:+6+.C8 4Y_\XKV YP ,ORK`SY\KV RO KX^SZRK]O ]cXMR\YXSdK^SYX ]^K^O MKX LO WKNO ZY]]SLVO SX WO^\YXYWO] ]c]^OW OS^RO\ Lc OXRKXMSXQ ^RO NKWZSXQ aOKU MY_ZVSXQ K]]YMSK^ON aS^R ^RO LK]O WY^SYX Y\ Lc QYSXQ ^Y `O\c VK\QO Y]MSVVK^SYX P\O[_OXMSO]  3P ^RO XK^_\KV P\O[_OXMSO] YP ^RO ^aY ZOXN_V_W] NSɼO\ Lc WY\O ^RKX K POa ZO\MOX^ ]cXMR\YXSdK^SYX aSVV XY^ YMM_\  3P ^RO XK^_\KV P\O[_OXMSO] YP ^aY WO^\YXYWO] K\O NSɼO\OX^ ]SQXS¶MKX^Vc L_^ aS^RSX KX KMMOZ^KLVO \KXQO S^ VOKN] ^Y K ]cXMR\YXSdON ]^K^O L_^ aS^R MYX]^KX^ ^SWO VKQ LO^aOOX ^aY Y]MSVVK^Y\] E G +] aO aSVV ]OO SX ^RO XOb^ ]OM^SYX] WY]^ YP ^RO]O YL]O\`K^SYX] WKNO Lc :KX^KVOYXO K\O KV]Y `O\S¶ON SX Y_\ ObZO\SWOX^] =cXMR\YXSdK^SYX] K\O _LS[_S^Y_] SX XK^_\O 0\YW K LSYVYQSMKV `SOa ]cX MR\YXSdK^SYX SX MY_ZVON Y]MSVVK^Y\] MKX LO ]OOX ^R\Y_QR Y_^ ^RO XK^_\KV aY\VN KXN ]ZOMSKVVc MYX]ZSM_Y_] SX VS`SXQ ^RSXQ] + QYYN ObKWZVO YP ]cXMR\YXSdK^SYX SX XK^_\O S] ^RO ·K]RSXQ YP ¶\O·SO] >RY_]KXN] YP MY_ ZVON Y]MSVVK^Y\ MKX LO ]OOX SX ^RO PY\W YP ¶\O·SO] ZK\KNSQW YP Z_V]O MY_ZVON Y]MSVVK^Y\] +^ XSQR^ ^RO]O SX]OM^] QK^RO\ K^ ]YWO ZVKMO KXN ·K]R SX ]cXMR\YXc /KMR SX]OM^ RK] S^] YaX \Rc^RW ^ROc SX^O\KM^ OKMR Y^RO\ YXVc aROX YXO ]OO] ^RO ]_NNOX ·K]R YP KXY^RO\ KXN ]RSP^ S^] \Rc^RW EG -\SMUO^ MRS\ZSXQ S] KXY^RO\ ObKWZVO YP Z_V]O MY_ZVON Y] MSVVK^Y\] EG 7KXc ObKWZVO] YP ^RO MY_ZVON Y]MSVVK^Y\ MKX LO PY_XN SX LSYVYQSMKV ]c]^OW]$ SX ^RO OSQR^OOX MOX^_\c 4OKX4KM[_O] .Y\^Y_] NO 7KS\KX NS]MY`O\ON ^RO MS\MKNSKX \Rc^RW Lc YL]O\`SXQ ^RO NKc KXN XSQR^ Y]MSVVK^SYX YP RK\SMY^ LOKX VOK`O] EG% ,S\N] SX ·YMU] ]cXMR\YXSdO ^KUO Yɼ KXN VKXNSXQ EG% 7KVO KXN POWKVO WY][_S^YO] ]cXMR\YXSdO aSXQ LOK^] EG% ZKMOWKUO\ MOVV] LOK^ SX ^RO ROK\^ ^YQO^RO\ EG O^M



=cXMR\YXc SX ]YMSKV SX^O\KM^SYX]

=cXMR\YXc SX NcXKWSM ]c]^OW] ]_MR K] ]YMSKV ]c]^OW] S] K \OMSZ\Y MKV KNKZ^K^SYX YP LORK`SY\] ^OWZY\KV ]^\_M^_\O] LO^aOOX SX^O\KM^KX^] E G ._\SXQ R_WKX MYWW_XSMK^SYX ]YWO XYX`O\LKV VKXQ_KQO] VSUO QO]^_\O] PKMSKV ObZ\O]]SYX] KXN XYNNSXQ K\O KV]Y SX`YV`ON +] ]^K^ON Lc 6 1VK]]$ o-YWZVOb LYNSVc \Rc^RW] K\O _LS[_S^Y_] SX VS` SXQ Y\QKXS]W]o E!G ,_^ ^RO K\S]SXQ [_O\c S] RYa ^RO]O ]cXMR\YXSdON LORK`SY\] Y\ \Rc^RW] SX^O\KM^ SX R_WKX ]YMSKV MYWW_XSMK^SYX] E"G 0Y\ SX]^KXMO ]SXQSXQ SX _XS]YX S] K RSQRVc ]cXMR\YXSdON PY\W YP ]YMSKV SX^O\KM^SYX @SU^Y\ O^ KV ObKWSXON KXN _X`OSVON Q\Y_Z NcXKWSM] _X NO\VcSXQ ^OWZY\KVVc MYY\NSXK^ON KM^SYX] MRYS\ ]SXQSXQ >RO K_^RY\] \O`OKVON ^RK^ ZRK]O ]cXMR\YXSdK^SYX SX ROK\^ \K^O `K\SKLSVS^c 2ROc MYXMV_NON ^RK^ \O]ZS\K^Y\c KXN MK\NSKM MY_ZVSXQ ZK^^O\X] \OXNO\ ^RO ZRc]SYVYQSMKV PY_XNK^SYX PY\ SX^O\ZO\]YXKV ^OWZY\KVVc MYY\NSXK^ON KM^SYX] E#G 0\ON -_WWSX] K_QWOX^ON ^RS] NS]M_]]SYX aS^R KX SX]^KXMO YP KZO\SYNSM ]cXMR\YXSdK^SYX YP MYWZVOb KM^SYX 2O PY_XN KZO\SYNSM ]cXMR\YXSdK^SYX YP MYWZVOb KM^SYX SX RS] ObZO\SWOX^KV ^K]U YP ]cXMR\YXY_] ]ZOKUSXQ EG =O`O\KV \O]OK\MRO] SX Z]cMRYVYQc ^YYU SX^Y KMMY_X^ ^RO MYXMOZ^ YP ]cX MR\YXc SX OK\Vc R_WKX ]YMSKV SX^O\KM^SYX] ^ROc ]^_NSON ^RO ^OWZY\KV MY Y\NSXK^SYX YP XYX`O\LKV LORK`SY\] K] LYNc WY`OWOX^] `YMKVSdK^SYX] QKdO KXN WKXc Y^RO\] EG +MMY\NSXQ ^Y Z]cMRYVYQS]^] aROX ^aY R_

