all those things,
all those things operating on the same sort of temporal scales, and if not,
can we get round that other some approximations that we can take to think about that?
So one of the things we can do is we can say, well, actually, maybe the resource dynamics are operating faster.
Maybe this plant is just going through its lifecycle a lot faster than the moth.
And I can't really drop my tummy and pop my head at the same time. But you get the idea this hand is going faster than this.
And so the resource dynamics are going faster than than the moth dynamics.
And the idea then is that the moth only experiences on average what the resources are doing.
So we can then approximate those resource dynamics with a little bit with with a little bit of, with a bit of math.
So we're going to assume that actually, on average, those resource dynamics are not changing.
The D, the d r by d t is set at zero.
So if we do that, I can then sort of set that first equation equal to zero and I can send re-arrange it just algebraically for R.
They're making a couple of assumptions in that and doing that. Firstly, as I said, the resource dynamics are going faster than the math dynamics,
but the little things are going to make an assumption about is the loss rate, this decay rate of the resource.
So we're just going to see that the models are really good eating their resource and not so good and they eat lots of it.
So there's very little left around so that actually the resource consumption rate is actually exceeding these loss rates.
And if we do that, we can make this some very simple approximation. And so if we make this rearrangement to the average,
the average resource dynamics are all now a given by lambda divided by this this the rate of consumption and the number of that around.
And that makes pretty good sense. Basically, if there's lots of resource flowing into the system,
that means the resource is going to increase and as the numbers of of the month increase than the amount of resources going to go down.
So it captures in some and some sense what might happen to the resource up our average amount of resource that's around in the system.
But we're not really interested in that.
I'm trying to think about what would we do when we when we plug that into the to the second equation to do with to do with the to the moth.
So I what I'm going to do is going to replace this term here by the approximate that by that approximation.
So if I do that, I'm going to take this average result some place replace the all in in that second equation.
And it's quite we get we get a much simpler expression in terms of the moth dynamic.
So I can basically say I can cancel these two terms and I'm left with lambda minus new G.
So basically so the. So in this in this slow, fast dynamic, I have the amount of resource flowing into the system.
I know the only thing the other thing that's happening is that the moths are just dying.
Just a nice, simple equation. So nice, simple differential equation.
And on on the right hand side of this and I'm pleasing my schoolteacher very much
because she or she used to tell me that I should show my workings a lot more. What we can do is that we can solve that equation.
I can go from this equation to knowing what was this function actually look like for all time?
So how many generalist moths, how many moths would be around at each point in time?
So I could work through some calculus? Is this this is this equation written small?
And here is my solution. At the end, it's a bit of calculation.
You need to know how to take the integral of one over G and something else and it all falls out.
Basically, I have this sort of function at the end. Okay, so that as well it may mean something to somebody, but we can do some things with this now.
We can I can draw it on the board or we can just do some thought experiments about what
happens for different values of T and the best values of T to pick are zero and infinity.
So if we set t, if we set T equal to zero and we work through this and everything else is just we work through to see what happens.
What how many generations do we expect to be around? At times zero.
It turns out to be zero. So we are down at the origin of a graph on an x y graph.
We're down in that. So when there's no time, there's no generate, there's no models around.
If we set T to infinity,
well basically this then approximate this this term here then is essentially zero and we're left with one -0 lambda over you, lambda over anew.
And what that basically tells me is that there's then there's a fixed state.
So if I know the value of the resource input, which is lambda and I know this decay rate, then I should be able to say how many generations around.
But a very cool thing about that, this approximation is that if we setting t to infinity and infinity goes on forever,
then I go from zero to some sum fixed constant value a ratio between London and a new.
And that's really cool because that's telling me that the dynamic should be pretty stable, right?
So for all time beyond zero days, this graph is going to go up and it's going to stay flat for all time.
And that tells me that actually with this approximation, my slow, fast dynamics,
I should expect to see resource dynamics being the most dynamic, being relatively stable.
Why is that? Well, there's an ecological process, so the resources are flowing into the system and it's called the donor control.
That's a bit of a jargon term,
but it basically means that the resources are flowing into the system and then modulating the dynamics between the moth and the resource.
And they're giving rise to these very stable dynamics. So the next thing we have to ask is, well, how true is that?
Well, this is where data gets in the way and we have the moss.
So it's a pretty pictures of the moss.
And this is the Time series that was collected from the 1970s to the early 2000s of the cinnabar moth interaction.
So I've got the log numbers on the on the Y axis and that on time, every year,
every year this person went out based in the Netherlands, just just south of Leiden.
It's a very nice habitat. They're counting the number of regular plants and the number of moths on those plants over that time.
So and we see that the black line is the moths and the red dotted line is the plants.
Okay. It doesn't really show that sort of that sort of expected dynamic.
So we've got to think why why might that not be the case?
And we'll get onto that very shortly. But it's actually sort of like brings in thinking about other bits of of biology and maths.
And we want to think about what we don't see that nice stable interaction, not nice stable dynamics of the moss.
And why is that? Well, we can we can appeal.
We've got an idea about the sorts of dynamics that we have in these these time series that we have,
allow us to be able to do some statistical analysis.
So I can ask basically what determines what numbers in this place in the Netherlands and how is it dependent?
Is it dependent on the market? Is it dependent on the resources? And how do those two things interact?
So the black line is the as I said, is the is the most numbers.
And I'm asking basically, how do those numbers change over time from one time steps?
And what are the drivers?
How are the numbers in the previous time step important or two time time steps back really important in driving these sorts of population dynamics.
And we see two things. One, we see that the models affect themselves.
