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

Mar 17, 2023 •150 likes •313 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

  1. 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 Emergent Systems, Jonny Pettersson, UmU 1 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 Emergent Systems, Jonny Pettersson, UmU 2 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 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 3 1

  2. 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 Emergent Systems, Jonny Pettersson, UmU 4 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 of neighbors (and own state) � Neighborhood � r = radius � neighborhood = 2 r + 1 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 5 1-Dim CA - Example � k = 2, r = 1 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 6 2

  3. 1-Dim CA - General � Neighborhood size = 2 r + 1 � k different states � � a rule table with k 2 r + 1 entries � � number of legal rule tables, k ^ k 2 r + 1 � Usually wrap-around of the linear grid � Initial population? � Random or a few ”on” 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 7 Wolfram’s CA Classification � Stephen Wolfram � 1980s � Resurrected cellular automata research � A ring of n cells with k possible states � k n different configurations of a row � Four different CA classes 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 8 Wolfram’s CA – Class 1 � Static � Compared to fractals – fixed point 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 9 3

  4. Wolfram’s CA – Class 2 � Periodic � Compared to fractals – limit cycles 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 10 Wolfram’s CA – Class 3 � Random-like � Compared to fractals – chaos, instable limit cycles 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 11 Wolfram’s CA – Class 4 � Complex patterns with local structures � Can perform computation, some even universal computation � Not regular, periodic or random � Between chaos and periodicity 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 12 4

  5. 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 = k 2 r + 1 � Number of entries in a rule table that map to the quiescent state, n q λ = ( N – n q )/ 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 Emergent Systems, Jonny Pettersson, UmU 13 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 Emergent Systems, Jonny Pettersson, UmU 14 Langton’s λ - Parameter 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 15 5

  6. Langton’s λ - Problems One rule set can have a high λ but still 1. 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 Emergent Systems, Jonny Pettersson, UmU 16 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 Emergent Systems, Jonny Pettersson, UmU 17 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 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 18 6

  7. Conway’s Game of Life – Universal computation � Static objects � memory 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 19 Conway’s Game of Life – Universal computation � Periodic objects � counters 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 20 Conway’s Game of Life – Universal computation � Moving objects � moving information 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 21 7

  8. Conway’s Game of Life – Universal computation � Breeders, glider guns � collide to make new moving objects 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 22 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 Emergent Systems, Jonny Pettersson, UmU 23 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 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 24 8

  9. 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 Emergent Systems, Jonny Pettersson, UmU 25 D. Discoideum – Life Cycle 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 26 D. Discoideum – Amoebae Stage � Growth phase � Free moving single cell � Lives in soil engulfs bacteria � Divides asexually � Doubling time ~ 3h (Picture from dictybase.org) 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 27 9

  10. D. Discoideum – Amoebae (Video from dictybase.org) 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 28 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 (Video from Zool. Inst. Univ. München) µm/min) 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 29 D. Discoideum – Spiral Waves � Spiral formation unclear, involves symmetry breaking � 1 frame/10 sec (Video from Zool. Inst. Univ. München) 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 30 10

  11. D. Discoideum – Stream Formation Stage � Streams depends on movement and symmetry breaking � Begin to form slug (Picture from R. Firtel, UCSD (dictybase.org)) 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 31 D. Discoideum – Mound Stage � 10000 – 100000 cells � Cells begin to differentiate � 1 frame/20 sec (Video from Zool. Inst. Univ. München) 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 32 D. Discoideum – Multi-armed spirals � Up to 10 spirals have been observed � This mound has 5 spirals (Video from Zool. Inst. Univ. München) 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 33 11

  12. 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 Emergent Systems, Jonny Pettersson, UmU 34 D. Discoideum – Culmination Stage � Cells differentiate into base, stalk and spores � 1 frame/5 sec (Video from Zool. Inst. Univ. München) 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 35 D. Discoideum – Fruiting Body Stage � Spores are dispersed, wind or animal � If sufficient moisture, spores germinate, release amoebas � Cycle begins again 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 36 12

  13. D. Discoideum – Life cycle (Picture from Zool. Inst. Univ. München) 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 37 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 Emergent Systems, Jonny Pettersson, UmU 38 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 of the cAMP receptor 9/11 - 05 Emergent Systems, Jonny Pettersson, UmU 39 13

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