arXiv:0901.4466v1  [cs.RO]  28 Jan 2009
Physarum boats: If plasmodium sailed it would
never leave a port
Andrew Adamatzky
University of the West of England, Bristol BS16 1QY, United Kingdom
andrew.adamatzky@uwe.ac.uk
October 25, 2018
Abstract
Plasmodium of Physarum polycephalum is a single huge (visible by naked
eye) cell with myriad of nuclei. The plasmodium is a promising substrate
for non-classical, nature-inspired, computing devices. It is capable for ap-
proximation of shortest path, computation of planar proximity graphs and
plane tessellations, primitive memory and decision-making. The unique
properties of the plasmodium make it an ideal candidate for a role of
amorphous biological robots with massive parallel information process-
ing and distributed inputs and outputs. We show that when adhered to
light-weight object resting on a water surface the plasmodium can propel
the object by oscillating its protoplasmic pseudopodia.
In experimen-
tal laboratory conditions and computational experiments we study phe-
nomenology of the plasmodium-floater system, and possible mechanisms
of controlling motion of objects propelled by on board plasmodium.
Keywords: Physarum polycephalum, motility, biological robots
1
Introduction
A plasmodium, or vegetative state, of Physarum polycephalum, also known as
a true or multi-headed slime mold, is a single cell with many diploid nuclei; it
behaves like a giant amoeba. In 2000 Nakagaki et al [17, 18, 19] shown that
topology of the plasmodium’s protoplasmic network optimizes the plasmodium’s
harvesting on distributed sources of nutrients and makes more efficient flow and
transport of intra-cellular components. Thus the field of Physarum computing
emerged [20]. The problems solved by the plasmodium include approximation of
shortest path in a maze [18, 19], computation of proximity graphs [8], Delaunay
triangulation [21], construction of logical gates [22], robot control [23], imple-
mentation of storage-modification machines [7], and approximation of Voronoi
diagram [21].
1
The plasmodium can be seen as a network of biochemical oscillators [14, 16],
where waves of excitation or contraction originate in several sources, travel
along the plasmodium, and interact one with another in collisions. The oscil-
latory cytoplasm of the plasmodium is a spatially extended nonlinear excitable
media. Previously we shown that the plasmodium of Physarum is a biolog-
ical analogue of chemical reaction-diffusion system encapsulated in an elastic
membrane [6]. Such encapsulation allows the plasmodium to act as a massive-
parallel reaction-diffusion computer [4] with parallel inputs and outputs. This
ensures the plasmodium explores environment in a distributed manner and re-
sponds to changes in surrounding conditions as an amorphous decentralized but
yet internally coordinated entity. Thus we can envisage that the plasmodium of
Physarum polycephalum is a promising biological substrate for designing amor-
phous robots with embedded reaction-diffusion intelligence.
An engineering
background for such amoeboid machines was developed by Yokoi et al [24, 25],
however no viable non-silicon implementations are known so far.
There were just two attempts to integrate spatially extended non-linear
chemical or biological systems with silicon hardware robots. In 2003 a wheeled
robot controlled by on board excitable chemical system, the Belousov-Zhabotinsky
medium, was successfully tested in experimental arena [3]. The on board chem-
ical reaction was stimulated by silver wire and motion vector towards source
of stimulation was exctracted by robot from topology of the excitation waves
in the Belousov-Zhabotinsky medium. In 2006 Tsuda et al [23] designed and
successfully tested in real-world conditions a Physarum contoller for a legged
robot. The controller’s functioning was based on the fact that light inhibits
oscillations in illuminated parts of the plasmodium. Thus a direction towards
light source can be calculated from phase differences between shaded and illu-
minated parts of the plasmodium controller. In both instances non-linear media
controllers were coupled with conventional hardware and relied upon additional
silicon devices to convert spatio-temporal dynamics of excitation and oscillation
to the robots motion.
The only known up to date result of self-propelled excitable-medium devices
can be attributed to Kitahata [13] who in 2005 experimentally shown that a
droplet of Belousov-Zhabotinsky medium is capable for translation motion due
to changes in interfacial tension caused by convention forces inside the droplet.
The convention forces are induced by excitation waves, propagating inside the
droplet [12].
Recently we experimentally demonstrated [10] that the plasmodium can act
as a distributed manipulator when placed on a water surface with light-weight
objects distributed around. We shown that the plasmodium senses data-objects,
calculates shortest path between the objects, pushes and pulls light-weight ob-
jects placed on a water surface. We also found that motility of the plasmodium
placed directly on the water surface is restricted [10]. The plasmodium does
propagate pseudopodia and forms protoplasmic trees, however it does not travel
as a localized entity. To achieve true mobility of a plasmodium we decided to at-
tach it to a floater in a hope that due to mechanical oscillations of propagating
pseudopodia the plasmodium will be capable for applying enough propulsive
2
force to the floater to propel the floater on the water surface. In laboratory
experiments we proved feasibility of the approach. In present paper we report
our findings on motion patterns observed in experiments and simulations.
The paper is structured as follows.
Experimental techniques and basics
of simulating a plasmodium-floater system in mobile cellular automaton lat-
tices are presented in Sect. 2.
In Sect. 3 we discuss modes of movement of
plasmodium-floater systems, observed in laboratory experiments.
Results of
computer simulations of the mobile excitable lattices, that mimic the plasmod-
ium behavior, are shown in Sect. 4. We overview the results and discuss our
findings in a context of unconventional robots and robotic controllers in Sect. 5.
2
Methods
Plasmodium of Physarum polycephalum was cultivated on a wet paper towels in
dark ventilated containers. Oat flakes were supplied as a substrate for bacteria
on which the plasmodium feeds. Experiments were undertaken in Petri dishes
with base diameter 35 mm, filled by 1/3-1/5 with distilled water. Variously
sized (but not exceeding 5-7 mm in longest dimension) pieces of plastic and
foam were used as floaters residing on the water surface. Oat flakes occupied by
the plasmodium were placed on top of the floaters. Behavior of the plasmodium-
floater systems was recorded using QX-5 digital microscope, magnification ×10.