So they compete, if you like, for access to this plant in some way.
But there's also a dependency of the plant.
There's actually a link that that that the idea that they are specialists and they do need this plant and that plant that drives their dynamics.
But it's more interesting than that. It isn't actually the number of plants that are around in the last year.
It's the number of plants to around two years back that drives these sorts of dynamics of the moth.
And we can do the same thing for the plants. We can do the time series and statistics on the plant so we can say, how do the plant numbers change?
And it turns out, well, plants affect themselves both now.
So the number of plants now are affected by the numbers that were around last year and the numbers are around and two years back,
but also affected by the number of moths that are around as well.
So we really there's a really dynamic interaction there between the plants and the moths and the moth in the plant.
And this approximation,
while it's really good in theory to solve doesn't it doesn't really hold when we think when we compare to data because of this inherent link.
And if I just flip back really quickly, is this is this idea here,
this idea of this can this this response and this link between what gets eaten and what gets and what translates into new,
new moths in the next in the next time step. So that brings me naturally, really naturally onto this idea.
We call that link in biology, in ecology a functional response.
If you're a biochemist, you'll know it as a Michalis mentum function.
It takes so many different words or phrases to explain what this what this is,
but it's a link between two different states of the system and it's a really great way and we'll get into it surely about how you can,
how you can approximate individual behaviour and individual performance into dynamics.
And I want to sort of explain that is with a completely different bit of biology,
and that is to do with the the blood stem cell system, the haematopoietic stem cell system.
So we're really thinking about how we can how the the individual response is
needed by we needed to make a blood system are dependent on what's around.
It's called the demand controls it's called the demand control system.
And your blood system is just fantastically fascinating.
In your bone marrow, you have stem cells that are constantly making new blood cells.
And they can make two broad types of of stem cells.
They can make two types of blood cells.
The lymphoid cells going down this sort of root are all the sorts of things that make your immune system, make it work.
And the other ones are those or myeloid cells which go on to make of a whole set of red blood cells.
And we just think about we just kind of stylise that sort of interaction into thinking about myeloid cells,
those that make red blood cells and lymphoid cells that make immune system.
We're not. But there's immense amount of complexity in terms of the different types of cells that can be made.
And obviously when it goes wrong, it goes wrong and it can make people very sick.
And we're trying to understand how we can use a mouse to actually understand the dynamics of this,
this really incredibly diverse system and how we can use these scaling as these functional spot response scaling to understand that.
So what we can do is that we can stylise it by going from that to a very complicated bit of biology to sound like.
It's like saying that I could write, you know, just write out in terms of just a schematic diagram.
So basically we were thinking about transition. So you can go, how do these how do these stem cells give rise to new stem cells?
How do they give rise to progenitor cells? And how do those progenitor cells give rise to those myeloid or those red blood cells or white blood cells?
And what tends that what the sort of ideas behind this that sort of so reach back to
the 1960s we really just thought about two types of transitions that could happen.
So this this this particular little blob represents a stem cell.
And the things that can happen in the next time step in the next event are that I get two new stem cells,
or I can go from a stem cell to a progenitor cell that's going to give rise to a red blood cell or a white blood cell.
But actually, the system's a lot more complicated than that. Like, if these are the man control system, it's basically your body.
Say, I need more. I need more red blood cells or I need more immune cells to fight off this disease.
There has to be other things going on and we can sort of extend that out.
So a stem cell could say there's also like an emergency response that says, actually,
this stem cell now needs to not only it needs to sort of like die and die and make
tuned you to new progenitor cells that then would make red or white blood cells,
depending on how the stem cell could die,
the progenitor cell could differentiate even further and make two new two different types of progenitor cells, two more progenitor cells.
Because that's what happens in these in these stem cell systems,
the progenitor cell could pass away or it could differentiate into something that's really specialised,
making a platelet or a T cell or whatever it might be.
So we can really think about these sorts, this sort of system in many different steps, so many, many different sort of event, event driven steps.
And by that, I mean that in any small interval of time, only one of these is allowed to occur.
And the way that we I'll show you what I mean by that shortly.
But what we can do is that we can we're getting toward some more mass. We can sort of start to put some some detail on to that.
So here's the stem cell, here's a progenitor cell and has a differentiate itself.
So a red or white blood cell. And these are sort of intermediate cells.
So I have some background that affects whether the stem cells are going to give birth or they're going to die or what controls their their activity.
But the important thing to note here is this these these arrows, this one.
So where we have this this this particular these two particular these two particular functions,
this negative signal, that's the demand control that we're interested in.
We're really thinking about what is it that how do we link this,
what's needed from the body in the peripheral blood to to tell the stem cells what to do?
Because they don't really know.
They're just going to just say they're just going to keep making things and they need to be told what to do and they need some way of linking.
And that's where that functional response actually comes in.
So we've got a progression of births, if you like, of progenitor cells, births of of differentiation, differentiate themselves.
But then when they've differentiated, we need to be able to understand,
we need to be able to counter this idea of this negative feedback, this negative signal.
It says, make more or make less, do something, basically.
We can capture that again with a set of ordinary differential equations, all of that biology and are more complicated than the ones I showed earlier.
So again, we've got this body t that's the change in stem cells through time.
I've got deep body. T That's the progenitors through time.
And Dee dee dee dee is the change in differentiated cells and they've got bits of biology that capture this.
They're actually the ideas of what we expect that happens in a stem cell system.
So this first term, this term here, this term here, depending on where you see, my point is just telling us that the stem cell has a niche.
It has a limited amount of size that it can occupy your. Your bone marrow is limited in space, the space it occupies.
