Int. Journ. of Unconventional Computing, Vol. 6, pp. 79–88
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Computation of Voronoi Diagram and
Collision-free Path using the Plasmodium
of Physarum polycephalum
Tomohiro Shirakawa1 and Yukio-Pegio Gunji1,2
1 Graduate
School of Science & Technology, Kobe University, Nada, Kobe, 657-8501, Japan
E-mail: [email protected]
2 Department of Earth & Planetary Sciences, Faculty of Science, Kobe University,
Nada, Kobe, 657-8501, Japan
Received: August 16, 2007. Accepted: September 2, 2007.
The plasmodium of Physarum polycephalum is a large, unicellular and
multinuclear organism whose computational ability is attracting a lot of
attention in the field of nature-inspired unconventional computing. To test
the computational capability of the organism and its utility for future applications, we have implemented the computation of a Voronoi diagram and
a collision-free path in an experimental system using Physarum plasmodium. Attractants and repellents for the plasmodium were arranged in the
experimental system to induce the plasmodium to form the graphs. The
plasmodium solved the complex problem and successfully formed Voronoi
diagrams and collision-free paths, demonstrating its computational ability.
Keywords: Physarum polycephalum, true slime mold, plasmodium, nature inspired
computing, Voronoi diagram, collision-free path.
1 INTRODUCTION
Currently, the plasmodium of Physarum polycephalum, a species of true slime
mold, is receiving much attention as a material for unconventional computing.
To date, the organism has been experimentally used in various kinds of computation such as maze-solving [1–3], calculation of efficient networks [4–6],
construction of logical gates [7] and robot control [8]. These experiments have
shown that the plasmodium is very useful in computation if we appropriately
interpret the problem in terms of behaviors of the organism.
The ability of the plasmodium derives from its particular kind of cellular structure. The plasmodium is a very large, unicellular and multinuclear
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FIGURE 1
The plasmodium of Physarum polycephalum crawling on a 1.5% agar plate. The sheet-like structure is indicated by black arrow and the tubular structure is indicated by white arrow. (Bar: 2 cm).
organism, which is visible to the naked eye (Figure 1). At one stage of the
life cycle of Physarum polycephalum, a large number of unicellular amoebae
aggregate and fuse together, forming the plasmodium. The cell body of the
plasmodium roughly consists of two parts: sheet-like structures at the locomotive front with tubular structures at the rear (Figure 1). The sheet-like parts
actively crawl on a plane surface searching for food sources, taking advantage
of the fact that its scale is much larger than that of the unicellular amoebae.
At the same time, tubes connect all the parts of the cell body to maintain the
integrity of the plasmodium as a single cell. The plasmodium tends to keep
the volume of tube construction as small as possible, and this enables the plasmodium to solve mathematical problems. When two food sources are provided
for a plasmodium spreading in a maze, the plasmodium connects them by the
shortest path giving a solution for the maze [1–3]. Similarly, for multiple food
sources, the plasmodium connects all of them providing an efficient network
structure such as minimum spanning tree or Steiner minimum tree [4–6]. In
this study, we applied the computational ability of the plasmodium to the
computation of a Voronoi diagram and a collision-free path.
When a non-empty finite set of planar points (sites) is given, the Voronoi
diagram partitions the plane such that each partition (Voronoi cell) includes
the set of planar points closest to the site in that Voronoi cell [9,10]. Thus a
Voronoi diagram is a subset of bisectors between all pairs of sites in the plane
(Figure 2(b)). A collision-free path is a subgraph of a Voronoi diagram that
connects two points by the shortest path between them: if we regard the sites
as obstacles, the collision-free path provides the shortest route between two
points, avoiding the obstacles effectively. In the context of unconventional
computing, the computation of Voronoi diagrams has been implemented in
chemical-based reaction-diffusion processors [10–12]. In such computations,
chemical waves propagate from each site and the waves collide with each other
at points equidistant from each site, constructing a set of lines equidistant from
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(b)
FIGURE 2
(a) Schema of the experimental system used in this study. The shaded area indicates the experimental field for the plasmodium. The squares indicate the position of the repellent containing
cubes, which are regarded as sites for the Voronoi diagram. (b) The accurate Voronoi diagram for
this arrangement of sites. (Bar: 2 cm).
each site, that is, the edges of Voronoi cells. In biology, the Voronoi diagram
was found in the physiology of plants, and is known as the area potentially
available [13] or the plant polygon [14]. The algorithm for Voronoi diagram
formation by plants would be similar to that of the chemical reaction-diffusion
computer: roots start from the stem and grow until they collide with the roots
from another stem. The plasmodium in our experiment formed the Voronoi
diagram in a different way, based on changes in its morphology, and the
cellular tubes themselves formed the boundaries of the Voronoi cells. In other
words, the plasmodium in this study demonstrated a new kind of algorithm
for the computation of Voronoi diagrams or collision-free paths and suggests
the presence of a highly sophisticated computational ability in the organism.
