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Chaos, Solitons and Fractals xxx (2006) xxx–xxx www.elsevier.com/locate/chaos
Binary collisions between wave-fragments in a sub-excitable Belousov–Zhabotinsky medium Andrew Adamatzky a
a,*
, Benjamin de Lacy Costello
b
Faculty of Computing, Engineering and Mathematical Sciences, University of the West of England, Bristol BS16 1QY, United Kingdom b Faculty of Applied Sciences, University of the West of England, Bristol BS16 1QY, United Kingdom Accepted 17 March 2006
Communicated by Professor M.S. El Naschie
Abstract In numerical studies we describe the phenomenology of interactions between localized, shape-preserving, wave-fragments in the sub-excitable Belousov–Zhabotinsky medium and build a representative catalog of wave-fragment collisions that include annihilation, fusion, and quasi-elastic types. We envisage the phenomena discovered will be used in signal tuning and general programming of collision-based [Adamatzky A, editor, Collision-based computing. Springer-Verlag, 2002] excitable chemical computers. 2006 Elsevier Ltd. All rights reserved.
1. Introduction In our previous work, see overview in [3], we demonstrated that reaction–diffusion chemical systems in general, and the Belousov–Zhabotinsky (BZ) medium [10] in particular, are capable of implementing various kinds of computational procedures. We showed that it is possible to construct a reaction–diffusion processor from a real chemical medium (such as a thin-layer solution, gel or film) that transforms data to results in a pre-programmable way. In the reaction–diffusion processor data are represented by perturbations of the chemical medium’s state (concentration profile of reagents). The perturbations can initiate—diffusive or phase waves—that travel in the medium, and interact with each other, to produce either stationary (e.g. precipitate concentration profile) or dynamic (e.g. oscillatory field) structures. The final state of the medium’s spatial dynamics and chosen transient states represent the results of the reaction–diffusion computation. We provided computational [2] and experimental [5] evidence of collision-based [1] computation in sub-excitable BZ media. We demonstrated that under carefully controlled conditions compact wavefragments develop in the medium, they travel and implement logical gates when they collide with each other. In [5] we have shown how to embed non-trivial logical circuits in an experimental BZ medium. The medium realizes an architecture-less computation because there are no pre-determined stationary wires, a trajectory of the traveling wave-fragment is a momentary wire: almost any part of the medium’s space can be used as a wire at some stage of *
Corresponding author. Tel.: +44 117 344 2662; fax: +44 117 344 3636. E-mail addresses:
[email protected] (A. Adamatzky),
[email protected] (B. de Lacy Costello).
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.095
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the computation. To realize a Boolean logical gate we represent logical TRUTH by presence of a wave-fragment and logical FALSE by absence of a wave-fragment. When two or more wave-fragments collide, they may fuse, annihilate, generate new wave-fragments or at least change their trajectories or velocity vectors. Thus values of Boolean variables represented by the fragments are changed and the computation is implemented. This means that by colliding wave-fragments we can implement a functionally complete set of logical operations and thus build a universal architecture-less computer in the sub-excitable BZ medium. However, our previous designs (see e.g. [2]) of BZ-based collision-based computing devices were not absolutely uniform and homogeneous—a number of artificial impurities (in the form of patches of non-uniform illumination) were introduced to manipulate the traveling wave-fragments in order to realize certain types of behavior. This imperfection is corrected in the present paper: we demonstrate that the trajectories of traveling wave-fragments can be dynamically changed, adjusted and tuned by colliding other control wave-fragments with them. We employ a two-variable Oregonator model of the light-sensitive BZ reaction held in sub-excitable mode (where open ended wave-fragments are supported) and study the principle scenarios of binary interactions between wave-fragments.
2. Methods We simulate the excitable chemical system using the two-variable Oregonator model [6,9] modified to describe the light-sensitive analogue of the BZ reaction under applied illumination [4,7] ou 1 uq ¼ u u2 ðfv þ /Þ þ Du r2 u ot uþq ov ¼uv ot
Fig. 1. Wave-fragment generated by a strip of nine excited sites collapses. Time steps (a)–(h) are 51, 351, 651, . . . , 2151 and (i) overlapped images of the collapsing wave-fragment.
