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Rules for Biologically Inspired Adaptive Network Design


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Transport networks are ubiquitous in both social and biological systems. Robust network performance involves a complex trade-off involving cost, transport efficiency, and fault tolerance. Biological networks have been honed by many cycles of evolutionary selection pressure and are likely to yield reasonable solutions to such combinatorial optimization problems. Furthermore, they develop without centralized control and may represent a readily scalable solution for growing networks in general. We show that the slime mold Physarum polycephalum forms networks with comparable efficiency, fault tolerance, and cost to those of real-world infrastructure networks—in this case, the Tokyo rail system. The core mechanisms needed for adaptive network formation can be captured in a biologically inspired mathematical model that may be useful to guide network construction in other domains.
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DOI: 10.1126/science.1177894
, 439 (2010); 327Science et al.Atsushi Tero,
Rules for Biologically Inspired Adaptive Network
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Supporting Online Material
SOM Text
Figs. S1 to S8
Tables S1 and S2
References and Notes
1 June 2009; accepted 1 December 2009
Published online 10 December 2009;
Include this information when citing this paper.
Rules for Biologically Inspired
Adaptive Network Design
Atsushi Tero,
Seiji Takagi,
Tetsu Saigusa,
Kentaro Ito,
Dan P. Bebber,
Mark D. Fricker,
Kenji Yumiki,
Ryo Kobayashi,
Toshiyuki Nakagaki
Transport networks are ubiquitous in both social and biological systems. Robust network performance
involves a complex trade-off involving cost, transport efficiency, and fault tolerance. Biological
networks have been honed by many cycles of evolutionary selection pressure and are likely to yield
reasonable solutions to such combinatorial optimization problems. Furthermore, they develop without
centralized control and may represent a readily scalable solution for growing networks in general. We
show that the slime mold Physarum polycephalum formsnetworkswithcomparableefficiency,fault
tolerance, and cost to those of real-world infrastructure networksin this case, the Tokyo rail system.
The core mechanisms needed for adaptive network formation can be captured in a biologically
inspired mathematical model that may be useful to guide network construction in other domains.
Transport networks are a critical part of the
infrastructure needed to operate a modern
industrial society and facilitate efficient
movement of people, resources, energy, and
information. Despite their importance, most net-
works have emerged without clear global design
principles and are constrained by the priorities
imposed at their initiation. Thus, the main motiva-
tion historically was to achieve high transport
efficiency at reasonable cost, but with correspond-
ingly less emphasis on making systems tolerant to
interruption or failure. Introducing robustness
inevitably requires additional redundant pathways
that are not cost-effective in the short term. In recent
years, the spectacular failure of key infrastructure
such as power grids (1,2), financial systems (3,4),
airline baggage-handling systems (5), and railway
networks (6), as well as the predicted vulnerability o f
systems such as information networks (7)orsupply
networks (8) to attack, have highlighted the need to
develop networks with greater intrinsic resilience.
Some organisms grow in the form of an inter-
connected network as part of their normal forag-
ing strategy to discover and exploit new resources
(912). Such systems continuously adapt to their
environment and must balance the cost of produc-
ing an efficient network with the consequences of
even limited failure in a competitive world. Unlike
anthropogenic infrastructure systems, these biolog-
ical networks have been subjected to successive
rounds of evolutionary selection and are likely to
have reached a point at which cost, efficiency, and
resilience are appropriately balanced. Drawing in-
spiration from biology has led to useful approaches
to problem-solving such as neural networks, ge-
netic algorithms, and efficient search routines de-
veloped from ant colony optimization algorithms
(13). We exploited the slime mold Physarum
polycephalum to develop a biologically inspired
model for adaptive network development.
Physarum is a large, single-celled amoeboid
organism that forages for patchily distributed
food sources. The individual plasmodium ini-
tially explores with a relatively contiguous for-
aging margin to maximize the area searched.
