harris - cellular automata

8/8/2019 Harris - Cellular Automata

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Biology 166

ellular Automata

ellular automata are a category of mathematical system in which numerical values are assigned toatial positions and then repeatedly re-calculated based on the values of each location's neighboring

ositions according to a set of rules.

typical example would be an array of squares, arranged like those of a chessboard, with each squaring assigned a value of either one or zero (instead of "one", you can say "filled", and instead of "ze

ou can say "empty"). You start out with an initial pattern of some squares filled and other squaresmpty, and you have a set of rules for calculating the next "generation". One set of rules would be th

y square that has two filled neighbors will stay the same in the next generation as it is now; and anuare that has 3 filled neighbors will become filled (whether it was filled in the current generation, o

ot); but any square that has 0 filled neighbors, or 1, or 4, 5, 6, 7 or 8 filled neighbors will becomempty, whether or not it is now filled. This set of rules was applied to the pattern on the extreme left

low, in order to generate the pattern on the extreme right. Do you see why the pattern will "die oute next generation is this set of rules is applied one more time?

fact, this particular set of rules (3->filled; 2->same; any other number->empty) are those of what iobably the best-known (and most studied) of all cellular automaton, namely the "Game of Life",vented by the British mathematician John Conway and popularized by Martin Gardner's

Mathematical Games" column in Scientific American magazine in 1970-1. A share-ware program foaying this game is included on the program disk. This game can be surprisingly addictive; in the ea

970s it is estimated that between 2 and 3 billion dollars a year worth of main-frame computer time wed up by programmers exploring this game without their bosses' permissions.

his clandestine programming led to the discovery of such things as gliders, glider guns and manyher strange beasts. Conway claims to have chosen these rules to mimic some ecological phenomenhers have subsequently found other combinations of rules which really do simulate certain interacti

mong animal populations. Other cellular automata are used in computer simulations of chemicalactions, gas flow ("lattice gases"), neutron chain reactions and who knows what else. The topic waintly invented at Los Alamos by two of the brightest and most infuential men of this century, Johnon Neumannand Stanslaw Ulam (the former invented the CPU style of computer architecture; thecond held the patent on the hydrogen bomb!).

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ozens of books and thousands of research papers have been written about cellular automata. There any variant forms: you can have hexagonal and triangular neighborhoods as well as square ones, yo

n decide to count only the 4 orthogonal neighbors (above, below, left, right) and omit the diagonalnes; or you can also count positions some distance away; for example, one of the programs on the dunts 24 neighbors for each square. In addition, the number of different alternative values that eachell" can be more than 2 (filled or empty); they can have values of 0, 1, 2, 3 etc. Furthermore, thecalculation rules can be based on something other than simply adding up filled neighbors. Computee good at this sort of thing: indeed, many things that one tries to simulate almost "try" to turnemselves into cellular automata. For example, if the numerical value of each square tends toproximate the average value of its immediate neighbors, then the result will be a simulation of dinary diffusion. If the calculation of new generations is based on the ratio of near neighbors to farighbors (with near ones stimulating and far ones inhibiting), then the system's behavior will generatterns like those of reaction diffusion systems, such as those of Turing that will be discussed below

onsider this set of rules: each square can have a value of either 0 or 1 or 2 or 3; any square having alue of 1 will always become a 2 in the next generation; any 2 will become a 3; and any 3 will beco

0; but any 0 that is next to a 1 will itself become a 1 in the next generation. Can you see what the nehavior will be? Can you see why it will "simulate" propagation of nerve impulses? (What correspothe refractory period?) It can also simulate forest fires!

ellular automata can also be one-dimensional (i.e. along a line, usually with sequential generatioing represented below each other down the page), two-dimensional (like the game of life) or 3-mensional (with neighbors above and below the plane of the page).

he two examples above were obviously initiated at single "filled" squares. But what if the initial patntained some kind of random mixture of filled and empty squares? Here is an example:




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hich of the sets of rules does the cellular automaton above seem to be obeying?what way is the initial pattern non-random?

he color patterns on several species of cone shells happens to have very much this particular patternhat do you conclude from this? What might be the explanation? Are there alternatives?

he major guru of cellular automata these days, especially one-dimensional cellular automata is a Brhiz kid named Stephen Wolfram (who is also the author/owner/president of Mathematica, a set ofpensive software for doing calculus and other higher math on a computer, which costs >$700!).you experiment with different sets of rules, you find hundreds or even thousands of different

haviors, but these fall into a few main patterns including

a) homogeneityb) patterns with internal self-similarity, analogous to fractalsc) chaos.