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where variables u and v represent local concentrations of bromous acid HBrO2 and the oxidized form of the catalyst ruthenium Ru(III) respectively, 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. Variable / is a light-induced bromide production rate proportional to the intensity of illumination and controls the excitability of the medium. The excitability is modified by light because the excited ruthenium catalyst reacts with bromo-malonic acid to produce bromide, which is an inhibitor of autocatalysis. The activator diffusion coefficient is Du, whilst the diffusion term for v is absent as it is assumed that the catalyst is immobilized in a thin-layer of gel. To integrate the system we used a Euler method with a five-node Laplasian operator, time step Dt = 103 and grid point spacing Dx = 0.25, with the following parameters: /0 = 0.076044, = 0.02272, f = 1.4, q = 0.002. These parameters roughly correspond to the region of higher excitability of the sub-excitable regime [8]. The waves were initiated by locally disturbing the initial concentration of species u, e.g. a few grid sites in a chain are each given a value u = 1.0. Such a segment-perturbation generates two localized wave fragments. The wave-fragments travel—a finite distance—preserving their shape like quasi-particles or dissipative solitons. To record images of the wave-fragments we saved the configuration of the excitable medium every 400th step (approximately 8 s of real time) of the simulation, as follows: a site (i, j) where concentration uij exceeds 0.2 is assigned a pixel (i, j) with gray value 250 · uij. The chosen parameters mean the model is very sensitive to initial conditions and therefore has maximum effectiveness for demonstrating the effects of fragment collisions. For example, adding just one site of seed-excitation to the initial perturbation makes a difference between explosive growth of the original excitation fragment and eventual extinction. Thus if a strip of seed-excitation consists of nine sites, then the resulting wave-fragment will propagate for only a finite number of steps, and after a certain period of simulation, the fragment starts to shrink and eventually collapses (Fig. 1). However, when the strip of seed-excitation consists of 10 sites the wave-fragment starts to expand (Fig. 2) and eventually results in a double-spiral wave.
Fig. 2. Wave-fragment generated by strip of ten excited sites expands. Time steps (a)–(j) are 51, 351, 651, . . . , 2751 and (k) overlapped images of the expanding wave-fragment.
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However, this said the life-time of the wave-fragments (period before the fragment starts to either expand or collapse) is sufficient in order to enable the study of the interaction of the wave-fragments.
3. Phenomenology of collisions To study the head-on collisions between two fragments we positioned two sets of nine-site long seed-excitation opposite each other with a consistent spacing. The study of offset head-on collisions was realized by shifting the center of the excitation segment on the east side r grid sites north related to the center of the excitation segment on the west side. When r = 12 (for the separation chosen) there was no interaction between the wave-fragments as from the point of initiation they start to collapse due to the size of the initial excitation and the offset of the original fragments is such that this enables them to pass each other (Fig. 3g). For r = 0, . . . , 4 the opposite is true and the offset of the original fragments is small thus the fragments simply collide and annihilate (Fig. 3a–c). Parts of the fragments only survive collision when the shift is substantial, e.g. r = 6 or 8, however the resulting daughter fragments are short-lived and soon collapse (Fig. 3d and e). The scenario becomes interesting when the spacing is r = 10 and the wave-fragments just brush past each other with a minimal area collision (Figs. 3f and 4). In this case two daughter fragments are formed (Fig. 4c) and these fragments start to expand. The fact that two daughter fragments which subsequently expand are formed from an offset collision between two collapsing parent fragments is interesting. It suggests that the effective size of the excitation making up each daughter fragment is increased. This is due to a fusion between the two collapsing parent fragments, followed by refocusing of the excitation before splitting into the two resulting daughter fragments. The expansion of the daughter fragments is limited by the catalyst trails in the refractory domains left by the parent fragments (Fig. 4d), therefore the daughter fragments start to expand asymmetrically outwards from their original trajectories (Fig. 4e and f). The effect of the refractory domains on the trajectories of the daughter fragments was observed and discussed in our previous paper dealing with collisions of excitation fragments in an experimental version of a subexcitable BZ system [5]. The overall effect of this type of collision between two excitation fragments is an impression that the wave-fragments have reflected from each other. When a wave-fragment traveling on a North-West trajectory collides with another wave-fragment traveling West, they usually fuse into one daughter fragment traveling on a North-West West trajectory (Fig. 5). The newly formed wave-fragment expands unlimitedly (Fig. 5e and f). However, when wave-fragments collide at right angles, as shown in Fig. 6 where a fragment traveling North collides with a fragment traveling West then two extreme cases Fig. 6a and j can be observed. If the offset spacing is large enough between the original sites of initiation then the wave-fragments collapse by themselves Fig. 6a. Alternatively if the initiation sites are too close then the collision of one is close to the initiation site of the other and annihilation
Fig. 3. Overlapped images of wave-fragments colliding head-on. The center of the seed-excitation on the east side of the simulation is shifted north in relation to the center of the seed-excitation on the west side by (a) 0 sites, (b) 2 sites, . . ., (g) 12 sites.
