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 http://www.agnld.uni-potsdam.de/~juergen/juergen.html Toolbox TOCSY Jkurths@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

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 Ecological Webs Protein interaction Neural Networks Genetic Networks Metabolic Networks

Technological Networks World-Wide Web Internet Power Grid

Transportation Networks Airport Networks Local Transportation Road Maps

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 or - minimum/ maximum intensities

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 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)

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 Dynamic Clusters

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

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

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

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

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

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

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

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)