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Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 1

Spreading Phenomena Spreading Phenomena and and Excitable Media Excitable Media

AI AI

Christian Jacob

Department of Computer Science University of Calgary

CPSC 565 — Winter 2003

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 2

Two-Dimensional Two-Dimensional Cellular Automata Cellular Automata

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 3

Formally, we can characterize the 2D Moore neighbourhood Nij(r) of radius r in a cell lattice L for a cell cij by Nij(r) = {(k, l) Œ L | |k-i| ≤ r and |l-j| ≤ r }.

Two-dimensional CA Neighbourhoods Two-dimensional CA Neighbourhoods

von Neumann r = 1 Moore r = 1 Extended Moore r = 1, 2, 3

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 4

Totalistic Cellular Automata Totalistic Cellular Automata

• Consider the following 1D CA rule:

111 110 101 100 011 010 001 000 0 1 1 0 1 0 0 0

• This automaton changes ci into a 1 only if the sum of the

cells ci-1, ci, and ci+1 is exactly 2: This is an example of a totalistic rule. ci(t+1) = 1, if ci-1(t) + ci(t) + ci+1(t) = 2 0, otherwise

{

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 5

Encoding of Totalistic Rules Encoding of Totalistic Rules

• We first look at one-dimensional, binary CA rules (k = 2):

ci(t+1) = f ( ∑j Œ Ni(2) ci+j (t) ), where f : {0, 1, …, 2r +1} Æ {0, 1}.

• We can make a table of f in the form of a tuple

( f (0), f (1), f (2), …, f (2r + 1) ).

• We can encode this tuple by the following formula:

Cf = ∑j=0, …, 2r+1 f ( j ) · 2j.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 6

Encoding of Totalistic Rules (2) Encoding of Totalistic Rules (2)

• With this encoding all legal rule codes are even numbers.

– All 32 rules for k = 2 and r = 2 are encoded by all even numbers between 0 and 62. Note: In the previous lecture we used a different notation, with V instead of k, and K = 2r +1. – Example: mod-2 rule Sum: 1 2 3 4 5 f (Sum): 1 1 1 Cf : 0 · 20 + 1 · 21 + 0 · 22 + 1 · 23 + 0 · 24 + 1 · 25 = 42

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 7

Encoding of Totalistic Rules (3) Encoding of Totalistic Rules (3)

• This encoding for binary, totalistic CAs (k = 2)

Cf = ∑j=0, …, 2r+1 f ( j ) · 2j can be generalized to any number of k values per cell: Cf = ∑j=0, …, 2r+1 f ( j ) · kj , where f maps to any of the k values for a neighbourhood of radius r: f : {0, 1, …, 2r +1} Æ {0, 1, …, k}.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 8

Some Example 2D Cellular Automata: Some Example 2D Cellular Automata: 1022 1022

5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic

0-39 40 30 20

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 9

Some Example 2D Cellular Automata: Some Example 2D Cellular Automata: 510 510

5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic

0-39 40 30 20

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 10

Some Example 2D Cellular Automata: Some Example 2D Cellular Automata: 374 374

5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic

0-39 40 30 20

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 11

Some Example 2D Cellular Automata: Some Example 2D Cellular Automata: 614 614

5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic

0-39 40 30 20

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 12

Some Example 2D Cellular Automata: Some Example 2D Cellular Automata: 174 174

5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic

0-39 40 30 20

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 13

Some Example 2D Cellular Automata: Some Example 2D Cellular Automata: 494 494

5-neighbourhood, outer totalistic 5-neighbourhood, outer totalistic

0-39 40 30 20

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Patterns from Seed Regions: Patterns from Seed Regions: 143954 143954

9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic

Single-Cell Seed Random Disordered Seed

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 15

Patterns from Seed Regions: Patterns from Seed Regions: 50224 50224

9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic

Single-Cell Seed Random Disordered Seed

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 16

Patterns from Seed Regions: Patterns from Seed Regions: 15822 15822

9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic

Single-Cell Seed Random Disordered Seed

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 17

Patterns from Seed Regions: Patterns from Seed Regions: 85507 85507

9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic

Single-Cell Seed Random Disordered Seed

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 18

Patterns from Seed Regions: Patterns from Seed Regions: 191044 191044

9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic

Single-Cell Seed Random Disordered Seed

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 19

Patterns from Seed Regions: Patterns from Seed Regions: 93737 93737

9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic

Single-Cell Seed Random Disordered Seed

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 20

Spreading Phenomena Spreading Phenomena

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 21

Spreading Phenomena Spreading Phenomena

• Spreading is the process in which an object extends itself
• ver an increasingly larger area by incorporating regions

adjacent to itself.