However, behind the margin, this is resolved into
a tubular network linking the discovered food
sources through direct connections, additional in-
termediate junctions (Steiner points) that reduce
the overall length of the connecting network,
and the formation of occasional cross-links that
improve overall transport efficiency and resil-
ience (11,12). The growth of the plasmodium is
influenced by the characteristics of the sub-
strate (14) and can be constrained by physical
barriers (15) or influenced by the light regime
(16), facilitating experimental investigation of
the rules underlying network formation. Thus,
for example, Physarum can find the shortest
path through a maze (1517) or connect dif-
ferent arrays of food sources in an efficient
manner with low total length (TL) yet short
average minimum distance (MD) between pairs
of food sources (FSs), with a high degree of
fault tolerance (FT) to accidental disconnection
(11,18,19). Capturing the essence of this sys-
tem in simple rules might be useful in guiding
the development of decentralized networks in
other domains.
We ob s erv e d Physarum connecting a template
of 36 FSs that represented geographical locations
of cities in the Tokyo area, and compared the result
with the actual rail network in Japan. The
Physarum plasmodium was allowed to grow from
Tokyo and initially filled much of the available
land space, but then concentrated on FSs by
thinning out the network to leave a subset of larger,
interconnecting tubes (Fig. 1). An alternative
protocol, in which the plasmodium was allowed
to extend fully in the available space and the FSs
were then presented simultaneously, yielded sim-
ilar results. To complete the network formation, we
allowed any excess volume of plasmodium to
Research Institute for Electronic Science, Hokkaido University,
Sapporo 060-0812, Japan.
PRESTO, JST, 4-1-8 Honcho,
Kawaguchi, Saitama, Japan.
Graduate School of Engineering,
Hokkaido University, Sapporo 060-8628, Japan.
Department of
Plant Sciences, University of Oxford, Oxford OX1 3RB, UK.
Department of Mathematical and Life Sciences, Hiroshima
University, Higashi-Hiroshima 739-8526, Japan.
Sanbancho, Chiyoda-ku, Tokyo, 102-0075, Japan.
*To whom correspondence should be addressed. E-mail: SCIENCE VOL 327 22 JANUARY 2010 439
on January 26, 2010 www.sciencemag.orgDownloaded from
accumulate on a large FS outside the arena (LFS
in Fig. 2A).
A range of network solutions were apparent
in replicate experiments (compare Fig. 2A with
Fig. 1F); nonetheless, the topology of many
Physarum networks bore similarity to the real rail
network (Fig. 2D). Some of the differences may
relate to geographical features that constrain the rail
network, such as mountainous terrain or lakes.
These constraints were imposed on the Physarum
network by varying the intensity of illumination, as
the plasmodium avoids bright light (16). This
yielded networks (Fig. 2, B and C) with greater
visual congruence to the real rail network (Fig. 2D).
Networks were also compared with the minimal
spanning tree (MST, Fig. 2E), which is the shortest
possible network connecting all the city positions,
and various derivatives with increasing numbers of
cross-links added (e.g., Fig. 2F), culminating in a
fully connected Delaunay triangulation, which rep-
resents the maximally connected network linking
all the cities.
The performance of each network was char-
acterized by the cost (TL), transport efficiency
(MD), and robustness (FT), normalized to the
corresponding value for the MST to give TL
. The TL of the Tokyo rail
network was greater than the MST by a factor
of ~1.8 (i.e., TL
1.8), whereas the average
for Physarum was 1.75 T0.30 (n= 21).
Illuminated networks gave slightly better clus-
tering around the value for the rail network (Fig.
3A). For comparison, the Delaunay triangulation
was longer than the MST by a factor of ~ 4.6.
Thus, the cost of the solutions found by Physarum
closely matched that of the rail network, with
about 30% of the maximum possible number of
links in place. The transport performance of the
two networks was also similar, with MD
0.85 and 0.85 T0.04 for the rail network and the
Physarum networks, respectively. However, the
Physarum networks achieved this with margin-
ally lower overall cost (Fig. 3A).
The converse was true for the fault tolerance
) in which the real rail network showed
marginally better resilience, close to the lowest
level needed to give maximum tolerance to a single
random failure. Thus, only 4% of faults in the rail
whereas 14 T4% would disconnect the illuminated
Physarum networks, and 20 T13% would
disconnect the unconstrained Physarum networks.
In contrast, simply adding additional links to the
MST to improve network performance resulted
in networks with poor fault tolerance (Fig. 3B).
