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

Our research interest

Statistical problems in mesoscopic chemical systems

Nonlinear Dynamics on complex networks

Complexity + Nonlinearity

A Neuron

Diversity: Morphology + Physiology

Oscillation

Spiking

Bursting

Chaos

Neuron Network

Human Brain: 1011 and 104

Complex

Network

Small-World

Scale-Free

Big Challenge : Dynamics + Functioning

An interesting phenomenon ...

Central Pattern Generator Small microcircuits Rhythmic motor commands Striking feature

Individual: irregular,chaotic bursts Ensemble: regular, rhythmic bursting

Mechanism ?

Related study

Chaos Regularization

N.F.Rulkov, PRL 86,183(2001)

Related study

Ordering Chaos by Random Shortcuts

F. Qi, Z.Hou, H. Xin, PRL 91, 064102 (2003)

Related Study

M. Wang, Z.Hou, H.Xin. ChemPhysChem 7 ， 579( 2006)

Ordering Bursting Chaos

Hindmarsh-Rose (HR) model system

Synchronization of Bursting System

Beyond complete synchronization

Spike Syn...

Burst Syn...

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

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

Transition to BS0.0002, 0.0028, 0.0044, 0.0052, 0.0066, 0.013

Phase Trajectories

Spike

Adding

Bursting

Bifur...

Phase Transitions

, , 1 ,2i i k i k i kt t T T T 1

jN i

jR e N

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

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

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

Cluster separation

Valid + Robust

Remarks

1 2ik

3 4ik

4

2/i

c

k

Easy

Hard

Easier5 SPB

6 SPB

FH

(Homogeneous)

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

Thank you !