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Lectures on Cellular Automata Continued. Modified and upgraded slides of Martijn Schut [email protected] Vrij Universiteit Amsterdam Lubomir Ivanov Department of Computer Science Iona College and anonymous from Internet . Morphogenetic modeling . Dynamical Systems

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Slide1 l.jpg

Lectures on

Cellular Automata Continued

Modified and upgraded slides of

Martijn Schut

[email protected]

Vrij Universiteit Amsterdam

Lubomir Ivanov

Department of Computer Science

Iona College

and anonymous from Internet


Morphogenetic modeling l.jpg

Morphogenetic modeling

Dynamical Systems

and Cellular Automata


Slide3 l.jpg

Organisms are dissipative processes in space and time

matter

energy

matter

heat

entropy dissipator


Dynamical system l.jpg

Model animals as dynamical systems

Tissues

Groups

Space

Dynamical system

Energy

Time


Which space which geometry l.jpg

morphometric space

S

A

Q=S/A

Which space? - which geometry?

geometric space

y

time

x

?

state space

parameter space

velocity

b

a

q


Which space which geometry6 l.jpg

morphometric space

morphology

Which space? - which geometry?

geometric space

ontogeny/phenotype

?

parameter space

state space

genotype

environment

GA evolution environment

phylogeny/ontogeny

Dynamics of species in environments.


Discretization has many aspects l.jpg
Discretization has many aspects

t

  • discrete time stepsT=0,1,2,3,…

  • discrete cells at X,Y,Z=0,1,2,3,...

  • discretize cell statesS=0,1,2,3,...

t+1

We can mix discrete and continuous values in some models

t+2


Cellular automaton l.jpg
Cellular Automaton

In standard CA the values of cells are discretized

t

neighbours

update rules

t+1


A simple cellular automaton l.jpg
A simple Cellular Automaton

  • Reduce dimensions: D=1, i.e. array of cells

  • Reduce # of cell states: binary, i.e. 0 or 1

  • Simplify interactions: nearest neighbours

  • Simplify update rules: deterministic, static


Effects of simplification l.jpg
Effects of simplification

CA

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Wolfram s 1d binary ca l.jpg
Wolfram's 1D binary CA

  • cell array

  • binary states

  • 5 nearest neighbours

  • „sum-of-states“ update rule:

S

Wolfram, S., Physica10D (1984), 1-35

Observe importance of symmetry of logic function

Number of “alive” neighbors


Slide12 l.jpg

Wolfram‘s 32 „Symmetry-based“ CA rules in 5-neighborhood

S{1,2,3,4,5}

S

S

S{0,1}

 25=32 different rules

Number of “alive” neighbors


Code for rule 6 00110 l.jpg
Code for rule 6 (00110)

S neighbours

1

2

3

4

5

S(i,T+1)

0

0

1

1

0

Number of “alive” neighbors


Rule codes l.jpg
Rule codes

0

0

0

0

0

rule 0

0

0

0

0

1

rule 1

0

0

0

1

0

rule 2

0

0

0

1

1

rule 3

0

0

1

0

0

rule 4

0

0

1

0

1

rule 5

1

1

1

1

1

rule 32

Number of “alive” neighbors


Temporal evolution l.jpg
Temporal evolution

initial configuration (t=0)

e.g. random 0/1

rule n

t=1

rule n

t=2

rule n

t=3



Rule 6 00110 l.jpg
Rule 6: 00110

t=0

t=100


Rule 10 01010 l.jpg
Rule 10: 01010

t=0

t=100


Ca a metaphor for morphogenesis l.jpg

static binary pattern

translated into

state transition rule

rule iteration in configuration space

genotype

translation,epigenetic rules

morphogenesis of the phenotype

1

2

3

4

5

0

1

0

1

0

0

1

0

1

0

CA: a metaphor for morphogenesis


Ca a model for morphogenesis l.jpg

update rule

iterative application of the update rule

CA state pattern

state space

genotype

epigenetic interpretation of the genotype, morphogenesis

phenotype

morphospace

CA: a model for morphogenesis


Ca rule mutations l.jpg

0

1

0

1

0

CA rule mutations

1

0

0

1

0

swap

genotype

negate

development

phenotype

0

1

0

1

1

How genotype mutations can change phenotypes?


Ca rule mutations22 l.jpg

0

1

1

1

0

1

1

0

1

0

1

0

CA rule mutations

0

1

0

1

0

1

0

0

1

0

rule

0

1

1

0

0

1

0

1

0

0

1

0

1

swap

negate

0

1

0

0

1

0

0

0

1

1


Predictability l.jpg
Predictability?

pattern at t

?

explicit simulation:

D iterations of rule n

hypothetical

"simple algorithm" ?

pattern at t+Dt

How to predict a next element in sequence? Tough!


Predictability24 l.jpg
Predictability

Sum of natural numbers 1… N

algorithm 1:

1+2+…+N

algorithm2:

N(N+1)/2

(Gaussian formula)

Nth prime number

algorithm 1:

trial and error

algorithm2: ?

no general formula

easy

Very tough!


Ca no effective pattern prediction l.jpg
CA: no effective pattern prediction

pattern at t

pattern at t+Dt

behaviour may be determined only by explicit simulation


Ca deterministic unpredictable irreversible l.jpg
CA: deterministic, unpredictable, irreversible

  • Simple rules generate complex spatio-temporal behavior

  • For non-trivial rules, the spatio-temporal behaviour is computable but not predictable

  • The behavior of the system is irreversible

  • Similarity of rules does not implysimilarity of patterns


Aspects of ca morphogenesis l.jpg
Aspects of CA morphogenesis

  • complex relationship between „genotype“ and „phenotype

  • effects of „genes“ are not localizable in specific phenes (pleiotropy)

  • phenes cannot be traced back to specific single genes (epistasis)

  • phenetic effects of "mutations" are not predictable


Slide28 l.jpg

Meinhardt, H. (1995). The Algorithmic Beauty of Sea Shells. Berlin: Springer


Patterns and morphology l.jpg

Pattern: a spatially and/or temporally ordered distribution of a physical or chemical parameter

Pattern formation

Form (Size and Shape)

Morphogenesis: The spatiotemporal processes by which an organism changes is size (growth) and shape (development)

Patterns and morphology


Where is the phene l.jpg
Where is the phene? of a physical or chemical parameter

phenotype A

  • Typification?

  • Comparative measures?

    • length

    • density

    • fractal dimension

  • Spatio-temporal development?

phenotype B


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