Network Neuroscience入門

ܭࢉ࿦తਆܦՊֶษڧձᶄ /FUXPSL/FVSPTDJFODFೖ໳ d!,FSOFM ౦ژେֶେֶӃ૯߹จԽݚڀՊ. ࢢ઒ߤฏ

ࠓ೔ͷ༧ఆ w ͳͥωοτϫʔΫʁ w ωοτϫʔΫՊֶೖ໳ʢ༻ޠ΍ఆٛͷಋೖʣ w ωοτϫʔΫղੳͷྫ w ωοτϫʔΫ͔ΒԿ͕Θ͔Δʁ w
࣮ݧ࿦จ঺հ w ߏ଄͔ΒμΠφϛΫε΁

ࢀߟจݙ w 0MBG4QPSOT/FUXPSLTPGUIF#SBJO  ˠجຊ͸͜ΕʹԊͬͯਐΊΔɻ෼͔Γ΍ͯ͘͢໘ന͍͕਺ࣜ͸ग़ͯ͜ͳ͍ɻ w .JLBJM3VCJOPW 0MBG4QPSOT$PNQMFYOFUXPSLNFBTVSFTPG CSBJODPOOFDUJWJUZ6TFTBOEJOUFSQSFUBUJPOT  ˠ਺ֶతͳఆٛͳͲ͸ͪ͜ΒΛࢀরͨ͠ɻ w
#BTTFUU %4 #VMMNPSF &4NBMMXPSMECSBJOOFUXPSLT  ˠ೴ͷεϞʔϧϫʔϧυੑʹؔ͢ΔϨϏϡʔ࿦จɻ

ͳͥωοτϫʔΫʁ OFUXPSLTDJFODF CSBJOTDJFODF VOEFSTUBOEJOUFHSBUJWFGVODUJPOTPGUIFCSBJO UIPVHIU NFNPSZ DPOTDJPVTOFTT FUDʜ 4JOHMFOFSWFDFMM CSBJOOFUXPSLT
DPNQMFYBOEJSSFEVDJCMFEZOBNJD QBUUFSOTUIFZDSFBUF

ωοτϫʔΫՊֶೖ໳ w ڭՊॻͷষʹରԠ w ஌͍ͬͯΔਓʹ͸ୀ۶͔΋ɾɾɾ

ωοτϫʔΫՊֶೖ໳ جૅ֓೦ w ωοτϫʔΫʢάϥϑʣͱ͸ɿ  ϊʔυʢ௖఺ʣͱͦͷؒͷ࿈݁ؔ܎Λද͢ΤοδʢลʣͰߏ੒͞ΕΔσʔλ ܕ͕άϥϑͰɺϊʔυ΍Τοδʹ෺ཧతҙຯΛ࣋ͨͤͨ΋ͷ͕ωοτϫʔΫ w άϥϑ͸େ͖͘෼ྨ͢Δͱ༗޲ແ޲άϥϑɺόΠφϦॏΈ෇͖άϥϑͷ ͭʹ෼͚ΒΕΔɻ w
೴ͷωοτϫʔΫΛߟ͑Δͱ͖͸෺ཧత઀ଓʹΑͬͯߏ੒͞ΕΔ BOBUPNJDBMOFUXPSLͱμΠφϛΫεͷ૬͔ؔΒܾ·ΔGVODUJPOBMOFUXPSL ͕͋ΓɺOFUXPSLOFVSPTDJFODFͰ͸ͲͪΒ΋ѻ͏ɻ w άϥϑͷ౳Ձͳදݱͱͯ͠ྡ઀ߦྻ͕͋Δɻ ༗޲όΠφϦάϥϑͷྫ

ωοτϫʔΫՊֶೖ໳ EFHSFFʢ౓਺ʣ ྡ઀ߦྻ Aij ϊʔυK͔ΒϊʔυJ΁઀ଓ͢ΔΤοδͷ༗ແΛද͢ߦྻɻ όΠφϦάϥϑͷ৔߹͸੒෼͸͔ɻॏΈ෇͖άϥϑͷ৔߹͸೚ҙͷ࣮ ਺ΛऔΔʢͨͩ͠OFUXPSLOFVSPTDJFODFͰ͸ࠓ·Ͱͷͱ͜ΖෛͷॏΈ ͸ѻΘΕ͍ͯͳ͍ʣɻແ޲άϥϑͷ৔߹͸ରশߦྻɻ w ͋ΔϊʔυJΛߟ͑ͨ࣌ɺͦΕʹ઀ଓ͍ͯ͠ΔΤοδͷຊ਺
ki = ∑ j∈N Aij w 0VUEFHSFF w *OEFHSFF kout i = ∑ j∈N Aij kin i = ∑ j∈N Aji w ౓਺෼෍͸ωοτϫʔΫͷάϩʔόϧͳಛ௃Λද͍ͯ͠Δʢޙड़ʣ

