1 Cellular Automata A dynamic system Invented by John von Neumann - - PDF document

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1 Cellular Automata A dynamic system Invented by John von Neumann - - PDF document

Mar 17, 2023 150 likes •317 views

Last time Nonlinear dynamic systems The Logistic map Strange attractors The Hnon attractor The Lorenz attractor Producer-consumer dynamics Equation-based modeling Individual-based modeling 9/11 - 05 Emergent

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9/11 - 05 1 Emergent Systems, Jonny Pettersson, UmU

Last time

Nonlinear dynamic systems

The Logistic map

Strange attractors

The Hénon attractor The Lorenz attractor

Producer-consumer dynamics

Equation-based modeling Individual-based modeling

9/11 - 05 2 Emergent Systems, Jonny Pettersson, UmU

Outline for today

Cellular automata

One-dimensional Wolfram’s classification Langton’s lambda parameter Two-dimensional

  • Conway’s Game of Life

Pattern formation in slime molds

Dictyostelium discoideum Modeling of pattern

9/11 - 05 3 Emergent Systems, Jonny Pettersson, UmU

Complex System

Things that consist of many similar and

simple parts

Often easy to understand the parts The global behavior much harder to explain On many levels Some are capable of universal computation

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9/11 - 05 4 Emergent Systems, Jonny Pettersson, UmU

Cellular Automata

A dynamic system Invented by John von Neumann

With help from Stanislaw Ulam 1940s Wanted to understand the process of

reproduction

The essence

9/11 - 05 5 Emergent Systems, Jonny Pettersson, UmU

One-Dimensional CA

Linear grid of cells Each cell can be in one of k different states Next state is computed as an function of the states

  • f neighbors (and own state)

Neighborhood

r = radius neighborhood = 2r + 1 9/11 - 05 6 Emergent Systems, Jonny Pettersson, UmU

1-Dim CA - Example

k = 2, r = 1

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9/11 - 05 7 Emergent Systems, Jonny Pettersson, UmU

1-Dim CA - General

Neighborhood size = 2r + 1 k different states a rule table with k2r + 1 entries number of legal rule tables, k^k2r + 1 Usually wrap-around of the linear grid Initial population?

Random or a few ”on” 9/11 - 05 8 Emergent Systems, Jonny Pettersson, UmU

Wolfram’s CA Classification

Stephen Wolfram 1980s Resurrected cellular automata research A ring of n cells with k possible states kn

different configurations of a row

Four different CA classes

9/11 - 05 9 Emergent Systems, Jonny Pettersson, UmU

Wolfram’s CA – Class 1

Static Compared to fractals – fixed point

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9/11 - 05 10 Emergent Systems, Jonny Pettersson, UmU

Wolfram’s CA – Class 2

Periodic Compared to fractals

– limit cycles

9/11 - 05 11 Emergent Systems, Jonny Pettersson, UmU

Wolfram’s CA – Class 3

Random-like Compared to fractals – chaos, instable limit cycles

9/11 - 05 12 Emergent Systems, Jonny Pettersson, UmU

Wolfram’s CA – Class 4

Complex patterns with

local structures

Can perform

computation, some even universal computation

Not regular, periodic

  • r random

Between chaos and

periodicity

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9/11 - 05 13 Emergent Systems, Jonny Pettersson, UmU

Langton’s Lambda Parameter

Chris Langton – ”Founder” of Artificial Life Was searching for a virtual knob to control the

behavior of a CA

Quiescent state – inactive, off Number of entries in a rule table, N = k2r + 1 Number of entries in a rule table that map to the

quiescent state, nq λ = (N – nq )/N

λ = 0 the most homogeneous rule table λ = 1 all rules map to non-quiescent states λ = 1 – 1/k the most heterogeneous

9/11 - 05 14 Emergent Systems, Jonny Pettersson, UmU

Langton’s λ - Example

Comparing with the examples of Wolfram’s

classes

The most heterogeneous:

λ = 1 - 1/k = 1 – 1/5 = 0.8

Class 1: λ = 0.22823267 (average) Class 2: λ = 0.43941967 (average, biased) Class 3: λ = 0.8164867 (average) Class 4: λ = 0.501841 (average)

9/11 - 05 15 Emergent Systems, Jonny Pettersson, UmU

Langton’s λ - Parameter

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9/11 - 05 16 Emergent Systems, Jonny Pettersson, UmU

Langton’s λ - Problems

1.