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Fig. 4. Expanded scheme showing minimal area collision where the offset spacing of the parent fragments traveling east and west was r = 10. (a) 51 steps after initial excitation, (b) 1051 steps, (c) 1201, (d) 1551 steps, (e) 2051 steps and (f) 3051 steps.
Fig. 5. Example of a side-on ‘‘fusion’’ collision between a wave-fragment traveling West and a wave-fragment traveling North-West. (a) 51st step, (b) 551st step, (c) 1051st step, (d) 1551st step, (e) 2051 step, (f) 2551st step and (g) overlapped images of traveling wavefragments.
of the fragments also results Fig. 6j. However, if the seed-excitation used to initiate the wave fragment traveling North is moved four sites closer, along the horizontal axis, toward the opposing initiation site of the West traveling fragment, we observe an effect termed ‘‘pulling’’—whereby the wave traveling North moves to the East as a result of the ensuing collision (Fig. 6b); whereas the fragment traveling West is annihilated. This effect arises because the wave traveling West is on the verge of collapse at the point of collision. When the wave traveling West is further from collapsing at the point of collision, it repels ‘‘pushes’’ the wave-fragment traveling North (Fig. 6c and d) so that the daughter fragment that results travels on a trajectory deviating slightly to the West. Where both the wave-fragments traveling West and North are at the same point of collapse when the collision takes place (the ‘relative zero shift’) this results in a symmetric fusion forming a daughter wave-fragment traveling North-West (Fig. 6e). If the initiation point of the North traveling fragment is moved past the proportionate collision point (Fig. 6e) further in an Easterly direction then the trajectory of the fused daughter fragment is deviated from NW toward W Fig. 6f and g. Eventually the wave traveling West is only
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Fig. 6. Wave-fragment moving North collides with a wave-fragment traveling West. The relative neutral position for the collision point is when the distance between the centers of the seed-excitations used to initiate the wave-fragments is the same along both the horizontal and vertical axes, this case is shown in (e). In sub-figures (a)–(d) the seed-excitation was shifted West by 14, 10, 8, and 6 sites respectively in relation to (e). Whereas in sub-figures (f)–(j) the seed-excitation was shifted East from the case of (e) by 4, 6, 10, 12, and 16 sites respectively.
minimally deviated by the collapsing fragment moving North either deflected (‘‘pushed’’) very slightly to the North (Fig. 6h) or ‘‘pulled’’ slightly in a southerly direction (Fig. 6i) by the wave moving North. When wave-fragments traveling South-West and South-East collide (the centers of their seed-excitations used for initiation are at the same level along the vertical axis) they fuse into a singular expanding excitation fragment traveling South (Fig. 7a). When we shift the center of the seed-excitation for the wave-fragment moving South-West in a southerly direction, then the wave-fragment traveling South East collides into the refractory tail of this wave-fragment (Fig. 7d–j), and is guided North-East (Fig. 8). Collisions between a fragment moving South-East and a fragment traveling West are shown in Fig. 9. When the seed-excitations of the waves are placed at the same distance from the collisions center (Fig. 9h) the colliding fragments fuse into an expanding fragment traveling South-East with a slight deviation to the West. If the center of the seeded initiation for the South-East moving fragment is moved slightly to the South then when it collides with the fragment traveling West the resulting fragment deviates slightly eastwards (Fig. 9i and j) whereas if it is moved to the North then
Fig. 7. Wave-fragment moving South-West collides with wave-fragment traveling South-East. In (a) the centers of the seed-excitations share the same coordinate on the vertical axis. Then we shift the center of the initial excitation of the South-West traveling wavefragment southwards by (b) 2 sites, (c) 4 sites, (d) 8 sites, (e) 14 sites, (f) 16 sites, (g) 18 sites, (h) 20 sites, (i) 22 sites, and (j) 24 sites.
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Fig. 8. Snapshots of wave-fragment moving South-West colliding with a wave-fragment traveling South-East. Center of the initial excitation of the South-West wave-fragment is shifted southwards by 22 sites. See overlapped images in (Fig. 7i).
