Possibility of Controlling Self-Organized Patterns with Totalistic Cellular Automata Consisting of Both Rules Like Game of Life and Rules Producing Turing Patterns

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Article Possibility of Controlling Self-Organized Patterns with Totalistic Cellular Automata Consisting of Both Rules like Game of Life and Rules Producing Turing Patterns

Takeshi Ishida ID Department of Ocean Mechanical Engineering, National Fisheries University, Shimonoseki 759-6595, Japan; [email protected]; Tel.: +81-832-86-5111

Received: 20 April 2018; Accepted: 28 June 2018; Published: 3 July 2018

Abstract: The basic rules of self-organization using a totalistic cellular automaton (CA) were investigated, for which the cell state was determined by summing the states of neighboring cells, like in Conway’s Game of Life. This study used a short-range and long-range summation of the cell states around the focal cell. These resemble reaction-diffusion (RD) equations, in which self-organizing behavior emerges from interactions between an activating factor and an inhibiting factor. In addition, Game-of-Life-type rules, in which a cell cannot survive when adjoined by too many or too few living cells, were applied. Our model was able to mimic patterns characteristic of biological cells, including movement, growth, and reproduction. This result suggests the possibility of controlling self-organized patterns. Our model can also be applied to the control of engineering systems, such as multirobot swarms and self-assembling microrobots.

Keywords: cellular automata; Game of Life; reaction-diffusion system; self-organization; Turing pattern model; Young model

1. Introduction Self-organization phenomena, in which global structures are produced from purely local interactions, are found in fields ranging from biology to human societies. If these emergent processes were to be controlled, various applications would be possible in engineering fields. For this purpose, the conditions under which they emerge must be elucidated. This is one of the goals of the study of complex systems. Such control methods may allow the automatic construction of machines. They could also be applied to the control of engineering systems, such as controlling robot swarms or self-assembling microrobots. In addition, the methods allow for the growth of artificial organs or the support of ecosystem conservation, as well as clarifying the emergence of the first life on ancient earth. Mathematical modeling of self-organization phenomena has two main branches: the mathematical analysis of reaction-diffusion (RD) equations, and discrete modeling using cellular automata (CA). The Turing pattern model is one type of RD model. This was introduced by Turing in 1952 [1], where he treated morphogenesis as the interaction between activating and inhibiting factors. Typically, this model achieves self-organization through the different diffusion coefficients for two morphogens, equivalent to an activating and an inhibiting factor. The general RD equations can be written as follows:

∂u = d ∇2u + f (u, v), ∂t 1 ∂v = d ∇2v + g(u, v), ∂t 2

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where u and v are the morphogen concentrations, functions f and g are the reaction kinetics, and d1 and d2 are the diffusion coefficients. Previous studies have considered a range of functions f and g, and models such as the linear model, the Gierer–Meinhardt model [2], and the Gray–Scott model [3] have been used to produce typical Turing patterns. By contrast, CA models are discrete in both space and time. The state of the focal cell is determined by the states of the adjacent cells and the transition rules. The advantage of CA models is that they can describe systems that cannot be modeled using differential equations. Historically, various interesting CA patterns have been discovered. In the 1950s, von Neumann introduced the mathematical study of self-reproduction phenomena, and proved, theoretically, that a self-replicating machine could be constructed on a two-dimensional (2D) grid using 29 cell states and transition rules [4]. Von Neumann’s self-reproducing machine was too large to be implemented until powerful computers became available in the 1990s [5]. Codd [6] showed that the number of cell states could be reduced to eight, and proved the possibility of a self-reproducing machine. However, this model was too complex to be applied to biological processes. In the 1970s, the Game of Life (GoL), invented by Conway [7], became popular. It is a simple CA model that is defined by only four rules, yet produces complex dynamic patterns. Wolfram [8] investigated the one-dimensional (1D) CA model and identified four categories of pattern formation. One of these was the chaotic behavior group. Langton [9] developed a simple self-reproducing machine that did not require von Neumann’s machine completeness. While having a very simple shape, Langton’s machine’s transition rules were complex and produced only specific circling shapes. Byl [10] simplified the Langton model, and Reggia [11] identified the simplest possible reproducing machine. Ishida [12] demonstrated a self-reproducing CA model that resembled a living cell, with DNA-like information carriers held inside a cell-like structure. On the other hand, many studies have applied CA models to biological phenomena, such as the generation of seashell-like structures by Wolfram [8], the development by Young of an RD model in a CA setting that reproduced the patterns of animal coloration [13], the application of the Belousov–Zhabotinskii reaction CA model by Madore and Freeman [14], the catalytic modeling of proteins by Gerhardt and Schuster [15], the modeling of the immune system by De Boer and Hogeweg [16] and by Celada and Seiden [17], the tumor growth model of Moreira and Deutsch [18], the genetic disorder model of Moore and Hahn [19], the modeling of the hippocampus by Pytte et al. [20], and the dynamic modeling of cardiac conduction by Kaplan et al. [21]. Furthermore, Markus [22] demonstrated that a CA model could produce the same output as RD equations. In addition, CA has two types of model. The first one is a totalistic CA, for which each cell state is determined by summing the states of the neighboring cells. Conway’s GoL is one such CA. GoL produces the three typical patterns from the particular initial cell state configurations shown in Figure1: “still”, “moving”, and “oscillating”. Meanwhile, most of the initial cell state configurations reach extinction, i.e., all cell states become 0. However, it is impossible to deduce the transition rules to make specific shapes automatically. In contrast with the totalistic CA, the second type of CA model counts all the neighbor cells’ state patterns. Wolfram’s 1D CA is one such CA, in which there are two states (0 and 1) of three inputs and one output. As there are 23 = 8 inputs, and each input has two outputs (0 or 1), there are 28 = 256 transition rules. Wolfram researched all these transition rules and divided the output patterns into four classes. Class 1 covers the steady states, Class 2 the periodical patterns, Class 3 the random patterns (edge of chaos), and Class 4 the chaotic patterns. Figure2 shows one result of the Class 3 rules, which produce unique patterns like that of a certain type of seashell. However, in a Wolfram-type model, increasing the inputs to five produces an explosive increase in the number of output combinations, to 232. It therefore becomes impossible to investigate all outputs. A similar situation occurs in the case of a 2D CA model, making it, again, impossible to investigate all possible patterns. Langton proposed the λ parameter [23] to elucidate the condition to give rise Micromachines 2018, 9, x FOR PEER REVIEW 3 of 19 Micromachines 2018, 9, 339 3 of 18 Micromachines 2018, 9, x FOR PEER REVIEW 3 of 19 give rise to the diversity of patterns—edge of chaos. However, this could not derive the rule to constructtogive the diversity rise favorite to the of patterns. diversity patterns—edge of patterns of— chaos.edge of However, chaos. However this could, this not could derive not the derive rule the to constructrule to favoriteconstruct patterns. favorite patterns.

FigureFigure 1. 1. 1.PatternsPatterns Patterns from from from Conway’s Conway’s Conway’s Game Game Game of Life of Life under und fourerer fourrules: four rules: rules:(1) a cell( 1)(1) witha a cell cell one with with or oneno oneliving or or no neighborsno living living neighborsdies;neighbors (2) a celldies dies with; (;2) (2) a fouracell cell with or with more four four living or or moremore neighbors living neighbors dies; (3) dies adies cell; ;( 3)( with3) a a cell cell three with with living three three living neighbors living neighbors neighbors is born; isand bornis born (4); aand; celland (4) with(4) a acell twocell with with or threetwo two or livingor threethree neighbors livingliving neighbors survives. survives.survives. The typical The The typical typical patterns patterns patterns that emergethat that emerge emerge can be cancategorizedcan be be categorized categorized into three into into groups: three three groups groups still, moving,:: still,still, moving,moving, or oscillation. or oscillation.oscillation.

