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Phototaxis of mobile excitable lattices
Chaos, Solitons and Fractals 13 (2002) 171±184
www.elsevier.com/locate/chaos
Phototaxis of mobile excitable lattices Andrew Adamatzky *, Chris Melhuish Intelligent Autonomous System Laboratory, University of the West of England, Frenchay Campus, DuPont Building, Coldharbour Lane, Bristol BS16 1QY, UK Accepted 10 October 2000
Abstract A two-dimensional lattices, cells of which excite depending on exact numbers of their excited neighbors, exhibit rich patterns of space-time behavior: from chaotic activity to traveling waves to swarms of self-localized excitations. What happens if we supply each cell of a lattice with its own cilium, which beats in a direction determined by local dynamic of excitation around the cell, and make edge cells of the lattice excite light sensitively? Do lattices move toward a source of light? What types of lattice motility are observed? What classes of local excitation functions can we de®ne from trajectories of lattice movement? We answer the questions by means of computer experiments. We demonstrate that a set of local excitation rules can be classi®ed into four groups depending on how often the lattices hit light target and how they move toward the light source. Each class of the functions is studied in respect to lattice activity levels, inhomogeneity and sensitivity of excitation rules, and eciency of lattice motion. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Active non-linear media, both reaction±diusion and excitable, are proved to be suitable to a role of massively parallel processors [8], either universal [7,16,24] or specialized [8±10,22,23]. In these processors, both data and results are represented by spatial patterns of excitation, and computation is implemented by waves of excitation spreading in the medium and interacting one with another. Most types of excitable media are well simulated in discrete lattices [27], and thus, may be thought of as possessing cellular automata architecture [8]. Despite wide-spread interest in unconventional information processing in non-linear media and their cellular automata models (see e.g. [14,15]) few to no results are obtained in locomotor activity of homogeneous excitable media with decentralized propulsive devices. If we imagine an excitable sheet then how could we make it moving sensibly and how could we couple excitation dynamic to sheet motility; one could also think about it as a mobile cellular automaton. In the paper, we explore one of the possible options: coupling of excitable medium with an array of actuators. Most ideas of the paper are derived from the following components: · our previous ®ndings on phenomenology of excitation and parameterization of excitable lattices [1,2]; · classi®cation of vector-based rules of pattern formation in automata models of granular materials [5];
*
Corresponding author. Tel.: +44-0-117-344-2662; fax: +44-0-117-344-3636. E-mail addresses: [email protected] (A. Adamatzky), [email protected] (C. Melhuish).
0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 2 3 3 - 2
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· results of our early experiments on guiding real robots by simulated excitable media [3,4]; · development of the ®eld of distributed arrays of sensors [18] and a theory of programmable force ®elds [13]. So, we supply each cell of a two-dimensional lattice with its own actuator, let edge cells be excited by light, excitation spread on the lattice and each actuator beat in the direction from which locally come. Behavior of such lattice is under investigation in the paper. The paper is organized as follows. Two-dimensional excitable lattices are de®ned in Section 2. In Section 3, we discuss how a lattice motility can be incorporated in the model in case of light following. Section 4 oers formal description of the mobile excitable lattice model. There we de®ned state transition rules of internal and external lattice cells as well as actuators attached to the internal cells. A classi®cation of excitation rules based on success rates of lattice hitting a light target is presented in Section 5. The phenomenology of locomotor activity of lattices governed by excitation rules from dierent classes is studied in Section 6. Section 7 deals with eciency of lattices, determined via activity levels and lengths of lattice travel paths. Two structural, intrinsic, parameters, potentially suitable for classi®cation of excitation rules are oered in Section 8. Section 9 of the paper tries to build some parallels between excitable lattice computational and phototactic abilities; a biological analog of mobile excitable lattice is also discussed there.
2. Excitable lattices The paper deals with cellular automata models of excitable media. We consider a two-dimensional cellular automaton, a lattice L of ®nite locally connected automata (cells). Every cell of the lattice L has eight closest neighbors, takes three states and changes its states in a discrete time. All cells of cellular automaton have the same neighborhood template and the same state transition function. The cells update their states simultaneously. Three elements of the cell state transition set Q f; ; g represent the rest , excited , and refractory , states of a cell. As usual, the decentralized mapping u : L ! L8 associates each cell of the lattice with its eight neighbors:
u
}
;
where s are the neighbors of cell }. A cell state transition function f : Q8 Q ! Q determines the cell state transition rule xt1 t t f xt ; u x , where xt1 and xt are the states of cell x 2 L at time steps t 1 and t, and u x is the state of the cell x's decentralized u x at time step t. We employ the following rule of excitation: P 8 t t > < ; x ^ y2u x v y ; 2 S; 1 ; xt ; xt1 > : ; otherwise; where v y t ; 1 if y t , and is zero otherwise; S is subset of f1; . . . ; 8g, S 2 2f1;...;8g . Given the set S a resting cell is excited if a number of its excited neighbors belongs to the set S. The cell changes its excited state to refractory state and then refractory state to resting state unconditionally, i.e., independently on dynamic of excitation in cell's local decentralized. The excitation rule (1) generalizes all common rules such as a rule of threshold excitation and a rule of interval excitation [1,2]. This allows us to consider all possible sensitivity modes of the lattice cell.
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3. Lattice motility To make an excitable lattice moving toward a source of stimulation its necessary to inform cells of the lattice the stimulation source's whereabouts and to make lattice generating any kind of propulsive forces that cause its movement in desired direction. We inform cells about position of the source using approach developed for computing of the shortest path in non-linear active media [6,9,10, 22,23]. Most techniques for the shortest path computation in non-linear active media employ an ability of excitation waves to transmit information across the medium. Supplying each element of the medium with some kind of very limited memory one can enable elements of the medium to know roughly a direction toward a source of wave generation. Quite commonly a velocity ®eld is calculated in the result of wave traveling [9,23]. The ®eld may represent an optimal path toward an exit of a labyrinth [23] or a shortest obstacle avoiding path between two points of the medium [9]. Real life experiments with Belousov± Zhabotinsky reaction [9,22,23] proved the viability of the technique. Consider a source of stimulation somewhere outside a lattice. Edge cells of the lattice close to the stimulation source are excited and traveling waves are generated. The waves travel on the lattice and inform the internal cells of the lattice about approximate relative localization of the stimulation source. If we want the lattice to execute a positive taxis and to move toward the stimulation source, then we need to supply the lattice with propulsive devices, cilia or actuators. To reach a state of ``perfect parallelism'', we supply each cell of the lattice with its own actuator, a cilium. Thus, we have a lattice system with all parallel sensing, computing and actuating. To make our discussion more objective we have chosen the task of phototaxis: if a lattice is placed in a room with a source of light the lattice must move toward the source of light. Let us also assume that only edge cells of the lattice are light sensitive. When edge cells are excited they transmit excitation to their neighbors, these neighbors to their neighbors and so on. We also assume that the actuators, attached to the cells, can beat in one of eight directions. An actuator beats opposite to the direction from which excitation wave came. Each cell calculates direction of actuator beating from the topology of excitation in its local decentralized. When the actuator beats it produces a local propulsive force, vector of which is oriented toward local maximum of excitation. 4. Sensing, decision and locomotion All cells of the excitable lattice update their states in parallel. They also implement decentralized control of the underlying actuator array. Internal cells of the lattice update their states by the rule (1). The edge cells of the lattice are excited probabilistically, depending on intensity of their illumination 8 with probability p x; > < ; t1 x ; xt ; > : ; otherwise p x n 1 f x , where n a a if a > 0 and n a 0, otherwise; f x d x m=s, where m minz fd zg, where d z is a distance from the point source of light to the cell z and s is a maximal distance between cells of the array; 0 < 1. Thus, the model incorporates deterministic internal cells of the lattice and deterministic actuators however sensor elements are unreliable. A local force vector at the cell x, produced by an actuator attached to the cell x, is represented as 2 vx vix ; vjx , due to two coordinate/index axes i and j, and takes its states from f 1; 0; 1g . The components are recalculated every step of simulation time as
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( vt1 x
rti ; rtj if xt1 ;
vtx ; P
otherwise;
where rtz y2u x:y t xk yk =jxk yk j for k i; j. To translate local forces to lattice motion we introduce, this was previously done in [4], a so-called global vector V Vi ; Vj . The vector represents the estimated direction of thePlight sourcePwith respect to the lattice. It is updated every step of the simulation time as follows: V t r x2L vtxi ; r x2L vtxj , where r acts as above. The components of the global vector are transformed into the rotation angles as shown in the table below.