The trade-off between fault tolerance and cost
was captured in a single benefit-cost measure, ex-
pressed as the ratio of FT/TL
=a. In general,
the Physarum networks and the rail network had
a benefit/cost ratio of ~0.5 for any given TL
(Fig. 3B). The relationship between different a
values and transport efficiency (Fig. 3C) high-
lighted the similarity in aggregate behavior of the
Physarum network when considering all three per-
formance measures (MD
Fig. 1. Network formation in Physa-
rum polycephalum.(A)Att=0,a
small plasmodium of Physarum was
placed at the location of Tokyo in an
experimental arena bounded by the
Pacific coastline (white border) and
supplemented with additional food
the region (white dots). The horizontal
widthofeachpanelis17cm.(Bto F)
The plasmodium grew out from the
initial food source with a contiguous
margin and progressively colonized
each of the food sources. Behind the
growing margin, the spreading myce-
lium resolved into a network of tubes
interconnecting the food sources.
0 hr
11 hr
5 hr
16 hr
8 hr
26 hr
Fig. 2. Comparison of the Physarum
networks with the Tokyo rail network.
(A) In the absence of illumination, the
Physarum network resulted from even
exploration of the available space. (B)
Geographical constraints were imposed
on the developing Physarum network
by means of an illumination mask to
restrict growth to more shaded areas
corresponding to low-altitude regions.
The ocean and inland lakes were also
given strong illumination to prevent
growth. (Cand D) The resulting network
(C) was compared with the rail network
in the Tokyo area (D). (Eand F)The
minimum spanning tree (MST) con-
necting the same set of city nodes (E)
and a model network constructed by
adding additional links to the MST (F).
22 JANUARY 2010 VOL 327 SCIENCE www.sciencemag.org440
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The rail network was embedded in the cluster of
results for the Physarum networks with a margin-
ally higher avalue for the same transport effi-
ciency (Fig. 3C).
Overall, we conclude that the Physarum net-
works showed characteristics similar to those of
the rail network in terms of cost, transport efficien-
cy, and fault tolerance. However, the Physarum
networks self-organized without centralized con-
trol or explicit global information by a process of
selective reinforcement of preferred routes and
simultaneous removal of redundant connections.
We developed a mathematical model for adapt-
ive network construction to emulate this behavior,
based on feedback loops between the thickness of
each tube and internal protoplasmic flow (1822)
in which high rates of streaming stimulate an in-
crease in tube diameter, whereas tubes tend to de-
cline at low flow rates (23). The initial shape of a
plasmodium is represented by a randomly meshed
lattice with a relatively fine spacing, as shown in
Fig. 4 (t= 0). The edges represent plasmodial
tubes in which protoplasm flows, and nodes are
junctions between tubes. Suppose that the pres-
sures at nodes iand jare p
and p
, respectively,
and that the two nodes are connected by a cyl-
and radius r
. Assuming that
flow is laminar and follows the Hagen-Poiseuille
equation, the flux through the tube is
Qij ¼
where his the viscosity of the fluid, and D
/8his a measure of the conductivity of the
tube. As the length L
is a constant, the behavior
of the network is described by the conductivities,
, of the edges.
At each time step, a random FS (node 1) is
selected to drive flow through the network, so the
flux includes a source term S
. A second
random FS is chosen as a sink (node 2) with a
corresponding withdrawal of I
such that S
. As the amount of fluid must be conserved,
the inflow and outflow at each internal node must
balance so that i(i1, 2), S
= 0. Thus, for a
given set of conductivities and selected source
and sink nodes, the flux through each of the
network edges can be computed.
To accommodate the adaptive behavior of the
plasmodium, the conductivity of each tube evolves
according to dD
/dt =f(|Q
|) D
on the right side describes the expansion of tubes in
response to the flux. The second term represents
the rate of tube constriction, so that in the absence
of flow the tubes will gradually disappear. The
functional form f(|Q|) is given by f(|Q|) = |Q|
/(1 +
), which describes a sigmoidal response where g
is a parameter that controls the nonlinearity of feed-
back (g> 0). A typical simulation result with I
and g= 1.8 (Fig. 4) gave a network with features
similar to those of both the Physarum system and
the rail network (Fig. 2, C and D, respectively).
