Fermat's Library | Rules for Biologically Inspired Adaptive Network Design annotated/explained version.

Here are some fun discussions of this research after it won the "ig...

Tshiyuki Nakagaki ia a Japanese professor and biologist, famous for...

> "Transport networks are ubiquitous in both social and biological ...

> "Drawing inspiration from biology has led to useful approaches to...

>"“Overall, we conclude that the Physarum networks showed character...

> "Our biologically inspired mathematical model can capture the bas...

16. J. A. Burns, P. L. Lamy, S. Soter, Icarus 40, 1 (1979).

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Physics, Chemistry, and Dynamics of Interplanetary

Dust, ASP Conference Series 104, B. A

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M. S. Hanner, Eds. (Astronomical Society of the Pacific,

San Francisco, 1996), pp. 179–182.

18. B. J. Buratti, M. D. Hicks, K. A. Tryka, M. S. Sittig,

R. L. Newburn, Icarus 155, 375 (2002).

19. F. Tosi et al., preprint available at http://arxiv.org/abs/

0902.3591 (2009).

20. Besides Iapetus, Hyperion, Titan, and outer-satellite

impacts were suggested; see also (12).

21. Reference (18)mentionsanincreaseofthedustfluxby~20%,

whereas (35)findsasmuchasafactorof3forsomecases.

22. D. J. Tholen, B. Zellner, Icarus 53, 341 (1983).

23. The leading sides of the moons beyond Mimas and inside

Titan should be substantially coated by E-ring particles

(24, 36, 37), making them less useful for this argument.

24. B.J.Buratti, J. A. Mosher, T. V. Johnson, Icarus 87,339(1990).

25. J. A. Burns et al., Science 284, 1146 (1999).

26. S. S. Sheppard, www.dtm.ciw.edu/users/sheppard/

satellites/satsatdata.html (2009).

27. D. Nesvorný, J. L. A. Alvarellos, L. Dones, H. F. Levison,

Astron. J. 126, 398 (2003).

28. D. Turrini, F. Marzari, H. Beust, Mon. Not. R. Astron. Soc.

391, 1029 (2008).

29. A. J. Verbiscer, M. F. Skrutskie, D. P. Hamilton, Nature

461, 1098 (2009).

30. This idea was develop ed in several papers

(18, 38, 39), but under the assumption that dust

from the outer saturnian moons formed Iapetus’

albedo dichotomy.

31. T. V. Johnson et al., J. Geophys. Res. Solid Earth 88, 5789

(1983).

32. T. Denk, R. Jaumann, G. Neukum, in Lisbon

Euroconference Jupiter After Galileo and Cassini,

Abstracts Book 17 to 21 June 2002, Lisbon, Portugal,

abstr. no. P-4.1.18, 2002, p. 118.

33. B. J. Buratti, J. A. Mosher, Icarus 90, 1 (1991).

34. M. E. Davies, F. Y. Katayama, Icarus 59, 199 (1984).

35. K. J. Zahnle, P. Schenk, H. Levison, L. Dones, Icarus 163,

263 (2003).

36. K. D. Pang, C. C. Voge, J. W. Rhoads, J. M. Ajello,

J. Geophys. Res. Solid Earth 89, 9459 (1984).

37. D. P. Hamilton, J. A. Burns, Science 264, 550 (1994).

38. P. C. Thomas, J. Veverka, Icarus 64, 414 (1985).

39. K. S. Jarvis, F. Vilas, S. M. Larson, M. J. Gaffey, Icarus

146, 125 (2000).

40. G. Neukum, B. A. Ivanov, in Hazards Due to Comets and

Asteroids, T. Gehrels, Ed. (Univ. of Arizona Press, Tucson,

AZ, 1994), pp. 359–416.

41. T. Roatsch et al., Planet. Space Sci. 57, 83 (2009).

42. We acknowledge the individuals at CICLOPS (at the Space

Science Institute in Boulder, CO) and JPL (Pasadena, CA), as

well as the members and associates of the Imaging Team for

the successful conduct of the ISS experiment onboard the

Cassini spacecraft. This paper is dedicated to Steve Ostro,

whose work helped considerably to explain the nature of

Iapetus’ dark terrain. This work has been funded by the

German Aerospace Center (DLR) and NASA/JPL.