And that means that the number of stem cells that can live in that space is going to be limited.
And that's set by K. We call that the carrying capacity in ecology, but it's just a fixed amount of space that the stem cells can live in.
And then then those things that can happen, they can they can give birth to new stem cells.
All they can do. They can change these. They make progenitor cells, which is this term here.
This is the second picture I had where a stem cell which replaces a stem cell and also makes a
presenter cell or this one here where there's a two says there's a more emergency signal that says,
make me two. That's that. That's the use of stem cell.
Need to now make me two progenitor cells. Okay. And then there's and then there's lost terms that's decay term to death terms of stem cells and death,
terms of progenitor cells and transfer of progenitor cells into into differentiate yourselves.
But it's these two functions that are really critical, this function response at linking these dynamics,
linking the individual performance so that to the to the population dynamics of the system.
So if I if I use some numerical optimisation methods, I can I can iterate that model.
So here's the number of differentiate yourself through time. The red line is the deterministic dynamic.
So if I just run that model is just as a is an all set of a coupled order and the differential equations, I end up with this red line.
As I said, it's a stochastic system, so it's a random system.
So if I make this event driven, if I make those, if I make each of those transitions an event,
I can only operate in a small, small amount, a small time stab.
Then I get a whole raft I can. So look at this of repertoire of dynamics that I see out of this.
What do I expect from this system,
given that that is that it is these random to knowing whether stem cell is going to make another stem cell or a progenitor cell.
And while this captures those actually the variability that we could see,
it shows us is bounded in some sense that actually the system doesn't go off to infinity and it doesn't go to zero.
There's actually some bounds. So we can run this many times and we can we can do some fancy maths and we can quantify the variability
that we see in this in the number of differentiated cells produced by any single one stem cell.
And this is just a spatial plot, if you like, of just a heap liposome.
Well, here's the number of stem cells, so of around about ten, and here is the number of differentiate ourselves on average around about 40.
But we see some spread around that that function response is, you know, not demand control is not perfect, actually.
And I'll come on to this. Surely there's we need to understand in a little bit more detail this idea of homeostasis.
And this idea of feedback. So how this demand feedback plays out and how it can can occur in all three Broadway's so really thinking about.
The number of lymphoid saw a number of cells that are improved for blood.
So we'll say the number of and the number of sort of immune cells that are around.
And we think, well, what proportion of myeloid cells should be produced given this number of lymphoid cells?
Well, the first one is could be completely independent. These two things in the skull could just operate independently of each other.
So we just have the number myeloid cells. It doesn't really care how many immune cells, how many immune cells.
So then I'm just going to make more red blood cells because that's what I set up to do.
All that could be some relationship and proportional to need.
So is the is that is that as there's more lymphoid cells, there's there's a need for more myeloid cells.
So it could be non-linear, it could actually start to increase and then plateau out.
We need to be able to understand that. We need to understand how this is so peripheral blood.
How this peripheral blood feeds back is information to the stem cells.
And this is the crux of it. You can write down a functional response.
We can write down a mathematical expression that says, how do we link the the the proportion,
the number or the proportion of cells that need to be made to what's already there?
And we start by doing that in homeostasis, where the system is not being perturbed.
So we think of this row H as the number of myeloid cells or the number of lymphoid
cells that are being produced in homeostasis to maintain the body in a fixed state.
And it's given by. So how many should I produce is in proportion to how many of that.
This is the ratio of myeloid cells that are there to lymphoid cells that are there.
And then there's some parameters, alpha and gamma,
that we want to be able to understand what they look like and what values should they have that give us a homeostatic state.
Well, if you want to know what values well, we can. So if this is homeostasis, that's a fixed state.
We know what that is because it's homeostasis. It's not changing over time.
So I can then sort of rearrange this expression and say, well, what values of alphas give me this?
What values? What values of alpha should I have that give me this homeostatic state?
And what values of gamma also give me this homeostatic state. And it turns out there's a whole family of them.
They all look like this. There's a whole set of values.
So here's the values of alpha and the values of Gamma at every single point on that line represents homeostasis.
So there's a whole different value. There isn't one set of alpha, and there's not one gamma value that's going to tell me that this is homeostasis.
It's just this whole family. Every value along this line is a point where we identify a home that will give us a homeostasis.
Give us homeostasis.
Well, that's pretty cool, because actually what that means is I can pick of I can pick a value of alpha and gamma and actually ask,
what will the functional response actually look like?
It doesn't actually look, you know, how how many so what proportion of myeloid lymphoid cells should I produce, given how many are there?
Basically, from each of those choosing two of those choosing an alpha and an associated gamma gives me a whole set of family of of outcomes to this.
So the functional response is it demand control? Is it that is in proportion to what's already there or is it independent of what's already there?
Really depends on the parameter values that we have to all along that line.
We can get a whole set of different dynamical outcomes from this, and I'm about to explain what I mean by that.
And when we talk about predator prey interactions, but the but we have a whole family,
a whole set of different shape functions that relate individual performance,
the demand required so of what's in peripheral blood to what the stem cells, what those stem cells should actually do, which is pretty cool.
And I'm going to get on to talk about predator prey interactions and I'm going to talk about the same sort of thing.
So the same sort of thing plays out. We're really thinking about how do individual predators, moths, eating plants or whatever it might be,
polar bears eating seals or dinosaurs eating small rats or whatever it might be.
How do they how does that how does that behaviour of eating those things translate
into a population dynamic outcome that affects these things through time?
So notice that differential equations. Oh, no, no, not differential equations, but similar to the ones I introduced before.
So I have different letters, but the same sort of biology. We're really good at this.