In general, increasing I
promoted the for-
mation of alternative routes that improved per-
formance by reducing MD
and made the
network more fault-tolerant, but with increased
cost (Fig. 3, A to C, and fig. S1I). Low values of g
also gave a greater degree of cross-linking with
an increased number of Steiner points (fig. S2, A
and B). Conversely, decreasing I
(fig. S1A) or
increasing g(fig. S2I) drove the system toward a
low-cost MST (Fig. 2E), but with an inevitable
decrease in resilience (Fig. 3B). The final net-
work solution also depended slightly on the
stochastic variation assigned to the starting values
of D
. Judicious selection of specific parameter
combinations (I
= 0.20, g= 1.15) yielded net-
works with remarkably similar topology and
metrics to the Tokyo rail network (fig. S2B). How-
ever, by increasing I
to 2 and gto 1.8, the simula-
tion model also achieved a benefit/cost ratio (a=
) that was better than those of the rail or
Physarum networks, reaching a value of 0.7 with
an almost identical transport efficiency of 0.85
(Fig. 3C). Conversely, the consequence of the in-
creased TL
observed in the rail or Physarum
networks would be to confer greater resilience to
Fig. 3. Transport performance,
resilience, and cost for Physa-
rum networks, model simula-
tions, and the real rail networks.
(A) Transport performance of
each network, measured as the
minimum distance between all
pairs of nodes, normalized to
the MST (MD
against the total length of the
network normalized by the MST
Black circles and blue squares
represent results obtained from
Physarum in the absence or
presence of illumination, respectively. The green triangle represents the actual
rail network. Open red circles represent simulation results as I
was varied from
0.20 to 7.19 at a fixed g( = 1.80) and initial random fluctuations of D
tolerance (FT), measured as the probability of disconnecting part of the network
with failure of a single link. Crosses represent results for reference networks; other
symbols as in (A). Different values of the benefit/cost ratio, a=FT/TL
shown as dashed lines. (C) Relationship between MD
and a. Although the
overall performance of the experiment and that of the real rail network are
clustered together, the simulation model achieves better fault tolerance for the
same transport efficiency.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Performance (MDMST)
1.0 1.5 2.0 2.5 3.0
Fault tolerance (FT)
1.0 1.5 2.0 2.5 3.0
Performance (MDMST)
Cost (TLMST)Cost (TLMST)Efficiency (FT / TLMST)
Fig. 4. Network dynamics for the
simulation model. In this typical time
course for evolution of the simula-
tion, time (t)isshowninarbitrary
was modeled as a single FS, apart
from Tokyo, which was an aggregate
of seven FSs to match the importance
of Tokyo as the center of the region.
At the start (t= 0), the available
space was populated with a finely
meshed network of thin tubes. Over
time, many of these tubes died out,
whilst a limited number of tubes be-
came selectively thickened to yield
a stable, self-organized solution. g=
1.80, I
t=29950 SCIENCE VOL 327 22 JANUARY 2010 441
on January 26, 2010 www.sciencemag.orgDownloaded from
multiple simultaneous failures at the expense of
increased cost, rather than tolerance to a single
disconnection that is evaluated by FT
Our biologically inspired mathematical model
can capture the basic dynamics of network
adaptability through iteration of local rules and
produces solutions with properties comparable to
or better than those of real-world infrastructure
networks. Furthermore, the model has a number
of tunable parameters that allow adjustment of
the benefit/cost ratio to increase specific features,
such as fault tolerance or transport efficiency, while
keeping costs low. Such a model may provide a
useful starting point to improve routing protocols
and topology control for self-organized networks
such as remote sensor arrays, mobile ad hoc net-
works, or wireless mesh networks (24).
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Supporting Online Material
Figs. S1 and S2
17 June 2009; accepted 20 November 2009
Measurement of Universal
Thermodynamic Functions for a
Unitary Fermi Gas
Munekazu Horikoshi,
*Shuta Nakajima,
Masahito Ueda,
Takashi Mukaiyama
Thermodynamic properties of matter generally depend on the details of interactions between its
constituent parts. However, in a unitary Fermi gas where the scattering length diverges,
thermodynamics is determined through universal functions that depend only on the particle
density and temperature. By using only the general form of the equation of state and the
equation of force balance, we measured the local internal energy of the trapped gas as a
function of these parameters. Other universal functions, such as those corresponding to the
Helmholtz free energy, chemical potential, and entropy, were calculated through general
thermodynamic relations. The critical parameters were also determined at the superfluid
transition temperature. These results apply to all strongly interacting fermionic systems,
including neutron stars and nuclear matter.
