PACS: a, L

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1 * 1)2) 1) 2) 1) (, ) 2) (, ) ( ; ) k k + 1(k = 1, 2), k k + 1 Chay k k + 1, ; 0 1,., :,,, PACS: a, L 1,,, [1]., [2].,,, [3 26].,, [3 5].,, [6,7],,, [6] [7]., [9 13], 2 3 [9,10 13] 3 4 [11 13], [14 18], [7,14], [17,18], [19 26],, [2,27], 1 [28 32]., * ( : , , , ) ( : GK ) guhuaguang@263.net c 2012 Chinese Physical Society

2 [2,27]., Hopf [30 34],, [31], [32]. [9,11 13,35 37]. [9 13], 1 2., [35]., k k + 1 (k = 1, 2, 3, 4, 5), k k + 1, [10,12,13,36,37] , 2 3., [10,36,37] ;, k k + 1 [10,12]. Chay, k k + 1 [36,37].,, [10,12]., P-2 2, P-3 3,,, k k + 1, (ISI), [32],. k k + 1 ISI,. Chay, k k k + 1 k + 1 k k + 1,,,,,

3 Sprague- Dawley, g, [7 13] (40 mg/kg),,, 6 14 d,,,, C,, 20 mm,, C,,,, A δ, Powerlab, 10.0 khz, ISI., ([Ca 2+ ] o, o ), 2.2 (Chay ) Chay [16], [7 13,30,33,36,37]. Chay, : dv dt =g Im 3 h (V I V ) + g kv (V k V )n 4 C + g kc 1 + C (V k V ) + g l (V l V ), (1) dn dt =n n, τ n (2) dc dt =τ c(m 3 h (V c V ) k c C). (3) V, n C K + Ca 2+ ([Ca 2+ ]). (1) Na + -Ca 2+ ( ) K + [Ca 2+ ] K + g I, g kv, g kc, g l, V I, V k V l Na + -Ca 2+ K +, m h Na + -Ca 2+, τ n K +, n K +, V c Ca 2+, k c Ca 2+. τ n = 1/(λ n (α n + β n )), s. m, h, n, α n β n [16]. Chay : g I = 1800 ms/cm 2, g kv = 1700 ms/cm 2, g kc = 20 ms/cm 2, g l = 7 ms/cm 2, V I = 100 mv, V k = 75 mv, V l = 40 mv, τ c = 0.27, k c = 3.3/18. [Ca 2+ ] o, V c, λ n, ξ(t),, ξ(t) :, ξ(t) = 0;, ξ(t), ξ(t ) = 2Dδ(t t ), D, Mannella [38] Chay, s. 2.3 [39] [38] Theiler [39], ( ).,,,,.,,,,, [40] : t i (i = 1, 2,, N), m R m, N m + 1 R m V 0 = (t i0,, t i0 +m 1), n = b(n m + 1), U k = (t jk,, t jk +m 1)(1 k n), V 0 n h p j0 +h = 1/n t jk +h, V 0 h k=1 t i0 +h, (NPE)

4 t = 1/N e NP (h) = (p i 0 t i0 +h) 2 1/2 ( t t i ) 2 1/2. (4) N t i, e NP (h) 1.0 i=1, h, ; e NP (h) < 1, h, m = 4, b = 1%. y 1, y 2,, y M, ρ(τ) = ȳ = 1/M (y i ȳ)(y i+τ ȳ) M τ i=1 M τ, (5) (y i ȳ) 2 i=1 M t i, τ(τ = 0, 1,, M 1) i=1. τ = 0, ρ(τ) = 1; τ 0, ρ(τ) τ. ρ(τ) = 0, ; ρ(τ) > 0, ; ρ(τ) < 0,, τ 0, ρ(τ) = 0;, τ 0, ρ(τ) 0. 2(a) ; 2 3 2(b) , 3, 3 [Ca 2+ ] o 1.2 mmol/l 0 mmol/l, k k + 1 (k = 1, 2) (a) 1 2 ; (b)