ne of my favorite facts about cellular automata is that if the rules are these:ll if you have an odd number of neighbors; Empty if you have an even number (or zero neighbors)he result is to duplicate the original pattern infinite numbers of times!!fact, this is only a special case of a more general theorum which is that if the rules are of this formll if sum of neighbors modulo any prime number; otherwise become empty.he "fill if odd, empty if even" rule above is the special case where the prime number is 2)ules of this class will multipley duplicate any initial pattern, for 1, 2 or 3 dimensional automata, andy definition of what is a neighborhood.

pontaneous pattern generation by Reaction Diffusion Systems:




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y itself, diffusion tends to even-out the concentrations of chemicals, but can produce the opposite repontaneous generation of spatial patterns) when combined with chemical reactions that have the rigoperties of positive-feedback (in which the rate of synthesis and/or breakdown of the chemicals vaproportion to their existing local concentrations) A further requirement is that there be two suchemicals, with one of them diffusing faster than the other. The great British mathematician Alan

uring invented such systems and proposed that they might be the cause of various embryological

tterns. We will begin with one of the simplest versions (which I have borrowed from Maynard-Smook on mathematical modeling in biology). Suppose that there are two chemicals (morphogens "A"B"), which obey the following combination of rules:

1) A catalyses its own synthesis and also catalyses the synthesis of B.

2) B catalyses its own breakdown and also catalyses the breakdown of A.

3) B diffuses faster than A.

or this particular combination of rules, there is the additional requirement that the initial

ncentrations of the two substances should be equal everywhere. These rules will cause themplification of any small perturbation in the concentrations of either one of the two chemicals. Thesult of this autocatalytic magnification of perturbations is to form either a local peak or valley of bstance concentration; these changes will then propagate to adjacent locations in such a way as tooduce a relatively regular, periodic alternation of peaks and valleys.

e can follow, step by step, how such patterns are generated by considering the result of a small, loc

crease in the concentration of the A substance. Since A catalyses its own synthesis, a local increaseconcentration will lead to a further increase; so a peak in A concentration will grow spontaneously

e site of the original perturbation. In addition, since A also catalyses the synthesis of B, a peak of Bncentration will also form spontaneously at this same location, the two peaks being superimposed ch other. However, because the B substance diffuses faster than A, this B peak will spread itself ouoader than the A peak. For the same reason, the B peak will also be somewhat less tall that the A p

his greater broadening of the B peak (resulting from the faster diffusion rate of the B substance)complishes several things. For one thing, it limits the autocatalytic growth of the width of the A peis is because the areas immediately around the A peak come to have a higher concentration of B thA, so that there is therefore a net breakdown of the A substance in these areas (by rule #2, B cataly

e breakdown of A). This results in the formation of a valley around the A peak; we might even calllley a "moat" since in 2 dimensions it extends completely around the peak; in these locations, thencentration of A becomes lower and lower. Not only does this limit the lateral expansion of any gi




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ak in substance A, it actually leads to the induction of secondary peaks in the concentration of A ineas just beyond the moat. These diagrams are all taken from an actual computer simulation of thisocess.

he propagated induction of secondary peaks is probably the most difficult aspect of this mechanismsualize. It results from the faster diffusion rate of B relative to A, combined with the existence of a

moat" in the concentration of B superimposed on the moat in A concentration. The essential fact is toth A and B diffuse into the moat from the surrounding areas; but since B diffuses faster, this pullsown the B concentration in the surrounding areas slightly more than it does the A concentration. Th

turn, is what initiates the additional peaks; wherever there is more A than B, there will be a netnthesis of both A and B (because of rule #1 above). So a new peak of A (and of B) will formontaneously on the far side of the "moat". Of course, this new, secondary peak will lead to thermation of another moat on its own far side; the new moat will induce a new peak, and the processntinue.

this example, pattern formation was initiated by a single perturbation at which the concentration oas raised slightly above that of B. But you could just as well have initiated the process by decreasinlow B, or by either increasing or decreasing B relative to A. Any of these changes will initiate the

me sequence of events, although in some cases a valley would form at the site of the originalrturbation, instead of a peak. Patterns will also be formed if many different perturbations are madeany different sites, such as if one made random changes in substance concentrations. In that case, th




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sult will also be waves of alternating peaks and valleys of both substance, but the spacing betweenem will be much less regular than if there had only been a single initiating perturbation so that all taks but one had been secondarily induced. Such a time sequence of randomly initiated waves is shthe diagram below.