In biology it does changing things with letters and not really changing much.
But we've got predators and prey.
So n is my numbers of prey and P's, my number of predators and damned by d d invites the change in imprinting p by d t is the change in predators.
And I got some biology on the right hand side. So I've got a birth rate. Oh, I've got a consumption rate by predators on prey.
I've got now I've got a conversion rate, so I'm converting dead prey into new predators, and then I've got a death rate of predators.
So let's just scale a lot back and think, well, what sort of dynamics should I expect to have if the predators aren't there?
So let's set pizza zero and see what happens here by six.
So we've got DRM by D, T, R and basically what sort of dynamics do we expect to see?
So again, we can do some maths this time. I'm not showing my workings, but I am.
Show you a picture that shows the sorts of dynamics we get. We expect if our is positive.
So there's positive birth as the R is greater than zero,
then the population of rabbits in this case just will grow exponentially unbounded and they'll keep doing that right.
So that's all that's going to us. All we expect to see in this is predator prey.
And in this in this interaction, when the predators are not around,
the population of rabbits, in my case will just continue to expand through time forever.
Well, that's not going to really happen, because what's going to happen is that the predators are going to come out and start consuming rabbits.
Basically, I'm going to think, well, actually, we really want to understand how does it as I said,
it was really thinking about how does the individual predator eating a rabbit,
which is a behavioural response, the predators are going to find a rabbit and catch it and eat it.
How does that translate into population dynamics?
Okay. So we are going to work out what this functional response looks like and do some analysis.
So here's the opportunity. I have six bags of chocolate.
You don't know is how many chocolates are in each bag. And will I like to do this sort of experiment?
It's a little exercise if you do. So I need six volunteers to put your hands up.
I will pass you a tray and I. And I'll pass your tray and a bag of chocolate, and we'll work through this.
All right? Oh, I can guarantee the trays came from the cafeteria, and they're still warm, so they're clean.
But I also bought clean pieces of paper as well, so feel free.
I'm just going to put those there, remember? Just put those I remember behind our trays.
First, take a tray. Who else can I. Can I give you a tray?
Excellent. Oh. Can I give you a try?
Can I give you a shake? Thanks. All right.
Anybody on this side want to try? Excellent. Well, yeah, they're still wet.
Well, that's disgusting. I'm sorry. You're going to get a wet tray.
Sorry. I've got one more. I'll go over this side of the room.
Bear with me a moment. Somebody over here is interesting enough.
Excellent. Take your tray. It's a bit wet. Sorry. I'll be back with some chocolate in a moment.
Right. All right. Right.
Now, remember, I gave my trace back a piece of paper. Thank you.
And I'm going to give you a random bag of sweets. You've probably got about 25 in there.
You need to open them and tip onto the piece of paper. So I'm going to be a piece of paper and a bag of chocolates.
I want I need you to do when you have a piece of paper to open the bag.
So guess how many chocolates you have. I'll try and tell you. And just spread them out on the piece of paper.
Probably about five or ten in there. What's your opinion?
Somebody's got to get really lucky because one of these bags has got about 100 sweets in them.
Oh, it's going to be a good. Oh, I'm sorry, Doctor.
You got five. I'm sorry. Don't worry. It could get worse.
You go back in there. Where else is my tray?
Oh, one here. Excellent. So it's put about ten or 25 in there.
Okay. Sorry. I have one more to do over here.
You get the bag of Eminem's that's left in here. Well, you can't use them all.
I'm afraid I only need you to take two out of the bag, but you can share them with the people that are around you if you like.
So there's the chocolate, and here's the back of them. So just take two out and put them on the piece of paper.
So if you've opened up your. So what we're going to do is we're going to do this idea of a functional response.
And the idea is that you to you are going to be predators and you're going to find your prey on the M&Ms that you've cast out on your piece of paper.
So take them out onto the piece of paper. If you don't know the person next to you,
it's a great opportunity to introduce yourself and just help get them to spread them at random so you can't see where they are.
Because what we do, we have to close our eyes. You're going to take your your your pride claw, which is actually your index finger.
This is your index finger. I have to say this every time this is your index finger and this is what you're going to use to search for your prey,
not your hand, your index finger. Okay.
What are you going to do is you're going to close your eyes and you're going to tap on the board on the piece of paper when you find a mate.
Please don't eat it just yet. Put it on the side because we need to know how many you've got.
So if you've all got a good idea of how many you've got on the paper. Yeah, how many you've got to start with yet.
You've got ten. Somebody's got 100. Yeah.
Somebody got 50. Who's got 50? Stick your hand up.
He's got 50. Somebody must have quite a lot out there.
Yeah. Yeah. So. So you've got some idea about how many you've got out there, right?
Okay, so we're going to do this. So what? The idea is that you're going to close your eyes, but it's not for a minute.
You're going to see how many you find. Okay. All right. We're going to do is going to do some analysis.
I'm going to see what how your behaviour of tapping relates to population dynamics.
Okay. So very happy with that. Yeah. Cool. Go happy at the back.
Yeah. Yeah. Everybody happy? I need to hear everybody's happy. Wave if you're happy.
Yeah. Go. I'll have you go watch Ireland.
Have you got what? Can you take me a minute? Yeah.
Yeah. Well, I know. Really. Excellent. Okay. Everybody ready? So you going to close your eyes?
Index fingers, people. Hold on. Let me see an index finger.
Index fingers. Index fingers. Thanks.
Okay. Close your eyes. Ready? We're going to talk for a minute.
Go. How we do.
10 seconds. Excellent. All three.
What? Stop. Okay, please stop.