ωοτϫʔΫՊֶೖ໳ w DMVTUFSJOHDPF⒏DJFOU w ͋ΔϊʔυJΛߟ͑ͨ࣌ɺͦΕʹ઀ଓ͍ͯ͠Δϊʔυؒͷ઀ଓ͕औΓಘ Δ઀ଓͷ͏ͪͲͷఔ౓઎Ί͍ͯΔ͔Λද͢ྔ w ωοτϫʔΫʹ͍ͭͯߟ͑Δͱ͖͸શϊʔυͷฏۉΛͱΔ C =
1 n ∑ i∈N Ci = 1 n ∑ i∈N 2ti ki (ki − 1) ti = 1 2 ∑ j,h∈N aij aih ajh ࠨਤͷྫͩͱɺCʹ઀ଓ͍ͯ͠ΔϊʔυؒͷՄೳͳ઀ଓ͸ ͋Γɺͦͷ͏࣮ͪࡍʹଘࡏ͢ΔΤοδ͸ຊͷͨΊɺ $ʢ༗޲άϥϑͷͨΊ্ͷఆٛͱ͸ҟͳΔ͜ͱʹ஫ҙʣ

ωοτϫʔΫՊֶೖ໳ w QBUIMFOHUI w TIPSUFTUQBUIMFOHUI  ϊʔυJͱϊʔυKΛ݁Ϳ࠷୹ܦ࿏ɻJ KؒʹΤοδ͕͋Ε͹ͰɺJͱK͕ ܨ͕͍ͬͯͳ͍৔߹͸㱣ʹͳΔɻ dij =
∑ auv ∈gi↔j auv ͸JͱKΛ݁Ϳ࠷୹ܦ࿏ʣ w DIBSBDUFSJTUJDQBUIMFOHUI  TIPSUFTUQBUIMFOHUIΛωοτϫʔΫશମʹ౉ͬͯฏۉͨ͠΋ͷɻ L = 1 n ∑ i∈N Li = 1 n ∑ i∈N ∑ j∈N,j≠i dij n − 1 gi↔j

ωοτϫʔΫՊֶೖ໳ w NPEVMBSJUZ w ӈਤͷΑ͏ʹɺωοτϫʔΫʹΑͬͯ͸Կݸ͔ͷϞδϡʔϧͷू·Γ ʹΑͬͯશମͷωοτϫʔΫ͕ߏ੒͞Ε͍ͯΔͱΈͳ͢ͷ͕ଥ౰ͳ΋ ͷ͕͋Δɻ w ࣍ͷNPEVMBSJUZͱݺ͹ΕΔྔ͕࠷େʹͳΔΑ͏ͳ෼ղΛߟ͑Δ͜ͱ͕ ଟ͘ɺ࠷దͳϞδϡʔϧͷݸ਺΋͔͜͜Βܾ·Δɻ

ωοτϫʔΫՊֶೖ໳ w NPEVMBSJUZ w Ϟδϡʔϧͱશࣸ૾  ɹɹɹɹɹɹʹΑͬͯϞδϡʔϧ෼ղ͸Ұҙʹఆ·Δɻ w ͜ͷ෼ղʹରͯ͠NPEVMBSJUZ2͸ M ∈
{1,2,...,m}(m < n) f : N → M Q = ∑ u∈M [ euu − (∑ v∈M euv) 2 ] w ͜͜Ͱɺ͸ϞδϡʔϧVʹؚ·ΕΔϊʔυͷΤοδͷ͏ͪϞδϡʔϧ Wʹؚ·ΕΔϊʔυʹ઀ଓ͍ͯ͠ΔΤοδͷׂ߹Λද͢ɻ euv w BSHNBY2ͷܭࢉ͸/1IBSEͷͨΊɺ༷ʑͳώϡʔϦεςΟΫε͕ఏҊ ͞Ε͍ͯΔɻ

ωοτϫʔΫՊֶೖ໳ w DFOUSBMJUZ w ೔ຊޠͰ͸த৺ੑई౓ɻωοτϫʔΫ಺Ͱݸʑͷϊʔυ͕࣋ͭॏཁੑ Λද͢ྔɻ w DMPTFOFTTDFOUSBMJUZଞͷ఺ͱڑ཭͕͍ۙ΄Ͳத৺ੑ͕ߴ͍ͱ͢Δɻ QBUIMFOHUIͷٯ਺ɻ L−1
i = n − 1 ∑ j∈N,j≠i dij w CFUXFFOOFTTDFOUSBMJUZͦͷ఺Λ௨Δܦ࿏͕ଟ͍΄Ͳɺத৺ੑ͕ߴ ͍ͱ͢Δɻ bi = 1 (n − 1)(n − 2) ∑ h,j∈N h≠j,h≠i,j≠i ρhj (i) ρhj ʢ͜͜Ͱɺ͸I͔ΒK΁ͷ࠷୹ܦ࿏ͷݸ਺ɺ͸ͦͷ͏ͪϊʔυJΛ ௨Δ΋ͷͷݸ਺ʣ ρhj ρhj (i)