WKX] SX^O\KM^ Y\ MYWW_XSMK^O aS^R OKMR Y^RO\ ^ROc NY XY^ YXVc _]O ]ZOOMR ^Y MYX`Oc ^RO MYX^OX^ YP K WO]]KQO L_^ KV]Y OWZVYc K VK\QO `K\SO^c YP XYX`O\LKV LORK`SY\] EG PY\ ObKWZVO$ RKXN WY`OWOX^] ZK_]O] N_\SXQ NS]M_]]SYX PKMSKV ObZ\O]]SYX] ^RK^ ]RYa ^ROS\ K^^S^_NO KXN ^ROS\ VO`OV YP K^^OX^SYX ^YaK\N] ^RO ZK\^XO\ O^M +X SWZY\^KX^ ZK \KWO^O\ YP XYX`O\LKV MYWW_XSMK^SYX] S] ^RO ^OWZY\KV MY\\OVK^SYX Y\ ]cXMR\YXc LO^aOOX ^RO LORK`SY\KV ]^K^] YP ^RO SX^O\KM^KX^] EG +X SX^O\O]^SXQ K]ZOM^ YP ^RO]O ]cXMR\YXSdON LORK`SY\] N_\SXQ R_WKX SX ^O\KM^SYX] S] S^] _XSX^OX^SYXKV XK^_\O 7Y\OY`O\ NO`OVYZWOX^KV Z]cMRYVYQc KV]Y MYX]SNO\ON ]cXMR\YXc K] KX O]]OX^SKV ZK\KWO^O\ PY\ SX^O\KM^SYX] LO^aOOX WY^RO\] KXN ^ROS\ MRSVN\OX 3X PKM^ SP ^RO WY^RO\ VY]O] ]cXMR\YXc SXPKX^ ]^\_QQVO] ^Y ]_]^KSX ^RO SX^O\KM^SYX EG 3XPKX^] ]cXMR\YXSdO ^ROS\ VOQ] WY^SYX aS^R KN_V^ ]ZOOMR EG 3X KNNS^SYX ]cXMR\YXc NO^OM^SYX WOMRKXS]W SX cY_XQ SXPKX^] ZVKc] K ZO\`K]S`O \YVO SX VOK\XSXQ KXN MYQXS^S`O NO`OVYZWOX^ E G aY\N VOK\XSXQ E!G YLTOM^ SX^O\KM^SYX ]USVV] E"G ]OVPKaK\OXO]] KXN MYX^\YV E#G VOK\XSXQ \OVK^ON ^Y ]OVP EG O^M 0\YW K XO_\YLSYVYQSMKV ZYSX^ YP `SOa XO_\Y SWKQSXQ ^OMRXS[_O] OXKLVO _] ^Y YL]O\`O ]cXMR\YXc P\YW VYMKV ]MKVO L\KSXp] VYMKV ¶OVN ZY^OX^SKV KXN LO^aOOX NS]^KX^ L\KSX \OQSYX] ^Y SX^O\SXNS`SN_KV ]MKVO SX K ]YMSKV ]O^^SXQ EG =O`O\KV ]^_NSO] _]ON P7RO \O]_V^] NOWYX]^\K^ON ^RK^ ZK^SOX^] SX^OX^SYXKV WY^Y\ MYY\NSXK^SYX aK] KV^O\ON aRSVO ^ROS\ _XSX^OX ^SYXKV WY^Y\MYY\NSXK^SYX aK] \O^KSXON >RS] ]^_Nc MYXMV_NO] ^RK^ _X SX^OX^SYXKV WY^Y\MYY\NSXK^SYX Z\O]O\`O] O`OX PY\ ]_LTOM^] KɼOM^ON Lc ]YMSKV SX^O\KM^SYX NS]Y\NO\ E!G

157

:+6+.C8 4Y_\XKV YP ,ORK`SY\KV RO :6@ PY\ ^aY ]SQXKV] S] NO¶XON Lc

0SQ_\O  /bZO\SWOX^KV ]O^_Z K 8KY L K LK]SM K_^YWK^YX  NOQ\OO YP P\OO NYW M  N 9`O\KVV ]O^_Z PY\ R_WKX\YLY^ SX^O\KM^SYX

PLVx,y =

1 ! | exp(i(φx − φy ))| N t=1 N

(1)

ARO\O N S] ^RO X_WLO\ YP ]KWZVO] KXN φx −φy S] ^RO ZRK]O NSɼO\OXMO LO^aOOX ^aY ]SQXKV] >RO :6@ `KV_O S] MVY]O ^Y  PY\ ]cXMR\YXSdON ]SQXKV] KXN KZZ\YKMRO]  Y^RO\aS]O @SNOY] YP Y_\ ObZO\SWOX^] MKX LO PY_XN YX$ R^^Z$aaaO^S]OX]OKP\XO_\YMcLO\@SNOY]]cXMR\Y

 2_WKX RO]O O]^SWK^ON `OM^Y\] KM^ K] `S]_KV ]^SW_VS KXN SXZ_^] ^Y ^RO Z\YZY]ON K\MRS^OM^_\O

 >RO Y]MSVVK^Y\ 7YNOV +] ]RYaX SX 0SQ_\O K NY^^ON LYb SX Y_\ K\MRS^OM^_\O KX Y]MSVVK^Y\ WYN_VO EG MYX^\YV] ^RO 8KYp] K\W WY`OWOX^] 3^ MKX KV]Y LO ]OOX K] ^RO ]SQXKV \OZ\O]OX^SXQ ^RO 8KYp] SX^O\XKV NcXKWSM] 3^ MYX]S]^] ^aY XO_\YX] N1 KXN N2 SXRSLS^SXQ OKMR Y^RO\ Z\YZY\^SYXKVVc ^Y ^RO `K\SKLVO β  >RO Y]MSVVK^SXQ P\O[_OXMc S] K P_XM^SYX YP ^RO `K\SKLVO] α1 α2 KXN β$

N1 (n + 1) = N1 (n) − βN2 (n) + α1 N2 (n + 1) = N1 (n) + βN2 (n) + α2

158

(2) (3)

:+6+.C8 4Y_\XKV YP ,ORK`SY\KV RO\O K\O ^aY WY`SXQ YLTOM^] SX ^RO ¶OVN YP `SOa YP 8KY 9XO WY`O] _ZaK\N KXN SXN_MO] ZY]S^S`O KM^S`S^SO] ]RYaX Lc ¶VVON LVKMU MYVY\ ZSbOV] aRO\OK] ^RO Y^RO\ WY`O] NYaX aK\N KXN SXN_MO] XOQK^S`O KM^S`S^SO] _X¶VVON ZSbOV] >RO]O ZY]S^S`O KXN XOQK^S`O KM^S`S^SO] MKX LO VOK\X^ Lc ^RO \YLY^ KXN WYNSPc ^RO Y] MSVVK^Y\ KMMY\NSXQVc AROX KX KQOX^ SX^O\KM^] aS^R K WY^SYX P\O[_OXMc MVY]O ^Y 8KYp] P\O[_OXMc 8KYp] Y]MSVVK^Y\ MKX LO WYNS¶ON P\O[_OXMc KXN ZRK]O aS^RSX MO\^KSX VSWS^] 9^RO\aS]O S^ MYX^SX_O] ^Y S^] NOPK_V^ P\O [_OXMc >RO WK^ROWK^SMKV O[_K^SYX YP ^RO Y]MSVVK^Y\ MKX LO \OZR\K]ON K]$

N1 (n + 1) = N1 (n) − βN2 (n) + α1 + f #

(4)