One rule set can have a high λ but still produce very simple behavior

  • 2. Little information in a singular value
  • 3. λ says nothing certain about the long-

term behavior

Dangerous to map to a single scalar

number

9/11 - 05 17 Emergent Systems, Jonny Pettersson, UmU

2-Dim CA – Conway’s Game of Life

John Conway, 1960s Wanted to find the simplest CA that could

support universal computation

k = 2, very simple rules 1970, Martin Gander described Conway’s

work in his Scientific American column

A global collaborative effort succeeded

9/11 - 05 18 Emergent Systems, Jonny Pettersson, UmU

Conway’s Game of Life

2-dimensional 8 neighbors Rules:

If a cell is alive and exactly two or three of its eight

neighbors are alive, it stays alive; otherwise it dies

If a cell is dead and exactly three of its neighbors are

alive, it comes to life

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9/11 - 05 19 Emergent Systems, Jonny Pettersson, UmU

Conway’s Game of Life – Universal computation

Static objects memory

9/11 - 05 20 Emergent Systems, Jonny Pettersson, UmU

Conway’s Game of Life – Universal computation

Periodic objects counters

9/11 - 05 21 Emergent Systems, Jonny Pettersson, UmU

Conway’s Game of Life – Universal computation

Moving objects moving information

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9/11 - 05 22 Emergent Systems, Jonny Pettersson, UmU

Conway’s Game of Life – Universal computation

Breeders, glider guns

collide to make new moving objects

9/11 - 05 23 Emergent Systems, Jonny Pettersson, UmU

Conway’s Game of Life – Universal computation

To implement universal computation

NOT and (AND or OR)

Computing science theorist: The rest are boring

details

9/11 - 05 24 Emergent Systems, Jonny Pettersson, UmU

Natural CA-like Phenomena

Describes phenomena that occur on

radically different time and space scales

Statistical mechanical systems

Lattice-gas automaton

Autocatalytic chemical sets

The Belousov-Zhabotinsky reaction

Gene regulation Multicellular organisms Colonies and ”super-organisms” Flocks and herds Economics and Society

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9/11 - 05 25 Emergent Systems, Jonny Pettersson, UmU

Pattern Formation in Slime Molds

Self-organization resulting in patterns The Belousov-Zhabotinsky reaction Honeybees Patterns generated by organisms midway in

complexity

Dictyostelium discoideum

9/11 - 05 26 Emergent Systems, Jonny Pettersson, UmU

  • D. Discoideum – Life Cycle

9/11 - 05 27 Emergent Systems, Jonny Pettersson, UmU

  • D. Discoideum – Amoebae Stage

Growth phase Free moving single cell Lives in soil engulfs

bacteria

Divides asexually

Doubling time ~ 3h (Picture from dictybase.org)

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9/11 - 05 28 Emergent Systems, Jonny Pettersson, UmU

  • D. Discoideum – Amoebae

(Video from dictybase.org) 9/11 - 05 29 Emergent Systems, Jonny Pettersson, UmU

  • D. Discoideum – Aggregation Stage

When starving

developmental phase

Aggregate by chemotaxis Multiple concentric

circles and spirals

Up to 100000 individuals 1 frame/36 sec Wave propagation 60 –

120 µm/min

Spiral accelerate cell

aggregation (18 vs 3 µm/min)

(Video from Zool. Inst. Univ. München) 9/11 - 05 30 Emergent Systems, Jonny Pettersson, UmU

  • D. Discoideum – Spiral Waves

Spiral formation unclear, involves symmetry breaking 1 frame/10 sec

(Video from Zool. Inst. Univ. München)

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9/11 - 05 31 Emergent Systems, Jonny Pettersson, UmU

  • D. Discoideum – Stream Formation

Stage

Streams depends

  • n movement and

symmetry breaking

Begin to form slug

(Picture from R. Firtel, UCSD (dictybase.org)) 9/11 - 05 32 Emergent Systems, Jonny Pettersson, UmU

