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Transition to Burst Synchronization onComplex Neuron Networks
Zhonghuai Hou( 侯中怀 )2007.9 Nanjing
Department of Chemical PhysicsHefei National Lab of Physical Science at Microscale
University of Science and Technology of China
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Our research interest
Statistical problems in mesoscopic chemical systems
Nonlinear Dynamics on complex networks
Complexity + Nonlinearity
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A Neuron
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Diversity: Morphology + Physiology
Oscillation
Spiking
Bursting
Chaos
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Neuron Network
Human Brain: 1011 and 104
Complex
Network
Small-World
Scale-Free
Big Challenge : Dynamics + Functioning
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An interesting phenomenon ...
Central Pattern Generator Small microcircuits Rhythmic motor commands Striking feature
Individual: irregular,chaotic bursts Ensemble: regular, rhythmic bursting
Mechanism ?
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Related study
Chaos Regularization
N.F.Rulkov, PRL 86,183(2001)
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Related study
Ordering Chaos by Random Shortcuts
F. Qi, Z.Hou, H. Xin, PRL 91, 064102 (2003)
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Related Study
M. Wang, Z.Hou, H.Xin. ChemPhysChem 7 , 579( 2006)
Ordering Bursting Chaos
Hindmarsh-Rose (HR) model system
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Synchronization of Bursting System
Beyond complete synchronization
Spike Syn...
Burst Syn...
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The present work
Fixed Network + increased coupling Transition from chaos to BS Different types of BS-states Spike-adding Bursting bifurcation Dynamic cluster separation Homoclinic orbits shrinking Local mean field analysis
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The model
Coupled HR system
3 2
2
0
j j j e i i j
j j j
j j j
x y ax bx z I x x
y c dx y
z r s x x z
SW Network:
N neurons
M added links
Parameters:
0.006
3.0125
r
I
Chaotic
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Transition to BS0.0002, 0.0028, 0.0044, 0.0052, 0.0066, 0.013
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Phase Trajectories
Spike
Adding
Bursting
Bifur...
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Phase Transitions
, , 1 ,2i i k i k i kt t T T T 1
jN i
jR e N
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Bursting Mechanism
Fast sub-system:
3 2extx y ax bx z I
2y c dx y
Slow Parameter:
z
Fold-Homoclinic(FHC) Fold-Hopf(FH)
Homoclinic Shrinking
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Local Mean Field
3 2i i i i i i i ext i ix y ax bx k x z I k x
• Fluctuate
• Close to 0
• Depend weakly on i
: i iKey k
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3 20
2
0
extx y ax bx x z I x
y c dx y
z r s x x z
Perturbed HR system
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
0
2
4
6
8
0.00 0.04 0.08 0.12 0.16-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
G
SP
B n
um
be
r
B
12
43
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Cluster separation
Valid + Robust
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Remarks
1 2ik
3 4ik
4
2/i
c
k
Easy
Hard
Easier5 SPB
6 SPB
FH
(Homogeneous)
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Conclusion
Transition to BS is investigated Two distinct types of transition Neuron degree is important Local mean field approximation
Large, Homogeneous HR network with many random links in between can show transition from spatiotemporal chaos to BS-states with FHC- and FH-bursting
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Thank you !