0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 00 00 00 0 10 1 10 11 10 11 10 11 10 1 1 1 01 11 01 11 00 00 00 00 0 0 0 00 00 00 01 1 01 01 01 01 01 01 01 01 0 1 0 10 10 10 11 11 00 00 00 0 0 0 00 00 01 1 1 10 00 00 00 00 00 00 00 0 0 0 00 00 01 01 11 11 00 00 0 0 0 00 01 11 10 01 1 10 00 00 00 00 00 00 0 0 0 00 01 01 10 10 11 11 00 0 0 0 01 11 11 01 10 1 11 10 00 00 00 00 00 0 0 0 01 01 11 11 01 01 11 11 0 0 0 11 10 10 10 01 01 01 11 10 00 00 00 00 0 0 1 01 10 10 10 10 10 10 11 1 10 0 11 01 00 0 0 0 10 11 11 10 00 00 00 1 0 1 11 11 00 00 00 00 01 01 10 01 1 11 11 01 0 0 10 11 01 01 11 10 00 10 1 1 0 10 11 11 00 00 01 01 11 1 10 0 10 10 11 01 10 1 11 10 10 11 11 10 11 1 1 1 01 01 11 11 01 01 10 10 10 01 1 00 01 11 10 01 01 01 01 01 01 01 01 01 0 1 0 10 10 10 10 10 11 11 00 0 0 0 01 11 11 01 0 0 00 00 00 00 00 00 00 0 0 0 00 00 00 00 01 01 11 11 0 0 0 11 10 10 1 10 0 00 00 00 00 00 00 00 0 0 0 00 00 00 01 01 10 10 11 1 10 0 11 01 01 1 1 10 00 00 00 00 00 00 00 0 0 0 00 00 01 01 11 11 01 01 10 01 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 0 0 0 0 1 1 0 0 1 1 1 1 1 1 1 10 10 00 0 1 1 11 11 10 10 00 00 00 1 0 1 01 11 11 11 10 00 00 10 10 0 0 10 10 10 1 1 0 00 01 01 10 10 00 10 1 0 0 10 10 00 01 11 10 10 10 10 01 1 00 00 01 01 1 1 01 01 11 11 10 10 10 1 1 1 10 10 11 11 11 01 00 00 0 0 0 00 01 01 10 01 10 10 10 10 01 01 00 01 0 1 1 01 01 01 00 10 11 01 00 0 0 0 01 01 11 1 1 1 00 00 11 11 01 01 01 1 1 1 10 00 01 11 11 11 11 01 0 0 0 01 10 10 0 0 1 11 11 11 10 10 00 10 1 0 1 11 11 11 10 00 00 10 11 01 10 0 11 11 10 10 1 1 00 00 01 01 10 10 10 0 1 0 10 00 01 11 10 10 11 11 10 0 0 11 01 01 01 10 1 11 01 01 11 11 00 11 1 1 0 10 11 11 11 01 01 01 01 1 1 1 10 10 00 0 01 01 10 10 10 10 11 11 11 1 0 1 01 01 00 10 10 00 00 10 10 0 0 10 10 10 10 01 1 11 00 00 01 01 00 01 0 1 0 00 01 11 10 10 10 10 10 10 01 1 00 00 00 0 10 10 11 11 01 01 01 01 01 0 1 0 11 11 11 01 00 00 00 00 0 0 0 00 00 00 01 10 01 01 10 10 00 00 00 00 0 0 1 01 00 10 11 01 00 00 00 0 0 0 00 00 01 01 1 1 11 11 10 10 00 00 00 1 0 1 01 11 11 11 11 01 00 00 0 0 0 00 01 01 10 01 0 00 01 01 10 10 00 10 1 0 0 10 10 00 00 10 11 01 00 0 0 0 01 01 11 1 01 01 01 01 11 11 10 10 10 1 1 1 10 10 10 10 11 11 11 01 0 0 0 01 10 10 01 10 0 00 10 10 01 01 00 01 0 1 1 01 00 00 01 01 00 10 11 01 10 0 11 11 11 1 10 10 10 10 11 11 01 01 01 1 1 1 11 01 01 01 01 11 11 11 10 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0

Figure 2. One of the patterns produced by Wolfram’s 1D cellular automaton (CA) model. Rule 126: Figurewhite 2. cell One indicates of the the patterns 0 state, produced while the by black Wolfram’s cell indicates 1D cellularthe 1 state. automaton The upper ( (CA)CA line) model. corresponds Rule Rule to126: 126: whitethe cell initial indicates condition. the 0 Applying state, while this the rule, black the cell time indicates series pattern the 1 state. produces The u upper pper a range line ofcorresponds triangular to theshapes, initial initial in condition. condition. this case resembling Applying Applying this a thiscertain rule, rule, thetype the time of timeseashell. series series pattern pattern produces produces a range a ofrange triangular of triangular shapes, shapes,in this case in this resembling case resembling a certain a typecertain of seashell.type of seashell. CA models reduce the computational load by removing the need to solve differential equations numerically.CA model models s However, reduce the it computational is necessary to load translate by removing the partial the need differential to solve equations differen differential tial of theequations RD numerically.equations into However, However, the transition it itis necessary is rules necessary of tothe translate toCA translate model. the partialIn the a study partial differential of up differential to equationspresent, equations no of general the RD ofmethod equations the RD equationsintocurrently the transition into exists the for rules transition doing of the this, CArules with model. of the In excCA aeption studymodel. ofof In upthe a to ultrastudy present,-discrete of up no generalto systems present, method [2 4no] used general currently in certain method exists currentlyforsoliton doing equations.exists this, with for Normally,doing the exception this, the with transition of the ultra-discreteexc ruleseption driving of the systems the ultra emergent- [discrete24] used phenomena systems in certain [2must4 soliton] used be found equations.in certain by trial and error. Studies that attempted to derive the transition rules using genetic algorithms [11,25] solitonNormally, equations. the transition Normally, rules the driving transition the rules emergent driving phenomena the emergent must phenomena be found bymust trial be andfound error. by demonstrated the difficulty of deriving generalized rules. The Ishida cell division model [12] is a trialStudies and that error. attempted Studies tothat derive attempted the transition to derive rules the using transition genetic rules algorithms using genetic [11,25 ]algorithms demonstrated [11, the25] hybrid of the Wolfram- and Conway-type CAs, in which part of the rules are totalistic transition demonstrateddifﬁculty of deriving the difficulty generalized of deriving rules. The generalized Ishida cell rules. division The model Ishida [12 cell] is division a hybrid model of the Wolfram-[12] is a rules, and the others are head-to-head-type transition rules, and these rules were discovered by trial hybridand Conway-type of the Wolfram CAs,- inand which Conway part-type of the CAs, rules in arewhich totalistic part of transition the rules rules, are totalistic and the otherstransition are rulesand, anderror. the others are head-to-head-type transition rules, and these rules were discovered by trial head-to-head-typeThe Young model transition [13] is rules, one of and the these 2D totalistic rules were models discovered that bridges by trial the and RD error.equations and CA and error. model;The Young this model model is [used13] is to one produce of the Turing2D totalistic patterns. models Some that other bridges examples the RD to equationsproduce Turing and CA The Young model [13] is one of the 2D totalistic models that bridges the RD equations and CA model;patterns this model are below is used. Adamatzk to producey Turing[26] patterns. studied Some a binary other-cell examples-state eight to produce-cell neighborhood Turing patterns model; this model is used to produce Turing patterns. Some other examples to produce Turing are below. Adamatzky [26] studied a binary-cell-state eight-cell neighborhood two-dimensional cellular patterns are below. Adamatzky [26] studied a binary-cell-state eight-cell neighborhood

