Ginzburg-Landau A A Kyoto Univ. Kobe Design Univ. A N. Tsukamoto, H. Fujisaka, K. Ouchi A Ginzburg-Landau ψ = ψ + (1 + ic 1 ) 2 ψ (1 + ic 2 ) ψ 2 ψ (1

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1 Ginzburg-Landau A A Kyoto Univ. Kobe Design Univ. A N. Tsukamoto, H. Fujisaka, K. Ouchi A Ginzburg-Landau ψ = ψ + (1 + ic 1 ) 2 ψ (1 + ic 2 ) ψ 2 ψ (1) Hopf [1] c 1, c 2 (1) { } ψ n+1 (r) = dr J(r r )ψ n (r ) ψ n (r ) 2 (1+ic2 )/2 + δ δ > J(r) = e (1+ic 1) 2 δ(r) (1) Ref.[2] ψ n 2 δ (2) δ θ n = arg ψ n c 2 log ψ n (2) e iθn+1(r) = dr J(r r )e iθ n(r )/ dr J(r r )e iθ n(r ) (3) 1+ic2 θ n (3) Benjamin-Feir (1 + c 1 c 2 = ) (1) (1) ψ ( ) (2) (2) (1) (2) (3) (1) (2) [1] I. S. Aranson, L. Kramer, Rev. Mod. Phys., 74, 99 (22). [2] S. Uchiyama, H. Fujisaka, Phy. Rev. E, 56, 99 (1997).

2 Synchronization of excitatory neurons with strongly heterogeneous phase response Yasuhiro Tsubo, Jun-nosuke Teramae, and Tomoki Fukai Laboratory for Neural Circuit Theory, RIKEN BSI 大脳皮質神経細胞の同期発火現象は, 脳の高次機能において重要な役割を果たすと考えられている. 同期発火現象は神経系の回路構造とともに, 位相応答などの神経細胞固有の性質によって影響を受ける. 現在までの研究から, 興奮性神経細胞である錐体細胞の位相応答は, あるクラスの関数で記述されること, またそのクラスの中で定性的に異なる2つのタイプに分類されることが示されてきた. 我々の最近の研究から, 錐体細胞の位相応答はその細胞が属する層 ( 場所 ) に依存してそのタイプが異なることがわかった. さらに, 同じ層に属する錐体細胞であっても, その位相応答の形は同じタイプでも均質ではなく不均質な性質をもつことがわかった. このように, 神経細胞の位相応答は不均質であると考えられるが, 不均質な位相応答をもつ振動子集団がどのような挙動を示すのかは, 明確にはわかっていない. そこで, 我々は不均質な位相応答をもつ振動子集団の挙動を, 大域的にパルス結合された位相振動子モデルを用いて, 解析的, 数値的に調べた. その結果, 均質な場合に同期傾向が弱い位相応答をもつ振動子は, 不均質にしても定性的に変化がなかった. 一方で, 均質な場合に同期傾向が強い位相応答をもつ振動子は, 不均質性を強めると同期, 部分同期, 非同期と転移することがわかった.

3 1 Physarum polycephalum

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{ÊÈÉÇÊÍÆÊÎÆÍÇÉÉÂÊÊ yçzçíçéîœêèâìéâêííîìâ wuéê ÎsÆÂ :~ÊÖ ÑÕ ÓÏÑÎÆÍÇÉÊÍÍ{ÊÂŒ ÊÊÊÇÈÍÉÆÍÇÂÈÊÍÇÉÉÉÇÈÍÉÉÇ ÍÂÊÊÈÍÈÍ ÊŒÉÍÍÉ ÇÊÉsÈÉÆÍÂ [1] Z. F.

6 y ÎÆÉÂÓÕ Ö ÊÑÏÕ ÐÊÖ ÑÕÉÆÍÇÂ ~Ê ÎÆÍÉÑÏÕ ÐÇÈÍÆÇÉÇÔÏÓ ÍÊÍÉÉ ŒÊÈÍÉÆÍ[1]ÂÇÊÍÆÊÊÊ~ÊÎÆÍ ÇÉÉÇÈÍÇÉÇÆÍÂÇÊÍÆÊyÊÇÉ ÆÍÉÇÊwuÊÍÊÍÉÉÊÈÍÉÆÍÇ[2]ÂÿjÊuÊ ÉÆÍÂ ÇÍÌÉÊÂFitzHugh-Nagumo ÕÒÖÊ ÎÔÎÑ ~ÉsÇÈÍ ÎÆÍÇÉÉÂÇÊÍÆÊyÇzÇÍÇÉÇvŒÊÍÉ ÉvÈÍÉÆÍÂÇÊyÊÊÍÉÉÇzÇÈÍÍÊ ÎÊÉÆÂÈÊÔÒÓ ÐÊÖÎÔÓÔÎvÈÍÇ ÉÊÍÉÉˆdÊuÉÇÍ[3]Â {ÊÈÉÇÊÍÆÊÎÆÍÇÉÉÂÊÊ yçzçíçéîœêèâìéâêííîìâ wuéê ÎsÆÂ :~ÊÖ ÑÕ ÓÏÑÎÆÍÇÉÊÍÍ{ÊÂŒ ÊÊÊÇÈÍÉÆÍÇÂÈÊÍÇÉÉÉÇÈÍÉÉÇ ÍÂÊÊÈÍÈÍ ÊŒÉÍÍÉ ÇÊÉsÈÉÆÍÂ [1] Z. F. Mainen and T. J. Sejnowski, Science 268, [2]J. Teramae, and D. Tanaka, Phys. Rev. Lett. 93, 2413 (24) [3]K. Nagai, H. Nakao, and Y. Tsubo, Phys. Rev. E, 71, (25)

