Excitable Delaunay

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Excitable Delaunaytriangulations

Andrew AdamatzkyUniversity of the West of England, Bristol, UK

Abstract

Purpose Delaunay triangulations provide a good alternative to existing approaches of modellingunstructured unconventional computers. Experimental research in novel and emerging computingparadigms and materials shows a great progress in designing laboratory prototypes of spatiallyextended computing devices. In these devices, computation is implemented by excitation waves andlocalisations in reaction-diffusion chemical media, geometrically constrained and compartmentalizedexcitable substrates, organic molecular assemblies, and gas-discharge systems. These unconventionalcomputing substrates can be formally represented by Delaunay triangulations with excitable nodes.Thus, it is important to uncover most common types of excitation dynamics on the Delaunaydiagrams. The purpose of this paper is to define excitable automata on Delaunay triangulation anddemonstrate how to control a space-time dynamics of excitation on the triangulation using absoluteand relative excitability thresholds.

Design/methodology/approach The paper modifies classical Greenberg-Hasting model totopology of Delaunay triangulations and considers not only a threshold of excitation but also a ratio ofexcited neighbours as an essential factor of nodes activation. Delaunay triangulations for variousdensities of nodes packaging are considered.

Findings The paper defines excitable automata on Delaunay triangulation and demonstrates howto control a space-time dynamics of excitation on the triangulation using absolute and relativeexcitability thresholds. The paper uncovers several interesting phenomena ranging from reaction ofexcitation waves by edge of triangulation to branching domains of activity guided by travellinglocalized excitations.

Originality/value The findings reported in the paper will contribute towards designs of novelcomputing substrates in non-crystalline structure. Also, automaton interpretation of activity dynamicson Delaunay triangulation can make a viable model of automaton-network approaches to design ofnano-computing devices.

Keywords Automation, Cybernetics, Modelling, Topology, Nanotechnology

Paper type Research paper

1. IntroductionGiven a finite set of planar points Delaunay triangulation is a planar proximity graphwhich subdivides the space onto triangles with nodes in the given set such that thecircumcircle of any triangle contains no points of the given set other than the trianglesvertices (Delaunay, B. Sur la sphere vide, Izvestia Akademii Nauk SSSR, 1934).Delaunay triangulation is a graph-theoretic dual of Voronoi (1907) diagrams.It represents connectivity of Voronoi cells. Voronoi diagram and its dual Delaunaytriangulation are widely used in studies related to filling a space with connectedstructural units. Voronoi diagram and Delaunay triangulation are used to approximate

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/0368-492X.htm

The work is part of the European Project 248992 funded under 7th FWP (Seventh FrameworkProgramme) FET Proactive 3: Bio-chemistry-based Information Technology CHEM-IT(ICT-2009.8.3). Many thanks to Julian Holley for commenting on the paper.

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arrangements of discs (Gervois et al., 1995), sphere packing (Lochmann et al., 2006;Filatovs, 1998; Luchnikov et al., 2002), to make structural analysis of liquids and gases(Anikeenko et al., 2004), and protein structure (Poupon, 2004), and to model dense gels(Zarzycki, 1992) and inter-atomic bonds (Hobbs, 1995).

We are interested in studying excitable Delaunay triangulation because they mayprovide a good alternative to existing approaches of modelling unstructuredunconventional computers (Teuscher and Adamatzky, 2005). Experimental research innovel and emerging computing paradigms and materials shows a great progress indesigning laboratory prototypes of spatially extended computing devices. In thesedevices, computation is implemented by excitation waves and localisations inreaction-diffusion chemical media (Adamatzky et al., 2005), geometrically constrainedand compartmentalized excitable substrates (Gorecki and Gorecka, 2006; Gorecka et al.,2009; De Lacy Costello et al., 2009; Kaminaga et al., 2006), organic molecular assemblies(Bandyopadhyay et al., 2010), and gas-discharge systems (Astrov Yu, 2010). Theseunconventional computing substrate can be formally represented by Delaunaytriangulations with excitable nodes. Thus, it is important to uncover most commontypes of excitation dynamics on the Delaunay diagrams.

Despite being a ubiquitous graph representation of wide range of naturalphenomena the Delaunay triangulation was not studied from automaton point of view.Will excitable Delaunay triangulation behave as a conventional excitable cellularautomata or there will be some unusual phenomena? We answer the question byslightly modifying classical Greenberg-Hasting model (Greenberg and Hastings, 1978)and considering not only a threshold of excitation but also a ratio of excited neighboursas an essential factor of nodes activation.

