An improved biologically plausible trajectory generator

HAL Id: inria-00072049

https://hal.inria.fr/inria-00072049

Submitted on 23 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub-

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,

An improved biologically plausible trajectory generator

Thierry Viéville

To cite this version:

Thierry Viéville. An improved biologically plausible trajectory generator. [Research Report] RR-4539, INRIA. 2002. �inria-00072049�

a p p o r t

d e r e c h e r c h e

Thème BIO

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

An improved biologically plausible trajectory generator

T. Viéville

N° 4539 – version 2

version initiale Septembre 2002 – version révisée Février 2006

/1032#46587&4:969:;

<-=?>@BACDFEHGJILKNMPOF>Q@RASM-TVUXWZYXWZ[ZU]\_^VAQM

`bacWedfAOgEihVKLMcMFAA

jkelVlmWZaFOhNA,acAQno=VAQacno=VAqpsr't_uZvZwyx{zZAQacMFU|W8p}

∗G~z8AaoMPUXWZp U|p?UOcUXkZY|AILAQlNOFAQ@TVacA}e€8€8}iGzZAaoMFU|W8p ac‚zLUXMF‚A.ƒ'‚zLacU|AQag}e€Z€8„ vZt l…ke[ZASM

†ˆ‡'‰QŠ‹ŒVŽZŠZ

 W8p?MPU]hNAQaFUXpV[ROF=VATVU|W8Y|W8[ZU]nkZYmW8agkeaFOFU|‘…nUXkZYsOFaokdfAQnOFW8aFK’[8ApVAQackeOFUXWZplVaFW8TVYXA@#“V”A,lVaFW8l…W_MPA,kBTVUXWe•

YXWZ[8UXnQkeYXY|KilVY]ke^?MFUXTVY|A–@BWLhVAYLWe—VOc=VAY|UXpV˜.T…AOf”–AQApOF=VA.™šUœ›*YXWNnkZYžP”=VAacA•:OFWZ•:[8We•ŸpVA LOP¡'YXWLnQkeYL@BAQno=?kZpVU]MP@

UXp{k.YXWLn^?M3@’kel{kZp?h¢OF=VA¢™šUXU]›[8Y|W8T?keYVOFaokdfAQnOFW8aFK[8ApVAQackeOFUXWZpRlVacWZTVYXA@#£<-=VUXMbA NlVY]keUXp?M3=VW”¤OF=VAg”–AQY|Y

˜LpVW”p¥=VUXlVlmWLnQke@Bl?kZYgacAY]kOFASh¤kZaFASkZM’nWZp_OckZU|pVUXpV[¦YXWLn^?MP•:@’kel…MB”UOc=§Y|WNnkZY@BAQno=?kZpVU]MP@’M{kZaFA1keY]MFW

z8AacK{YXU|˜8AYXKˆkeTVYXA.OFW’MFWZYXzZAiOF=?A,[ZYXWZT?kZY¨lVaFW8TVYXA@#“VlVacWzLUXhNUXpV[¢OF=VApVA^VaokeY¨@’kZl#k8n"OcU|zLU|OfK{UXMaFAQYXkeOFAQhˆOFW

=…keac@RW8pVU]nglmWeOcAp_OFU]keY]M£

IL^…no= acAlVacAQMFAp_OokOFUXWZp{k8MFMF^V@BA)OF=?keOW8T?MfOokZnY|ASMbOcWRkžz8WZU]h©™šW8a-nWZp?MPOFaokeUXp_OcM–pVWeOOFWRzLU|W8YXkeOFAž›+nWZacaFA•

MFlmWZp?h OFWˆ@Bke NU|@’kBWe—bk’MFWe•ªnkZY|YXAQh l…WZOFAQp8OcUXkZY6“…”=VUXY|A,Oc=VA[8W8keY!nWZacacAQMFl…W8p?hVM–OFW{U|OcMg@BU|pVUX@^?@£<-=VA

nWZacaFASMPlmWZp…hNU|p?[#kZY|[8WZacUOc=V@«Oc=_^…MTmA=?kžz8AQMk8M,U—W8pVA’OF=?aFW”Mk1MF=VAQAOW8p8OcW#OF=VU]MMPOckeOFAˆMFl?k8nAZ“!OF=?UXM

=LKLlmAaF•ŸMF^VaF—¬kZnA¢aFAQY|UXA—-TmAUXpV[#AYXAzkOcAQh­WZp¦WZT?MPOck8nYXAQMQ“¨”UOc=®k =?WZYXABkeO.OF=VA’[8W8keYYXWNnkOcU|W8p*£’ƒ'U|p?hVU|pV[

k OcackedfAQn"OcWZacK1OF=L^?MaFAShN^?nASMiOFWˆPaFW8Y|YbhVW”pL¡BkZY|W8pV[OF=VU]M,acAYXU|A—f“&UXp¦OF=VA{hNUXacAQn"OcU|W8p¦We—OF=VA’lmWeOcAp_OFU]keY

[8ack8hNUXAp_OQ£

<-=VA­WZacU|[8U|p…keYXUOfK¥We—,Oc=VA¯lVacAQMFAp_Oˆ”WZac˜¥UXMˆOFW¤TV^VUXYXhHkep§=?kZaF@BWZp?UXn1lmWeOcAp_OFU]keYB™œOc=L^?M”UOc=VWZ^NO

YXWNnkZY…@BUXpVU|@¢^V@{›–—šaFW8@°k¢MFl?kZacMFAgkZhVl?kOcU|z8AinWZ@T?U|p?keOFUXWZp{WZ—&AYXA@BAQp8OokeacK’Y|WNn^?M“Nk8M–UXp?MFlVUXaFASh{TLKBOF=VA

T?U|W8Y|W8[ZU]nkeYs@BWNhNAYXUX±QkOcU|W8p WZ—&Oc=VA,=VUXlVl…WNnQke@Bl?keY!MfOcaF^?nOF^VacAQMQ£

<-=VU]M’YXAQk8hVMROFW²kZp¤UXp_OFAacp?kZYaFAQlVaFASMPAQp_OckOcU|W8p³WZ—)Oc=VA1lVacWZTVYXA@´k8M’k¦pVWZpV•6OcWZlmWZ[Zaokel?=VUXnQkeY@’kel*“

UXp?naFAQ@RAQp_OckeYXYXK{TV^VUXYOghV^VaFUXpV[BOc=VAMPKNMPOFAQ@°A NlVYXWZaokOcU|W8p*£3µgMMF^?no=*“VU|OlVacWzLUXhNASMkR˜ZAKˆAYXA@BAp_O—šWZak

T?U|W8Y|W8[ZU]nkeYXYXKil?YXkZ^?MPUXTVYXA-@RWNhNAQYVWe—…OF=VAacAY]kOcAQh¢=VUXlVl…WNnQke@BlV^?M@BAQno=…kepVU]MP@’MUXpRnWZ=VAQaFAQp?nA”U|OF=B^?MF^?keY

T?U|W8Y|W8[ZU]nkeY¨k8MFMF^V@BlNOFUXWZp…MkZT…W8^NOgMF^?no= TmA=…kžz_UXWZaS£

¶!·F¸f¹ªº»Q¼’½)»¾ ¿+¿¤ÀžÁQÂQà ÄbºÆÅfÇ¢È,»Q¸f·–·cÉZÊN·c¸fºÆÈ,·o¼ÅPËÅfºÆ»S¼?Ì8¹ª»QÈ,·-·cÉeÅf·o¼8¹ªº»Q¼_¹3»¾sÅfÇ_·–È,·cÅfÇ_»eÍËQ¼_ÍRËg¾X»Q¸fÈËÎ

Í8·oÈ,»Q¼_¹ŸÅª¸PËÅfºÆ»Q¼R»Q¾mÅfÇ_·ËSÍ8ËQÊZÅfºÏS·ËQÎÆÐS»¸fºÑÅfÇ_ÈÓÒo»S¼eϞ·F¸fÐS·o¼8Òo·SÔ

Õ1Ö_×sØcÙÚ ‹Û'‰Z <!aokdfAQnOFW8aFK’[8ApVAQackeOFUXWZp*“NÜgkZaF@BW8pVUXnilmWeOcAp_OFU]keY:“VCU|W8Y|W8[ZU]nkeYs@BWNhNAY

∗Ý&ÞSßÑà*áâfã¬à:ßÑä"å)ލæoà*çä"ã6ââªèQéeâPã¬ßÑçâfåê:æcê6ßÑä"å_ëeà:ä"çâ!âfèê6âPåSà:ßÑä"åžà!äoìNê:ޞâ3çâªê6ޞäíiæoåžíiæ+ì|ä"ã6çæoî8ížâfçä"åžà6ê6ã6æcê:ßÑä"å)äoì ê:ޞâbæoížæoéSê:ßáâ+æoîÑï"ä"ã¬ßê:ÞSçñðfä"åáâfã¬ï"âPåSðPâ"ò

¢,

‰ 

 W8p?MPU]hN‚QackZp8O.YXA{nWZp_Oca8Y|A¢TVU|W8Y|W8[ZU]\_^VABWZ^¦keaFOFU|‘…nUXAY3hNA’Y]k[Z‚p?‚aokOFUXWZp h^VpVAROFaokdfASn"OFW8U|acAZ“¨p?WZ^?M

l?aF‚SMPAQp8OcWZp?M)U]nU^Vp @RWNhN>QY|ABTVUXWZYXWZ[8UX\_^VAQ@BAp_O)lVY]ke^…MPUXTVYXA¢T?k8MP‚BMF^Va.^VpVAR@R‚OF=VWNhNA’hNU|OFA’hNABlmWeOcAp_OFUXAY

=…keac@RW8pVU]\8^?AZ£

&AQM'@B‚OF=?WLhVAQMT?k8MPA-hNAl…WZOFAQp8OcU|AQYNMP^Vl?l…W_MPAQp8O\_^VA–YXAQM'W8T?MfOokZnY|ASMg‚zLUOcAa™šW8^YXAQMnW8p8OcackZU|p_OcAQM

p?Agl?k8M+AQpN—šaFAQU|p…hNaFAž›bnWZacaFASMPlmWZp?hVAp_OhNASM+@’ke LUX@’khNAinAglmWeOcAp_OFUXAYmOokep?hNU]M\8^?AgYXAgTV^NOkeOPOFAQU|p…hNaFA

ASMfOR^Vp³^?pVUX\_^VA@BU|p?U|@¢^V@#£kZY|[8WZacUOc=V@BAˆMFA nW8@Bl…W8aPOcA nW8@B@RAMPUYW8p1dfAOokeU|OR^Vp¥hNaokel³MF^VaBnAO

ASMPl…kZnA8“'—šWZac@BkZp_O¢hNAQMT…W_MFMFAQM¢ke^N ¦WZT…MfOokZnYXAQMAOR^Vp³nacA^V  z8AaoMYXA{TV^NObOFacWZ^?zZAa¢^VpVAˆOFaokdfAQnOFW8U|acA

acAzLUXAp_OkeYXWZaoMRMPAžY]keU]MFMFAa[ZYXU]MFMFAaBYXA)YXWZpV[BhNA,nA.hNaokel zZAaoM–Y|Ail…W8U|p_OhNA.=?kZ^NOFAQ^Va-@RUXpVUX@’keYXAZ£



AQM

@B‚OF=VWNhNASMlmAac@RAOPOcAp_O)hNAl…kZMcMPAQahN^ žYXWLnQkeY'ke^#[ZYXWZT?kZY! AQp©nAMFAp?Mg\8^^Vp?A@BU|pVUX@BUXMckOcU|W8p1Y|WNnQkeYXA

hV^ˆlmWeOFAQp_OFUXAYml…AQaF@BAO‘?p?kZY|AQ@BAp_OhNA)OFacWZ^?zZAa–^VpVAgOFaokdfAQnOFW8U|acA[ZYXWZT…keYXAZ“Lno=VWZU]MPUXa+^?pˆU|OFUXpV‚aokeUXaFA8“8AOcnZ£|£

