[PDF] - 1 1-Dim CA - Gener al Wolf rams CA Class 2 Neighbor hood size = 2 PDF Document

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16/11 - 04 1 Emergent Systems, Jonny Pettersson, UmU

Last t ime

❒ Nonlinear dynamic syst ems

❍ The Logist ic map

❒ St range at t ract or s

❍ The Hénon at t ract or ❍ The Lor enz at t r act or

❒ Producer-consumer dynamics

❍ Equat ion-based modeling ❍ I ndividual-based modeling

16/11 - 04 2 Emergent Systems, Jonny Pettersson, UmU

Out line f or t oday

❒ Cellular aut omat a

❍ One-dimensional ❍ Wolf r am’s classif icat ion ❍ Langt on’s lambda par amet er ❍ Two-dimensional

Conway’s Game of Lif e

❒ Pat t ern f ormat ion in slime molds

❍ Dict yost elium discoideum ❍ Modeling of pat t er n

16/11 - 04 3 Emergent Systems, Jonny Pettersson, UmU

Complex Syst em

❒ Things t hat consist of many similar and

simple par t s

❍ Of t en easy t o under st and t he part s ❍ The global behavior much harder t o explain ❍ On many levels ❍ Some ar e capable of univer sal comput at ion

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Cellular Aut omat a

❒ A dynamic syst em ❒ I nvent ed by J ohn von Neumann

❍ Wit h help f r om St anislaw Ulam ❍ 1940s ❍ Want ed t o under st and t he pr ocess of

r eproduct ion

❍ The essence

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One-Dimensional CA

❒ Linear gr id of cells ❒ Each cell can be in one of k dif f erent st at es ❒ Next st at e is comput ed as an f unct ion of t he st at es

f neighbors (and own st at e)

❒ Neighborhood

❍ r = radius neighborhood = 2r + 1 16/11 - 04 6 Emergent Systems, Jonny Pettersson, UmU

1-Dim CA - Example

❒ k = 2, r = 1

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16/11 - 04 7 Emergent Systems, Jonny Pettersson, UmU

1-Dim CA - Gener al

❒ Neighbor hood size = 2r + 1 ❒ k dif f er ent st at es ❒ a rule t able wit h k2r + 1 ent r ies ❒ number of legal r ule t ables, k^k2r + 1 ❒ Usally wr ap-ar ound of t he linear gr id ❒ I nit ial populat ion?

❍ Random or a f ew ”on” 16/11 - 04 8 Emergent Systems, Jonny Pettersson, UmU

Wolf ram’s CA Classif icat ion

❒ St ephen Wolf ram ❒ 1980s ❒ Resurrect ed cellular aut omat a research ❒ A ring of n cells wit h k possible st at es kn

dif f erent conf igurat ions of a row

❒ Four dif f erent CA classes

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Wolf ram’s CA – Class 1

❒ St at ic ❒ Compared t o f ract als – f ixed point

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Wolf ram’s CA – Class 2

❒ Per iodic ❒ Compared t o f ract als

– limit cycles

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Wolf ram’s CA – Class 3

❒ Random-like ❒ Compared t o f ract als – chaos, inst able limit cycles

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Wolf ram’s CA – Class 4

❒ Complex pat t erns wit h

local st r uct ures

❒ Can perf or m

comput at ion, some even universal comput at ion

❒ Not regular, periodic

r r andom

❍ Bet ween chaos and

periodicit y

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16/11 - 04 13 Emergent Systems, Jonny Pettersson, UmU

Langt on’s Lambda Par amet er

❒ Chr is Langt on – ”Founder ” of Ar t if icial Lif e ❒ Was sear ching f or a vir t ual knob t o cont rol t he

behavior of a CA

❒ Quiescent st at e – inact ive, of f ❒ Number of ent r ies in a r ule t able, N = k2r + 1

λ = (N – nq )/ N

❒ λ = 0 t he most homogeneous r ule t able ❒ λ = 1 all r ules map t o non-quiescent st at es ❒ λ = 1 – 1/ k t he most het erogeneous

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Langt on’s λ - Example

❒ Comparing wit h t he examples of Wolf ram’s

classes

❒ The most het erogeneous:

❍ 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 (aver age)