For ease of use, rotations were represented as multiples of a basic unit of rotation, the angle a. As we see from the rules every internal cell is excited when a number of excited neighbors belongs to the set S, S 2 2f1;...;8g . Varying on the exact structure of the set S we can study 256 dierent rules of local lattice excitation dynamic. For every excitation rule, we conducted computer experiments. In every experiment, a lattice is put somewhere in a plane, at a ®xed distance from light source. The two main ®xed parameters of lattice motion are the basic angle of rotation a 0:5 and the motion increment d 0:5. What happens when we put a lattice at some distance from the light source? The lattice starts to wander around and then begins to move toward the light source along a trajectory, the form of which is determined by the regime of excitation dynamic. Several following processes determine the lattice behavior. The edge cells of the array are excited by light; a probability of cell excitation is proportional to a degree of cell illumination. The patterns of excitation move inward the lattice. The movement of excitation patterns modi®es the local orientations of the actuators, attached to the cells. A ®eld of local propulsive forces is formed. Physical integration of the local forces results in the lattice subsequent rotation and movement. A couple of examples of excitation dynamic of mobile lattices is shown in Fig. 1. In the ®rst example, (Fig. 1(a)), the lattice exhibits mostly self-localized patterns of excitation. These self-localized excitations move across the lattice and leave behind straight trails of coherently beating cilia (Fig. 1(a)). Space-time dynamic of excitation in the lattices from the second example (Fig. 1(b)), is more familiar for anyone. Several wave generators are formed. They produce target-waves. Such wave dynamic is clearly re¯ected in con®gurations of local forces produced by cilia (Fig. 1(b)). 5. Hit-or-miss based classi®cation For each function, we calculated a lattice travel time in the following manner. We have chosen a ``ceiling travel time'' s . To give lattices a plenty of time to ful®ll their tasks we take s equal 2 104 in our experiments, which is several orders more than time (this equals 930 time units) of straightforward traveling from a starting point to a source of light. If a lattice does not reach target in s steps, its attempt is counted as unsuccessful. For every rule, we conducted 30 trials and recorded the number of failures and successes. In each trial, a lattice was put at the same site but turned randomly; the sequence of initial random rotations of a lattice was the same for each tested rule of lattice excitation. The only travel times of successful trials are taken into account. From the distribution of success rates, calculated in computer experiments, we classi®ed excitation rules into the four following classes: · C0: Lattices, cells of which are governed by the functions of the group, never exhibit excitation patterns, and, therefore, do not move at all.
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Fig. 1. Instant patterns of excitation of lattices from the classes C1 (a) and C3 (c) and con®gurations of local forces produced by actuators (b) and (d).
· C1: Majority of the lattices, namely 98% of all excitation functions, hit the target in every trial (the success rate 1.0). Remaining 2% of rules cause lattice to reach the target with the success rate 0.99. · C2: Almost all rules have the success rates either 1.0 (44%) or 0.99 (37%). Tail of the distribution represents functions with success rates 0.97 (12%) and 0.96 (7%). · C3: Distribution of the success rates in this class C3 is shown in Fig. 2. As we see, neither of the functions have a success rate exceeding 0.74. 12% of all functions have the rate 0.39. Other functions have the rates 0, 0.3 or 0.6. To describe rule classes compactly we recall that the set S may be represented by such a vector v that the cell x, resting at time step t, excites at time step t 1 if vP
y2u x
v y t ;
1:
This means indexes of non-zero entries of the vector v represent numbers of excited neighbors of the cell x that cause excitation of the cell x. Using this vector-representation we are able to represent groups of excitation function by Boolean expressions, which employ the notations x _ y for disjunction xy for conjunction. Consider, for example, the set S f2; 5; 7; 8g. This is represented by the vector v 01001011, or in Boolean notations in the following expression v1 v2 v3 v4 v5 v6 v7 v8 . Thus, we can represent a class C by a minimal Boolean formula P C which variables are entries of the vector v. These representations are shown below: P C0 v1 v2 v3 v4 v5 v6 v7 v8 ; P C1 v1 v2 v3 _ x3 v4 v5 v6 v7 v8 _ v1 v2 _ v2 v3 v4 v5 v6 v7 v8 _ v1 v3 v4 v5 v6 v7 v8 ;
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Fig. 2. Distribution of success rates amongst functions of the class C3.
P C2 v1 v2 v3 v4 v5 v6 v7 v8 _ v3 v4 v5 v6 v7 v8 ; P C3 v2 v3 v1 v4 _ v1 v4 _ v4 v5 v6 v7 v8 :
6. Phenomenology Building a correspondence between morphology of trajectories of lattice motion on one side and space-time dynamic of excitation on another side allows us to give a deep insight on the intrinsic processes of excitable lattice motility. We will discuss the phenomenology of the three classes: C1, C2 and C3. We do not consider the class C0 because no long term excitation occurs in lattices of the class due to v1 0 and v2 0. In examples of several trajectories of lattices from each class (Fig. 3(a)) we see that all lattices of the class C1 move toward the target straightforwardly, subject to probabilistic excitation of edge nodes and quasirandom blasts of excitation caused by collisions of mobile self-localized excitations. In Fig. 4, we see that excitation is born on lattice edges relatively close to the light target. Due to the speci®cs of cell state transition functions, which constitute the class, no long wave fronts are formed but swarms of mobile clusters of excitations emerge. These breather-like structures travel across the lattice unchanged until they reach absorbing boundaries or collide with other mobile excitations. In our previous papers (see e.g. [7]), we have shown that two basic types of mobile self-localizations can be detected in the lattice of the class C1, particularly for excitation function represented by the vector v 0100000: 2 particles, which move along rows and columns of the lattice, and 3 -particles, which run along the diagonals. In most cases, velocity vectors of localizations are collinear (subject to discreteness) to the normal from the target to the lattice edge. Thus, the mobile patterns cause cilia to generate a quite perfect propulsive ®eld that pushes the lattice toward the target (Figs. 3(a) and 4). The space-time dynamics of excitation in the class C2 exhibit no long-spreading clusters of excitation, because newly formed wave fronts are immediately broken due the values of third and fourth entries of the vector v : v3 0 or v4 0 (Fig. 5). Dynamic inside, the lattice is chaotically homogeneous and, therefore, mostly excitation of cells near edges of the lattice contributes to formation of the local forces ®eld. This leads to losses in lattice stability to random ¯uctuations, which result in curvy long trajectories (Fig. 5). Remarkably, lattices of the class C3 spend a lot of time circling either near their initial positions or around the target, however not hitting it (Fig. 3, b).
