June 24, 2013 12:14 WSPC/S0219-5259 169-ACS 1250034

Advances in Complex SystemsVol. 16, Nos. 2 & 3 (2013) 1250034 (28 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0219525912500348

BIO-DEVELOPMENT OF MOTORWAY NETWORKIN THE NETHERLANDS: A SLIME

MOULD APPROACH

ANDREW ADAMATZKY

University of the West of England,Bristol BS16 1QY, United Kingdom

MICHAEL LEES

Nanyang Technological University,Singapore

PETER SLOOT

University of Amsterdam,Amsterdam, The Netherlands

Received 31 October 2011Revised 2 February 2012Published 31 March 2012

Plasmodium of a cellular slime mould Physarum polycephalum is a very large eukaryoticmicrobe visible to the unaided eye. During its foraging behavior the plasmodium spanssources of nutrients with a network of protoplasmic tubes. In this paper we attempt toaddress the following question: Is slime mould capable of computing transport networks?By assuming the sources of nutrients are cities and protoplasmic tubes connecting thesources are motorways, how well does the plasmodium approximate existing motorwaynetworks? We take the Netherlands as a case study for bio-development of motorways,while it has the most dense motorway network in Europe, current demand is rapidlyapproaching the upper limits of existing capacity. We represent twenty major cities withoat flakes, place plasmodium in Amsterdam and record how the plasmodium spreadsbetween oat flakes via the protoplasmic tubes. First we analyze slime-mould-built andman-built transport networks in a framework of proximity graphs to investigate if theslime mould is capable of computing existing networks. We then go on to investigate ifthe slime mould is able calculate or adapt the network through imitating restructuring

of the transport network as a response to potential localized flooding of the Netherlands.

Keywords: Bio-inspired computing; Physarum polycephalum; pattern formation; TheNetherlands motorways; road planning.

1. Introduction

The approximation or computation of shortest path transportation networks hasdrawn significant attention from the field of Unconventional Computing Sciences.

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A. Adamatzky, M. Lees and P. Sloot

Nature-inspired computing paradigms and experimental implementations have beensuccessfully applied to calculation of a minimal-distance path between two givenpoints in a space or a road network. Computational models of ant-based opti-mization have been shown to be an effective way of developing novel approachestowards load-balancing of telecommunications [12], which indeed involves dynamicaldesign of transport links for packets. Other works include a shortest-path problemsolved in experimental reaction-diffusion chemical systems [1], gas-discharge ana-log systems [24], spatially extended crystallization systems [6], formation of funginetworks [17] and plasmodium of Physarum polycephalum [20].

Amongst all experimental prototypes of path-computing devices slime mouldPhysarum polycephalum is perhaps the most cost efficient biological substrate avail-able, coupled with fact that it is both easy to cultivate and observe, it makes anexcellent computational substrate. These are the main reasons we adopt it for thiswork.

A cellular slime mould Physarum polycephalum has quite a sophisticated lifecycle [27], which includes various stages such as: fruit bodied, spores, single-cellamoebas, and plasmodium. Plasmodium is a vegetative stage of Physarum poly-cephalum, it is a syncytium, a single cell, where many nuclei share the same cyto-plasm. The plasmodium consumes microscopic particles, and during its foragingbehavior the plasmodium spans scattered sources of nutrients with a network ofprotoplasmic tubes.

During its development, growth and colonization of surrounding environmentslime mould of P. polycephalum shows articulated traits of guided self-organization.The plasmodium spreads its protoplasmic network in the environment and thusmaximizes receptivity of its distributed sensorial inputs. Exact structure of the plas-modial networks depends on physical and nutritional characteristics of the growthsubstrate: the plasmodium propagates as omni-directional pattern on a nutrient-agar and grows as a tree-like structure on non-nutrient agar or non-humid sub-strates, e.g., aluminium foil or glass [9]. Growth of the slime mould is guided bygradients of attractants and repellents, including chemical factors, temperature andillumination. These gradients represent an “information field” which drives devel-opment of the slime mould. The protoplasmic network is usually optimized to coverall sources of food while still managing to guarantee robust and quick distributionof nutrients in the plasmodium body. Plasmodium’s foraging behavior can be inter-preted as computation, with data represented by spatial distribution of attractantsand repellents, and results represented by the structure of protoplasmic networks [9].Plasmodium is capable of solving computational problems with natural parallelism,namely shortest path [20] and hierarchies of planar proximity graphs [4], computa-tion of plane tessellations [26], execution of basic logical computing schemes [8, 29],and natural implementation of spatial logic and process algebra [25]. For furtherexamples see the overview of Physarum-based computers in [9].