Videos of experiments are available at [11].
The Petri dish with plasmodium was illuminated by LED (minimal light set-
ting for the camera) by white light which acted as a stimulus for the plasmodium
phototaxis. The illuminating LED was positioned 2-4 cm above North-East part
of the experimental container. The microscope was placed in the dark box to
keep illumination gradient in the dish safe from disturbances.
Plasmodium-floater systems were simulated by mobile two-dimensional cel-
lular automata, with eight cell neighbourhood, which produce force fields [2].
We employed the model of retained excitation [5]. Every cell of the automaton
takes three states — resting, excited and refractory, and updates its state de-
pending on states of its eight closest neighbors. A resting cell becomes excited
if number of its excited neighbors lies in the interval [θ1, θ2], 1 ≤θ1, θ2 ≤8.
An excited cell remains excited if number of excited neighbors lies in the in-
terval [δ1, δ2], 1 ≤δ1, δ2 ≤8, otherwise the cell takes refractory state. A cell
in refractory state becomes resting in the next time step. This is a model of
’retained excitation’ [5] used to imitate Physarum foraging behavior [6] because
this cellular automaton exhibits ‘amoeba-shaped’ patterns of excitation. We
denote the local transitions functions as R(θ1, θ2, δ1, δ2).
The reaction to light stimulus was implemented as follows. Cells of lattice
edges most distant from the light were excited with probability 0.15. The illu-
mination dependent excitation of edge cells corresponds to stimulus-dependent
changes in plasmodium oscillation frequencies [15, 16]. Namely, the parts of the
plasmodium closest to positive stimulus (e.g. sources of nutrients) periodically
contract with higher frequencies than parts closest to negative stimulus (e.g.
3
high illumination).
We derived motion from excitation dynamics by supplying every cell of the
lattice with a virtual local force vector. The local vectors are updated at every
step of the simulation.
A vector in each cell becomes oriented towards less
excited part of the cell’s neighborhood. An integral vector, which determines
lattice rotation and translation at each step of simulation time, is calculated as
a sum of local vectors over all cells. The approach is proved to be successful in
simulation of mobile excitable lattices, see details in [2].
3
Experimental results
In experiments, we observed five types of movements executed by plasmodium-
floater system: random wandering, quick sliding, pushing, and directed pro-
pelling.
The random wandering is caused by sudden movements, and associated vi-
brations, of plasmodium’s protoplasm. Movement of protoplasm is caused by
peristaltic contraction waves generated by disordered network of biochemical
oscillators. The higher the frequency of contractions in a particular domain of
the plasmodium, the more protoplasm is attracted into the domain [15, 16].
Relocations of protoplasmic mass change mass center of the plasmodium-floater
system, thus causing the floater to wander. The floater influenced by the pro-
toplasmic vibration may exhibit a random motion (Fig. 1a–e), which usually
persists till some part of the plasmodium propagates beyond edges of the floater
(after that another types of motion take place).
The quick sliding motion occurs when substantial amount of protoplasm,
e.g.
a propagating pseudopodium, relocates to one edge of the floater and
this pseudopodium penetrates water surface. The floater then become partly
submerged. This changes the way water surface forces act on the floater, and
sets the floater in a motion. The floater usually slides till collide with an obstacle
or a wall of the container.
The random wandering and sliding are rather types of motion with non
controllable and unpredictable trajectories. These motions unlikely to be used
in future architectures of biological robots so we not discuss them further.
The pushing motion can be observed in situations when plasmodium propa-
gates from the floater onto the water surface and develops a tree of protoplasmic
tubes. At some stage of the development tips of the growing protoplasmic tree
reach walls of the water container and adhere to the walls. An example is shown
in Fig. 2: three-four hours after being placed on a floater the plasmodium starts
propagation (Fig. 2bc) and a protoplasmic tree emerges (Fig. 2f). Already at
this stage the floater comes into motion (Fig. 2fg), traveling in the direction op-
posite to propagation of the protoplasmic tree. The motion becomes pronounced
when a part of the tree attaches itself to a wall of the container (Fig. 2h), a
force of protoplasm pumped into the tubes causes the floater to noticeably move
outwards the growing tubes (Fig. 2i).
A position of the floater can be ‘fine-tuned’ by several protoplasmic trees,
4
Figure 1:
Transition between types of motion of physarum-floater system.
Random wandering, or vibration-based, movement (a)–(e), 90o clockwise ro-
tation (e)–(f), and directed propelling (f)–(h). Positions of the floater, at time
steps corresponding to snapshots (a)–(h) and trajectory the floater’s centre are
shown in (i). See video at [11].
5
Figure 2: Pushing the floater by growing protoplasmic tubes.
(a)–(h) Pho-
tographs of experimental container. (i) Time-lapsed contours of the floater. See
video at [11].
6
Figure 3: Collective movement of the floater by several protoplasmic trees. (a)–
(h) Photographs of the experimental container. (i) Time-lapse contours of the
floater. See video at [11].
7
which attached themselves to different sides of experimental container (Fig. 3).
In experiment shown in Fig. 3 protoplasmic trees grow along North-West to
South-East arc and to the direction North-East (Fig. 3bc). The sub-tree growing
in the North-East direction diminishes with time, because it happen to be in
the most illuminated area of the container, the only branch heading South
survives. The protoplasmic tree becomes stronger in the North-West direction
(Fig. 3bc)g–h). The dynamical reallocations of the trees cause the floater to
rotate and then gradually move South-West (Fig. 3i).
The last and the most interesting type of motion observed is direct pro-
pelling of floater by on board plasmodium. This type of motion occurs when
pseudopodia propagate beyond the edge of the floater but then partly sank.