5 2 3 ISI 3, 4(a). ISI, 4, P N1, P N2, P N3 P N4, 4(b)., n ISI 9, 4(c).,,, ISI., P N2 P N4 2, P N1, P N3 P N4 3, ISI., ISI, 1,,, 4(d)., 2 3 ISI (a) ISI ; (b) ISI ; (c) ISI ; (d) 2 3, 2 0, 3 1,, (a),,, ISI 2 3 ISI. ISI ,, 2 3 ISI

6 1. 4 Chay, λ n = 227.5, V c mv, : (V c = mv), 2, 6(a). λ n = 230, V c mv, (V c = mv), 3, 6(b) (a) ; (b) [32], R 1, 0 R 0 ; 01, 10, R 01, R 10, R 00 R 11. : R 1 = 0.606, R 0 = 0.394; R 01 = 0.077, R 10 = 0.077, R 00 = 0.318, R 11 = , R 1 R 0 R 1 R 0 = 0.239, R1 2 = 0.279, R0 2 = , R 1 R 0 R 01, R 2 1 R 11, R 2 0 R , 2, 2 (R 2 0 < R 00 ), 3, 3 (R 2 1 < R 11 ), 01 ρ, 5(b), τ = 1 τ = ,, 1 6 Chay. (a) (λ n = 227.5); (b) (λ n = 230) Chay, ( D = ), :

7 , 2, 7(a) ; 2, 3, 7(b). 2 3 (V c = mv) 8, 2 3,, 2 3 ISI 9. 9 ( 4). Chay, 2 3 ISI., ISI (a). 10(a), 01, 2 3 ISI. 01 : R 0 = 0.708, R 1 = 0.292; R 00 = 0.637, R 01 = 0.071, R 10 = 0.071, R 11 = R 2 0 = 0.501, R 1 R 0 = 0.207, R 2 1 = R 10 R 1 R 0, R 2 1 R 11, R 2 0 R , 01, 10(b), τ = 1 5, ρ 0,, 7 Chay (a) 1 2 (λ n = 227.5); (b) 2 3 (λ n = 230) 8 Chay

8 9 Chay 2 3 (a) ISI ; (b) ISI ; (c) ISI ; (d) ISI 5 10 Chay (a) ; (b), [10,12,13,36,37],, k k k + 1 k + 1 (k = 1, 2), k k + 1, [9 13] [10,12,13,35 37], k k + 1 [10,12,13,36,37], ; k k + 1 k