In previous work [3] we have evaluated the road-modeling potential ofP. polycephalum, however, previous results were inconclusive. A step forward

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Bio-Development of Motorway Networks in The Netherlands

biological-approximation, or evaluation, of man-made road networks was done inour previous work on approximation of United Kingdom motorways and Mexi-can Federal highways by plasmodium of Physarum polycephalum [7, 10]. In bothcases it was shown that transportation links constructed by plasmodium suffi-ciently determines man-made motorways, with some pernicious differences. Com-paring results for United Kingdom and Mexico we found that shape of a countryand spatial configuration of urban areas or cities sufficiently determines behav-ior of the plasmodium. More experiments are necessary to provide generaliza-tion, in order to develop a theory of slime-mould based road planning and urbandevelopment.

In this paper we hope to move towards a more general understanding of slimemoulds capability to compute road networks by investigating the roads in theNetherlands. The Netherlands presents an excellent case study for the two followingreasons.

First, the Netherlands is amongst top countries with highest population densitybecause only 12% of territory is allocated to build-up areas [11]. The country has thehighest density motorway network in Europe. Moreover, the demand on the systemis at levels which are reaching current limits, with a total length of 132,397 kmand usage of 140 × 109 people per km per year.a Such high-occupancy may posea need for urgent expansion of the transport networks and a better understandingof the limitations to that growth. As analyzed in [11], the Dutch national spatialstrategy put a stress on developing transport networks amongst the major cities.Six national “Networks of Cities” might lead to increased congestion and increasedtravel times. Sprawling of urban areas along major transport arteries is a commonfeature, which also could be imitated by slime mould due to its active branchingbehavior.

Second, The Netherlands is also at risk of significant floodingb because over40% of the country’s land lies below sea level. The largest flooding-disaster in 1953costed almost two thousands human lives [14], and only by chance larger cities asAmsterdam, Rotterdam, and The Hague were not flooded at that time [32]. Despitea wide range of technologically advanced systems the Netherlands are not immunefrom mass-scale floods. The chance flood defense systems are small

“However, on a human time scale it means a chance of at least 1% of thepopulation having to face flooding once in its lifetime a tiny percentage,yes, but not negligible. More importantly, if things do go wrong, in manyplaces they are apt to go badly wrong. Millions of people live several metersbelow sea level. In the worst scenario, parts of these areas will be belowseveral meters of water within a few hours, and many casualties will result.Economic, cultural, and societal damage will be unimaginable.” [32]

awww.autosnelwegen.nlbhttp://urbanflood.eu/default.aspx

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A. Adamatzky, M. Lees and P. Sloot

This is why it is important to imitate a worse-case scenario when substantial areaof the Netherlands is flooded and mass-migration of those who manage to stay alivestarts. We imitate this mass-migration of slime mould.

The remainder of this paper is structured as follows. We delineate the experi-mental method and setup in Sec. 2. Section 3 presents the principal experimentalresults, which are then analyzed in a framework of proximity graphs in Sec. 4.Restructuring of Physarum-approximated transport links for the case of partialflooding of the Netherlands is described in Sec. 5. The paper then concludes witha summary of the work and ideas for further studies in Sec. 6.

2. Methods

All experiments are conducted with plasmodium of P. polycephalum that is cul-tivated in a plastic container. The plasmodium are first placed on paper kitchentowels, sprinkled with still water and fed with oat flakes.c The experiments areconducted in 120 × 120 mm polystyrene square Petri dishes with rounded corners.The plasmodium will eventually grow on Agar plates, which are cut into the shapeof the Netherlands. The Agar plates are formed using 2% agar gel (Select agar,Sigma Aldrich).

All experiments consider the twenty one most populous urban areas in theNetherlands [Fig. 1(a)]:

(a) Leeuwarden(b) Groningen(c) Den Helder(d) Lelystad(e) Zwolle(f) Haarlem(g) Amsterdam(h) Utrecht(i) Amersfoort(j) Apeldoorn(k) Enschede(l) Den Haag

(m) Rotterdam(n) Dordrecht(o) Nijmegen(p) Hertogenbosch(q) Breda(r) Tilburg(s) Middelburg

cAsda’s Smart Price Porridge Oats.

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Bio-Development of Motorway Networks in The Netherlands

(a) (b)

(c) (d)

Fig. 1. Experimental basics. (a) Outline map of the Netherlands with twenty one sources ofnutrients indicated. (b)–(d) Snapshots typical setups: urban areas are represented by oat flakes,plasmodium is inoculated in Amsterdam, the plasmodium spans oat flakes by protoplasmic trans-port network.

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(t) Eindhoven(u) Maastricht.

Further we refer to the urban regions as U. The regions in U are projected ontothe gel in the following manner: oat flakes are placed in the positions of each region[Fig. 1(b)]. At the beginning of each experiment a piece of plasmodium, usuallyalready attached to an oat flake, is placed in Amsterdam [region 7 in Fig. 1(a)].