Degenerate two-component Fermi systems
with large scattering lengths are of great
interest in diverse settings such as neutron
stars (13), quark-gluon plasma (4), high critical
temperature (T
) superconductors (5), and reso-
nantly interacting cold Fermi gases near Feshbach
resonances (618). Even though the temperature
of these systems ranges widely from 10
cold atoms to more than 10
K for quark-gluon
plasma, they exhibit remarkably similar behav-
ior at the unitarity limit. As the scattering length
diverges, the universal thermodynamics that de-
scribes these systems depends only on the particle
density, n, and temperature, T. This assumption is
referred to as the universal hypothesis (UH)
In the context of cold atoms, two fermionic
alkali elements,
Li and
K, have been suc-
cessfully used to explore the physics of the uni-
tarity limit (618). This was possible because
of the tunability of the fermion-fermion interac-
tion and the stability of ultracold fermionic
gases near Feshbach resonances (21,22).
Recently, a comparison of the entropy-energy
relations extracted from experimental measure-
ments on both
Li and
K provided evidence of
universal thermodynamics at the unitarity limit
(23). However, because a unitary Fermi gas is
realized in a harmonic trap, the inhomogeneous
atomic density distribution causes the thermo-
dynamic quantities to be position-dependent.
Therefore, integration over the entire cloud pro-
vides only indirect information on the relation-
ship between each individual thermodynamic
quantity and the particle density. To determine
the universal thermodynamic functions using such
an inhomogeneous system, the thermodynamic
Japan Science and Technology Agency, Exploratory Research for
Advanced Technology (ERATO), Macroscopic Quantum Control
Project, 2-11-16 Yayoi, Bunkyo-ku, Tokyo 113-8656, Japan.
Department of Physics, University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113-0033, Japan.
Center for Frontier Science
and Engineering, University of Electro-Communications, 1-5-1
Chofugaoka, Chofu, Tokyo 182-8585, Japan.
*To whom correspondence should be addressed. E-mail:
Fig. 1. Universal function of the internal en-
ergy. Universal functions of the internal energy
) plotted for an ideal Fermi gas
(green diamonds) and for a unitary Fermi gas
(red circles). The data are averaged over a suit-
able temperature range. The error bars show
the data spread of one standard deviation
originating mainly from statistical errors. The
green dashed curve shows the theoretical uni-
versal function for the ideal Fermi gas, whereas
the red solid curve shows the measured univer-
sal function for the unitary Fermi gas. The red
solid curve is obtained by fitting the data repre-
sented by red circles so that it levels off at f
[0] =
3(1 + b)/5 = 0.25 at the low-temperature limit,
where bis the universal parameter (15), and ap-
proaches the theoretical value obtained at the
high-temperature limit (20). The blue square cor-
responds to the critical point.
on January 26, 2010 www.sciencemag.orgDownloaded from
... This is a well-studied problem and a variety of approaches have been suggested, such as shortest-path minimization [4,5] and assignment strategies [6]. Other approaches that are based on adaptation dynamics [7][8][9] have also been proposed to model biological distribution networks. ...
... However, these approaches fall short on describing realistic scenarios where transport flows are limited by constraints, requiring a more general theory of optimal transport (OT). OT has been used to model and optimize various aspects of transport networks such as network design [7,[9][10][11] and traffic flows [12][13][14][15][16]. These approaches guarantee a principled and computationally efficient way of solving transportation problems on networks. ...
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Finding optimal trajectories for multiple traffic demands in a congested network is a challenging task. Optimal transport theory is a principled approach that has been used successfully to study various transportation problems. Its usage is limited by the lack of principled and flexible ways to incorporate realistic constraints. We propose a principled physics-based approach to impose constraints flexibly in such optimal transport problems. Constraints are included in mirror descent dynamics using the principle of D'Alembert-Lagrange from classical mechanics. This leads to a sparse, local and linear approximation of the feasible set leading in many cases to closed-form updates.
... This is a well-studied problem and a variety of approaches have been suggested, such as shortest-path minimization [4,5] and assignment strategies [6]. Other approaches that are based on adaptation dynamics [7][8][9] have also been proposed to model biological distribution networks. ...
... However, these approaches fall short on describing realistic scenarios where transport flows are limited by constraints, requiring a more general theory of optimal transport (OT). OT has been used to model and optimize various aspects of transport networks such as network design [7,[9][10][11] and traffic flows [12][13][14][15][16]. These approaches guarantee a principled and computationally efficient way of solving transportation problems on networks. ...