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Langt on’s λ - Paramet er

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Langt on’s λ - Problems

1. One rule set can have a high λ but st ill produce very simple behavior

2. Lit t le inf ormat ion in a singular value
3. λ says not hing cert ain about t he long-

t erm behavior

❒

Dangerous t o map t o a single scalar number

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2-Dim CA – Conway’s Game of Lif e

❒ J ohn Conway, 1960s ❒ Want ed t o f ind t he simplest CA t hat could

support univer sal comput at ion

❒ k = 2, very simple rules ❒ 1970, Mart in Gander described Conway’s

work in his Scient if ic American column

❒ A global collaborat ive ef f ort succeeded

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Conway’s Game of Lif e

❒ 2-dimensional ❒ 8 neighbor s ❒ Rules:

❍ Loneliness: Less t han t wo neighbors, die ❍ Overcrowding: More t han t hree neighbors, die ❍ Reproduct ion: Empt y cell wit h t hree neighboors, live ❍ St asis: Exact t wo neighboors, st ay t he same

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Conway’s Game of Lif e – Univer sal comput at ion

❒ St at ic obj ect s memor y

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Conway’s Game of Lif e – Univer sal comput at ion

❒ Per iodic obj ect s count ers

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Conway’s Game of Lif e – Univer sal comput at ion

❒ Moving obj ect s moving inf ormat ion

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Conway’s Game of Lif e – Univer sal comput at ion

❒ Br eeder s, glider guns

collide t o make new moving obj ect s

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Conway’s Game of Lif e – Univer sal comput at ion

❒ To implement univer sal comput at ion

❍ NOT and (AND or OR)

❒ Comput ing science t heorist : The rest ar e bor ing

det ails

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Nat ur al CA-like Phenomena

❒ Describes phenomena t hat occur on

radically dif f erent t ime and space scales

❒ St at ist ical mechanical syst ems

❍ Lat t ice-gas aut omat on

❒ Aut ocat alyt ic chemical set s

❍ The Belousov-Zhabot insky react ion

❒ Gene regulat ion ❒ Mult icellular organisms ❒ Colonies and ”super-organisms” ❒ Flocks and herds ❒ Economics and Societ y

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P at t ern Format ion in Slime Molds

❒ Self -organisat ion r esult ing in pat t er ns ❒ The Belousov-Zhabot insky react ion ❒ Honeybees ❒ Pat t erns gener at ed by or ganisms midway in

complexit y

❒ Dict yost elium discoideum

16/11 - 04 26 Emergent Systems, Jonny Pettersson, UmU

D. Discoideum – Lif e Cycle

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D. Discoideum – Amoebae St age

❒ Growt h phase ❒ Fr ee moving single cell ❒ Lives in soil engulf s

bact er ia

❒ Divides asexually

❍ Doubling t ime ~ 3h (Picture from dictybase.org) 16/11 - 04 28 Emergent Systems, Jonny Pettersson, UmU

D. Discoideum – Amoebaes

(Video from dictybase.org) 16/11 - 04 29 Emergent Systems, Jonny Pettersson, UmU

D. Discoideum – Aggr egat ion St age

❒ When st ar ving

development al phase

❒ Aggregat e by chemot axis ❒ Mult iple concent r ic

cir cles and spirals

❒ Up t o 100000 individuals ❒ 1 f r ame/ 36 sec ❒ Wave pr opagat ion 60 –

120 µm/ min

❒ Spiral accelerat e cell

aggregat ion (18 vs 3 µm/ min)

(Video from Zool. Inst. Univ. München) 16/11 - 04 30 Emergent Systems, Jonny Pettersson, UmU

D. Discoideum – Spir al Waves

❒ Spir al f or mat ion unclear, involves symmet r y br eaking ❒ 1 f r ame/ 10 sec

(Video from Zool. Inst. Univ. München)

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16/11 - 04 31 Emergent Systems, Jonny Pettersson, UmU

D. Discoideum – St ream Format ion

St age

❒ St r eams depends

n movement and

symmet r y br eaking

❒ Begin t o f orm slug

(Picture from R. Firtel, UCSD (dictybase.org)) 16/11 - 04 32 Emergent Systems, Jonny Pettersson, UmU