• The spreading or kinetic growth (KG) models describe a

wide variety of natural processes:

– Tumor growth – Epidemic spread – Gelation – Rumor-mongering – Fluid flow through porous media – …

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 22

Slow Diffusive Growth: Slow Diffusive Growth: 256746 256746

9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 23

Slow Diffusive Growth: Slow Diffusive Growth: 736 736

9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 24

Slow Diffusive Growth: Slow Diffusive Growth: 174826 174826

9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 25

Slow Diffusive Growth: Slow Diffusive Growth: 175850 175850

9-neighbourhood, outer totalistic 9-neighbourhood, outer totalistic

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 26

Eden Model Eden Model

• The original kinetic growth model was the Eden model,

introduced by the biologist M. Eden in the 1960s.

• The Eden model simulates accretive growth of a cell

cluster (originally for simulating the spread of tumor cells) in a square lattice by sequentially adjoining randomly selected cells from the perimeter.

• In more detail:

– We start with a cluster list consisting of a seed site. – A site is randomly chosen from the perimeter list, consisting of the nearest sites adjacent to the seed site (v. Neumann neighbourhood). – The cluster and perimeter lists are updated. – Another site is randomly selected from the perimeter list, and so on.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 27

Variations of the Eden Model Variations of the Eden Model

• We look at two variations of the Eden model:

– The Invasion Percolation Model – The Single Percolation Cluster Model

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 28

The Invasion Percolation Model The Invasion Percolation Model

• The invasion percolation model describes the flow of fluid

through porous media.

• Invasion Percolation Algorithm:

– Each site in the perimeter list has a random number associated with it. – The cluster spreads by incorporating the perimeter site with the lowest associated random number.

• This model describes a process of spreading that “follows

the path of least resistance.”

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 29

Invasion Percolation Model: Spreading Invasion Percolation Model: Spreading

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 30

Invasion Percolation Model: Spreading (2) Invasion Percolation Model: Spreading (2)

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 31

The Single Percolation Cluster Model The Single Percolation Cluster Model

• The single (or random) percolation cluster model describes

the epidemic spread of disease.

• Single Percolation Cluster Algorithm:

– Each randomly selected perimeter site has a probability p of joining the cluster. – If the selected site is placed in the cluster, it is removed from the perimeter list. – Even if the selected site is not placed in the cluster, it is removed from the perimeter list so that it cannot be chosen again later.

• When p = 1, this model reduces to the Eden model.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 32

Single Percolation Cluster Model: Single Percolation Cluster Model: p p = 1.0 = 1.0

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 33

Single Percolation Cluster Model: Single Percolation Cluster Model: p p = 1.0 = 1.0

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 34

Single Percolation Cluster Model: Single Percolation Cluster Model: p p = 0.5 = 0.5

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 35

Invasion vs. Single Percolation Invasion vs. Single Percolation

Invasion Percolation Single Percolation

p = 1.0 p = 0.5

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 36

Excitable Media Excitable Media

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 37

Excitable Media Excitable Media

• Systems involving excitable media:

– Slime mold growth, – Star formation in spiral galaxies, – Cardiac tissue contraction, – Reaction-diffusion chemical systems, – Infectious disease epidemics

• In each of these cases various spatially distributed patterns

are formed (such as concentric and spiral waves).

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 38

Excitable Media (2) Excitable Media (2)

• Self-organized, self-propagating structures consist of

spatially distributed elements which

– can be excited as a result of interacting with neighbouring elements, – Subsequently returning incrementally to the quiescent state in which they are again receptive to being excited.

• The excited-refractory-receptive cycle characterizes these

systems.

• These can be modeled using multi-state cellular automata.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 39

2D CA: Emergent Pattern Formation in Excitable Media 2D CA: Emergent Pattern Formation in Excitable Media

Neuron excitation Neuron excitation Cyclic Space Cyclic Space Hodgepodge Hodgepodge

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 40

Neuron Excitation Neuron Excitation

• Electrical activity in biological systems can exhibit

excitable media behaviour.

• Example: contraction of cardiac tissue (heart muscle)
• The two-dimensional Greenberg-Hastings CA has been

used to model neuron excitation and recovery in a network

• f neurons.

– Lattice site (neuron) values range from 1 to r. – A neuron with value 1 is said to be excited. – A neuron with value r is said to be rested. – A neuron with an intermediate value is said to be in a state of recovery.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 41

Neuron Excitation: Rules Neuron Excitation: Rules

• In the neuron excitation-recovery process, an excited

neuron goes through a sequence of recovery states until it is rested.

• A rested neuron becomes excited if at least one nearest

neighbour (v. Neumann) neuron is excited.

• Update rules:

– A rested neuron (having value v = r) with at least one excited nearest neighbour neuron (having value v = 1) becomes excited (r Æ 1). – A rested neuron (having value v = r) with no excited nearest neighbour remains rested (r Æ r). – A neuron that is in a recovery state (1 ≤ v ≤ r-1) goes to the next recovery state (v Æ v +1).

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 42

Neuron Excitation: Rule Implementation Neuron Excitation: Rule Implementation

• Here is an example of how to implement the Neuron

Excitation rules through pattern matching in Mathematica:

neuron[a_, b_, r, d_, e_]:= 1 /; MatchQ[1,a|b|d|e] neuron[a_, b_, r, d_, e_]:= r neuron[_, _, c_, _, _] := c + 1 Attributes[neuron]= Listable

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 43

Neuron Excitation: Visualization Neuron Excitation: Visualization

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 44

Neuron Excitation: Visualization Neuron Excitation: Visualization

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 45

Cyclic Space Cyclic Space

• In the Greenberg-Hastings CA, the recovery of a site

proceeds independently of the state of neighbouring sites.