Micromachines 2018, 9, 339x FOR PEER REVIEW 44 of of 19 18 two-dimensional cellular automaton model with semitotalistic transitions rules. Dormann [27] also usedautomaton a 2D model outer-totalistic with semitotalistic model with transitions threes states rules. to Dormann produce [ 27 a ] Turing also used-like a 2D patter outer-totalisticn. Tsai [28] analyzedmodel with a threes self-replicating states to produce mechanism a Turing-like in Turing pattern. patterns Tsai with [28] analyzed a minimal a self-replicating autocatalytic monomermechanism‐dimer in Turing system. patterns with a minimal autocatalytic monomer-dimer system. Young’s CA model uses a real number function to derive the distance effects, with two constant values within within a grida grid cell: cell: u1 (positive) u1 (positive and) u 2 and(negative). u2 (negative). The calculation The calculation begins by beginsrandomly by distributing randomly distributingblack cells on black a rectangular cells on a grid. rectangular Then, for grid. each Then, cell at for position each cell R0 atin position 2D ﬁelds, R the0 in next2D fields, cell state the of next R0, celldue state to all of nearby R0, due black to all cells nearby at position black Rcellsi, are at added position up. R Ri,i areis assumed added up. to beRi ais blackassumed cell withinto be a radius black cellr2 from within R0 cell. radius The r function2 from R v(r)0 cell. is a The continuous function step v(r) function, is a continuous as shown step in Figure function,3b. The as activation shown in Figurearea, indicated 3b. The activation by u1, has area, a radius indicated of r1 byand u1 the, has inhibition a radius of area, r1 and indicated the inhibition by u2, area, has a indicated radius of by r2 (ru22, >has r1 ). a Atradius position of r2 R(r02, when> r1). At Ri positionis assumed R0, towhen be a R gridi is assumed within r2 ,to function be a grid v(|R within0 − R ri|)2, function value is v(|addedR0 − up. Ri|) Function value is |R added0 − R i| up. indicates Function the | distanceR0 − Ri| between indicates R 0 theand distance Ri. If ∑ i betweenv (|R0 − RR0 i|)and > 0, Ri the. If ∑gridi v (| cellR0 at− R pointi|) > 0, R0 theis markedgrid cell as at a point black R cell.0 is marked If ∑i v (|R as 0a −blackRi|) cell. < 0, If the ∑igrid v (|R cell0 − becomesRi|) < 0, the a white grid cell cell. becomesIf ∑i v (|R a0 white− Ri|) cell. = 0, Ifthe ∑ gridi v (| cellR0 − does Ri|) not = 0, change the grid state cell [13 does]. Using not these change functions, state [13 Young]. Using described these functions,that a Turing Young pattern described can be that generated. a Turing Spot pattern patterns can orbe stripedgenerated. patterns Spot patterns can be created or striped with patterns relative canchanges be created between with u1 relativeand u2. changes between u1 and u2.

Figure 3. Outline of Young’s model. ( a) The activation area has a radius r1 and the inhibition area has an outer radius r 2;;( ( bb)) Function Function v(r) v(r) is is a a continuous continuous step step function function representing representing the the activation activation area area and inhibition area.

In this Young model, let u1 = 1, u2 = w (here 0 < w < 1), and further, if the state of the cell is set to 0 In this Young model, let u1 = 1, u2 = w (here 0 < w < 1), and further, if the state of the cell is set to 0 (white) and 1 (black), this model can be arranged as follows. The state of cell i is expressed as (white) and 1 (black), this model can be arranged as follows. The state of cell i is expressed as ci(t) 푐 (푡) (푐 (푡) = [0, 1]) at time t. The following state 푐 (푡 + 1) at time 푡 + 1 is determined by the states (c푖 i(t) =푖 [0, 1]) at time t. The following state ci(t + 1푖 ) at time t + 1 is determined by the states of the of the neighboring cells. Here, N1 is the sum of the states of the domain within S meshes of the focal neighboring cells. Here, N1 is the sum of the states of the domain within S meshes of the focal cell. cell. Similarly, N2 is the sum of the states of the domain within t meshes of the focal cell, assuming Similarly, N2 is the sum of the states of the domain within t meshes of the focal cell, assuming that thatS < t. S < t. S 푆 N1 = ∑ ci(t) 푁1 = ∑i=1 푐푖(푡) 푖=1 T 푇 N2 = ∑ ci(t) 푁2 = ∑i=1 푐푖(푡) Here, S (T) is the number of cells within s (t) meshes푖=1 from focal cell. Figure4 shows the schematic of theHere, total sumS (T) of is states the N number1 and N of2. The cells next within time s state (t) meshes of the focal from cell focal is determined cell. Figure by the 4 shows following the schematicExpression of (1): the total sum of states N1 and N2. The next time state of the focal cell is determined by the following Expression (1):

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Cell state at the next time step 1: if 푁> 푁 1× : 푤 if N1 > N2 × w 1 2 (1) Cell state at the next = time{Unchange step=: if 푁Unchange1 = 푁2 × :푤 if N1 = N2 × w (1)  0: if 푁1 < 푁2 0× : 푤 if N1 < N2 × w

Figure 4. Schematic of the summing of states N1 and N2. Each grid cell has state 0 (white) or 1 (black). Figure 4. Schematic of the summing of states N1 and N2. Each grid cell has state 0 (white) or 1 (black). The inner area has a domain within s grids of the focal cell and the outer area a domain within t grids. The inner area has a domain within s grids of the focal cell and the outer area a domain within t grids.

TheThe essence of of the the Turing Turing pattern pattern model model is isthat that the the short short-- and and long long-range-range spatial spatial scales scales are each are eachaffected affected by separate by separate factors factors [29], and [29 pattern], and pattern formation formation emerges emerges from nonlinear from nonlinear interactions interactions between betweenthe two the factors. two factors. Turing Turing used twoused chemicals two chemicals with with different different diffusion diffusion coefficients coefficients to to create create these short-short- andand long-rangelong-range spatialspatial effects.effects. However,However, asas longlong asas therethere existsexists aa differencedifference betweenbetween long-long- andand short-rangeshort-range effects,effects, otherother implementationimplementation methodsmethods cancan bebe applied.applied. This model usedused two rangesranges of ss meshmesh andand ttmesh mesh toto create create a a difference. difference. It It is is therefore therefore thought thought that that this this model model is essentiallyis essentially the the same same as anas an RD RD model. model. FigureFigure5 5shows shows the the results results for for the the Expression Expression (1) (1) model model using using a a square square grid. grid. As As the the initial initial conditions,conditions, 300300 statestate 11 cellscells werewere setset randomlyrandomly inin thethe calculationcalculation fieldfield 8080× × 80.80. Turing-likeTuring-like patternspatterns emergedemerged as as parameter parameterw changed.w changed. When Whenw was w greaterwas greater than 0.4, than spot 0.4, patterns spot patterns were observed, were observed, whereas atwhereasw = 0.3, at stripe w = 0.3, patterns stripe emerged.patterns emerged. When w wasWhen 0.20 w orwa lower,s 0.20 allor cellslower, in all the cells field in had the afield value had of 1a (shownvalue of in 1 black).(shown Changing in black). the Changing parameters the Nparameters1 and N2 merely N1 and changed N2 merely the scalechanged of the the patterns. scale of the patterns.On the other hand, the GoL is one of the 2D totalistic CAs to emerge patterns of diversity. In this model,On the the state other of hand, the focal the GoL cell is is activatedone of the when 2D totalistic the total CAs number to emerge of surrounding patterns of statesdiversity. is within In this a certainmodel, range.the state Therefore, of the focal we cell considered is activated the extendedwhen the model total number of GoL. of Whereas surrounding the GoL states uses is the within states a ofcertain the eight range. surrounding Therefore, cellswe considered (the Moore the neighborhood), extended model an of extended GoL. Whereas neighborhood the GoL is uses possible the states that considersof the eight two surrounding or more meshes cells (the from Moore the focalneighborhood), cell like Young an extended model. neighborhood Such models are is possible expected that to generateconsiders a two similar or more variety meshes of patterns from tothe the focal GoL. cell like Young model. Such models are expected to generateIn natural a simi phenomena,lar variety of short-distance patterns to the influences GoL. play a stronger role than more distant influences. UnderIn this natural circumstance phenomena, (1) The short influence-distance decreases influences as the distance play a fromstronger the focal role cell than increases, more distant which isinfluences.N1 + N2 × Underw type this model circumstance (w: weight (1) parameter, The influence 0 < w

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Figure 5. Results for the model of ExpressionExpression (1) with s = 5 and t = 9 in a square grid. As the initial condition, 300 state 1 cells were set randomly in the calculation field. ﬁeld. Turing Turing-like-like patterns emerged as parameter w changed. When w was greater than 0.4, spot patterns were observed, whereas at w = 0.3, 0.3, stripe patterns emerged. When When ww waswas 0.20 0.20 or or lower, lower, all c cellsells in the fiel ﬁeldd had a value of 1 (shown in black).

