1. 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, Russia
³ Jacobs University Bremen, Germany
Toolbox TOCSY

2. 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

3. 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)

4. 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!

5. 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, )
Parkinson (PRL, )
Epilepsy (Lehnertz...)
Kidney (Mosekilde...)
Population dynamics (Blasius, Stone)
Cognition (PRE, 2005)
Climate (GRL, 2005)
Tennis (Palut)

6. 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

7. 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

8. Complex networks – a fashionable topic or a useful one?

9. 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.

10. Biological Networks Ecological Webs Protein interaction
Neural Networks
Genetic Networks
Metabolic Networks

11. Technological Networks
World-Wide Web
Internet
Power Grid

12. Transportation Networks
Airport Networks
Local Transportation
Road Maps

13. 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, ...

14. 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)

15. 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, , 2006)

16. 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)

17. Main results Synchronizability universally determined by:
- mean degree K and
- heterogeneity of the intensities or
- minimum/ maximum intensities

18. Hierarchical Organization of Synchronization in Complex Networks
Homogeneous (constant number of connections in each node)
vs.
Scale-free networks
Zhou, Kurths: CHAOS 16, (2006)

20. Identical oscillators

21. Transition to synchronization

23. Clusters of synchronization

24. 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

25. Cat Cerebal Cortex

26. Connectivity
Scannell et al.,
Cereb. Cort., 1999

27. Modelling
Intention:
Macroscopic  Mesoscopic Modelling

28. Network of Networks

29. Hierarchical organization in complex brain networks
Connection matrix of the cortical network of the cat brain (anatomical)
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)

30. Density of connections between the four com-munities
Anatomic clusters
Connections among the nodes: 2-3 … 35
830 connections
Mean degree: 15

31. Model for neuron i in area I
Fitz Hugh Nagumo model – excitable system

32. Transition to synchronized firing
g – coupling strength – control parameter

33. Network topology vs. Functional organization in networks
Weak-coupling dynamics  non-trivial organization
 relationship to underlying network topology

34. Functional vs. Structural Coupling
Dynamic
Clusters

35. Intermediate Coupling
3 main dynamical clusters

36. Strong Coupling

37. 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 t1 t2
Correct words (Priester)
Pseudowords (Priesper)
time
Conventional ERP Analysis
Dynamical Network Approach

38. Initial brain states influence evoked activity
+ corr, significant
trial3
non significant
- corr, significant
trial13
On-going
fluctuations =
single trial EEG
minus
average ERP
trial15

39. Identification of connections – How to avoid spurious ones?
Problem of multivariate statistics: distinguish direct and indirect interactions

40. Case: multivariate system of linear stochastic processes
Linear Processes
Case: multivariate system of linear stochastic processes
Concept of Graphical Models (R. Dahlhaus, Metrika 51, 157 (2000))
Application of partial spectral coherence

41. 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

42. Partial Phase Synchronization
Synchronization Matrix
with elements
Partial Phase Synchronization Index

43. Example

44. Only bidirectional coupling 1 – 2; 1 - 3
Example
Three Rössler oscillators (chaotic regime) with additive noise; non-identical
Only bidirectional coupling 1 – 2; 1 - 3

46. Take home messages: Summary
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

47. Our papers on complex networks
Europhys. Lett. 69, 334 (2005)
Phys. Rev. E 71, (2005)
CHAOS 16, (2006)
Physica D 224, 202 (2006)
Physica A 361, 24 (2006)
Phys. Rev. E 74, (2006)
Phys: Rev. Lett. 96, (2006)
Phys. Rev. Lett. 96, (2006)
Phys. Rev. Lett. 96, (2006)
Phys. Rev. Lett. 97, (2006)
Phys. Rev. Lett. 98, (2007)
Phys. Rev. E 76, (2007)
Phys. Rev. E 76, (2007)
Phys. Rev. E 76, (2007)
New J. Physics 9, 178 (2007)