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Fig. 3. Sets of seven exemplar trajectories of robots guided by excitable lattice from dierent classes (a). The class C1, exemplar function v 00111111 (b). The class C2, exemplar function v 11100101 (c). The class C3, exemplar function v 01110100. Each lattice starts its motion at the left bottom corner of the arena; the light sources is placed at the right top corner.
Fig. 4. Snapshots of lattice, governed by the function v 00111111 of the class C1, which travels toward the light target. Con®guration of the lattice is recorded every 100th step of the experiment. The lattice travels from the left to the right.
Typical waves of excitation are observed in the lattices of the class C3. Initially, the waves are generated on the edges of the lattice and move centripetally. Pretty soon wave generators, represented by compact groups of excited and refractory cells, are formed in internal parts of the lattice. These generators produce
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Fig. 5. Snapshots of lattice, governed by the function v 11101101 of the class C2, which travels toward the light target. Con®guration of the lattice is recorded every 100th step of the experiment. The lattice travels from the left to the right.
Fig. 6. Snapshots of lattice, governed by the function v 01111101 of the class C3, which travels toward the light target. Con®guration of the lattice is recorded every 100th step of the experiment. The light target is shown by circles.
excitation waves, which are usually annihilate waves emitted by light-sensitive edges of the lattice. These wave generators are responsible for lattice losing its stability. It is very dicult to destroy generators. Because of periodic emission of waves by the generators, the lattice sometimes starts to rotate or circle (Figs. 3(c), and 6). Eventually, the lattice either stop motion towards the target (Fig. 6) or move in circles (Fig. 7). 7. Eciency If we look at excitable lattices from a practical point of view, e.g., would think about chemical or silicon implementation of mobile excitable media, we may be particularly concerned with eciency. An eciency
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Fig. 7. Snapshots of lattice, governed by the function v 11111001 of the class C3, which travels toward the light target. Con®guration of the lattice is recorded every 100th step of the experiment. The lattice travels from the left to the right.
of mobile lattice is determined by a length of travel path from start site to light target and also by an average energy lattice consumes per time step. In the paper, we consider actuators as abstract entities and, therefore, do not take them into consideration when analyzing eciency. So, we deal only with lattice cells. In our model, both lattice cells and cilia update their states simultaneously in discrete time, therefore we measure a travel time in discrete time steps of lattice updates. Assuming a ®xed amount of energy is released when a cell is excited we can evaluate energy as a lattice activity level. For each rule f we calculate an activity level a f as a number of excited cells summed over whole period of robot traveling from initial point to target and normalized to the travel time and number of cells in the lattice. A distribution of travel times of lattice successful trials is shown in Fig. 8. Lattices of the class C1 reach the target in the shortest time. For majority of the excitation functions the lattices from the class C3 travel 1.5 times slower than the lattices of the class C1. We should remember that lattices from the class C1 reach the target in all trials however only few trials may be successful for the lattices of the class C3. For any excitation function f: a f 6 1=3. Every cell of the lattice updates its state in the cycle ! !
! ! ! :
Fig. 8. Distribution of travel times, counted only for successful trials, for lattices of the classes C1 }; C2 and C3 (). Horizontal axis represents travel times, vertical ± the normalized number of functions.
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Fig. 9. Distribution of activity levels a for the functions of the classes C1 (solid lines), C2 (dash lines), and C3 (dot lines). Horizontal axis represents activity levels, vertical ± the number of functions.
The transitions ! and ! are unconditional; they happen anyway. Therefore, even in the ideal situation when all cells of the lattice are excited synchronously during an evolution period 0; t the maximal sum of excited cells is close to a~ t=3n, where n is a number of cells. Thus, we have maximal value of activity level a f a~= nt 1=3. Distributions of activity levels for all the possible rules are shown in Fig. 9. Distribution of activity levels of the functions from C1 has two characteristic peaks at a f 0:075 and a f 0:265; whereas almost all functions of other two classes have near maximal activity levels: 0.315 for C2 and 0.325 for C3. So, we may speculate that Excitable lattices with highest levels of activity may be less suitable for phototaxis task. From the distributions of activity levels, travel times and success rates we can order the classes of mobile excitable lattices in the following hierarchy of eciency: C1 > C2 > C3:
8. Intrinsic parameters In the previous sections, we classi®ed excitation rules into several groups depending on the behavior of mobile lattices governed by the rules and common sense suitability of the lattices to execute phototactic behavior. We also gave some hints on how to select ``good navigation'' rules via analysis of excitable lattice activity. Obviously both trajectory-based classi®cation and activity-based selection require experimenting with excitable lattice. Is there any way to decide on the suitability of an excitation rule without experimenting with the lattice but simply studying the rule's structure? This is generally a problem of parameterization of cellular automata rules, formulated and investigated in [17,25] and applied to excitable lattices and lattice swarms in [2,5]. Based on the results of our previous works (see e.g. [2,5]) we could oer two characteristics, which may be utilized in express design of mobile excitable lattices. The ®rst one represents some kind of inhomogeneity of cell state transition rule; the second deals with rather a sensitivity of the excitation rule. For every excitation function f, we can calculate a static parameter k as follows: P w2Q8 v f w; : k 38 Given excitation rule f, the parameter k f indicates a ratio of cell neighborhood con®gurations (every con®guration is represented as a string of eight symbols of the ternary alphabet) which may excite the central cell (owner of the neighborhood).