Supporting Online Material

www.sciencemag.org/cgi/content/full/science.1177088/DC1

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;

10.1126/science.1177088

Include this information when citing this paper.

Rules for Biologically Inspired

Adaptive Network Design

Atsushi Tero,

1,2

Seiji Takagi,

Tetsu Saigusa,

Kentaro Ito,

Dan P. Bebber,

Mark D. Fricker,

Kenji Yumiki,

Ryo Kobayashi,

5,6

Toshiyuki Nakagaki

1,6

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.

ransport networks are a critical part of the

infrastruc ture needed to operate a modern

industrial society and facilitate efficient

movement of people, resources, energy, and

informati on. Despite their importance, most net-

works have emerged without clear global design

principles and are constrained by the priorities

impose d at thei r initia tion. Thus, the main motiva -

tion historical ly was to achieve high transpo rt

efficiency at reasonable cost, but with correspond-

ingly less emphasi s on makin g system s toler ant to

interruption or failure. Introducing robustness

inevit ably requ ires add ition al redund ant pathw ays

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 baggag e-handling systems (5), and railway

networks(6), as well asthepredictedvulnerabi lityof

systems such as information networks (7)orsupply

networks (8)toattack,havehighlightedtheneedto

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

(9–12 ). Such systems continuously adapt to their

environment and must balance the cost of produc-

ing an efficie nt 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

atubularnetworklinkingthediscoveredfood

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 experiment al investigat ion of

the rules underlying network formation. Thu s,

for example, Physarum can find the shortest

path through a maz e (15–17) 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 observed Physarum connecting a template

of 36 FSs that represented geographical locations

of cities in the T okyo 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. T o 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,

Hokkai do Universi ty, 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.

JST, CREST, 5

Sanbancho, Chiyoda-ku, Tokyo, 102-0075, Japan.

*To whom correspondence should be addressed. E-mail:

nakagaki@es.hokudai.ac.jp

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accumulate on a large FS outside the arena (LFS

in Fig. 2A).

Arangeofnetworksolutionswereapparent

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 geographica l 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 Delaun ay triangulati on, 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 robustnes s (FT), normalized to the

corresponding value for the MST to give TL

MST

,andFT

MST

.TheTLoftheTokyorail

network was greater than the MST by a factor

of ~1.8 (i.e., TL

MST

≈ 1.8), whereas the average

MST

for Physarum was 1.75 T 0.30 (n =21).

Illuminated networks gave slightly better clus-

tering around the value for the rail network (Fig.

3A). For comp ar is o n, th e De la un ay triang u la tio n

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

MST

0.85 and 0.85 T 0.04 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

(FT

MST

)inwhichtherealrailnetworkshowed

margi nally better resilie nce, c lose to th e lowes t

level needed to give maximum tolerance to a single

random failure. Thus, only 4% of faults in the rail

network would lead to isolation of any part,

whereas 14 T 4% would disconnect the illuminated

Physarum networks, and 20 T 13% would

disconnect the uncons trained 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

MST

= a.Ingeneral,

the Physarum networks and the rail network had

abenefit/costratioof~0.5foranygivenTL

MST

(Fig. 3B). The relationship between different a

values and transport efficiency (Fig. 3C) high-

lighted the similarity in aggr egate beha vior of the

Physarum network when considering all three per-

formance measures (MD

MST

,TL

MST

,andFT

MST

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

sources at each of the major cities in

the region (white dots). The horizontal

width of each panel is 17 cm. (B to 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)Intheabsenceofillumination,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. (C and D)Theresultingnetwork

in the Tokyo area (D). (E and F)The

minimum spanning tree (MS T) con-

necting the same set of city nodes (E)

and a model network constructed by

adding additional links to the MST (F).

LFS

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 a value 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, transp ort effic ien-

cy, and fault tolerance. However , the Physarum

networks self-organ ized 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 constr uction to emula te this behavior,

based on feedback loops between the thickness of

each tube and internal protoplasmic flow (18–22)

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).Theedgesrepresentplasmodial

tubes in which protoplasm flows, and nodes are

junctions between tubes. Suppose that the pres-

sures at nodes i and j are p

and p

,respectively,

and that the two nodes are connected by a cyl-

inder of length L

and radius r

.Assumingthat

flow is laminar and follows the Hagen-Poiseuille

equation, the flux through the tube is

ðp

− p

8hL

ðp

− p

ð1Þ

where h is the viscosity of the fluid, and D

/8h is 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,

,oftheedges.