  • D. Discoideum – Mound Stage

10000 – 100000 cells Cells begin to differentiate 1 frame/20 sec

(Video from Zool. Inst. Univ. München) 9/11 - 05 33 Emergent Systems, Jonny Pettersson, UmU

  • D. Discoideum – Multi-armed

spirals

Up to 10 spirals have been observed This mound has 5 spirals

(Video from Zool. Inst. Univ. München)

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9/11 - 05 34 Emergent Systems, Jonny Pettersson, UmU

  • D. Discoideum – Slug Stage

Behaves as single organism Migrates; seeks light, seeks or avoids heat No brain, no nervous system 1 frame/10sec

(Video from Zool. Inst. Univ. München) 9/11 - 05 35 Emergent Systems, Jonny Pettersson, UmU

  • D. Discoideum – Culmination Stage

Cells differentiate into base, stalk and spores 1 frame/5 sec

(Video from Zool. Inst. Univ. München) 9/11 - 05 36 Emergent Systems, Jonny Pettersson, UmU

  • D. Discoideum – Fruiting Body

Stage

Spores are dispersed,

wind or animal

If sufficient moisture,

spores germinate, release amoebas

Cycle begins again

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9/11 - 05 37 Emergent Systems, Jonny Pettersson, UmU

  • D. Discoideum – Life cycle

(Picture from Zool. Inst. Univ. München) 9/11 - 05 38 Emergent Systems, Jonny Pettersson, UmU

Modeling the waves

The waves are probably the result of spiral

waves of cyclic adenosine 3’,5’- monophosphate (cAMP) in an excitable medium

Spirals rotate fastest, push other

activities to the border Bigger slugs Better dispersal Selective pressure

9/11 - 05 39 Emergent Systems, Jonny Pettersson, UmU

Biological Basis of Aggregation in

  • D. discoideum

Not exactly clear how cAMP plays a vital role Starved D. discoideum produce cAMP

release into the environment

cAMP released in two ways

Oscillatory release (with a period of 5-10 min) Relay

Positive and negative feedback at the level

  • f the cAMP receptor
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9/11 - 05 40 Emergent Systems, Jonny Pettersson, UmU

What to model?

Cells moves toward higher concentration

  • f cAMP, approximately one-tenth the

speed of cAMP waves

Divide the modeling into two parts

1.

Let the cells be static and model the cAMP waves without diffusion of cAMP

2.

Add diffusion of cAMP and cell movement

9/11 - 05 41 Emergent Systems, Jonny Pettersson, UmU

Equation-Based Modeling

Modeling oscillations and relay The cAMP receptor has four states The equations... What happens when the extracellular

concentration of cAMP suddenly increase?

Diffusion give a wave of cAMP that

stimulate cells to move

A spiral wave may be generated when

individual cells spontaneously fire off pulses of cAMP in a random manner

9/11 - 05 42 Emergent Systems, Jonny Pettersson, UmU

A Model for Cell Movement in Response to cAMP

Three assumptions:

Each cell can detect the cAMP

gradient and move in the direction of increasing cAMP concentration

A prolonged cAMP stimulus

decreases the cells’ ability to detect cAMP gradients

Cell-cell adhesion comes into play

  • nce the cells are close enough

to one another, and thus once a clump forms it cannot quickly disperse

(Picture from R. Firtel, UCSD (dictybase.org))

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9/11 - 05 43 Emergent Systems, Jonny Pettersson, UmU

Individual-Based Modeling

A Cellular Automata Model of Spiral Waves Two end states A positive feedback mechanism A negative feedback mechanism The NetLogo Model: B-Z Reaction

9/11 - 05 44 Emergent Systems, Jonny Pettersson, UmU

Summary

Cellular automata

One-dimensional Wolfram’s classification Langton’s lambda parameter Two-dimensional

  • Conway’s Game of Life

Pattern formation in slime molds

Dictyostelium discoideum Modeling of pattern

9/11 - 05 45 Emergent Systems, Jonny Pettersson, UmU

Next time

Autonomous Agents Self-Organization Ant Algorithms