Hopefully the person who found two managed to find them all.
Yeah. Excellent. That's a good start. Okay, so we're going to do the analysis now you found.
So I just need you to count up how many you found. So somebody had five.
How many did you get? Well done. You're a good predator. Excellent.
Top of the chain. Excellent. Somebody had ten on the board.
How many did you find? Three. You're not so good.
At least you got them to eat for the rest of their lecture, even if I'm boring. So that's quite good.
Somebody had a 25 on the board. How did you two get?
22. Did you use your index finger or a whole hand? Excellent.
Good one. Always good to know. 22. All right.
Somebody had 50. 3838.
Wow. My God. This is going to be interesting. All right.
So and so. The person in the back. 42. 42.
Excellent. All right. Let's see how this works.
So what we're going to do. I am going to. Quickly make sure I've done this right.
Oh, there we go. Always an issue when it doesn't like the numbers, but, hey, he's got a bit cranky.
Can I go to work? Okay, it's not gonna work. So, what, we. Okay, I'm just going to fudge the numbers, but you'll see just a little bit.
Just to make it look. Let's see if we can make this. Just to give you an idea of what happens.
So the two things that two things that go, the two types of responses that go on.
So I have a picture somewhat here. So you guys were really great.
Without a shadow of a doubt, my analysis doesn't take into account the fact that you could be linear.
The fact is that what we have on this graph sorry, the font is not great on a but on the x axis I have the number of M&M that are
on the board and on the y axis I have the number that were caught in a minute.
And what you guys have done is managed to find whether you have 100 on the board.
You found half of them when you found when you have 50 on the board, you got quite a lot.
I know the idea is that that response is linear.
It will go the I've I've fudged the points because we're trying to fit a curve through this.
We you would have a nice linear relationship there of the number court and the number the number on the board and the number that you caught.
What tends to happen when we do this and we replicate and then obviously I can only do this with six of you if you all
were able to do this and we were in a little bit more of a controlled environment when we were doing this experiment.
What would happen is that over time, even when you had how many did you find when you had 100 on the board?
40 to 42. Right. So you don't find them.
All right. You find about half of them. And the really important thing there is that actually what we're doing is we're linking the behaviour,
this idea of, of, of, of foraging for the prey to the population dynamics.
And what we expect to see is that as you increase, as you increase on the X axis,
you can't actually get many more than about 40 out of 30 or something out there.
So it doesn't there's a finite amount of time that you have to actually be able
to take these Eminem's off the board and put them on the side or as a predator,
kill your zebra and convert that zebra into a new lion.
That takes time. You have to eat the thing. You have to digest it. You have to you have to give birth to new lions.
That's a has a time dependent response and it has this particular shape.
Okay. So as the number increase, as I have numbers up on this axis at the moment against the numbers on the on on the on the board.
But if I think instead of having numbers, if I have the proportion,
what happens with this graph is that actually down at the low end, most things get called.
You know, that might be counterintuitive, but actually if there's a small number of prey, most of them are going to get captured.
But when there's a large number of prey. They don't all get caught.
And as the number as the number increases, they continue to.
They continue to they continues to go down.
So we have this negative relationship. So the proportion eaten against the number on the board goes down.
And that basically means out here, when we have lots of prey, then we expect to see many of those prey.
Giving are escaping predation and are able then to do whatever they do give birth.
So we expect to see that actually when there's a large number of prey,
this when there's when the small number of prey are being caught, lots of them will be able to increase in the absence of predation.
As we talk and as we showed earlier, they all increase exponentially.
So what is that? What does that what is it how does that translate to my moth dynamics?
So I have this sort of idea, I have this sort of relationship that actually when there's when we have this dynamic,
when they have this functional response between behaviour and dynamics that give rise to this idea that when there's lots of them around,
very few, proportionately, very few of them are going to get eaten.
Lots of them will escape. Predation means that they can grow exponentially.
And then what happens is that the predator catch up is a numerical game. So there's more prey around.
There's going to be eventually more predators around. And that's going to turn over.
And we get these predator prey type cycles.
And that's what we see in this system, basically, that we see the again, this is the the moth numbers through time.
The black line is the most the morphs, the red dotted line is the plants.
And we see this cyclic dynamic playing out. And that's because of the relationship.
The link between the scaling between individual performance,
how a predator or a moth in this case performs, and how that translates into the population dynamics.
And that's pretty cool. It's a pretty cool way of which linking, you know, different,
different aspects of behaviour operates on a different temporal scale to population dynamics.
And when we can take that, we can translate it into these sorts of population dynamics.
So to finish, I want to sort of like talk about this sort of like the third way that we can sort of think about mass gatherings.
And it's a bit mathematical, but it's about approximations,
how we can take approximate how we can approximate dynamics to operate on a different time scale.
It's like the fast, slow dynamics, but we can do it in a lot better and a little bit more rigorous way.
And I want to also motivate this by a really useful and so really cool way in which we approach this sort of analysis.
We are interested and a group of colleagues in Oxford and elsewhere.
We're interested in being able to understand the mood dynamics associated with patients with bipolar disorder.
So bipolar are very, very common mental illness that occurs in late teens, early twenties,
and is so a lifelong chronic condition where you get episodes of mania and depression.
But in between, those episodes of mania and depression are really,
really rare compared to the mood fluctuations that go on on a day to day or a week to week basis.
And what we can use there is a set of standardised psychological scoring system sets, statistic sets,
standardised psychological scoring metrics that can be used to quantify mood, and they're called quints.
So it's a quick inventory of depressed symptom, symptom or symptoms, and we can use that as a self-reporting system.
