Network of Networks – a Model for Complex Brain

Dynamics

Jürgen Kurths¹, C. Hilgetag³, G. Osipov², G. Zamora¹, L.

Zemanova¹, C. S. Zhou¹ ¹University Potsdam, Center for

Dynamics of Complex Systems (DYCOS), Germany

² University Nizhny Novgorod, Russia³ Jacobs University Bremen, Germany

http://www.agnld.uni-potsdam.de/~juergen/juergen.htmlToolbox TOCSYJkurths@gmx.de

Outline

• Introduction• Synchronization in complex

networks via hierarchical (clustered) transitions

• Application: structure vs. functionality in complex brain networks – network of networks

• Retrieval of direct vs. indirect connections in networks (inverse problem)

• Conclusions

Types of Synchronization in Complex Processes

- phase synchronization (1996)

phase difference bounded, a zero Lyapunov exponent becomes

negative (phase-coherent)- generalized

synchronization (1995)a positive Lyapunov

exponent becomes negative, amplitudes and

phases interrelated- complete synchronization

(1984)

Identification of Synchronization from Data

• Complete Synchronization: simple task

• Phase and Generalized Synchronization: highly non-trivial

• correlation and spectral analysis are often not sufficient (may even lead to artefacts);

• special techniques are necessary!

Applications in various fields

Lab experiments:• Electronic circuits (Parlitz, Lakshmanan, Dana...)• Plasma tubes (Rosa)• Driven or coupled lasers (Roy, Arecchi...)• Electrochemistry (Hudson, Gaspar, Parmananda...)• Controlling (Pisarchik, Belykh)• Convection (Maza...)Natural systems:• Cardio-respiratory system (Nature, 1998...)• Parkinson (PRL, 1998...)• Epilepsy (Lehnertz...)• Kidney (Mosekilde...)• Population dynamics (Blasius, Stone)• Cognition (PRE, 2005)• Climate (GRL, 2005)• Tennis (Palut)

Ensembles: Social Systems

• Rituals during pregnancy: man and woman isolated from community; both have to follow the same tabus (e.g. Lovedu, South Africa)

• Communities of consciousness and crises

• football (mexican wave: la ola, ...)• Rhythmic applause

Networks with complex topology

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

• Small-world networks (Watts, Strogatz, 1998)

• Scale-free networks (Barabasi, Albert, 1999)

• Applications: neuroscience, cell biology, epidemic spreading, internet, traffic, systems biology

• Many participants (nodes) with complex interactions and complex dynamics at the nodes

Networks with Complex Topology

Complex networks – a fashionable topic or a

useful one?

Hype: studies on complex networks

• Scale-free networks – thousands of examples (log-log plot with „some plateau“ SF, similar to dimension estimates in the 80ies…)

• Application to huge networks (number of different sexual partners in one country SF) – What to learn from this?

• Many promising approaches leading to useful applications, e.g. immunization problems, functioning of biological/physiological processes as protein networks, brain dynamics (Hugues Berry), colonies of thermites (Christian Jost) etc.

Biological Networks

Neural Networks

Genetic Networks

Protein interactionEcological Webs

Metabolic Networks

Technological Networks

World-Wide Web

Power Grid

Internet

Transportation Networks

Airport Networks

Road Maps

Local Transportation

Scale-freee 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, ...

Synchronization in such networks

• Synchronization properties strongly influenced by the network´s structure (Jost/Joy, Barahona/Pecora, Nishikawa/Lai, Timme et al., Hasler/Belykh(s), Boccaletti et al., etc.)

• Self-organized synchronized clusters can be formed (Jalan/Amritkar)

• Previous works mainly focused on the influence of the connection´s topology (assuming coupling strength uniform)

Universality in the synchronization of weighted

random networks

Our intention:

Include the influence of weighted coupling for complete synchronization

(Motter, Zhou, Kurths, 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)

Main results

Synchronizability universally determined by:

- mean degree K and

- heterogeneity of the intensities

- minimum/ maximum intensities

or

Hierarchical Organization of Synchronization in Complex

Networks

Homogeneous (constant number of connections in each node)

vs.

Scale-free networks

Zhou, Kurths: CHAOS 16, 015104 (2006)

Identical oscillators

Transition to synchronization

Clusters of synchronization

Transition to synchronization in complex networks

• Hierarchical transition to synchronization via clustering

• Hubs are the „engines“ in cluster formation AND they become synchronized first among themselves

Cat Cerebal Cortex

Connectivity

Scannell et al.,

Cereb. Cort., 1999

Modelling

• Intention:

Macroscopic Mesoscopic Modelling

Network of Networks

Hierarchical organization in complex brain networks

a) Connection matrix of the cortical network of the cat brain (anatomical)

b) Small world sub-network to model each node in the network (200 nodes each, FitzHugh Nagumo neuron models - excitable)

Network of networks

Phys Rev Lett 97 (2006), Physica D 224 (2006)

Density of connections between the four com-munities

Anatomic clusters

•Connections among the nodes: 2-3 … 35

•830 connections

•Mean degree: 15

Model for neuron i in area I

Fitz Hugh Nagumo model – excitable system

Transition to synchronized firing

g – coupling strength – control parameter

Network topology vs. Functional organization in networks

Weak-coupling dynamics non-trivial organization

relationship to underlying network topology

Functional vs. Structural Coupling

DynamicClusters

Intermediate Coupling

Intermediate Coupling:

3 main dynamical clusters

Strong Coupling

Inferring networks from EEG during cognition

Analysis and modeling of Complex Brain Networks

underlying Cognitive (sub) Processes Related to Reading, basing on single trial evoked-activity

time

Dynamical Network Approach

Correct words (Priester)Pseudowords (Priesper)

Conventional ERP Analysis

t1 t2

Initial brain states influence evoked activity

+ corr, significant

- corr, significant

non significant

On-going fluctuations =

single trial EEG

minus

average ERP

trial3

trial13

trial15

Identification of connections – How to avoid spurious ones?

Problem of multivariate statistics: distinguish direct and indirect interactions

Linear Processes

• Case: multivariate system of linear stochastic processes

• Concept of Graphical Models (R. Dahlhaus, Metrika 51, 157 (2000))

• Application of partial spectral coherence

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

Partial Phase Synchronization

Synchronization Matrix

with elements

Partial Phase Synchronization Index

Example

Example

• Three Rössler oscillators (chaotic regime) with additive noise; non-identical

• Only bidirectional coupling 1 – 2; 1 - 3

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 of cognition

• The identification of direct connections among nodes is non-trivial

Our papers on complex networks

Europhys. Lett. 69, 334 (2005)Phys. Rev. E 71, 016116 (2005)CHAOS 16, 015104 (2006)Physica D 224, 202 (2006)Physica A 361, 24 (2006)Phys. Rev. E 74, 016102 (2006)Phys: Rev. Lett. 96, 034101 (2006)Phys. Rev. Lett. 96, 164102 (2006)Phys. Rev. Lett. 96, 208103 (2006)Phys. Rev. Lett. 97, 238103 (2006)Phys. Rev. Lett. 98, 108101 (2007)Phys. Rev. E 76, 027203 (2007)Phys. Rev. E 76, 036211 (2007)Phys. Rev. E 76, 046204 (2007)New J. Physics 9, 178 (2007)