The paper is structured as follows. We introduce automata on triangulations andexcitation rules in Section 2. In Section 3, we discuss structural properties of automatatriangulations. Sections 4 and 5 present classification of space-time dynamics of

excitation for absolute (based on a number of excited neighbours) and relative(based on a ratio of excited neighbours) rules of excitation. Results are discussed inSection 6.

2. Delaunay automata and excitationsGiven a planar finite set V the Delaunay triangulation (Delaunay, B. Sur la sphere vide,Izvestia Akademii Nauk SSSR, 1934) DV kV; El is a graph subdividing the spaceonto triangles with vertices in V and edges in E where the circumcircle of any trianglecontains no points of V other than its vertices. Neighbours of a node v [ V are nodesfrom V connected with v by edges from E.

The set V is constructed as follows. We take a disc container of radius 480 and fill it

with up to 15,000 disc nodes. We assume that each disc node has radius 2.5, thus aminimal distance between any two nodes is 5. The Voronoi diagram and its dualtriangulation are appropriate representations of such identical sphere packing on 2Dsurface, where planar points of V represent centres of the spheres.

We define density f as a ratio of areas occupied by discs to area of the disccontainer. Examples of triangulations for densities f 0.0027, 0.027, 0.136, and 0.407,corresponding to number of nodes packed 100, 1,000, 5,000, and 15,000, are shown inFigure 1.

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The Delaunay automaton is defined in the following way. A node v [ V is a finite-statemachine. Every node updates its state in discrete time depending on states of itsneighbours. All nodes update their states simultaneously. Nodes can have differentnumber of neighbours, therefore, we better use totalistic node-state update function,where a node updates its state depending on just the numbers of different node statesin its neighbourhood.

Here, we are concerned only with modelling excitation on Delaunay triangulations.Thus, we assign three states resting (+), excited () and refractory (2) to nodes ofV.

Figure 1.Examples of

triangulations fordensities (a) f 0.0027;

(b) f 0.027,(c) f 0.136,(d) f 0.407,

(e) fragment oftriangulation (a) is

zoomed on

(a) (b)

(c) (d)

(e)

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We assume that a resting node excites depending on a number of excited neighbours.If a node is excited at time t, the node takes refractory state at time step t 1,independently on states of its neighbours. Transition from refractory to resting state isalso unconditional.

Let nv {u [ V : vu [ E} be node vs neighbourhood, vt

a state of node v attime step t, st(v) a number of excited neighbours ofv at step t, and d(v) be a degree, or anumber of neighbours jn(v)j, of node v. Then, the node-state transition functions can bedefined as follows:

. Absolute excitability:

v t1

; if stv [ S

2; if v t

+; if v t 2

8>>>:

1

Node v excites ifstv [ S, where S is a set of natural numbers. For example,S {2, 4} means a resting node excites if it has two or four excited neighbours.The rule includes threshold excitation stv $ u, u is a natural number. Forexample, u 2 means a resting node excites if it has at least two excitedneighbours, i.e. S {2, 3, 4, . . .}.

. Relative excitability:

v t1

; ifstv

dv. e

2; if v t

+; if v t 2

8>>>>>>>:

2

Node v excites if:

rtv stv

dv. e;

where 0 # e# 1. This condition brings more fairness in excitation process.Some nodes can have less neighbours than other nodes, thus measuringexcitation of neighbourhood just by number of excited neighbour would not befair. It is feasible to calculate a ratio rt(v) of excited neighbours st(v) to a totalnumber of neighbours d(v).

3. Structural properties of Delaunay automataPropagation of excitation in Delaunay automata depends on structural properties oftheir graphs. We found that in case of sparse distribution of disc nodes (Figure 1(a)),f 0.0027, majority of nodes have five then six and four neighbours each (Figure 2(a)).With increase of the density (Figure 1(b)-(d)), maximum of degree distribution is shiftedto six neighbours per node (with probability p(6) 0.45), five nodes (p(5) 0.27) andseven nodes (p(7) 0.22) (Figure 2(a)). The deviation of the distribution significantlydecreases with increase of the density (Figure 2(b)).

The degree distributions found experimentally conform to classical results onpacking of equal spheres which is between 5.5 for sparse packing and 6.4 for a close yet

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random packing (Bernal and Mason, 1960), and mean degree of 6 for randomly packedsphere with density 0.64 (Gotoh and Finney, 1974).