The first first type type of of the the model model is isexpected expected to produce to produce results results similar similar to those to those of other of other totalistic totalistic CAs. UnderCAs. Under specific specific conditions, conditions, GoL-like GoL-like patterns patterns will be willfound. be The found. second The model second is model expected is expected to produce to produceresults similar results tosimilar those to of those a Young of a Young model, model, which which similarly similarly takes takes account account of of the the nearness nearness of of the activating and inhibiting factors.factors. This studystudy adoptedadopted thethe second second type type of of the the model, model, and and the the emergence emergence of a of variety a variety of patterns, of patterns, such suchas the as Turing the Turing pattern, pattern, were confirmed.were confirmed. In addition, In add byition, applying by applying GoL-type GoL rules-type to rules the secondto the second model modeltype, more type, complex more complex patterns patterns are expected, are expected such as, such self-reproducing as self-reproducing and growth and growth patterns. patterns. Our study Our studymodel model was able was to ableproduce to produce not only not a Turing-like only a Turing pattern-like while pattern using while a simple using transition a simple expression, transition expression,but also the modelbut also produces the model various produces patterns various such as patterns still, moving, such asand still, oscillation. moving, Furthermore, and oscillation. this Furthermoremodel can control, this these model patterns can control with two these parameters, patterns whichwith two means parameters, the possibility which of meanscontrolling the possibilityself-organized of control patternsling with self- aorganized totalistic CApattern model.s with a totalistic CA model.

2. M Modelodel

We investigatedinvestigated thethe resultsresults ofof anan NN11 −− N22 ×× ww typetype totalistic totalistic CA CA model model in in a a 2D 2D field ﬁeld that was implemented in Java. The The GoL GoL-like-like rules rules assumed in in this model which limit the range of survival as state 1.1. ToTo addressaddress this, this, by by extending extending Expression Expression (1), (1), we we introduced introduced Expression Expression (2). (2 This). This expression expression was wasobtained obtained by adding by adding Rule 4 Rule to Expression 4 to Expression (1). When (1).N 2When= 0, it N is2 assumed= 0, it is that assumed Rule 4that takes Rul precedence.e 4 takes precedence.   0 : if N1 > N2 × w2 (Rule 4) Cell state of the next time step  1 : if N1 > N2 × w1 (Rule 3) Cell state of the next time step0: if= 푁1 > 푁2 × 푤2 (푅푢푙푒 4) (2) Unchange : if N1 = N2 × w1(Rule 2) 1: if 푁1 > 푁2 × 푤1 (푅푢푙푒 3) (2) = {  N < N × w (Rule ) Unchange: if 푁1 =0푁 :2 if× 푤11(푅푢푙푒2 2) 1 1 0: if 푁1 < 푁2 × 푤1(푅푢푙푒 1) S N1 = ∑푆 ci(t) i=1 푁1 = ∑ 푐푖(푡) 푖=1

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MicromachinesMicromachines 2018 2018, 9, 9x, FORx FOR PEER PEER REVIEW REVIEW 7 of7 of19 19 T 푇 N2 =푇∑ ci(t) 푁 = ∑i=1 푐 (푡) 푁2 2= ∑ 푐푖(푖푡) Here, 1 > w2 > w1 > 0, and N1 is the sum of푖= the푖1=1 states of the domain within s meshes of the focal cell. Similarly,Here, 1 N> w2 2is > thew1 > sum 0, and of theN1 statesis the sum of the of domain the state withins of the t meshesdomain ofwithin the focal s meshes cell, assumingof the focal that Here, 1 > w2 > w1 > 0, and N1 is the sum of the states of the domain within s meshes of the focal s

Figure 6. Hexagonal grid field. Here, N1 is the domain within s meshes of the focal cell. Similarly, N2 FigureFigure 6. Hexagonal grid grid field. ﬁeld. Here Here,, NN1 1isis the the domain domain within within s meshes s meshes of of the the focal focal cell. cell. Similarly, Similarly, N2N 2 is

theis isthe sumthe sum sum of of the ofthe statesthe states states of of the ofthe the domain domain domain within within within tt meshesmeshes t meshes of ofthe the focal focal focal cell, cell, cell, assuming assuming assuming that thatthat s s

FigureFigure 7. 7.Initial Initial conditions conditions of ofhexagonal hexagonal grids grids field. field. The The black black cell cell indicates indicates the the 1 state1 state and and the the white white Figure 7. Initial conditions of hexagonal grids field. The black cell indicates the 1 state and the white cellcell indicates indicates the the 0 state.0 state. These These configurations configurations of of state state 1s 1s were were set set in in the the center center of ofthe the calculation calculation cell indicates the 0 state. These configurations of state 1s were set in the center of the calculation fields. fields.fields.

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~~3. Results~~

MicromachinesFigure 82018 maps, 9, 339 the model results against parameters w1 and w2. Initial condition is based on8 300 of 18 randomly arranged state 1 cells. Different patterns were observed at different values of w1 and w2. As in the Young-type model, the parameter w1 determined the pattern type, whether spotted or striped. 3. Results As w2 became smaller, pulsing and unsteady white spots were born within the black pattern. The vibrationFigure of 8these maps white the modelspots became results more against intense parameters as the valuew1 and of w2 .was Initial reduced. condition Figure is based9 shows on the300 result randomlys when arranged varying state the number 1 cells. Differentof states patterns1 (black werecell) to observed place them at different randomly values in the of winitial1 and state.w2. As The in calculation the Young-type conditions model, were the parameters = 5, t = 9,w w11 determined= 0.40, and w the2 = pattern0.80 in type,a hexagonal whether grid. spotted Based or onstriped. the result As ws 2ofbecame this figure smaller,, if there pulsing areand more unsteady than a certain white spots number were of born states within 1, it is the evident black pattern.that a similarThe vibration pattern offormation these white develops. spots Furthermore, became more Figure intense 10 as shows the value the transition of w2 was of reduced.the ratio of Figure states9 1shows and 2, the and results the ratio when of varyingthe applied the numbercells of Rule of states 4 at 1each (black time cell) step. to placeThis result them randomlywas calculated in the when initial wstate.2 was Thechanged calculation (s = 5, t conditions = 9, and w1 were = 0.35s )=. When 5, t =9, ww2 was1 =0.40, 1, Rule and 4 wwas2 = not 0.80 applied, in a hexagonal and the grid.result Based was equivalenton the results to the of Young this ﬁgure, model. if thereAs w2are was more lowered, than the a certain application number ratio of of states Rule 1, 4 itincrease is evidentd. At that this a time,similar white pattern region formations were develops.generated Furthermore, in the black Figure spots, 10and shows the theinstability transition increased of the ratio and of became states 1 pulsatingand 2, and. This the phenomenon ratio of the applied is attributed cells of to Rule Rule 4 4. at each time step. This result was calculated when w2 wasFigure changed 11 shows (s = the 5, t effect= 9, and of thew1 initial= 0.35). condition When w which2 was 1,is Rulecentrally 4 was arranged not applied, on four and black the resultcells. Thewas instability equivalent became to the more Young pronounced model. As asw 2wwas2 decreased, lowered, and the the application central spotted ratio ofpattern Rule 4grew increased. as an unstableAt this time, wave white in the regions calculation were generatedfield. By contrast, in the black when spots, w2 was and relatively the instability large, increased the central and spot became did notpulsating. spread. This phenomenon is attributed to Rule 4.

Figure 8. Results for the model with s = 5 and t = 9 in a hexagonal grid. The results image of each Figure 8. Results for the model with s = 5 and t = 9 in a hexagonal grid. The results image of each w1 w1 and w2 value is based on 300 randomly arranged state 1 cells. Instability became stronger as w2 and w2 value is based on 300 randomly arranged state 1 cells. Instability became stronger as w2 decreased to w1. decreased to w1.