ωοτϫʔΫՊֶೖ໳ w &SEPT3FOZJ SBOEPNOFUXPSL w ͍ΘΏΔϥϯμϜωοτϫʔΫɻϊʔυ਺/ Τοδͷੜ੒֬཰QΛ༩͑ ͯɺશϊʔυؒͷϖΞʹ͍ͭͯ֬཰QͰΤοδΛுΔɻQͷ࣌͸શ݁ ߹ωοτϫʔΫʹͳΔɻ w
DIBSBDUFSJTUJDQBUIMFOHUIখ w DMVTUFSJOHDPF⒏DJFOUখ

ωοτϫʔΫՊֶೖ໳ w 8BUUTBOE4USPHBU[ TNBMMXPSMEOFUXPSL w ͍ΘΏΔεϞʔϧϫʔϧυωοτϫʔΫͷੜ੒Ϟσϧɻ  ϊʔυ਺/ ฏۉ౓਺L Τοδͷܨ͔͗͑֬཰QΛ༩͑ͯɺ࣍ݩ֨ࢠά ϥϑΛ࡞੒ͨ͠ޙ֬཰QͰΤοδͷܨ͔͗͑Λߦ͏ɻQͷ࣌͸ϥϯμ
ϜωοτϫʔΫʹͳΔɻ w DIBSBDUFSJTUJDQBUIMFOHUIখ w DMVTUFSJOHDPF⒏DJFOUେ  ˠεϞʔϧϫʔϧυੑ w εϞʔϧϫʔϧυੑ͸༷ʑͳωοτϫʔΫʹ  ݟΒΕΔීวతͳੑ࣭ɻ

ωοτϫʔΫՊֶೖ໳ w #BSBCBTJ TDBMFGSFFOFUXPSL w εέʔϧϑϦʔωοτϫʔΫͷੜ੒Ϟσϧɻ খ͞ͳωοτϫʔΫ͔Βελʔτͯ͠ɺͦ ͜ʹ࣍ʑͱϊʔυΛՃ͑Δ͜ͱͰ࡞͍ͬͯ ͘ɻͦͷࡍʢطଘͷάϥϑͷʣͲͷϊʔυ ʹ઀ଓ͢Δ͔͸ϊʔυͷ౓਺ʹൺྫ͢Δ֬
཰ʹΑܾͬͯΊΒΕΔɻ w ౓਺෼෍͕΂͖৐෼෍ʹै͍ɺಛ௃తͳε έʔϧ͕ݟΒΕͳ͍ͨΊεέʔϧϑϦʔͱ ݺ͹ΕΔɻ w εέʔϧϑϦʔੑ΋888ͳͲ༷ʑͳωο τϫʔΫʹݟΒΕΔීวతͳੑ࣭͕ͩɺ෺ ཧత੍໿ͷͨΊ೴Ͱ͸΄ͱΜͲݟΒΕͳ͍ɻ

ωοτϫʔΫղੳͷྫ ڭՊॻͷQd ˠKVQZUFSOPUFCPPLͰઆ໌ IUUQTHJUIVCDPNUSJQEBODFSOFUXPSL@OFVSPTDJFODF

ωοτϫʔΫ͔ΒԿ͕෼͔Δʁ ࿦จ঺հᶃ  %ZOBNJDSFDPOpHVSBUJPOPGIVNBOCSBJOOFUXPSLT EVSJOHMFBSOJOH #BTTFUU %4 8ZNCT /' 1PSUFS ."
.VDIB 1+ $BSMTPO +. (SBGUPO 45 IUUQTEPJPSHQOBT w .PEVMBSJUZʹண໨ͯ͠ɺλεΫͷशख़౓ͱG.3*͔ΒଌఆՄೳͳྔͱͷ ؒͷ૬ؔΛࣔͨ͠ɻ w ಈతͳωοτϫʔΫͷղੳʹΑͬͯɺ೴಺ͷϞδϡϥʔߏ଄͸ֶश࣌ʹ มԽ͍ͯ͠Δ͜ͱ͕෼͔ͬͨɻ