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This parameter is derived from the k parameter, which was developed in [17] to link the intrinsic characteristics of cell state transition rules with the spatio-temporal behavior of certain classes of cellular automata. It was demonstrated in [17] that arranging cell state transition rules by the values of k one can achieve a transition between ordered, complex and chaotic dynamic. In our case, the parameter k can be readily interpreted as the intrinsic probability of excitation. Further, since only a cell at rest can become excited, we have 0 6 k 6 1=3. It is true in general that the less k the more probably system will ®nish evolution in the entirely resting state, the more k the more possible that the whole lattice will be ®lled with excitation patterns after some period of its evolution. There are, as usual, some exclusions. Distribution of excitation rules on their values of the parameter k is shown in Fig. 10. We see that more than a third of all functions of the class C0 has values of k concentrated in the lower part of k scale. The class C2 occupies median of the k scale. k values of almost half of members of the class C3 are grouped in the higher part of the k scale. Basing on the results of previous section and this k distribution we can assert that Mobile excitable lattices governed by rules with highest values of k are less suitable for executing the phototaxis. The sensitivity parameter m f is measured as a number of changes from 0s to 1s and from 1s to 0s in the vectors v of local excitation function f. To appreciate the parameter m we must recall that usually threshold excitation is employed in models of excitable media: a cell is excited if at least h of its neighbors are excited. In the vector v representation of the threshold function the entries with indexes are less than h are zero valued, others take the value 1. If the function f is threshold, its m parameter equals 1. Quite recently, we investigated so-called interval excitations (see e.g. [1,4]), where every lattice cell is excited if the sum of its excited neighbors belongs to the interval h1 ; h2 , where 1 6 h1 6 h2 6 8 in the case of eight-cell decentralized. Obviously, the function f represented by excitation interval h1 ; h2 has such a vector v that every entry with index h1 6 j 6 h2 takes the value 1, and others are 0. Thus, we have m f 2. Generally, we can say the more the value of m f the more sensitive are cells of an excitable lattice governed by f. As we see in the distribution of functions on the parameter m (Fig. 11) the classes C3 and C0 have lowest values of m while the classes C1 and C2 occupy the top of m hierarchy. Given a choice one must prefer an excitable lattice with most sensitive function of local excitation to design mobile excitable lattices.
Fig. 10. Normalized distribution of excitation functions on k values: } C1; C2, and (C3). Only non-zero entries are shown. Horizontal axis represents values of k , vertical ± normalized number of functions.
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Fig. 11. Normalized distribution of excitation functions on m values: } C0; C1, C2; C3. Horizontal axis represents values of m, vertical ± normalized number of functions.
Both parameters k and m are structural in the sense that they are calculated on cell state transition rules of cellular automata and parameters themselves do not take into account any kinds of dynamical computation or analysis of space and time con®gurations of the evolving systems.
9. Discussion In the result of computer experiments with two-dimensional excitable lattices coupled with actuators, we found that a set of 256 rules of cell excitation can be roughly split into four main classes. The lattices of the ®rst class, C0, do not exhibit persistent patterns of excitation; therefore, they do not react on a stimulation and do not move. Members of the second class, C1, show excellent phototactic response. Their trajectories are impeccable. The lattices of the third class, C2, do well in general. However, their trajectories are lengthy and certainly not optimal. The class C2 is rather transitional from lattices behaving straightly to lattices behaving weirdly. The fourth class, C3, is constituted of lattices, which also burst in excitation when illuminated, however speci®c of their space-time excitation dynamic is not re¯ected in perfect locomotor activity. The present phenomenological ®nding on space-time dynamic of mobile excitable lattices is consistent with our previous results (see e.g. [2]) on vector-based rules of excitation. In our early paper, [2] we classi®ed lattices on their computational abilities. Do computational abilities imply good phototactic behavior? Everyone would be most interested in whether excitable lattices which play a role of massively parallel universal computers or at least specialized image processing devices show signs of strong phototactic behavior. There is no certain answer. Some lattices of the class C1 produce self-localized excitations. Therefore, they can implement universal collision based computing [7]. They also move straightforwardly toward the light source. However, signi®cant number of members of C1 do not do any computation (at least for known problems). Nevertheless they are good navigators. Inversely, the lattices of the class C3 can be employed in image processing and computational geometry [8]; however, these lattices are very poor navigators. Ideally, it would be impressive to verify our theoretical results with real life implementations. Unfortunately, at this stage we could not oer any arti®cially designed mobile arrays of actuators with cellular automata architecture. Fabrication of the arrays, possibly based on mono-molecular arrays, such as Scheibe aggregates [11,12], may be a task of the near future. Luckily, we have a living analog of arrays of cilia, which exhibit positive taxis. This is a phylum Ciliophora, or ciliates ± protists, bodies of which are covered with cilia (one could refer to Paramecium caudatum as a typical example) [21,26]. Each cilium, attached to a membrane of a protist, sweeps with a power stroke in the direction opposite to intended direction of organism movement. The cilia are usually arranged in rows; this arrangement forms a longitudinal axis of beating. Cilia beat coherently in waves and propel the individual forward. A physical integration of strokes of many cilia propel the protist in the direction opposite to the direction of beating.
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There are still limited data on coordination of cilia. The control may be implemented by either information transfer via sub-membrane network of microtubules or coordinated contractions of the membrane traveling along the protist's body. In any case, we can speculate about direct analogies between wave of cilia beating, caused by excitation waves of the lattice, and waves of co-beating of cilia in protists, that move from front to rear parts of the organism. The absence of any directional photoreceptors in ciliates [19] matches even closer our theoretical models. Thus, for example, it is proved in real life experiments [19,20] that only the frequency of directional changes together with frequency distribution of an angle of directional changes are aected by light; the speed is not aected at all. This was an intuitive assumption in our model of mobile excitable lattices. So, both excitable lattice and real protists are rather oriented by steering toward a direction of maximal light intensity [19]. Also this may be quite useful to interpret our models in terms of a theory of programmable force ®elds [13], which deal with small actuators arranged in two-dimensional array and coupled with arrays of sensors. The actuators execute a few simple actions in the reaction on sensorial inputs. A force ®eld is formed by the actuator array that surrounds an object put on the array and causes the object to move [13]. What types of object transportation can be achieved in cilia supplied excitable lattice? This may be a subject of future studies.
