CHAOS 2876
5 January 2004 Disk used
ARTICLE IN PRESS
No. of Pages 6, DTD = 4.3.1
Chaos, Solitons and Fractals xxx (2004) xxx–xxx
F
www.elsevier.com/locate/chaos
Collision-based computing in Belousov–Zhabotinsky medium
3
Andrew Adamatzky*
OO
2
PR
4 Faculty of Computing, Engineering and Mathematical Sciences, University of the West of England, Bristol BS16 1QY, United Kingdom
5 Abstract
TE
D
A photosensitive sub-excitable Belousov–Zhabotinsky medium exhibits propagating wave fragments that preserve
their shapes during substantial periods of time. In numerical studies we show that the medium is a computational
universal architectureless system, if presence and absence of wave fragments are interpreted as truth values of Boolean
variable. When two or more wave fragments collide they may annihilate, fuse, split or deviate from their original paths,
thus values of the logical variables are changed and certain logical gates are realized in result of the collision. We
demonstrate exact implementation of basic operations with signals and logical gates in Belousov–Zhabotinsky dynamic
circuits. The findings provide a theoretical background for subsequent experimental implementation of collision-based,
architectureless, dynamical computing devices in homogeneous active chemical media.
2003 Published by Elsevier Ltd.
16 1. Introduction
CO
RR
Certain families of thin-layer reaction-diffusion (RD) chemical media can implement sensible transformation of
initial (data) spatial distribution of chemical species concentrations to final (result) concentration profile [1,32]. In these
RD computers a computation is realized via spreading and interaction of diffusive or phase waves. Specialized, intended
to solve a particular problem, experimental RD processors implement basic operations of image processing [2,18,24,25],
computation of optimal paths [3,7,26,33] and control of mobile robots [4]. A number of computationally universal RD
chemical devices were implemented, the findings include logical gates [31,34] and diodes [12,19,21] in Belousov–
Zhabotinsky (BZ) medium, and X O R gate in palladium processor [5]. All the known so far experimental prototypes of
RD processors exploit interaction of wave fronts in a geometrically constrained chemical medium, i.e. the computation
is based on a stationary architecture of medium’s inhomogeneities. Constrained by a stationary wires and gates RD
chemical universal processors pose a little computational novelty and none dynamical reconfiguration ability because
they simply imitate architectures of silicon computing devices.
To appreciate in full massive-parallelism of thin-layer chemical media and to free the chemical processors from
limitations of fixed computing architectures we adopt an unconventional paradigm of dynamical, architectureless, or
collision-based, computing. The paradigm originates from computational universality of Game of Life [9], conservative
logic and billiard-ball model [14] and their cellular-automaton implementations [20]. A collision-based (CB) computation employs mobile compact patterns, in our particular case they are self-localized excitations in active non-linear
medium. The localizations travel in space and perform computation (implement logical gates) when they collide with
each other. There are no predetermined stationary wires––a trajectory of the traveling pattern is a momentarily wire––
almost any part of the medium’s space can be used as a wire. Truth values of logical variables are given by either
absence or presence of a localization or by various types of localizations. State of the art of CB computing is presented
in [6].
UN
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
EC
6
7
8
9
10
11
12
13
14
15
*
Tel.: +44-117-965-6261; fax: +44-117-976-2150.
E-mail address: [email protected] (A. Adamatzky).
0960-0779/$ - see front matter 2003 Published by Elsevier Ltd.
doi:10.1016/j.chaos.2003.12.068
CHAOS 2876
5 January 2004 Disk used
2
No. of Pages 6, DTD = 4.3.1
A. Adamatzky / Chaos, Solitons and Fractals xxx (2004) xxx–xxx
PR
OO
F
Solitons, defects in tubulin microtubules, excitons in Scheibe aggregates and breather in polymer chains are most
frequently considered candidates for a role of information carrier in nature-inspired CB computers, see overview in [1].
It is experimentally difficult to reproduce all these artifacts in natural systems, therefore existence of mobile localizations
in an experiment-friendly RD media would open new horizons for fabrication of CB computers. Until recently we have
a little if any information about interaction of mobile localizations in 2D or 3D RD media. However the works [10,29]
demonstrated existence and rich interaction of quasi-particles (dissipative solitons) in a three-component RD system.