The Petri dishes with plasmodium are kept in darkness, at a temperature ofbetween 22 and 25◦C, except for short periods of observation and image record-ing. Periodically the dishes are scanned using an Epson Perfection 4490 scanner.Scanned images of dishes are enhanced to increase readability of the image, thisis done by increasing saturation and contrast (saturation is increased to 55 andcontrast to 40). A total of 62 experiments were conducted. To ease understandingof experimental images we provide complementary binary version of each image,where appropriate.

3. Transport Links Via Foraging

In the following we present experimental results which show that the plasmodiumis capable of computing, or calculating, the transport links between each of thetwenty one most populous areas of the Netherlands.

In a laboratory experiment, illustrated in Fig. 2, the following chain of eventsunfolds (dynamics of colonization is schematically represented in Fig. 3). An oatflake colonized by plasmodium was placed on top of the oat flake representingAmsterdam. In 12 hours the plasmodium follows gradients of chemoattractants,

(a) t = 12 h (b) t = 34 h

Fig. 2. (Color online) Illustrative example of plasmodium development on configuration of cities

represented by oat flakes: (a)–(c) Scanned image of experimental Petri dish. Time elapsed frominoculation is shown in the sub-figure captions, (d)–(f) binary images, Θ = (100, 100, 100).

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Bio-Development of Motorway Networks in The Netherlands

(c) t = 57 h (d) t = 12 h

(e) t = 34 h (f) t = 57 h

Fig. 2. (Continued )

links Amsterdam with Haarlem, and propagates towards Utrecht and Amersfoort,spreading in all directions except north-west [Figs. 2(a) and 2(d)]. After 34 hours theplasmodium colonizes Leeuwarden and Groningen. It develops clearly visible proto-plasmic tubes, which represent a transport link Amersfoort – Lelystad – Leeuwarden– Groningen [Figs. 2(b) and 2(e)]. In the same time interval the plasmodium col-onizes Apeldoorn and start propagations towards Zwolle and Enschede [Figs. 2(b)and 2(e)].

After a total of 57 hours the plasmodium connects Apeldoorn with Enschede andZwolle by protoplasmic tubes and colonized south-west part of the country. Name-lym, the plasmodium links Haarlem and The Hague and builds a route from TheHague to Middelburg and a link Hague – Rotterdam – Dordrecht – Breda – Tilburg– Hertogenbosch [Figs. 2(c) and 2(f), Fig. 3(a), green dashed lines]. At the sametime the plasmodium forms a protoplasmic tube directly connecting Amsterdam

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(a) (b)

Fig. 3. (Color online) Diagram of colonization dynamics derived from experiments Figs. 2(a) and4(b): Links developed in 12 hours after inoculation are shown by red solid arrows, in 34 hours byblue dotted arrows, in 57 hours by green dashed lines, in 80 hours by dash-dotted lines. Largemesh-patterned arrows indicate migration of plasmodium outside the country.

Den Helder, and The Hague with Hertogenbosch, and develops the links Hertogen-bosch – Nijmegen and Tilburg – Hertogenbosch – Eindhoven – Maastricht [Figs. 2(c)and 2(f)].

We observe that the dynamics of colonization is non-uniform [Fig. 3(a)]. Theplasmodium does not spread or diffuse in all directions simultaneously but rathercolonizes north-north-west part of the country first and only then explores south-south-west. This may be due to the fact that centers of activity (biochemical oscilla-tors) form during propagation, and the contractive waves evoked by the oscillatorsthat force the protoplasm to move towards the oscillators. Therefore, if an oscillatoris formed in the north part of plasmodium, then propagation in all other directionswould be suppressed.

The plasmodium of Physarum polycephalum rarely repeats itself in experimen-tal trials. The overall or average pattern, as we will discuss further in the paper,may be the same but a myriad of variations are possible in the course of plasmod-ium’s spatial development. Outperforming (spreading out of the dedicated area)and under-performing (not colonizing the whole area) are typical examples of thevarieties in plasmodium behavior. These two examples are illustrated and discussedbelow.

In a substantial number of laboratory experiments, the plasmodium did not stopits foraging activity even when all sources of nutrients were occupied and the whole

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Bio-Development of Motorway Networks in The Netherlands

agar plate was explored. As shown in Fig. 4 a vigorous plasmodium can spread oversurrounding Petri dishes, trying to settle on bare plastic.

In this experiments plasmodium starts its colonization in Amsterdam as before.It’s colonization is more aggressive in this case and it colonizes Haarlem, Den Haag,Rotterdam, Dordrecht, Breda, Tilburg, Hertogenbosch, Eindhoven within the firsttwelve hours. A pronounced protoplasm transport link is established connecting

(a) t = 12 h (b) t = 34 h

(c) t = 57 h (d) t = 80 h

Fig. 4. Plasmodium spreads beyond “dedicated” experimental domain: (a)–(d) Scanned image

of experimental Petri dish. Time elapsed from inoculation is shown in the sub-figure captions.(e)–(h) Binary images, Θ = (100, 100, 100).