References [1] Adamatzky A, Holland O. Phenomenology of excitation in 2D cellular automata and lattice swarm. Chaos, Solitons & Fractals 1998;9:1233±65. [2] Adamatzky A, Holland O. Edges and computation in excitable media. In: Adami C, et al., editors. Arti®cial Life VI: Proceedings of the Sixth International Conference on Arti®cial Life. A Bradford Book, 1998. p. 379±384. [3] Adamatzky A, Holland O, Rambidi NG, Win®eld A. Wet arti®cial brains: towards the chemical control of robot motion by reaction±diusion and excitable media. Lecture Notes Comp Sci 1999;1674:304±13. [4] Adamatzky A, Melhuish C. Parallel controllers for decentralized robots: toward nano design. Kybernetes 2000;29:733±45. [5] Adamatzky A. Choosey hot sand: re¯ection of grain sensitivity on pattern morphology. Int J Modern Phys C 2000;11: 47±68. [6] Adamatzky A. Computation of shortest path in cellular automata. Math Comput Modelling 1996;23:105±13. [7] Adamatzky A. Universal dynamical computation in multidimensional excitable lattices. Int J Theoret Phys 1998;37: 3069±108. [8] Adamatzky A. Reaction±diusion and excitable processors: a sense of the unconventional. Parallel and Distributed Computing Practices, in print. [9] Agladze K, Magome N, Aliev R, Yamaguchi T, Yoshikawa K. Finding the optimal path with the aid of chemical wave. Phys D 1997;106(3-4):247±54. [10] Babloyantz A, Sepulchre JA. Front propagation into unstable media: a computational tool. In: Holden AV, Markus M, Othmer HG, editors. Nonlinear wave processes in excitable media. Plenum Press: New York; 1991. p. 343±5. [11] Bartnik EA, Blinowska KJ, Tuszynski JA. Analytical and numerical modelling of Scheibe aggregates. Nanobiology 1992;1:239± 50. [12] Bartnik EA, Tuszynski JA. Theoretical models of energy transfer in two-dimensional molecular assemblies. Phys Rev E 1993;48:1516±28. [13] B ohringer K-F, Donald BR, MacDonald NC. Programmable force ®elds for distributed manipulation, with applications to MEMS actuator arrays and vibratory parts feeders. Int J Robotics Res 1999;18:168±200. [14] Calude CS, Casti JL, Dineen MJ, Unconventional models of computation. Berlin: Springer; 1999. [15] Gramss T, Bornholdt S, Gross M, Mitchell M, Pellizzari T. Nonstandard computation. New York: Wiley; 1998. [16] Jakubowski MH, Steiglitz K, Squier R. Information transfer between solitary waves in the saturable Schr odinger equation. Phys Rev E 1997;56:7267±72. [17] Langton C. Computation at the edge of chaos: phase transitions and emergent computation. Phys D 1990;42:12±27. [18] Li P, Wen Y. An arbitrarily distributed tactile piezoelectric sensor array. Sensors and Actuators A 1998;65:141±6. [19] Marangoni R, Preosti G, Colombetti G. Phototactic orientation mechanism in the ciliate Fabred salina, as inferred from numerical simulations. J Photochem Photobiol B 2000;54:185±93. [20] Marangoni R, Batistini A, Puntoni S, Colombetti G. Temperature eects on motion parameters and the phototactic reaction of the marine ciliate Fabrea salina. J Photochem Photobiol B: Biol 1995;30:123±7. [21] Melkonian M, Anderson RA, Schnepf E. The cytoskeleton of ¯agellate and ciliate protists. Berlin: Springer; 1991. [22] Rambidi NG, Yakovenchuck D. Finding path in a labyrinth based on reaction±diusion media. Adv Mat Optics Electron 1999:67±72. [23] Steinbock O, T oth A, Showalter K. Navigating complex labyrinths: optimal paths from chemical waves. Science 1995;267: 868±71.
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www.elsevier.com/locate/chaos
Phototaxis of mobile excitable lattices Andrew Adamatzky *, Chris Melhuish Intelligent Autonomous System Laboratory, University of the West of England, Frenchay Campus, DuPont Building, Coldharbour Lane, Bristol BS16 1QY, UK Accepted 10 October 2000
Abstract A two-dimensional lattices, cells of which excite depending on exact numbers of their excited neighbors, exhibit rich patterns of space-time behavior: from chaotic activity to traveling waves to swarms of self-localized excitations. What happens if we supply each cell of a lattice with its own cilium, which beats in a direction determined by local dynamic of excitation around the cell, and make edge cells of the lattice excite light sensitively? Do lattices move toward a source of light? What types of lattice motility are observed? What classes of local excitation functions can we de®ne from trajectories of lattice movement? We answer the questions by means of computer experiments. We demonstrate that a set of local excitation rules can be classi®ed into four groups depending on how often the lattices hit light target and how they move toward the light source. Each class of the functions is studied in respect to lattice activity levels, inhomogeneity and sensitivity of excitation rules, and eciency of lattice motion. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Active non-linear media, both reaction±diusion and excitable, are proved to be suitable to a role of massively parallel processors [8], either universal [7,16,24] or specialized [8±10,22,23]. In these processors, both data and results are represented by spatial patterns of excitation, and computation is implemented by waves of excitation spreading in the medium and interacting one with another. Most types of excitable media are well simulated in discrete lattices [27], and thus, may be thought of as possessing cellular automata architecture [8]. Despite wide-spread interest in unconventional information processing in non-linear media and their cellular automata models (see e.g. [14,15]) few to no results are obtained in locomotor activity of homogeneous excitable media with decentralized propulsive devices. If we imagine an excitable sheet then how could we make it moving sensibly and how could we couple excitation dynamic to sheet motility; one could also think about it as a mobile cellular automaton. In the paper, we explore one of the possible options: coupling of excitable medium with an array of actuators. Most ideas of the paper are derived from the following components: · our previous ®ndings on phenomenology of excitation and parameterization of excitable lattices [1,2]; · classi®cation of vector-based rules of pattern formation in automata models of granular materials [5];
*
Corresponding author. Tel.: +44-0-117-344-2662; fax: +44-0-117-344-3636. E-mail addresses: [email protected] (A. Adamatzky), [email protected] (C. Melhuish).
0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 2 3 3 - 2
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· results of our early experiments on guiding real robots by simulated excitable media [3,4]; · development of the ®eld of distributed arrays of sensors [18] and a theory of programmable force ®elds [13]. So, we supply each cell of a two-dimensional lattice with its own actuator, let edge cells be excited by light, excitation spread on the lattice and each actuator beat in the direction from which locally come. Behavior of such lattice is under investigation in the paper. The paper is organized as follows. Two-dimensional excitable lattices are de®ned in Section 2. In Section 3, we discuss how a lattice motility can be incorporated in the model in case of light following. Section 4 oers formal description of the mobile excitable lattice model. There we de®ned state transition rules of internal and external lattice cells as well as actuators attached to the internal cells. A classi®cation of excitation rules based on success rates of lattice hitting a light target is presented in Section 5. The phenomenology of locomotor activity of lattices governed by excitation rules from dierent classes is studied in Section 6. Section 7 deals with eciency of lattices, determined via activity levels and lengths of lattice travel paths. Two structural, intrinsic, parameters, potentially suitable for classi®cation of excitation rules are oered in Section 8. Section 9 of the paper tries to build some parallels between excitable lattice computational and phototactic abilities; a biological analog of mobile excitable lattice is also discussed there.
2. Excitable lattices The paper deals with cellular automata models of excitable media. We consider a two-dimensional cellular automaton, a lattice L of ®nite locally connected automata (cells). Every cell of the lattice L has eight closest neighbors, takes three states and changes its states in a discrete time. All cells of cellular automaton have the same neighborhood template and the same state transition function. The cells update their states simultaneously. Three elements of the cell state transition set Q f; ; g represent the rest , excited , and refractory , states of a cell. As usual, the decentralized mapping u : L ! L8 associates each cell of the lattice with its eight neighbors:
u
}
;
where s are the neighbors of cell }. A cell state transition function f : Q8 Q ! Q determines the cell state transition rule xt1 t t f xt ; u x , where xt1 and xt are the states of cell x 2 L at time steps t 1 and t, and u x is the state of the cell x's decentralized u x at time step t. We employ the following rule of excitation: P 8 t t > < ; x ^ y2u x v y ; 2 S; 1 ; xt ; xt1 > : ; otherwise; where v y t ; 1 if y t , and is zero otherwise; S is subset of f1; . . . ; 8g, S 2 2f1;...;8g . Given the set S a resting cell is excited if a number of its excited neighbors belongs to the set S. The cell changes its excited state to refractory state and then refractory state to resting state unconditionally, i.e., independently on dynamic of excitation in cell's local decentralized. The excitation rule (1) generalizes all common rules such as a rule of threshold excitation and a rule of interval excitation [1,2]. This allows us to consider all possible sensitivity modes of the lattice cell.