• In the Cyclic Space CA, the recovery process for a site

proceeds only when a neighbouring site is slightly more relaxed.

• The behaviour of this model has been related to the

changes that occur in phase transitions (e.g., water to ice).

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 46

Cyclic Space: Rules Cyclic Space: Rules

• Lattice sites have values between 0 and n-1.
• Update rules:

– A cell in state n-1 goes to state 0 if any of its nearest neighbour cells are in state 0. That is, a cell in state n-1 is “eaten” by a nearest neighbour cell in state 0. – A cell in state v, 0 ≤ v ≤ n-1, goes to state v +1 if any of its nearest neighbour cells is in state v +1. – A cell in state v, 0 ≤ v ≤ n -1, remains in state v if it has no nearest neighbour cells in state v +1.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 47

Cyclic Space: Rule Implementation Cyclic Space: Rule Implementation

• Here is an example of how to implement the Cyclic Space

rules through pattern matching in Mathematica:

cyclicSpace[a_, b_, (n-1), d_, e_]:= 0 /; MatchQ[0, a | b | d | e ] cyclicSpace[a_, b_, c_, d_, e_]:= (c+1) /; MatchQ[(c+1), a | b | d | e ] cyclicSpace[_, _, c_, _, _] := c Attributes[cyclicSpace]= Listable

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 48

Cyclic Space: Visualization Cyclic Space: Visualization

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 49

Cyclic Space: Visualization Cyclic Space: Visualization

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 50

Hodgepodge Machine Hodgepodge Machine

• Hodgepodge = heterogeneous mixture
• The Hodgepodge Machine models autocatalytic chemical

reactions.

• Two or more compounds combine, dissociate, and

recombine in the presence of a catalyst.

• Examples:

– Combination of carbon monoxide and oxygen to form carbon dioxide while these molecules are absorbed on the surface of dispersed palladium crystallites (car catalysers!). – Belousov-Zhabotinsky reaction: malonic acid is oxidized by potassium bromate in the presence of iron or cerium. – Spreading of infectious diseases.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 51

Hodgepodge: Rules Hodgepodge: Rules

• s ¥ s square lattice with periodic boundary conditions.
• Lattice site values range from 0 to r.
• Cells having a value of 0 are said to be healthy.
• Cells having a value r are said to be ill.
• All other cells (0 < v < r) are said to be infected.

The higher the value, the more infected the cell is.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 52

Hodgepodge: Rules Hodgepodge: Rules

• Update rules:

– If a cell is ill (v = r), it becomes healthy (v Æ 0). – If a cell is healthy (v = 0), it becomes infected with the new v determined by:

Min[r, Floor[# infected NN cells / k1 ] + Floor[# ill NN cells / k2 ]

where k1 and k2 are constants. Note: The use of Min ensures that v does not exceed r. – If a cell is infected (0 < v < r), it becomes more infected with the new v determined by:

Min[r, g + Floor[sum of NN cell values / # infected NN cells ]

where g is an integer known as the speed of infection constant.

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 53

Hodgepodge: Rule Implementation Hodgepodge: Rule Implementation

• Here is an example of how to implement the Hodgepodge

rules through pattern matching in Mathematica:

hodgepodge[r, _, _, _]:= 0 hodgepodge[0, x_, y_, _]:= Min[r, Floor[y/k1] + Floor[x/k2]] hodgepodge[c_, x_, y_, z_] := Min[r, g + Floor[(c+(r x) + z)/(y+1)]] Attributes[hodgepodge]= Listable

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 54

Hodgepodge: Visualization Hodgepodge: Visualization

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 55

Hodgepodge Machine: Visualization Hodgepodge Machine: Visualization

Christian Jacob, University of Calgary Emergent Computing — CPSC 565 — Winter 2003 56

References References

• Eden, M. (1960). A two-dimensional growth process . In Proceedings of

Fourth Berkeley Symposium on Mathematics, Statistics, and Probability, volume 4, pages 223-239, Berkeley, 1960. University of California Press.

• Gaylord, R. J. and P. R. Wellin (1995). Computer Simulations with

MATHEMATICA: Explorations in Complex Physical and Biological Systems. New York, Springer. (in particular: chapters 4 and 11).

• Gaylord, R. J. and K. Nishidate (1996). Modeling Nature: Cellular

Automata Simulations with Mathematica. New York, TELOS/Springer.

• Toffoli, T. and N. Margolus (1987). Cellular Automata Machines: A New

Environment for Modeling, MIT Press.

• Wolfram, S. (1994). Cellular Automata and Complexity. Reading,

Addison-Wesley.

• Wolfram, S. (2002). A New Kind of Science. Champaign, IL, Wolfram

Media.