D. Discoideum – Mound St age

❒ 10000 – 100000 cells ❒ Cells begin t o dif f erent iat e ❒ 1 f r ame/ 20 sec

(Video from Zool. Inst. Univ. München) 16/11 - 04 33 Emergent Systems, Jonny Pettersson, UmU

D. Discoideum – Mult i-ar med

spir als

❒ Up t o 10 spirals have been observed ❒ This mound has 5 spirals

(Video from Zool. Inst. Univ. München) 16/11 - 04 34 Emergent Systems, Jonny Pettersson, UmU

D. Discoideum – Slug St age

❒ Behaves as single organism ❒ Migrat es; seeks light , seeks or avoids heat ❒ No brain, no nervous syst em ❒ 1 f rame/ 10sec

(Video from Zool. Inst. Univ. München) 16/11 - 04 35 Emergent Systems, Jonny Pettersson, UmU

D. Discoideum – Culminat ion St age

❒ Cells dif f erant iat e int o base, st alk and spor es ❒ 1 f r ame/ 5 sec

(Video from Zool. Inst. Univ. München) 16/11 - 04 36 Emergent Systems, Jonny Pettersson, UmU

D. Discoideum – Fr uit ing Body

St age

❒ Spor es ar e disper sed,

wind or animal

❒ I f suf f icient moist ure,

spor es germinat e, release amoebaes

❒ Cycle begins again

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D. Discoideum – Lif e cycle

(Picture from Zool. Inst. Univ. München) 16/11 - 04 38 Emergent Systems, Jonny Pettersson, UmU

Modeling t he waves

❒ The waves are probably t he result of spiral

waves of cyclic adenosine 3’,5’- monophosphat e (cAMP) in an excit able medium

❒ Spirals rot at e f ast est , push ot her

act ivit ies t o t he border Bigger slugs Bet t er dispersal Select ive pressure

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Biological Basis of Aggr egat ion in

D. discoideum

❒ Not exact ly clear how ❒ cAMP plays a vit al role ❒ St ar ved D. discoideum produce cAMP

release int o t he envir onment

❒ cAMP released in t wo ways

❍ Oscillat or y r elease (wit h a per iod of 5-10 min) ❍ Relay

❒ Posit ive and negat ive f eeback at t he level

f t he cAMP recept or

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What t o model?

❒

Cells moves t oward higher concent rat ion

f cAMP, appr oximat ely one-t ent h t he

speed of cAMP waves

❒

Divide t he modeling int o t wo part s

1.

Let t he cells be st at ic and model t he cAMP waves wit hout dif f usion of cAMP

2.

Add dif f usion of cAMP and cell movement

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Equat ion-Based Modeling

❒ Modeling oscillat ions and relay ❒ The cAMP recept or has f our st at es ❒ The equat ions... ❒ What happens when t he ext racellular

concent rat ion of cAMP suddenly increase?

❒ Dif f usion give a wave of cAMP t hat

st imulat e cells t o move

❒ A spiral wave may be gener at ed when

individual cells spont aneously f ire of f pulses of cAMP in a random manner

16/11 - 04 42 Emergent Systems, Jonny Pettersson, UmU

A Model f or Cell Movement in Response t o cAMP

❒ Three assumpt ions:

❍ Each cell can det ect t he cAMP

gr adient and move in t he dir ect ion of incr easing cAMP concent r at ion

❍ A prolonged cAMP

st imulus decreases t he cells’ abilit y t o det ect cAMP gradient s

❍ Cell-cell adhesion comes int o play

nce t he cells ar e close enough

t o one anot her , and t hus once a clump f orms it cannot quickly disper se

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

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I ndividual-Based Modeling

❒ A Cellular Aut omat a Model of Spiral Waves ❒ Two end st at es ❒ A posit ive f eedback mechanism ❒ A negat ive f eedback mechanism ❒ The Net Logo Model: B-Z React ion

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Summar y

❒ Cellular aut omat a

❍ One-dimensional ❍ Wolf r am’s classif icat ion ❍ Langt on’s lambda par amet er ❍ Two-dimensional

Conway’s Game of Lif e

❒ Pat t ern f ormat ion in slime molds

❍ Dict yost elium discoideum ❍ Modeling of pat t er n

16/11 - 04 45 Emergent Systems, Jonny Pettersson, UmU