The pseudopodia oscillate, their oscillations are clearly visible on the video
recordings in [11]. These oscillations cause the floater to move in the direction
opposite to submersed pseudopodia (in contrast to sliding motion, the floater
does not submerse at all during the propulsive motion). The examples of direct
propelling are shown in Figs. 1 and 4.
In experiment shown in Fig. 1 plasmodium propagates a pseudopodium in
South-West direction (Fig. 1de). The pseudopodium does not stay on the water
surface but partly sinks. The initial wetting of the pseudopodium causes the
floater to rotate clockwise with the protruding pseudopodium facing now North-
West (Fig. 1de). The immersed pseudopodia oscillate and propel the floater in
the South-Easterly direction (Fig. 1g–i).
An example of collective steering of the floater by several pseudopodia is
shown in Fig. 4. After initial adaptation period (Fig. 4ab), associated with vi-
brational random motion, the plasmodium propagates pseudopodia in Westward
and South-Easterly directions (Fig. 4de). At the beginning only South-Eastern
pseudopodia contribute to propelling of the floater, with Western pseudopo-
dia are merely balancing action of their South-Eastern counterparts. Thus the
floater travels towards North (Fig. 4fg). Later Westerly pseudopodia increase
frequency of their oscillations (see [11]) and the floater turn more towards North-
East (Fig. 4hi).
4
Results of simulations
Proposition 1. Given large enough container and a stationary source of light
a floater with plasmodium on board will move in irregular cycles around the
source.
By assuming the container is large enough we secured the situation when
it is impossible for pseudopodia, or protoplasmic trees, to reach walls of the
container and push the floater. The wandering motion only occurs at the very
beginning of plasmodium’s development on the floater. Thus we are left with
propelling motion.
Let the floater be stationary.
Due to negative phototaxis, plasmodium’s
pseudopodia propagate towards the less illuminated parts of the container.
8
Figure 4: Direct propelling of floater by oscillating pseudopodia. (a)–(i) Pho-
tographs of the experimental container. (j) Time lapses contours of the floater.
See video at [11].
9
When the pseudopodia spawn from the floater to the water surface, their os-
cillatory contractions propel the floater towards source of illumination.
The
pseudopodia continue their action till the floater pass the highly illuminated
part. Then the pseudopodia happens to be on the most illuminated side of the
floater. They retract and new pseudopodia are formed on the less illuminated
part of the floater.
They propel the floater towards the source of illumina-
tion again. Exact positions of growing pseudopodia depends on distribution of
biochemical sources of oscillations and interactions between traveling waves of
contraction. New pseudopodia may not emerge at the same place as old pseu-
dopodia. Thus each trajectory of the floater towards the source of illumination
will be different from previous ones.
To verify the proposition we simulated plasmodium-floater system by mobile
cellular automata with various cell-state transition functions. An illustrative
series of snapshots is shown in Fig. 5. The automaton lattice was placed at some
distance from the source of light (Fig. 5a). Edges of the lattice oriented towards
less illuminated area of experimental container became excited (Fig. 5b). The
excitation propels the lattice towards the source of light (Fig. 5e–f).
When
lattice passes the site with highest illumination, another edge of the lattice
excites and propels the lattice back to the domain of high illumination (Fig. 5gh).
The sequence of excitations continues and the lattice travels along irregular
cyclic trajectories around the source of illumination (Fig. 5i–k).
Exact pattern of lattice cycling around the source of illumination is deter-
mined by particulars of the local excitation dynamics. For example, trajectories
of a lattice with threshold excitation (Fig. 6a) are very compactly arranged
around the source: as soon as the lattice approaches the source of illumination
it stays nearby, mostly turning back and force. When we impose upper bound-
ary of the excitation, e.g. when a resting cell is excited only if one or two of its
neighbors are excited, the lattice trajectory loosens (Fig. 6b).
Let us discuss now few examples of lattices governed by functions R(θ1θ2δ1δ2),
where a resting cell excites if number σ of excited neighbors lies in the interval
[θ1, θ2] and excited cell can stay excited if σ ∈[δ1, δ2] (Fig. 6c–f). We focus on
excitation interval [2, 2] because this type of local transition is typical for media
with traveling localized excitations [1], which does closely relate to propagation
activities of Physarum polycephalum [9]. We observed that lattices with very
narrow intervals of excitation and retained excitation demonstrate a combina-
tion of compact and lose trajectories (Fig. 6c). Widening interval of retained
excitation disperses lattice trajectories in space (Fig. 6de), thus emulating de-
layed responses to changes in sensorial background. When lower, δ1, and upper,
δ2, boundaries of retained excitation increase to 4 and 6, respectively, a lattice
starts to exhibit a quasi-ordered behavior: combination of long runs away from
the source and tidy loops of rotations (Fig. 6f).
10
Figure 5: Snapshots of cellular automaton model (two-dimensional lattice of
200 × 200 cells) of plasmodium-floater system. The local excitation dynamics is
controlled by function R(2201). Source of illumination is shown by solid black
disc. Each snapshot is supplied with trajectory of the center of the floater from
the beginning of simulation. See exemplar videos at [11].
11
Figure 6:
Trajectories of the center of cellular-automaton lattice traveling
around a source of illumination.
The lattice’s behavior is governed by local
transition functions as follows: (a) R(1899), threshold excitation, a resting cell
excites if there is at least one excited neighbor, values 9 for δ1 and δ2 mean that
the cell never stays excited longer then one step of a time, (b) R(1299), a rest-
ing cell excites if it has one or two excited neighbors. (c) R(2222), (d) R(2201),
(e) R(2211), (f) R(2246),
12
5
Summary
In laboratory experiments and computer simulations we demonstrated that plas-
modium of Physarum polycephalum can act as an ‘engine’ or a propelling agent
for light-weight floating objects. We found that most typical type of movement
generated by the plasmodium is a propulsive forward motion. This motion is
caused by pseudopodia protruding beyond the the floater and oscillating. In
small container, when growing tree of plasmodium’s protoplasmic tubes can
reach sides of the container, the plasmodium pushes floater by increasing length
of the tubes connecting the float and the container’s sides.