Let every node of a triangulation takes just two states: 0 (unoccupied) and 1(occupied). A resting node becomes occupied if it has at least hneighbours in state 1.

Initially just one (for h 1) or two (for h 2) nodes are assigned state 1. Dynamicsof occupation, measured in a ratio a of nodes occupied by time step t, is shown inFigure 3.

A speed of propagation of state 1 measured in a ratio of nodes occupied in step tincreases with decrease of occupation threshold h. If we measure speed in discretetime steps, we will see that it also decreases with increase of the density f. However, thedistance covered in any period of time remains comparable between triangulations withdifferent number of nodes:

Figure 2.Distribution of degree d

probabilities p(d ) (a) andstandard deviation sofdistribution p(d) (b) for

several densities fof discnodes

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

p(d)

d

(a)

0

0.01

0.02

0.03

s

0.04

0.05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14d

(b)

f= 0.027f= 0.027f= 0.136

f= 0.407

f= 0.027f= 0.027f= 0.136f= 0.407

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Finding 1. Threshold h of node occupation determines a shape of propagatingoccupation fronts.

For h 2 occupation wave front is convex, shaping into planar by the end ofpropagation (Figure 4(b)). In case of lower threshold of occupation, h 1, propagationwave near the edges of triangulation moves quicker then inside the triangulation core(Figure 4(a)). Thus, wave front becomes concave. The part of wave front travellingalong edges of triangulation reaches the side of triangulation opposite to the initialperturbation side quicker then the front propagating inside the triangulation. Thus,wave front becomes closed (Figure 4(a)). The domain surrounded by a nested group oftarget waves, in the western part of the triangulation in Figure 4(a), is the place wherethe occupation waves travelling from east collapse:

Figure 4.Time-lapsed contours ofactivation in triangularlattices with densityf 0.407 and activationthreshold (a) h 1 and(b) h 2

(a) (b)

Notes: Propagating wave front is converted to a contour every tenth step of simulation;

initially eastmost nodes are activated

Figure 3.Dynamics of occupation aof triangulations fromsingle-node perturbation;a is a ratio of occupiednodes at time step t to atotal number of nodes inthe triangulation

1

0.9

0.8

0.7

0.6

a 0.5

0.4

0.3

0.2

0.1

00 20 40 60 80 100

t

120 140 160 180 200

f= 0.136, h =2

f= 0.407, h =2

f= 0.136, h =1

f= 0.407, h =1

Note: Graph presents dynamics of occupation for densities f= 0.136 and f= 0.407,

and occupation thresholds h = 1 and h = 2

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Finding 2. Perturbation propagates faster along edges of triangulation whenthreshold of node perturbation is low.

An edge of triangulation is a set of nodes lying on segments of the convex hull of V.

Spatial structures of node neighbourhoods at the edge of triangulation may beresponsible for the particulars of propagations described above.Typically, neighbours of each node are distributed more or less equally around the

node while nodes belonging to the edge of triangulation have their neighbours locatedtowards inside of the triangulation or on the edge (Figure 5(a)). We can integrate spatialdistribution of neighbours of node v as a vector p vg from the node v to geometriccentre g of the node vs neighbourhood. Edge nodes of triangulation have usuallylonger vectors p (Figure 5(b)). This is why a perturbation propagates faster on or nearedges of triangulation for a low threshold of occupation (Figure 4(a)). Increase ofoccupation threshold is disadvantageous for edge nodes which are reflected in changedshape of the growing pattern (Figure 4(b)).

4. Absolute excitabilityIf a resting node v excites when it has at least one excited neighbour, rule (1) st(v) $ 1,classical excitation waves are observed (Figure 6(a) and (b)). By classical we meanthat excitation waves annihilate when they reach edges of triangulation, two wavesmerge when they collide one with another, a single-site perturbation initiates acircularly propagating excitation wave, and refractory state is a necessary componentin seeds of target-wave generators.

The excitable triangulation behaves less conventionally when nodes are selective intheir excitability. Consider the rule (1) S {1}: a resting node excites if it has exactlyone excited neighbour. Single-site excitation leads to formation of circular waves.The waves do merge when collide with each other and also they may form generatorsof target waves at the sites of their collision (Figure 6(c)):

Figure 5.Geometry of

neighbourhoods

(a) (b)

Notes: (a) A fragment of triangulation, density f= 0.136,

with visible neighbourhoods of edge nodes, edge of

triangulation is on the right; (b) vectors from nodes to their

geometric centres, only vectors which length exceeding

eight units are shown; density is f= 0.136

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Finding 3. Let a resting node of Delaunay triangulation excites if exactly oneneighbour is excited, then a generator of target waves can be producedby a single-site excitation.