2 MATERIALS AND METHODS
2.1 Culture of Physarum plasmodia
We cultivated Physarum plasmodia using the method of Camp [15]. Briefly,
glass Petri dishes were thickly arranged in a plastic box and paper towels were
laid out on the dishes. Tap water filled the space below the towels to supply
moisture. The plasmodia were cultured on the towels at 24◦ C. Oatmeal was fed
daily.
2.2 Induction of cellular tube network with a shape of Voronoi diagram
by a spatial distribution of repellent-containing agar cubes
To provide an experimental field for the computation of a Voronoi diagram
and a collision-free path, a circle 8 cm in diameter was cut out from a plastic
film. From the circular plastic film, another concentric circle 6 cm in diameter was cut out producing a ring-shaped plastic film. Then the ring-shaped
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film was put on 1.5% agar gel in a 9 cm Petri dish. The plasmodium tends to
avoid the dry surface of the plastic film, thus the experimental set up described
above provides a circular experimental field for the plasmodium within the
6 cm diameter circle (Figure 2(a)). The frontal parts of the plasmodium were
cut into 6 mm square pieces for experimental use. Eight pieces of the plasmodium were symmetrically arranged at the edge of the experimental field
(Figure 3(a)). They were incubated in the dark at 24◦ C for 6–8 hours, until
they fully spread into the field and fused together to form a single cell, uniformly spread within the field (Figure 3(b)). Then 7 pieces of 1.5% agar gel
cube (approximately 5 mm on a side) containing 500 mM potassium chloride
(a repellent for the plasmodium) were arranged on the experimental field. At
the same time, 1.5% agar gel cubes containing 50 mg/ml crushed oatmeal (food
source for the plasmodium) were thickly placed at the edge of the experimental
field, so that they were in contact with the plasmodium at the boundary of the
field. The sample was again incubated in the dark at 24◦ C, for an additional
2–3 hours until the plasmodium formed a tubular network. Photographs were
taken using a digital camera (EOS30-D, Canon, Japan) and enhanced using
Adobe Photoshop CS2 (Adobe, CA, USA).
2.3 Conversion of network shape from Voronoi diagram to collision-free
path by re-arrangement of distribution of attractant agar cubes
After the plasmodium in the experimental system formed a Voronoi diagram,
all the food sources enclosing the experimental field were removed. Then, two
food sources containing agar cubes were placed at arbitrarily chosen points
on the edge of the Voronoi diagram. The system was incubated in the dark
at 24◦ C for another 2–3 hours, until the plasmodium formed a new network.
Photographs were taken using a digital camera (EOS30-D, Canon, Japan) and
enhanced using Adobe Photoshop CS2 (Adobe, CA, USA).
3 RESULTS
In this study, we have tested 20 samples of Voronoi diagram and collision-free
path computation using the plasmodium of Physarum polycephalum, and got
similar results from almost all of the samples. Thus we picked up the results of
5 samples and illustrated them in Figures 3, 4, 5, and 6. When the plasmodium
uniformly spread in the field, repellent containing agar cubes were supplied.
Then the plasmodium formed cellular tubes avoiding the repellent cubes and
tubes laid between the cubes formed a Voronoi diagram. Therefore, the position of the repellent cubes in this experiment is regarded as the position of
sites in a Voronoi diagram. Cubes containing a food source were arranged at
the boundary of the experimental system to promote tubulogenesis midway
between the repellent cubes by drawing the extra cytoplasm to the periphery. The experiment led to the formation of a Voronoi diagram. Figure 2(b)
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(b)
(c)
(d)
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FIGURE 3
Photographs of sample No. 1 during the computation of a Voronoi diagram. (a) The experimental
set up for computation of the Voronoi diagram and the initial arrangement of Physarum plasmodia.
Eight pieces of plasmodia (arrowhead) were symmetrically arranged at the edge of the circular
experimental field. (b)Aplasmodium spread over the field, the arrangement of repellent containing
cubes (black arrow) and food source containing cubes (white arrow). (c) The Voronoi diagram
formed in this experiment. (d) A schema of the Voronoi diagram in (c). The tubes that corresponded
to the Voronoi edges are illustrated here. Though major tubes were preferentially picked up, the
choice of the tubes here is somewhat arbitrary. (Bar: 2 cm).
illustrates the mathematically accurate solution of the Voronoi diagram for
the arrangement of sites provided in this study. Compared with this accurate
solution, the Voronoi diagrams formed by the plasmodia (Figures 3(c), 3(d)
and 4) have some deviations in their position, and the branches that do not
correspond to Voronoi edges are still remaining in the network. However,
except for sample No. 5 in Figure 4, which lacks one of the Voronoi edges,
the plasmodia successfully constructs all of the Voronoi edges and correctly
divides the plane providing all of the Voronoi cells.
