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/juergen.kurths@pik-potsdam.de

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

• 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 SynchronisationNREM 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

• Random graphs/networks (Erdös, Renyi, 1959)

• Small-world networks (Watts, Strogatz, 1998F. Karinthy hungarian writer – SW hypothesis, 1929)

• Scale-free networks (Barabasi, Albert, 1999;D. de Solla Price – number of citations – heavy tail distribution, 1965)

Networks with Complex Topology

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 and a few long-range

connections

Nearest neighbourconnections

Regular 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 lawRandom - 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 networkslarge 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

- minimum/ maximum intensities

or

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

ApplicationNeuroscience

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 com-munities

•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 com- munities

• 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)