ARO\O f # S] ^RO SXN_MON OXO\Qc MYWZ_^ON Lc KX SX^OQ\K^SXQ Y`O\ ^SWO KVV ^RO KM^S`O ZSbOV] SX ^RO SWKQO >RS] P_XM^SYX MKX LO NO¶XON K] K ^SWO \OZ\O]OX^K^SYX YP ^RO [_KX^S^c YP OXO\Qc Z\YN_MON Lc ^RO ZY]S^S`O KXN XOQK^S`O KM^S`S^SO] SX ^RO SWKQO f # WKc LO XOQK^S`O Y\ ZY]S^S`O 3^p] aY\^R XY^SMSXQ ^RK^ ^RS] NS\OM^ POONSXQ YP ^RO WY^Y\ MYX^\YVVO\ Lc f #  \O·OM^] ^RO SX·_OXMO YP ^RO `S]_KV ]^SW_VS YX 8KYp] KM^SYX] aRSMR WKUO] ^RO \YLY^ MRKXQSXQ S^] LORK`SY\] \OQK\NSXQ ^Y ^RO WY^SYX SX ^RO `S]_KV ¶OVN SX KX _XSX^OX^SYXKV WKXXO\

3^p] SWZY\^KX^ ^Y XY^O ^RK^ ^RO SX·_OXMO YP ^RO `S]_KV ]^SW_VS f #  S] aOSQR^ON Lc K MY_ZVSXQ PKM^Y\ Y\ MY_ZVSXQ ]MKVSXQ PKM^Y\ Sf ]OO 0SQ _\O L 2SQRO\ `KV_O] YP Sf SXN_MO K LSQQO\ SX·_OXMO YP ^RO `S]_KV ]^SW_VS YX ^RO \YLY^ WY^Y\ MYX^\YVVO\ -YX]O[_OX^Vc ^RO \YLY^ aSVV LO KLVO ^Y ]cXMR\YXSdO aS^R K VK\QO\ \KXQO YP P\O[_OXMSO]

0SQ_\O  KXN 0SQ_\O  SVV_]^\K^O ^RO \O]_V^] YP Y_\ ObZO\SWOX^] PY\ ^aY MK]O]$ ¶\]^ PY\ 2_WKX8KY SX^O\KM^SYX KXN ]OMYXN PY\ +_^YWK^YX8KY SX^O\KM^SYX 0SQ_\O K NO^KSV] ^RO WY^SYX ]SQXKV] YP LY^R 8KYp] K\W KXN ^RO R_WKX K\W Y\ RKXN aRSVO OX^\KSXSXQ OKMR Y^RO\ 0S\]^ WY`OWOX^] YP LY^R KQOX^] K\O XY^ ]cXMR\YXSdON -YX]O[_OX^Vc :6@ WOK]_\O YP ]cXMR\YXc ]RYa] VYaO]^ `KV_O ]OO 0SQ_\O L +] ]RYaX SX 0SQ _\O K KXN L N_\SXQ ^RO SX^O\KM^SYX LO^aOOX 8KY KXN ^RO R_WKX LY^R K\O ]cXMR\YXSdSXQ ]_MMO]]S`OVc K] ^SWO ZK]]O] >RO SXM\OK]SXQ ^OXNOXMc SX :6@ `KV_O] \O·OM^] ^RO OWO\QSXQ ]cXMR\YXc LO^aOOX ^RO ^aY KQOX^] 0SQ_\O] K KXN L KV]Y SVV_]^\K^O ^RK^ KP^O\ K MO\^KSX ^SWO LY^R SX^O\KM^KX^] K\O P_VVc ]cXMR\YXSdON KXN ^RO MY\\O]ZYXNSXQ :6@ `KV_O] K\O K^ S^] WKbSW_W 6S]]KTY_] M_\`O] ]RYa ^RO NcXKWSM] YP ^RO ^aY ]SQXKV] 3P ^RO ]SQXKV] RK`O ^RO ]KWO P\O[_OXMSO] ^ROS\ 6S]]]KTY_] M_\`O aSVV LO K ]^\KSQR^ VSXO aS^R KX KXQVO YP  NOQ\OO aS^R ^RO RY\SdYX^KV KbS] 3P ^RO ]SQXKV] RK`O ^RO ]KWO P\O[_OXMSO] L_^ aS^R K ]WKVV ZRK]O ]RSP^ ^RO VS]]]KTY_] M_\`O ^KUO ^RO PY\W YP KX OVVSZ]O 1\KN_KVVc SXM\OK]O SX ^RO ZRK]O ]RSP^ WKUO] ^RO OVVSZ]O aSNO\ aS^R K ZRK]O ]RSP^ YP # NOQ\OO ^RO M_\`O LOMYWO] K MS\MVO 3X Y_\ MK]O SP ^RO ^aY ]SQXKV] K\O SNOX^SMKV KXN ]cXMR\YXSdON aS^R OKMR Y^RO\ aS^R ]WKVV ZRK]O ]RSP^ ^ROX ^RO 6S]]KTY_] M_\`O] LO^aOOX ^RO ]SQXKV] YP \YLY^ KXN ^RO SX^O\KM^KX^ ]RY_VN LO KX OVVSZ]O 0SQ_\O M ]RYa] 6S]]KTY_] M_\`O LO^aOOX N(t) 8KYp] Y]MSVVK^SYX KXN H(t) R_WKXp] WY`OWOX^] >RO OVVSZ^SM ]RKZO YP ^RO M_\`O SXNSMK^O] ^RK^ LY^R ]SQXKV] K\O KVWY]^ SNOX^SMKV 0Y\ ^RS] ObZO\SWOX^ 8KYp] ]^KXNK\N P\O[_OXMc aK] 0.428 2d KXN R_WKX Y]MSVVK^SYX] aO\O LO^aOOX 0.4615 2d 7.8 RSQRO\ ^RKX 8KYp] P\O[_OXMc ^Y 0.476 2d 11 RSQRO\ KXN ^RO ]MKVSXQ W_V^SZVcSXQ PKM^Y\ Sf  PY\ f # aK]  0SQ_\O  ]RYa] ^RO SX^O\KM^SYX KXN ]cXMR\YXSdK^SYX LO^aOOX 8KY KXN ^RO K_^YWK^YX 0SQ_\O K ]RYa] ^RO WY^SYX ]SQXKV] YP LY^R K_^YWK ^YX KXN 8KYp] K\W =SWSVK\Vc VSUO PY\ R_WKX \YLY^ SX^O\KM^SYX +_^YWK