The basis for CB universality of RD chemical media was finally laid when Sendi}
na-Nadal et al [30] experimentally
proved existence of localized excitations––traveling wave fragments which behave like quasi-particles––in photosensitive sub-excitable BZ medium.
In present paper we aim to computationally demonstrate how logical circuits can be fabricated in a sub-excitable BZ
medium via collisions between traveling wave fragments. While implementation CB logical operations themselves is
relatively straightforward, more attention should be paid to control of signal propagation in the homogeneous medium.
For example, to realize a (non-conservative) analog of Fredkin–Toffoli–Margolus billiard-ball model of interaction
logic [14,20] we must somehow Ôfabricate’ a reflector to control information quanta trajectories. It has been demonstrated widely that applying light of varying intensity we can control excitation dynamic in BZ medium [8,15,16,22],
wave velocity [28], patter formation [36]. Of particular interest are experimental evidences of light-induced back
propagating waves, wave-front splitting and phase shifting [37]; we can also manipulate medium’s excitability by
varying intensity of the medium’s illumination [11]. Basing on these facts we show how to control signal-wave fragments
by varying geometric configuration of excitatory and inhibitory segments of impurity-reflectors.
D
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
ARTICLE IN PRESS
57 2. Methods
TE
58
We based our model on a two-variable Oregonator equation [13,35] adapted to a light-sensitive BZ reaction with
59 applied illumination [8,17]:
ov
¼uv
ot
CO
RR
where variables u and v represent local concentrations of bromous acid HBrO2 and the oxidized form of the catalyst
ruthenium Ru(III), sets up a ratio of time scale of variables u and v, q is a scaling parameter depending on reaction
rates, f is a stoichiometric coefficient, / is a light-induced bromide production rate proportional to intensity of illumination (an excitability parameter––moderate intensity of light will facilitate excitation process, higher intensity will
produce excessive quantities of bromide which suppresses the reaction). We assumed that the catalyst is immobilized in
a thin-layer of gel, therefore there is no diffusion term for v. To integrate the system we used Euler method with fivenode Laplacian operator, time step Dt ¼ 103 and grid point spacing Dx ¼ 0:15, with the following parameters:
/ ¼ /0 þ A=2, A ¼ 0:0011109, /0 ¼ 0:0766, ¼ 0:03, f ¼ 1:4, q ¼ 0:002. When adjusting parameters of the model we
took into account that a decrease in results in unbounded growth of excitation activity, while by increasing f we may
roughly control outcomes of wave collision [27]. Chosen parameters correspond to a region of ‘‘higher excitability of the
sub-excitability regime’’ outlined in [30] (see also how to adjust f and q in [23]) that supports propagation of sustained
wave fragments (Fig. 1a). These wave fragments are used as quanta of information in our design of CB logical circuits.
The waves were initiated by locally disturbing initial concentrations of species, e.g. 10 grid sites in a chain are given
value u ¼ 1:0 each, this generated two or more localized wave fragments, similarly to counter-propagating waves induced by temporary illumination in experiments [37]. The traveling wave fragments keep their shape for around 4 · 103 –
104 steps of simulation (4–10 time units), then decrease in size and vanish. The wave’s life-time is sufficient however to
implement logical gates; this also allows us not to worry about Ôgarbage collection’ in the computational medium.
UN
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
EC
ou 1
uq
¼
u u2 ðfv þ /Þ
þ Du r2 u
ot uþq
79 3. Results
80
We model signals by traveling wave fragments [8,30]: a sustainably propagating wave fragment (Fig. 1a) represents
8 1 T R U E value of a logical variable corresponding to the wave’s trajectory (momentarily wire). To demonstrate that a
82 physical systems is logically universal it is enough to implement negation and conjunction or disjunction in spatio83 temporal dynamics of the system. To realize a fully functional logical circuit we must also know how to operate input
CHAOS 2876
5 January 2004 Disk used
ARTICLE IN PRESS
No. of Pages 6, DTD = 4.3.1
3
RR
EC
TE
D
PR
OO
F
A. Adamatzky / Chaos, Solitons and Fractals xxx (2004) xxx–xxx
and output signals in the system’s dynamics, namely to implement signal branching and routing; delay can be realized
via appropriate routing.