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(e) t = 12 h (f) t = 34 h

(g) t = 57 h (h) t = 80 h

Fig. 4. (Continued )

these cities in a chain (first 12 hours from the moment of inoculation, Figs. 4(a)and 4(e)). 34 hours after inoculation the plasmodium sprawls from Hertogenbosch toUtrecht, Amersfoort and Apeldoorn, and then builds a transport link Amersfoort-Lelystad-Zwolle-Enschede [Figs. 4(b) and 4(f)].

Protoplasmic tubes connecting Haarlem, Amsterdam, Lelystad with Den Helderare grown simultaneously after 57 hours of the experiment. By the same time plas-modium also connects Nijmegen with Apeldoorn Figs. 4(c) and 4(g). Protoplas-mic transport links Den Helder – Leeuwarden – Groningen, Rotterdam – Middel-burg and Eindhoven – Maastricht are developed by the 80th hour of plasmodium’s

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foraging activity [Figs. 4(d) and 4(h)]. A Schematic illustration of the colonizationdynamics is shown in Fig. 3(b).

Plasmodium starts to show over-performance after 57 hours of the experiment. Itsprawls from Den Helder north-westward and from Enschede south-eastward ontobare plastic of the experimental container Figs. 4(c) and 4(g). The plasmodiumdoes not propagate on the plastic long enough and retracts in few hours [this canbe seen in Figs. 4(d) and 4(h)]. Another sprawling takes place by the 80th hour ofexperimentation, when plasmodium propagates westward of Middelburg and south-eastward of Maasrticht Figs. 4(d) and 4(h). See also diagrams of sprawling outsidethe county in Fig. 3(b). Also notice how the plasmodium dynamically changes itsforaging strategy (Fig. 4). It first attempts to colonize cities in north-east part ofthe country but then abandons the attempt and move to north-east later via theIJsselmeer lake.

In some experiments the plasmodium never manages to span all cities, and failsto colonize some oat flakes. An example is shown in (Fig. 5), 65 hours after inoc-ulation the plasmodium colonizes the majority of the Netherlands and establishesa network of protoplasmic tubes over most of the cities represented by oat flakes[Figs. 5(c) and 5(f)]. Later it goes into a hibernation stage and forms a sclerotium.However, at no moment of its development does the plasmodium even approachMiddelburg. Such situations are rather atypical and did not happen often in ourexperiments.

As illustrated above, plasmodium is quite an unpredictable creature and thepatterns formed by its protoplasmic networks in any two experiments rarely matcheach other exactly. Thus we generalize results of our experiments by constructing

(a) t = 22 h (b) t = 43 h

Fig. 5. Plasmodium does not always span all cities (sources of food): (a)–(d) Scanned image

of experimental Petri dish. Time elapsed from inoculation is shown in the sub-figure captions.(e)–(h) Binary images, Θ = (100, 100, 100).

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(c) t = 65 h (d) t = 22 h

(e) t = 43 h (f) t = 65 h

(g) θ = 0 (h) θ = 162

(i) θ = 262

(j) θ = 362

Fig. 5. (Continued ) Configurations of threshold Physarum-graph P(θ) for θ = 0, 162

, . . . , 1562

.Thickness of an edge is proportional to the edge’s weight.

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(k) θ = 462

(l) θ = 562

(m) θ = 662

(n) θ = 762

(o) θ = 862

(p) θ = 962

(q) θ = 1062

(r) θ = 1162

(s) θ = 1262

(t) θ = 1362

(u) θ = 1462

(v) θ = 1562

Fig. 5. (Continued )

a probabilistic Physarum graph. A Physarum graph is a tuple P = 〈U,E, w〉,where U is a set of 21 cities, E is a set edges, and w : E → [0, 1] is a probability-weights of edges from E. For every two cities a and b from U there is an edgeconnected a and b if a plasmodium’s protoplasmic link is recorded at least in one of k

experiments, and the edge (ab) has a probability calculated as a ratio of experimentswhere protoplasmic link (ab) occurred to the total number of experiments k. Wedo not take into account exact configuration of the protoplasmic tubes but merelytheir existence, e.g., protoplasmic tubes linking Eindhoven with Maastricht alwayspositioned inside the Netherlands territory but corresponding edge in Physarumgraph represents the tubes by straight line. We also consider threshold θ ∈ [0, 1]Physarum graphs P(θ), defined as follows: for a, b ∈ U, (ab) ∈ E if w(ab) > θ. Theparameter θ is a ratio of occurrences of an edge in all laboratory experiments tothe total number of experiments.