W8aFUX[ZUXp?keYXU|OF‚ hVAY]k­lVaF‚SMPAQp_OFAˆ@B‚OF=VWNhNA#AQMPO’hNA1nWZp…MfOcaF^VUXacAˆ^Vp¤l…WZOFAp_OcU|AQY=?keac@BWZpVU]\_^VA­™6hNWZp?n

Mckep…M,@BU|pVUX@^?@%YXWNnkeYœ›,nWZ@B@BA{^VpVAˆnWZ@¢TVU|p…keU]MPW8p­YXU|p?‚QkeUXacAB‘?pVUXAˆhNA’—šW8p?n"OcU|W8p?M=?kZaF@BW8pVUX\_^VASM.‚YX‚•

@BAQp8OokeUXaFASM£"Ap…MPAQ@TVYXAihVA.nAQM‚YX‚@BAQp8OoM–—šWZac@RAi^Vp‚Qno=?kZp_OFUXY|YXWZpVp…ke[ZAihNAiYXk—šaFW8p_OFUX>acAihN^#hNW8@BkZU|p?A

k8hN@BUXMcMFU|TVYXAZ“ZT…W8achN‚l…kea3YXAQMWZT…MfOokZnYXAQMW8^’nWZp_OcackZU|p_OFASM£  AY]k,nWZp…hN^VU|O)^?pVAaFAQlVac‚QMFAp_OckeOFUXWZpUXp_OFAQaFpVA

hV^ˆlVacWZTVYX>@BAiMPW8^?MYXk—šWZac@RA.h^VpVA.nkZaPOcA)pVW8pN•:OFWZlmWZ[8ackZlV=VU]\_^VAnW8p?MfOcaF^?UOcAikZ^{—š^Va-AO¢@RASMP^?aFAihNAQM

‚QzZW8Y|^NOcU|W8p?MhN^ MFKNMfOc>@BA)AOacAYXUX‚A.hNA)@’kZpVU|>QaFA)pVWZpV•:YXU|p?‚QkeUXacA#¢YXk¢[Z‚QWZ@B‚OcaFUXA)hN^ lVacWZTVYX>@BAgac‚AQY6£$bp

Ookep_O\_^VAgOFAQY6“NnAQYXk—šWZ^VacpVU|O–^Vp @BWLhV>YXA)\_^?kep_OcUOokOFU|—*TVUXWZYXWZ[8UX\_^VAQ@RAQp_O+lVY]ke^?MFU|T?Y|AihN^{—šW8p?n"OcU|W8pVpVA@BAQp8O

hVA+Y=VUXlVlmWNnke@BlmA–AOhVAQM'nkeaFOFASM&nWZ[Zp?UOcU|z8AQM&\_^VULK,MFWZp_O!acAlVac‚QMFAp_Oc‚AQMQ“žnWZ=?‚acAp_O'kžz8AQn3YXAQM!=LK_lmWeOc=V>QMFAQM

T?U|W8Y|W8[ZU]\8^?AQM^?MF^VAYXY|ASM£

¶!·F¸f¹ªº»Q¼#½Í&% ¿+¿³ÀÁQÂQà Ë"Ϟ·oÒgÊ_Î'%8¹Í)(·cÉZÊ+*c¸fºÆÈ,·o¼ÅPËÅfºÆ»Q¼…Ì-,.%_·cÎ/,.%8·o¹·cÉeÅf·o¼8¹ªºÆ»S¼_¹-Í8·gÎ|ËÈ0*cÅfÇ8»eÍ8·g·cÅ1%8¼_·

Í&*oÈ,»Q¼_¹ŸÅª¸PËÅfºÆ»Q¼¢¾]»¸fÈ,·oÎÆÎÆ·Í8·ÎËiÒo»Q¼ZÏS·c¸fÐS·o¼8Òo·Í8·Î2(ËÎÐQ»Q¸fºÑÅfÇ_È,·gËQÍ_ËQÊZÅPËÅfºÑ¾:Ô

3 Ú ŠS‰ Ø Ž54 ‰ 76 ‚p?‚aokOFUXWZphNA.OFaokdfASn"OcWZUXaFA8“_`WZOFAp_OcU|AQY*=?keac@BWZpVU]\_^VAZ“+81WNhN>QY|A,T?U|W8Y|W8[ZU]\8^?A

- $'$'

.0/21

ŠQ‹ Ú Û'ŽeŠ43Ú 1 5

687

ŒN‹. Ú 1

3¬Ž 4Ú Ž&!‰ ²Œ:9‰,; <

5>=?

9 Ö ‹3 Ö 1

Š@3

1BA

Š‹ŒC Ö ŽeŠ Ú ‹ × A Ö 1 Ö ‹žŒVŠ@3Ú 1 6D6

E F Ú 1

Ž&4'‰@3Ú 1 6DG

†IH‰@3 1BA Œ 1KJ ŒN‹. Ú 1 3¬ŽL9 Ú Š Ö 1 Š43¬Œ-4 Ù 3šŠ J 1 ‡ Ú 1 Û Ö ÛNMVŒ)4 Ö ‰ 5!6

O P

¢$& ’ +$!

QSRTNU$V:WYX[Z\T^]!V:_`W:acbdT^eT[WfXBVYgh_dedikjWY_`lmhT^npo

q A{nW8p?MFUXhNAQa.=VAacAROF=VABTVUXWZYXWZ[8UXnQkeYWZakZaPOcU‘…nUXkZY[ZAp?AaokOFUXWZp­We—-kˆOcackedfAQnOFWZacK²™šl?Y|ASkZMFARacA—šAQaiOFW#AZ£[?£

rs w t&—šWZa)kep#U|p_OcaFWNhN^?nOFUXWZp…›-kZp?h#”WZ^VY]hY|UX˜ZA.OcW’lVaFW8l…W_MPAkBT?U|W8Y|W8[ZU]nkeYXYXK{lVY]ke^?MFU|T?Y|A,@BWNhNAY!We—3MP^…no=#k

@BASno=?kepVU]MF@£

ƒ'UXp?hNUXpV[Rk.fOFaokdfAQnOFW8aFK8¡-—šaFW8@ kZp{UXpVUOcUXkZY¨MfOokOFA8“_OFWBk,‘?p?kZY¨MfOokOFA8“LUXMkz8AacKR[8ApVAQackZY?lVacWZT?Y|AQ@vu'U|—

Oc=VA,MfOokOcA)U]Mk¢v,w WZa}w lmW8MFUOcU|W8pˆUXp W8^Va1$b^?nY|U]hNAQkZpMFl?k8nAiWZaWZp#k[8AWZ[8ackZlV=VU]nkZY?@’kel*“NU|OU|p…hNAASh

@BASkep?M-‘…p?hNUXpV[{U|OcM”-kžKˆUXp©k’nYX^NOFOFAacAQh1MFl?k8nA,—šW8aU|p…MfOokep?nAZ“…kžz8WZU]hNUXpV[ROFWZ^…no=VU|p?[BOF=VAacW8k8hT…W8achNAQaQ“

no=?W_W_MPUXpV[TmAOf”AAQp{Of”W¢l…kOF=…M“Nkep…hˆMFWWZp&£b<-=VA)TmWZ^?p?hVkeacKBWe—!kZpˆW8T?MfOokZnY|A)@BkžKBTmA,hNA‘?p?AQh{TLK{kZp

AS\_^?kOcU|W8p*“*MFkžK

c(x) = 0“¨”U|OF=­OF=VA’nW8p?MPOFaokeUXp8O

c(x) >0—šWZaFkZhN@BU]MFMFUXTVY|A¡l…W8U|p_OoM

x WZ^NOoMPU]hNAROF=?A W8T?MPOckZnY|A8£3<-=VA.[ZW8kZY¨@’kžK{MPUX@BU|Y]keacY|KˆT…AhNA‘?pVASh TLKˆkZp UXpVAS\8^…keYXUOfK’OcW’T…AMckOcUXMP‘?ASh¨£

C^NOOcackedfAQnOFWZacK{[8ApVAQackeOFUXWZp#keY]MPWB@BAQkZp?MMFWZ@BAOc=VU|p?[’@RW8aFA,[8ApVAQackZY6£–ƒVWZagU|p?MPOckZp?nA8“VYXAOg^?MgpVW”

nWZp?MFU]hNAa'OF=?Al…W_MPU|OFUXWZp

xkep?h¢zZAQY|WNnU|OfK

˙

xWe—mOF=VAhVAzLUXnAZ£  WZp?MPOFaokeUXp_OcM'We—…OF=VA-—šW8aF@

c(x,x)˙ >0^@’kžK

hVA‘?pVA,=?W” OcW{kžzZWZU]hMP˜LU]h¨“VWZaTmWZ^?p?hOF=VAU|p?MPOckZp_Ockep?AWZ^…M–AQpVAac[ZKnW8p?MP^?@RlVOFUXWZp*£8©WZacAi[8ApVAQackZY|YXKZ“

nWZp?MPOFaokeUXp_OcM”=VU]no=1nWZacacAQMFl…W8p?hBOFW’l?=_KNMFUXnQkeY¨Y]kž”MpVWeOOcW’z_UXWZY]kOcA,nkepTmA,”aFU|OPOcAp#U|p1MF^?no=#k—šW8aF@#£

ƒ?WZa’UXp?MPOckep…nAZ“-”=VAQpñ@BkZpVA^?zZAacUXpV[¯OcW®l?keac˜³k®nQkea©™š”=?UXno=§nkZpñT…A©hNA‘?pVASh§kZM’MF@BkZY|Y)MPAS\8^?Ap?nA

WZ—bY|WNnQkOFUXWZp…MOcWkeOPOckZU|pm›”ARnQkepVpVWZO)@BWzZAifY]kOcAaokeYXY|K8¡TV^NO.=?kžzZAOFW —šWZYXY|W”ÓkˆaFASkeYXUXMPOFU]n,OcackedfAQn"OcWZacKu

Oc=VU]M)nQkep©keY]MFW’T…ARMPlmAQnU‘?ASh©kZM)k{nW8p?MPOFaokeUXp8O rx tŸ£iINU|@BUXYXkZaFYXKZ“mhN^VacU|pV[ˆaFW8T…WZOFU]nMP^?aF[8AacK­™¬MFAA¢AZ£[ rs € t

—šW8aikZp¯A NkZ@BlVY|ARacAY]kOFASh1OcWnQkeaohNU|W8Y|W8[ZKV›k hNAzLU]nAR”UOc= k YXkZaF[8ApL^V@TmAa.WZ—–hNA[8aFAQAQM)We—b—šacAAShNWZ@#“

=…kZMgOFWˆ[8ApVAQackeOFARk{=VUX[Z=VYXK©nW8p?MPOFaokeUXpVAQh{fOFaokdfASn"OFW8aFK8¡.—šaFW8@OF=VARUXpVUOcUXkZY3nWZpV‘?[Z^VaokOcU|W8p#OFW”-keaohOF=VA

MF^Vac[ZAQaFKBYXWLnQkOcU|W8pˆUXp?MPU]hNA.Oc=VA.T…WNhNK8£bµqMfOcAl#ke=VASkZh¨“NkRlVY]kepVpVUXpV[ROck8MP˜{MF^?no=#kZM–OF=?A,aFASkeYXU|±SkOFUXWZpˆWe—'k

P[ZASMfOc^VacA"¡)@’kžK{TmAMFl…ASnU|‘?AQh#kZMgkep{PkZT?MPOFaokZn"OP¡gOFaokdfASn"OcWZacK’[ZAp?AaokOFUXWZplVacWZTVYXA@#“VhNA‘?pVASh#T_KMPW8@BA

