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Welcome back to this course on modelling and simulation of natural processes.

So we continue our discussion on modelling and

the module now that I'm going to discuss is about the fact that

is meant as a mathematical abstraction of a physical system or a real system.

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And you have the macroscopic scale,

where usually you would like to have some answer.

You have the microscopic scale, where you have atoms.

And with [INAUDIBLE] you have maybe something we call

mesoscopic in the sense that it's an abstraction of the macroscopic

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So why such simple, discreet dynamical system can be a good model of reality?

That's certainly a question you may ask after seeing the model or

the length model, you can, if you wonder what makes this system close to reality,

and one important response to that is that,

we know from statistical physics that microscopic behaviour of

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world or the macroscopic level.

So what really makes the macroscopic system what they are is the symmetry or

the conservation law that prevail at the microscopic scale among

the interaction with the elementary.

So the idea would be why don't we imagine

a fictitious world, which would be very easy

to implement on a computer, which has the proper symmetries and conservation law.

And then yield the expected macroscopic behavior.

So as an example, we can remember that water, air,

they're all made of molecules, the interaction between water molecules and

air molecules might be quite different.

But anyway both system they evolve or

they obey to [INAUDIBLE] equation for hydrodynamics.

Even sand sometimes can be described by the same equation even though

the interaction between grain of sands might be very different

from water molecules, so you see that even though at a small scale,

the interaction may be different, at the large scales,

everything fade out and you may recover the same microscopic behavior.

So that's the idea, we can invent our own microscopy.

So that it gives you a right, large-scale effect.

So that's a reason why several [INAUDIBLE] are good.

You abstract the microscope at the mesoscopic level by simple discrete

system, and then you hope that you can reproduce something at the large scale.

So it's a very simple and intuitive approach, and that's interesting

because sometimes it's very hard to write equation, but in a discreet universe it's

maybe easier to implement some [COUGH] rule describing nature.

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So, I'm trying to illustrate this idea with this simple example,

so this a ceilometer rune and I'm saying that it's a caricature of reality.

So, if I'm asking you what this is,

I guess most people they immediately say, it's a snow flake.

Okay?

So our eyes recognize a variety of snowflakes out of this image.

So this is three images because it's just a gross evolution of the model

from an initial small system to a big and

fully develop snowflakes.

I'm saying it's just a caricature of reality because it you look at real

snowflakes, they are all different, first, and

none is really like the one we've seen.

So how come we can recognize a snowflake out of

this because this one probably never have been shown anywhere.

Anyways, because it has some symmetry, and

some characteristic that makes us recognize it immediately.

What are these specific feature that have

been captured that goes by nature and

how our accelerometer we can try to explain in this way.

So in real life, in physics,

a snowflake's built out of a vapor which can solidify.

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So it vapor turning to ice.

And the way it is possible is that

the vapor should be close to an initial piece of ice,

or an initial dust in the atmosphere.

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But not close to too many of these ice particle.

Because if it's their close to too many, it will just not solidified.

It will just spread to all the neighbor and

we will not see a new element building up snowflakes.

So, that's the idea that vapor molecules become ice if it's neighbor of only one

already frozen molecules, but if it's neighbor of two it will not, okay.

So that's the first ingredient that is directly taken from

what we understand from the real snowflakes.

And the second is the symmetry, the geometry of the system,

which mean that growth can only 60 degrees because that's

what a water molecule like, they like this angle of 60 degrees.

So you put two, these two ingredient in the discrete model and

you get essentially what I showed you on this slides.

Which of course different from the real stuff but

still close enough that we recognize it.

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So now I like to cover quickly a few more of this

discrete rules which have a very interesting global behaviour and

this one is a rule where you reproduce the gross

of a system made of two possible components.

So initially you have a soup of random molecule

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either blue or red and, I showed you, this molecule they can

change color and as time goes on you build a much bigger cluster.

So what is this very simple rule?

We call it biased majority rule.

So for each cells you just compute.

How many neighbors you have of one given color?

So you can have zero neighbor, which is on the given color.

One neighbor, two neighbor, or all the nine neighbor, around the same column.

So basically at the next stage, you do the same as the majority of your neighbors.

So if you have few guys which are one or zero,

if you have many guys around you which one you get one.

But here you have this weird idea that for the middle you just

do the opposite of what it would naturally be, to put the one here and the zero here.

So that's what we call the Biased Majority Rule because for

the middle entry you do the opposite of what you would like.

And that gives you this very nice picture.

So I will show you an animation.

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So, you see here the system evolving.

And actually, there were at two different scale first.

It was the initial situation from random to something which start building up.

Then there's a jump in time where we see the final stage.

And actually, it's not the final stage because if you would keep evolving this,

you would finally find one big red cluster and one big blue cluster.

And what is actually interesting here is that you can show

that the speed at which this domain,

the growth is proportional to the curvature of the domain,

which is exactly what we know from physics.

So, this very simple bias majority rule captures the growth which

is proportional to the curvature of the domain, which of course,

is looks a bit magical on the first side that you see there.