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Threshold Physarum-graphs extracted from 62 laboratory experiments areshown in Figs. 5 and 6. The graph becomes planar only for θ = 9

62

[Fig. 5(p)], i.e., when edges occurred in over 15% of the experiments. We cantherefore infer that the Physarum graph is planar. However, with acquiring

(a) θ = 1662

(b) θ = 1762

(c) θ = 1962

(d) θ = 2162

(e) θ = 2262

(f) θ = 2362

(g) θ = 2562

(h) θ = 2662

(i) θ = 2762

(j) θ = 3062

(k) θ = 3162

(l) θ = 3462

(m) θ = 3562

(n) θ = 3762

Fig. 6. Configurations of threshold Physarum-graph P(θ) for θ = 1662

, 1762

, . . . , 3762

. Thickness ofan edge is proportional to the edge’s weight.

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planarity the graph becomes disconnected: one node, Den Helder city, becomesisolated.

The Physarum-graphs become acyclic for θ = 2662 [Fig. 6(h)], i.e., when its edges

appear as protoplasmic tubes in over 41% of the experiments. When the graphbecomes acyclic it is split into a set of isolated nodes: Den Helder, Leeuwarden,Haarlem, Amsterdam, Utrecht, Amersfoort, Enschede, Middelburg, and two addi-tional components. One component is a chain of three cities: Lelystad, Zwolle andGroningen. The second component is a tree routed in Tilburg. The tree has threelinear branches:

• Tilburg – Breda – Dordrecht – Rotterdam – Den Haag• Tilburg – Hertogenbosch – Nijmegen – Apeldoorn• Tilburg – Eindhoven – Maastricht.

This tree is a characteristic feature of the Physarum-graph and it appears in over60% of experiments. The tree is “destroyed” when θ ≥ 30

62 [Fig. 6(j)], then onlychains remain, which give away isolated nodes with further increase of θ. Somechains are more stable than others. Thus, the chain Breda – Dordrecht – Rotter-dam – Den Haag appears in over 54% of experiments [Fig. 6(l)]. While the chainDordrecht – Rotterdam – Den Haag appears in almost 60% of experiments.

The experiments have now provided a reasonably consistent set of connectionsbetween the various urban centers in the Netherlands. The next question is to assesshow well these Physarum graphs approximate the Netherlands motorway network.A graph H of Dutch motorways is Fig. 7(a). We construct the motorway graph Has follows. Let U be a set of urban regions/cities, for any two regions a and b fromU, the nodes a and b are connected by an edge (ab) if there is a motorway startingin vicinity of a and passing in vicinity of b and not passing in vicinity of any otherurban area c ∈ U. If there is a branching motorways, which e.g., starts in a goes inthe direction of b and c and at some point branches towards b and c, we then addtwo separate edges (ab) and (ac) to the graph H.

The intersection P(θ) ∩ H of Physarum and motorways graphs is shown inFigs. 7(b)–7(d) for θ = 0, 8

16 and 1516 . A relaxed probabilistic Physarum graph P(0),

where an edge appears in the graph if it is recorded in at least one experiment,matches the motorway graph H almost perfectly. Just three edges of H are notpresented in P(0) ∩ H:

• (Amsterdam, Der Helden),• (Zwolle, Apeldoorn),• (Rotterdam, Dordrecht) (Fig. 7(b)).

θ = 816 is the highest value for which Physarum graph P(θ) remains connected

(Fig. 5). Graph P( 862 ) ∩ H loses few more edges (presented in P( 0

62 ) ∩ H):

• (Den Helder, Lelystad),• (Den Helder, Haarlem),

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(a) H (b) P(0)T

H

(c) P( 862

)T

H (d) P( 1562

)T

H

Fig. 7. Graph H of man-made motorway network is shown in (a). Intersection P(θ)T

H, θ = 1,816

and 1516

, of Physarum P and motorways H graphs is shown in (b)–(d).

• (Leeuwarden, Lelystad),• (Utrecht, Dordrecht),• (Utrecht, Nijmegen),• (Breda, Enschede),• (Utrecht, Enschede) [Fig. 7(c)].

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As soon as θ reaches value 1516 the graph P(θ) ∩ H becomes separated on isolated

node Den Helder, and there components:

• chain Enschede – Den Haag – Rotterdam,• cycle Nijmegen – Breda – Middelburg with branches Breda – Hertogenbosch

– Tilburg, Middelburg – Eindhoven and Nijmegen – Dordrecht – Apeldoorn –Amersfoort,

• cycle Leeuwarden – Groningen – Zwolle with branches Zwolle – Enschede andZwolle – Lelystad – Amsterdam – Haarlem, Utrecht [Fig. 7(d)].

4. Comparing Motorway and Physarum Graphswith Proximity Graphs

We conjectured that plasmodium of Physarum polycephalum constructs planarproximity graphs by its protoplasmic network [4]. A protoplasmic network con-structed in any particular experiment is planar, a generalized Physarum graphP may be non-planar. A planar graph consists of nodes which are points on aEuclidean plane with edges which are straight segments connecting the points, andedges cross each other. A planar proximity graph is a planar graph where twopoints are connected by an edge if they are close in some sense. Usually a pair ofpoints are assigned a certain neighborhood, and the pair are connected by an edge iftheir neighborhood is empty. Relative neighborhood graph [16], Gabriel graph [19],β-skeletons [18] and spanning tree are most known examples of proximity graphs.