A plasmodium shows negative phototaxis. It tries to evade regions illumi-
nated by non-yellow light and grows towards more shaded areas. When the
plasmodium attached to a floating object, plasmodium–floater system exhibits
positive phototaxis. Due to growth, and associated oscillations, of pseudopo-
dia on the less illuminated side of a floater, the floater moves towards light.
We mimicked this phenomenon in cellular automaton models of mobile light-
sensitive lattices. We found that in ideal conditions the “plasmodium–floater”
system will wander around the site of highest illumination, often following quasi-
chaotic trajectories due to many sources of excitation competing with one an-
other in the protoplasm.
The phenomena and primitive constructs of the plasmodium-floaters will
be employed in future designs of amorphous decentralized robots operating on
water-surface. The robots will be capable to search for objects, travel towards
the objects’ location, implement manipulation and sorting. The robots will be
controlled by gradient fields of illumination and chemo-attractants.
Acknowledgement
The works is supported by the Leverhulme Trust Research Grant F/00577/I.
References
[1] Adamatzky A. Computing in Nonlinear Media and Automata Collectives.
IoP Publishing, 2001.
[2] Adamatzky A. and Melhuish C. Phototaxis of mobile excitable lattices
Chaos, Solitons, & Fractals, 13 (2002) 171–184.
[3] Adamatzky A., De Lacy Costello B., Melhuish C., RatcliffN. Experimental
implementation of mobile robot taxis with onboard BelousovZhabotinsky
chemical medium Materials Science and Engineering: C 24 (2004) 541–548.
[4] Adamatzky A., De Lacy Costello B., Asai T. Reaction-Diffusion Comput-
ers, Elsevier, Amsterdam, 2005.
13
[5] Adamatzky A. Phenomenology of retained excitation. Int. J. Bifurcation
and Chaos (IJBC) 17 (2007) 3985–4014.
[6] Adamatzky A. Physarum machines: encapsulating reaction-diffusion to
compute spanning tree. Naturwisseschaften 94 (2007) 975–980.
[7] Adamatzky A. Physarum machine:
Implementation of a Kolmogorov-
Uspensky machine on a biological substrate. Parallel Processing Letters
17 (2007) 455–467.
[8] Adamatzky A. Developing proximity graphs by Physarum polycephalum:
does the plasmodium follow the Toussaint hierarchy? Parallel Processing
Letters (2009), in press.
[9] Adamatzky A., De Lacy Costello B., Shirakawa T. Universal computation
with limited resources: Belousov-Zhabotinsky and Physarum computers.
Int. J. Bifurcation and Chaos (2008), in press.
[10] Adamatzky A. and Jones J. Towards Physarum robots: computing and
manipulating on water surface. J Bionic Engineering 5 (2008) 348–357.
[11] Adamatzky
A.
2008.
Supplementary
materials
to
present
paper:
http://uncomp.uwe.ac.uk/adamatzky/physarumboat.
[12] Kitahata H., Aihara R., Magome N., Yoshikawa K., Convective and peri-
odic motion driven by a chemical wave. J. Chem. Phys. 116 (2002) 5666–
5672.
[13] Kitahata H. Spontaneous motion of a droplet coupled with a chemical re-
action. Prog. Theor. Phys. Suppl., 161 (2006) 220–223.
[14] Matsumoto K, Ueda T, Kobatake Y. Reversal of thermotaxis with oscilla-
tory stimulation in the plasmodium of Physarum polycephalum. Journal of
Theoretical Biology 130 (1980) 175–182.
[15] Miyake Y.,
Tabata S.,
Murakami H.,
Yano M. and Shimizu H.,
Environment-dependent self-organization of positional information field in
chemotaxis of Physarum Plasmodium. J. Theor. Biol. 178 (1996) 341–353.
[16] Nakagaki T., Yamada H. and Ueda T. Modulation of cellular rhythm and
photoavoidance by oscillatory irradiation in the Physarum plasmodium.
Biophys. Chem. 82 (1999) 23–28.
[17] Nakagaki T., Yamada H., Ueda T. Interaction between cell shape and con-
traction pattern in the Physarum plasmodium, Biophysical Chemistry 84
(2000) 195–204.
[18] Nakagaki T., Smart behavior of true slime mold in a labyrinth. Research
in Microbiology 152 (2001) 767-770.
14
[19] Nakagaki T., Yamada H., and Toth A., Path finding by tube morphogenesis
in an amoeboid organism. Biophysical Chemistry 92 (2001) 47-52.
[20] Nakagaki T. Special Issue of the Int. J. Unconventional Computing on
Physarum Computing (2009), in press.
[21] Shirakawa T., Adamatzky A., Gunji Y.-P., Miyake Y. On simultaneous
construction of Voronoi diagram and Delaunay triangulation by Physarum
polycephalum (2008), submitted.
[22] Tsuda S., Aono M., Gunji Y.-P. Robust and emergent Physarum logical-
computing. Biosystems 73 (2004) 45–55.
[23] Tsuda S, Zauner K.-P., Gunji Y.-P. Robot control with bio-logical cells.
BioSystems 87 (2007) 215–223.
[24] Yokoi H., Kakazu Y. Theories and applications of autonomic machines
based on the vibrating potential method. Proc. Int. Symp. Distributed
Autonomous Robotics Systems, 1992, 31–38.
[25] Yokoi H., Nagai T., Ishida T., Fujii M., Iida T. Amoeba-like robots in the
perspective of control architecture and mor-phology/materials. In: Hara F,
Pfeifer R (eds.), Morpho-Functional Machines: The New Species, Springer-
Verlag, Tokyo, 2003, 99-129.