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

= I

.Asecond

random FS is chosen as a sink (node 2) with a

corresponding withdrawal of I

such that S

–I

.Astheamountoffluidmustbeconserved,

the inflow and outflow at each internal node must

balance so that i (i ≠ 1, 2), S

=0.Thus,fora

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

.Thefirstterm

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 +

|Q|

), which describes a sigmoidal response where g

is a parameter that controls the nonlinearity of feed-

back (g >0).AtypicalsimulationresultwithI

and g =1.8(Fig.4)gaveanetworkwithfeatures

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

MST

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

.Judiciousselectionofspecificparameter

combinations (I

=0.20,g =1.15)yieldednet-

works with remarkably similar topology and

metrics to the Tokyo rail network (fig. S2B). How-

ever, by increasing I

to 2 and g to 1.8, the simula-

tion model also achieved a benefit/cost ratio (a =

FT/TL

MST

)thatwasbetterthanthoseoftherailor

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

MST

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)Transportperformanceof

each network, measured as the

minimum distance between all

pairs of nodes, normalized to

the MST (MD

MST

)andplotted

against the total length of the

network normalized by the MST

(TL

MST

)asameasureofcost.

Black circles and blue squares

represent results obtaine d 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)andinitialrandomfluctuationsofD

.(B)Fault

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

MST

,are

shown as dashed lines. (C)RelationshipbetweenMD

MST

and a.Althoughthe

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.

0.75

0.8

0.85

0.9

0.95

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Performance (MD

MST

)

0.75

0.8

0.85

0.9

0.95

1.0 1.5 2.0 2.5 3.0

0.3

0.6

0.7

=0.2

0.4

0.2

0.4

0.6

0.8

Fault tolerance (FT)

1.0 1.5 2.0 2.5 3.0

Performance (MD

MST

)

Cost (TL

MST

)

Cost (TL

MST

)

Efficiency (FT / TL

MST

)

Fig. 4. Network dynamics for the

simulation model. In this typical time

course for evolution of the simula-

tion, time (t)isshowninarbitrary

units; cities are blue dots. Each city

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),theavailable

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

astable,self-organizedsolution.g =

1.80, I

=2.00.

t=0

t=1000

t=3000

t=29950

www.sciencemag.org SCIENCE VOL 327 22 JANUARY 2010 441

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multipl e simulta neous failur es at the expense of

increased cost, rather than tolerance to a single

disconnection that is evaluated by FT

MST

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 tho se of rea l-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 toleranc e or transport effic iency, 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).

References and Notes

1. R. Albert, I. Albert, G. Nakarado, Phys. Rev. E 69,

025103R (2004).

2. R. V. Solé, M. Rosas-Casals, B. Corominas-Murtra,

S. Valverde, Phys. Rev. E 77, 026102 (2008).

3. R. M. May, S. Levin, G. Sugihara, Nature 451,893(2008).

4. J. Kambhu, S. Weidman, N. Krishnan, Econ. Policy Rev.

13, 1 (2007).

5. House of Commons Transport Committee, The Opening of

Heathrow Terminal 5 HC 543 (Stationery Office, London,

2008).

6. Train Derailment at Hatfield (Independent Investigation

Board, Office of Rail Regulation, London, 2006).

7. R. Albert, H. Jeong, A.-L. Barabási, Nature 406,378(2000).

8. R. Carvalho et al., http://arxiv.org/abs/0903.0195 (2009).

9. D. Bebber, J. Hynes, P. Darrah, L. Boddy, M. Fricker,

Proc. R. Soc. London Ser. B 274, 2307 (2007).

10. J. Buhl et al., Behav. Ecol. Sociobiol. 63, 451 (2009).

11. T. Nakagaki, H. Yamada, M. Hara, Biophys. Chem. 107,

1 (2004).

12. T. Nakagaki, R. Kobayashi, Y. Nishiura, T. Ueda,

Proc. R. Soc. London Ser. B 271, 2305 (2004).

13. A. Colorni et al., Int. Trans. Oper. Res. 3, 1 (1996).

14. A. Takamatsu, E. Takaba, G. Takizawa, J. Theor. Biol. 256,

29 (2009).