For self-consistency we provide a brief definition of the following graphs:

• RNG: Points a and b are connected by an edge in RNG if no other point c iscloser to a and b than dist(a, b) [28].

• GG: Points a and b are connected by an edge in GG if a disc with diameterdist(a, b) centered in middle of the segment ab is empty [13, 19].

• BS(β): A β-skeleton, β ≥ 1, is a planar proximity undirected graph of anEuclidean point set where nodes are connected by an edge if their lune-basedneighborhood contains no other points of the given set; parameter β determinessize and shape of the nodes’ neighborhoods [18].

• MST: The Euclidean minimal spanning tree (MST) [21] is a connected acyclicgraph which has minimal possible sum of edges’ lengths.

The graphs are related as MST ⊆ RNG ⊆ BS ⊆ GG [16, 19, 28].We constructed a relative neighborhood graph [28] RNG [Fig. 8(a)], a Gabriel

graph [13, 19] GG [Fig. 8(c)], a β-skeleton [Fig. 8(b)] and a minimum spanningtree MST [Fig. 8(d)] (we rooted MST in Amsterdam) over nodes correspondingto centers of urban areas. We then calculated intersections of these graphs with thePhysarum graph P(0) and the motorway graph H, see Fig. 9.

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(a) RNG (b) BS(1.5)

(c) GG (d) MST

Fig. 8. Proximity graphs constructed on regions U: (a) Relative neighborhood graph RNG,(b) β-skeleton with control parameter 1.5, BS(1.5) (c) Gabriel graph GG, (d) Minimum spanningtree MST.

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(a) PT

RNG (b) PT

BS(1.5) (c) PT

GG

(d) PT

MST (e) HT

RNG (f) HT

BS(1.5)

(g) HT

GG (h) HT

MST

Fig. 9. Intersection of Physarum graph P(0) (a)–(d) and motorway graph H (e)–(h) with prox-imity relative neighborhood graph RNG, β-skeleton BS(1.5), Gabriel graph GG and minimumspanning tree MST.

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The following is a list of edges of the proximity graphs that are not present inPhysarum or motorway graphs:

• P ∩ MST = MST − {(Den Helder, Haarlem), (Zwolle, Apeldoorn)}• P ∩ RNG = RNG− {(Den Helder, Haarlem), (Zwolle, Apeldoorn)}• P∩BS(1.5) = BS(1.5)−{(Den Helder, Haarlem), (Zwolle, Apeldoorn), (Utrecht,

Den Haag)}• P ∩ GG = GG − {(Den Helder, Haarlem), (Zwolle, Apeldoorn), (Utrecht, Den

Haag), (Den Helder, Amsterdam), (Den Helder, Lelystad)}• H ∩ MST = MST− {(Haarlem, Enschede), (Rotterdam, Hertogenbosch)}• H ∩ RNG = RNG− {(Haarlem, Enschede), (Rotterdam, Hertogenbosch)}• H ∩ BS(1.5) = BS(1.5) − {(Haarlem, Enschede), (Rotterdam, Hertogenbosch)}• H ∩ GG=GG−{(Haarlem, Enschede), (Rotterdam, Hertogenbosch), (Ensch-

ede, Tilburg), (Rotterdam, Nijmegen), (Amersfoort, Dordrecht), (Dordrecht,Tilburg)}.

The motorway graph closely matches the spanning tree, relative neighborhoodgraph and β-skeleton. Only two edges (Haarlem, Enschede), (Rotterdam, Herto-genbosch), presented in RNG, MST, BS(1.5) do not exist in H. The fact thatthe relative neighborhood graph is almost a sub-graph of the motorway graph indi-cates intrinsically logical organization of the transport networks in the Netherlands.This is because a relative neighborhood graph is commonly considered to be opti-mal in terms of total edge length and travel distance, and is known to be a goodapproximation of road networks [30, 31].

Said that the transport networks in the Netherlands are redundant — fromslime mould’s point of view — because there is a substantial number of edges of Hnot presented in RNG.

The same can be said about the Physarum graph, because only edges (DenHelder, Haarlem) and (Zwolle, Apeldoorn) of RNG are not represented by proto-plasmic tubes. Under-representations of β-skeleton and GG in between P(0) aremuch more substantial: three and five edges, respectively.

5. Flooding

In this section we describe the experiments which investigate the capability ofthe plasmodium to adapt its network during induced flooding of the Petri dish. Inessence we investigate how the plasmodium would calculate and adapt the transportnetwork, if the Netherlands were to suffer similar flooding of its cities and roads.Experiments on flooding were conducted in 12 × 12 cm Petri dishes. A Petri dishwas raised by 1–2 cm in its south-east corner and partly filled with liquid (distilledwater, either pure or colored with one drop of food coloring) [Fig. 10(a)]. The areaflooded roughly corresponds to the territories of the Netherlands with 1/10,000–1/4000 chances of the protection from flooding failure [15, 32].