Figure 11 shows the effect of the initial condition which is centrally arranged on four black cells. The instability became more pronounced as w2 decreased, and the central spotted pattern grew as an unstable wave in the calculation ﬁeld. By contrast, when w2 was relatively large, the central spot did not spread.

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Figure 9. Results when varying the number of states 1 (black cell) to place them randomly in the FigureFigure 9. 9. RResultsesults when when varying varying the the number number of of states states 1 1 (black (black cell) cell) to place them randomly in thethe initial initial states. The calculation conditions were s = 5, t = 9, w1 = 0.40, and w2 = 0.80 in a hexagonal grid. initialstates. state Thes. calculationThe calculation conditions conditions were weres = 5,s =t 5,= 9,t =w 9,1 w=1 0.40,= 0.40, and andw 2w=2 = 0.80 0.80 in in a a hexagonal hexagonal grid.grid. Based Based on this result, if there are more than a certain number of states 1, it is evident that a similar Basedon this onresult, this result, if there if there are more are more than than a certain a certain number number of statesof states 1, it1, is it evidentis evident that that a similara similar pattern pattern formation develops. patternformation formation develops. develops.

~~Time step Time step~~

Figure 10. Transition ratio of states 1 and 2, and the ratio of the applied cells of Rule 4 at each time FigureFigure 10 10.. TransitionTransition ratio ratio of of states states 1 1and and 2, 2, and and the the ratio ratio of of the the applied applied cells cells of of Rule Rule 4 4at at each each time time step. step. This result was calculated when w2 was changed in s = 5, t = 9, and w1 = 0.35. When w2 was 1, step.This This result result was was calculated calculated when whenw2 wasw2 was changed changed in sin= s 5,= t5,= t 9,= and9, andw1 w=1 0.35.= 0.35. When Whenw w2 2was was 1, 1, Rule 4 Rule 4 was not applied, and it was equivalent to the Young model. As w2 was lowered, the Rulewas not4 was applied, not applied, and it was and equivalent it was equivalent to the Young to the model. Young As model.w2 was As lowered, w2 was the lowered, application the ratio application ratio of Rule 4 increased. In comparison with Figure 8, white regions were generated in applicationof Rule 4 increased. ratio of Rule In comparison4 increased. withIn comparison Figure8, white with Figure regions 8, were white generated regions were in the generated black spots, in and the black spots, and instability increased and began pulsating. This phenomenon is attributed to Rule 4. theinstability black spots increased, and instability and began increased pulsating. and began This pulsating phenomenon. This isphenomenon attributed tois attributed Rule 4. to Rule 4.

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Figure 11. Results for the model with s = 5 and t = 9 in a hexagonal grid. The image of each w1 and w2 Figure 11. Results for the model with s = 5 and t = 9 in a hexagonal grid. The image of each w1 and w2 value is based on centrally arranged four black cells as shown in Figure7. Instability became stronger value is based on centrally arranged four black cells as shown in Figure 7. Instability became stronger as w2 decreased. The initial central spotted pattern grew as an unstable wave in the ﬁeld. When w2 as w2 decreased. The initial central spotted pattern grew as an unstable wave in the field. When w2 was relatively large, the central spot did not spread. was relatively large, the central spot did not spread.

FigureFigure 12 shows the time time series series results results of of the the model model with with ss == 4, 4, t t= =8, 8, ww1 =1 0.40,= 0.40, and and w2w =2 0.75.= 0.75. In Inthis this figure, figure, i showsi shows the the iterat iterationion number number of of the the calculation. calculation. Th Thee video video of of F Figureigure 1212 cancan be viewed in the VideoVideo S1S1 ofof SupplementarySupplementary Materials.Materials. As the initial condition, four state-1state-1 cells (black) were set inin thethe centercenter ofof thethe fieldfield shownshown inin FigureFigure7 7.. TheThe centralcentral spottedspotted patternpattern waswas divideddivided intointo twotwo spots,spots, and these these divisions divisions spread spread until until the the field field was was filled filled with with spotted spotted patterns. patterns. Figure Figure 13 shows 13 show the resultss the whenresults the when initial the condition initial conditi of stateon 1of was state subjected 1 was subjected to the same to conditionsthe same conditions as in Figure as 12 in( s Figure= 4, t = 12 8, w(s1 == 4, 0.40, t = 8, and w1 =w 0.40,2 = 0.75 and in w a2 hexagonal= 0.75 in a hexagonal grid). When grid). there When was onlythere one was state only 1 one in the state initial 1 infield, the initial Rule 4field was, Rule applied 4 was around applied this around state 1, whichthis state disappeared 1, which disappear at the nexted step. at the If therenext step. are two If there or more are statestwo or 1 asmore the states initial state,1 as the it was initial observed state, it that was the observed state 1 survived, that the andstate pattern 1 survive formationd, and similar pattern to formation Figure 12 occurred.similar to F Inigure the 12 case occur of severalred. In the state case 1s, of due several to the state balance 1s, due between to the balance Rules 3 between and 4,complicated Rules 3 and patterns4, complicate wered observedpatterns were in two observed or three in steps twofrom or three the beginningsteps from ofthe the beginning calculation, of the and calculation, thereafter, aand self-replicating thereafter, a self pattern-replicating was observed. pattern was observed. In addition, addition, Figure Figure 14 14 shows shows the the calculation calculation results results when when increasing increasing the black the black region region as the as initial the condition,initial condition under, theunder same the conditions same conditions as in Figure as in 12 Figure( s = 4, t12= 8,(s w= 14,= t 0.40, = 8, andw1 =w 0.40,2 = 0.75 and in w a2 hexagonal= 0.75 in a grid).hexagonal The pointgrid).symmetrical The point symmetrical initial shape initial tended shape to be tend a fixeded to pattern be a fixed due pattern to the balance due to ofthe the balance rules. Onof the the rule others. On hand, the ifother the blackhand, area if the was black larger area than was a larger certain than size, aRule certain 1 was size, mostly Rule 1 applied, was mostly and disappearedapplied, and atdisappear the seconded at step. the second To produce step. aT self-replicatingo produce a self pattern,-replicating an asymmetric pattern, an shapeasymmetric must initiallyshape must occur. initially Figure occur. 15 shows Figure the 15 transition shows the of transition the ratios of of the states ratios 1 and of states 2, and 1 theand ratio2, and of the applied ratio cellsof applied of Rule cells 4 per of time Rule step 4 per under time the step same under conditions the same as conditions in Figure 12 as( s in= Figure 5, t = 9, 12w 1 (s= = 0.40, 5, t and= 9, w21 = 0.40, 0.75). and In 90 w2 to = 1000.75 steps,). In 90 the to frequency 100 steps, of the pattern frequency division of pattern became division high, and bec theame application high, and ratio the ofapplication Rule 4 increased. ratio of RuleRule 44 generatedincreased. a R whiteule 4 areagenerated in the blacka white spot area patterns, in the andblack instability spot patterns increased;, and hence,instability the symmetricalincreased; hence, shape the collapsed symmetrical and becameshape collapsed a trigger forand division. became a trigger for division. Figures 16 and 17 show the results with an initial pattern of a symmetrical ring with state 1s. In the case where the initial shape was symmetrical, the still pattern or the oscillation pattern often appeared in a wide range of w1 and w2. When w1 = 0.45 and w2 = 0.55, the ring pattern spread while dividing spots. In an opposite way, as w2 was approached as w1, the instability increased and the pattern disappeared.

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1 2 FigureFigureFigure 12. 1122.. Results ResultsResults for forfor the thethe model modelmodel with withwiths ss= == 4, 44,,t tt= == 8, 8,8,w ww11 = == 0.40,0.40,0.40, andandand w ww22 == 0.750.75 inin aa hexagonalhexagonalhexagonal grid.grid.grid. AsAsAs thethethe initialinitialinitial condition,condition, fourfour statestate state 11 1 cellscells cells (black)(black) (black) werewere were setset set inin in thethe the centercenter center ofof the ofthe the fieldfield field asas showshow as shownnn inin FigureFigure in Figure 7.7. TheThe7. Thecentralcentral central spottedspotted spotted patternpattern pattern waswas wasdivideddivided divided intointo intotwotwo spots, twospots, spots, whichwhich which thenthen then spreadspread spread untiluntil until thethe fieldfield the fieldwaswas filled wasfilled filled withwith withspottedspotted spotted papatterns.tterns. patterns.