he argument has sometimes been made (for example, see Bard and Lauder, 1976) that mechanisms is kind would be inherently incapable of producing such regular patterns as one often finds in theatomical or color patterns of animals. In other words, such mechanisms can produce the irregularotchy pattern of a Jersey cow, but could they produce anything as regular as zebra's stripes? Oneunter-argument which can be made to this is that the degree of regularity can be increased bycreasing the number of initiating perturbations, perhaps even reducing this number to only one.

nother type of possibility involves changing the rules of the reactions themselves in such a way thaey tend to regularize the patterns, so that even when the perturbations are widely and irregularlystributed the eventual pattern which results will have a much greater regularity of spacing than didese perturbations. One approach is to introduce certain non-linearities into the rates by which A andfluence each other's synthesis and breakdown. Another important criticism concerns the demonstraility of actual embryos to "regulate", that is to adjust their spacing mechanisms to match the size ofaterial available to them for subdivision ("size invariance"). Is it possible for such mechanisms asuring proposed to make the "wavelengths" of their resulting patterns vary in proportion to the total




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mount of material available?ÝIn fact, we will see that there are actually several ways to accomplishis, although critics may well doubt if any of these are biologically realistic.

ere are some important general principles:

rst, what matters are the rules of behavior, not the actual chemical nature of the substances. Forample, substance A might be a protein, or an ion, or a nucleotide, or a sugar, or anything else;

kewise, B could be any of these. In fact, one can imagine a pattern generating mechanism in which riables A and B were something other than chemical concentrations: they might be the populationnsities of two cell types, which stimulated each other's growth and death rates, with one of themawling faster than the other! Or A could be cell number and B could be physical tension, and so onWhat matters is the rules A and B obey, not their chemical nature.

Secondly, we should not be prematurely concerned with the specific reasons why the molecularncentration might obey these different rules. For example, A might catalyze its own synthesis bytivating the expression of a gene, or it might be autocatalytic because it activates an enzyme, or eve

a result of having A molecules somehow convert other molecules into additional A. The resultingtterns would be the same.

Thirdly, the set of rules described above is not unique in its morphogenetic powers. Many other,ternative sets of rules of interaction can produce essentially the same effects. The following is anotht of rules:

1) A is made at constant rate

2) B is destroyed at constant rate

3) A catalyses its own synthesis, but catalyses the breakdown of B.

4) B catalyses its own synthesis, but catalyses the breakdown of A.

his set of rules happens to be closer to Turing's original ones.ÝThe diagram below was generated bmputer program obeying this latter set of rules; again, sequential times are shown as successive linots displaced toward the lower right.

otice that the location of the peak in A concentration corresponds to a valley in the concentration of




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dÝvice versa. (rather than the morphogen waves being in phase with each other, as was the case we patterns generated by the first set of rules above). In other respects the resulting patterns are almoentical.

ere is still another set of pattern-generating rules:

1) If the concentration of A is greater than that of B, then both A and B are synthesized at a

constant rate

2) Otherwise, both A and B are destroyed at a constant rate.

3) B diffuses faster than A.

fact, there are an unlimited number of other combinations of rules which also have this capacity tonerate wave-like patterns.ÝSuch systems can also vary in other respects.ÝFor example, instead of itiating the process of wave-generation by having random fluctuations in the local concentrations oe morphogen substances, one can just as well make the reaction equations themselves vary slightlyrandom way, from point to point in space.ÝRandomness in the reaction rates can provide the "seedgrow and propagate peaks and valleys.ÝIn fact, in Turing's original model, the necessary randomn

as introduced in the form of fluctuations in the reaction rates, rather than random variations inorphogen concentrations.

nly certain combinations of reaction rules have an ability to generate patterns. And not all of theseve yet been identified; but there are systematic mathematical methods have been developed by wh

ne can use to look for new ones. Unfortunately, we do still lack experimental criteria for determininhether a given biological pattern was produced by a reaction-diffusion system (or something logicauivalent). This is one of the most serious unsolved problems in the field currently. Another questiohether mechanical instabilities, instead of chemical ones, may produce at least some anatomicaltterns directly. The mechanical forces that the cells exert might themselves produce physicalstabilities that would produce the anatomical patterns in one step. Increasing numbers of papers haven written on this subject. If you want to know more about any of these questions, please ask.

the next section, Curvature, contractility, tension and the shaping of surfaces

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http://www.bio.unc.edu/Faculty/harris/Courses/biol166/curvature.html

harris - cellular automata

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