The human eye - an evolutionary look

A short introductionA short introductionThe goal of this project was predicting the The goal of this project was predicting the number of generations required for an eye number of generations required for an eye to evolveto evolve . .

I used a genetic algorithm on a population of I used a genetic algorithm on a population of individual eyes, until I got the desired eyeindividual eyes, until I got the desired eye..

A sort of Darwin principle: the strongest eye A sort of Darwin principle: the strongest eye survives each timesurvives each time..

For those of you who forgot For those of you who forgot – some basic theory– some basic theory……

• There are two parameters (which I dealt There are two parameters (which I dealt with) that make one eye fitter than with) that make one eye fitter than another: the first is the shape of the eye another: the first is the shape of the eye and the second is the lens.and the second is the lens.

• The shape of the initial eye is a flat The shape of the initial eye is a flat patch of light-sensitive cells sandwiched patch of light-sensitive cells sandwiched between a transparent protective layer between a transparent protective layer and a layer of dark pigment. and a layer of dark pigment.

Our goal is to achieve better Our goal is to achieve better direction resolution. This is direction resolution. This is achieved by two successive achieved by two successive operations:operations:

•Forming a depression in the Forming a depression in the patchpatch

•Closing the “hole” formedClosing the “hole” formed

Forming a depression in Forming a depression in the patchthe patch

This deepening reduces the angle This deepening reduces the angle at which each cell receives light.at which each cell receives light.

But we still have a But we still have a problemproblem

The problem comes The problem comes from the fact that from the fact that each point in the each point in the object (here - the object (here - the dolphin) sends light dolphin) sends light to many different to many different cells on the retina. cells on the retina. We know that the We know that the object is on our right object is on our right side, but we're not side, but we're not exactly sure where.exactly sure where.

Closing the “hole” Closing the “hole” formedformed To solve this problem To solve this problem

the “hole” is closed:the “hole” is closed:

And then we get a And then we get a clearer picture:clearer picture:

• when the pit depth equals its when the pit depth equals its width, deepening is no longer width, deepening is no longer helpful, and closure of this pit is helpful, and closure of this pit is more important.more important.

• there is a limit to the size of the there is a limit to the size of the opening. As this opening gets opening. As this opening gets smaller the optical image smaller the optical image becomes well resolved but also becomes well resolved but also much dimmer, which leads to much dimmer, which leads to noises.noises.

• The theory of The theory of Snyder tells us that Snyder tells us that the maximum the maximum detectable special detectable special frequency is:frequency is:

• We can then see We can then see the resolution as the resolution as function of A and P:function of A and P:

2max 0.375 ln(0.736 )

Pv A I

A

Resolution is optimal when a/p=0.19

The lensThe lensIn a lensless eye each In a lensless eye each spot is blurred and has spot is blurred and has the size of the aperture.the size of the aperture.

A lens situated in the A lens situated in the aperture will converge the aperture will converge the light into one place, light into one place, making the object making the object sharper, without sharper, without decreasing the brightness decreasing the brightness of the image, which would of the image, which would have happened if we tried have happened if we tried to achieve sharpness by to achieve sharpness by further closure of the further closure of the aperture.aperture.

Genetic algorithmGenetic algorithm

• A genetic algorithm is an iterative A genetic algorithm is an iterative procedure that consists of a constant-procedure that consists of a constant-size population of individuals.size population of individuals.

• Every evolutionary step, which is Every evolutionary step, which is called generation, the individuals are called generation, the individuals are evaluated according to some pre-evaluated according to some pre-defined quality criterion called fitness defined quality criterion called fitness

Genetic algorithmGenetic algorithm

Forming a new generation consists of Forming a new generation consists of three major stages:three major stages:

• SelectionSelection – individuals are selected with a – individuals are selected with a probability proportional to their relative probability proportional to their relative fitness. fitness.

• Reproduction Reproduction – two selected individuals – two selected individuals change parameters by some probability.change parameters by some probability.

• MutationMutation – there is a small chance for – there is a small chance for each individual to mutate. That means each individual to mutate. That means change a parameter by chance.change a parameter by chance.

Fitness functionFitness function

The heart of the algorithm - the The heart of the algorithm - the “fitness” function.“fitness” function.

This function should be the quality of This function should be the quality of vision, because that should be vision, because that should be reason for an eye to evolve in a reason for an eye to evolve in a certain way. certain way. 1 1

fitness=5.263-(p/a) p-lens

The algorithm’s The algorithm’s parametersparameters

The eye has 3 parameters: The eye has 3 parameters:

• the length of the light-sensitive patch the length of the light-sensitive patch – it is constant. – it is constant.

The next 2 contribute to the fitness:The next 2 contribute to the fitness:

• radius (which is the inverse of the radius (which is the inverse of the curvature of the patch) curvature of the patch)

• the lens focal length. the lens focal length.

How the algorithm worksHow the algorithm works• We start with a population of 100 eyesWe start with a population of 100 eyes• The eyes are sorted by fitnessThe eyes are sorted by fitness• Each time 2 eyes are selectedEach time 2 eyes are selected

The selection function:The selection function:: : 99-(random^2)*10099-(random^2)*100

• These eyes reproduce and have a childThese eyes reproduce and have a child• This child mutatesThis child mutates

after a 100 children are born, they are the after a 100 children are born, they are the new generationnew generation

The different stagesThe different stages

The different stagesThe different stages

The different stagesThe different stages

ResultsResultsThe results depend on a lot of different The results depend on a lot of different choices made: choices made:

• What is the probability for mutationWhat is the probability for mutation• By how much to mutate each timeBy how much to mutate each time• What are the initial factors givenWhat are the initial factors given• What is the percentage of children that What is the percentage of children that

look exactly like their parentslook exactly like their parents• The different possible options for a The different possible options for a

selection functionselection function• And – of course, the fitness function.And – of course, the fitness function.

Results for different Results for different mutation probabilitymutation probability

mutationmutationAverage Average number of number of generatiogeneratio

nsns

MaximuMaximum errorm error

0.0010.001 1727417274 6%6%

0.0040.004 60406040 5%5%

0.0070.007 42364236 3%3%

Results for a different Results for a different amount of mutation each amount of mutation each timetime

Amount Amount of changeof change

Average Average number of number of generatiogeneratio

nsns

MaximuMaximum errorm error

1%1%1727417274 6%6%

00 . .1%1%173063173063 ~~0%0%

conclusionsconclusions

• The results are a very rough estimation, of The results are a very rough estimation, of course, but they let us see that it is course, but they let us see that it is possiblepossible

• the number of generations it takes the eye the number of generations it takes the eye to evolve varies a lot by the parameters to evolve varies a lot by the parameters introduced introduced

• in every group where the parameters in every group where the parameters stayed the same, the results were very stayed the same, the results were very close to one another close to one another

The results show the The results show the eye’s structure up to eye’s structure up to now.now.We’ll leave the future We’ll leave the future to evolutionto evolution!!