1. Hiroki Sayama sayama@binghamton.edu 2nd Annual French Complex Systems Summer School Continuous Space, Lattices and Swarms: Pattern Formation in Complex Systems Lattice Models

2. 2 Continuous-time/space models are computationally not handy It takes huge computational resource (time, memory, or money) to simulate large-scale systems with high accuracy © Climateprediction.net Columbia at NASA Ames Research Center

3. 3 Alternative approach For complex systems studies: –Rapid model development for qualitative understanding of problems and proof of concepts is more important –Biological/social systems are often discrete in their nature –Underlying rules may be more complex than those assumed in classical models  Discrete-time/space/state models

4. 4 Discrete model Easy to understand, develop and simulate –Doesn’t require an expression for the rate of change (derivative) –Can model abrupt changes and/or chaotic dynamics using fewer variables –Directly implementable as a stepwise simulation in computer –Experimentally, we often have samples of system states at specific points of time

5. Cellular Automata: A Simplified Discrete-State Model

6. 6 Cellular automata (CA) A regular grid model made of many “automata” whose states are finite and discrete (  nonlinearity) Their states are simulta- neously updated by a uniform state-transition function that refers to states of their neighbors s t+1 (x) = F ( s t (x+x 0 ), s t (x+x 1 ),..., s t (x+x n-1 ) )

7. 7 Some terminologies Configuration –A mapping from spatial coordinates to states (s t (x)); global arrangement of states at time t Local situation –A specific arrangement of states within a local neighborhood, to be given to the state- transition function as an input Quiescent state ( ) –A state that represents “empty” space; never changes if surrounded by other quiescent states Some CA have no quiescent state

8. 8 How CA works Neighborhood State set {,} T CR B L State-transition function CTRBLCTRBLCTRBLCTRBL

9. 9 Boundary conditions Periodic boundary condition –Cells at the edge of the space is connected to the cells at the other edge Cut-off boundary condition –Cells at the edge of the space do not have neighbors outside the space Fixed boundary condition –Cells at the edge of the space are fixed to specific states

10. 10 Typical 2-D neighborhood shapes von Neumann neighborhoodMoore neighborhood

11. 11 In Mathematica… Mathematica has a built-in function CellularAutomaton[] –If you want to know why, see the profile of its creator Stephen Wolfram A “wrapper” code CA2DSimulator[] with improved interface is available –See the Mathematica notebook; this will be used in class

12. 12 Modeling example: Panic in a gym

13. 13 Fire alarm causes initial panics

14. 14 Rules of local interaction With four or more panicky persons around you With two or fewer panicky persons around you

15. 15 Exercise What happens if you change the initial ratio of panicky people? What happens if you change the state transition rules? Can you modify the code so that it produces time series of the number of panicky people as well as the visual plot?

16. 16 Droplet rule Majority rule Parity rule (XOR rule, MOD rule) Game of Life –See the Mathematica notebook for details Exercise: Typical binary-state CA

17. Properties of Cellular Automata

18. 18 Phase space of cellular automata For a fixed number of cells, the total number of possible configurations is also fixed and finite: |S| N –S: State set –N: Total number of cells For deterministic CA, each configuration is always mapped to just one configuration –You can draw a state-transition diagram

19. 19 Exercise Draw a phase space of the binary- state 1-D majority CA with 5 cells and periodic boundary conditions –Use symmetries to reduce the number of configurations

20. 20 Features in phase space Self-loop → fixed point Cycle → periodic attractor Configuration with no predecessor → “Garden of Eden” states Just one big basin of attraction → Initial configurations don’t matter Many separate basins of attraction → Sensitive to initial configurations

21. 21 Example |S|=2, N=16 © Andy Wuenshe http://www.ddlab.com/

22. 22 Wolfram’s classification Wolfram (1984) classified binary- state 1-D CA based on their typical attractors’ properties –I: Fixed point of homogeneous states –II: Fixed point or small cycle involving heterogeneous states –III: Chaotic attractor (very long cycle) –IV: Complex, long-lived localized structures (attractor preceded by very long tree-like basins of attraction; computational universality?)

23. 23 Wolfram’s classification

24. 24 Some CA are computationally universal as a Universal Turing Machine can be built on those CA spaces Minsky Register Machine implemented on GoL CA (P. Chapman 2002) Computational universality You can’t predict the behavior of CA more efficiently than explicit simulation!!

25. 25 Asynchronous CA CA where cells are updated asynchronously –Random updating, sequential updating, etc. –Can simulate any synchronous CA Global synchrony is not strictly required for the dynamics of CA!!

26. Discretizing PDE-Based Models into Lattice Models

27. 27 PDEs and lattice models PDE-based models may be simulated using lattice models by discretizing time and space Each site on a lattice has a real- valued state (scalar or vector)

28. 28 Discretizing time Discretize time at interval  t:  f/  t = F(f, x, t) f t+  t (x) = f t (x) +  f/  t  t = f t (x) + F(f t, x, t)  t –Spatial derivatives in F should also be replaced with their discrete equivalents

29. 29 Discretizing spatial derivatives  f/  x = lim h  0 ( f(x+h,t) – f(x,t) ) / h If we make a discrete analog of this by letting h =  x:  f/  x ~ ( f t (x+  x) – f t (x) ) /  x If both sides of neighbors are used:  f/  x ~ ( (f t (x+  x) – f t (x)) + (f t (x) – f t (x-  x)) ) / 2  x = ( f t (x+  x) – f t (x-  x) ) / 2  x

