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Synchronization and Complex Networks:
Are such Theories Useful for Neuroscience?
Jürgen Kurths¹ ², N. Wessel¹, G. Zamora¹, and
C. S. Zhou³
¹Potsdam Institute for Climate Impact Research, RD
Transdisciplinary Concepts and Methods and
Institute of Physics, Humboldt University, Berlin,
Germany
² King´s College, University of Aberdeen, Scotland
³ Baptist University, Hong Kong
http://www.pik-potsdam.de/members/kurths/
[email protected]
Outline
• Introduction
• Synchronization of coupled complex systems and
applications
• Synchronization in complex networks
• Structure vs. functionality in complex brain
networks – network of networks
• How to determine direct couplings?
• Conclusions
Nonlinear Sciences
Start in 1665 by Christiaan Huygens:
Discovery of phase synchronization,
called sympathy
Huygens´-Experiment
Modern Example: Mechanics
London´s Millenium Bridge
- pedestrian bridge
- 325 m steel bridge over the Themse
- Connects city near St. Paul´s Cathedral with Tate
Modern Gallery
Big opening event in 2000 -- movie
Bridge Opening
• Unstable modes always there
• Mostly only in vertical direction considered
• Here: extremely strong unstable lateral
Mode – If there are sufficient many people
on the bridge we are beyond a threshold and
synchronization sets in
(Kuramoto-Synchronizations-Transition,
book of Kuramoto in 1984)
Supplemental tuned mass dampers to reduce the oscillations
GERB Schwingungsisolierungen GmbH, Berlin/Essen
Examples: Sociology, Biology, Acoustics, Mechanics
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Hand clapping (common rhythm)
Ensemble of doves (wings in synchrony)
Mexican wave
Organ pipes standing side by side – quenching or
playing in unison (Lord Rayleigh, 19th century)
• Fireflies in south east Asia (Kämpfer, 17th century)
• Crickets and frogs in South India
Types of Synchronization in Chaotic Processes
• phase synchronization
phase difference bounded, but amplitudes may remain
uncorrelated (Rosenblum, Pikovsky, Kurths 1996)
• generalized synchronization
a positive Lyapunov exponent becomes negative,
amplitudes and phases interrelated (Rulkov, Sushchik,
Tsimring, Abarbanel 1995)
• complete synchronization (Fujisaka, Yamada 1983)
Phase Definitions
Analytic Signal Representation (Hilbert Transform)
Direct phase
Phase from Poincare´ plot
(Rosenblum, Pikovsky, Kurths, Phys. Rev. Lett., 1996)
(Phase) Synchronization – good or bad???
Context-dependent
Application:
Cardiovascular System
Cardio-respiratory System
Analysis technique: Synchrogram
Schäfer, Rosenblum, Abel, Kurths: Nature, 1998
Cardiorespiratory Synchronisation
NREM
REM
Synchrogram
5:1 synchronization during NREM
Testing the foetal–maternal
heart rate synchronization
via model-based analyses
Riedl M, van Leeuwen P, Suhrbier A, Malberg H, Grönemeyer D, Kurths J, Wessel N. Testing the fetal maternal heart rate
synchronisation via model based analysis. Philos Transact A Math Phys Eng Sci. 367, 1407 (2009)
Distribution of the synchronization epochs (SE)
over the maternal beat phases in the original and
surrogate data with respect to the n:m
combinations 3:2 (top), 4:3 (middle) and 5:3
(bottom) in the different respiratory conditions.
For the original data, the number of SE found is
given at the top left of each graph. As there were
20 surrogate data sets for each original, the
number of SE found in the surrogate data was
divided by 20 for comparability. The arrows
indicate clear phase preferences. p-values are
given for histograms containing at least 6 SE.
(pre, post: data sets of spontaneous breathing
prior to and following controlled breathing.)
Special test statistics: twin surrogates
van Leeuwen, Romano, Thiel, Kurths, PNAS
(2009)
Networks with Complex
Topology
Basic Model in Statistical Physics and
Nonlinear Sciences for ensembles
• Traditional Approach:
Regular chain or lattice of coupled oscillators;
global or nearest neighbour coupling
• Many natural and engineering systems more
complex (biology, transportation, power grids
etc.)  networks with complex topology
Regular Networks – rings, lattices
Networks
with Complex Topology
Networks with complex topology
• Random graphs/networks (Erdös, Renyi, 1959)
• Small-world networks (Watts, Strogatz, 1998
F. Karinthy hungarian writer – SW hypothesis, 1929)
• Scale-free networks (Barabasi, Albert, 1999;
D. de Solla Price – number of citations – heavy tail
distribution, 1965)
Types of complex networks
fraction of nodes in the network having at
least k connections to other nodes have
a power law scaling
Warning: do not forget the log-log-lies!
Small-world Networks
Nearest neighbour
connections
Regular

Nearest neighbour and a
few long-range
connections
Complex Topology
Basic Characteristics
• Path length between nodes i and j:
- mean path length L
• Degree connectivity – number of connections
node i has to all others
- mean degree K
- degree distribution P(k)
Scale-free - power law
Random - Poisson distribution
Basic Characteristics
Clustering Coefficient C:
How many of the aquaintanences (j, m) of a given person i,
on average, are aquainted with each other
Local clustering cofficient:
Clustering Coefficient
Properties
• Regular networks
large L and medium C
• Random networks (ER)
rather small L and small C
• Small-world (SW)
small L and large C
• Scale-free (SF)
small L and C varies from cases
Basic Networks
Betweenness Centrality B
Number of shortest paths that connect nodes j and k
Number of shortest paths that connect nodes i and j AND
path through node i
Local betweenness of node i
(local and global aspects included!)