Fig. 9. Overlapped images of wave-fragment traveling South-East colliding with a wave-fragment traveling West. (h) centers of seedexcitations are at the same distance from the collision point. (a)–(g) center of South-East seed-excitation is shifted northward by (a) 22 sites, (b) 20 sites, (c) 16 sites, (d) 12 sites, (e) 8 sites, (f) 4 sites, (g) 2 sites. (i)–(j) center of South-East seed-excitation is shifted southward by (i) 2 sites, (j) 6 sites.
the fragment deviates westwards (Fig. 9f and g). In Fig. 9a and e the wave-fragment moving South-East hits the tail (Fig. 9c–e) or a refractory trail (Fig. 9a and b) of the wave traveling West. When the South-East fragment hits the refractory trail it is reflected North-East, loses stability and expands (Fig. 9a and b). In contrast when the tail of the West fragment is hit by the South-East moving fragment, the West traveling fragment is shifted off course and begins moving in a South-Westerly direction (Fig. 9d and e). The most remarkable example is shown in Fig. 10 (and overlapped images of traveling fragments in Fig. 9): whereby the South-East traveling wave-fragment undergoes a minimal collision with the West moving fragment. The result of this minimal collision is that the South-East fragment is deflected East (with a minor North-East inclination mainly due to the subsequent expansion of the daughter fragment), whereas the fragment previously traveling West is ‘shifted’ northwards but the orientation of its velocity vector remains intact.
Fig. 10. Snapshots of wave-fragment moving South-East colliding to wave-fragment traveling West. See overlapped images in (Fig. 9c).
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Fig. 11. Schematic representation of interaction between wave-fragments. (a) reflection, (b) fusion, (c) attraction, (d) repulsion, (e) sliding, (f) shifting–sliding.
The collision types discussed fit in the following principal classifications (Fig. 11): quasi-elastic reflection of wavefragments (Fig. 11a), fusion of two fragments into one fragment, deviation in the form of ‘‘pulling’’ and ‘‘pushing’’ of a traveling wave-fragment via interaction with another wave-fragment (Fig. 11b and c), deviation via ‘‘sliding’’ of one wave-fragment along the refractory trail of another fragment (Fig. 11d), and translation of a wave-fragment along one axis by another wave-fragment (Fig. 11e). These cover the main types of deviation imparted via the binary collisions of excitation fragments. Also observed during this work is the interesting phenomenon that the collision between two collapsing wave-fragments may give rise to two expanding wave-fragments. It is not clear why this fusion followed by subsequent splitting due to the directly opposed velocity fields should give rise to the observed effect. However, we can postulate that the geometry of each fragment (as well as the trajectory) is altered in the collision and there is evidence from previous theoretical studies [2] that this does give rise to a switch from collapsing to expanding wave forms.
4. Discussion In computational experiments we demonstrated how to tune the trajectories of traveling wave-fragments in a subexcitable chemical medium by colliding ‘‘control’’ wave-fragments. In this work we dealt with the simplest case where the two original fragments are generated from original sources of seeded excitation of approximately the same size. Also we did not deal with the cases of secondary collisions or multiple collisions of two or more fragments and/or their respective daughter fragments. However, even taking the case of relatively simple binary collisions between fragments of excitation in a model chemical system a wide range of interactions were uncovered. These included fragment fusion, reflection, sliding and shifting. We also showed that the outcome of any collision strongly depends on whether the wavefragments collide front on (at least in part), or one fragment hits a tail and or edge of another fragment. We also showed that it is not necessary for two fragments to actually collide for there to be an effect. A common interaction of two fragments resulting in substantial deviation was observed where one fragment collided with the refractory trail of the other fragment. In this case the former fragment may have already collapsed or traveled on unhindered but still exerts an influence over the trajectory of the other fragment. This effect mirrors the findings from our previous experimental study dealing with collisions of fragments in a homogeneous sub-excitable BZ system. The main contribution of our results in the field of collision-based computing is that now we have the capability of designing flexible computing architectures where the propagating signals are manipulated themselves by other propagating signals, i.e. it is not necessary to incorporate stationary impurities (heterogeneous illumination) in the excitable logical computer as we did in our earlier constructions [2]. Ideally we would wish to validate these results experimentally however, whilst it is relatively easy to obtain single open ended fragments of excitation that travel across a homogeneous light field and either expand or contract it is challenging to obtain two fragments initiated in exact locations and with specific trajectories and appropriate spacing. This
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has inevitably led to an experimental methodology whereby pseudo wired, e.g. light modulation [8], or hard wired, e.g. continuous heterogeneous light fields are employed to control the fragments trajectory and stability. As stated however, we were keen to explore collisions in a homogeneous chemical system which we believe gives a richer dynamic and thus computational outcome thus the experimental challenges to implement this type of chemical collision-based computation still remain.
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