People use their cell phones or the computer to answer 16 questions that give a give an answer to, and we can create an aggregated score of that.
So this is what this is. That's what these are the scores. Those are the mood scores that we get out of patients over a period of time.
And we can do that. We can do that on very different temporal scales.
We can do that on a daily basis. We can do it on a weekly basis. We can do it before and after intervention analysis.
This is just a sort of cohort experiment that we had for 14, 15 patients or so where we monitor that mood,
which is the, the, the solid black line over a 28 day period before and after a treatment intervention.
My colleagues I work with with this are a clinical psychologist and they use behavioural interventions.
So cognitive behavioural therapy to treat people and help people with bipolar disorder.
What we can do is we can ask the question is does the mood fluctuations change before or after treatment?
So we can quantify it before they go into treatment for four weeks and we can come out.
They come out the other side and we can characterise their mood dynamics.
So this is what we see.
We're able to sort of like characterise their these mood profiles and just collect them in a, in a temporal time series sequence.
So again, it's just, it's just population dynamics, it's just population ecology, the sort of stuff I grew up doing.
So with a very clinical relevance here.
So what we can do is that we can sort of like pick that apart a little bit and we can ask basically how can we understand mood dynamics?
How can we get to the mass mood, mass we call it, how can we unpick the dynamic, those things that drive mood fluctuations?
And we've done a bunch of things. We've used a whole set of different mathematical methods.
And I'm just going to so I'll talk to you about this one and how we can then link this to some scaling dynamics,
how we scale between different things that may be going on.
So again, another just so a differential equation,
we're going to split mood up into sort of like take them, we'll take the words, we've got change in mood.
So from one time step to the next, very small time step, I'm not going to be explicit about what that is at the moment.
And then we can split up and we can use a total derivative approach. We can split it up into different component parts, essentially.
So I've got some sort of like partial derivative mood respect to time.
So that might just be baseline mood. What happens, you know, this just happens to your mood and just baseline and then we can.
So spitting out further into something to do with fluctuations some things driving this mood fluctuations in our in our heads.
And we've got to really think about how we understand that.
And we can so split that we can sort of think of this the X by D, T and D, Z by T as oscillator, something that's driving that.
So they may be innervate, inhibitory and accumulating oscillators that drive dynamics.
And then there's all these sorts of things. The partial change in mood with respect to X is just a slope.
It's just a regression on how mood changes with respect to the size of the the oscillator that we're
although the the inhibitor or excitatory effect that we're trying to measure so we can write out this.
So this differential equation in terms of the change in mood is just baseline mood that's just going to be flat.
So some constant that's going to change over that's going to that's not going to change over time.
And then again, these things are going to be so there's going to be fluctuations to do this oscillator and this oscillator with some with some
magnitude associated with the parameters in front of that and what we wanted to be able to do and we can do that statistically.
We've done that statistically. And that was what was on the previous slide is what we can try and fit these sorts of models to the Time series.
But I want to get into a little bit more about this. And there's a lot of words and there's a lot of detail on this slide.
And what I want to just explain is that we can we can write that this differential equation or this differential equation as a as an oscillator.
So if thinking about things that inhibit and things that accumulate sort of like this, these dynamical fluctuations and that really cool way,
one of the really cool ways to be able to capture that and phenomenological very broadly without too much mechanistic detail,
is with what we call a relaxation oscillator. And a relaxation oscillator is a really cool thing.
It's a to say it's going to oscillate, it's going to go up and down through time and it's relaxation because actually I'm going
to spend a long time in one state and I'm going to relax very quickly to another state.
So it's just like a friend to price cycle might be. One way to think of it is I'm going to stay in a high state and I'm going to stay there.
And then I go to relaxed very quickly to a low state, and then I go relax out of that state to a high state again.
And we can capture that. We can capture that phenomena logically, very broadly with this this this ordinary differential equation.
So we can couple two things together.
We can put some complicated nonlinear function in there, and then we can capture this idea of a relaxation oscillator.
Well, for bipolar disorder, that's pretty cool, because then we can think about that as I see you're in a state,
maybe you're in a high mood state and you can relax to a baseline state.
You may be in a low mood state and you can relax to a sort of baseline state.
So we are capturing that idea. One idea that might be one idea that we're trying to capture.
We may also be trying to capture this idea that there's inhibition and excitatory processes going on within our heads to sort of drive the dynamics.
We can do other things. So it's a little bit more complicated stuff on this slide.
There's some stuff at the end. So this stuff here and this stuff here is capturing the idea.
We can make these things stochastic so we can kick the system with a little bit of randomness each time.
So that just to add some just to add some additional fluctuations in that that may not be captured very broadly by this by this oscillator.
And the other cool thing we can do is that we can couple these things together.
We can say, actually, what's happening to one isn't independent of what is happening in the other.
So we can really make this these sources, this sort of scaling. But when we can, as I said,
we can fit that to the Time series so we can introduce noise and we can do or we can look at all of these couplings and we can see we
can challenge these models against the data and look at what the what the explanations might be that best describe individual patients.
So you as an individual patient with bipolar you need to oscillators.
Do they have to be couple? Do they have to be noisy? And it really is.
And it really helps us understand the temporal dynamics of bipolar.
So it's just what we do. We consult those three patients just picked at random.
And here's the oscillators that are all oscillating.
They don't have to oscillate. The parameter values, the parameters in the in that oscillator could actually give rise to stable dynamics.
It's just what values they choose choose to take on when we fit them to the Time series.
But each of them, as you can see, everybody each of these three patients has an idiosyncratic.
So a light oscillator system. Sometimes they have to be coupled.