ωοτϫʔΫ͔ΒԿ͕෼͔Δʁ w /PEFqFYJCJMJUZUIFOVNCFSPGUJNFTUIBUFBDIOPEF DIBOHFTNPEVMFBMMFHJBODF OPSNBMJ[FECZUIFUPUBMQPTTJCMF OVNCFSPGDIBOHFT   w /PEFqFYJCJMJUZDIBOHFEEVSJOHUIFMFBSOJOHQSPDFTT  pSTUJODSFBTJOHBOEUIFOEFDSFBTJOH
w 5IFqFYJCJMJUZPGBQBSUJDJQBOUJO POFTFTTJPODPVMECFVTFEBTB QSFEJDUPSPGUIFBNPVOUPG MFBSOJOHJOUIFGPMMPXJOHTFTTJPO

ωοτϫʔΫ͔ΒԿ͕෼͔Δʁ ࿦จ঺հᶄ #SBJODPOOFDUJWJUZEZOBNJDTEVSJOHTPDJBMJOUFSBDUJPO SFqFDUTPDJBMOFUXPSLTUSVDUVSF 4DINÅM[MF 3 #SPPL0`%POOFMM . (BSDJB +0
$BTDJP $/ #BZFS + #BTTFUU %4 ʜ'BML &# IUUQTEPJPSHQOBT w ࣾձతഉআ͞Ε͍ͯΔঢ়گʹ߹͏ͱɺ.FOUBMJ[JOHOFUXPSLͱݺ͹ΕΔ ෦Ґͷ݁߹ڧ౓͕ڧ͘ͳΔɻ

ωοτϫʔΫ͔ΒԿ͕෼͔Δʁ w ͜ͷ࣌ͷ݁߹ڧ౓ͷมԽʹ͸ݸਓ͕ࠩ͋Δ͕ɺͦΕ͕ݸਓͷ࣋ͭ GSJFOETIJQOFUXPSLͷಛ௃ͱ૬͍ؔͯ͠Δ͜ͱ͕ࣔ͞Εͨɻ w GSJFOETIJQOFUXPSLͷ`%FOTJUZ` ඃݧऀʹ ର͢ΔDMVTUFSJOHDPF⒏DJFOU ͕খ͍͞΄ Ͳ݁߹ڧ౓ͷมԽ͕େ͖͍ɻ
w ͜͜ͰɺݸਓͷGSJFOETIJQOFUXPSL͸ GBDFCPPL͔ΒಘͨσʔλΛ࢖༻ɻ  ˠʮ೴಺ͷʯωοτϫʔΫͰ͸ͳ͍ͨΊɺ ڭՊॻͷѻ͏ର৅͔Β͸֎ΕΔ͕ɾɾɾ

ߏ଄͔ΒμΠφϛΫε΁ w ཧ࿦తΞϓϩʔν͔Β͸Կ͕ݴ͑Δ͔ʁ w ϚΧΫβϧͰܭࢉͨ͠Α͏ʹɺ೴಺ͷωοτϫʔΫʹ͓͍ͯ΋εϞʔϧ ϫʔϧυੑ͕޿͘ݟΒΕΔɻ a small-world topology can
support both segregated/specialized and distributed/integrated information processing. #BTTFUU %4 #VMMNPSF & 4NBMMXPSMECSBJOOFUXPSLT/FVSPTDJFOUJTU r IUUQTEPJPSH ͜ΕΛܭࢉϞσϧ͔Β໌Β͔ʹ͍ͨ͠ɻ ˠ$PNQMFYJUZͷಋೖʢڭՊॻୈষʹରԠʣ

ߏ଄͔ΒμΠφϛΫε΁ $PNQMFYJUZ 4QPSOT 5POPOJ &EFMNBO In brains of higher vertebrates,
the functional segregation of local areas that diﬀer in their anatomy and physiology contrasts sharply with their global integration during perception and behavior. 5POPOJ ( 4QPSOT 0 &EFMNBO (. "NFBTVSFGPSCSBJODPNQMFYJUZSFMBUJOHGVODUJPOBMTFHSFHBUJPOBOEJOUFHSBUJPOJOUIFOFSWPVTTZTUFN1SPDFFEJOHTPG UIF/BUJPOBM"DBEFNZPG4DJFODFT rIUUQTEPJPSHQOBT ෦෼ಉ࢜Ͱ͸ಠཱʢ෼཭ʣ͍ͯ͠Δ͕ɺશମͱͯ͠ΈͨΒ౷߹͍ͯ͠ΔγεςϜ ͕ཧ૝తɻͦͷࢦඪͱͯ͠ͷ$PNQFYJUZɻ