15
arXiv:0901.4466v1  [cs.RO]  28 Jan 2009
Physarum boats: If plasmodium sailed it would
never leave a port
Andrew Adamatzky
University of the West of England, Bristol BS16 1QY, United Kingdom
andrew.adamatzky@uwe.ac.uk
October 25, 2018
Abstract
Plasmodium of Physarum polycephalum is a single huge (visible by naked
eye) cell with myriad of nuclei. The plasmodium is a promising substrate
for non-classical, nature-inspired, computing devices. It is capable for ap-
proximation of shortest path, computation of planar proximity graphs and
plane tessellations, primitive memory and decision-making. The unique
properties of the plasmodium make it an ideal candidate for a role of
amorphous biological robots with massive parallel information process-
ing and distributed inputs and outputs. We show that when adhered to
light-weight object resting on a water surface the plasmodium can propel
the object by oscillating its protoplasmic pseudopodia.
In experimen-
tal laboratory conditions and computational experiments we study phe-
nomenology of the plasmodium-floater system, and possible mechanisms
of controlling motion of objects propelled by on board plasmodium.
Keywords: Physarum polycephalum, motility, biological robots
1
Introduction
A plasmodium, or vegetative state, of Physarum polycephalum, also known as
a true or multi-headed slime mold, is a single cell with many diploid nuclei; it
behaves like a giant amoeba. In 2000 Nakagaki et al [17, 18, 19] shown that
topology of the plasmodium’s protoplasmic network optimizes the plasmodium’s
harvesting on distributed sources of nutrients and makes more efficient flow and
transport of intra-cellular components. Thus the field of Physarum computing
emerged [20]. The problems solved by the plasmodium include approximation of
shortest path in a maze [18, 19], computation of proximity graphs [8], Delaunay
triangulation [21], construction of logical gates [22], robot control [23], imple-
mentation of storage-modification machines [7], and approximation of Voronoi
diagram [21].
1
The plasmodium can be seen as a network of biochemical oscillators [14, 16],
where waves of excitation or contraction originate in several sources, travel
along the plasmodium, and interact one with another in collisions. The oscil-
latory cytoplasm of the plasmodium is a spatially extended nonlinear excitable
media. Previously we shown that the plasmodium of Physarum is a biolog-
ical analogue of chemical reaction-diffusion system encapsulated in an elastic
membrane [6]. Such encapsulation allows the plasmodium to act as a massive-
parallel reaction-diffusion computer [4] with parallel inputs and outputs. This
ensures the plasmodium explores environment in a distributed manner and re-
sponds to changes in surrounding conditions as an amorphous decentralized but
yet internally coordinated entity. Thus we can envisage that the plasmodium of
Physarum polycephalum is a promising biological substrate for designing amor-
phous robots with embedded reaction-diffusion intelligence.
An engineering
background for such amoeboid machines was developed by Yokoi et al [24, 25],
however no viable non-silicon implementations are known so far.
There were just two attempts to integrate spatially extended non-linear
chemical or biological systems with silicon hardware robots. In 2003 a wheeled
robot controlled by on board excitable chemical system, the Belousov-Zhabotinsky
medium, was successfully tested in experimental arena [3]. The on board chem-
ical reaction was stimulated by silver wire and motion vector towards source
of stimulation was exctracted by robot from topology of the excitation waves
in the Belousov-Zhabotinsky medium. In 2006 Tsuda et al [23] designed and
successfully tested in real-world conditions a Physarum contoller for a legged
robot. The controller’s functioning was based on the fact that light inhibits
oscillations in illuminated parts of the plasmodium. Thus a direction towards
light source can be calculated from phase differences between shaded and illu-
minated parts of the plasmodium controller. In both instances non-linear media
controllers were coupled with conventional hardware and relied upon additional
silicon devices to convert spatio-temporal dynamics of excitation and oscillation
to the robots motion.
The only known up to date result of self-propelled excitable-medium devices
can be attributed to Kitahata [13] who in 2005 experimentally shown that a
droplet of Belousov-Zhabotinsky medium is capable for translation motion due
to changes in interfacial tension caused by convention forces inside the droplet.
The convention forces are induced by excitation waves, propagating inside the
droplet [12].
Recently we experimentally demonstrated [10] that the plasmodium can act
as a distributed manipulator when placed on a water surface with light-weight
objects distributed around. We shown that the plasmodium senses data-objects,
calculates shortest path between the objects, pushes and pulls light-weight ob-
jects placed on a water surface. We also found that motility of the plasmodium
placed directly on the water surface is restricted [10]. The plasmodium does
propagate pseudopodia and forms protoplasmic trees, however it does not travel
as a localized entity. To achieve true mobility of a plasmodium we decided to at-
tach it to a floater in a hope that due to mechanical oscillations of propagating
pseudopodia the plasmodium will be capable for applying enough propulsive
2
force to the floater to propel the floater on the water surface. In laboratory
experiments we proved feasibility of the approach. In present paper we report
our findings on motion patterns observed in experiments and simulations.
The paper is structured as follows.
Experimental techniques and basics
of simulating a plasmodium-floater system in mobile cellular automaton lat-
tices are presented in Sect. 2.
In Sect. 3 we discuss modes of movement of
plasmodium-floater systems, observed in laboratory experiments.
Results of
computer simulations of the mobile excitable lattices, that mimic the plasmod-
ium behavior, are shown in Sect. 4. We overview the results and discuss our
findings in a context of unconventional robots and robotic controllers in Sect. 5.
2
Methods
Plasmodium of Physarum polycephalum was cultivated on a wet paper towels in
dark ventilated containers. Oat flakes were supplied as a substrate for bacteria
on which the plasmodium feeds. Experiments were undertaken in Petri dishes
with base diameter 35 mm, filled by 1/3-1/5 with distilled water. Variously
sized (but not exceeding 5-7 mm in longest dimension) pieces of plastic and
foam were used as floaters residing on the water surface. Oat flakes occupied by
the plasmodium were placed on top of the floaters. Behavior of the plasmodium-
floater systems was recorded using QX-5 digital microscope, magnification ×10.