Sometimes they don't. Sometimes they're not. Sometimes they have to be noisy. Sometimes they don't.
And we can fit those to the times, as you can see how good a fit they are, which is the purple dots in this and this in this case,
it's really good technical challenge because actually bipolar patients are pretty bad reporting every time.
So there's lots of missing values in this and we have to be able to deal with those statistically.
And that's all pretty cool, too. Well, we can do that. That's pretty nice. But that's not what I want to get to.
I want to take you to where we are with all of this, because I think it's really cool because what we're trying to do is really understand.
So I've explained this. This is my total derivative. I've got the baseline mood, I've got some parameters associated with these fluctuations,
but we're really trying to do is to step down into heads and think about how the neuronal firing patterns drive mood dynamics.
Well, it's a huge ask, right?
We're actually things no neurone firing dynamics are operating on such a fast time scale compared to how fast your mood changes.
And I like my mood, but my moves, well, I don't know, up and down at the moment, cause I'm standing here talking.
But, you know, it changes, you know, not as fast as a my neurones are going like, oh my God, I've got to keep talking.
They're really firing off in my head to of hopefully make me lucid.
But what we really want to do is to link these sort of these sorts of neurone firing patterns to the to these to this population dynamic,
this, this dynamic of mood change over time. And these are operating on different time scales.
And the way that we're going, the way we're thinking about approaching that is actually approximating these functions.
So these are the these two differential equations are describing the inhibitory and excitatory processes that go on in your own firing patterns.
And we can aggregate those together and we can really think about how we how we can describe we can just take those,
those two differential equations, these two here, and really get into the neurobiology and the theoretical neurobiology and neurobiology of that.
Well, we want to take you up a scale, but these are operating on different time scales.
And the way that we can do that is to approximate these things with a Taylor series.
And a Taylor series is just a way of writing out a function in an infinite sum,
and we can write it out in terms of what at a particular point in time.
And we can take that function and we can just say, Well, what is it, what is it?
What is its linear component and what is its quadratic component?
What is the cubic component? What is its quartette component? And we can just keep adding on it forever and we can get an understanding.
We can approximate these neurone dynamics at a particular point in the mood dynamics with this Taylor expansion, and that holds out great possibility.
It's where we are. We're at this step, and this is where I'll take you to where we are in this sort of problem.
I don't have any answers to that yet,
but we think it's a really cool approach to be able to approximate the neurone dynamics in terms of using these these sorts of Taylor expansions.
So let me finish up with another sort of implication about scaling a different one, completely different one.
Just another little game we'll play very shortly. What I should do is sort of like thinking about bio control people in my group.
I guess a sat around here are interested in thinking about how we control pests,
insect pests, vectors that spread disease, how we can think about the biology,
biological control of doing that,
and how maths helps us understand that there's a very old method in insect control called the sterile insect technique.
And the idea is that you would mass these insects that lower radiation, make them sterile,
release them out into the environment, and when they make, they make no viable offspring.
So they don't so they don't hatch or the insects die or whatever it might be.
So there's the idea being that you're aiming to sort of like to send the population into decline because the meetings are not successful.
Well, there are many new twists on that. And one of the latest ones is being able to use genetics to do that.
So you can genetically modify your favourite agricultural pest or your mosquito spreading disease.
You can change its genes so that when it goes and you must rear them again,
throw them out when they mate with wild types, they don't make any viable offspring.
Those those larvae may die or the caterpillars may die or the eggs may die.
So the idea being that you're trying to suppress the population, we can do it both conventionally with radiation or with genetics.
So the idea is essentially to suppress the population.
There's a whole set of cool maths behind all of that, and it's to do with what we think about as optimal control.
We can do optimisation, we can do static optimisation, we can say, actually, we want to just know what this is for some fixed values,
but actually this is a dynamic problem and somebody said You should shock people.
Mike So here's the shocking bit of this still.
So this is a bit of an old poster, but actually still something like 2000 kids die every day from malaria.
Since I've been speaking over the last hour, probably 50 kids have died from malaria.
What we have is a completely preventable disease.
And using these sorts of methods, these bio control methods, we have an opportunity to be able to do that on our mats.
Really helps us think about that because actually we've got a dynamic.
So this equation at the top DV by d t is the change in the vector numbers, the change in the number of mosquitoes over time.
And it's something to do. There's some bursts of mosquitoes and some deaths of mosquitoes.
And we can fact this thing in brackets is the function that says, I'm going to do this style insect release.
I'm either going to do it genetically or I'm going to do it conventionally, and I'm going to release a number of modified mosquitoes.
I will if i0vtv over v t, it's just going to be is going to be one.
So it's going to cancel out. But it's actually as I increase the number of these star releases,
this term is going to become more important and this term is going to end in a whole term on the first term,
on the right hand side is going to get smaller.
We're aiming to reduce the number of births, but it's not not all that what's happening in this system when we're thinking about malaria,
that we've got people being infected, we've got infected vector.
So we've got a dynamic situation there. We've got dynamics operating in terms of the changes in the vector numbers, being born and being controlled.
What we're trying to do, we're trying to stop people getting sick. So actually we can think about this as an optimal control problem.
This is a dynamic control problem in that we're thinking about actually what we want to do is we want to control this problem over time,
a fixed period of time, we're going to discount it. That's an economics term.
It's a way of scaling thinking in the future.
And we're going to think about what we want to do is we want to minimise this function in terms of the number of people
that are getting sick and the number of the so the number of these sterile insects that we have to put out there.
So also optimisation, we need to this has to be bigger than zero, but we don't want it to be too bigger than zero because that's going to cost us.