159

:+6+.C8 4Y_\XKV YP ,ORK`SY\KV Y SWZ\Y`O ^RO 67= MYX`O\QOXMO N_\SXQ YXVSXO VOK\XSXQ VOK\XSXQ WYN_VK^SYX η RK] LOOX SX^\YN_MON 3^ WYN_VK^O] \KXNYWVc ^RO VOK\XSXQ ]ZOON Lc SX^\YN_MSXQ K \KXNYWSdK^SYX OɼOM^ ^RK^ ]_ZZ\O]]O] ^RO XOQK^S`O OɼOM^]" YP ^RO ^OWZY\KV \OQ_VK\S^SO] YP ^RO SXZ_^ NK^K >RO XY\WKVSdK^SYX ^O\W kεX Xk (t)2 + σ 1 S] ]ZOMS¶M ^Y ^RO 867= σ 1 S] K ]WKVV `KV_O _]ON ^Y K`YSN ^RO NS`O\QOXMO YP ^RO ]cXKZ^SM aOSQR^] SP ^RO `S]_KV KM^S`S^SO] X SWKQO `KV_O] K\O ^YY ]WKVV >RO _]O YP ^RO 867= S] WY^S`K^ON Lc ^RO PKM^ ^RK^ ^RO XY\WKVSdK^SYX ^O\W ]_ZZ\O]] ^RO OɼOM^ YP \KZSN `K\SK^SYX] SX ^RO SXZ_^ NK^K N_\SXQ ^RO YXVSXO VOK\XSXQ + PK]^O\ MYX`O\QOXMO S] ^ROX YL^KSXON 0Y\ ^RO ]OVOM^SYX YP ZK\^XO\ ^RO K\MRS^OM^_\O aY\U] SX ^aY ]_MMO]]S`O ZRK]O]$ K VOK\XSXQ ZRK]O KXN K ^O]^SXQ ZRK]O ._\SXQ ^RO VOK\XSXQ ZRK]O 8KY Y]MSVVK^O] KMMY\NSXQ ^Y S^] ]^KXNK\N P\O[_OXMc >Y VOK\X S^] YaX NcXKWSM] 8KY VYYU] K^ S^] YaX RKXN >RO ]SQXKVZ\ONSM^SYX WYN _VO aRSMR aK] dO\Y N_O ^Y ^RO XYX K`KSVKLSVS^c YP `S]_KV ]^SW_VS ]^K\^] XYa VOK\XSXQ ^RO \YLY^p] WYNS¶KLVO Y]MSVVK^Y\ ]SQXKV K] K aOSQR^ON ]_W YP ^RO `S]_KV ]^SW_VS SXN_MON Lc S^] YaX KM^SYX] 7Y\O Z\OMS]OVc ^RO \YLY^ VOK\X] ^RO K]]YMSK^SYX LO^aOOX S^] WY`OWOX^] ]OX]Y\SWY^Y\ SX PY\WK^SYX KXN ^RO SXN_MON `S]_KV `OVYMS^c `OM^Y\] YZ^SMKV ·Ya +] K MYX]O[_OXMO K] NO]M\SLON SX ]OM^SYX  S^ KV]Y WYNS¶O] ^RO 8KYp] Y]MSVVK^Y\ 8KYp] K\W WY`OWOX^ >RS] Z\YMO]] YP WYNSPcSXQ VOK\XSXQ KXN KNKZ^SXQ MYX^SX_O] KXN MYX`O\QO KP^O\ ]YWO ^SWO >RS] KNT_]^ WOX^ MKX LO K]]_WON K] K LK]SM Z\YMO]] Lc aRSMR SXPKX^] QKSX ]OVP \O·OM^S`O KLSVS^SO] K] _XNO\VSXON Lc RO ]KWO S] ^\_O PY\ ^RO W_V^SZVO KQOX^] MK]O +WYXQ ^aY SX^O\KM^KX^] YXVc ^RO KQOX^ RK`SXQ K ]SWSVK\ P\O[_OXMc K] 8KY S] ]OVOM^ON 3X ^RS] ObZO\SWOX^ ^R\OO KQOX^] K\O SX`YV`ON SX KNNS^SYX ^Y 8KY KXN R_WKX K LK]SM K_^YWK^YX S] SX^\YN_MON 0SQ_\O N >RO MY_ZVSXQ PKM^Y\ aK] ! 8KYp] NOPK_V^ P\O[_OXMc aK] 0.407 2d ^RO K_^YWK ^YX ]cXMR\YXSdON P\O[_OXMc aK] 0.4318 6% RSQRO\ KXN ^RO R_WKX ]cXMR\YXSdON P\O[_OXMc aK] 0.36 11% VYa ^Y 0.38 6% VO]] ^RKX NO PK_V^ P\O[_OXMc AROX K ]_LTOM^ SX^O\KM^] aS^R K P\O[_OXMc MVY]O ^Y ^RO VOK\X^ YXO ^RS] ]OVOM^SYX YP ZK\^XO\ KVQY\S^RW ]OVOM^] ^RS] KQOX^ K] K QYYN SX^O\KM^SXQ ZK\^XO\ KXN 8KYp] WYNS¶KLVO Y]MSVVK^Y\ ]cXMR\YXSdO] aS^R S^ 3XS^SKVVc LY^R KQOX^] WY`O aS^R MVY]O P\O[_OXMSO] aS^RSX KX KVVYaKLVO \KXQO L_^ KP^O\ ]YWO ^SWO YP SX^O\KM^SYX 8KY KNYZ^] ^RO R_ WKX WY`OWOX^] KXN LY^R Y]MSVVK^O aS^R ObKM^Vc ^RO ]KWO P\O[_OXMc MY\\O]ZYXNSXQ ^Y ^RO R_WKX WY^SYX 1YYN \O]_V^] K\O YL^KSXON aS^R ^RS] K\MRS^OM^_\O ^RO]O K\O MYVVOM^S`OVc ]RYaX SX ^RO XOb^ ]OM^SYX] 3X ^RO Z\O]OX^ON aY\U Y_\ KSW S] ^Y _]O ^RO XY^SYX YP _XSX^OX^SYXKV ]cXMR\YXc ^Y K_^YWK^SMKVVc SXS^SK^O ^RO R_WKX\YLY^ SX^O\KM^SYX VOKNSXQ ^Y VOK\X MYWZVOb ^K]U] SX K NO`OVYZWOX^ aKc N_\SXQ SWS^K^SYX QKWO]

163

:+6+.C8 4Y_\XKV YP ,ORK`SY\KV RO QVYLKV K\MRS^OM^_\O K .cXKWSMKV 3X^O\KM^SYX WYNOV L =OVOM^SYX YP :K\^XO\$ ]OVOM^ KX SX^O\KM^SXQ ZK\^XO\ YX ^RO LK]S] YP ]cXMR\YXc NO^OM^SYX KWYXQ

`K\SY_] SX^O\KM^SXQ KQOX^] M 0YM_] YP +^^OX^SYXKV$ ^RS] K\MRS^OM^_\O MKZKLVO YP VYMK^SXQ KXN ZYSX^SXQ ^RO ]cXMR\YXSdON SX^O\KM^SXQ KQOX^ NO¶XON Lc ^RO ]OVOM^SYX YP ZK\^XO\ KVQY\S^RW

0SQ_\O " =RYa] RYa ^RO NSɼO\OX^ Y]MSVVK^Y\] + , KXN - K\O ]OVOM^ON KMMY\NSXQ ^Y ^RO `S]_KV ]^SW_VS ._O ^Y ^RO VKMU YP SX^O\KM^KX^ \KXNYW ]OVOM^SYX YP ^RO

Y]MSVVK^Y\] S] YL]O\`ON +P^O\ "   ]OM ^SWO _XS^] Y]MSVVK^Y\] K\O MRY]OX KMMY\NSXQ ^Y ^RO P\O[_OXMc YP ^RO KQOX^] 8KYp] Y]MSVVK^Y\ ]OVOM^SYX NY^ON VSXO KMMY\NSXQ ^Y R_WKX P\O[_OXMc ]YVSN VSXO