FigureFigureFigure 13. 1313.. ResultsRResultsesults withwith thethethe initialinitialinitial conditionconditioncondition ofofof statestate 111 underunderunder the thethe samesamesame conditionsconditionsconditions asasas in inin FigureFigureFigure 12 1212( ((sss = == 4,4, 1 2 ttt = == 8,8, ww11 == 0.40,0.40, and and ww2 == 0.750.75 0.75 inin in aa a hexagonalhexagonal grid).grid). IfIf therethere isis only only one one state state 1 1 in in the the initial initial ﬁeld,field,field, RuleRule 44 waswaswas applied appliedapplied at atat that thatthat location, locationlocation which,, whichwhich disappeared disappeareddisappeared at atat the thethe next nextnext step. step.step. If there IfIf therethere are areare two twotwo or more oror moremore state statestate 1s as 11s thes asas initialthethe initial initial state, state, state, state 1 state state was 1observed 1 waswas observed observed to survive, to to survive andsurvive a pattern,, andand a formationa patternpattern similarformationformation to Figure similar similar 12 to tooccurred. Figure Figure 12 12 occurred.occurred. Figures 16 and 17 show the results with an initial pattern of a symmetrical ring with state 1 s. In the case where the initial shape was symmetrical, the still pattern or the oscillation pattern often appeared in a wide range of w1 and w2. When w1 = 0.45 and w2 = 0.55, the ring pattern spread while dividing spots. In an opposite way, as w2 was approached as w1, the instability increased and the pattern disappeared.

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Figure 14. Calculation results, when increasing the black region as the initial condition, under the FigureFigure 14. 14Calculation. Calculation results, results when, when increasingincreasing the black region region as as the the initial initial condition condition,, under under the the same conditions as in Figure 12 (s = 4, t = 8, w1 = 0.40, and w2 = 0.75 in a hexagonal grid). The point samesame conditions conditions as inas Figurein Figure 12 12( s (=s = 4, 4,t =t = 8, 8,w w11 == 0.40,0.40, and ww22 == 0.75 0.75 in in a ahexagonal hexagonal grid). grid). The The point point symmetrical initial shape tended to be a fixed pattern, due to the balance of each of the rules. On the symmetricalsymmetrical initial initial shape shape tended tended to to be be a a ﬁxed fixed pattern,pattern, due to to the the balance balance of ofeach each of ofthe the rules. rules. On Onthe the other hand, if the black area is larger than a certain size, Rule 1 is mostly applied, and disappeared at otherother hand, hand, if the if the black black area area is is larger larger than than a a certaincertain size, Rule Rule 11 is is mostly mostly applied applied,, and and disappeared disappeared at at the second step. To produce a self-replicating pattern, an asymmetric shape must initially occur. the secondthe second step. step. To T produceo produce a self-replicatinga self-replicating pattern, pattern, an asymmetric shape shape must must initially initially occur. occur.

FigureFigure 15. 15Transition. Transition of of thethe ratios of of states states 1 and 1 and 2, and 2, andthe ratio the ratioof the ofapplied the applied cells of Rule cells 4 of per Rule time 4 per Figure 15. Transition of the ratios of states 1 and 2, and the ratio of the applied cells of Rule 4 per time timestep step under under the the same same conditions conditions as in as Figure in Figure 12 (s 12= 5(,s t == 5,9, tw=1 = 9, 0.40,w1 = and 0.40, w2 and= 0.75).w2 =From 0.75). 90 Fromto 100 90 to step under the same conditions as in Figure 12 (s = 5, t = 9, w1 = 0.40, and w2 = 0.75). From 90 to 100 Micromachinessteps, the 2018 frequency, 9, x FOR of PEER pattern REVIEW division became high, and the application ratio of Rule 4 increased. 13 of 19 100 steps,steps, the frequency frequency of of pattern pattern division division became became high, high, and and the application the application ratio ratioof Rule of 4 Rule increased. 4 increased.

Figure 18 shows the initial pattern of a deformed ring with a thick upper boundary. When w1 = Figure 18 shows the initial pattern of a deformed ring with a thick upper boundary. When w1 = 0.45 and w2 = 0.65, the ring pattern was divided while moving. When w1 = 0.45 and w2 = 0.72, a 0.45 and w2 = 0.65, the ring pattern was divided while moving. When w1 = 0.45 and w2 = 0.72, a constantly moving ring-like pattern formed. When w1 = 0.45 and w2 = 0.75, a single fixed symmetrical constantly moving ring-like pattern formed. When w1 = 0.45 and w2 = 0.75, a single fixed symmetrical ringring pattern pattern formed. formed. Thus, Thus, when when the the initial initial shape shape is is asymmetric, asymmetric, a a mov movinging pattern pattern or or dividing dividing pattern tends to be generated from the breaking of symmetry. As w2 approaches 1, it approaches pattern tends to be generated from the breaking of symmetry. As w2 approaches 1, it approaches fixedfixed patterns patterns, ,and and results results in in a a pattern pattern close close to to the the Turing Turing pattern. pattern.

Figure 16. Results for the model with s = 5 and t = 8 in a hexagonal grid. The initial pattern was a Figure 16. Results for the model with s = 5 and t = 8 in a hexagonal grid. The initial pattern was a symmetrical ring. When w1 = 0.45 and w2 = 0.65, the ring pattern became an oscillation pattern with symmetrical ring. When w1 = 0.45 and w2 = 0.65, the ring pattern became an oscillation pattern with period 2. When w1 = 0.45 and w2 = 0.72, the ring pattern likewise became an oscillation pattern with period 2. When w1 = 0.45 and w2 = 0.72, the ring pattern likewise became an oscillation pattern with period 2. When w = 0.45 and w = 0.75, a still pattern was formed. period 2. When1 w1= 0.45 and w22 = 0.75, a still pattern was formed.

~~Figure 17. Results for the model with s = 5 and t = 8 in a hexagonal grid. The initial pattern was a~~

symmetrical ring. When w1 = 0.45 and w2 = 0.48, the ring pattern disappeared. When w1 = 0.45 and w2 = 0.55, the ring pattern spread while breaking apart. When w1 = 0.45 and w2 = 0.60, still patterns were formed.

~~Micromachines 2018, 9, x FOR PEER REVIEW 13 of 19~~

~~Figure 16. Results for the model with s = 5 and t = 8 in a hexagonal grid. The initial pattern was a~~

symmetrical ring. When w1 = 0.45 and w2 = 0.65, the ring pattern became an oscillation pattern with period 2. When w1 = 0.45 and w2 = 0.72, the ring pattern likewise became an oscillation pattern with Micromachines 2018, 9, 339 13 of 18 period 2. When w1= 0.45 and w2 = 0.75, a still pattern was formed.

Figure 1 17.7. ResultsResults for for the the model model with with s =s =5 and 5 and t =t 8= in 8 ina hexagonal a hexagonal grid. grid. The Theinitial initial pattern pattern was a was a symmetrical ring. ring. When When ww1 1= =0.45 0.45 and and w2w =2 0.48,= 0.48, the the ring ring pattern pattern disappeared. disappeared. When When w1 = w0.451 = and 0.45 and w2 == 0.55, 0.55, the the ring ring pattern pattern spread spread while while breaking breaking apart. apart. When When w1 w= 10.45= 0.45 and andw2 =w 0.60,2 = 0.60,still pattern still patternss were formed.

Figure 18 shows the initial pattern of a deformed ring with a thick upper boundary. When w1 = 0.45 and w2 = 0.65, the ring pattern was divided while moving. When w1 = 0.45 and w2 = 0.72, a constantly moving ring-like pattern formed. When w1 = 0.45 and w2 = 0.75, a single ﬁxed symmetrical ring pattern formed. Thus, when the initial shape is asymmetric, a moving pattern or dividing pattern tends to be generated from the breaking of symmetry. As w2 approaches 1, it approaches ﬁxed patterns, Micromachines 2018, 9, x FOR PEER REVIEW 14 of 19 and results in a pattern close to the Turing pattern.