For st

(v) $ 2 no excitation persists. However, when resting node excites if exactly twoor three of its neighbours are excited, we obtain a quasi-chaotic excitation dynamics(Figure 6(d)). This happens when nodes follow state update rules (1) for S 2 andS {2, 3}.

A delay in recovery from refractory states modifies space-time dynamics ofexcitation. Assume that if a node took refractory state then the node stays in therefractory state for d time steps. In automata governed by rule (1) with excitability

st(v) $ 1, initial random disturbances nuclei of excited and refractory states will

Figure 6.Examples of excitationdynamics for rule (1)stv $ 1 (a)-(c) and rule(1) s

t(v) 1 (d)

(a) (b)

(c) (d)

Notes: (a, b) Waves generated by singular excitations merge when collide; (b) random

initial configuration (at the beginning of simulation a node got excited state with

probability 0.1, refractory state with probability 0.1, and resting state with probability

0.8) develops into a configuration of several target-wave generators; (c) configuration

of excitation developed in 300 time steps from initial configuration where all but one

nodes are resting; density of disc nodes in triangulation is f= 0.407

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not lead to formation of wave generators (as in rule st(v) $ 1 without node-staterecovery delay). However, a phenomenon of excitation backfiring is still observed.Travelling excitation wave can backfire with localized excitations. These localizedexcitations pass through the waves refractory tail and initiate generation of new wave

fronts (Figure 7):

Finding 4. Delaunay excitable automata governed by rules of absolute excitabilityexhibit the following phenomena:

. Threshold activation rules causes formation of classical excitation wave fronts.

. Rules relying on exact number of excited neighbours show formation of targetwave generators during collision between ordinary circular waves.

. When recovery of a node from refractory state to resting state is delayedpropagating wave fronts backfire with localized excitations, which causeformation of target-wave generators.

Figure 7.Example of wave

generators formed due tobackfiring of excitation

from a single propagatingwave front

(a) t = 35 (b) t = 52

(c) t = 65 (d) t = 148

Notes: Rule st(u) 1 with delayed (by ten-time steps) recovery from refractory state;

density of disc nodes in triangulation is f= 0.407

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5. Relative excitationWe found that we can initiate a persistent excitation patterns for 0 # e, 0.25. Fore$ 0.25 any initially invoked excitation quickly ceases. Automata excited by rule (2)with e 0 exhibit persistent global oscillations because every node autonomously

follows the cycle +!!2! . For excitability range 0 , e, 0.09 Delaunayautomata exhibit classical excitation waves. Single excitation generates single circularwave, clusters of excited and refractory states may become generators of target wave.When a propagating wave front hits edge of triangulation, the wave disappears. Twocolliding wave fronts merge or annihilate.

When threshold of excitation increases to e 0.09 edges of the triangulation maybecome reflective for excitation waves. If an excitation wave front hits an edge of thetriangulation the front reverses it velocity vector and again propagates inside thetriangulation. An example is shown in Figure 8. Initially graph is in resting state. Weexcited one node. A circular wave of excitation propagates outwards the initialperturbation (Figure 8(a)). When the wave front reaches edge of the triangulation itstarts to disappear and almost annihilate. However, in some part of triangulation edgedomains of centrifugal-wave fragments evoke centripetal-wave fragments (Figure 8(b)and (c)). Fragments of the centripetal wave closer to edge of triangulation propagatequicker then those inside the triangulation. Therefore, the wave encircles thetriangulation (Figure 8(d) and (e)) and collapses inside the triangulation (Figure 8(f)).

The reflection of waves can be observed forvalues 0.09 , e, 0.11.Further increase ofexcitation threshold brings up the phenomenon of excitation backfiring again.

Triangulations, which nodes are excited if ratio e is at least 0.11, exhibit backfiring generators of target waves are formed behind the wave front initiated by a single

Figure 8.Dynamics of excitabletriangulation for rule (2)e 0.091

(a) t = 20 (b) t =30 (c) t = 41

(d) t = 67 (e) t = 100 (f) t = 132

Notes: Initially, just a single node is excited; density of disc nodes in triangulation is

f= 0.407

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excitation (Figure 9). Owing to inhomogeneous structures of node neighbourhoods, thewave front (Figure 9(a)) breaks up (Figure 9(b)). A gap is formed and some part ofthe front folds backward. A spiral wave or waves are formed. They give rise to asuccession of target waves (Figure 9(c)-(e)). Eventually, original wave front disappears

at the edge of triangulation but the triangulation remains filled with sources of targetwaves (Figure 9(f)).