164

:+6+.C8 4Y_\XKV YP ,ORK`SY\KV RO]O Y] MSVVK^SYX P\O[_OXMSO] K\O VOK\X^ Lc ^R\OO NSɼO\OX^ ]SQXKVZ\ONSM^SYX WYN _VO] 8KYp] Y]MSVVK^SXQ P\O[_OXMc S] ]OVOM^ON KWYXQ ^RO]O ^R\OO Y]MSV VK^Y\] NOZOXNSXQ YX ^RO `S]_KV ]^SW_VS 3X ^RO KL]OXMO YP `S]_KV ]^SW _VS ^RO Y]MSVVK^Y\] MYX^\YVVSXQ 8KY] K\W K\O ]OVOM^ON \KXNYWVc O`O\c  ]OMYXN] +] ^RO R_WKX SX^O\KM^] aS^R K MO\^KSX P\O[_OXMc YXO YP ^RO ]SQXKVZ\ONSM^SYX WYN_VO MVY]O ^Y ^RO SX^O\KM^KX^p] P\O[_OXMc ]cXMR\Y XSdO] aS^R S^ 9_\ K\MRS^OM^_\O ]OVOM^] KWYXQ ^R\OO Y]MSVVK^Y\] ^RO YXO RK`SXQ ^RO WSXSWKV O\\Y\ aS^R ^RO `S]_KV ]^SW_V_] /bZO\SWOX^KV \O]_V^] K\O ]RYaX SX 0SQ_\O " 3X ^RO YX]O^ ^RO\O S] XY `S]_KV ]^SW_V_] ^SVV 26.6 ]OMYXN] Y\ 800 ^SWO _XS^] -YX]O[_OX^Vc YXO YP ^RO ^R\OO Y]MSVVK ^Y\] K\O ]OVOM^ON \KXNYWVc +P^O\ ]YWO ^SWO 800 ^SWO _XS^] KX KQOX^ ]^K\^] SX^O\KM^SXQ aS^R ^RO Y]MSVVK^SYX] YP 0.81 2d XOK\ ^Y ^RO Y]MSVVK^Y\ , aRSMR S] MYX]O[_OX^Vc ]OVOM^ON K] ^RO \YLY^p] M_\\OX^ NcXKWSM +P^O\ 58 ]OMYXN] 1750 ^SWO _XS^] ^RO KQOX^ MRKXQO] RS] Y]MSVVK^SYX] MVY]O ^Y ^RO P\O[_OXMc YP ^RO Y]MSVVK^Y\ - ]Y ^RO Y]MSVVK^Y\ - S] ]OVOM^ON >RO Y]MSVVK^Y\ + S] ]OVOM^ON SX K ]SWSVK\ aKc 3^p] aY\^R XY^SMSXQ ^RK^ N_\SXQ ObZO\SWOX^] aS^R XKS`O SX^O\KM^KX^] aO K]UON ^RO ZK\^SMSZKX^] ^Y WY`O ^ROS\ K\W K^ ^ROS\ YaX Z\OPO\\ON P\O [_OXMc KXN ]^cVO ]YWO YP ^RO SX^O\KM^KX^] WY`ON aS^R MYWZVOb QO] ^_\O] SWZVcSXQ ^RO Z\O]OXMO YP W_V^SZVO P\O[_OXMSO] 3X ^RS] MK]O SP ^RO P_XNKWOX^KV P\O[_OXMc YP ^RO SX^O\KM^SXQ KQOX^ aK] MVY]O ^Y ^RO \YLY^p] SX^O\XKV NcXKWSM] 8KY aK] KLVO ^Y ]cXMR\YXSdO aS^R RSW ^RO Y^RO\ RK\ WYXSM] K\O XOQVOM^ON Y^RO\aS]O 8KY aK] WY`SXQ K^ S^] YaX NOPK_V^ P\O[_OXMc >RO]O \O]_V^] K\O K`KSVKLVO PY\ ZK\^XO\ NS]^KX^ P\YW ^RO \YLY^ Lc 0.5 ^Y 3.5 WO^O\]

 +^^OX^SYXKV 7OMRKXS]W

9XO YP ^RO WKTY\ MYXMO\X] YP SX^O\KM^S`O \YLY^SM] S] RYa ^Y PYM_] YX ]KVSOX^ POK^_\O] KWYXQ `K\SY_] `S]_KV ]^SW_VS 3X PKM^ PYM_]SXQ K^^OX^SYX KXN NS]M\SWSXK^SXQ _]OP_V NK^K P\YW ^RO Y^RO\] \ON_MO ]SQXS¶MKX^Vc ^RO LSQ KWY_X^ YP SXMYWSXQ SXPY\WK^SYX P\YW ]OX]Y\] KXN UOOZ MYWZ_^K ^SYXKV \O]Y_\MO] K`KSVKLVO PY\ Y^RO\ SWZY\^KX^ ^K]U] -_\\OX^ KZZ\YKMRO] YP K^^OX^SYXKV WOMRKXS]W] K\O _]_KVVc LK]ON YX ^RO ]YVO `S]_KV SXPY\WK^SYX AO Z\YZY]O RO\O ^Y MYX^\YV ^RO K^^OX^SYXKV WOMRKXS]W P\YW K VYa VO`OV WY^Y\ MYX^\YVVO\ ?]SXQ ^RO ]OVOM^SYX YP ZK\^XO\ K\MRS^OM^_\O SX ^RO Z\O]OXMO YP ^aY `S]_KV ]^SW_VS SX ^RO 8KYp] `S]_KV ¶OVN ^RO \YLY^ aSVV ]cXMR\YXSdO aS^R ^RO oSX^O\ KM^SXQo ZK\^XO\ RK`SXQ K WY^SYX P\O[_OXMc MVY]O ^Y XKYp] YaX NcXKWSM 2YaO`O\ 8KY aSVV XY^ LO KLVO ^Y VYMK^O ^RO QYYN SX^O\KM^SXQ ZK\^XO\ SX S^] `S]_KV ¶OVN LOMK_]O Y_\ KVQY\S^RW aY\U] YX ^RO ZO\MOS`ON OXO\Qc S\\O]ZOM^S`O YP ^RO ]ZK^SKV SXPY\WK^SYX KQOX^ VYMK^SYX 0SQ_\O !M ]RYa] ^RO K\MRS^OM^_\O ^Y MYX^\YV ^RO 0YM_] YP +^^OX^SYX 09+ >RO 09+ K\MRS^OM^_\O P_XM^SYX] SX ^aY ]^OZ]$ VOK\XSXQ KXN ^O]^ SXQ AROX ^RO\O S] XY KQOX^ ^RO 09+ WY`O] K\Y_XN \KXNYWVc NOZOXN SXQ YX ^RO XYS]O +] ^RO R_WKX ZK\^XO\ ]^K\^ WY`SXQ RS] K\W aS^R K P\O[_OXMc MVY]O ^Y 8KYp] NcXKWSM] ^RO SWKQOZ\ONSM^SYX WYN_VO X ##  VOK\X] ^RO ]ZK^SKV VYMK^SYX SX S^] `S]_KV ¶OVN YP ^RO SX^O\KM^SXQ ZK\^XO\ K] K aOSQR^ON ]_W YP 8KYp] ]cXMR\YXSdON P\O[_OXMc >RO K\MRS^OM^_\O S] KLVO ^Y Z\ONSM^ ^RO VYMK^SYX YP ^RO ]cXMR\YXSdON ZK\^XO\ 3P KXY^RO\ KQOX^ MYWO] KXN SX^O\KM^] aS^R K NSɼO\OX^ P\O[_OXMc VYaO\ Y\ RSQRO\ ^RKX 8KY K] ]RYaX SX 0SQ_\O M X ## aRSMR KV\OKNc VOK\X^ ^RO ]ZK^SKV ZY]S^SYX YP ^RO ]cXMR\YXSdON \Rc^RWSM WY`OWOX^] ]^\YXQVc Z\ONSM^] ^RO VYMK^SYX YP ^RO ]cXMR\YXSdON KQOX^ O`OX SX ^RO Z\O]OXMO SX 8KYp] `S]_KV