ߏ଄͔ΒμΠφϛΫε΁ ४උ O੒෼Λ࣋ͭఆৗଟ࣍ݩ֬཰աఔʹ͋ΔχϡʔϥϧωοτϫʔΫΛߟ͑ɺͦͷ֬཰ม਺ Λ9Ͱද͢ɻϊʔυͷ஋Λද͢ϥϯμϜม਺Λ" ɺ઀ଓߦྻΛ$@\JK^ɺΨ΢γΞϯϊΠζ Λ3Ͱද͢ͱఆৗঢ়ଶ͕࣍ͷࣜΛຬͨ͢Α͏ͳઢܗաఔΛߟ͑Δɻ A = Cij ⋅
A + R ఆৗաఔͱ3ͷੑ࣭͔Βɺ"ͷڞ෼ࢄߦྻ͸͙͢ʹܭࢉͰ͖ͯͱ͢Δͱ Q = [1 − Cij ]−1 COV(X) = ⟨AT ⋅ A⟩ = ⟨QT ⋅ RT ⋅ R ⋅ Q⟩ = QT ⋅ Q

ߏ଄͔ΒμΠφϛΫε΁ NVUVBMJOGPSNBUJPOɾJOUFHSBUJPO ܥΛͭͷ෦෼ܥʹ෼͚ɺͦΕΒΛͱද͢ʢL͸੒෼ͷ਺ɺK͸Մೳͳݸ ͷ෼ׂʹ͚ͭΒΕͨJOEFYʣɻ ෦෼ܥಉ࢜ͷಠཱੑ͸NVUVBMJOGPSNBUJPO͔Β෼͔Δɻ Xk j , X −
Xk j n Ck MI(Xk j ; X − Xk j ) = H(Xk j ) + H(X − Xk j ) − H(X) ʢͨͩ͠ɺ)͸Τϯτϩϐʔʣ ͞ΒʹɺJOUFHSBUJPOͱݺ͹ΕΔ࣍ͷྔΛಋೖ͢Δɻ I(X) = n ∑ i=1 H(xi ) − H(X) JOUFHSBUJPOͱNVUVBMJOGPSNBUJPO͸࣍ͷؔ܎Ͱ݁͹Ε͍ͯΔɻ I(X) = n−1 ∑ i=1 MI({xi }; {xi+1,...,n })

ߏ଄͔ΒμΠφϛΫε΁ $PNQMFYJUZ ٻΊΔࢦඪ͸ɺʮܥ͕શମͱͯ͠͸౷߹͞Ε͍ͯΔ͕ɺҰ෦෼ʹண໨͢Δͱಠཱ͍ͯ͠ Δ࣌ʹେ͖͘ͳΔΑ͏ͳ஋ʯɻ શମͱͯ͠ͷ౷߹۩߹͸ɺJOUFHSBUJPO* 9 Λߟ͑Ε͹ྑ͍ɻ ·ͨJOUFHSBUJPO͸γεςϜαΠζʹґଘ͢Δ͔ΒɺγεςϜαΠζLͷ෦෼ܥͷ౷߹۩ ߹ͱͯ͠Մೳͳ෦෼ܥશͯʹ͍ͭͯͷJOUFHSBUJPOͷฏۉΛߟ͑Δɻ ͜͜ͰɺJOUFHSBUJPOͷੑ࣭͔ΒͱͳΔ͜ͱ͕ࣔͤΔɻ
⟨I(Xk j )⟩ ⟨I(Xk+1 j )⟩ ≥ ⟨I(Xk j )⟩ /FVSBM $PNQMFYJUZΛ࣍ͷΑ͏ʹఆٛ͢Δɻ CN (X) = ∑ k k n I(X) − ⟨I(Xk j )⟩ Πϝʔδਤ

ߏ଄͔ΒμΠφϛΫε΁ ΑΓܭࢉྔ͕খ͍͞ࢦඪͱͯ͠ɺ࣍ͷͭͷ౳ՁͳදݱΛ࣋ͭDPNQMFYJUZ΋ఏҊ͞Εͯ ͍Δʢಋग़͸লུɺOFVSBMDPNQFYJUZͱ౳ՁͰ͸ͳ͍͕ඇৗʹ͍ۙʣɻ C(X) = H(X) − ∑ i H(xi
∣ X − xi ) = ∑ i MI(xi ; X − xi ) − I(X) = (n − 1)I(X) − n⟨I(X − xi )⟩ ઢܗఆৗ֬཰աఔΛߟ͑ͨͨΊɺ) 9 ͕ղੳతʹղ͚Δʢಋग़͸লུʣ H(X) = 1 2 ln((2πe)n ⋅ ∣ COV(X) ∣ )