We can branch a signal using two techniques. Firstly, we can collide a smaller auxiliary wave to a wave fragment
representing the signal, the signal-wave will split then into two signals (these daughter waves shrink slightly down to
stable size and then travel with constant shape further 4 · 103 time steps of the simulation) and the auxiliary wave will
annihilate (Fig. 1b). Secondly, we can temporarily and locally apply illumination impurities on a signal’s way to change
properties of the medium and thus cause the signal to split (Fig. 1c and d). We must mention, it was already demonstrated in [37], that wave front influenced by strong illumination (inhibitory segments of the impurity) splits and its
ends do not form spirals, as in typical situations of excitable media. A control impurity, or reflector, consists of a few
segments of sites which illumination level is slightly above or below overall illumination level of the medium. Combining excitatory and inhibitory segments we can precisely control wave’s trajectory, e.g. realize U-turn of a signal (Fig.
1e and f).
UN
84
85
86
87
88
89
90
91
92
93
94
95
CO
Fig. 1. Basic operations with signals. Overlay of images taken every 0.5 time units. Exciting domains of impurities are shown in black,
inhibiting domains of impurities are shown in gray. (a) Wave fragment traveling north. (b) Signal branching without impurities: a wave
fragment traveling east splits into two wave fragments (traveling south-east and north-east) when collides to a smaller wave fragment
traveling west. (c) Signal branching with impurity: wave fragment traveling west is split by impurity (d) into two waves traveling northwest and south-west. (e) Signal routing (U-turn) with impurities: wave fragment traveling east is routed north and then west by two
impurities (f). An impurity-reflector consists of inhibitory (gray) and excitatory (black) chains of grid sites.
CHAOS 2876
5 January 2004 Disk used
4
No. of Pages 6, DTD = 4.3.1
A. Adamatzky / Chaos, Solitons and Fractals xxx (2004) xxx–xxx
TE
D
PR
OO
F
A typical billiard-ball model interaction gate [14,20] has two inputs––x and y, and four outputs––xy (ball x moves
undisturbed in absence of ball y), xy (ball y moves undisturbed in absence of ball x), and twice xy (balls x and y change
their trajectories when collide to each other). We were unable to make wave fragments implement exact billiard-ball
gate, because the interacting waves either fuse or one of the waves annihilates in result of the collision to another wave.
However, we have implemented a BZ (non-conservative) version of billiard-ball gate with two inputs and three outputs,
i.e. just one xy output instead of two. This BZ collision gate is shown in Fig. 2.
Rich dynamic of BZ medium allows us also to implement complicated logical operations just in a single interaction
event. An example of a composite gate with three inputs and six outputs is shown in Fig. 3. As we see in Fig. 3, some
outputs, e.g. xyz, are represented by gradually vanishing wave fragments. The situation can be dealt with by either using
very compact architecture of the logical gates or by installing temporary amplifiers made from excitatory fragments of
illumination impurities.
CO
RR
EC
Fig. 2. Two wave fragments undergo angle collision and implement interaction gate hx; yi ! hxy ; xy; xyi. (a) In this example x ¼ 1 and
y ¼ 1, both wave fragments are present initially. Overlay of images taken every 0.5 time units. (b) Scheme of the gate. In upper left and
bottom left corners of (a) we see domains of wave generation, two echo wave fragments are also generated, they travel outwards gate
area and thus do not interfere with computation.
UN
96
97
98
99
100
101
102
103
104
105
106
ARTICLE IN PRESS
Fig. 3. Implementation of hx; y; zi ! hxy ; y z; xyz; xyz; xyz; xyzi interaction gate. Overlay of images of wave fragments taken every 0.5
time units. The following combinations of input configuration are shown: (a) x ¼ 1, y ¼ 1, z ¼ 0, north–south wave collides to
east–west wave. (b) x ¼ 1, y ¼ 1, z ¼ 1, north–south wave collides to east–west wave, and to west–east wave. (c) x ¼ 1, y ¼ 0, z ¼ 1,
west–east and east–west wave fragments pass near each other without interaction. (d) x ¼ 0, y ¼ 1, z ¼ 1, north–south wave collides to
east–west wave. (e) Scheme of the gate.