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(a) (b)

Fig. 10. (Color online) Flooding setup: (a) Array of Petri dishes during imitated flooding,(b) Flooding scheme. Flooded part is filled is shaded (green), direction of outward migrationare shown by arrows, each large arrow represents 20%, while small arrow 10%.

In most cases the flooded area included Middelburg on the south-west andGroningen on the north-east. In the central part of the country the flooding oftenreached Zwolle. The exact flooded area varied between experiments due to slightvariations in the thickness of agar gel substrate and minor differences in inclinationsof Petri dishes, the flooding scheme shown in (Fig. 10) is rather indicative.

Initially plasmodium reacts to flooding with increased activity. During the firstfew hours of flooding the plasmodium typically increases its branching at the bound-ary of the flooded area [Fig. 11(a)]. Often there are indications of indiscriminateincrease of foraging, panic foraging. For example, in Fig. 11(b) we can see activesprawling of plasmodium in the areas around Apeldoorn, Dordrecht and Enschede:no protoplasmic tubes are formed but rather uniform sheets of plasmodium prop-agate in these areas. Eventually the activity ceases and flooded transport linksbecome abandoned [Fig. 11(c)].

In some cases no “panic” branching occurs, but non-flooded protoplasmic tubesbecome thicker due to increased transport of nutrients and relocation of masses ofprotoplasm from areas affected by flooding [Fig. 12(a)]. Often only protoplasmictubes located along flood line are hypertrophied, for example there is an increasedthickness of tubes along the route Hertogenbosch – Apeldoorn – Zwolle – Groningenin Fig. 12(b).

With time, water is absorbed by the gel and is sucked under the gel plate dueto capillary forces, which in turn causes the overall humidity to increase. This isassociated with a reduced concentration of nutrients, and an increased concentrationof metabolites ejected in the agar plate, force the plasmodium to abandon the agar

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(a) t = 0 h (b) t = 30 h

(c) t = 46 h

Fig. 11. Illustration of plasmodium behavior during experiment on flooding. Photographs aremade at different angles.

(a) (b)

Fig. 12. Compensation of transport networks (a) and increase in borderline transport (b).

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(a) (b)

Fig. 13. Examples of evacuation.

plate and migrate beyond. Often the plasmodium attempts to complete evacuationby crawling onto the water surface (for example see Fig. 13).

Outward migration, calculated in nine experiments, are shown in Fig. 10(b). Weconclude that if a flooding of the Netherlands were to happen the major impactof migration (60%) will be felt by Western Germany, with just a minor impacton Northern France (20%) and a very slight impact on Belgium (10%). If theNetherlands were flooded the Western Germany would accept the largest impact ofmass-migration.

6. Discussion

In laboratory experiments with plasmodium of Physarum polycephalum we discov-ered that the Physarum protoplasmic network forms a sub-network of man-mademotorway networks, i.e., every transport link represented by Physarum can alsobe found as a segment of the motorway network. However, the converse does nothold. The motorway network is not a sub-graph of the Physarum network; threeare edges of motorway graph not represented by edges Physarum graph. Transportlinks Amsterdam to Der Helden, Zwolle to Apeldoorn, and Rotterdam to Dordrechtare never presented in Physarum graphs. This slight redundancy of the man-mademotorway network — comparing to Physarum — could be due to several reasons.For example, transport networks in many countries are based on economical andpolitical factors, which do not necessarily possess a feature of optimality. Also,some motorway links were designed to reduce congestion and to reduce commutingfor people living outside major urban areas. Congestion per se is not an issue forP. polycephalum whose protoplasmic tubes are elastic and adaptable to almost allacceptable degree of throughput. Moreover, the branching of tubes and formationof new tubes happen only when new sources of nutrients are detected (via chemo-attractants) by the slime mould or existing source of nutrients are depleted and themould switches to its exploration mode.

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Also, in many experiments Den Helder city remains disconnected from othercities. This is either a mishap of the Physarum approach, e.g., plasmodium prefersnot to enter narrow peninsula of North Holland, or an indication of a somewhatinefficient location of Den Helder.

The most robust component of the Physarum graph, the component which ispresent in the majority of experiments is a tree with three linear branches. Firstbranch is Tilburg – Breda – Dordrecht – Rotterdam – Den Haag, second is Tilburg– Hertogenbosch – Nijmegen – Apeldoorn, and third is Tilburg – Eindhoven –Maastricht. A possible explanation would be that relative positions and distancesbetween the cities in this tree is optimal for plasmodium physiological functioning.Cities are not close enough to be contaminated by products of plasmodium activitybut close enough not to put significant strain on the pumping of nutrients betweendistant parts of plasmodium’s body.

