Complex periodic spiral waves

Transcription

1 Complex periodic spiral waves (in a BZ system & in vitro cardiac system) Kyoung J. Lee NCRI Center for Neuro-dynamics and Department of Physics, Korea University 1 Coupled network of nonlinear oscillators in biology 1) Bacterial colony 2) Firefly population 3) amoeba population 4) cardiac tissue 5) neural networks: eg eg., suprachiasmatic nucleus (the master biological clock) 2 1

2 Simple system but complex dynamics Cz eg. Rössler system cx= -cy-cz cy=cx+acy cz=czcx-ccz+b period-1 cycle (C=C 1 ) Cx Cy period-2 cycle (C=C 2 ) Cx time period-4 cycle (C=C 3 ) 3 camp waves in signaling amoeba populations spiral wave circular waves 2.0x2.8 cm 2 K. Lee et al., PRL (1996); PRL (2001); Nonlinearity (2002) 4 2

3 Period-1 spiral wave in a diffusively coupled Rössler system 5 Period-2 spiral wave with one line-defect in a Rössler system Parameters: A=0.2, B=0.2, C=3.84, D t/( r) 2 =1.6x10-2 Period-2 spiral line-defect τ 1 V ( x, t) = [ G( x, t + t' ) G( x, t + t' τ )] dt' τ 0 [References] Goryachev et al.,prl 76 (1996), PRE 54 (1996), PRL 80 (1998); PRL 83 (1999); Chaos 10 (2000). 6 3

4 Period-2 spiral wave in the coupled Rössler system 7 P1-P2 P2 spiral wave transition in the coupled Rössler system 8 4

5 A pair of period-2 spiral waves in BZ EXPERIMENT J. Park and K. Lee, PRL 83, 5393 (1999) Disk diameter: 20 mm One-side fed reactor: 0.89 M H 2 SO 4, 0.06 M KBrO 3, 0.27 M melonic acid, 2.5 M ferroin, 0.1 mm NaBr, 0.1 mm SDS 9 Schematic diagram of two-side fed gel reactor 10 5

6 Period-1 to Period-2 spiral wave transition in a BZ system [BrO 3- ]=0.75 M J. Park & K. Lee, PRL (2002) [H + ]=3.02 M 2.40 M 2.12 M Raw images Processed Images showing local periodicity P-1 in space P-1 in time P-1 in space P-2 in time Dia. 8.2 mm P-2 in space P-2 in time 11 Intermittent behaviors at a P-1 P 1 to P-2 P spiral wave transition [H + ]=3.02 M 2.80 M 2.48 M phase slips!! 2.12 M 12 6

7 Breathing line-defect of a spiral pair at P-1 P 1 to P-2 P 2 transition (a) (b) [H + ]=2.80M [BrO 3- ]=0.75M Field of view: 8.2 x 4.8 mm^2 13 Line-defect density fluctuation during a P-1 to P-2 spiral wave transition 14 7

8 Dynamic shape transition of a spiraling line-defect [H + ]: 2.48 M 2.22 M at t = 0 J. Park et. al., JKPS 41, 4 (2002) 15 Transverse instability of a line-defect Diameter: 5.6 mm Two-side fed reactor: SIDE A 0.90 M H 2 SO 4, M KBrO 3, SIDE B 0.53 M MA, 2.5 M ferroin, 0.1 mm NaBr, 0.1 mm SDS [BrO 3 - ] = 0.52 M Flow rate = 120 ml/hr, Volume = 10 ml, Temp 22 o C 0.54 M 0.58 M 0.60 M 16 8

9 Analogy with bistable systems Iodate-ferrocyanidesulfite reactiondiffusion system, SCIENCE 1993, K. Lee et al. FitzHugh-Nagumo type model reaction-diffusion system, PRL 1994, A. Hagberg and E. Meron 17 Dynamics of LINE-DEFECT in a Williamoskii- Rössler system ) ls e ix ( p n tio r a a p e S. im m bif.. Parameter k5 Simulation by S. Woo (2002) 18 9

10 Heuristic picture & quantifications λ λ 2 λ 1 A 19 Line-defect proliferation & chaotic domains sec Diameter = 24 mm sec [BrO 3- ]: 0.60 M 0.64 M f.o.v.: 5.4x 4.0 mm

11 Similar observations in a Williamoskii- Rössler system λ 1 λ 2 P-2 regime ~ P-4 regime 21 Complex periodic spiral waves: summary - So far, we discussed several phenomena associated with diffusively (i.e., linearly) coupled nonlinear oscillators, mostly, in Period-1 & Period-1/Period-2 (mixed) regimes. - Obviously, the system will become more complex, if it1) moves into more complex periodic regime, 2) involves a more complex coupling mechanism, or 2) becomes three-dimensional in space. - Any complex periodic spiral wave in biology? 22 11

12 Students Jin-sung Park (BZ) Sung-jae Woo (Model) Seong-min Hwang (Cardiac waves) 23 12