\_^?kZp_OFU|OckOcU|z8A l…keaoke@BAOcAaoM™6k¦MfOokOFA#z8AQn"OcWZa"›¢”UOc=¥kZp¤UXpVU|OFU]keYgMPOckeOFA1kep?h¥k [ZW_keYOFW²keOPOckZU|p¥”=VUXY|A

z8AacU—šKLUXpV[BnWZp…MfOcackZU|p_OcMQ£

IL^…no=#kRlVacWZT?Y|AQ@°=?kZMOcW’T…AMFWZYXzZAShyu

™¬U]›+PhN^?aFUXpV[{kZnOFUXWZpL¡MFU|p?nA,OF=?A,OFaokdfASn"OFW8aFKˆ[ZAp?AaokOFUXWZp#U]MkeY]MFW{kep1A NlVYXWZaokOcU|W8p WZ—OF=VAMPOckeOFAMFl?k8nAZ“

Oc=L^?M,”U|OF=¦OF=?A’l…WZOFAp_OcUXkZY–hNU]MFnWzZAQaFK1WZ—-kZpVWeOc=VAanWZp?MPOFaokeUXp_OcM’™¬keO,OF=VA hNU{zsAacAp…nA’We—k#lVacAQhNA‘?pVASh

P@BWeOcWZaF•:lVacWZ[8ackZ@¢¡-”=VU]no=#aFASkeYXU|±SkOcU|W8pˆU]M-Y|ASMFM-UXp#aFAQYXkeOFUXWZp”UOc=A NlVY|W8ackeOFUXWZp…›“

™¬U|Uœ›PkeO-k¢[ZYXWZT…keYmY|AQzZAY¡iMFU|p…nA)OF=VA.[ZW_keY…OFWBkOFOckeUXpˆU]M-kacA@BWZOFAil?keaFOWZ—*OF=?A.MPOckeOFA.MPl…kZnAi”=VUXY|A.YXWNnkeY

@BASno=?kepVU]MF@BM–keacAgpVWZOMF^{nUXAp_OOFW¢MFWZYXzZAiMP^…no=ˆklVacWZT?Y|AQ@™6kOkhNUzsAacAp?nAgWe—'k)PaFA ?A _¡WZakYXWNnkeY

f—šAAShNT?k8no˜Z¡iY|WLWZlm›

™¬U|UXUœ›)nWZp?MFU]hNAacU|p?[MPOckeOFABMPl?k8nASM)”U|OF=1PYXkZaF[8A"¡¢hNUX@BAp?MFU|W8p?MiT…ASnkZ^?MPAROc=VA’hNU|@BAQp?MPUXWZp­OFW#nW8p?MFUXhNAQa.UXM

Oc=VA,pL^V@TmAaWZ—hVA[ZacAAiWe—'—šacAAQhVWZ@’MWe—'OF=VA,lVacWZT?Y|AQ@£

µ)Mk’nW8@Rl?^NOckeOFUXWZp?kZY¨lVaFW8TVYXA@

rs w “ s “ w t&UOU]M^?MF^?kZY|YXKˆz8AacK’=?keaoh OFW{MFWZYXzZAiT…ASnke^…MPA.Oc=VUXM \

Y#+![f UXMikeYX@BW8MPO)keYX”kžKNM-pVW8pN•ŸY|UXpVAQkZaQ£<-=?AWZTVdfAQn"OgOFW{‘…p?h1U]M)kFn^?aFz8A"¡,U6£AZ£

kZp­W8TNdfAQnO”=VU]no= kZp UXpN‘?p?UOcA{hNU|@BAQp?MPUXWZp&£Lq§=VAp®kZlVlVacWž LUX@’kOcAQh©TLK kMFAOWZ—-MFA[8@BAp_OcMQ“¨‘?p?hVU|pV[

Oc=VU]Mn^VaczZAiAQk8MPUXYXK{Y|ASkZhVMOcW’kRnWZ@¢TVU|p…kOFW8aFU]keYsA Ll?Y|W_MPUXWZp*£

l _ V l g_dmh_`bdgh]X m V:WYX[Z\T^]!V:_`WYa b`T^e T[WYX V:_`W!o

 W8p_OFaokeacK,OFWOF=VU]M–kZlVl?kZaFAQp8O+nWZ@BlVYXA NU|OfKZ“LMP@’keYXY?@’kZ@R@’kZY|U]kep’z8AacKAQk8MPUXYXKRMFWZYXzZA-OcackedfAQn"OcWZacK[8ApVAQacke•

OcU|W8p{lVacWZTVYXA@’MQ“LU|p?nY|^…hNU|p?[¢”U|OF=nWZ@¢TVUXp?kOcWZacUXkZY?nW8@Rl?Y|A LU|OfK1™šA8£[…£'aokOoM+UXpˆY]keTLKLacU|p_OF=…M+kZMhNU]MFn^?McMPASh

UXp

r

} “ } tš›+uU|O–U]M+W8TLz_UXWZ^…MOF=?keO–Oc=VA)MF^VaczLU|zkeY…We—&k,T?U|W8Y|W8[ZU]nkeY…MPKNMPOFA@#“LlVaokžKW8a+lVacAQhVkeOFW8aQ“8UXMhNUXaFASn"OFYXK

acAY]kOcAQhOFW,U|OcM3nkZl?kZnUOfK.OcWnW8aFacAQnOFYXK,A NAQn^NOFAk)OFaokdfASn"OFW8aFK.UXp’k)˜LpVW”pRWZa^Vp?˜_p?W”p¢AQp_zLUXaFW8pV@BAp_O@u

‘…p?h©U|OcM)”kžK OFW#k{”-kOcAaiMPW8^VacnAZ“…—šWZYXY|W”qkZp¯WZY|—¬kZnOFW8aFK OFaokZno˜#kžzZW8UXhNUXpV[’pVAQ[8keOFUXzZAMF@BAYXYXMiWZa)WNhNWZaoM“

AOcnZ£|£

<-=VA­=VUXlVl…WNnQke@BlV^?MU]M

s

MfO{Oc=VA¦MP^VTV•ŸnWZaFOFU]nkeY)MfOcaF^?nOF^VacA¯”=?UXno= =…kZM{TmAAQpHMF=VW”p r} s

t.OFW¤T…A

hVU|acAQnOFYXK¥UXpLzZW8Y|z8AQhñUXp MP^…no= @BAQno=?kZpVUXMF@’M £ µ)MˆacAzLUXA”AQhñUXp*“gA8£[…£

r} tŸ“Oc=VA¯—¬k8n"O=VU|l?l…WNnkZ@Bl?keY

YXAQMFUXWZp?M)hNack8MfOcUXnQkeYXY|KUX@Rl…keUXaOF=?A¢keT?U|YXUOfK#WZ—baFWNhNAQp8OoMOcWˆp?kžzLUX[8kOcA,=?kZM)YXAQh#OcW{OF=?A¢kZMcMF^V@BlNOFUXWZp1Oc=?kO

=?U|lVlmWNnkZ@Rl?^?M-lVY]kžKLMkepUX@RlmWZaFOckZp_OacWZYXA)UXpp?kžzLU|[_kOFUXWZpˆT_K’l?aFWzLU]hNU|p?[RkRnW8[ZpVU|OFUXzZAi@’kel MPOFW8aFASh{TLK

lmWeOcAp_OFU]keY3MFKLp?kelNOcUXn”–AQU|[8=_OcMiaFAQlVacAQMFAp_OFUXpV[{TmWeOc=­MFl?kOcUXkZYkZp?h1OcA@BlmWZaokeYnWZacacAY]kOFUXWZp…M£.µnnWZaohNU|p?[

OcW

r} tŸ“?Oc=VA¢MPOFW8aFASh#U|pN—šW8aF@’keOFUXWZp1U]MgacAY]kOcAQhOFWˆY|WNnkeOFUXWZp?M)kep?h#lmW8MFU|OFUXWZp¯MF=VU|—œOcMinWNhNAQh#TLK#lVY]kZnAnAQY|Y

k8n"OcU|zLU|OfK’OF=?keOgkeacUXMFA)—šaFW8@ MPKLp?kZlNOFU]nil…WZOFAQp8OcUXkeOFUXWZp*£

<-=VA=?U|lVlmWNnkZ@Rl?^?Mbkep…h¢acAY]kOcAQhRkeacAQk8M3keacAUXp?hNAQAQhRpVWZO3OF=VAW8pVY|K¢WZpVAMPOFac^?n"Oc^VacA”UOc=,PYXWLn^?Mf¡WZa

l?YXk8nAnAYXYXM)WZaiU|pLz8WZYXzZAQhUXp¯lVY]kep?pVU|p?[?£)ÜgW”–AQzZAQaQ“…”–A¢aFASMfOcaFU]n"OgW8^Va)hVUXMcn^?McMFU|W8p1OFWˆOF=VASMPARMfOcaF^?nOF^VacAQM

TmAQnQke^?MFA”A”-kep_OgOcW hNU]Mcn^?McMaokOc=VAailVacAQnU]MFAlVacWZlmAaFOFUXAQMWZ—+MP^…no=©@’kel?MQ“mkZp?h1acAY]kOcAOF=?A@OFWˆOF=VA

l?aFW8l…W_MPASh{@BWNhNAY:“?kZM-l?aFASMPAQp8OcAQh pVW”HkZp?hhNUXMcn^…MFMFAQh U|pOc=VAMPAS\_^VAY:£

81W8aFA¢lVacAQnU]MFAYXKZ“m”–ARnWZp…MPU]hNAagOF=VABkeaono=VU|OFASn"OF^?aFAWZ—3OF=VAR=VUXlVlmWLnQke@Bl?kZY'MPOFac^?n"Oc^VacAQMQ“…@’keUXpVYXK#OF=VA

AQp_OFWZac=VUXp?keY!nWZaFOFA 

→ hNAp_OcackZY&[ZKLac^?M

→ ^{ µgv  µ s ™ } WZaclV^?Mgµg@B@RW8pVU]MgU]MgkZpVWeOc=VAa)p?ke@BAWZ—3OF=VA

=?U|lVlmWNnkZ@Rl?^?Mc›+UXp’OF=VAi=VUXlVlmWLnQke@BlV^?M

→lVacA—šacWZp_OckZYeMfOcaFU]kOc^V@ykZaFASkZMg™6MPAQA—šW8a–UXp?MPOckep…nA

rs } t…—šW8akZp

A LOFAp…hNAQh hNUXMcn^…MFMFU|W8pˆkeTmWZ^VO+OF=?A.MFl?keOFU]keYskZMFl…ASn"OcM–We—&OF=VAi=VUXlVl…WNnQke@BlV^?M–—š^Vp?nOFUXWZp?kZY|U|OfK{kep…h

r

}Z€ tm—šWZa

kBhNU]Mcn^?McMPUXWZp#keTmWZ^VO-WZOF=VAQal…AQacnAlNOc^?keY!kep?h @BWeOFW8a-Ock8MP˜NM-We—!OF=VA,=?U|lVlmWNnkZ@Rl?^?Mc›“?kZM-UXYXY|^?MPOFaokOcAQhUXp

ƒ'UX[?£

s £

"!$#&%'#)($!*!%,+

Ý*ޞâ-"â/.âfèQéeâfã6ßÑçâfåêŸæoîVì]æoðªê3ßÑà!ê:ޞâížßÑà:ðfäFáâfã0.äoì216îÑäð,3žà3ðfâfîÑîÑà4657 89?ßÑå,ê:ޞâ–ÞžßÑéžéeäðFæoçé 3žàFò

:

3žççæoã¬ß<;PßÑåSïžëžà3SðŸÞ.ðfâfîÑîÑà!ލæfáâ+ê6ޞâ+éžã¬ä"éZâfãšê..ê6ä>=Zâfðfä"çâfàæoðªê6ßáâ@?&ޞâfåiê6ޞâ+æoåžßÑçæoîLßÑà!æcêæ-ï"ßáâfå.îÑäðPæcê:ßÑä"å,ßÑågê:ÞSâ

âPåáQßÑã6ä"åžçâfåêæoåSíA

B

ßDCsðfä"ã6ã¬âfà:éeä"åžížà!ê:äæ–à:éæoðfâ3îÑäðFæcê6ßÑä"å?&ޞßÑðŸÞ)ßÑà&æ–ä"ã¬ížâfã'äoìVçæoï"åžßêE3žížâ3îÆæoã¬ï"âfã&ê6ލæoåiê6ޞâbæoåžßÑçæoîZà:ß<;fâ"ë

B

ßÑßDCmðfä"åê:ßÑå3žâbê:äGFžã6â3âªáâfå.ßìNê6ޞâ3âfåáQßÑã6ä"åSçâPåê&ê:ä-"âfåžà'æoã6â3ã¬âfçäFáâfí.ä"ã*ßìNê:ÞSâbæoåžßÑçæoî_ßÑà¨ßÑågê:ޞâ3ížæoãE-eë

B

ßÑßÑßDCsæoã¬âH3žà:âfí)ì|ä"ã*ížßIZâfã6âfåê'îÑäðPæcê:ßÑä"åžà&ßÑåiíSßI8âfã¬âfåê!âfåáQßÑã6ä"åSçâPåê6àFë

BßáJCsލæPáâ3á"æoã¬ßÑä3žà&à:ލæoéeâfàK?!ßê6ޞä3Sê'æoåL.)íSßÑã6âfðªê'îÑßÑå -'?!ßê6Þ)ê6ޞâ3âfåáSßÑã¬ä"åžçâfåê!çâªê:ã¬ßÑð"ë

B

áJCLލæPáâ"ë"ì|ä"ã…æ!à3J= M]à:âªêVäoìeà3žð:ÞðfâfîÑîÑàFëæoåæoðfê6ßáQßê.–åSäoêmä"åžî.bã¬âfîÆæcê:âfíê6ä'ê6ޞâ*îÑäQðPæcê6ßÑä"åG=J3QêmæoîÑà6ä!ê6ä'ê6ޞâ*îÑäQðPæoîQê6ã6æON6âfðªê6ä"ã.