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3. Lattice motility To make an excitable lattice moving toward a source of stimulation its necessary to inform cells of the lattice the stimulation source's whereabouts and to make lattice generating any kind of propulsive forces that cause its movement in desired direction. We inform cells about position of the source using approach developed for computing of the shortest path in non-linear active media [6,9,10, 22,23]. Most techniques for the shortest path computation in non-linear active media employ an ability of excitation waves to transmit information across the medium. Supplying each element of the medium with some kind of very limited memory one can enable elements of the medium to know roughly a direction toward a source of wave generation. Quite commonly a velocity ®eld is calculated in the result of wave traveling [9,23]. The ®eld may represent an optimal path toward an exit of a labyrinth [23] or a shortest obstacle avoiding path between two points of the medium [9]. Real life experiments with Belousov± Zhabotinsky reaction [9,22,23] proved the viability of the technique. Consider a source of stimulation somewhere outside a lattice. Edge cells of the lattice close to the stimulation source are excited and traveling waves are generated. The waves travel on the lattice and inform the internal cells of the lattice about approximate relative localization of the stimulation source. If we want the lattice to execute a positive taxis and to move toward the stimulation source, then we need to supply the lattice with propulsive devices, cilia or actuators. To reach a state of ``perfect parallelism'', we supply each cell of the lattice with its own actuator, a cilium. Thus, we have a lattice system with all parallel sensing, computing and actuating. To make our discussion more objective we have chosen the task of phototaxis: if a lattice is placed in a room with a source of light the lattice must move toward the source of light. Let us also assume that only edge cells of the lattice are light sensitive. When edge cells are excited they transmit excitation to their neighbors, these neighbors to their neighbors and so on. We also assume that the actuators, attached to the cells, can beat in one of eight directions. An actuator beats opposite to the direction from which excitation wave came. Each cell calculates direction of actuator beating from the topology of excitation in its local decentralized. When the actuator beats it produces a local propulsive force, vector of which is oriented toward local maximum of excitation. 4. Sensing, decision and locomotion All cells of the excitable lattice update their states in parallel. They also implement decentralized control of the underlying actuator array. Internal cells of the lattice update their states by the rule (1). The edge cells of the lattice are excited probabilistically, depending on intensity of their illumination 8 with probability p x; > < ; t1 x ; xt ; > : ; otherwise p x n 1 f x , where n a a if a > 0 and n a 0, otherwise; f x d x m=s, where m minz fd zg, where d z is a distance from the point source of light to the cell z and s is a maximal distance between cells of the array; 0 < 1. Thus, the model incorporates deterministic internal cells of the lattice and deterministic actuators however sensor elements are unreliable. A local force vector at the cell x, produced by an actuator attached to the cell x, is represented as 2 vx vix ; vjx , due to two coordinate/index axes i and j, and takes its states from f 1; 0; 1g . The components are recalculated every step of simulation time as
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( vt1 x
rti ; rtj if xt1 ;
vtx ; P
otherwise;
where rtz y2u x:y t xk yk =jxk yk j for k i; j. To translate local forces to lattice motion we introduce, this was previously done in [4], a so-called global vector V Vi ; Vj . The vector represents the estimated direction of thePlight sourcePwith respect to the lattice. It is updated every step of the simulation time as follows: V t r x2L vtxi ; r x2L vtxj , where r acts as above. The components of the global vector are transformed into the rotation angles as shown in the table below.
For ease of use, rotations were represented as multiples of a basic unit of rotation, the angle a. As we see from the rules every internal cell is excited when a number of excited neighbors belongs to the set S, S 2 2f1;...;8g . Varying on the exact structure of the set S we can study 256 dierent rules of local lattice excitation dynamic. For every excitation rule, we conducted computer experiments. In every experiment, a lattice is put somewhere in a plane, at a ®xed distance from light source. The two main ®xed parameters of lattice motion are the basic angle of rotation a 0:5 and the motion increment d 0:5. What happens when we put a lattice at some distance from the light source? The lattice starts to wander around and then begins to move toward the light source along a trajectory, the form of which is determined by the regime of excitation dynamic. Several following processes determine the lattice behavior. The edge cells of the array are excited by light; a probability of cell excitation is proportional to a degree of cell illumination. The patterns of excitation move inward the lattice. The movement of excitation patterns modi®es the local orientations of the actuators, attached to the cells. A ®eld of local propulsive forces is formed. Physical integration of the local forces results in the lattice subsequent rotation and movement. A couple of examples of excitation dynamic of mobile lattices is shown in Fig. 1. In the ®rst example, (Fig. 1(a)), the lattice exhibits mostly self-localized patterns of excitation. These self-localized excitations move across the lattice and leave behind straight trails of coherently beating cilia (Fig. 1(a)). Space-time dynamic of excitation in the lattices from the second example (Fig. 1(b)), is more familiar for anyone. Several wave generators are formed. They produce target-waves. Such wave dynamic is clearly re¯ected in con®gurations of local forces produced by cilia (Fig. 1(b)). 5. Hit-or-miss based classi®cation For each function, we calculated a lattice travel time in the following manner. We have chosen a ``ceiling travel time'' s . To give lattices a plenty of time to ful®ll their tasks we take s equal 2 104 in our experiments, which is several orders more than time (this equals 930 time units) of straightforward traveling from a starting point to a source of light. If a lattice does not reach target in s steps, its attempt is counted as unsuccessful. For every rule, we conducted 30 trials and recorded the number of failures and successes. In each trial, a lattice was put at the same site but turned randomly; the sequence of initial random rotations of a lattice was the same for each tested rule of lattice excitation. The only travel times of successful trials are taken into account. From the distribution of success rates, calculated in computer experiments, we classi®ed excitation rules into the four following classes: · C0: Lattices, cells of which are governed by the functions of the group, never exhibit excitation patterns, and, therefore, do not move at all.
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Fig. 1. Instant patterns of excitation of lattices from the classes C1 (a) and C3 (c) and con®gurations of local forces produced by actuators (b) and (d).
· C1: Majority of the lattices, namely 98% of all excitation functions, hit the target in every trial (the success rate 1.0). Remaining 2% of rules cause lattice to reach the target with the success rate 0.99. · C2: Almost all rules have the success rates either 1.0 (44%) or 0.99 (37%). Tail of the distribution represents functions with success rates 0.97 (12%) and 0.96 (7%). · C3: Distribution of the success rates in this class C3 is shown in Fig. 2. As we see, neither of the functions have a success rate exceeding 0.74. 12% of all functions have the rate 0.39. Other functions have the rates 0, 0.3 or 0.6. To describe rule classes compactly we recall that the set S may be represented by such a vector v that the cell x, resting at time step t, excites at time step t 1 if vP
y2u x
v y t ;
1:
This means indexes of non-zero entries of the vector v represent numbers of excited neighbors of the cell x that cause excitation of the cell x. Using this vector-representation we are able to represent groups of excitation function by Boolean expressions, which employ the notations x _ y for disjunction xy for conjunction. Consider, for example, the set S f2; 5; 7; 8g. This is represented by the vector v 01001011, or in Boolean notations in the following expression v1 v2 v3 v4 v5 v6 v7 v8 . Thus, we can represent a class C by a minimal Boolean formula P C which variables are entries of the vector v. These representations are shown below: P C0 v1 v2 v3 v4 v5 v6 v7 v8 ; P C1 v1 v2 v3 _ x3 v4 v5 v6 v7 v8 _ v1 v2 _ v2 v3 v4 v5 v6 v7 v8 _ v1 v3 v4 v5 v6 v7 v8 ;
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Fig. 2. Distribution of success rates amongst functions of the class C3.