30. 30 Discretizing Laplacians (1)  2 f/  x 2 = lim h  0 ( f’(x+h,t) – f’(x-h,t) ) / 2h = lim h  0 { lim k  0 (f(x+h+k,t) – f(x+h-k,t)) / 2k – lim k  0 (f(x-h+k,t) – f(x-h-k,t)) / 2k } / 2h = lim h  0 ( f(x+2h,t) + f(x-2h,t) – 2f(x,t) ) / (2h) 2 If we make a discrete analog of this by letting h =  x /2:  2 f/  x 2 ~ ( f t (x+  x ) + f t (x-  x ) – 2f t (x) ) /  x 2

31. 31 Discretizing Laplacians (2) Similarly, for 2-D space:  2 f =  2 f/  x 2 +  2 f/  y 2 ~ ( f t (x+  x,y )+f t (x-  x,y )–2f t (x,y) ) /  x 2 +( f t (x,y+  y )+f t (x,y-  y )–2f t (x,y) ) /  y 2 = ( f t (x+  k,y ) + f t (x-  k,y ) + f t (x,y+  k ) + f t (x,y-  k ) – 4f t (x,y) ) /  k 2 (if  x =  y =  k)

32. 32 CA with vector states Extended cellular automata in which states are defined as vectors (rather than finite discrete states) VectorState [0.1, 1.5] VectorState [0.1, 2.3] VectorState [0, 1.9] VectorState [-0.1, 1.8] VectorState [-0.1, 1.3] VectorState [0.2, 0.7] VectorState [0.3, 1.1] VectorState [0.2, 1.4] VectorState [0, 1.7] Can simulate discretized PDEs with multiple variables Some tricks for encoding and visualization needed (see notebook)

33. 33 Exercise Convert the Keller-Segel slime mold aggregation model into a lattice model with vector-valued states Implement the simulation code in Mathematica and conduct simulations

34. Applications to the Modeling of Biological Systems

35. 35 Several biological models on CA Waves in excitable media Host-pathogen models Epidemic / forest fire models

36. 36 Waves in excitable media Propagation of signals over tissues made of nerve or muscle cells that are “excitable” –Stochastic excitation of resting cells by excited neighbors –Excitation followed by refractory states, eventually going back to resting

37. 37 Host-pathogen models Propagation of pathogens over dynamically growing hosts –Spatial growth of hosts –Infection of pathogens to nearby hosts –Death of hosts caused by infection

38. 38 Epidemic / forest fire models Propagation of disease or fire over statically distributed hosts –Propagation of disease or fire to nearby hosts –Death or breakdown of hosts caused by propagation

39. Evolutionary Patterns in CA-Based Artificial Life

40. 40 CA used as a logical device to reconstruct lifelike behaviors in spatio-temporal patterns of “virtual state machines” State-transition rules and initial configurations designed… –by hand –through trial and error with computer simulations –with help of evolutionary algorithms Artificial life based on CA

41. 41 Emergent evolution on CA Evoloops (Sayama 1998) Self-replicating worms (Sayama 2000) Ecology of self- replicating worms (Suzuki et al. 2003, 2006) Genetic evolution of self-replicating loops (Salzberg et al. 2003, 2004) http://bingweb.binghamton. edu/~sayama/

42. 42 Model: Evoloop An evolvable loop constructed on 9- state 5-neighbor CA (Sayama 1999) Obtained by improving the robustness of state-transition rules of Langton’s loop (Langton 1984) Evolutionary process spontaneously emerges from local rules only

43. 43 Evolutionary paths Populations spontaneously evolve toward small species Dynamics were originally believed to be limited in diversity and adaptation

44. 44 Detailed tracking of evolutionary patterns Every birth/death event pinpointed in space and time by finding specific local situations during CA updates Genotype, phenotype and genealogical information accumulated in “Species Database”genotype GGGGCGCGTT GCCCCG pheno- type 8 8 071: G (“growth”) 041: T (“turning”) 1 : C (“core”)

45. 45 Detailed tracking of evolutionary patterns [demo]

46. 46 Findings Diversities in macro-scale morphologies and mutational biases Genetic adaptation Genetic diversification and continuing exploration

47. 47 Huge genetic possibility space Permutation of genes (G, T) and core states (C) under constraints estimates # of viable genotypes to be n# of speciesn n 415911,440149,657,700 5561043,7581537,442,160 621011167,96016145,422,675 779212646,64617565,722,720 83,003132,496,144182,203,961,430 2n-2n-2

48. 48 Diversity in growth patterns (size-4)

49. 49 Diversity in growth patterns (size-6)

50. 50 Diversity in mutational biases (size-6)

51. 51 (Ancestor: GGCCGGCGCGTTGCGC/8x8; grid: 800x800) Complex genealogy (Salzberg et al. 2006)

52. 52 Survival rate –Characterized by the average relative population ratio after a given time period in competition with other species Colony density index –Characterized by the quadratic coefficient of a parabola fitted to the population growth in an infinite space Measuring “fitness”

53. 53 Distribution and correlation (size-4)

54. 54 Evolution in vivo (from size-8)

55. 55 Evolution improves “fitness” Evolutionary transition actually observed in the previous slide

56. 56 Certain subsequences are found to be non- mutable in our model: G{C*}T{C*}TG A long non-mutable subsequence in ancestor reduces size-based selection pressure Such “GMO” loops show long-lasting evolutionary exploration processes GGGGCGC GCCTCCTG G Genetic modification to indirectly control selection pressure

57. 57 Enhanced diversity control with long non-mutable subsequences

58. 58 Experiments with hostile environments Space partially cleared at regular intervals leaving only a fraction of survivors Effects of mutational biases further manifested

59. 59 control with long non-mutable subsequences with subsequences + hostile environment

60. 60 Summary Lattice models can demonstrate strikingly rich behaviors –Static/dynamic pattern formation –Phase transition –Darwinian evolution CA-based artificial life demonstrates hierarchical emergence where population-level evolutionary changes arise from interactions between sub- individual-level elements