ížßÑã6âfðªê6ßÑä"å_ë

B

áQßDCsà3žðŸÞ)ðfâPîÑîÑà*à¬êŸæoãšê:àPFã¬ßÑåžïæoà&à:ää"å.æoà*ê:ޞâbæoåSßÑçæoî_âfåê6âfã6âfíiê6ޞâbðfä"ã6ã¬âfà:éeä"åžížßÑåSïgâfåáSßÑã¬ä"åžçâfåêFò

10 Entorhinal Cortex

5

C A 3 −> C A 1105 Dental Gyrus

Prefrontal cortex

Environment loci

Shifts of the present location Present location

Goal location

Sparse locus map(s)

motor command sensori inputs

Ventral striatum 106

ƒ'UX[Z^VacA

s

u)µgacno=?UOcAQn"Oc^VacA¢WZ—+OF=?AR@’kZU|p¦MfOcaFASke@«kep…h­MPOckeOFARzžkZaFU]keT?Y|ASM)WZ—bOF=?ARacAY]kOcAQh¯=VU|l?l…WNnkZ@Bl?keY

MPOFac^?nOF^VacAQMQ“VMPAQAiOFA LO—šW8ahNAOokeUXYXMQ£

Dªp¢OF=VU]M3keaono=VU|OFASn"OF^?aFA.™¬MFAAAZ£[?£

r

v8€ tN—šWZa3k)[ZAQpVAaokeYLhNU]MFn^?MFMFUXWZp¢kZp?h¢AZ£[?£

r

}Zu tN—šWZa3kgacAQnAp_OaFAQz_UXA”g›

UXpN—šW8aF@’kOcU|W8p­kZT…W8^NOY|WNnU3UXM.UXpVlV^NO,—šacWZ@ OF=VA’AQp8OcWZac=VUXp?keY3nW8aPOcA ¯”=?UXno= U|p_OcA[ZaokOcAQM.MFAzZAQackZY'MFWZ^?acnA

WZ—U|pN—šW8aF@’keOFUXWZp*“UXp?nYX^?hNUXpV[­MFl?keOFU]keY+W8pVAQMQ£©<-=VA A Nl?kep…MPUXWZp®Y]kžKZAQaaFW8Y|A’U]M¢Y|UX˜ZAQY|K­OFW¯hVUXMFl?kOono=®OF=?UXM

UXpN—šW8aF@’kOcU|W8p*“FkeYXY|WNnkeOFUXpV[¡Y|WNn^…M'nAYXYXM™6MPAQA-AZ£[?£

rs u tN—šWZabk.hNAOckeUXYXAQhRhNUXMcn^…MFMFU|W8p…›"£'<-=VAnW8@BlVaFASMFMFUXWZp

Y]kžKZAQa–acWZYXA.UXM-YXUX˜ZAYXK’OcW{nWZ@¢TVUXpVA,kep?hMP^?@¢•Ÿ^Vl OF=VA,YXWNn^?MnAYXY&nW8@TVUXp?keOFUXWZp?MTV^VUXY]hNU|p?[BOF=VA,YXWNnkOcU|W8p

MF=VU|—œObWZ^NOclV^NObUXpROF=VA  µ s

MfOcaF^?nOF^VacAOFW”-keaohVM!@RWZOFW8abkeacAQkZM'zLU]kgOc=VAzZAQp_OFaokeYNMfOcaFU]kOc^V@#£'<-=?AMP^VTVdfAQn"O

YXWNnkeOFUXWZpUXMkZYXMFW’nWNhNAQh UXp Oc=VAQMFA.l?keaFOcM-WZ—!Oc=VA,=VUXlVlmWLnQke@Bl?kZYs—šWZac@BkeOFUXWZp*£

Dªp­aFASnAp_O.TVUXWZYXWZ[ZU]nkZY&@BWNhNAY]MQ“sOF=VARlVacA—šacWZp_OckZY'MPOFac^?n"Oc^VacAU]MgOc=VWZ^?[Z=¯acAQMFlmWZp?MFU|T?Y|A—šWZaiAp?nWLhVU|pV[

[8W8kZY?YXWLnQkOcU|W8p*“_kZYOc=VWZ^?[Z=’Oc=VAacA)kZaFAikeY]MPWAQzLUXhNAQp?nASMbOc=?kO-=VUXlVlmWLnQke@Bl?kZY…YXWNn^?MnAQY|Y]M+acAY]kOcAQhBOcW[8W8keY

YXWNnkeOFUXWZp?M-UXp?nacAQk8MPA.Oc=VAUXagk8n"OFUXzLUOfK8£<-=?UXMg@BkžK T…A¢hN^VA,OcW’U|p_OFAQack8n"OcU|W8p?MTmAOf”AAp#TmWeOc=1@BkZl?MQ£<-=VA

˜8AK’lmWZUXp_O=VAacA.UXMOc=?kO@BWZacAiOF=?kZp W8pVA.@BkZl YXUX˜ZAYXK{UXp_OFAaokZnOU|p OF=VAMFAp…MPW8aFU|•:@BWZOFWZaYXW_W8lˆUXpLzZWZYXzZASh

UXp®OcackedfAQn"OcWZacK¯[8ApVAQackeOFUXWZp*£©µ nWZ@BlV^VOckOcU|W8p?keY–@BWLhVAY@^?MPOOoke˜ZA MP^?no=®YXWNn^?MP•Ÿ@BkZl²nW8@TVUXp?kOcU|W8p

UXp_OFW’kZnQnW8^Vp_OQ£

µgpVWeOc=VAanaF^?nUXkZY3kZMFl…ASn"O,UXMiOc=VAB—¬kZnO.OF=?keO.YXWNnUbkZaFABpVWZO.W8aF[_kepVUX±ASh#OFW8l…W8[ZaokelV=?UXnQkeYXY|K8“m”=?AacAQkZM

—šW8a–k,[8U|z8Ap{MFl?kOcUXkZY•ªkeacAQkiYXWNnU…kZaFAfacAQnaF^VU|OFASh_¡”UOc=VWZ^NO-kepLKBMPl?keOFU]keYmMPW8aPOcAacU|pV[,”=?Ap{acAQ\_^VUXaFASh¨£ƒVWZa

k¯[ZUXzZAQp²MPl?keOFU]keY|•ŸkZaFASkV“keTmWZ^NORt8€«WZ—OF=VAnAQY|Y]M¢kZaFA aFASnac^VU|OFAQh*£­<-=VU]Mz8AacK­UX@Bl…W8aPOokep_O¢l?aFW8l…AQaPOfK

@¢^?MPO–TmAglVacAQMFAp_O+UXp{knWZ@BlV^NOokOFUXWZp…keY?@BWNhNAQY6£3<-=VAYXWNnkeOFUXWZp’U]M+acAY]kOcAQhROFWOF=VAiMPl?keOFU]keY|•ŸkZaFASk)—šackZ@RA

WZ—3aFA—šAacAp?nAZ“shN^VAOcW{TmWeOF=¯A LOFAQaFp?kZY!MFAp?MFWZacK n^VAQMR™š@’kZU|pVYXKz_U]MF^?keY'TV^NOipVWeOiWZpVYXKV›kep?h©UXhNUXWeOc=VAOcUXn

™6MPAQY—œ•Ÿ@BWeOcU|W8p1aFAQYXkeOFAQh…›UXp8OcAaokZnOFUXpV[’”UOc=¯ASkZno=N•ŸWeOc=VAaS£<-=VA¢MF^VTNdfASn"O)YXWNnkeOFUXWZp1U]M–FnkeYXUXTVackeOFASh_¡.”U|OF=

acAQMFlmAQn"O-OcWBOF=VU]MMPl…kOFU]keY¨@’kZl*£

w)^VacU|pV[#Oc=VA’OcackZp?MPU|OFUXWZp TmAOf”AAp Of”W1MPl?keOFU]keY|•ŸkZaFASkZMQ“sOF=VA{MFAOWZ—YXWNnU+kZaFABacA•ªkZnOFUXzžkeOFASh¯UXp¦OF=VA

YXWNn^?M–@’kel*“_Az8Ap_OF^…keYXY|KBacA•Ÿ^?MPUXpV[klVaFAQzLU|W8^?MPYXKRaFAQ[ZU]MfOcAacAQhBYXWLn^?M–@BkZlˆkep…h’Az8Ap_OF^?kZY|YXKR^Vlmh?kOFUXpV[RUO

”U|OF=pVA”®UXp?nW8@BU|pV[)U|pV—šWZac@BkeOFUXWZp*£'IL^?no=YXWNn^?M!@’kel,U]M*Oc=_^…M&acA@’kep?Ap_O&”U|OF=acAQMFlmAQn"O&OFWOc=VA-MPAQp?MPW8aFK

UXpVlV^VO¢™¬MPOFW8aFAShˆacAlVacAQMFAp_OokOFUXWZp We—'OF=VA¢MP^VacacWZ^Vp?hVU|pV[_Mc›£`wg^?aFUXpV[’p?kžzLUX[8kOcU|W8pˆTmAOf”AAQp^?p?nW8pVpVAQnOFASh

lmWZaFOFUXWZp…M)WZ—–MFl?kZnAZ“sOF=?AacA¢U]M,k MF”U|Ocno=¯TmAOf”AAQp­YXWNn^?Mi@’kel…M£<-=?AR”-kžK1k nWZ@BlV^NOokOcU|W8p?keY@BWNhNAY

nQkep1UX@BlVY|AQ@BAp_O)MF^?no=©k’lVacWZlmAaFOfKˆU]Mgk{kZYXMFW{kB˜ZAQK U]McMP^VA8£ÜAacAZ“…Y|WNn^?MgnAYXYXM)keacAkZn"OcU|z8AYXKˆUXpLzZWZYXzZASh

UXp#aFAQYXkeOFUXWZp pVWeOOcWRW8TNdfAQnOUXhNAQp_OFU|OfKˆTV^NOYXWNnkeOFUXWZp*£

µgpVWeOc=VAa#kZMFl…ASn"OS“gkelVlmAQkZaFUXpV[®UXp lVacAzLUXWZ^?MˆaFA—šAacAp?nAQMQ“UXM{OF=?keO pVAQ^VackZYiMFU|OFAQMˆaFAQYXkeOFASh¥OFW³YXWNnU

nWZacaFASMPlmWZp…hNU|p?[¢OcWRPWZT?MPOck8nYXAQMf¡’™¬U6£AZ£)kOgOc=VATmWZaohNAQaWe—3Oc=VA¢MFl?kOcUXkZY•ªkeacAQk_›–=?kžzZAAQY|W8pV[8keOFAShMF=?kZl…ASM£