Figure 18. Results for the model with s = 5 and t = 8 in a hexagonal grid. The initial pattern was Figure 18. Results for the model with s = 5 and t = 8 in a hexagonal grid. The initial pattern was a a deformed ring with a thick upper boundary. When w1 = 0.45 and w2 = 0.65, the ring pattern was deformed ring with a thick upper boundary. When w1 = 0.45 and w2 = 0.65, the ring pattern was divided while moving. When w1 = 0.45 and w2 = 0.72, a moving ring-like pattern formed. When divided while moving. When w1 = 0.45 and w2 = 0.72, a moving ring-like pattern formed. When w1 = 0.45 and w2 = 0.75, a single ﬁxed symmetrical ring pattern formed. w1 = 0.45 and w2 = 0.75, a single fixed symmetrical ring pattern formed.

4. Discussion4. Discussion InIn this this study, study we, we constructed constructed a modela model in in which which the the focal focal state state 1 survives1 survives only only when when a moderatea moderate rate of therate numberof the number of state of 1 state cells surrounds1 cells surround the focals the cell. focal This cell. model This model created created a central a central white white part (state part 0) (state 0) within the black domain by changing parameter w2. The patterns then became increasingly unstable over the calculation field, and this instability increased the rate of pattern emergence, including self-reproducing spot patterns and stripe patterns. The self-reproducing spot pattern of Figure 12 seems to have some association with the Gray–Scott model, which is a type of RD model. Generally, the Gray–Scott model produces a self-reproducing spot pattern when the parameters are adjusted. In future work, we will investigate the relationship between the current model and the Gray–Scott model. Particularly, in the case of the model with s = 5 and t = 8 shown in Figures 16–18, the result suggests the possibility of controlling self-organized patterns. When w1 is increased, a spot shape can be generated, and by decreasing w1, a stripe pattern can be grown from spotted seeds. In the symmetric shape, when w2 is large, it is stable in many cases. However, by making the value of w2 small and making it close to w1, the spotted pattern becomes unstable, and asymmetric shapes are born. Furthermore, these spots are moving or divided by the controlling w2 parameter, as shown in Figure 19. The robustness of the initial conditions was found to have the following tendencies. (1) If state 1 is one, it disappears; (2) When state 1 is randomly arranged, if two or more states 1 are distributed in the range for counting the number of surrounding states, the development of the pattern is recognized. Next, depending on the degree of instability due to the w2 parameter, it is decided whether the pattern develops spatially. Due to random initial placement, an identical pattern does not occur, but qualitatively similar patterns can be stably generated; (3) When several states 1 are gathered, complicated pattern changes occur in the first two or three steps due to the balance between Rules 3 and 4, and if a shape that is not point symmetric is subsequently generated, a pattern spreading in the space is generated. When a point symmetric shape is born, a circular or ring-fixed pattern is generated; (4) When the black (state 1) area is large, the black area disappears due to Rule 1.

~~Micromachines 2018, 9, x FOR PEER REVIEW 15 of 19~~

In a future study, if CA rules can translate to a chemical reaction process, a control technology for nanomachines becomes feasible. Thus, the simple totalistic CA presented in this paper allows the emergence of a wide range of self-organization patterns to be observed. MicromachinesBased2018 on, 9 ,these 339 results, if the parameters w1 and w2 are changed at each time step, it is possible14 to of 18 generate a continuously changing pattern. Figure 20 shows a calculation case in which w1 and w2 are changed in time over time. The video of Figure 20 can be viewed in the Video S2 of Supplementary withinMaterials. the black As domainin this result, by changing it is possible parameter to createw ch2.anges The patternsby changing then w became1 and w2; increasinglybirth of annulus unstable -> overgrowth the calculation -> division ﬁeld,-> move and -> decline this instability -> growth increased -> annihilation. the rate of pattern emergence, including self-reproducingFigure 21 spot shows patterns the transition and stripe of the patterns. ratio of states 1 and 2, and the ratio of the applied cells of RuleThe self-reproducing4 in each time step spot during pattern the ofcalculation Figure 12 ofseems Figure to have20. Rule some 4 was association applied withmore the frequently Gray–Scott when spot patterns replicated, as compared to when the spot moved or vibrated. Furthermore, model, which is a type of RD model. Generally, the Gray–Scott model produces a self-reproducing when the pattern disappeared, the application frequency of Rule 4 increased. Thus, it is suggested spot pattern when the parameters are adjusted. In future work, we will investigate the relationship that pattern stability can be controlled by changing the application ratio of Rule 4. betweenSince the current this research model and is still the in Gray–Scott its first stage, model. discussions are limited mainly to qualitative considerations,Particularly, in but the in case future of the work, model it withwill bes = necessary 5 and t = to 8 shown investigate in Figures further 16 quantitative–18, the result suggestsassessments. the possibility There ofis controlling a possibility self-organized of finding an patterns. index like When Langtonw1 is’s increased, λ parameter a spot with shape the can be generated,evaluation andof many by decreasing calculationw cases.1, a stripe Furthermore, pattern can since be it grown has rules from similar spotted to seeds.the Game of Life, basicallyIn the symmetric like the Game shape, of Li whenfe, manyw2 trialsis large, are necessary it is stable to in find many the desired cases. However,pattern. Slight by makingchanges the valuein of thew 2 initialsmall value and making may develop it close into to differentw1, the spotted patterns. pattern In the becomes future, it unstable, is necessary and to asymmetric make a shapescatalog are born. of pattern Furthermore, formation these on the spots relation are moving between or the divided initial bycondition the controlling and patternw2sparameter, based on as shownmany in Figurecalculation 19. results.

FigureFigure 19. 1Possibility9. Possibility of of self-organized self-organized ring pattern patternss with with one one parameter. parameter. The results The results are a part are of a partthe of the modelmodel with s = 5 and ss = 8 8 in a hexagonal grid, grid, as as shown shown in in Figure Figure 13 13. The. The ring ring pattern pattern can can be be controlled with parameter w2. When w2 = 0.65, the ring pattern was divided while moving. When controlled with parameter w2. When w2 = 0.65, the ring pattern was divided while moving. When w2 = 0.72, a moving ring pattern formed. When w2 = 0.75, a fixed ring pattern formed. w2 = 0.72, a moving ring pattern formed. When w2 = 0.75, a ﬁxed ring pattern formed.

The robustness of the initial conditions was found to have the following tendencies. (1) If state 1 is one, it disappears; (2) When state 1 is randomly arranged, if two or more states 1 are distributed in the range for counting the number of surrounding states, the development of the pattern is recognized. Next, depending on the degree of instability due to the w2 parameter, it is decided whether the pattern develops spatially. Due to random initial placement, an identical pattern does not occur, but qualitatively similar patterns can be stably generated; (3) When several states 1 are gathered, complicated pattern changes occur in the ﬁrst two or three steps due to the balance between Rules 3 and 4, and if a shape that is not point symmetric is subsequently generated, a pattern spreading in the space is generated. When a point symmetric shape is born, a circular or ring-ﬁxed pattern is generated; (4) When the black (state 1) area is large, the black area disappears due to Rule 1. Micromachines 2018, 9, 339 15 of 18