ߏ଄͔ΒμΠφϛΫε΁ wϚΧΫβϧͷൽ࣭ؒ઀ଓωοτϫʔΫ͔Β$PNQMFYJUZΛܭࢉ͢Δ ͱɺϥϯμϜωοτϫʔΫΑΓ༏Ґʹେ͖͘ͳͬͨɻ

ߏ଄͔ΒμΠφϛΫε΁ w·ͨɺٯʹϥϯμϜͳωοτϫʔΫ͔Βग़ൃͯ͠ɺ$PNQMFYJUZ͕ େ͖͘ͳΔΑ͏ʹ.$.$ͰωοτϫʔΫΛൃలͤ͞ΔͱεϞʔϧ ϫʔϧυੑ͕ݱΕͨɻ ʢਤ͸લϖʔδͱ΋4QPSOT 0 5POPOJ ( &EFMNBO (.
5IFPSFUJDBMOFVSPBOBUPNZBOEUIFDPOOFDUJWJUZPGUIF DFSFCSBMDPSUFY#FIBWJPVSBM#SBJO3FTFBSDI r rIUUQTEPJPSH4 ΑΓʣ

ͦͷଞͷཧ࿦Ϟσϧ w*OUFHSBUFE*OGPSNBUJPO5IFPSZ  5POPOJΒ͕$PNQMFYJUZ͔Βൃలͤͨ͞ɻఆৗ֬཰աఔҎ֎΋ѻ͑Δ͕ɺ ඇৗʹෳࡶͰܭࢉྔ΋๲େɻ  w/FUXPSL$POUSPM5IFPSZ  #BSBCBTJΒ͕ಋೖɻ੍ޚཧ࿦ͷ؍఺͔ΒωοτϫʔΫΛଊ͑Α͏ͱ͍͏ ߟ͑ํɻ࠷ۙ#BTTFUUΒͷάϧʔϓ͕OFVSPTDJFODF΁ͷద༻Λੵۃతʹ ਐΊ͍ͯΔɻ

·ͱΊ w ࠓ೔΍ͬͨ͜ͱ w ωοτϫʔΫղੳͷجૅʢωοτϫʔΫΛಛ௃෇͚Δछʑͷྔͷಋೖͱͦͷ ܭࢉํ๏ʣΛಋೖͨ͠ɻ w ؆୯ͳωοτϫʔΫΛ࢖࣮ͬͯࡍʹܭࢉͯ͠Έͨɻ w ʢ࠷ۙͷʣωοτϫʔΫՊֶͷχϡʔϩαΠΤϯε΁ͷԠ༻ྫΛ঺հͨ͠ɻ
w $PNQMFYJUZͱݺ͹ΕΔྔΛಋೖͯ͠ɺOFUXPSLUPQPMPHZͱμΠφϛΫεʢ৘ ใॲཧʣͱͷؔ܎ੑΛٞ࿦ͨ͠ɻ w ࠓ೔ѻ͑ͳ͔ͬͨ͜ͱ w ೴ͷ؍ଌσʔλ͔ΒωοτϫʔΫΛߏங͢Δํ๏ w ωοτϫʔΫͷʢ෺ཧతʣޮ཰ੑɺؤ݈ੑ w FUDʜ

Network Neuroscience入門

Network Neuroscience入門

Kohei Ichikawa

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ܭࢉ࿦తਆܦՊֶษڧձᶄ /FUXPSL/FVSPTDJFODFೖ໳ d!,FSOFM ౦ژେֶେֶӃ૯߹จԽݚڀՊ. ࢢ઒ߤฏ

ࠓ೔ͷ༧ఆ w ͳͥωοτϫʔΫʁ w ωοτϫʔΫՊֶೖ໳ʢ༻ޠ΍ఆٛͷಋೖʣ w ωοτϫʔΫղੳͷྫ w ωοτϫʔΫ͔ΒԿ͕Θ͔Δʁ w

ࢀߟจݙ w 0MBG4QPSOT/FUXPSLTPGUIF#SBJO  ˠجຊ͸͜ΕʹԊͬͯਐΊΔɻ෼͔Γ΍ͯ͘͢໘ന͍͕਺ࣜ͸ग़ͯ͜ͳ͍ɻ w .JLBJM3VCJOPW 0MBG4QPSOT$PNQMFYOFUXPSLNFBTVSFTPG CSBJODPOOFDUJWJUZ6TFTBOEJOUFSQSFUBUJPOT  ˠ਺ֶతͳఆٛͳͲ͸ͪ͜ΒΛࢀরͨ͠ɻ w

ͳͥωοτϫʔΫʁ OFUXPSLTDJFODF CSBJOTDJFODF VOEFSTUBOEJOUFHSBUJWFGVODUJPOTPGUIFCSBJO UIPVHIU NFNPSZ DPOTDJPVTOFTT FUDʜ 4JOHMFOFSWFDFMM CSBJOOFUXPSLT

ωοτϫʔΫՊֶೖ໳ w ڭՊॻͷষʹରԠ w ஌͍ͬͯΔਓʹ͸ୀ۶͔΋ɾɾɾ

ωοτϫʔΫՊֶೖ໳ w DMVTUFSJOHDPF⒏DJFOU w ͋ΔϊʔυJΛߟ͑ͨ࣌ɺͦΕʹ઀ଓ͍ͯ͠Δϊʔυؒͷ઀ଓ͕औΓಘ Δ઀ଓͷ͏ͪͲͷఔ౓઎Ί͍ͯΔ͔Λද͢ྔ w ωοτϫʔΫʹ͍ͭͯߟ͑Δͱ͖͸શϊʔυͷฏۉΛͱΔ C =