IN^?no= kZp kep?UXMFWeOcaFW8lVUXngMP=?kZl…Ai@BU|[8=_OTmA)acAY]kOcAQhBOcWOc=VAi—¬kZnOYXWNnU¨kOOF=VAiTmWZaohNAaWe—*Oc=VA,MPl…kOFU]keY|•ªkeacAQk

acAL …AQn"OoM'OF=VATmWZaohNAQaOFWZlmWZYXWZ[8KZ“eUXpROF=VAMFAp…MPAOF=…kO+”=?keO3UXM3W8pROF=VAWZOF=VAQabMPU]hNAWe—mOF=?AT…W8achNAQa3UXMYXUX˜ZAYXK

p?WeO-aFAQlVaFASMPAQp_OFAQh{WZp OF=VAi@’kel*£–E)pˆOF=VA,nW8p_OFaokeacK¢YXWNnU*nW8aFacAQMFlmWZp?hNUXpV[,OcW¢UXp_OFAQaF@BAQhVUXkeOFAi[ZW8kZYXM–=?kžz8A

kZp U]MFWeOFacWZl?UXniMP=…kelmAZ£

&WLn^?MnAYXYXM,acAY]kOcAQh­OFW1UXp?MFUXhVABOF=VA{MFl?kOcUXkZY•ªkeacAQkkeacABY|ASMFMlVaFASMPAQp_O.T?^NOU]MPWZOFacWZlVU]nu.Oc=VA{acWZYXARWZ—

MF^?no=®YXWNnU@BkžK¦TmAˆacAY]kOcAQh¦OcW©U|p_OcAac@RAShNU]kOFA WZT?MPOck8nYXAQMTV^NO’keY]MPW©UXp²acAY]kOcU|W8p®”UOc=²U|p_OcAac@RAShNU]kOFA

[8W8kZY:™6Mc›£



W8@Rl…keacU|pV[¯[ZW8kZYbWZacUXAp_OFASh OckZMF˜NMacAY]kOcAQh OFW¯p?kžzLU|[_kOFUXWZp…keYbTmA=?kžzLUXWZaoM,”U|OF=²acAQMFl…ASn"OOcW¯”-kepN•

hVAacU|pV[¢Ock8MP˜NM”U|OF=VW8^NOklVacAQnUXMFAi[ZW8kZYmOFWBkOPOokeUXp*“LOF=?AacA.kZaFA)WZpVYXKˆMF^VTNOcY|A,hNUzmAQaFAQp?nASMU|p

 µ s

‘?acU|p?[

no=…keaokZn"OcAacUXMPOFU]nM.™¬@BkZU|p?Y|KkB@BkZaF[8U|p…keY¨UXp?nacAQk8MPA.We—'OF=?A.‘?aFUXpV[’aokOcA,U|pOF=?A

s

MPOgnk8MPAž›"£+<-=VU]MacAQMF^VY|OUXM

nWZ@Bl?keOFUXTVY|A,”U|OF=Oc=VA,—¬kZnOOF=?keOOF=?A[ZW_keYsY|WNnQkOFUXWZp#U]MAp…nWNhNAQhAQYXMFA”=VAQaFA{™¬AZ£[?£bUXp#OF=?A,lVaFA—šaFW8p_OckeY

nWZaFOFA V›kZp?h MF=VW”M–OF=…kOTmWeOF=#˜LUXp?hWe—!TmA=…kžz_UXWZaoMY|UX˜ZAQY|Kˆ^?MPA.Oc=VAMcke@BA.@RASno=?kep?UXMF@’M£

81WNhNAQaFpUXhNASkZM'keTmWZ^NO'pVA^?ackZY8T?k8MPASM&WZ—Vp?kžzLU|[_kOcU|W8pMPAQA@’M*OcW)hNA—šAQkeO'MFWZ@BA+WZ—VOF=VA–^?MF^?keYLnW8p?nAQlNOcM

kZT…W8^NO{acAY]kOFAShñnWZ[8pVU|OFUXzZA1@’kel?M kep…hñUOˆU]M’pVWZO nYXAQkZa’U—kZlVl?keacAp_OcY|K³hVU{zsAacAp_OˆTmA=?kžzLUXWZaoM1™š[8W8keY

W8aFUXAp_OcAQh­zZAaoMF^?M.”kZp?hNAQaFUXpV[A NlVYXWZaokOFUXWZp&“&MPl…kOFU]keY+zZAaoMF^?MiT…ASkZnW8p§™šWZa.OcWZ˜8Ap¦T?kZMFAQh…›ip?kžzLU|[_kOFUXWZp&“

@BAOFacUXnizZAQacMF^?M–OFW8l…W8Y|W8[ZU]nkZYmp…kžz_UX[8keOFUXWZp…›–keacA.pVWeOghVU{zsAacAp_O-—¬kZnAOcM-WZ—!Oc=VAMcke@BA.@RASno=?kep?UXMF@ u

•iT…ASkZnWZp¦p?kžzLU|[_kOcU|W8p nQkep®TmAˆzLU|AQ”–ASh¦kZMk#[8W8kZYbaFAQYXkeOFASh­p…kžz_UX[8keOFUXWZp¦”U|OF=²MFAzZAQackZY3U|p_OcAac@RAShNU]kOFA

MF^VTN•Ÿ[ZW_keY]M

•BA NlVYXWZaokOcU|W8p§nkep T…A­z_UXA”AQh kZMˆk³Y|UX@BUO1nk8MPA¯We—[ZW_keYghVU|acAQnOFAQh p?kžzLU|[_kOFUXWZp§”U|OF=Hk²”AQke˜¥WZa

AQzkepVASMFnAp_O[ZW8kZY

•@BAOcaFU]n¢p?kžzLU|[_kOFUXWZp1@’kžK1TmARknW8p?MFAQ\_^VAp…nAWZ—bOF=?AR[8ApVAQackeOFUXWZp©WZ—–kep­WZlNOcU|@’kZY'l…kOF=¯OFW”-keaohVMgk

[8W8kZYmT?^NO^?MFU|pV[’kZp_KˆpVWZpN•Ÿ@BAOcaFU]niaFAQlVaFASMPAQp_OckOcU|W8p*“LAOone£X£

K#* (%

ò S

ê:æcèQä"åžä"ç.gäoìNê6ޞâ3ížßIZâfã6âfåê'à¬ê:ã6æcê:âfï"ßÑâfà'ã¬âfîÆæcê:âfíiê6ä-àE3žðŸÞiðfä"ï"åžßê6ßáâbçæoéžà¨ßÑà*ížâªêŸæoßÑîÑâfí

ßÑå 5€ 9_ì|ã¬ä"ç¥ê:ÞSâ+à6ßÑçéžîÑâ'ã¬âféJ3žîÑà6ßÑä"åoæcê6ê6ã6æoðfê6ßÑä"åçâPð:ލæoåžßÑà:çà

B

â"òïžò*æG=eâfâbä"ã6ßÑâfåêŸæcê6ßÑä"å ?&ßê:Þ)ã6âfà:éeâfðªêê6äê:ÞSâ+àE3žåCsê:ä

à:ä"éžÞžßÑà¬ê:ßÑðPæcê6âfí¢çâªê6ã6ßÑð+çæoé,åæPáQßÑïæcê6ßÑä"å

B

â"òïžò–æÞ3žçæoå,ßÑåæðPæoã)CŸòmâfã0. =žã¬ßÑâ .ëLæoðfðPä"ã¬ížßÑåžï)ê:äê:ÞSâ-æ 3Sê6ޞä"ã¬àFëLàE3žðŸÞ

=ZâfލæfáSßÑä"ã!ã6â3žßÑã¬âPà!ê:ޞâ+æ 3Sê6ä"åžä"çä3žà'îÑäQðfä"çäoê6ßÑä"åiäoìVê:ÞSâæoï"âfåê3æoåžíiê6ޞâbåžäoê6ßÑä"å,äoì…æcê6ê6ã:æoðªê6ßÑä"åcã¬âPé 3žîÑà:ßÑä"å¢æoåží,çæ$.

=Zâ3à¬ê:ã3žðªêE3žã6âfí,ßÑågê:âfã¬çà&äoì A

LL( !*!"

ë?!ÞSâPã¬â–ê6ޞâ–à3J=&N6âfðªêbæcê¬ê:âfçéSê6à'ê6ä)çâfçä"ã¬ß<;Pâ3ê6ޞâ+à:â&3SâPåSðPâäoìséeâfã6ðfâféSê6ßÑä"åžàbã6âfðfâfßáâPí

íJ3žã¬ßÑåžïê6ޞâbéžã6âªáQßÑä3žà&åæPáQßÑïæcê:ßÑä"å.æoåží.îÑäðFæoîÑî.çäížßFâfà*ßê6àP=eâfލæfáSßÑä"ã*ê:äã6âféžã¬äQí 3žðfâ–ê6ޞßÑà*à:â&3SâPåSðPâ"ë

#

L%%/#$% (& !*!"

ë?!ÞSâPã¬â)ê6ޞâ)à3J=&N6âfðªêçâfçä"ã6ß<;fâfà–ì|ä"ãâPæoðŸÞ 16îÑäðPæcê:ßÑä"å4.æ 16åSâ,?³ížßÑã¬âfðªê:ßÑä"å&4.æoåží

åæPáQßÑïæcê:âH=L.géæoà:à6ßÑåžïì|ã6ä"ç³ä"åžâbîÑäQðPæcê6ßÑä"å)ê6äæoåžäoê6ޞâfã3žà6ßÑåžïê6ޞâP16ížßÑã¬âfðªê:ßÑä"åFžâPîÑí4ªë

'

&!()*#!* +,)L%!"

ë ?&ޞâfã¬â-ê:ÞSâgàE3J= N¬âfðªê–Þžæoà+çâfçä"ã¬ß<;fâPí¢æ)ï"ã6æoéžÞBäoì¨ê6ޞâîÑäðPæcê:ßÑä"åžà

B

åžäQíSâPà–äoì¨ê:ÞSâ

ï"ã:æoéžÞ C…æoåží–äoìZê6ޞâ*éæcê6ޞà

B

áâfãšê:ßÑðfâfàsäoìeê:ÞSâ&ï"ã6æoéžÞCNìXã¬ä"ç ä"åSâ¨îÑäQðPæcê6ßÑä"åê:äbæoåSäoê:ޞâfãfëL?!ÞSßÑîÑâ¨ßê…åžæPáQßÑïæcê:âfà =L.–ßÑížâfåê6ßì*.QßÑåžï

ê:ޞâ3à6â&3žâfåžðfâ+äoìVîÑäQðPæcê6ßÑä"åžà*ìXã¬ä"ç¥ßê6à*îÑäQðPæcê6ßÑä"åiê:ä–ê:ޞâbï"äæoî]ë

-

'($.#/ +,)L%!"

ë?&ޞâfã¬â!ê6ޞâ'àE3J= N¬âfðfêmލæoàsåžä$?#æ3çâªê6ã6ßÑð"ëß]òâ"òLê:ä"éeä"ï"ã6æoéžÞžßÑð"ëžçæoé-æoåžíßÑížâfåê6ßìD.-åžäoê¨ä"åSî.

ê:ޞâ&à:â3žâfåžðfâ'äoì8îÑäQðPæcê6ßÑä"åžà = 3Sê¨æoîÑà6ä3ðPä"åSà:ßÑížâfãmê6ޞâ&îÑâPåSïoê:ÞäoìZê:ÞSâ&ê6ã:æN6âfðªê:ä"ã0.ëžâªáâfåêE3æoîÑî.