P C2 v1 v2 v3 v4 v5 v6 v7 v8 _ v3 v4 v5 v6 v7 v8 ; P C3 v2 v3 v1 v4 _ v1 v4 _ v4 v5 v6 v7 v8 :
6. Phenomenology Building a correspondence between morphology of trajectories of lattice motion on one side and space-time dynamic of excitation on another side allows us to give a deep insight on the intrinsic processes of excitable lattice motility. We will discuss the phenomenology of the three classes: C1, C2 and C3. We do not consider the class C0 because no long term excitation occurs in lattices of the class due to v1 0 and v2 0. In examples of several trajectories of lattices from each class (Fig. 3(a)) we see that all lattices of the class C1 move toward the target straightforwardly, subject to probabilistic excitation of edge nodes and quasirandom blasts of excitation caused by collisions of mobile self-localized excitations. In Fig. 4, we see that excitation is born on lattice edges relatively close to the light target. Due to the speci®cs of cell state transition functions, which constitute the class, no long wave fronts are formed but swarms of mobile clusters of excitations emerge. These breather-like structures travel across the lattice unchanged until they reach absorbing boundaries or collide with other mobile excitations. In our previous papers (see e.g. [7]), we have shown that two basic types of mobile self-localizations can be detected in the lattice of the class C1, particularly for excitation function represented by the vector v 0100000: 2 particles, which move along rows and columns of the lattice, and 3 -particles, which run along the diagonals. In most cases, velocity vectors of localizations are collinear (subject to discreteness) to the normal from the target to the lattice edge. Thus, the mobile patterns cause cilia to generate a quite perfect propulsive ®eld that pushes the lattice toward the target (Figs. 3(a) and 4). The space-time dynamics of excitation in the class C2 exhibit no long-spreading clusters of excitation, because newly formed wave fronts are immediately broken due the values of third and fourth entries of the vector v : v3 0 or v4 0 (Fig. 5). Dynamic inside, the lattice is chaotically homogeneous and, therefore, mostly excitation of cells near edges of the lattice contributes to formation of the local forces ®eld. This leads to losses in lattice stability to random ¯uctuations, which result in curvy long trajectories (Fig. 5). Remarkably, lattices of the class C3 spend a lot of time circling either near their initial positions or around the target, however not hitting it (Fig. 3, b).
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Fig. 3. Sets of seven exemplar trajectories of robots guided by excitable lattice from dierent classes (a). The class C1, exemplar function v 00111111 (b). The class C2, exemplar function v 11100101 (c). The class C3, exemplar function v 01110100. Each lattice starts its motion at the left bottom corner of the arena; the light sources is placed at the right top corner.
Fig. 4. Snapshots of lattice, governed by the function v 00111111 of the class C1, which travels toward the light target. Con®guration of the lattice is recorded every 100th step of the experiment. The lattice travels from the left to the right.
Typical waves of excitation are observed in the lattices of the class C3. Initially, the waves are generated on the edges of the lattice and move centripetally. Pretty soon wave generators, represented by compact groups of excited and refractory cells, are formed in internal parts of the lattice. These generators produce
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Fig. 5. Snapshots of lattice, governed by the function v 11101101 of the class C2, which travels toward the light target. Con®guration of the lattice is recorded every 100th step of the experiment. The lattice travels from the left to the right.
Fig. 6. Snapshots of lattice, governed by the function v 01111101 of the class C3, which travels toward the light target. Con®guration of the lattice is recorded every 100th step of the experiment. The light target is shown by circles.
excitation waves, which are usually annihilate waves emitted by light-sensitive edges of the lattice. These wave generators are responsible for lattice losing its stability. It is very dicult to destroy generators. Because of periodic emission of waves by the generators, the lattice sometimes starts to rotate or circle (Figs. 3(c), and 6). Eventually, the lattice either stop motion towards the target (Fig. 6) or move in circles (Fig. 7). 7. Eciency If we look at excitable lattices from a practical point of view, e.g., would think about chemical or silicon implementation of mobile excitable media, we may be particularly concerned with eciency. An eciency
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Fig. 7. Snapshots of lattice, governed by the function v 11111001 of the class C3, which travels toward the light target. Con®guration of the lattice is recorded every 100th step of the experiment. The lattice travels from the left to the right.
of mobile lattice is determined by a length of travel path from start site to light target and also by an average energy lattice consumes per time step. In the paper, we consider actuators as abstract entities and, therefore, do not take them into consideration when analyzing eciency. So, we deal only with lattice cells. In our model, both lattice cells and cilia update their states simultaneously in discrete time, therefore we measure a travel time in discrete time steps of lattice updates. Assuming a ®xed amount of energy is released when a cell is excited we can evaluate energy as a lattice activity level. For each rule f we calculate an activity level a f as a number of excited cells summed over whole period of robot traveling from initial point to target and normalized to the travel time and number of cells in the lattice. A distribution of travel times of lattice successful trials is shown in Fig. 8. Lattices of the class C1 reach the target in the shortest time. For majority of the excitation functions the lattices from the class C3 travel 1.5 times slower than the lattices of the class C1. We should remember that lattices from the class C1 reach the target in all trials however only few trials may be successful for the lattices of the class C3. For any excitation function f: a f 6 1=3. Every cell of the lattice updates its state in the cycle ! !
! ! ! :
Fig. 8. Distribution of travel times, counted only for successful trials, for lattices of the classes C1 }; C2 and C3 (). Horizontal axis represents travel times, vertical ± the normalized number of functions.
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Fig. 9. Distribution of activity levels a for the functions of the classes C1 (solid lines), C2 (dash lines), and C3 (dot lines). Horizontal axis represents activity levels, vertical ± the number of functions.
The transitions ! and ! are unconditional; they happen anyway. Therefore, even in the ideal situation when all cells of the lattice are excited synchronously during an evolution period 0; t the maximal sum of excited cells is close to a~ t=3n, where n is a number of cells. Thus, we have maximal value of activity level a f a~= nt 1=3. Distributions of activity levels for all the possible rules are shown in Fig. 9. Distribution of activity levels of the functions from C1 has two characteristic peaks at a f 0:075 and a f 0:265; whereas almost all functions of other two classes have near maximal activity levels: 0.315 for C2 and 0.325 for C3. So, we may speculate that Excitable lattices with highest levels of activity may be less suitable for phototaxis task. From the distributions of activity levels, travel times and success rates we can order the classes of mobile excitable lattices in the following hierarchy of eciency: C1 > C2 > C3:
8. Intrinsic parameters In the previous sections, we classi®ed excitation rules into several groups depending on the behavior of mobile lattices governed by the rules and common sense suitability of the lattices to execute phototactic behavior. We also gave some hints on how to select ``good navigation'' rules via analysis of excitable lattice activity. Obviously both trajectory-based classi®cation and activity-based selection require experimenting with excitable lattice. Is there any way to decide on the suitability of an excitation rule without experimenting with the lattice but simply studying the rule's structure? This is generally a problem of parameterization of cellular automata rules, formulated and investigated in [17,25] and applied to excitable lattices and lattice swarms in [2,5]. Based on the results of our previous works (see e.g. [2,5]) we could oer two characteristics, which may be utilized in express design of mobile excitable lattices. The ®rst one represents some kind of inhomogeneity of cell state transition rule; the second deals with rather a sensitivity of the excitation rule. For every excitation function f, we can calculate a static parameter k as follows: P w2Q8 v f w; : k 38 Given excitation rule f, the parameter k f indicates a ratio of cell neighborhood con®gurations (every con®guration is represented as a string of eight symbols of the ternary alphabet) which may excite the central cell (owner of the neighborhood).