Videos of experiments are available at [11].
The Petri dish with plasmodium was illuminated by LED (minimal light set-
ting for the camera) by white light which acted as a stimulus for the plasmodium
phototaxis. The illuminating LED was positioned 2-4 cm above North-East part
of the experimental container. The microscope was placed in the dark box to
keep illumination gradient in the dish safe from disturbances.
Plasmodium-floater systems were simulated by mobile two-dimensional cel-
lular automata, with eight cell neighbourhood, which produce force fields [2].
We employed the model of retained excitation [5]. Every cell of the automaton
takes three states — resting, excited and refractory, and updates its state de-
pending on states of its eight closest neighbors. A resting cell becomes excited
if number of its excited neighbors lies in the interval [θ1, θ2], 1 ≤θ1, θ2 ≤8.
An excited cell remains excited if number of excited neighbors lies in the in-
terval [δ1, δ2], 1 ≤δ1, δ2 ≤8, otherwise the cell takes refractory state. A cell
in refractory state becomes resting in the next time step. This is a model of
’retained excitation’ [5] used to imitate Physarum foraging behavior [6] because
this cellular automaton exhibits ‘amoeba-shaped’ patterns of excitation. We
denote the local transitions functions as R(θ1, θ2, δ1, δ2).
The reaction to light stimulus was implemented as follows. Cells of lattice
edges most distant from the light were excited with probability 0.15. The illu-
mination dependent excitation of edge cells corresponds to stimulus-dependent
changes in plasmodium oscillation frequencies [15, 16]. Namely, the parts of the
plasmodium closest to positive stimulus (e.g. sources of nutrients) periodically
contract with higher frequencies than parts closest to negative stimulus (e.g.
3
high illumination).
We derived motion from excitation dynamics by supplying every cell of the
lattice with a virtual local force vector. The local vectors are updated at every
step of the simulation.
A vector in each cell becomes oriented towards less
excited part of the cell’s neighborhood. An integral vector, which determines
lattice rotation and translation at each step of simulation time, is calculated as
a sum of local vectors over all cells. The approach is proved to be successful in
simulation of mobile excitable lattices, see details in [2].
3
Experimental results
In experiments, we observed five types of movements executed by plasmodium-
floater system: random wandering, quick sliding, pushing, and directed pro-
pelling.
The random wandering is caused by sudden movements, and associated vi-
brations, of plasmodium’s protoplasm. Movement of protoplasm is caused by
peristaltic contraction waves generated by disordered network of biochemical
oscillators. The higher the frequency of contractions in a particular domain of
the plasmodium, the more protoplasm is attracted into the domain [15, 16].
Relocations of protoplasmic mass change mass center of the plasmodium-floater
system, thus causing the floater to wander. The floater influenced by the pro-
toplasmic vibration may exhibit a random motion (Fig. 1a–e), which usually
persists till some part of the plasmodium propagates beyond edges of the floater
(after that another types of motion take place).
The quick sliding motion occurs when substantial amount of protoplasm,
e.g.
a propagating pseudopodium, relocates to one edge of the floater and
this pseudopodium penetrates water surface. The floater then become partly
submerged. This changes the way water surface forces act on the floater, and
sets the floater in a motion. The floater usually slides till collide with an obstacle
or a wall of the container.
The random wandering and sliding are rather types of motion with non
controllable and unpredictable trajectories. These motions unlikely to be used
in future architectures of biological robots so we not discuss them further.
The pushing motion can be observed in situations when plasmodium propa-
gates from the floater onto the water surface and develops a tree of protoplasmic
tubes. At some stage of the development tips of the growing protoplasmic tree
reach walls of the water container and adhere to the walls. An example is shown
in Fig. 2: three-four hours after being placed on a floater the plasmodium starts
propagation (Fig. 2bc) and a protoplasmic tree emerges (Fig. 2f). Already at
this stage the floater comes into motion (Fig. 2fg), traveling in the direction op-
posite to propagation of the protoplasmic tree. The motion becomes pronounced
when a part of the tree attaches itself to a wall of the container (Fig. 2h), a
force of protoplasm pumped into the tubes causes the floater to noticeably move
outwards the growing tubes (Fig. 2i).
A position of the floater can be ‘fine-tuned’ by several protoplasmic trees,
4
Figure 1:
Transition between types of motion of physarum-floater system.
Random wandering, or vibration-based, movement (a)–(e), 90o clockwise ro-
tation (e)–(f), and directed propelling (f)–(h). Positions of the floater, at time
steps corresponding to snapshots (a)–(h) and trajectory the floater’s centre are
shown in (i). See video at [11].
5
Figure 2: Pushing the floater by growing protoplasmic tubes.
(a)–(h) Pho-
tographs of experimental container. (i) Time-lapsed contours of the floater. See
video at [11].
6
Figure 3: Collective movement of the floater by several protoplasmic trees. (a)–
(h) Photographs of the experimental container. (i) Time-lapse contours of the
floater. See video at [11].
7
which attached themselves to different sides of experimental container (Fig. 3).
In experiment shown in Fig. 3 protoplasmic trees grow along North-West to
South-East arc and to the direction North-East (Fig. 3bc). The sub-tree growing
in the North-East direction diminishes with time, because it happen to be in
the most illuminated area of the container, the only branch heading South
survives. The protoplasmic tree becomes stronger in the North-West direction
(Fig. 3bc)g–h). The dynamical reallocations of the trees cause the floater to
rotate and then gradually move South-West (Fig. 3i).
The last and the most interesting type of motion observed is direct pro-
pelling of floater by on board plasmodium. This type of motion occurs when
pseudopodia propagate beyond the edge of the floater but then partly sank.