Full-text available
Finding optimal trajectories for multiple traffic demands in a congested network is a challenging task. Optimal transport theory is a principled approach that has been used successfully to study various transportation problems. Its usage is limited by the lack of principled and flexible ways to incorporate realistic constraints. We propose a principled physics-based approach to impose constraints flexibly in such optimal transport problems. Constraints are included in mirror descent dynamics using the principle of D'Alembert-Lagrange from classical mechanics. This leads to a sparse, local and linear approximation of the feasible set leading in many cases to closed-form updates.
Electrical activity of fungus Pleurotus ostreatus is characterised by slow (hours) irregular waves of baseline potential drift and fast (minutes) action potential likes spikes of the electrical potential. An exposure of the mycelium colonised substrate to a chloroform vapour lead to several fold decrease of the baseline potential waves and increase of their duration. The chloroform vapour also causes either complete cessation of spiking activity or substantial reduction of the spiking frequency. Removal of the chloroform vapour from the growth containers leads to a gradual restoration of the mycelium electrical activity.
We propose that fungi Basidiomycetes can be used as computing devices: information is represented by spikes of electrical activity, a computation is implemented in a mycelium network and an interface is realised via fruit bodies. In a series of scoping experiments we demonstrate that electrical activity recorded on fruits might act as a reliable indicator of the fungi’s response to thermal and chemical stimulation. A stimulation of a fruit is reflected in changes of electrical activity of other fruits of a cluster, i.e. there is distant information transfer between fungal fruit bodies. In an automaton model of a fungal computer we show how to implement computation with fungi and demonstrate that a structure of logical functions computed is determined by mycelium geometry.
Bionic trabecular bone scaffolds (Bio-Tb-S), which mimic the human cancellous bone, are widely recognized as the most effective repair material for large bone defects. Nevertheless, in situations with bone defects located in high-stress areas with intricate and varying mechanical environments, the scaffold necessitates reinforcement to withstand the limit stress. Achieving a balance between the “mechanical load-bearing properties, elastic modulus, and permeability” represents a substantial challenge. A scaffold with locally personalized reinforced was proposed based on slime mould algorithm (SMA), called bionic trabecular bone scaffold with slime mould algorithm (Bio-Tb-S-SMA), that is capable of echoing hard tissue characteristics. Additive manufacturing (AM) was utilized to fabricate the scaffold owing to its superior manufacturing flexibility. Compression tests, friction tests and bidirectional fluid–structure interaction experiments to evaluate the mechanical load-bearing properties, service stability, and permeability of scaffold. The results demonstrate that Bio-Tb-S-SMA effectively adapts to the complex in vivo stress environment, exhibiting strong bone-bearing capacity and service stability, while simultaneously retaining excellent permeability. Bio-Tb-S-SMA has broad potential for use as a bone repair material, and the bionic design and bionic reinforcement algorithm inject new vitality into the field of tissue engineering.
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Fungal mycelial networks are essential for translocating and storing water, nutrients, and carbon in forest ecosystems. In particular, wood decay fungi form mycelial networks that connect various woody debris on the forest floor. Understanding their foraging strategies is crucial for complehending the role of mycelium in carbon and nutrient cycling in forests. Previous studies have shown that mycelial networks initiate migration from the original woody resource (inoculum) to a new woody resource (bait) if the latter is sufficiently large but not if it is small. However, the impact of energetic costs during foraging, such as the distance to the bait, has not been considered. In the present study, we conducted full-factorial experiments with two factors, bait size (4 and 8 cm ³ ) and distance from the inoculum (1 and 15 cm). An inoculum wood block, colonized by the wood decay fungus Phanerochaete velutina , was placed in one corner of a bioassay dish (24 cm × 24 cm) filled with unsterilized soil. Once the mycelium grew onto the soil to a distance >15 cm from the inoculum, a sterilized new bait wood block (of either size) was placed on the soil at one of the two distances to be colonized by the mycelia from the inoculum. After 50 days of incubation, the baits were harvested, and their dried weight was measured to calculate the absolute weight loss during incubation. The inoculum wood blocks were retrieved and placed on a new soil dish to determine whether the mycelium would grow out onto the soil again. If no growth occurred within 8 days of additional incubation, we concluded that the mycelium had migrated from the inoculum to the bait. The results showed that mycelia in inocula coupled with baits positioned 1 cm away migrated to the baits more frequently than those with baits positioned 15 cm away. A structural equation model revealed that bait weight loss (energy gain) and hyphal coverage on the soil (foraging cost) significantly influenced mycelial migration decisions. These findings suggest that fungal mycelia may employ their own foraging strategies based on energetic benefits.