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This parameter is derived from the k parameter, which was developed in [17] to link the intrinsic characteristics of cell state transition rules with the spatio-temporal behavior of certain classes of cellular automata. It was demonstrated in [17] that arranging cell state transition rules by the values of k one can achieve a transition between ordered, complex and chaotic dynamic. In our case, the parameter k can be readily interpreted as the intrinsic probability of excitation. Further, since only a cell at rest can become excited, we have 0 6 k 6 1=3. It is true in general that the less k the more probably system will ®nish evolution in the entirely resting state, the more k the more possible that the whole lattice will be ®lled with excitation patterns after some period of its evolution. There are, as usual, some exclusions. Distribution of excitation rules on their values of the parameter k is shown in Fig. 10. We see that more than a third of all functions of the class C0 has values of k concentrated in the lower part of k scale. The class C2 occupies median of the k scale. k values of almost half of members of the class C3 are grouped in the higher part of the k scale. Basing on the results of previous section and this k distribution we can assert that Mobile excitable lattices governed by rules with highest values of k are less suitable for executing the phototaxis. The sensitivity parameter m f is measured as a number of changes from 0s to 1s and from 1s to 0s in the vectors v of local excitation function f. To appreciate the parameter m we must recall that usually threshold excitation is employed in models of excitable media: a cell is excited if at least h of its neighbors are excited. In the vector v representation of the threshold function the entries with indexes are less than h are zero valued, others take the value 1. If the function f is threshold, its m parameter equals 1. Quite recently, we investigated so-called interval excitations (see e.g. [1,4]), where every lattice cell is excited if the sum of its excited neighbors belongs to the interval h1 ; h2 , where 1 6 h1 6 h2 6 8 in the case of eight-cell decentralized. Obviously, the function f represented by excitation interval h1 ; h2 has such a vector v that every entry with index h1 6 j 6 h2 takes the value 1, and others are 0. Thus, we have m f 2. Generally, we can say the more the value of m f the more sensitive are cells of an excitable lattice governed by f. As we see in the distribution of functions on the parameter m (Fig. 11) the classes C3 and C0 have lowest values of m while the classes C1 and C2 occupy the top of m hierarchy. Given a choice one must prefer an excitable lattice with most sensitive function of local excitation to design mobile excitable lattices.
Fig. 10. Normalized distribution of excitation functions on k values: } C1; C2, and (C3). Only non-zero entries are shown. Horizontal axis represents values of k , vertical ± normalized number of functions.
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Fig. 11. Normalized distribution of excitation functions on m values: } C0; C1, C2; C3. Horizontal axis represents values of m, vertical ± normalized number of functions.
Both parameters k and m are structural in the sense that they are calculated on cell state transition rules of cellular automata and parameters themselves do not take into account any kinds of dynamical computation or analysis of space and time con®gurations of the evolving systems.
9. Discussion In the result of computer experiments with two-dimensional excitable lattices coupled with actuators, we found that a set of 256 rules of cell excitation can be roughly split into four main classes. The lattices of the ®rst class, C0, do not exhibit persistent patterns of excitation; therefore, they do not react on a stimulation and do not move. Members of the second class, C1, show excellent phototactic response. Their trajectories are impeccable. The lattices of the third class, C2, do well in general. However, their trajectories are lengthy and certainly not optimal. The class C2 is rather transitional from lattices behaving straightly to lattices behaving weirdly. The fourth class, C3, is constituted of lattices, which also burst in excitation when illuminated, however speci®c of their space-time excitation dynamic is not re¯ected in perfect locomotor activity. The present phenomenological ®nding on space-time dynamic of mobile excitable lattices is consistent with our previous results (see e.g. [2]) on vector-based rules of excitation. In our early paper, [2] we classi®ed lattices on their computational abilities. Do computational abilities imply good phototactic behavior? Everyone would be most interested in whether excitable lattices which play a role of massively parallel universal computers or at least specialized image processing devices show signs of strong phototactic behavior. There is no certain answer. Some lattices of the class C1 produce self-localized excitations. Therefore, they can implement universal collision based computing [7]. They also move straightforwardly toward the light source. However, signi®cant number of members of C1 do not do any computation (at least for known problems). Nevertheless they are good navigators. Inversely, the lattices of the class C3 can be employed in image processing and computational geometry [8]; however, these lattices are very poor navigators. Ideally, it would be impressive to verify our theoretical results with real life implementations. Unfortunately, at this stage we could not oer any arti®cially designed mobile arrays of actuators with cellular automata architecture. Fabrication of the arrays, possibly based on mono-molecular arrays, such as Scheibe aggregates [11,12], may be a task of the near future. Luckily, we have a living analog of arrays of cilia, which exhibit positive taxis. This is a phylum Ciliophora, or ciliates ± protists, bodies of which are covered with cilia (one could refer to Paramecium caudatum as a typical example) [21,26]. Each cilium, attached to a membrane of a protist, sweeps with a power stroke in the direction opposite to intended direction of organism movement. The cilia are usually arranged in rows; this arrangement forms a longitudinal axis of beating. Cilia beat coherently in waves and propel the individual forward. A physical integration of strokes of many cilia propel the protist in the direction opposite to the direction of beating.
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There are still limited data on coordination of cilia. The control may be implemented by either information transfer via sub-membrane network of microtubules or coordinated contractions of the membrane traveling along the protist's body. In any case, we can speculate about direct analogies between wave of cilia beating, caused by excitation waves of the lattice, and waves of co-beating of cilia in protists, that move from front to rear parts of the organism. The absence of any directional photoreceptors in ciliates [19] matches even closer our theoretical models. Thus, for example, it is proved in real life experiments [19,20] that only the frequency of directional changes together with frequency distribution of an angle of directional changes are aected by light; the speed is not aected at all. This was an intuitive assumption in our model of mobile excitable lattices. So, both excitable lattice and real protists are rather oriented by steering toward a direction of maximal light intensity [19]. Also this may be quite useful to interpret our models in terms of a theory of programmable force ®elds [13], which deal with small actuators arranged in two-dimensional array and coupled with arrays of sensors. The actuators execute a few simple actions in the reaction on sensorial inputs. A force ®eld is formed by the actuator array that surrounds an object put on the array and causes the object to move [13]. What types of object transportation can be achieved in cilia supplied excitable lattice? This may be a subject of future studies.
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