#) %(%L(+.01(+2&03&%4

(5!).(6.(+ * %)!7

æoåžíRæoîÑà:ä.ßÑåê:ã¬äíJ3žðfßÑåžï

%83 ./#&%>.6.L0)&EL&

B

à:âfâ 577 9…ì|ä"ã+æiéžã¬âPðfßÑà6âížßÑà:ð,3žà6à:ßÑä"åä"å¢ê6ޞßÑà

à3J=&N6âfðªê/CŸò

Q

å+ê:ޞßÑàVê:æcèSä"åžä"ç@.ë"ê:ޞâ¨ßÑíSâFæ!äoìJ1¬ðPä"ï"åSßê:ßáâ¨çæoéžà04¨æoà?éžã¬ä"éeä"à:âfí'=L. 57+ 9SßÑàVßÑåê6ã6äíJ3žðfâfí_ëL?&ޞßÑðŸÞ–ížâ/Fåžâfàsæoå–ßÑåê6âfã6åæoî

ã6âféžã¬âfà:âfåê:æcê6ßÑä"åäoìNê6ޞâbâfåáQßÑã¬ä"åžçâfåêP3žà6âPí=.ê6ޞâ+æoåžßÑçæoîeê:äGFžåží.ßê6àK?¨æ$.ò

!"$#&%' )(%* +&,! %*

$b@BAQaF[8U|pV[ nWZp…nAlVOcM rs x t'OFAQp?hOFWˆU|p_OFAQ[ZaokOcA,U|p©k{nWZ@B@BWZp#—šaoke@BA”WZac˜’Oc=VAQMFAkel?l?keacAp_OFYXK hVU—œ•

—šAQaFAQp_OT…AQ=?kžzLU|W8acM™6nW8p8OcackZaFK{OFW’zZAQaFKnW8@Rl?Y|A  @BAQno=?kZpVU]MP@’MMF^?no=©kZMOF=VW_MPAl?aFW8l…W_MPAShˆUXp1AZ£[?£ r}e„ t

UXp#WZaohNAaOcWRA NlVYXkZU|p OF=?AhNU{zsAacAp_Ogk8MPlmAQnOcM-We—'OF=?AQMFAiTmA=…kžz_UXWZaoMo›"£

ƒ?WZYXY|W”UXpV[1Oc=VUXMOFaokZno˜s“k©nW8@BlV^NOckeOFUXWZp?kZY+@BWNhNAY@^…MfORpVWeORWZpVYXK hVAQMcnacU|TmA{=VW” OFW­Y|ASkeacp¦OF=VA

MFl?keOFU]keY¨lVacWZlmAaFOfK’We—'OF=VA,AQpLz_UXacWZpV@BAp_OS“NTV^NOgkeY]MFWBk8hVhNacAQMcM–OF=VA,[8Y|W8T?keY¨lVacWZT?Y|AQ@°We—'OFaokdfASn"OFW8aFK’[8ApN•

AQackeOFUXWZpñTmAQnkZ^?MFA²™¬MFAA©A8£[…£ —šW8aˆk²hNAQ@RW8p?MPOFaokOFUXzZA#A NkZ@BlVY|A

r

}et tœ›’U|p_OFAQaFp…keYacAlVacAQMFAp_OokOFUXWZp…M’keacA

kZY|”-kžKNM-TV^VUXYOiU|p©hVAAl¯aFAQYXkeOFUXWZp#”UOc=1OF=?AUXagacAY]kOFASh#MPAQp?MPW8aFU|•Ÿ@RWZOFW8a-OckZMF˜u+=VAQaFAOF=VAMFl?keOFU]keY&@’kZl1UXM

T?^VU|Y|O)UXp1acAY]kOFUXWZp#”U|OF=#Oc=VAp?kžzLUX[8keOFUXWZp OckZMF˜¦™¬MFAAZ“?—šWZaA Vke@BlVYXA,OF=VA¢hNUXMcn^…MFMFU|W8p1U|p&“

r} x

tŸ“…”=?AacAUO

U]MikeY]MPWˆMF=VW”p1Oc=?kOi=VAQk8hNUXpV[ hNUXacAQn"OcU|W8p©U]MgacAQMFl…W8p?MFU|TVYXA—šW8agOc=VAUX@BlVYXA@BAp_OckeOFUXWZp­We—+kep?[Z^VY]kea)l?kOc=

UXp_OFAQ[ZaokOFUXWZpm›"£

µOOc=VUXMl…W8U|p_OUOgkZlVlmAQkeaoM–OF=?keOgkeYXY¨=VUXlVl…WNnQke@Bl?keY¨acAY]kOcAQh @BAQno=?kZpVUXMF@’M“VkZYOc=VWZ^?[Z=@^?no=@BWZacA

nWZ@BlVYXA ’Oc=?kep ”=?kOU]M–T?aFUXAL ?K’acAzLUXA”AQh’=?AacAZ“NkeacA)W8pVYXK’aFAQYXkeOFAQh’OFWROF=VA.YXWNnkeYNP”=VAacA•:OFWZ•:[8We•ŸpVA LOf¡

k8MPlmAQnO¢WZ—OF=VA OFaokdfASn"OFW8aFK­[ZAQpVAaokOcU|W8p*£­µgp?h®Oc=VA \_^VASMfOcU|W8p®OFW­aFAQYXkeOFA{Oc=VU]MYXWNnkeY@BAQno=?kZpVU]MP@#“bkZM

Mcno=VAQ@BkeOFUX±ASh²UXp¤Oc=VA1acU|[8=8OBl?kZaPO’We—.ƒ'U|[…£ } OcW OF=VA1[8Y|W8T?keYOFaokdfASn"OcWZacK [8ApVAQackeOFUXWZp¤lVaFW8TVYXA@ UXp¥k

MFAp…MPW8aFK_•Ÿ@RWZOFW8a+YXW_W8l#aFAQ@BkZU|p…MkZp U]McMP^VA8£

<-=VU]M UXMˆOF=VA¦kZMFlmAQn"O”–A¯”kZp8O OFW¥kZhVhNacAQMcM’=VAQaFA8“gMF=VW”U|p?[®OF=?keO ”U|OF= k²MP^?UOokeTVYXA¯YXWNn^?MnAQY|Y

acAl?aFASMPAQp8OokOcU|W8p­Oc=VAˆYXWNnkZYP”=VAacA•:OFWZ•:[8We•ŸpVA LOP¡¢Y|WNnQkeY@RASno=?kep?UXMF@ hVU|acAQnOFYXK acAY]kOcAQMOFW©Oc=VAˆ[8Y|W8T?keY

OcackedfAQnOFWZacKg[8ApVAQackeOFUXWZp.lVacWZTVYXA@qkZp?h,lVacWzLUXhNASM*kgMFWZYX^NOcU|W8p.OFWgU|OQ“ž—š^?Y|YXKnW8@Rl…kOFUXTVYXA+”U|OF=,Oc=VA–k8MPlmAQnOcM

acAzLUXA”AQhˆ=VAacAZ£"&AO^?MUXp_OFacWLhV^?nAiOF=?UXMU]hNASk¢p?W”,£

\geb j _ V:T^e VYg X m V:_ b`T^e T[WYX V:TKV:WYX[Z\T^]!V:_`WfghTP

Dªp#nW8@BlV^NOFAQaMFnU|AQp?nA8“NWZpVA)OFaokZno˜©™6MPAQA)—šW8a-U|p?MPOckZp?nA

rs “ w t¨—šW8a-”–W8aF˜NM-kOOF=VA,W8aFUX[ZUXp{WZ—&OF=?UXM-W8pVAS›+OFW

MFWZYXzZAOF=VARp?kžzLU|[_kOcU|W8p1lVaFW8TVYXA@UXM)OcW acAlVacAQMFAp_OOc=VA¢OFaokdfAQnOFW8aFK[ZAQpVAaokOcU|W8p1lVaFW8TVYXA@^?MFU|pV[#kˆMFWe•

nQkeYXY|ASh Y%*f%'£bIL^…no=R@BAOF=VWNhVM–kZMcMP^V@BAOF=…kO+W8T?MfOokZnY|ASM'OFWkžzZW8UXh™šW8a+nW8p?MPOFaokeUXp8OoMpVWeO3OcW,zLU|W8YXkeOFAS›

nWZacaFASMPlmWZp…h#OFW#@’k NUX@Bk We—–OF=VABlmWeOcAp_OFU]keY:“&”=VUXYXA¢Oc=VAB[ZW_keY3nW8aFacAQMFl…W8p?hVMgOFW1k ^VpVU]\8^?AR@BUXpVUX@^V@#£

<-=?A{nWZacacAQMFl…W8p?hNUXpV[kZY|[8WZacUOc=V@ Oc=L^?M,TmA=?kžz8AQM,kZMU—WZp?ABOF=VacW”M,k1MP=?AAOW8p_OFW#OF=?UXM¢MfOokOFA{MFl?k8nAZ“

Oc=VU]M–=LKLlmAaF•ŸMF^VaF—¬kZnAgaFAQY|UXA—*TmAUXpV[RAYXAzkOcAQh{W8p{WZT…MfOokZnYXAQMQ“_”UOc= k=?WZYXA)W8p’OF=VAi[ZW_keY:“_MFWOc=?kO‘?p?hVU|pV[

k¢OcackedfAQn"OcWZacK’aFAShN^?nAQMOFWPaFW8Y|Y¨hNW”pN¡iUXp Oc=VAhNUXaFASn"OcU|W8p WZ—&OF=?A,l…WZOFAp_OcUXkZY*[ZaokZhNUXAp_OS£

ÜgW”–AQzZAaS“”=VAp³TV^VUXYXhNUXpV[¦MP^…no=³k©lmWeOcAp_OFU]keY–OF=VA @’keUXp³lVacWZTVYXA@ U]MOc=VAˆWNnQn^VacaFAQp?nAˆWe—gYXWNnkeY

@BUXpVUX@BkH™¬MFAA­—šWZaU|p…MfOokep?nA

r

vZ„ ti—šWZa#k³MFU|@BlVYXA¦nYXW8MFAQhL•:—šWZac@ UXYXY|^?MPOFaokOcU|p?[®OF=VU]M#MfOokOcA@BAp_Oo›£ ƒVWZa

UXp?MPOckZp?nA8“k²nW8p?nkžz8A1WZT?MPOck8nYXA1@BkžK®P=VUXhVA"¡1OF=VA­[ZW_keYgkZp?hñ[ZAQpVAaokOcA©k®Y|WNnQkeYg@BUXpVUX@^V@#“g”=?AacA

Oc=VAROFaokdfASn"OFW8aFK1U]MiOFaokel?l…ASh¨£{IL^?no=®k#MPU|OF^?keOFUXWZp zZAQaFK1YXU|˜8AYXK©WNnn^VacM rs w tŸ£’<&ASno=VpVU]\_^VAQMiOcW@’kZp?ke[8A

Oc=VU]M’nkžzZASkORA NUXMPO@u —šWZaBUXp?MPOckep…nAU|p rs t:“l…W8U|p_OoM¢”=?AacAˆOc=VA#[ZaokZhNUXAp_OBUXMBkep¥AUX[ZAQpN•ŸzZAQnOFW8aWe—)OF=VA

ÜgAQMcMPU]kep.WZ—_Oc=VA+lmWeOcAp_OFU]keY_keYXY|W”MmOcWghNA‘?pVA+kacW8k8hL•Ÿ@BkZl)W8p_OFW”=VU]no=.OFaokdfASn"OFW8aFUXAQMs@’kžKiT…A+[ZAp?AaokOFASh

kžz8WZU]hNUXpV[gYXWNnkeYL@BUXpVU|@¢^V@#£3ÜW”AzZAQaQ“MF^?no=@BASno=?kepVU]MF@BMkZaFA–—¬kea'—šacWZ@ÓTmAUXpV[)OFacU|zLU]keYLOFWiUX@Rl?Y|AQ@RAQp_OQ£

ƒ?^VaFOF=VAQaF@BWZacAZ“_Oc=VAK =?kžzZAipVW’YXU|pV˜ˆ”UOc=T?U|W8Y|W8[ZU]nkeYs@BAQno=…kepVU]MP@’MQ£