ωοτϫʔΫՊֶೖ໳ w QBUIMFOHUI w TIPSUFTUQBUIMFOHUI  ϊʔυJͱϊʔυKΛ݁Ϳ࠷୹ܦ࿏ɻJ KؒʹΤοδ͕͋Ε͹ͰɺJͱK͕ ܨ͕͍ͬͯͳ͍৔߹͸㱣ʹͳΔɻ dij =

ωοτϫʔΫՊֶೖ໳ w NPEVMBSJUZ w Ϟδϡʔϧͱશࣸ૾  ɹɹɹɹɹɹʹΑͬͯϞδϡʔϧ෼ղ͸Ұҙʹఆ·Δɻ w ͜ͷ෼ղʹରͯ͠NPEVMBSJUZ2͸ M ∈

ωοτϫʔΫՊֶೖ໳ w DFOUSBMJUZ w ೔ຊޠͰ͸த৺ੑई౓ɻωοτϫʔΫ಺Ͱݸʑͷϊʔυ͕࣋ͭॏཁੑ Λද͢ྔɻ w DMPTFOFTTDFOUSBMJUZଞͷ఺ͱڑ཭͕͍ۙ΄Ͳத৺ੑ͕ߴ͍ͱ͢Δɻ QBUIMFOHUIͷٯ਺ɻ L−1

ωοτϫʔΫՊֶೖ໳ w &SEPT3FOZJ SBOEPNOFUXPSL w ͍ΘΏΔϥϯμϜωοτϫʔΫɻϊʔυ਺/ Τοδͷੜ੒֬཰QΛ༩͑ ͯɺશϊʔυؒͷϖΞʹ͍ͭͯ֬཰QͰΤοδΛுΔɻQͷ࣌͸શ݁ ߹ωοτϫʔΫʹͳΔɻ w

ωοτϫʔΫՊֶೖ໳ w 8BUUTBOE4USPHBU[ TNBMMXPSMEOFUXPSL w ͍ΘΏΔεϞʔϧϫʔϧυωοτϫʔΫͷੜ੒Ϟσϧɻ  ϊʔυ਺/ ฏۉ౓਺L Τοδͷܨ͔͗͑֬཰QΛ༩͑ͯɺ࣍ݩ֨ࢠά ϥϑΛ࡞੒ͨ͠ޙ֬཰QͰΤοδͷܨ͔͗͑Λߦ͏ɻQͷ࣌͸ϥϯμ

ωοτϫʔΫՊֶೖ໳ w #BSBCBTJ TDBMFGSFFOFUXPSL w εέʔϧϑϦʔωοτϫʔΫͷੜ੒Ϟσϧɻ খ͞ͳωοτϫʔΫ͔Βελʔτͯ͠ɺͦ ͜ʹ࣍ʑͱϊʔυΛՃ͑Δ͜ͱͰ࡞͍ͬͯ ͘ɻͦͷࡍʢطଘͷάϥϑͷʣͲͷϊʔυ ʹ઀ଓ͢Δ͔͸ϊʔυͷ౓਺ʹൺྫ͢Δ֬

ωοτϫʔΫղੳͷྫ ڭՊॻͷQd ˠKVQZUFSOPUFCPPLͰઆ໌ IUUQTHJUIVCDPNUSJQEBODFSOFUXPSL@OFVSPTDJFODF

ωοτϫʔΫ͔ΒԿ͕෼͔Δʁ ࿦จ঺հᶃ  %ZOBNJDSFDPOpHVSBUJPOPGIVNBOCSBJOOFUXPSLT EVSJOHMFBSOJOH #BTTFUU %4 8ZNCT /' 1PSUFS ."

ωοτϫʔΫ͔ΒԿ͕෼͔Δʁ w /PEFqFYJCJMJUZUIFOVNCFSPGUJNFTUIBUFBDIOPEF DIBOHFTNPEVMFBMMFHJBODF OPSNBMJ[FECZUIFUPUBMQPTTJCMF OVNCFSPGDIBOHFT   w /PEFqFYJCJMJUZDIBOHFEEVSJOHUIFMFBSOJOHQSPDFTT  pSTUJODSFBTJOHBOEUIFOEFDSFBTJOH

ωοτϫʔΫ͔ΒԿ͕෼͔Δʁ ࿦จ঺հᶄ #SBJODPOOFDUJWJUZEZOBNJDTEVSJOHTPDJBMJOUFSBDUJPO SFqFDUTPDJBMOFUXPSLTUSVDUVSF 4DINÅM[MF 3 #SPPL0`%POOFMM . (BSDJB +0