With e increasing from 0.11 to 0.17 a number of wave generators forming behindpropagating wave front increases considerably. Thus, in networks excited by rule (2)with e. 0.14, a generator is formed almost immediately after single-excitation wavestarts its propagation. The earlier in time a generator is born the more likely thegenerator will dominate the triangulation. Therefore, in many cases, we can observe

just few generators close to the site of original excitation (Figure 10).

Figure 9.Example of backfiring of

excitation wave frontswhich leads to formation

of generators of target

waves in excitabletriangulation, density

f 0.407, for rule (2) withe 0.11

(a) t = 18 (b) t = 28 (c) t = 37

(d) t = 45 (e) t = 53 (f) t = 83

Figure 10.Example of generators

formed close to a site oforiginal perturbation

Notes: Automaton is excited by rule (2) with e= 0.150; density of

disc nodes in triangulation is f= 0.407

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With e exceeding 0.17 the automaton approaches regime of sub-excitability. Single-siteexcitation no longer leads to formation of a circular wave. However, we observe travellingand stationary-localized excitations, distant analogous to wave fragments in sub-excitableBelousov-Zhabotinsky medium (De Lacy Costello and Adamatzky, 2005). Initial local

perturbations lead to propagating-localized excitations. The excitations later can form alocalized, and not changing it is outer shape, domain of activity (Figure 11) or a slowlygrowing domain, which may eventually occupy the whole triangulation (Figure 12).Usually, two or three wave fragments are formed near the site of initial activation ofresting triangulation. These wave fragments travel outward the perturbation loci.Sometime after their initiation, the wave-fragment backfires few excitedmicro-localizations which may form generators of wave fragments. Thus, the domain ofexcitation is born. Sometimes, generators of wave fragments emerge due to folding of thewings of a wave-segment backwards. Folded parts of wave fragment interact with eachother and thus they produce the generator.

Figure 11.Example of excitationdynamics for e 0.17

(a) t = 10

(c) t = 40 (d) t = 142

(b) t = 16

Notes: A local perturbation leads to localized excitations and slowly growing domains;

the domain shown in (d) has stationary boundaries, however, configuration of excited

and refractory states inside the domain is changing; density of disc nodes in

triangulation is f= 0.407

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The growing domains guided by mobile localisations are typical for excitability values0.17 , e, 0.2. When e exceed 0.2, the domains seize propagating. The excitation canstill persist but in a minuscule quantity. A random initial configuration, where everynode gets one of three states equiprobably, is either transformed to a totally restingstate or to a resting configuration with one or two tiny oscillators. Examples of mostcommon oscillators are shown in Figure 13.

What are degrees of nodes occupied by non-resting states of the oscillators?

We stimulated triangulations with random configurations of excited and refractorystates (each state is assigned with probability 1/3), waited till all activity but minimaloscillator ceases and recorded degrees of nodes occupied by the oscillator states.We found that nodes occupied by excited or refractory states in the minimal oscillator(Figure 13(c)) have 4, 6, 7 or 8 neighbours. In larger, oscillators nodes have degrees 4, 5,6, 7, 8 and 4, 5, 6, 7, 8, 9:

Proposition 1. Diversity of node degrees is a necessary requirement for existenceof an oscillator in a sub-excitable triangulation.

The oscillators survive for 0.2 # e, 0.25. For e$ 0.25 no excitation persist at all:

Finding 5. Delaunay excitable automata governed by rules of relative excitability

with dimensionless threshold of excitation e exhibit the followingphenomena.

The finding is shown in Figure 14. Is there any structural meaning of e boundariesbetween different classes of excitable triangulations? Value e 0.09 corresponds to asituation when a node with 11 neighbours has at least one excited neighbour. Nodes withdegree 11 are rare species in Delaunay triangulations (Figure 2). However, when suchnodes are deprived, i.e. for e. 0.09, from a chance to be excited at all by a minimalperturbation waves of excitation start to be reflected by edges of triangulation.