The pseudopodia oscillate, their oscillations are clearly visible on the video
recordings in [11]. These oscillations cause the floater to move in the direction
opposite to submersed pseudopodia (in contrast to sliding motion, the floater
does not submerse at all during the propulsive motion). The examples of direct
propelling are shown in Figs. 1 and 4.
In experiment shown in Fig. 1 plasmodium propagates a pseudopodium in
South-West direction (Fig. 1de). The pseudopodium does not stay on the water
surface but partly sinks. The initial wetting of the pseudopodium causes the
floater to rotate clockwise with the protruding pseudopodium facing now North-
West (Fig. 1de). The immersed pseudopodia oscillate and propel the floater in
the South-Easterly direction (Fig. 1g–i).
An example of collective steering of the floater by several pseudopodia is
shown in Fig. 4. After initial adaptation period (Fig. 4ab), associated with vi-
brational random motion, the plasmodium propagates pseudopodia in Westward
and South-Easterly directions (Fig. 4de). At the beginning only South-Eastern
pseudopodia contribute to propelling of the floater, with Western pseudopo-
dia are merely balancing action of their South-Eastern counterparts. Thus the
floater travels towards North (Fig. 4fg). Later Westerly pseudopodia increase
frequency of their oscillations (see [11]) and the floater turn more towards North-
East (Fig. 4hi).
4
Results of simulations
Proposition 1. Given large enough container and a stationary source of light
a floater with plasmodium on board will move in irregular cycles around the
source.
By assuming the container is large enough we secured the situation when
it is impossible for pseudopodia, or protoplasmic trees, to reach walls of the
container and push the floater. The wandering motion only occurs at the very
beginning of plasmodium’s development on the floater. Thus we are left with
propelling motion.
Let the floater be stationary.
Due to negative phototaxis, plasmodium’s
pseudopodia propagate towards the less illuminated parts of the container.
8
Figure 4: Direct propelling of floater by oscillating pseudopodia. (a)–(i) Pho-
tographs of the experimental container. (j) Time lapses contours of the floater.
See video at [11].
9
When the pseudopodia spawn from the floater to the water surface, their os-
cillatory contractions propel the floater towards source of illumination.
The
pseudopodia continue their action till the floater pass the highly illuminated
part. Then the pseudopodia happens to be on the most illuminated side of the
floater. They retract and new pseudopodia are formed on the less illuminated
part of the floater.
They propel the floater towards the source of illumina-
tion again. Exact positions of growing pseudopodia depends on distribution of
biochemical sources of oscillations and interactions between traveling waves of
contraction. New pseudopodia may not emerge at the same place as old pseu-
dopodia. Thus each trajectory of the floater towards the source of illumination
will be different from previous ones.
To verify the proposition we simulated plasmodium-floater system by mobile
cellular automata with various cell-state transition functions. An illustrative
series of snapshots is shown in Fig. 5. The automaton lattice was placed at some
distance from the source of light (Fig. 5a). Edges of the lattice oriented towards
less illuminated area of experimental container became excited (Fig. 5b). The
excitation propels the lattice towards the source of light (Fig. 5e–f).
When
lattice passes the site with highest illumination, another edge of the lattice
excites and propels the lattice back to the domain of high illumination (Fig. 5gh).
The sequence of excitations continues and the lattice travels along irregular
cyclic trajectories around the source of illumination (Fig. 5i–k).
Exact pattern of lattice cycling around the source of illumination is deter-
mined by particulars of the local excitation dynamics. For example, trajectories
of a lattice with threshold excitation (Fig. 6a) are very compactly arranged
around the source: as soon as the lattice approaches the source of illumination
it stays nearby, mostly turning back and force. When we impose upper bound-
ary of the excitation, e.g. when a resting cell is excited only if one or two of its
neighbors are excited, the lattice trajectory loosens (Fig. 6b).
Let us discuss now few examples of lattices governed by functions R(θ1θ2δ1δ2),
where a resting cell excites if number σ of excited neighbors lies in the interval
[θ1, θ2] and excited cell can stay excited if σ ∈[δ1, δ2] (Fig. 6c–f). We focus on
excitation interval [2, 2] because this type of local transition is typical for media
with traveling localized excitations [1], which does closely relate to propagation
activities of Physarum polycephalum [9]. We observed that lattices with very
narrow intervals of excitation and retained excitation demonstrate a combina-
tion of compact and lose trajectories (Fig. 6c). Widening interval of retained
excitation disperses lattice trajectories in space (Fig. 6de), thus emulating de-
layed responses to changes in sensorial background. When lower, δ1, and upper,
δ2, boundaries of retained excitation increase to 4 and 6, respectively, a lattice
starts to exhibit a quasi-ordered behavior: combination of long runs away from
the source and tidy loops of rotations (Fig. 6f).
10
Figure 5: Snapshots of cellular automaton model (two-dimensional lattice of
200 × 200 cells) of plasmodium-floater system. The local excitation dynamics is
controlled by function R(2201). Source of illumination is shown by solid black
disc. Each snapshot is supplied with trajectory of the center of the floater from
the beginning of simulation. See exemplar videos at [11].
11
Figure 6:
Trajectories of the center of cellular-automaton lattice traveling
around a source of illumination.
The lattice’s behavior is governed by local
transition functions as follows: (a) R(1899), threshold excitation, a resting cell
excites if there is at least one excited neighbor, values 9 for δ1 and δ2 mean that
the cell never stays excited longer then one step of a time, (b) R(1299), a rest-
ing cell excites if it has one or two excited neighbors. (c) R(2222), (d) R(2201),
(e) R(2211), (f) R(2246),
12
5
Summary
In laboratory experiments and computer simulations we demonstrated that plas-
modium of Physarum polycephalum can act as an ‘engine’ or a propelling agent
for light-weight floating objects. We found that most typical type of movement
generated by the plasmodium is a propulsive forward motion. This motion is
caused by pseudopodia protruding beyond the the floater and oscillating. In
small container, when growing tree of plasmodium’s protoplasmic tubes can
reach sides of the container, the plasmodium pushes floater by increasing length
of the tubes connecting the float and the container’s sides.