¶OVN YP KX _X]cXMR\YXSdON YXO LOMK_]O ^RO Z\ONSM^SYX S] WKNO Lc ^RO aOSQR^ON ]_W YP ^RO VOK\X^ P\O[_OXMc >Y NO^O\WSXO ^RO MY\\OM^ SX^O\KM^SXQ ZK\^XO\ KXN ^Y NS]M\SWSXK^O LO ^aOOX W_V^SZVO ]^SW_VS Y_\ KVQY\S^RW WYN_VK^O] ^RO M_\\OX^ `S]_KV ]^SW_VS aS^R ^RO SWKQOZ\ONSM^SYX YP ^RO WY`SXQ K\OK] X ##  + WO\QSXQ LVYMU S] _]ON ^Y MKVM_VK^O K aOSQR^ON K`O\KQO WYN_VK^SYX YP ^RO]O M_\\OX^ `S ]_KV WY^SYX KXN ^RO Z\ONSM^ON YXO] >RO RSQRO\ `KV_O] YP ^RS] WO\QSXQ LVYMU K\O ^ROX MY\\OVK^ON ^Y ^RO VYMK^SYX YP ]cXMR\YXY_] WY`OWOX^] +VV ^RO ZSbOV] YP ^RO WO\QSXQ LVYMU K\O Z\YTOM^ON YX ^RO x KbS] SO KVV ZSbOV] SX OKMR MYV_WX K\O KNNON + ASXXO\ >KUO] +VV A>+ ]OVOM^] ^RO RSQR O]^ KM^S`K^ON MYV_WX >RS] ]OVOM^ON MYV_WX SXNSMK^O] ^RO VYMK^SYX YP ^RO ]cXMR\YXSdON WY`OWOX^ KXN ^RO \YLY^ MKX ZYSX^ ^Y ^RO ]cXMR\YXSdON \OQSYX ^Y ]RYa ^RO M_\\OX^ 0YM_] 9P +^^OX^SYX 09+ >RO \YLY^ KVaKc] PYM_] aS^R ^RO ]cXMR\YXSdON ]^SW_V_] O`OX SP ^RO ZK\^XO\ MRKXQO] RS] VYMK^SYX 0Y\ ^RS] ObZO\SWOX^ ^RO \O]YV_^SYX YP ^RO Z\ONSM^ON SWKQO S] ×   MYV_WX] Y\ VYMK^SYX ^RO]O  ZY]]SLVO VYMK^SYX] K\O SX K \KXQO YP 600 KXQVO] −300 ^Y 300  Z\OMS]SYX YP 20  3X Y_\ ObZO\SWOX^] 8KYp] ROKN NS\OM^SYX S] _]ON ^Y ZYSX^ ^RO M_\\OX^ PYM_] YP K^^OX^SYX >RO VOK\XSXQ \_VO PY\ ^RO SWKQOZ\ONSM^SYX X ##  WYN_VO S] XOK\ ^Y 2OL LSKX VOK\XSXQ aS^R K XY\WKVSdK^SYX YP aOSQR^] >RO aOSQR^ XY\WKVSdK ^SYX K`YSN] ^RO NS`O\QOXMO KXN KVVYa] ^Y PY\QO^ VO]] _]ON K]]YMSK^SYX] >RO XO_\YX] KM^S`S^SO] SX X ##  MKX LO MYWZ_^ON _]SXQ ^RO y → X ## ]cXKZ]O] " >RO VOK\XSXQ YP y → X ## ]cXKZ^SM aOSQR^] MKX LO MYW Z_^ON Lc ^RO O[_K^SYX #

Xi## (t) =

kεy

Wyk →Xi## yk

Wyj →Xi## = "

aS^R$

KXN$

!

Wy∗j →X ##

kεy

i

Wy∗j →X ## i

Wy∗j →X ## (t + dt) = Wyi →Xi## (t) + ε yj Ui i

Ui =

! mεX

WXm →Xi## Xm

(8) (9) (10) (11)

ARO\O Xi## S] ^RO XO_\YX KM^S`S^cSX X ## Q\Y_Z yk S] K XO_\YX YP ^RO MYXNS ^SYXKV Q\Y_Z 9]MSVVK^Y\ KXN Ui S] ^RO SXMYXNS^SYXKV ]^SW_V_] 3WKQO B Xm S] ^RO KM^S`S^SO] SX ^RO SXMYXNS^SYXKV Q\Y_Z B Wyj →Xi## \OZ\O]OX^] ^RO ]cXKZ^SM aOSQR^] P\YW ^RO Y]MSVVK^Y\ c XO_\YX T ^Y ^RO SWKQOZ\ONSM^SYX Bo XO_\YX S _XXY\WKVSdON aOSQR^] K\O ]RYaX aS^R n ∗ o

! /bZO\SWOX^KV \O]_V^]

! =cXMR\YXc LK]ON PYM_] YP K^^OX^SYX YX K ]^K^SM ZVK^PY\W ºbON MKWO\K >RO PYM_] YP K^^OX^SYX WOMRKXS]W RK] LOOX ^O]^ON SX K ]SWZVO aKc KXN \O]_V^] K\O ]RYaX SX 0SQ_\O # 3X K 8KYp] Y]MSVVK^SYX] KVYXQ aS^R ^RO SX^O\KM^SXQ KQOX^ WY^SYX ]SQXKV K\O ]RYaX L SXNSMK^O] ^RO KXQVO YP 09+ KMMY\NSXQ ^Y ^RO \Rc^RWSM WY^SYX +XN M NS]ZVKc] ^RO SX^O\KM ^KX^p] VYMK^SYX] SX P\YX^ YP 8KY +^ ^RO LOQSXXSXQ XY `S]_KV ]^SW_VS S] Z\O]OX^ON ^Y 8KY ^RO 09+ WY`O] \KXNYWVc LO^aOOX −300 KXN 300  +P^O\  ^SWO _XS^]   ]OM K ]cXMR\YXSdON WY^SYX S] SX^\YN_MON P\YW ^RO VOP^ ]SNO −28.30  YP 8KY >RS] SX^O\KM^SYX \O]_V^] SX ^_\XSXQ ^RO