Figure 12.In triangulation, density of

disc nodes is f 0.407,excited by rule (2) with

e 0.17 a localperturbation leads to

localized excitations andslowly propagating

domains, which mayoccupy significant number

of nodes

(a) t = 7 (b) t = 23 (c) t = 33

(d) t = 53 (e) t = 119 (f) t = 278

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Excitation wave-fronts backfire when we make nodes with degree 9 non-excitable by

raising excitability to over 0.11. This is because e 0.11 describes a situation of a nodewith nine neighbours which has just one neighbour excited.

Localizations emerge in triangulations when e$ 0.17. In principle, this value of e

corresponds to two situations: a node with six neighbours has one excited neighbour and a

node with 12 neighbours has two excited neighbours. The latter situation is not important

because nodes with 12 neighbours are extremely rare. Thus, we can propose that

excitation becomes localized when six-neighbour nodes are activated by at least two

neighbours. Two excited neighbours are a minimal requirement for existence oflocalizations in orthogonal automaton networks with eight-cell neighbourhoods

(Adamatzky, 2001). In a regular network, or cellular automaton, a travelling

localization preserves its shape and velocity vector as long as necessary Table I.

Delaunay triangulations are irregular therefore localized wave fragments do not

survive for a long time. These travelling localizations frequently change directions of

their motion. Soon after its birth, a localization either perishes or expands, backfires

and forms a slowly growing cluster of activity (Figure 12). Any excitation ceases to

Figure 14.Visualisation of basictypes of excitation activity

parameterised by relativeexcitability threshold e

0 0.112 0.25

e

Figure 13.Examples of oscillators

(a)

(b)

(c)

Notes: (a) and (b) two oscillators for excitability e= 0.2; (c) minimal oscillator e= 0.21;

each sub-figure presents three consecutive states of an oscillator; density of disc nodes

in triangulation is f= 0.407

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sustain when e exceeds 0.25, which corresponds to one of four, two of eight, and threeof 12 excited neighbours.

Does density of disc nodes affect structure of behavioural space?For relative excitability 0 , e, 0.09 decrease of density f increases chances of

wave generators to be formed from a simple-propagating wave front. This possiblyhappens due to increased in homogeneity of node degrees and increasing role of eachparticular node in excitation propagation. In case of 0.09 # e, 0.11, decrease of fincreases opportunities for wave generators to form when a wave collides to edge oftriangulation.

For excitability 0.11 # e, 0.17 time of backfiring and formation of generatorsbehind wave fronts, is proportional to density f, higher the fthe longer it takes for asingular wave front to backfire. And finally, chances that an excitation domaingrowing from a single perturbation, in triangulations with excitability 0.17 # e, 0.2,occupies the whole triangulation are inversely proportional to density of disc nodes inthe triangulation.

6. DiscussionWe defined excitable automata on Delaunay triangulation and demonstrated how tocontrol a space-time dynamics of excitation on the triangulation using absolute andrelative excitability thresholds. We uncovered several interesting phenomena rangingfrom reflection of excitation waves by edge of triangulation to branching domains ofactivity guided by travelling-localized excitations.

Growing domains of localized excitation may be analogous to patterns of excitation inBelousov-Zhabotinsky reaction in micro-emulsion (Kaminaga et al., 2005). Our findingson reflection of waves by edge of triangulation support previously published results onactive waves reflection in spatially inhomogeneous non-linear media (Matsuoka et al.,

1998) and reflection of reaction-diffusion waves by no-flux boundary due toconsumption of reactant between a wave and a boundary (Mikhailov and Showalter,2006). Further studies of wave speed up along edge of triangulation could be enhancedby additional techniques of characterising topological and dynamical properties ofcomplex networks, e.g. a diversity entropy proposed in (Viana et al., 2010).

We believe our findings will contribute towards designs of novel-computingsubstrates in non-crystalline structure. Also, automaton interpretation of activitydynamics on Delaunay triangulation can make a viable model of automaton-networkapproaches to design of nano-computing devices (Isokawa et al., 2007).

e Phenomenon

0 Global oscillations0 , e, 0.09 Classical excitation waves0.09 # e, 0.11 Waves are reflected from edges0.11 # e, 0.17 Waves backfire with excitation0.17 # e, 0.2 Localized wave-fragments lead growing tips of branching domains0.2 # e, 0.25 Only tiny oscillating domains are present0.25 # e No excitation persists Table I.

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Corresponding authorAndrew Adamatzky can be contacted at: [email protected]

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