network of networks – a model for complex brain dynamics jürgen kurths¹, c. hilgetag³, g....
TRANSCRIPT
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 [email protected]
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)