165

:+6+.C8 4Y_\XKV YP ,ORK`SY\KV RO ]KWO ]O[_OXMO S] \OZOK^ON KQKSX ^Y `O\SPc ^RK^ 09+ KVaKc] PYVVYa ^RO SX^O\KM^KX^ >RO 0SQ_\O # MY\\O]ZYXN] ^Y ^RO ObZO\SWOX^ ]RYaX YX ^RO `SNOY K`KSVKLVO YX Y_\ aOL]S^O +P^O\ ^RS] ]SWZVO ObZO\SWOX^ aO ObKWSXO Y_\ ]OVOM^SYX YP ZK\^XO\ KVQY \S^RW KVYXQ aS^R 09+ K\MRS^OM^_\O Lc Ob^OXNSXQ ^RO ]^_Nc ^Y ^RO MK]O YP ^aY KQOX^]$ KX K_^YWK^YX YXO .Y0 KXN K R_WKX YXVc YXO YP ^ROW S] ]cXMR\YXSdON K^ K ^SWO RO ObZO\SWOX^KV \O]_V^] K\O ]RYaX SX 0SQ_\O  0SQ_\O K ]UO^MRO] ^RO M_\`O] YP ^RO K_^YWK^YX KXN 8KYp] WY^SYX ]SQXKV 3XS ^SKVVc ^RO K_^YWK^YX S] WY`SXQ P\YW ^RO VOP^ ]SNO YP 8KY KLY_^ −20◦  ]OO 0SQ_\O K ,Y^R KQOX^] ]cXMR\YXSdO] KP^O\ K ]RY\^ ZO\SYN Lc _]SXQ Y_\ ZK\^XO\ ]OVOM^SYX K\MRS^OM^_\O 0SQ_\O K NO]M\SLO] ^RO [_KVS^c YP ]cXMR\YXc LO^aOOX 8KY KXN ^RO K_^YWK^YX SX ^O\W] YP :6@ +^ ^RO LOQSXXSXQ :6@ aK] K^ S^] VYaO]^ `KV_O L_^ K] ^RO SX^O\KM^SYX MYX^SX_O] S^ SXM\OK]O] ]VYaVc ^Y ^RO RSQRO]^ YXO N_\SXQ ^RO ]cXMR\YXS ]K^SYX ZRK]O LO^aOOX ^RO KQOX^] 3XS^SKVVc ^RO\O S] XY Y^RO\ KQOX^ SX P\YX^ YP \YLY^ ObMOZ^ ^RO +_^YWK^YX ^RO\OPY\O 09+ ^_\X] ^YaK\N] ^RO +_^YWK^YX 0SQ_\O K +P^O\ ! ^SWO _XS^] 23.33 ]OMYXN] ^RO R_WKX ]^K\^] SX^O\KM^SXQ P\YW ^RO \SQR^ ]SNO YP ^RO \YLY^ aS^R K NSɼO\OX^ P\O[_OXMc L_^ RO PKSV] ^Y NS]^_\L 8KY+_^YWK^YX SX^O\KM^SYX ^RO :6@ MYX^SX_O] ^Y S^] RSQRO\ `KV_O] PY\ ^RO K_^YWK^YX KXN 09+ MYX^SX_O] ^Y ZYSX^ Y_^ ^Y K_^YWK^YX 8Ya ^RO \YVO] K\O ]aS^MRON 2_WKX S] KN`S]ON ^Y WKUO ]SWSVK\ WY`O WOX^] K] 8KY aRSVO ^RO K_^YWK^YX S] KNT_]^ON ^Y K VYaO\ P\O[_OXMc

0SQ_\O L KXN L -YX]O[_OX^Vc 8KY KV]Y ·SZ] S^] \YVO Lc ]cX MR\YXSdSXQ aS^R ^RO R_WKX KXN ]OVOM^] RSW K] K ZK\^XO\ :6@ SXM\OK]O] PY\ ^RO R_WKX KXN NOM\OK]O] PY\ ^RO K_^YWK^YX 0SQ_\O M +] ^RO ]cXMR\YXc OWO\QO] LO^aOOX R_WKX KXN 8KY 09+ KV]Y ^_\X] P\YW K_ ^YWK^YX ^Y R_WKX 0SQ_\O M >Y `KVSNK^O Y_\ ObZO\SWOX^ aO KQKSX ]aS^MR ^RO \YVO] YP ^RO ^aY SX^O\KM^SXQ KQOX^] KP^O\   ^SWO _XS^] "" ]OM -YX]O[_OX^Vc ^RS] SXN_MO] K ]aS^MR YP ^RO PYM_] YP K^^OX^SYX KXN ^RO ]cXMR\YXSdON KQOX^ 0SQ_\O L

! =cXMR\YXc LK]ON PYM_] YP K^^OX^SYX YX K WYLSVO ZVK^PY\W WY`SXQ MKWO\K >Y ]^_Nc ^RO ZY]]SLVO QOXO\KVSdK^SYX YP ^RO Z\O`SY_]Vc NO^KSVON WYNOV ^Y ^RO MK]O YP K WYLSVO ZVK^PY\W  aO OWLONNON ^RO K\MRS^OM^_\O YX K WYLSVO RS] S] N_O ^Y ^RO VSWS^K^SYX]

YP ^RO _]ON YZ^SMKV ·Ya KVQY\S^RW O`OX WY\O aS^R ^RO ]WKVV X_WLO\ YP S^O\K^SYX] _]ON SX ^RS] ObZO\SWOX^ ]O^ ^Y K ¶bON `KV_O YP 4 PY\ ]K`SXQ MYWZ_^K^SYXKV ^SWO >RO WYN_VK^ON YZ^SMKV ·Ya POON ^RO component direction-selective neurons >RO]O XO_\YX] ]SW_VK^O ^RY]O YP @ KXN 7> L\KSX K\OK] aRSMR K\O NO]SQXON Lc XO_\YLSYVYQSMKV \OMY\N] K] ]OX]S^S`O ^Y Z\OPO\\ON WY^SYX NS\OM^SYX] MKVVON component direction-selective neurons Lc 7Y`]RYX EG >RO]O MYWZYXOX^ NS\OM^SYX]OVOM^S`O XO_\YX] ¶V^O\ ^RO WYN_VK^ON YZ ^SMKV ·Ya ^RO ¶\SXQ YP OKMR YP ^RO]O XO_\YX Ai  S] Z\YZY\^SYXKV ^Y ^RO KXQ_VK\ NS]^KXMO LO^aOOX ^RO `S]_KV ]^SW_V_] YZ^SMKV ·Ya KXN S^] Z\O PO\\ON NS\OM^SYX aOSQR^ON Lc ^RO WY^SYX SX^OX]S^c K] LOVYa$ −(

Ai = exp

167

(β−βi )2 2τ12

)

2 −( V 2 ) 2τ

·(1 − exp

2

)

(12)

:+6+.C8 4Y_\XKV YP ,ORK`SY\KV RS] ]SWZVS¶MK^SYX aK] WKNO N_O ^Y ^RO PKM^ ^RK^ ^RO \YLY^ SX ^RS] ]ZOMS¶M KZZVSMK^SYX ^_\X] \SQR^ Y\ VOP^ ^Y PYM_] S^] K^^OX^SYX YX ^RO ZK\^XO\ SXN_MSXQ NSɼO\OX^ SX^OX]S^SO] L_^ YXVc SX ^RO ^aY RY\SdYX^KV NS\OM^SYX] YP ^RO YZ^SMKV ·Ya >RO Y_^Z_^] YP ^RO]O XO_\YX] K\O ^ROX SX^OQ\K^ON SX ^SWO Lc XO_\YX] aRSMR K\O ]OX]S^S`O ^Y NSɼO\OX^ `S]_KV WY^SYX SX^OX]S^SO] PY\ K QS`OX NS \OM^SYX YP WY^SYX >ROc K\O ]SWSVK\ ^Y K ]SWZVS¶ON `O\]SYX YP ^RO pattern direction-selective neurons aRSMR SX^OQ\K^O ^RO MYWZYXOX^ NS\OM^SYX]OVOM^S`O XO_\YX] SX ^RO 7> K\OK YP ^RO R_WKX L\KSX EG /KMR YP ^RO]O XO_\YX] MKX LO ]OOX K] KX RS]^YQ\KW] \OZ\O]OX^SXQ ^RO YMM_\\OXMO SX ^SWO YP `S]_KV `OVYMS^c `OM^Y\] RK`SXQ K QS`OX WY^SYX NS \OM^SYX RO\O 00 Y\ 1800  aS^R K QS`OX WY^SYX SX^OX]S^c SX Z\KM^SMO K QS`OX ]WKVV \KXQO YP SX^OX]S^SO] 0Y\ OKMR WY^SYX NS\OM^SYX aO _]O 20 ZK^^O\X NS\OM^SYX]OVOM^S`O XO_\YX] SX Y_\ WYNOV + aOSQR^ON WOKX YP ^RO Y_^Z_^] YP ^RO  XO_\YX] S] ^ROX MYWZ_^ON \OZ\O]OX^SXQ PY\ OKMR