µgp­AQY|AQ[8kep_O,MPW8Y|^VOFUXWZp¯=…kZMiT…AQAp U|p_OcaFWNhN^?nAQh¯T_K  W8pVpVW8Y|YXK©kep?h 6 aF^?l…AQp

rw “ x “ t3nWZp?MFU]hNAacU|p?[

+ D":!%Y# # f%)f%'# u’U|O¢U]M”AYXY+˜LpVW”p¦OF=…kO¢MF^?no=²k#—š^Vp…n"OFUXWZp³=?kZMpVW©@BU|p?U|@¢^V@

A VnAlVOW8pOc=VAhNWZ@’keUXpRT…W8^Vp?hVkZaFUXAQM-™¬MFAA-A8£[…£

rs v tVW8akZlVlmAp?hNU|  µ —šWZabkiMF=VWZaFO'acAzLUXA”²W8p¢=…keac@RW8pVU]n

—š^?p?n"OcU|W8p?M-lVacWZlmAaFOFUXAQMo›"£3<-=VU]MkelVlVacW8k8no=ˆkeY]MFWRkZY|YXW”MbOcWBU|p_OFacWNhN^?nA)pVW8pN•Ÿ=VWZYXWZpVW8@BUXninW8p?MfOcackZU|p_OoM rx t

™¬U6£AZ£{MFWZYXzZAROc=VABOFaokdfASn"OFW8aFK1[8ApVAQackeOFUXWZp­lVaFW8TVYXA@ AQzZAp¦U|—–Oc=VA’^Vp?hVAacY|KLUXpV[#@RASno=?kep?UXnQkeYbMFKLMPOFAQ@«OFW

nWZp_OFacWZY!UXMiMFU|pV[8^VY]keagUXp1Oc=VARMFAp…MPAOF=?keO)YXWLnQkeYnW8p8OcaFW8Y&YXWLWZl?M—¬keUXYXMihNacU|zLUXpV[ˆU|Oo›£i<-=VA



WZp?pVWZYXY|K#kep…h

6 ac^VlmApRkelVlVacW8k8no=keY]MPW)keYXY|W”M&OFW)aFAQYXkeOFA–MF^?no=RklmWeOFAQp_OFU]keY_OFWikepW8T?MPOckZnY|AnW8Y|YXUXMFUXWZplVacWZT…keTVUXY|U|OfK r t:“

[8ApVAQackeOFUXpV[BkOcackedfAQnOFWZacKB”UOc=1MPW8@RA,MPOckeOFU]MfOcUXnQkeY¨WZlVOFUX@BkZY|U|OfK’”U|OF=#aFASMPlmAQnO-OFW’WZT…MfOokZnYXA.kžzZWZU]hVkZp?nA8£

jAQnAQp_OFYXK ™¬AZ£[?£ r„ tš›"“Oc=VU]M [8aFW8^Vl =?k8M k¤hNAz8AYXWZlmAQh TVU|W8Y|W8[ZU]nkZYg@BWNhNAYiWe—TmA=…kžz_UXWZa ^?MFU|pV[¤OF=?UXM

kZlVlVacW8k8no=ˆkZMhNU]Mcn^?McMPAShˆUXpOF=VAMFAQ\_^VAQY6£

DªpBOc=VAUXa+UX@BlVY|AQ@BAp_OckeOFUXWZp*“8OF=VA)ke^NOc=VWZaoMbnkZYXn^VY]kOFA-Oc=VAlmWeOcAp_OFU]keY…WZp’k,acA[Z^?YXkZa3@RASMP=&“_Mcke@BlVYXU|p?[

Oc=VA­MPOckeOFA MPl?k8nA8£y<-=VA¦hNaokž”T?kZno˜¤We—MF^?no= kep U|@BlVYXA@BAp_OokOFUXWZp UXM U|p…hNAAShñOF=VA Y]keac[ZA©@BA@BW8aFK

nWZp?MF^V@BlNOcU|W8p kZp?h©Oc=VA’=L^V[ZA’nWZ@BlV^NOokOFUXWZp­OFUX@BAZ£ˆƒ?WZa,UXp?MPOckep…nAZ“*—šWZaklVacWZTVYXA@ ”U|OF=®„#hNAQ[ZacAAQM

WZ—.—šaFAQAQhNW8@~™šA8£[…£qk¦aFW8T…WZOFU]n1keac@’›’”UOc=§l?keaoke@BAOcAa’lVacAQnUXMFUXWZpñWe—f“MckžKNM“-€V£

s

Oc=VU]M{KLU|AQYXh?M’OFW

k#acA[8^VY]kea@BAQMF=®We—,£X£

s € s

^?pVUOoM£©<-=?UXMUXM.Oc=L^?MUXp_OFaokZn"OokeTVYXAZ£#Dªp?hVAAQh*“'MFAz8AaokeYb=VAQ^VacUXMPOFU]nM{™¬AZ£[?£

=?U|AQackZacno=VU]nkZYekZhVkZlNOFUXzZA

rs t tœ›!@’kžK.aFAShN^?nA–OF=?UXM3nWZ@BlVYXA NUOfK8“TV^NO3pVWZOU|OcMW8achVAa'We—?@’ke[8pVU|OF^?hNA8“e”=VUXY|A

kZpLK Mcke@BlVYXU|pV[ nWZ@BlVYXA NU|OfK =…kZMUXp_OFacU|p?MFU]nkeYXYXKkZp1A Nl…W8pVAp_OcUXkZY*[ZacW”-OF=#”U|OF=1Oc=VA¢MFl?kZnAhNUX@RAQp?MFU|W8p

™¬AZ£[?£

rv tV—šWZa–k.aFASnAp_O+hNU]Mcn^?McMPUXWZpRacA[8kZachVU|pV[iOcaFU]kepV[8^VY]kOFUXWZpRWe—s=VUX[Z=N•ªhNUX@BAp?MFU|W8p?keYVMFl?k8nAS›“Z”=?keOFAQzZAa

Oc=VA#MPl?k8nA’OcaFU]kep?[Z^VY]kOcU|W8p®WZaR@RASMP=³@RAOF=VWNh³UXMQ£ µ)M¢k­nW8p?MPAS\_^VAp?nAZ“Oc=VUXM¢@BAOc=VWNh²=?k8MTmAAQp*“bUXp

l?ack8n"OFU]nA8“LpVWeOacAQkeYXYXK’^?MPASh{—šW8ahNUX@RAQp?MFU|W8p?M=VU|[8=VAa-Oc=?kep

n= 2^WZa n= 3 ™6MPAQAi—šWZaUXp?MfOokep?nA rs “ x

tš›"£

ƒ?^VaPOc=VAac@BWZacAZ“NOc=VA,pL^V@BAacUXnQkeY&nWZpLzZAQaF[8Ap?nAg@’kžKˆTmAz8AacKˆMPYXW”,“NTmAQnkZ^?MFA.OF=VAUXp ?^?Ap?nA,We—OF=VA

[8W8kZY&™š”=VU]no=’UXp8OcaFWNhN^…nAQM–kMFY|W8l…AUXp’Oc=VAlmWeOFAQp_OFU]keYmaFAQY|UXA—¨UXp’WZaohNAa3OcW¢keOPOFaokZnO3OF=VAgOFaokdfASn"OFW8aFKV›'@¢^?MfO

TmA,lVacWZl?kZ[8keOFAQhˆOF=?WZ^V[8= Oc=VA”=VW8Y|A,MPOckeOFAMFl?kZnA.T_KˆOF=VAp_^?@RAQaFU]nkZY*MFno=?A@BA,^?MFAQh OFW{nWZ@BlV^NOcA,OF=VA

=…keac@RW8pVU]nglmWeOcAp_OFU]keY+™6MPAQA.AZ£[?£

rt t*—šWZahVAOckZU|Y]Mo›bOcW’T…A.Ooke˜8ApU|p_OFW’k8nnW8^Vp_OU|p#acA@BWeOcAiYXWNnkeOFUXWZp?MQ£

Dªp WZaohNAa OFWñkžzZW8UXh§OF=VA lVacAzLU|W8^?MT?kZhHMPU|OF^…kOFUXWZp&“OF=?A l?aFASMPAQp8O#”WZac˜ñkZU|@’MkO1TV^VUXY]hNU|p?[¥kZp

=…keac@RW8pVU]n¢lmWeOcAp_OFU]keY)™œOF=L^?M”UOc=VWZ^NOYXWNnkZYb@BU|pVUX@^?@’›kZM,k1MFl?keaoMPA¢‘?pVU|OFAˆY|UXpVAQkZanW8@TVUXp?kOcU|W8p­WZ—

AQY|AQ@BAp_OckZaFK1=?kZaF@BW8pVUXn¢—š^Vp…n"OFUXWZp…M£{DŸOkZlVlmAQkeaoMiOF=…kOMF^?no= k#nWZp?MPOFac^?n"OcU|W8p­U]M,UXp®hNU|acAQnOY|UXpV˜©”U|OF=

”=…kOUXMlVaFASMPAQp_OFYXK’^Vp?hNAQacMPOFWLWNhaFAQ[8keaohNUXpV[=?U|lVlmWNnkZ@Rl…keY¨acAY]kOFASh MPOFac^?nOF^VacAQMQ£

DªpˆOF=VA)—šWZYXY|W”UXpV[MFAQnOFUXWZp{”A—šW8aF@’kZY|YXKBaFAQz_UXA”ñOc=VAgl?aFW8TVY|AQ@ kep?hˆU|p_OFacWNhN^?nA)kMFl?kZacMFAgkZh?kelNOcU|z8A

—šW8aF@yWe—sl?keaoke@BAOFacU|±QAQh^?p_TmWZ^?p?hNAQhB=…keac@RW8pVU]n–lmWeOcAp_OFU]keY:“8nQkeYXY|AShBkep ,2 £ q­A)MP=VW”³OF=?keObOF=?UXM

—¬kZ@BU|YXK{We—lmWeOFAQp_OFU]keY*MFWZYXzZAQM–Oc=VA,lVacAQMFAp_OlVaFW8TVYXA@#£

q A)kZYXMFWhNU]MFn^?MFM+=VW”¤OF=VU]M+@’kžK¢Y|ASkZhROFW¢kep’UXp8OcAacp?keY?aFAQlVaFASMPAQp_OckOcU|W8pBWe—sOF=VA)lVacWZTVYXA@ kZM+k.p?WZpN•

OcWZlmWZ[8ackZlV=VU]n@’kel¦UXp¦aFAQYXkeOFUXWZp ”UOc=­Oc=VA{=VUXlVlmWNnke@BlV^…M,@BAQno=?kZpVUXMF@’M,kep?h­Oc=VA’aFAQYXkeOFASh­nWZ[Zp?UOcU|z8A

@’kZl?M£

Dªp¦Oc=VAˆpVA LO¢MPASn"OcU|W8p ”A{hNASMFnaFUXT…Aˆ=VW”y”A{nkZp®U|@BlVYXA@BAQp8ORMP^?no=²k#@BASno=?kepVU]MF@£#<!”–WˆfOFWK8¡

A Nl…AQaFUX@BAp_OcM,kZaFARaFAQl…W8aPOcAQh¨“¨”=?UXno= aFASMP^?YOoM,keYXY|W”qOcW1kep?kZY|KL±A¢OF=?A{\8^…keYXUOokOFUXzZARlVacWZlmAaFOFUXAQM.WZ—+OF=?UXM

@BASno=?kepVU]MF@£

%’– ,#

WY_`l mT^n _`Wfn X mhgyXBVYgh_de

&AO^?MnWZp?MFU]hNAa4u

™¬k_›-kBMPKNMPOFA@#“?hVA‘?pVAShˆTLKk #+%)%) 2%)

x∈ R^n“ n≥2

™šTm›kep f%'^#%)%) “N”aFU|OPOcAp

x₀∈ R^n“

™¬nS›

r ^{Y#+% f%#} hNA‘?pVASh TLKˆMcnkZYXkZa-U|pVAS\_^?keYXUOcU|ASM

ci(x)>0, i∈ {1..r}^“

™¬h…›k ,4 hNA‘?pVAQhTLK kZpnWZp?MPOFaokeUXp_OWZ—!Oc=VA.—šWZac@

c0(x)≤0^“

k8M-U|YXYX^?MfOcackeOFAShˆUXp1ƒ!UX[?£ } £

c (x) < 0 i

c (x) < 0j

x0

(t)

goal c (x) = 0 0

ƒ'UX[Z^VacA}:ubINno=VAQ@BkeOFU]n.zLU|AQ” We—!OF=VA.OcackedfAQnOFWZacK’[ZAQpVAaokOcU|W8pˆlVacWZT?Y|AQ@£