A transport network is basically defined as a set of connected nodes and links allowing the circulation of goods and/or individuals. Traditionally studied for itself and in rather abstract ways through graph‐theoretical and economic lenses, the transport network has increasingly been defined as one component only, albeit a crucial one, of wider logistical, territorial, and societal realities.
The universal scaling relationship between an attribute and the size of a system is widespread in nature and society and is known as allometric growth. Previous studies have explained that the allometric growth exponent of single-source systems is uniquely determined by the dimension. However, the phenomenon that the exponent shows diversity in some systems, such as rivers, freight transportation and gasoline stations, lacks a reasonable explanation. In this paper, we hold the view that allometric growth may originate from efficient delivery from sources to transfer sites in a system and propose a multisource transportation network model that can explain diversified allometric growth exponents. We apply this model to some multisource systems, and the results show that our model successfully reproduces the diversity of the allometric growth exponent.
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The plasmodium of the slime mould Physarum polycephalum is a large amoeba-like cell consisting of a dendritic network of tube-like structures (pseudopodia). It changes its shape as it crawls over a plain agar gel and, if food is placed at two different points, it will put out pseudopodia that connect the two food sources. Here we show that this simple organism has the ability to find the minimum-length solution between two points in a labyrinth.
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We measured the shape of the foraging trail networks of 11 colonies of the wood ant Formica aquilonia (Formica rufa group). We characterized these networks in terms of their degree of branching and the angles between branches, as well as in terms of their efficiency. The measured networks were compared with idealized model networks built to optimize one of two components of efficiency, total length (i.e., total amount of trail) and route factor (i.e., average distance between nest and foraging site). The analysis shows that the networks built by the ants obtain a compromise between the two modes of efficiency. These results are largely independent of the size of the network or colony size. The ants’ efficiency is comparable to that of networks built by humans but achieved without the benefit of centralized control.
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To evaluate performance in a complex survival task, we studied the morphology of the Physarum plasmodium transportation network when presented with multiple separate food sources. The plasmodium comprises a network of tubular elements through which chemical nutrient, intracellular signals and the viscous body are transported and circulated. When three separate food sources were presented, located at the vertices of a triangle, the tubular network connected them via a short pathway, which was often analogous to the mathematically shortest route known as Steiner's minimum tree (SMT). The other common network shape had high fault tolerance against accidental disconnection of the tubes and was known as cycle (CYC). Pattern selection appeared to be a bistable system involving SMT and CYC. When more than three food sources were presented, the network pattern tended to be a patchwork of SMT and CYC. We therefore concluded that the plasmodium tube network is a well designed and intelligent system.
We studied the effect of the size of food sources (FSs) presented to the true slime mould Physarum polycephalum on the tubular networks formed by the organism to absorb nutrient. The amount of plasmodium gathering at an FS was shown to be proportional to both the concentration of nutrient and the surface area of the FS. We presented two FSs to test which connection the organism selected in response to varying amounts of food and derived a simple rule for connection persistence: the longer connection collapses earlier. A mathematical model for tube selection in response to amount of food was derived and predicted our experimental findings regarding the choice of connection. When three FSs were presented to the organism, the longer tubes were also the first to collapse, explained by the relative probability of disconnection. The size of the FS is thus a key parameter determining network formation.
We review how soft matter is self-organized to perform information processing at the cell level by examining the model organism Physarum plasmodium. The amoeboid organism, Physarum polycephalum, in the class of true slime molds, exhibits the intelligent behavior of foraging in complex situations. When placed in a maze with food sources at two exits, the organism develops tubular structures with its body which connect the food sources along the shortest path so that the rates of nutrient absorption and intracellular communication are maximized. This intelligent behavior results from the organism's control of a dynamic network through which mechanical and chemical information is transmitted. We review experimental studies that explore the development and adaptation of structures that make up the network. Recently a model of the dynamic network has been developed, and we review the formulation of this model and present some key results. The model captures the dynamics of existing networks, but it does not answer the question of how such networks form initially. To address the development of cell shape, we review existing mechanochemical models of the protoplasm of Physarum, present more general models of motile cells, and discuss how to adapt existing models to explore the development of intelligent networks in Physarum.
