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  • Timing and stability in cellular automata

    Konstantin Klemm

    Bioinformatics GroupUniversity of Leipzig, Germany

    Konstantin Klemm (Leipzig) Timing & Stability in CA 1 / 25

  • Definition

    finite set of states S , often |S | = 2

    mapping (rule table) f : Sz S

    a lattice of n sites with coordination number z where site i hasneighbours a(i , 1), a(i , 2), . . . a(i , z)

    time-discrete dynamics of lattice site i

    si (t + 1) = f [sa(i ,1)(t), . . . sa(i ,z)(t)]

    All lattice sites are updated in synchrony.

    Konstantin Klemm (Leipzig) Timing & Stability in CA 2 / 25

  • Purpose

    Computer Science: Models of computation,e.g. Game of Life and rule 110 are Turing-complete.

    Artificial Life: Study of self-reproducing structures

    Physics: Abstractions of spatio-temporal dynamics, pattern formation

    Motto: Simplest rules may yield most complex patterns / computations.

    Konstantin Klemm (Leipzig) Timing & Stability in CA 3 / 25

  • Putative CA dynamics on plant leaves

    Peak et al., PNAS (2004)

    Konstantin Klemm (Leipzig) Timing & Stability in CA 4 / 25

  • CA model for morphogenesis in Hydrasp

    ace

    time

    Ped

    Ped/CnNK2

    1

    rump

    foot

    Rohlf & Bornholdt, JSTAT (2005)

    Konstantin Klemm (Leipzig) Timing & Stability in CA 5 / 25

  • Seashell

    Konstantin Klemm (Leipzig) Timing & Stability in CA 6 / 25

  • Seashell and elementary CA rule 22

    Konstantin Klemm (Leipzig) Timing & Stability in CA 7 / 25

  • Seashell and elementary CA rule 30

    Konstantin Klemm (Leipzig) Timing & Stability in CA 8 / 25

  • Role of noise?

    Are the observed patterns robust under small stochasticperturbations?

    Stability analysis in continuous dynamical systems: Make a smallperturbation and check if the system returns to the fixed point / limitcycle.

    Here: Discretized state space. What is a small perturbation?

    Typical implementation of noise in CA:

    Stochastic asynchronous update

    Konstantin Klemm (Leipzig) Timing & Stability in CA 9 / 25

  • Rule 22, comparison sync / async update

    synchronous update asynchronous update async, another realization

    Konstantin Klemm (Leipzig) Timing & Stability in CA 10 / 25

  • Rule 150, comparison sync / async update

    synchronous update asynchronous update async, another realization

    Konstantin Klemm (Leipzig) Timing & Stability in CA 11 / 25

  • Rule 90, comparison sync / async update

    synchronous update asynchronous update async, another realization

    Konstantin Klemm (Leipzig) Timing & Stability in CA 12 / 25

  • Loss of memory

    0 20 40 60 80 100 120 140 160 180 200time

    0.01

    0.1

    1in

    form

    atio

    n [b

    its]

    information of a cells state at time tmutual information between cells state at time tand initial condition

    Evolution of mutual information between one cells state at time t and thefull initial configuration.

    Konstantin Klemm (Leipzig) Timing & Stability in CA 13 / 25

  • Summary so far

    Deterministic CA:Complex (interesting) spatio-temporal pattern formation

    CA with noise implemented as stochastic update order:Largely irreproducible dynamics

    Devastating effect of stochastic asynchronous update known for long,cf. Ingerson and Buvel, Physica D (1984)

    Konstantin Klemm (Leipzig) Timing & Stability in CA 14 / 25

  • How to proceed

    Solution 0

    Use more states and more complicated interactions to implementadditional clock signalscf. Nehaniv, Int. J. Alg. Comp. (2004)

    Solution 1

    Stick to simple rules

    Consider a less destructive type of perturbation.

    Get happy.

    Konstantin Klemm (Leipzig) Timing & Stability in CA 15 / 25

  • Framework for perturbed timing

    Now time t R, no longer discrete.

    Expand initial condition s(0) to unit interval:

    si (t) =

    {

    fi(s(0)) , if t > 1 isi (0) , otherwise

    where i [0, ] is drawn randomly and independently for each node i .

    Let the system evolve according to

    si (t + 1) =

    [

    (2c)1

    t+c

    tc

    fi (s())d 1/2

    ]

    where is the step function and 1/2 > s > 0.

    Check, if system autonomously regains synchrony.

    Klemm & Bornholdt, PNAS 102, 18414 (2005); Phys. Rev. E 72, 055101(R) (2005).

    Konstantin Klemm (Leipzig) Timing & Stability in CA 16 / 25

  • CA rule 22, initial timing perturbation

    Konstantin Klemm (Leipzig) Timing & Stability in CA 17 / 25

  • CA rule 22, spatial coarsening of time lags

    Konstantin Klemm (Leipzig) Timing & Stability in CA 18 / 25

  • CA rule 22, timing perturbation heals

    t=240

    Konstantin Klemm (Leipzig) Timing & Stability in CA 19 / 25

  • Probability of healing

    0 200 400 600 8000

    0.2

    0.4

    0.6

    0.8

    1

    prob

    . tha

    t per

    turb

    atio

    n he

    als

    out

    CA rule 22

    0 200 400 600 800number of cells N

    CA rule 150

    0 200 400 600 800

    CA rule 110

    Konstantin Klemm (Leipzig) Timing & Stability in CA 20 / 25

  • Time until healing

    100 1000number of cells N

    102

    103

    104

    105

    rule 22rule 150slope 2slope 1

    Konstantin Klemm (Leipzig) Timing & Stability in CA 21 / 25

  • Unstable pattern in rule 110

    Konstantin Klemm (Leipzig) Timing & Stability in CA 22 / 25

  • Overview of results

    stable elementary rules0, 4, 22, 32, 36, 54, 72, 76, 104, 128, 132, 160, 164, 200, 204, 218,222, 236, 250,254

    partially stable el. rules (strong dependence on n)90, 122, 126, 150

    unstable elementary rules50, 94, 108, 110, 178

    Elementary CA fail to produce sustained synchronous blinking of allcells

    Game of Life: Blinkers, gliders, spaceships etc. are unstable

    Konstantin Klemm (Leipzig) Timing & Stability in CA 23 / 25

  • Further results

    Konstantin Klemm (Leipzig) Timing & Stability in CA 24 / 25

  • Summary

    Stable and unstable CA rules can be distinguished by theirresponse to minimal timing perturbations.

    Konstantin Klemm (Leipzig) Timing & Stability in CA 25 / 25

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