<-=VAˆlVaFW8TVYXA@ U]M,Oc=L^?MhNA‘?pVASh¦^?MPUXpV[1UX@BlVY|U]nU|O¢AS\_^?kOcU|W8p?M

ci : Rⁿ → R ™6kZMcMP^?@RASh­OFW©TmAˆkO YXAQk8MfO-Of”U]nA,hNUzmAQaFAQp_OFU]keTVYXAZ“VW8anWZp…MPU]hNAacUXpV[RhVUXMPOFacU|T?^NOFUXWZp?Mo›"£

DŸ—–OF=VAR[ZW_keY'UXM.MFU|@BlVYXK#OFWaFASkZno=¯kˆ‘?p?kZY3MfOokOFA

x_∞ ∈ R^{n ™šU:£}AZ£k lV^Vp?nOF^?kZY[ZW8kZY]›“mOF=VABacAY]kOFASh UX@BlVYXUXnUOAS\8^…kOFUXWZpU]M“?MckžKZ“

c0(x) =||x−x_∞||^£

<-=VU]MhNA‘?pVASMk’hNWZ@’keUXp[u

U ={x,∀i∈ {0..r}, ci(x)>0}

k8MFMF^V@BAQhˆOFWBTmA

f2%)

”UOc=

x0∈ U kep?h ”=VU]no=ˆ—šacWZp_OcU|AQaUXM@u

∂U ={x,∃i∈ {0..r}, ci(x) = 0}=∪^r_i=0∂Ui

”=?AacA

∂Ui = {x, ci(x) = 0}U]M.OF=?A{—šacWZp_OFUXAa,WZ—Oc=VA

iOF=²nWZp?MPOFaokeUXp_OWZT?MPOck8nYXA™¬U—

i >0›,WZa,Oc=VA

—šacWZp_OcU|AQaWZ—!Oc=VA,[ZW_keY3™šU|—

i= 0^›"£

q A,”aFU|OFA

U¯=U ∪∂U OF=?A.MPOckeOFAMFl?kZnA.We—&Oc=VU]MlVacWZTVYXA@#£

 W8p?nacAOcAYXKZ“

x nWZp_OckZU|p…MglV=LKNMFUXnQkeYl?keaoke@BAOFAaoMQ£.<-=?AK©keacA¢OF=L^?MiT…W8^Vp?hNASh­\_^?kZp_OFU|OFUXAQM.kep?h¯”–A nQkep#kZMcMP^?@RAiOc=?kO

U UXMTmWZ^?p?hNAQh*“?MPWROc=?kO

U¯^{UXM f%£}

ÜgAacA,”–A.YXW_W8˜’—šWZak¢OcackedfAQn"OcWZacK

γ:R⁺→ Rⁿ MP^?no= OF=…kO@u

•bUOgMPOckZaPOoMkO-OF=VA,UXpVU|OFU]keY*lmWZUXp_O

γ(0) =x₀^“

•bzZAacU|‘?AQMOc=VA.lVaFW8TVYXA@ nWZp?MPOFaokeUXp_OcM

∀t∈ R⁺, γ(t)∈ U^“

•baFASkZno=VASMi™6kOYXAQk8MfOkZMFKL@BlNOFWZOFU]nkeYXYXKN›+Oc=VA,[ZW_keY YXU|@

t→∞c0(γ(t)) = 0^“

Oc=VU]M-OFaokdfASn"OFW8aFKBTmAUXpV[{nW8p_OFUXp_^?WZ^?MkZp?h keOY|ASkZMPOOf”UXnA.hVU{zsAacAp_OFU]keT?Y|A8£

<-=VABhNUzmAQaFAQp_OFU]keTVUXY|U|OfK©nW8p?hNU|OFUXWZp¯kZY|YXW”MgOFW nWZacaFASn"OcY|K1nW8p_OFacWZY*Oc=VA¢OFaokdfAQnOFW8aFKA NAQn^NOFUXWZp­TLK#k

l?=_KNMFUXnQkeY¨hNAQz_U]nA8£3Dªp—¬kZn"OS“NOF=?A.MFWZYX^NOcU|W8p lVacWZlmW8MFAQh UXp Oc=VAMFAQ\_^VAY¨KLUXAY]hVM-UXpN‘?pVU|OFAhNUzsAacAp_OFU]keTVUXYXUOfK8£

€ 4

e V$Wf_ ]ygeb R X Wfn _de g] j _ V:T^e VYg X m

DªpWZaohNAQa–OcW’MPW8Y|z8AgOc=VA,lVacAQMFAp_Ol?aFW8TVY|AQ@°”A)UXp_OFacWLhV^?nA,kRl…WZOFAQp8OcUXkZY

V :U → R”=VU]no=@RUXpVUX@BU|±Sk•

OcU|W8pkZY|YXW”MOFWB[8ApVAQackeOFA)OF=VA,A LlmAQnOFAShˆOcackedfAQnOFWZacK

γ()^u

™šUœ›-MPOckeaFOFUXpV[R—šWZa-Oc=VA,UXpVUOcUXkZY¨l…W8U|p_O

™šUXUœ›Y|WLWZ˜LUXpV[R—šWZakBMF@BkZY|YXAa-zkeYX^VA,WZ—

V() kOAQk8no=MPOFAQl

™šUXUXU]›-MPOFW8lVlVUXpV[B”=VApOF=?A,@RUXpVUX@^V@ We—

V()U]M-acAQkZno=?AQh 81W8aFA.lVacAQnUXMFAYXKZ“N”A,nW8p?MPU]hNAQa–OcackedfAQnOFWZacUXAQM

γ(t) MP^?no= OF=…kO@u

γ⁰(t) =−∇V(γ(t)) ^{™ s ›}

”=?UXno=UXMgkRMPOckZp?hVkZach s MPOWZaohNAa-@BU|p?U|@BUX±QkOcU|W8pMcno=VAQ@RA8“?MFU|p?nAu

V⁰(γ(t)) =∇V(γ(t))^Tγ⁰(t) =−||∇V(γ(t))||²<0

µ)MgMF^?no=*“VOc=VU]MglmWeOcAp_OFU]keY*p?WeOgW8pVY|K#keYXYXW”MOFW’‘?p…h©kROFaokdfASn"OFW8aFK{—šaFW8@ kB[8U|z8Ap

x0

OcW’OF=VA[ZW8kZY6“

T?^NO.—šaFW8@ kepLK

x₀ ∈ U OFWOF=?AR[8W8kZY u)”–A¢OF=L^?M.=?kžz8AAQ@TmAQhVhNASh©Oc=VABlVacAQMFAp_Oil?aFW8TVY|AQ@ UXp­k@BWZacA [8ApVAQackZYsWZpVA8“NU|p#WZaohNAQa–OcW’T…AkZTVYXA)OcW’MPW8Y|z8AiU|OQ£

Dªp#WZaohNAQa–Oc=VU]Ml?keaokZhVU|[8@ OFW’TmA,zžkZY|U]h¨“N”A.U|p_OFacWNhN^?nA.OF=VA.—šW8Y|YXW”U|p?[RnWZp?MPOFaokeUXp_OcM@u

C0

<-=VA,[ZW_keY¨nW8aFacAQMFlmWZp?hVM–OcWR@BUXpVUX@BkBWZ—&Oc=VA,l…WZOFAQp8OcUXkZY6£bÜgAacA.”–A,no=VWLW8MFAu

YXU|@

x→∂U0V(x) =−∞ ^™:}Z›

Ci

E)T?MPOck8nYXAQM-keacA.@Bke NU|@’kRWe—!OF=VA,lmWeOcAp_OFU]keY:£bÜAQaFA.”A.no=?W_W_MPA,u

∀i,^Y|UX@ x→∂UiV(x) = +∞ ^™6v8›

Cc

<-=VAQaFA.U]M-pVW’YXWLnQkeY¨@BUXpVU|@¢^V@%™¬WZa' …keOaFAQ[ZUXWZp?Mo›–We—&Oc=VA,lmWeOFAQp_OFU]keY'u

∀x∈ U,||∇V(x)||>0

ÜAQaFA,”A,no=VWLW8MFAikZp + ff%)2D%u

∀x∈ U,∆V(x) = 0 ^™¬t_›

”=VU]no=*“?kZMhVAOckZU|YXAQh UXp#kZlVlmAp?hNU|  µ “?keYXYXW”M+OcWBzZAacU|—šK

Cc

£

 =VWLW_MPUXpV[1^VpLTmWZ^Vp?hVAQh¦zkeYX^VAQMWZp OF=?AˆhNWZ@’kZU|p®TmWZaohNAa@BkžK­Y|WLW8˜¯YXU|˜8A{k©MfOcackZpV[ZA’no=VW8UXnA’TV^NO

YXAQk8hVMgOFW#k MF^VaclVaFU]MFU|pV[8Y|K1MFU|@BlVYXA 2#2 + MFWZYX^NOcU|W8p¯hNAz8AYXWZlmAQh¯pVW”,£µO)Oc=VABU|@BlVYXA@BAQp8OokOcU|W8p

YXAz8AY:“LT…W8^Vp?hNAShˆlmWeOcAp_OFU]keY]MkeacA)[8WZUXpV[OcWBT…A.AQk8MPUXYXK{hNAacU|z8AQh{—šaFW8@ OF=VU]MnYXW8MFAQhN•6—šW8aF@ MFWZYX^NOFUXWZp*£3<-=VA

OcAQno=Vp?UXnQkeYVnW8p?MFAQ\_^VAp…nAQMWZ—mk8nnAQlNOFUXpV[,UXpN‘?pVU|OFAzkZY|^VASM3WZp¢OF=?AhNWZ@’keUXpBT…W8^Vp?hVkZaFUXAQMWZ—mkZpR=…keac@RW8pVU]n

lmWeOcAp_OFU]keYkeacA{hNU]MFn^?MFMFAQh¦UXp²kel?l…AQp?hNU|  µ £1<-=VUXMaFAQlVacAQMFAp_OckeOFUXWZp¦UXM@BWeOFUXzkOcAQh¦TLK¯Oc=VA’—¬k8n"OOc=?kO

‘…p?hNUXpV[’kepA NlVYXUXnUOMFWZYX^NOFUXWZp —šWZakRT…W8^Vp?hNAShˆ=…keac@RW8pVU]nglmWeOcAp_OFU]keY*MFAAQ@BMkZpˆUXp_OFaokZn"OokeTVYXA.lVacWZTVYXA@

rs v tb”=VU|YXA{kepLK1pL^V@BAacU]nkeYbMFWZYX^NOcU|W8p?M.keacARYXUX@RU|OFASh T_K©OF=VA{n^VacMFARWe—hVU|@BAp…MPUXWZp?kZY|U|OfK rs “ x

tbU|p OF=?UXM

nQkZMFAZ£



=VWLW_MPUXpV[{kep#=?kZaF@BW8pVUXn.lmWeOcAp_OFU]keY!UXM—¬kea—šacWZ@ T…AQU|p?[’OF=VAWZpVYXK W8pVAMFWZYX^NOFUXWZp®™šA8£[…£hNA‘?p?U|pV[{k

nWZpLzZA glmWeOFAQp_OFU]keYZU]M&kY|ASMFM*nWZp?MPOFaokeUXpVAShikZYOcAacp?kOcU|z8AS›mTV^VO&=?kZM!k-zZAQaFK)U|p_OFAQaFASMfOcU|p?[WZlNOcU|@’keY8lVacWZlmAaFOfK!u