So for those who are interested,

you can run many of these animation from this website.

It's actually the one I'm using now.

And you will have [COUGH] the same as I'm showing plus other one.

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differentiation which make that some cells become functional for some particular

function and the early stage you see for

real embryo that a bit less than 25% of the initial cells.

They become neural cells.

Okay?

And the rest they are used to build the body of the fly.

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that each cell they try two different shades and

they produce a substance That we call S because just a protein.

And this substance, if you see in large enough concentration in the cell,

you would just unload the differentiation to neuroblasts.

But then there's another mechanism, which is the competition which enable so

you would like to be the one selected to be nerveless and so you try to

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the production of this protein to your neighbors.

Okay?

So, that's what biologist think could be the mechanism of initiation but

of course it doesn't need it explained to 25%.

So let's see if we can translate that into a very discreet model,

where we need this competition and inhibition process to differentiate.

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So first we'll use a hexagonal lattice because we know that the cells are more,

like hexagonal than squares.

And yeah it is that in each cell you can have the value of the substance

which is either 0, meaning that it's inhibited, or 1,

meaning that it's active and ready to differentiate.

[COUGH] Now if you have not yet

produce the C, the S, sorry.

You want to do it, differentiate, so you want to produce some of this S.

So you wanna turn to S equal one, but

this is only possible if all your neighbors are also zero.

Because otherwise you keep feeling this inhibition.

So if you are 0 and all your neighbors are 0, you wanna turn to 1 and

you do it with some probability that we call here Pgrow.

And if you are in state 1 [COUGH] you may turn back to state 0 if your

neighbors already in state 1 because then you are in competition with your neighbor.

Okay, so the idea that if you 1 and nobody around you is 1, you stay 1.

But if they are enabled, which are 1, then you may turn back to 0.

And this is done with a probability that we call PDK,okay.

So if you do that, you realize that you have two possible stable states,

which are depicted on these two images, okay.

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And in each of these two situations you can see that

the black cells are the different shaped ones and

they are stable and the white cells they are inhibited and they are stable as such.

And they have density which is here one celled.

Sorry, one third here and one over seven here.

Okay.

So it's not really the 25% that we were looking at.

It's two extreme cases which are around that.

But this is really in extreme cases If you run the CA module with this probability,

you see that actually the steady state that you get as 23 person of

this black spot.

And that is exactly, or very close,

to what biologists measure in the cell embryo.

The good news is that it's almost independent on this known parameter which

I've been discussing before so it mean that this parameter which describe

the strength of the inhibition and the competition they are not known, but

they are totally irrelevant to get this Is value.

So our model is robust to the lack of detail and

it, on the other hand, needs very much the sectagonal lattice.

If you do the same on a square lattice, you get definitely another value.

So we see that again we had to put the right symmetry, and

the right conservation law, or mechanism of interaction to explain an observation.

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contagion model so epidemy propagation.

So, it's c model, where you have three possible states.

So, you can call the first normal, or resting, depending on your vocabulary.

The second one is the excited state.

That's value two, and it's also the state where you can propagate and

epidemic or you're contagious.

Stage three is called refractory.

Or it may only mean that you are not back to normal,

that you cannot get sick anymore, and you cannot propagate the sickness.

So, the typical rule that you can imagine about this is,

if you are in a excited state, you will turn into a refractory state.

If you're in a refractory state, you will end up in a normal state.

So that's the evolution of the illness, if you want.

But if you are in a normal state, you can stay normal unless you have

neighbors which are in this contagious state or excited state.

So normal can become excited if there are enough excited neighbors around you.

Okay, so that's the basic idea.

And this is well illustrated in this Greenberg-Hastings Model

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where the possible state goes from zero to n minus one.

And the normal state is x equals zero.

And then you have half of the state which are evolution of the sickness,

if you want.

And the rest are refractory.

So basically, the fact that you have more than just three state is to have some

time duration of the event.

And if you start with an initial state where you put some randomly

excited states in a background of normal states, you'll see that these dots,

they will start developing and they start building a very nice pattern like waves.

So, I wanna show you a demo for that.

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Another model, it's a bit more complex.

I'm not gonna go into the detail.

I just want to show you the demo.

It's supposed to represent the famous Belousov-Zhabotinski

reaction which has been coined tube worm by homologous, and

I'll let you read the rule if you want, but I just want now to run it.

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And last example of this type of excitable media is a model for forest fire.

So the idea that here you have a green state which are normal tree.

You have yellow, which is a burning tree, and you have black,

which is a burned tree or ashes.

And the dynamic is very simple.

If you're a normal tree, green tree,

next to a burning one then you catch fire.

If you catch fire then you turn into ashes, you'll become black.

And after sometimes from the ashes there's a new tree then that will grow.

And again we can see how here you have a snapshot of a situation, but

I want to run it again on the real time evolution system,

where you see how this pattern grow and [INAUDIBLE].

Okay, and with this I would like to finish this set of examples

where I showed you that a CA can be a mathematical abstraction of reality.

And in the next module I would like to illustrate three

rules to describe the car traffic.