15. T. Nakagaki, H. Yamada, Á. Tóth, Nature 407,470(2000).

16. T. Nakagaki et al., Phys. Rev. Lett. 99, 068104 (2007).

17. T. Nakagaki, H. Yamada, Á. Tóth, Biophys. Chem. 92,47

(2001).

18. A. Tero, K. Yumiki, R. Kobayashi, T. Saigusa, T. Nakagaki,

Theory Biosci. 127, 89 (2008).

19. T. Nakagaki, R. Guy, Soft Matter 4, 57 (2008).

20. T. Nakagaki, T. Saigusa, A. Tero, R. Kobayashi, in

Topological Aspects of Critical Systems and Networks:

Proceedings of the International Symposium,K.Yakubo

et al., Eds. (World Scientific, Singapore, 2007), pp. 94–100.

21. A. Tero, R. Kobayashi, T. Nakagaki, J. Theor. Biol. 244,

553 (2007).

22. A. Tero, R. Kobayashi, T. Nakagaki, Physica A 363, 115

(2006).

23. T. Nakagaki, H. Yamada, T. Ueda, Biophys. Chem. 84,

195 (2000).

24. I. Akyildiz, X. Wang, W. Wang, Comput. Netw. 47, 445

(2005).

25. Supported by MEXT KAKENHI grants 18650054 and

20300105, Human Frontier Science Program grant

RGP51/2007, EU Framework 6 contract 12999 (NEST),

and NERC grant A/S/882.

Supporting Online Material

www.sciencemag.org/cgi/content/full/327/5964/439/DC1

Figs. S1 and S2

17 June 2009; accepted 20 November 2009

10.1126/science.1177894

Measurement of Universal

Thermodynamic Functions for a

Unitary Fermi Gas

Munekazu Horikoshi,

* Shuta Nakajima,

Masahito Ueda,

1,2

Takashi Mukaiyama

1,3

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.

egenerate two-component Fermi systems

with large scattering lengths are of great

interest in diverse settings such as neutron

stars (1–3), quark-gluon plasma (4), high critical

temperature (T

)superconductors(5), and reso-

nantly interacting cold Fermi gases near Feshbach

resonances (6–18). Even though the temperature

of these systems ranges widely from 10

−7

Kfor

cold atoms to more than 10

Kforquark-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,andtemperature,T.Thisassumptionis

referred to as the “universal hypothesis (UH)”

(19, 20).

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 (6–18). 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

Kprovidedevidenceof

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 113-865 6, Japan.

Departm ent of Physics, Univer sity of Tokyo, 7-3- 1 Hong o,

Bunkyo-ku, Tokyo 113-0033, Japan.

Center for Frontier Science

and Engineering, University of Electro-Communications, 1-5-1

Chofuga oka, Chofu , Tokyo 182-8585, Japan.

*To whom correspondence should be addressed. E-mail:

hori@sogo.t.u-tokyo.ac.jp

Fig. 1. Universal function of the internal en-

ergy. Universal functions of the internal energy

[q]=E/Ne

) 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 da ta spread of on e 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 b is 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.

22 JANUARY 2010 VOL 327 SCIENCE www.sciencemag.org

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Discussion

> "Drawing inspiration from biology has led to useful approaches to problem-solving such as neural networks, genetic algorithms, and efficient search routines developed from ant colony optimization algorithms. We exploited the slime mold Physarum polycephalum to develop a biologically inspired model for adaptive network development." > "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 networks, or wireless mesh networks." >"“Overall, we conclude that the Physarum networks showed characteristics similar to those of the rail network in terms of cost, transport efficiency, and fault tolerance. However, the Physarum networks self-organized without centralized control or explicit global information by a process of selective reinforcement of preferred routes and simultaneous removal of redundant connections." Here are some fun discussions of this research after it won the "ignobel" prize! https://www.ox.ac.uk/news/science-blog/ig-nobel-slime-networks https://www.wired.com/2010/01/slime-mold-grows-network-just-like-tokyo-rail-system/ Tshiyuki Nakagaki ia a Japanese professor and biologist, famous for his experiments on slime mold, he is one of 9 people who has received multiple ignobel prizes. Source: https://en.wikipedia.org/wiki/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 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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