Wireless mesh networks (WMNs) consist of mesh routers and mesh clients, where mesh routers have minimal mobility and form the backbone of WMNs. They provide network access for both mesh and conventional clients. The integration of WMNs with other networks such as the Internet, cellular, IEEE 802.11, IEEE 802.15, IEEE 802.16, sensor networks, etc., can be accomplished through the gateway and bridging functions in the mesh routers. Mesh clients can be either stationary or mobile, and can form a client mesh network among themselves and with mesh routers. WMNs are anticipated to resolve the limitations and to significantly improve the performance of ad hoc networks, wireless local area networks (WLANs), wireless personal area networks (WPANs), and wireless metropolitan area networks (WMANs). They are undergoing rapid progress and inspiring numerous deployments. WMNs will deliver wireless services for a large variety of applications in personal, local, campus, and metropolitan areas. Despite recent advances in wireless mesh networking, many research challenges remain in all protocol layers. This paper presents a detailed study on recent advances and open research issues in WMNs. System architectures and applications of WMNs are described, followed by discussing the critical factors influencing protocol design. Theoretical network capacity and the state-of-the-art protocols for WMNs are explored with an objective to point out a number of open research issues. Finally, testbeds, industrial practice, and current standard activities related to WMNs are highlighted.
We have proposed a mathematical model for the adaptive dynamics of the transport network in an amoeba-like organism, the true slime mold Physarum polycephalum. The model is based on physiological observations of this species, but can also be used for path-finding in the complicated networks of mazes and road maps. In this paper, we describe the physiological basis and the formulation of the model, as well as the results of simulations of some complicated networks. The path-finding method used by Physarum is a good example of cellular computation.
We have studied how the plasmodium of Physarum polycephalum, a large amoeboid cell, is able to track the shortest path between two selected points in a labyrinth. When nutrients are supplied at these points to a sheet-like plasmodium extended fully in a maze, the organism forms a single tube which connects the two sites via the shortest route. During the path finding, plasmodial parts in dead ends of the maze shrink and finally the tube with the minimum-length is selected from the existing possibilities. A simple cellular mechanism based on interacting cellular rhythms may describe the experimental observations.
Branching network growth patterns, depending on environmental conditions, in plasmodium of true slime mold Physarum polycephalum were investigated. Surprisingly, the patterns resemble those in bacterial colonies even though the biological mechanisms differ greatly. Bacterial colonies are collectives of microorganisms in which individual organisms have motility and interact through nutritious and chemical fields. In contrast, the plasmodium is a giant amoeba-like multinucleated unicellular organism that forms a network of tubular structures through which protoplasm streams. The cell motility of the plasmodium is generated by oscillation phenomena observed in the partial bodies, which interact through the tubular structures. First, we analyze characteristics of the morphology quantitatively, then we abstract local rules governing the growing process to construct a simple network growth model. This model is independent of specific systems, in which only two rules are applied. Finally, we discuss the mechanism of commonly observed biological pattern formations through comparison with the system of bacterial colonies.
The relationship between cell shape and rhythmic contractile activity in the large amoeboid organism Physarum polycephalum was studied. The organism develops intricate networks of veins in which protoplasmic sol moved to and fro very regularly. When migrating on plain agar, the plasmodium extends like a sheet and develops dendritic veins toward the rear. After a particular stimulation, the vein organization changes into veinless or vein-network structures. In both structures, the mixing rate of the protoplasm, which is related to communication among contraction oscillators, decreased compared with that of the dendritic one. Accompanying these changes in vein structure, the spatio-temporal pattern of the rhythmic contraction changed into a small-structured pattern from a synchronized one. In the above process, cell shape affects the contraction pattern, but, conversely, the contraction pattern effects the cell shape. To demonstrate this, a phase difference in the rhythmic contraction was induced artificially by entraining the intrinsic rhythm to external temperature oscillations. New veins then formed along the direction parallel to the phase difference of the rhythm. Consequently, the vein organization of the cell interacts with the contractile activity to form a feedback loop in a mechanism of contraction pattern formation.