Figure 20.20. CCalculationalculation results results in which w1 and w2 areare changed over time at each step. The initial pattern was a hexagonal ring shape of state 1s, with s = 5 and t = 8 in a hexagonal grid. As As in this result,result, it is possible toto createcreate changeschanges byby changingchanging ww11 andand ww22;; whenwhen ii == 1–9,1–9, ww11 == 0.44,0.44, and ww22 = 0.98 0.98,,

birth of of annulus annulus was was observed observed;; when when i =i =11 11–19,–19, w1w =1 0.44= 0.44,, and and w2 =w 0.702 = 0.70,, an oscillating an oscillating spot spotpattern pattern was observedwas observed;; when when i = 2i0=–2 20–29,9, w1 =w 1 0.32= 0.32,, and and w2 w=2 0.50= 0.50,, a agrowing growing ring ring pattern pattern was was observed observed;; when ii = 3 30–39,0–39, w1 == 0.30 0.30,, and and ww22 == 0.70 0.70,, a a growing growing ring ring pattern pattern was was observed observed;; when ii = 4 40–99,0–99, w1 = 0.45 0.45,, and and

w22 == 0.70 0.70,, divide divide patterns patterns and and replication replication were observed observed;; when i = 100–139,100–139, w1 = 0.58 0.58,, and w2 == 0.70 0.70,, i w w pattern disappearance was observed;observed; when i == 140–259, 140–259, w11 = 0.470.47,, and w2 = 0.710.71,, moving spotted i w w patterns were observed;observed; when i = 26 260–279,0–279, w1 = 0.44 0.44,, and and w22 == 0.98 0.98,, still still circle circle patterns patterns were were observed; observed; when i = 280–319, w = 0.37, and w = 0.70, a growing ring pattern was observed; when i = 320–449, when i = 280–319, w1 = 0.37, and w22 = 0.70, a growing ring pattern was observed; when i = 320–449, w = 0.47, and w = 0.71, divide patterns and replication were observed; when i = 460–, w = 0.58, and w11 = 0.47, and w22 = 0.71, divide patterns and replication were observed; when i = 460–, w11 = 0.58, and w = 0.65, disappearance of all patterns was observed. w22 = 0.65, disappearance of all patterns was observed.

~~Micromachines 2018, 9, 339 16 of 18~~

Based on these results, if the parameters w1 and w2 are changed at each time step, it is possible to generate a continuously changing pattern. Figure 20 shows a calculation case in which w1 and w2 are changed in time over time. The video of Figure 20 can be viewed in the Video S2 of Supplementary Materials. As in this result, it is possible to create changes by changing w1 and w2; birth of annulus -> growth -> division -> move -> decline -> growth -> annihilation. Figure 21 shows the transition of the ratio of states 1 and 2, and the ratio of the applied cells of Rule 4 in each time step during the calculation of Figure 20. Rule 4 was applied more frequently when spot patterns replicated, as compared to when the spot moved or vibrated. Furthermore, when the pattern disappeared, the application frequency of Rule 4 increased. Thus, it is suggested that pattern stability can be controlled by changing the application ratio of Rule 4. Since this research is still in its first stage, discussions are limited mainly to qualitative considerations, but in future work, it will be necessary to investigate further quantitative assessments. There is a possibility of finding an index like Langton’s λ parameter with the evaluation of many calculation cases. Furthermore, since it has rules similar to the Game of Life, basically like the Game of Life, many trials are necessary to find the desired pattern. Slight changes in the initial value may develop into different patterns. In the future, it is necessary to make a catalog of pattern formation on theMicromachines relation between2018, 9, x FOR the PEER initial REVIEW condition and patterns based on many calculation results. 17 of 19

FigureFigure 21.21. TransitionTransition of the the ratio ratio of of states states 1 1and and 2, 2,and and the the ratio ratio of the of theapplied applied cells cells of Rule of Rule 4 in 4each in eachtime timestep stepduring during the calculation the calculation of Figure of Figure 20. Rule 20 .4 Rulewas applied 4 was applied more frequently more frequently when spot when patterns spot patternsreplicated replicated than when than the when spot the mov spoted moved or vibrate or vibrated.d. Furthermore, Furthermore, when when the pattern the pattern disappear disappeared,ed, the theapplication application frequency frequency of of Rule Rule 4 4 increased. increased. Thus Thus,, it it is is suggested suggested that that pattepatternrn stability stability can can be be controlledcontrolled by by changing changing the the application application ratio ratio of of Rule Rule 4. 4.

5.5. ConclusionsConclusions ToTo observe observe special special patterns patterns of of emergent emergent behavior behavior that that mimicmimic biological biological systems,systems, wewe developeddeveloped a totalistic CA model with an activating inner area and inhibiting outer area. This is a CA model, yet a totalistic CA model with an activating inner area and inhibiting outer area. This is a CA model, it works in a way that is equivalent to that of RD approaches like the Turing pattern model. If a yet it works in a way that is equivalent to that of RD approaches like the Turing pattern model. If a multiple CA model can be designed that allows the number of states to be increased, we may multiple CA model can be designed that allows the number of states to be increased, we may witness witness the emergence of more complex patterns, like those of living things. the emergence of more complex patterns, like those of living things. Our model also has potential applications in the engineering field. For example, it may be Our model also has potential applications in the engineering ﬁeld. For example, it may be possible possible to develop swarming robots that can work as a single unit or divide into groups. As shown to develop swarming robots that can work as a single unit or divide into groups. As shown in the in the result of Figure 13, spot-like regions can be formed in the space by this algorithm. If many result of Figure 13, spot-like regions can be formed in the space by this algorithm. If many robots are robots are dispersedly arranged in this space, it becomes possible to gather by grouping the robots dispersedly arranged in this space, it becomes possible to gather by grouping the robots within this within this spot-like rounded area. Furthermore, using this algorithm, it is also possible to divide spot-like rounded area. Furthermore, using this algorithm, it is also possible to divide one spot area one spot area into two. Therefore, robots can be divided into two groups simply by paying attention so that the robot that was in the first spot area does not come out of the area. The model can further be applied to information networks. If the transition rules can be applied to Internet of Things (IoT) devices through the Web, hierarchical structures of IoT devices may emerge. For example, if the IoT devices know their own position, this algorithm can be applied by transmitting tokens to the nearby devices. Then, the spotted structure can be formed in the network space of devices, and the devices can be divided into plural groups. Furthermore, if our method can be applied to microsystems, it may become possible to make a production process of self-organizing a microrobot from nanoparts automatically. The model also produces Turing-like patterns using a simple algorithm and without the need to solve complex simultaneous differential equations. However, a weak point is the necessity to add up the state quantities within a certain area. While these calculations are simple to perform using electronic devices, it is difficult to see how an individual living cell in an organism would be able to calculate the states of distant cells. To apply this model to morphogenesis, therefore, further modification is needed. Other outstanding questions include the mathematical relationship between this model and the RD equations, and the relationship with a Turing machine. It would also be interesting to consider the entropy of the model.

Micromachines 2018, 9, 339 17 of 18 into two. Therefore, robots can be divided into two groups simply by paying attention so that the robot that was in the first spot area does not come out of the area. The model can further be applied to information networks. If the transition rules can be applied to Internet of Things (IoT) devices through the Web, hierarchical structures of IoT devices may emerge. For example, if the IoT devices know their own position, this algorithm can be applied by transmitting tokens to the nearby devices. Then, the spotted structure can be formed in the network space of devices, and the devices can be divided into plural groups. Furthermore, if our method can be applied to microsystems, it may become possible to make a production process of self-organizing a microrobot from nanoparts automatically. The model also produces Turing-like patterns using a simple algorithm and without the need to solve complex simultaneous differential equations. However, a weak point is the necessity to add up the state quantities within a certain area. While these calculations are simple to perform using electronic devices, it is difficult to see how an individual living cell in an organism would be able to calculate the states of distant cells. To apply this model to morphogenesis, therefore, further modification is needed. Other outstanding questions include the mathematical relationship between this model and the RD equations, and the relationship with a Turing machine. It would also be interesting to consider the entropy of the model.

Supplementary Materials: The following are available online at http://www.mdpi.com/2072-666X/9/7/339/s1, Video S1: Video of Figure 12 (Results for the model with s = 4, t = 8, w1 = 0.40, and w2 = 0.75 in a hexagonal grid); Video S2: Video of Figure 20 (Calculation results in which w1 and w2 are changed over time at each step). Funding: This research was supported by grants from the Project of the NARO (National Agriculture and Food Research Organization) Bio-oriented Technology Research Advancement Institution (the special scheme project on regional developing strategy). Conﬂicts of Interest: The authors declare no conﬂict of interest.

~~References~~

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