CHAOS 2876
5 January 2004 Disk used
ARTICLE IN PRESS
No. of Pages 6, DTD = 4.3.1
A. Adamatzky / Chaos, Solitons and Fractals xxx (2004) xxx–xxx
5
OO
F
Fig. 4. Small perturbation of a traveling wave fragment––a decrease of light intensity in a local domain of five by nine grid sites
(domain of perturbation is indicated by arrow)––changes geometry of the wave fragment and thus cause uncontrollable growth of the
wave front.
107 4. Discussions
D
PR
In summary, we have demonstrated that sub-excitable light-sensitive BZ medium is capable to dynamical (architectureless, CB) computational universality. Signals are represented by traveling wave fragments. Operations with
signals are implemented by local changes in medium’s illumination. Logical operations, or gates, are realized at sites of
wave fragments collisions.
The studied medium is highly sensitive to local perturbations and even a tiny local change in illumination may lead
to drastic modification of the medium’s dynamics (Fig. 4). Thus the computation occurs at the edge of instability.
Finally, the Oregonator model satisfactory shows qualitative spatio-temporal dynamic of excitation, however the
model does not give us a realistic quantitative picture of all processes undergoing in BZ system. Therefore our further
studies will concern experimental implementation of CB logical circuits in BZ medium.
TE
108
109
110
111
112
113
114
115
116
117 References
CO
RR
EC
[1] Adamatzky A. Computing in nonlinear media and automata collectives. IoP Publishing; 2001.
[2] Adamatzky A, De Lacy Costello B, Ratcliffe NM. Experimental reaction-diffusion pre-processor for shape recognition. Phys Lett
A 2002;297:344–52.
[3] Adamatzky A, De Lacy Costello B. Reaction-diffusion path planning in a hybrid chemical and cellular-automaton processors.
Chaos, Solitons & Fractals 2003;16:727–36.
[4] Adamatzky A, De Lacy Costello B, Melhuish C, Ratcliffe N. Experimental reaction-diffusion chemical processors for robot path
planning. J Intell Rob Syst 2003;37:233–49.
[5] Adamatzky A, De Lacy Costello BPJ. Experimental logical gates in a reaction-diffusion medium: the XOR gate and beyond. Phys
Rev E 2002;66:046112.
[6] Adamatzky A, editor. Collision-based computing. Springer-Verlag; 2002.
[7] Agladze K, Magome N, Aliev R, Yamaguchi T, Yoshikawa K. Finding the optimal path with the aid of chemical wave. Physica D
1997;106:247–54.
[8] Beato V, Engel H. Pulse propagation in a model for the photosensitive Belousov–Zhabotinsky reaction with external noise. In:
Schimansky-Geier L, Abbott D, Neiman A, Van den Broeck C, editors. Noise in complex systems and stochastic dynamics. Proc
SPIE 2003;5114:353–62.
[9] Berlekamp ER, Conway JH, Guy RL. In: Winning ways for your mathematical plays, vol. 2. Academic Press; 1982.
[10] Bode M, Liehr AW, Schenk CP, Purwins H-G. Interaction of dissipative solitons: particle-like behaviour of localized structures in
a three-component reaction-diffusion system. Physica D 2000;161:45–66.
[11] Brandtst€
adter H, Braune M, Schebesch I, Engel H. Experimental study of the dynamics of spiral pairs in light-sensitive Belousov–
Zhabotinsky media using an open-gel reactor. Chem Phys Lett 2000;323:145–54.
[12] Dupont C, Agladze K, Krinsky V. Excitable medium with left–right symmetry breaking. Physica A 1998;249:47–52.
[13] Field RJ, Noyes RM. Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction. J Chem
Phys 1974;60:1877–84.
[14] Fredkin F, Toffoli T. Conservative logic. Int J Theor Phys 1982;21:219–53.