It's going to be really costly to do that economically. It's going to be costly to do that.
And this is where we're able to scale both biology and economics.
And what I'd like to do you just to finish up is just to sit on my group, on my on our on our group website.
It's just going to work. We have a game, we can play this game and we'll just play this game because it's really it's really instructions.
It's an ultimate optimal control game and it's so set out without any of the math detail.
It's got lots of the biology in there. So one of the first things you need to do when you're thinking about this,
there's this economics of biocontrol is to know the problem, know what the size of the problem is that you're trying to control.
And the way, way that we do that is we have to estimate, well, the big what the insect, the pest population of the vector population is.
How big is it? Well, ecologists do that with a method called the mark release recapture method.
So you mock a bunch of insects, you throw them out, you wait a bit time,
you go back and you collect a bunch and you work out the proportion that were marked,
which is a mouse, and it tells you what you can use as some wacky formula and it tells you the number of of of mosquitoes.
Oh, that was the wrong one, but. Yes.
Go back. Did you do the record?
No, I did not. Okay. So are you guys not going to do it?
Because we with kids, we get them little kids, we get them to run around and they can find mosquitoes and can do all of this.
So if I pick a number of how many mosquitoes we want in the population, there's some stuff about the biology.
Well, this is where it becomes important. We can now try to optimise this problem and trying to make it as cost effective as possible.
We're trying to scale the economics and the biology to make this as cost effective as possible.
And there are two things. The two things you can control.
Firstly, that I how many sterile mosquitoes, how many sterile insects am I going to put out there?
And then I got to think, how long can I run my program for? So the top one is the ecology, and the bottom one is the economics.
Okay, so on the top one, I can choose any number between one and ten.
Excuse me. A number between one and ten to ask a good number.
Okay, so that means I've got to pay for every wildtype mosquito I'm going to put out there.
I'm going to release two style insects. Okay.
How long do you wanna run your program for? You can have two, five or ten. Well, five was the first number I heard.
That's a good number. Okay, so this we'll see what happens.
So we're going at two graphs.
One of the top one, we're going to do the ecology of the mosquito, the dynamics, the population dynamics of the mosquito.
On the bottom one, we're going to see how many people you managed to save with your particular program.
Okay. And we're going to look for a dollar figure.
And we want that to be as small as possible. The bigger is, the worse it's going to be, essentially.
So I'm going to start on the other thing of ecology is that I need to know when in the season I'm going to start my control.
I'm just going to pick one of these. I'm going to see two lines, basically. And the black line on the top is what I did in the absence of control.
And the green line is what I do in the presence of control. And you can see that actually, I can reduce my.
So I'm doing a 2 to 1 release ratio. So two sterile mosquitoes to every wild type.
And on the bottom, I have the number of sick people in red that in the absence of control and in the presence of control in blue.
And it goes up and down because we're talking about mosquitoes. And they like it wet and dry or they like it.
They don't like it dry. So there's some seasonal dynamics that are essentially what I call things about this is actually when I got to the numbers.
Yeah, well let's talk about the dynamics is not on the top. I only suppressed my population a little bit.
There's 2 to 1 release ratio. It doesn't actually squash the population down very far at all.
Yeah, it's still it's only been suppressed a little bit. But actually I can reduce the disease dynamics by about half the number.
The average number between the red line and the blue line is gone down by half.
So that's pretty cool, right? Actually, I do know.
So one of the things is I don't actually need to eradicate my mosquitoes from the system to start to have an effect on disease control.
If I choose a bigger number, it will be more costly to do that.
And actually, I could get the same sort of phenomena. And as we know, this is a well-known this is a well known fact in epidemiology,
where if you suppress those agricultural pests or those disease vectors be a lower threshold, then, you know, you don't have to eradicate them.
You just have to push them down below that threshold. Then you can have a big effect on the disease dynamics.
We stop our control. After five years, we run out of cash. So that stops after five years.
And actually then because of reinvention, there's really good documented evidence of that happening throughout the world.
So actually we see the disease rebounding to pre control levels.
So what happens then? The most important thing is what this dollar figure is for the better.
The bottom. How many cases did we avert? How many cases do we avoid?
And how much did it cost? How much was the control enough?
What was the optimal value that we can get out of that turns out to 52 is not bad, but we can be better.
I had a seven year old girl do this one time. She spent 45 minutes playing on this, this, this, this game, and she came out with 229.
I've never got down to 229. So if anybody plays this game, it gets below to 29.
I want to know because that's really is really important thinking about how we are able to blend this idea of
economics and ecology in scaling these two things so that different products are operating on different time scales.
You've got to think about how long you run your program for, and that could be many years.
But these mosquitoes and people getting sick is on a daily basis.
And as I said, many, many people still die from from from these sorts of infectious diseases.
Okay. I I'm just about done. I think I've told you about three things, three ways that we can scale slow, fast dynamics.
I didn't do the scaling laws that I might have expected to do, but I've done some dynamical stuff, and I hope you found it interesting.
We can have slow, fast dynamics so we can set those things, one,
going fast and the other we can make really cool approximations and use some very, very cool master work that through.
We worked on the functional response for enterprise interactions in particular, and there's a ways in which we can approximate these things with,
with particular types of functions like the Taylor series, there's really nice ways and think linking things that operate on a different time scales.
We do a raft of things in my group, if anyone is interested. We have a great website, some of them around the room.
If they stick their hands up, they may show their faces, they may not.
They may run out the room thinking, Oh my God, my supervisor is embarrassing me, but that's perfectly fine.
And I'm more than happy to take some questions. Thank you.