NS\OM^SYX ^RO WOKX `S]_KV WY^SYX SX^OX]S^c Ob^\KM^ON P\YW ^RO `S]_KV ¶OVN Y`O\ ^SWO ._\SXQ ^RO VOK\XSXQ ZRK]O aO \Y^K^O ^RO \YLY^ aS^R  NSɼO\OX^ ]ZOON] YX ^RO VOP^ ]SNO KXN  NSɼO\OX^ YXO] YX \SQR^ ]SNO >RO ]ZOON] K\O MYX ^\YVVON Lc  WY^Y\ XO_\YX]  PY\ OKMR NS\OM^SYX K] ]RYaX 0SQ_\O  >RO]O XO_\YX] \OZ\O]OX^ ^RO _XMYXNS^SYXKV SXZ_^] YP ^aY 67= XO^aY\U YXO PY\ OKMR NS\OM^SYX aRSMR VOK\X ^Y K]]YMSK^O ^RO WY^Y\ MYX^\YVVO\ Y\NO\] WY^Y\ `OVYMS^SO] ^Y ^RO WOKX `S]_KV WY^SYX SX^OX]S^c Ob^\KM^ON P\YW ^RO ZK^^O\X NS\OM^SYX]OVOM^S`O XO_\YX] N_\SXQ ^RO ObZO\SWOX^ -YX]O[_OX^Vc KP^O\ ^RS] VOK\XSXQ ]^OZ aROX ^RO \YLY^ ]^K\^ WY`SXQ aS^R K QS`OX `OVYMS^c ^RO 67= aSVV LO KLVO ^Y ^\SQQO\ PY\ OKMR NS\OM^SYX ^RO MY\\OM^ `S]_KV WY^SYX SX^OX]S^c WOKX `KV_O aRSMR MKX LO ]_L^\KM^ON ]OO 0SQ_\O  P\YW ^RO M_\\OX^ YZ^SMKV ·Ya SX Y\NO\ ^Y MYWZOX]K^O ^RO SXN_MON OQYWY^SYX >RO\OPY\ ^RO ¶V^O\ON `S]_KV ]^SW_VS MKX LO _]ON K] KX 3XZ_^ PY\ ^RO Z\O`SY_] WYNOV ^Y ]OVOM^ KXN VYMK^O K ZK\^XO\ SX ^RO MK]O YP K WY`SXQ \YLY^ 6O^] XYa MYX]SNO\ ^RO MYWZVO^O ObZO\SWOX^KV ]MOXK\SY 0S\]^ K R_WKX ZK\^XO\ WY`O] RS] K\W SX P\YX^ YP ^RO \YLY^ ^Y ^OKMR K Z\OPO\\ON P\O [_OXMc YP SX^O\KM^SYX ^RO \YLY^ S] SX K ]^K^SM ZY]S^SYX +P^O\ ^RS] VOK\X SXQ ZRK]O ^RO \YLY^ ]^K\^] WY`SXQ \KXNYWVc >RO OQYWY^SYX SXN_MON Lc ^RO \YLY^p] WY`OWOX^] K\O MYWZOX]K^ON K] ObZVKSXON KLY`O >RO VOK\X^ P\O[_OXMc MKX LO YL]O\`ON YX ^RO \YLY^p] ZOXN_V_W ^KSV SX ^RO 0SQ_\O  ?]SXQ ^RO Z\O`SY_]Vc NO]M\SLON WYNOV] aROX K R_WKX SX ^O\KM^] aS^R ^RO \YLY^ aS^R K NcXKWSM MVY]O ^Y ^RO VOK\X^ YXO ^RO \YLY^ ]OVOM^] KXN VYMK^O] ^RS] Z\OPO\\ON ZK\^XO\ YX ^RO LK]S] YP ]cXMR\YXc NO^OM^SYX -YX]O[_OX^Vc ^RO ]cXMR\YXc LK]ON PYM_] YP K^^OX^SYX OXKLVO] ^RO VOK\XSXQ YP ^RO ]OVOM^ON ZK\^XO\ ]RKZO _]SXQ ^RO KVQY\S^RW NO]M\SLON SX E G >RO \O]_V^] YP ^RS] ObZO\SWOX^ K\O SVV_]^\K^ON SX 0SQ_\O  0S\]^ ^RO WY LSVO \YLY^ ]^K\^] WY`SXQ Lc PYM_]SXQ S^] K^^OX^SYX LVKMU VSXO YX \KXNYW \OQSYX] YP ^RO `S]_KV ¶OVN N_O ^Y ^RO VKMU YP ]KVSOX^ \OQSYX] YP SX^O\O]^ +^ ^SWO t = 40 ]OMYXN] K R_WKX ]^K\^] SX^O\KM^SXQ aS^R K P\O[_OXMc MVY]O ^Y ^RO YXO VOK\XON Lc ^RO \YLY^ -YX]O[_OX^Vc ^RO ]cXMR\YXc LK]ON PY M_] YP K^^OX^SYX ]OVOM^] KXN Z\ONSM^] ^RO VYMK^SYX YP ^RO R_WKX ZK\^XO\ LV_O VSXO SX 0SQ_\O  + XO_\KV 0SOVN MYX^\YVVSXQ ^RO \YLY^p] WY`OWOX^] ^_\X] ^RO  9X_US C 5YLKcK]RS KXN C 5_XY 7YNOV PY\ MYX^\YVVSXQ K ^K\QO^ R_WKXp] K^^OX^SYX SX W_V^SZK\^c ]O^^SXQ] 3/// \KX]KM^SYX] YX +ɼOM^S`O -YWZ_^SXQ @YV  8Y  :KQO] #     E!G 6 1VK]] =cXMR\YXSdK^SYX KXN \Rc^RWSM Z\YMO]]O] SX ZRc]SYVYQc 8K^_\O $ !!"  E"G RO ]Y_XN YP WKXc RKXN] MVKZZSXQ 8K^_\O  "  E!G 7 @K\VO^ 6 7K\SX = RO KXKVc]S] YP WY`SXQ `S]_KV ZK^^O\X] :K^^O\X \OMYQXS^SYX WOMR KXS]W] $! #" E G = 6OZ\O^\O : 1K_]]SO\ KXN 4: -YM[_O\Od 0\YW XK`SQK^SYX ^Y KM^S`O YLTOM^ \OMYQXS^SYX -S^O]OO\ 

171