A plasmodium shows negative phototaxis. It tries to evade regions illumi-
nated by non-yellow light and grows towards more shaded areas. When the
plasmodium attached to a floating object, plasmodium–floater system exhibits
positive phototaxis. Due to growth, and associated oscillations, of pseudopo-
dia on the less illuminated side of a floater, the floater moves towards light.
We mimicked this phenomenon in cellular automaton models of mobile light-
sensitive lattices. We found that in ideal conditions the “plasmodium–floater”
system will wander around the site of highest illumination, often following quasi-
chaotic trajectories due to many sources of excitation competing with one an-
other in the protoplasm.
The phenomena and primitive constructs of the plasmodium-floaters will
be employed in future designs of amorphous decentralized robots operating on
water-surface. The robots will be capable to search for objects, travel towards
the objects’ location, implement manipulation and sorting. The robots will be
controlled by gradient fields of illumination and chemo-attractants.
Acknowledgement
The works is supported by the Leverhulme Trust Research Grant F/00577/I.
References
[1] Adamatzky A. Computing in Nonlinear Media and Automata Collectives.
IoP Publishing, 2001.
[2] Adamatzky A. and Melhuish C. Phototaxis of mobile excitable lattices
Chaos, Solitons, & Fractals, 13 (2002) 171–184.
[3] Adamatzky A., De Lacy Costello B., Melhuish C., RatcliffN. Experimental
implementation of mobile robot taxis with onboard BelousovZhabotinsky
chemical medium Materials Science and Engineering: C 24 (2004) 541–548.
[4] Adamatzky A., De Lacy Costello B., Asai T. Reaction-Diffusion Comput-
ers, Elsevier, Amsterdam, 2005.
13
[5] Adamatzky A. Phenomenology of retained excitation. Int. J. Bifurcation
and Chaos (IJBC) 17 (2007) 3985–4014.
[6] Adamatzky A. Physarum machines: encapsulating reaction-diffusion to
compute spanning tree. Naturwisseschaften 94 (2007) 975–980.
[7] Adamatzky A. Physarum machine:
Implementation of a Kolmogorov-
Uspensky machine on a biological substrate. Parallel Processing Letters
17 (2007) 455–467.
[8] Adamatzky A. Developing proximity graphs by Physarum polycephalum:
does the plasmodium follow the Toussaint hierarchy? Parallel Processing
Letters (2009), in press.
[9] Adamatzky A., De Lacy Costello B., Shirakawa T. Universal computation
with limited resources: Belousov-Zhabotinsky and Physarum computers.
Int. J. Bifurcation and Chaos (2008), in press.
[10] Adamatzky A. and Jones J. Towards Physarum robots: computing and
manipulating on water surface. J Bionic Engineering 5 (2008) 348–357.
[11] Adamatzky
A.
2008.
Supplementary
materials
to
present
paper:
http://uncomp.uwe.ac.uk/adamatzky/physarumboat.
[12] Kitahata H., Aihara R., Magome N., Yoshikawa K., Convective and peri-
odic motion driven by a chemical wave. J. Chem. Phys. 116 (2002) 5666–
5672.
[13] Kitahata H. Spontaneous motion of a droplet coupled with a chemical re-
action. Prog. Theor. Phys. Suppl., 161 (2006) 220–223.
[14] Matsumoto K, Ueda T, Kobatake Y. Reversal of thermotaxis with oscilla-
tory stimulation in the plasmodium of Physarum polycephalum. Journal of
Theoretical Biology 130 (1980) 175–182.
[15] Miyake Y.,
Tabata S.,
Murakami H.,
Yano M. and Shimizu H.,
Environment-dependent self-organization of positional information field in
chemotaxis of Physarum Plasmodium. J. Theor. Biol. 178 (1996) 341–353.
[16] Nakagaki T., Yamada H. and Ueda T. Modulation of cellular rhythm and
photoavoidance by oscillatory irradiation in the Physarum plasmodium.
Biophys. Chem. 82 (1999) 23–28.
[17] Nakagaki T., Yamada H., Ueda T. Interaction between cell shape and con-
traction pattern in the Physarum plasmodium, Biophysical Chemistry 84
(2000) 195–204.
[18] Nakagaki T., Smart behavior of true slime mold in a labyrinth. Research
in Microbiology 152 (2001) 767-770.
14
[19] Nakagaki T., Yamada H., and Toth A., Path finding by tube morphogenesis
in an amoeboid organism. Biophysical Chemistry 92 (2001) 47-52.
[20] Nakagaki T. Special Issue of the Int. J. Unconventional Computing on
Physarum Computing (2009), in press.
[21] Shirakawa T., Adamatzky A., Gunji Y.-P., Miyake Y. On simultaneous
construction of Voronoi diagram and Delaunay triangulation by Physarum
polycephalum (2008), submitted.
[22] Tsuda S., Aono M., Gunji Y.-P. Robust and emergent Physarum logical-
computing. Biosystems 73 (2004) 45–55.
[23] Tsuda S, Zauner K.-P., Gunji Y.-P. Robot control with bio-logical cells.
BioSystems 87 (2007) 215–223.
[24] Yokoi H., Kakazu Y. Theories and applications of autonomic machines
based on the vibrating potential method. Proc. Int. Symp. Distributed
Autonomous Robotics Systems, 1992, 31–38.
[25] Yokoi H., Nagai T., Ishida T., Fujii M., Iida T. Amoeba-like robots in the
perspective of control architecture and mor-phology/materials. In: Hara F,
Pfeifer R (eds.), Morpho-Functional Machines: The New Species, Springer-
Verlag, Tokyo, 2003, 99-129.
15