[15] Flesselles J-M, Belmonte A, Gaspar V. Dispersion relation for waves in the Belousov–Zhabotinsky reaction. J Chem Soc Faraday
Trans 1998;94:851–5.
[16] Grill S, Zykov VS, M€
uller SC. Spiral wave dynamics under pulsatory modulation of excitability. J Phys Chem 1996;100:19082–8.
[17] Krug HJ, Pohlmann L, Kuhnert L. Analysis of the modified complete Oregonator (MCO) accounting for oxygen- and
photosensitivity of Belousov–Zhabotinsky systems. J Phys Chem 1990;94:4862–6.
[18] Kuhnert L, Agladze KL, Krinsky VI. Image processing using light-sensitive chemical waves. Nature 1989;337:244–7.
[19] Kusumi T, Yamaguchi T, Aliev R, Amemiya T, Ohmori T, Hashimoto H, et al. Numerical study on time delay for chemical wave
transmission via an inactive gap. Chem Phys Lett 1997;271:355–60.
[20] Margolus N. Physics-like models of computation. Physica D 1984;10:81–95.
UN
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
CHAOS 2876
5 January 2004 Disk used
6
No. of Pages 6, DTD = 4.3.1
A. Adamatzky / Chaos, Solitons and Fractals xxx (2004) xxx–xxx
CO
RR
EC
TE
D
PR
OO
F
[21] Motoike IN, Yoshikawa K. Information operations with multiple pulses on an excitable field. Chaos, Solitons & Fractals
2003;17:455–61.
[22] Petrov V, Ouyang Q, Swinney HL. Resonant pattern formation in a chemical system. Nature 1997;388:655–7.
[23] Qian H, Murray JD. A simple method of parameter space determination for diffusion-driven instability with three species. Appl
Math Lett 2001;14:405–11.
[24] Rambidi NG. Neural network devices based on reaction-diffusion media: an approach to artificial retina. Supramol Sci
1998;5:765–7.
[25] Rambidi NG, Shamayaev KR, Peshkov GYu. Image processing using light-sensitive chemical waves. Phys Lett A 2002;298:375–
82.
[26] Rambidi NG, Yakovenchuk D. Chemical reaction-diffusion implementation of finding the shortest paths in a labyrinth. Phys Rev
E 2001;63:0266071.
[27] Ramos JJ. Oscillatory dynamics of inviscid planar liquid sheets. Appl Math Comput 2003;143:109–44.
[28] Schebesch I, Engel H. Wave propagation in heterogeneous excitable media. Phys Rev E 1998;57:3905–10.
[29] Schenk CP, Or-Guil M, Bode M, Purwins H-G. Interacting pulses in three-component reaction-diffusion systems on twodimensional domains. Phys Rev Lett 1997;78:3781–4.
[30] Sedi}
na-Nadal I, Mihaliuk E, Wang J, Perez-Mu}
nuzuri V, Showalter K. Wave propagation in subexcitable media with periodically
modulated excitability. Phys Rev Lett 2001;86:1646–9.
[31] Sielewiesiuk J, Gorecki J. Logical functions of a cross junction of excitable chemical media. J Phys Chem A 2001;105:8189–95.
[32] Sienko T, Adamatzky A, Rambidi N, Conrad M, editors. Molecular computing. The MIT Press; 2003.
[33] Steinbock O, Toth A, Showalter K. Navigating complex labyrinths: optimal paths from chemical waves. Science 1995;267:868–71.
[34] T
oth A, Showalter K. Logic gates in excitable media. J Chem Phys 1995;103:2058–66.
[35] Tyson JJ, Fife PC. Target patterns in a realistic model of the Belousov–Zhabotinsky reaction. J Chem Phys 1980;73:2224–37.
[36] Wang J. Light-induced pattern formation in the excitable Belousov–Zhabotinsky medium. Chem Phys Lett 2001;339:357–61.
[37] Yoneyama M. Optical modification of wave dynamics in a surface layer of the Mn-catalyzed Belousov–Zhabotinsky reaction.
Chem Phys Lett 1996;254:191–6.
UN
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
ARTICLE IN PRESS