ωοτϫʔΫ͔ΒԿ͕෼͔Δʁ w ͜ͷ࣌ͷ݁߹ڧ౓ͷมԽʹ͸ݸਓ͕ࠩ͋Δ͕ɺͦΕ͕ݸਓͷ࣋ͭ GSJFOETIJQOFUXPSLͷಛ௃ͱ૬͍ؔͯ͠Δ͜ͱ͕ࣔ͞Εͨɻ w GSJFOETIJQOFUXPSLͷ`%FOTJUZ` ඃݧऀʹ ର͢ΔDMVTUFSJOHDPF⒏DJFOU ͕খ͍͞΄ Ͳ݁߹ڧ౓ͷมԽ͕େ͖͍ɻ

ߏ଄͔ΒμΠφϛΫε΁ w ཧ࿦తΞϓϩʔν͔Β͸Կ͕ݴ͑Δ͔ʁ w ϚΧΫβϧͰܭࢉͨ͠Α͏ʹɺ೴಺ͷωοτϫʔΫʹ͓͍ͯ΋εϞʔϧ ϫʔϧυੑ͕޿͘ݟΒΕΔɻ a small-world topology can

ߏ଄͔ΒμΠφϛΫε΁ $PNQMFYJUZ 4QPSOT 5POPOJ &EFMNBO In brains of higher vertebrates,

ߏ଄͔ΒμΠφϛΫε΁ ४උ O੒෼Λ࣋ͭఆৗଟ࣍ݩ֬཰աఔʹ͋ΔχϡʔϥϧωοτϫʔΫΛߟ͑ɺͦͷ֬཰ม਺ Λ9Ͱද͢ɻϊʔυͷ஋Λද͢ϥϯμϜม਺Λ" ɺ઀ଓߦྻΛ$@\JK^ɺΨ΢γΞϯϊΠζ Λ3Ͱද͢ͱఆৗঢ়ଶ͕࣍ͷࣜΛຬͨ͢Α͏ͳઢܗաఔΛߟ͑Δɻ A = Cij ⋅

ߏ଄͔ΒμΠφϛΫε΁ NVUVBMJOGPSNBUJPOɾJOUFHSBUJPO ܥΛͭͷ෦෼ܥʹ෼͚ɺͦΕΒΛͱද͢ʢL͸੒෼ͷ਺ɺK͸Մೳͳݸ ͷ෼ׂʹ͚ͭΒΕͨJOEFYʣɻ ෦෼ܥಉ࢜ͷಠཱੑ͸NVUVBMJOGPSNBUJPO͔Β෼͔Δɻ Xk j , X −

ߏ଄͔ΒμΠφϛΫε΁ ΑΓܭࢉྔ͕খ͍͞ࢦඪͱͯ͠ɺ࣍ͷͭͷ౳ՁͳදݱΛ࣋ͭDPNQMFYJUZ΋ఏҊ͞Εͯ ͍Δʢಋग़͸লུɺOFVSBMDPNQFYJUZͱ౳ՁͰ͸ͳ͍͕ඇৗʹ͍ۙʣɻ C(X) = H(X) − ∑ i H(xi

ߏ଄͔ΒμΠφϛΫε΁ wϚΧΫβϧͷൽ࣭ؒ઀ଓωοτϫʔΫ͔Β$PNQMFYJUZΛܭࢉ͢Δ ͱɺϥϯμϜωοτϫʔΫΑΓ༏Ґʹେ͖͘ͳͬͨɻ

ߏ଄͔ΒμΠφϛΫε΁ w·ͨɺٯʹϥϯμϜͳωοτϫʔΫ͔Βग़ൃͯ͠ɺ$PNQMFYJUZ͕ େ͖͘ͳΔΑ͏ʹ.$.$ͰωοτϫʔΫΛൃలͤ͞ΔͱεϞʔϧ ϫʔϧυੑ͕ݱΕͨɻ ʢਤ͸લϖʔδͱ΋4QPSOT 0 5POPOJ ( &EFMNBO (.

·ͱΊ w ࠓ೔΍ͬͨ͜ͱ w ωοτϫʔΫղੳͷجૅʢωοτϫʔΫΛಛ௃෇͚Δछʑͷྔͷಋೖͱͦͷ ܭࢉํ๏ʣΛಋೖͨ͠ɻ w ؆୯ͳωοτϫʔΫΛ࢖࣮ͬͯࡍʹܭࢉͯ͠Έͨɻ w ʢ࠷ۙͷʣωοτϫʔΫՊֶͷχϡʔϩαΠΤϯε΁ͷԠ༻ྫΛ঺հͨ͠ɻ