0:15

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

Â So we continue our discussion on modelling and

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

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

Â 0:51

And you have the macroscopic scale,

Â where usually you would like to have some answer.

Â You have the microscopic scale, where you have atoms.

Â And with [INAUDIBLE] you have maybe something we call

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

Â 1:21

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

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

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

Â and one important response to that is that,

Â we know from statistical physics that microscopic behaviour of

Â 2:00

world or the macroscopic level.

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

Â the conservation law that prevail at the microscopic scale among

Â the interaction with the elementary.

Â So the idea would be why don't we imagine

Â a fictitious world, which would be very easy

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

Â And then yield the expected macroscopic behavior.

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

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

Â air molecules might be quite different.

Â But anyway both system they evolve or

Â they obey to [INAUDIBLE] equation for hydrodynamics.

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

Â the interaction between grain of sands might be very different

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

Â the interaction may be different, at the large scales,

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

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

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

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

Â You abstract the microscope at the mesoscopic level by simple discrete

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

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

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

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

Â 4:13

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

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

Â So, if I'm asking you what this is,

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

Â Okay?

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

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

Â from an initial small system to a big and

Â fully develop snowflakes.

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

Â snowflakes, they are all different, first, and

Â none is really like the one we've seen.

Â So how come we can recognize a snowflake out of

Â this because this one probably never have been shown anywhere.

Â Anyways, because it has some symmetry, and

Â some characteristic that makes us recognize it immediately.

Â What are these specific feature that have

Â been captured that goes by nature and

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

Â So in real life, in physics,

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

Â 5:41

So it vapor turning to ice.

Â And the way it is possible is that

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

Â or an initial dust in the atmosphere.

Â 6:00

But not close to too many of these ice particle.

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

Â It will just spread to all the neighbor and

Â we will not see a new element building up snowflakes.

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

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

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

Â what we understand from the real snowflakes.

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

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

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

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

Â you get essentially what I showed you on this slides.

Â Which of course different from the real stuff but

Â still close enough that we recognize it.

Â 7:09

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

Â discrete rules which have a very interesting global behaviour and

Â this one is a rule where you reproduce the gross

Â of a system made of two possible components.

Â So initially you have a soup of random molecule

Â 7:35

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

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

Â So what is this very simple rule?

Â We call it biased majority rule.

Â So for each cells you just compute.

Â How many neighbors you have of one given color?

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

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

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

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

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

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

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

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

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

Â And that gives you this very nice picture.

Â So I will show you an animation.

Â 8:58

So, you see here the system evolving.

Â And actually, there were at two different scale first.

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

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

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

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

Â And what is actually interesting here is that you can show

Â that the speed at which this domain,

Â the growth is proportional to the curvature of the domain,

Â which is exactly what we know from physics.

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

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

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

Â So for those who are interested,

Â you can run many of these animation from this website.

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

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

Â 10:35

differentiation which make that some cells become functional for some particular

Â function and the early stage you see for

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

Â They become neural cells.

Â Okay?

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

Â 11:13

that each cell they try two different shades and

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

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

Â you would just unload the differentiation to neuroblasts.

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

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

Â 11:43

the production of this protein to your neighbors.

Â Okay?

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

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

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

Â where we need this competition and inhibition process to differentiate.

Â 12:11

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

Â like hexagonal than squares.

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

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

Â meaning that it's active and ready to differentiate.

Â [COUGH] Now if you have not yet

Â produce the C, the S, sorry.

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

Â So you wanna turn to S equal one, but

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

Â Because otherwise you keep feeling this inhibition.

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

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

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

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

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

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

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

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

Â which are depicted on these two images, okay.

Â 13:42

And in each of these two situations you can see that

Â the black cells are the different shaped ones and

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

Â And they have density which is here one celled.

Â Sorry, one third here and one over seven here.

Â Okay.

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

Â It's two extreme cases which are around that.

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

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

Â this black spot.

Â And that is exactly, or very close,

Â to what biologists measure in the cell embryo.

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

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

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

Â they are totally irrelevant to get this Is value.

Â So our model is robust to the lack of detail and

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

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

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

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

Â 15:24

contagion model so epidemy propagation.

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

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

Â The second one is the excited state.

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

Â epidemic or you're contagious.

Â Stage three is called refractory.

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

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

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

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

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

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

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

Â neighbors which are in this contagious state or excited state.

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

Â Okay, so that's the basic idea.

Â And this is well illustrated in this Greenberg-Hastings Model

Â 16:45

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

Â And the normal state is x equals zero.

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

Â if you want.

Â And the rest are refractory.

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

Â time duration of the event.

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

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

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

Â So, I wanna show you a demo for that.

Â 17:55

Another model, it's a bit more complex.

Â I'm not gonna go into the detail.

Â I just want to show you the demo.

Â It's supposed to represent the famous Belousov-Zhabotinski

Â reaction which has been coined tube worm by homologous, and

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

Â 18:46

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

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

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

Â which is a burned tree or ashes.

Â And the dynamic is very simple.

Â If you're a normal tree, green tree,

Â next to a burning one then you catch fire.

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

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

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

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

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

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

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

Â And in the next module I would like to illustrate three

Â rules to describe the car traffic.

Â