Brain Machine Interface (BMI) BMI () [1] (a) [, 2π] 3 5 15 (b) #1-551-111-15 Fig. 1. (a) x = y < x 2 15 (b)(a) REFERENCES [1] L. Shpigelman, Y. Singer, R.

8 Brain Machine Interface (BMI) BMI () [1] (a) [, 2π] (b) # Fig. 1. (a) x = y < x 2 15 (b)(a) REFERENCES [1] L. Shpigelman, Y. Singer, R. Paz, and E. Vaadia, Spikernels: Predicting arm movements by embedding population spike rate patterns in inner-product spaces, Neural Comput., vol. 17, pp , 25.

9 1) 2) 3) Time Dependent Ginzburg-Landau (TDGL) t ψ(r, t) = ψ + γψ ψ 2 ψ + 2 ψ + he iωt, (ψ = X + iy ) (1) 1 γ =.3, Ω =.5, h = (a) 1 (c) 1 (b) (d) ψ (t) = L ψ(z, t)dz/l (1 (b)) (1 (d)) γ γ 2 (a) θ = (2 (b)) X(z,t) X(z,t) -.54 (a) z (c) z Y Y (b) X 1 (d) X y (a) x Y (b) : γ =.5, Ω = 1., h =.5 (a) X(x, y) (b) X 1: (a) (b) (c) (d) 1) B. K. Chakrabarti, and M. Acharyya, Rev. Mod. Phys. 71, 847 (1999). 2) E. Machado et al., Phys. Rev. E 71, 1612 (25). 3) G. S. Jeon et al., Phys. Rev. B 65, (22).

10 () 1: (a) (c) (d) (f) (a) (d)(b) (e)(c) (f) (a)(d): (b)(e): (c)(f):

11 [ ] [ ] [ ] [ ] Wistar [ ] 1 ZT(zeitgeiber time)82 ZT233 ZT21 ZT12 [ ] [ ]

12 FitzHugh- (FHN) [1, 2] du/dt = u(u α)(1 u) v, (1) dv/dt = τ(u γv). (2) α, γ, τ u = u(t) () v = v(t) [3] du i /dt = u i (u i α)(1 u i ) v i + K N dv i /dt = τ(u i γv i ), κ i,j (u j u i ), i j K N {κ i,j } κ i,j = { 1, with probability 1 p 1, with probability p, p α =.1, τ =.1, γ = 1. p = (u i, v i ) = (, ) [4] p K p 1(a)

u u u i i i 1 1 1 t t t K p ( 1(c)) κ i,j N Hopf K c κ i,j K c (2p 1) = α + γτ. (3) [5] ( 1(b)) (a) (b) (c). 1. 5 5. 5 5. 5 5 1 1: (a) u i {κ i,j } (b)(p, K)- N 1 (Hopf ) (c) K [1] R. FitzHugh.

13 u u u i i i t t t K p ( 1(c)) κ i,j N Hopf K c κ i,j K c (2p 1) = α + γτ. (3) [5] ( 1(b)) (a) (b) (c) : (a) u i {κ i,j } (b)(p, K)- N 1 (Hopf ) (c) K [1] R. FitzHugh. Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1: , [2] J. Nagumo, S. Arimoto, and S. Yoshizawa. An active pulse transmission line simulating nerve axon. Proc. IRE, 5:261 27, [3] T. Aoyagi. Network of neural oscillators for retrieving phase information. Phys. Rev. Lett., 74: , [4] T. Yanagita, T. Ichinomiya, and Y. Oyama. Pair of excitable fitzhugh-nagumo elements: Synchronization, multistability, and chaos. Phys. Rev. E, 72(5):56218, Nov 25. [5] Y.Oyama, T.Yanagita, and T. Ichinomiya. Numerical analysis of fitzhugh-nagumo neurons on random networks. Prog. of Theor. Phys. Suppl., 161: , 26.

14 1 1 ( ) 1: Gaussian White Noise 1-3L & 1-3R events/s N= directional tuning curve 1ms time window 1-2L & 1-3L 1-2R & 1-3R events/s 1-2L & 1-2R 2: directional tuning curvel events/s 18 N= events/s N=9 R 18 N=9

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