If we prune the Physarum graph by removing edges which occur in less than aquarter of experiments and look at the intersection (i.e., the set of edges present inboth graphs) of this graph with a graph of motorways we find that intersection con-sist of three disconnected components. The first component is a chain Enschede –Den Haag – Rotterdam. The second component is a cycle Nijmegen – Breda – Mid-delburg with branches Breda – Hertogenbosch – Tilburg, Middelburg – Eindhovenand Nijmegen – Dordrecht – Apeldoorn – Amersfoort. Finally, the third componentis a cycle Leeuwarden – Groningen – Zwolle with branches Zwolle – Enschede andZwolle – Lelystad – Amsterdam – Haarlem, Utrecht.

To evaluate topological quality of approximation of the motorways with slimemould we calculate several integral indices for the motorway graph H and thePhysarum graph P( 8

62 ) [we have chosen highest value of θ which does not destroyconnectivity of the Physarum graph, Fig. 5(o)]:

• Harary index [22] is defined as follows: 12

∑ij ξ(D)ij , where i and j are indices of

a graph nodes, D is a graph distance matrix, where Dij is a length of a shortestpath between i and j, ξ(D)ij = D−1

ij if i �= j and 0, otherwise.• Randic index [23]

∑ij Cij ∗ ( 1√

(di∗dj)), where Cij is a connectivity matrix.

• Diameters of graphs measured in nodes and normalized lengths, i.e., lengths ofall edges are divided by length of a longest edge before calculation of index.

There is a very good match between indices of the motorway and thePhysarum graph (Table 1), especially for Randic index and graph diameter

Table 1. Integral indices calculated for motorway graph H andPhysarum graph P( 8

62).

Index Motorway graph H Physarum graph P( 862

)

Harary index 199.5 257.7Randic index 20.2 20.3Diameter in nodes 6 7Diameter in lengths 3.6 3.6

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Bio-Development of Motorway Networks in The Netherlands

in normalized lengths. This may act as an indication that despite occasionalmismatch between the graphs in position of particular edges, P. polycephalumprovides almost perfect topological approximation of the Netherlands transportnetwork.

With respect to the proximity graph the key finding is that relative neigh-borhood graphs (which is commonly recognized as a best approximation of urbanstreets and transport networks) are almost (apart of two edges) subgraphs of thePhysarum graph and motorway graph. Only two edges (Haarlem, Enschede), (Rot-terdam, Hertogenbosch), presented in relative neighborhood graph, minimum span-ning tree and β-skeleton do not exist in the motorway graph H. Physarum graphP(0) is not a sub-graph the minimum spanning tree. This means that despite beinggood approximation of the motorway network the slime mould — in the particularcase of the Netherlands — does not include minimum transport network, repre-sented by a spanning tree.

By physically imitating flooding of some parts of the Netherlands we predictedthat if a real flooding were to occur, the following events will take place: substantialincrease in traffic on the parts of motorway networks close to the boundary betweenflooded and non-flooded areas, propagation of the traffic congestion to all non-flooded parts of the country, complete paralysis and abandonment of transportnetwork, migration of population from the Netherlands to Germany and France,and Belgium.

The plasmodium is incredibly similar, especially in wave-like behavior, tosub-excitable non-linear media [2] and is mainly guided by gradients of chemo-attractants [5]. Based on these two facts we could assume the plasmodiummust spread from Amsterdam omni-directionally and then start forming branchesbetween cities, as e.g., illustrated in Fig. 14. Such phenomena do not occur, ratherthe plasmodium behaves more like a “single-headed” creature, it chooses one direc-tion of movement, explores it, then chooses another direction and explores it

(a) t = 1 (b) t = 2 (c) t = 3

Fig. 14. Snapshots of a growing spanning tree of U rooted in Amsterdam. Vertices active at timestep t′ are shown by black disc at snapshot t = t′.

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(d) t = 4 (e) t = 5 (f) t = 7

(g) t = 8 (h) t = 9

Fig. 14. (Continued )

again, see Fig. 3. Such behavior of the plasmodium may explain the differencesbetween ideal planar proximity graphs and protoplasmic networks constructed bythe plasmodium.

In future work we plan to undertake more experiments on Physarum-basedimitation of road formation in other European countries, and may be even makeexperiments at a large scale, where plasmodium grows over the whole Europe. Wealso plan to evaluate the city state of Singapore, this present a fairly unique roadnetwork that has been carefully planned and constructed within the last 50 years.One further objective would be to try and incorporate three-dimensional landscapeelevation into our laboratory experiments.

Acknowledgments

Peter M. A. Sloot would like to acknowledge the support of this work by the RussianFederation Leading Scientist Grant, contract: 11.G34.31.0019 and the EuropeanUnion Dynanets project: 233847.

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