Furthermore, the food sources were rearranged to induce the formation of
collision-free path, and as expected, the Voronoi diagrams were converted to
graphs of the collision-free path. The accuracy of the computation is lower
than in the experiment for the Voronoi diagram: sample No. 2 fails to form
a new graph, and in samples No. 4 and No. 5 redundant edges still remain.
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Sample No. 2
Sample No. 3
Sample No. 4
Sample No. 5
FIGURE 4
Photographs of 4 samples that formed Voronoi diagrams, and their schematic images. Sample
No. 5 lacks one of the Voronoi edges: two of the sites in the lower right part are in a same Voronoi
cell and there is no partition between them. (Bar: 2 cm).
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(b)
(c)
(d)
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FIGURE 5
Photograph of sample No. 1 during the computation of a collision-free path. (a) Initial condition
for the experiment. This is the same image as in Figure 2(c). (b) Immediately after the removal of
cubes containing the food source at the periphery of the experimental system. Newly supplied food
sources are indicated by the white arrow. (c) The collision-free path formed in this experiment.
(d) A schema of the collision-free path in (c). The tube that corresponded to the collision-free
path is illustrated here. (Bar: 2 cm).
However, sample No. 1 and sample No. 3 successfully form collision-free
paths that give routes for avoiding the obstacles (repellent cubes).
4 DISCUSSION
In this study, we have attempted to implement the computation of the Voronoi
diagram and the collision-free path in an experimental system, using the plasmodium of Physarum polycephalum. For a uniformly spreading plasmodium,
repellent containing agar cubes were supplied. Then by selecting edges from
initially existing network, the plasmodium formed Voronoi diagram. The initial distribution of plasmodial cell body was uniform enough, so it seemed that
the arrangement of initially existing network did not affect the result of computation. Compared with the reaction-diffusion chemical computation [10–12],
the plasmodium has a principally different way of computation, though they
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Sample No. 3
Sample No. 4
Sample No. 5
FIGURE 6
Photographs of 3 samples that formed collision-free paths, together with their schematic images.
(Bar: 2 cm).
give a same solution. In the first step of computation of Voronoi diagram by
reaction-diffusion processor, drops of a reagent that represent the positions of
site are put in the medium containing a substrate for the reagent. The reagent
then diffuses in the medium reacting with the substrate and forming precipitate. At the bisector points between the sites, the precipitate is not formed
because of competition for the substrate. A group of the precipitate around the
site represents a Voronoi cell, and the lines indicated by the absence of the precipitate represent a Voronoi diagram. In this computation, a Voronoi diagram
is formed first, and then a set of obstacle avoiding paths is found. On the other
hand, in the computation of the plasmodium, all possible obstacle avoiding
paths are searched, and edges for a Voronoi diagram are then chosen. The two
types of computation have reverse algorithm for the computation of Voronoi
diagram, i.e., they are complementary.
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In the experiment for the computation of Voronoi diagrams, there are three
requirements for the plasmodium: avoiding repellents; making contact with
the food sources at the periphery of the experimental field; and keeping, as
much as possible, the connections between separated parts of the cell body to
maintain integrity as a single cell. Satisfying these requirements, the plasmodium forms an approximate Voronoi diagram. In other words, the plasmodium
solves a very complex problem. In contrast to the maze-solving experiment in
which food sources are used as the only input for the plasmodium [1–3], our
experiment consisted of two types of input: food source and repellent. Furthermore, the experimental field for maze-solving is one dimensional, whereas
our experimental field is two dimensional. Therefore our experiment demonstrated that the plasmodium is able to deal with more complex situations. In
addition, the formation of collision-free paths based on the morphology of
Voronoi diagrams indicates that the plasmodium retains the information in its
field for a long time.
Although the plasmodium succeeds in computing Voronoi diagrams and
collision-free paths, the accuracy of these graphs is lower than that from
collision-based computing with inorganic chemicals [10–12]. However, this
does not necessarily indicate that the computational ability of the plasmodium
is also lower or that our experimental set up is incomplete. The plasmodium
autonomously creates complex structures showing cell motility on a broad
scale, even on a homogeneous plane without attractant or repellent (see Figure 1, for example). We assume that biological entities always compute their
own tasks and we can never completely exclude the effect of such inherent
processes in any experimental system. Thus we assume that inaccuracy in
the computations implemented in a biological experiment can arise from both
imperfections in the experimental system (settings of the boundary condition)
and built-in characteristics of the biological organism, and we would not be
able to specify which one mainly affects the computation in the experimental system. In conclusion, the plasmodium demonstrated its computational
ability by successfully computing the solutions for Voronoi diagrams and
collision-free paths in a relatively well-constructed experimental system.
ACKNOWLEDGEMENT
This work was supported in part by the Science and Technology Agency of
Japan through the 21st Century Center of Excellence (COE) Program, “Origin
and Evolution of Planetary Systems (Kobe University)”.
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