9 k + 1, k k + 1 k k + 1,, k k + 1 ISI, k k + 1 k k + 1 [10,12,13], k k + 1, ISI k k + 1,, ISI [10]. ISI,,,, [9 13], [10 13,36,37],,, k [10] [33],,,, [1] May R M 1976 Nature [2] Gammaitoni L, Hänggi P, Jung P, Marchesoni F 1998 Rev. Mod. Phys [3] Hayashi H, Ishzuka S, Ohta M, Hirakawa K 1982 Phys. Lett. A [4] Hayashi H, Ishzuka S, Hirakawa K 1983 Phys. Lett. A [5] Aihara K, Matsumoto G, Ikegaya Y 1984 J. Theor. Biol [6] Hayashi H, Ishzuka S 1992 J. Theor. Biol [7] Li L, Gu H G, Yang M H, Liu Z Q, Ren W 2004 Int. J. Bifur. Chaos [8] Wang D, Mo J, Zhao X Y, Gu H G, Qu S X, Ren W 2010 Chin. Phys. Lett [9] Gu H G, Zhu Z, Jia B 2011 Acta Phys. Sin (in Chinese) [,, ] [10] Yang M H, Liu Z Q, Li L, Xu Y L, Liu H J, Gu H G, Ren W 2009 Int. J. Bifur. Chaos [11] Ren W, Hu S J, Zhang B J, Xu J X, Gong Y F 1997 Int. J. Bifur. Chaos [12] Gu H G, Yang M H, Li L, Liu Z Q, Ren W 2004 Dynam. Contin. Dis. B [13] Zhao X Y, Song S L, Wei C L, Gu H G, Ren W 2010 Acta Biophys. Sin (in Chinese) [,,,, ] [14] Xie Y, Xu J X, Kang Y M, Hu S J, Duan Y B 2003 Acta Phys. Sin (in Chinese) [,,,, ] [15] Fan Y S, Holden A V 1993 Chaos Solitons Fract [16] Chay T R 1985 Physica D [17] Wu S G, He D R 2000 Chin. Phys. Lett [18] Wu S G, He D R 2001 Commun. Theor. Phys [19] Schiff S J, Jerger K, Duong D H 1994 Nature [20] Thomas E, William J R, Zbigniew J K, James E S, Karl E G, Niels B 1994 Physiol. Rev [21] Lovejoy L P, Shepard P D, Canavier C C 2001 Neuroscience [22] Kanno T, Miyano T, Tokudac I, Galvanovskisd J, Wakui M 2007 Physica D [23] Hu S J, Yang H J, Jian Z, Long K P, Duan Y B, Wan Y H, Xing J L, Xu H, Ju G 2000 Neuroscience [24] So P 1998 Biophys J [25] Jian Z, Xing J L, Yang G S, Hu S J 2004 Neurosignals [26] Wan Y H, Jian Z, Hu S J 2000 Neuroreport [27] Longtin A, Bulsara A, Moss F 1991 Phys. Rev. Lett [28] Braun H A, Wissing H, Schäfer K, Hirsch M C 1994 Nature [29] Xing J L, Hu S J, Xu H, Han S, Wan Y H 2001 Neuroreport [30] Gu H G, Ren W, Lu Q S, Wu S G, Yang M H, Chen W J 2001 Phys. Lett. A [31] Gong P L, Xu J X, Hu S J, Long K P 2002 Int. J. Bifur. Chaos [32] Gu H G, Jia B, Lu Q S 2011 Cogn. Neurodyn [33] Gu H G, Zhang H M, Wei C L, Yang M H, Liu Z Q, Ren W 2011 Int. J. Mod. Phys. B [34] Jia B, Gu H G, Li Y Y 2011 Chin. Phys. Lett [35] Huber M T, Krige J C, Braun H A, Pei X, Neiman A, Moss F 2000 Neurocomputing [36] Gu H G, Yang M H, Li L, Liu Z Q, Ren W 2003 Phys. Lett. A [37] Gu H G, Yang M H, Li L, Liu Z Q, Ren W 2003 Int. J. Mod. Phys. B [38] Mannella R, Palleschi V 1989 Phys. Rev. A [39] Theiler J, Eubank S, Longtin A, Galdrinkian B 1992 Physica D [40] Sauer T 1994 Phys. Rev. Lett

10 Identification of a stochastic neural firing rhythm lying in period-adding bifurcation and resembling chaos Gu Hua-Guang 1)2) Xi Lei 1) Jia Bing 2) 1) ( College of Physics and Information Technology, Shaanxi Normal University, Xi an , China ) 2) ( College of Life Sciences, Shaanxi Normal University, Xi an , China ) ( Received 23 August 2011; revised manuscript received 8 September 2011 ) Abstract To identify non-periodic neural rhythm to be chaos or stochasticity has been an important scientific thesis. A kind of nonperiodic spontaneous firing pattern, whose behavior is transition between period-k burst in a string and period-k + 1 burst in a string (k = 1, 2), lying between period-k bursting pattern and period-k + 1 bursting pattern, is found in the experimental neural pacemaker. The deterministic structures of the firing are identified by nonlinear prediction and first return map of the interspike intervals (ISIs) series. The co-existence of the period-k bursting and period-k + 1 bursting is manifested in the deterministic theoretical neuronal model, Chay model. Non-periodic firing patterns similar to the experimental observation are simulated in the co-existing parameter region, implying that the firing pattern is transition between two kinds of bursts induced by noise. A binary series can be acquired by transforming two kinds of bursts to symbols 0 and 1, respectively. The stochastic dynamics within the transitions between two kinds of bursts are detected by probability analysis on the binary series. It not only shows that the rhythm is stochastic firing with deterministic structures instead of chaos, but also provides the typical examples and effective methods to intensively identify the chaotic and stochastic firing patterns in a real nervous system. Keywords: chaos, stochastic rhythm, neural firing pattern, period-adding bifurcation PACS: a, L * Project supported by the National Natural Science Foundation of China (Grant Nos , , , ) and the Fundamental Scientific Research Foundation for the Central Universities of China (Grant No. GK ). guhuaguang@263.net