Betweenness Centrality B = <
>
Useful approaches with networks
• Immunization problems (spreading of
diseases)
• Functioning of biological/physiological
processes as protein networks, brain
dynamics, colonies of thermites and of social
networks as network of vehicle traffic in a
region, air traffic, or opinion formation etc.
Scale-freee-like Networks
Network resiliance
• Highly robust against random failure of a
node
• Highly vulnerable to deliberate attacks on
hubs
Applications
• Immunization in networks of computers,
humans, ...
Universality in the synchronization of
weighted random networks
Our intention:
What is the influence of weighted coupling for
complete synchronization
Motter, Zhou, Kurths: Phys. Rev. E 71, 016116 (2005)
Europhys. Lett. 69, 334 (2005)
Phys. Rev. Lett. 96, 034101 (2006)
Weighted Network of N Identical
Oscillators
F – dynamics of each oscillator
H – output function
G – coupling matrix combining adjacency A and weight W
- intensity of node i (includes topology and weights)
General Condition for Synchronizability
Stability of synchronized state
N eigenmodes of
ith eigenvalue of G
Main results
Synchronizability universally determined by:
- mean degree K and
- heterogeneity of the intensities
or
- minimum/ maximum intensities
Transition to synchronization in complex
networks
• Hierarchical transition to synchronization via
clustering (e.g. non-identical elements, noise)
• Hubs are the „engines“ in cluster formation
AND they become synchronized first among
themselves
Clusters of
synchronization
Application
Neuroscience
System Brain: Cat Cerebal Cortex
Connectivity
Scannell et al.,
Cereb. Cort., 1999
Modelling
• Intention:
Macroscopic  Mesoscopic Modelling
Network of Networks
Density of connections
between the four communities
•Connections among
the nodes: 2 … 35
•830 connections
•Mean degree: 15
Zamora, Zhou,
Kurths,
CHAOS 2009
Major features of organization of cortical
connectivity
• Large density of connections (many direct
connections or very short paths – fast
processing)
• Clustered organization into functional communities
• Highly connected hubs (integration of
multisensory information)
Model for neuron i in area I
FitzHugh Nagumo model
Transition to synchronized firing
g – coupling strength – control parameter
Possible interpretation: functioning of the brain near a
2nd order phase transition
Functional Organization vs. Structural (anatomical) Coupling
Formation of dynamical clusters
Intermediate Coupling
Intermediate Coupling:
3 main dynamical clusters
Strong Coupling
Network topology (anatomy) vs.
Functional organization in networks
• Weak-coupling dynamics  non-trivial
organization
• Relationship to the underlying network
topology
Cognitive Processes
• Processing of visual stimuli
• EEG-measurements (500 Hz, 30 channels)
• Multivariate synchronization analysis to
identify clusters
Kanizsa Figures
Challenges
• ECONS: Evolving COmplex NetworkS
connectivity is time dependent – strength of
connections varies, nodes can be born or die
out
• Directed Networks
directionality of the connections - not equal
in both directions in general
Identification of connections – How to avoid
spurious ones?
Problem of multivariate statistics: distinguish
direct and indirect interactions
Extension to Phase Synchronization Analysis
• Bivariate phase synchronization index (n:m
synchronization)
• Measures sharpness of peak in histogram of
Schelter, Dahlhaus, Timmer, Kurths: Phys. Rev. Lett. 2006
Summary
Take home messages:
• There are rich synchronization phenomena in
complex networks (self-organized structure
formation) – hierarchical transitions
• This approach seems to be promising for
understanding some aspects in neuroscience
and many others (climate, systems biology)
• The identification of direct connections
among nodes is non-trivial
Our papers on complex networks
Europhys. Lett. 69, 334 (2005) Phys. Rev. Lett. 98, 108101 (2007)
Phys. Rev. E 71, 016116 (2005) Phys. Rev. E 76, 027203 (2007)
CHAOS 16, 015104 (2006)
New J. Physics 9, 178 (2007)
Physica D 224, 202 (2006)
Phys. Rev. E 77, 016106 (2008)
Physica A 361, 24 (2006)
Phys. Rev. E 77, 026205 (2008)
Phys. Rev. E 74, 016102 (2006) Phys. Rev. E 77, 027101 (2008)
Phys: Rev. Lett. 96, 034101 (2006) CHAOS 18, 023102 (2008)
Phys. Rev. Lett. 96, 164102 (2006) J. Phys. A 41, 224006 (2008)
Phys. Rev. Lett. 96, 208103 (2006) Phys. Reports 469, 93 (2008)
Phys. Rev. Lett. 97, 238103 (2006) Europhys. Lett. 85, 28002 (2009)
Phys. Rev. E 76, 036211 (2007) CHAOS 19, 013105 (2009)
Phys. Rev. E 76, 046204 (2007)
Physica A 388, 2987 (2009)
Europ. J. Phys. B 69, 45 (2009)
PNAS (in press) (2009)