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E-raamat: Criticality in Neural Systems

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A conference was held during may 2012 in Bethesda, Maryland to highlight experimental findings predicted by the criticality theory in neurology, particularly the discovery of neuronal avalanches in 2001. From that gathering, 25 papers consider such topics as critical brain dynamics at large scale, a thermodynamic model of criticality in the cortex based on EEG/ECoG data, peak variability and optimal performance in cortical networks at criticality, cortical networks with lognormal synaptic connectivity and their implications in neuronal avalanches, and complex networks from social crises to neuronal avalanches. Annotation ©2014 Ringgold, Inc., Portland, OR (protoview.com)
List of Contributors xvii
1 Introduction 1(4)
Dietmar Plenz
Ernst Niebur
1.1 Criticality in Neural Systems
1(4)
2 Criticality in Cortex: Neuronal Avalanches and Coherence Potentials 5(38)
Dietmar Plenz
2.1 The Late Arrival of Critical Dynamics to the Study of Cortex Function
5(6)
2.1.1 Studying Critical Dynamics through Local Perturbations
7(1)
2.1.2 Principles in Cortex Design that Support Critical Neuronal Cascades
8(3)
2.2 Cortical Resting Activity Organizes as Neuronal Avalanches
11(9)
2.2.1 Unbiased Concatenation of Neuronal Activity into Spatiotemporal Patterns 11
2.2.2 The Power Law in Avalanche Sizes with Slope of -3/2
15(2)
2.2.3 Neuronal Avalanches are Specific to Superficial Layers of Cortex
17(1)
2.2.4 The Linking of Avalanche Size to Critical Branching
17(3)
2.3 Neuronal Avalanches: Cascades of Cascades
20(3)
2.4 The Statistics of Neuronal Avalanches and Earthquakes
23(1)
2.5 Neuronal Avalanches and Cortical Oscillations
23(5)
2.6 Neuronal Avalanches Optimize Numerous Network Functions
28(2)
2.7 The Coherence Potential: Threshold-Dependent Spread of Synchrony with High Fidelity
30(3)
2.8 The Functional Architecture of Neuronal Avalanches and Coherence Potentials
33(3)
Acknowledgement
36(1)
References
36(7)
3 Critical Brain Dynamics at Large Scale 43(24)
Dante R. Chialvo
3.1 Introduction
43(2)
3.1.1 If Criticality is the Solution, What is the Problem?
43(2)
3.2 What is Criticality Good for?
45(2)
3.2.1 Emergence
46(1)
3.2.2 Spontaneous Brain Activity is Complex
46(1)
3.2.3 Emergent Complexity is Always Critical
47(1)
3.3 Statistical Signatures of Critical Dynamics
47(8)
3.3.1 Hunting for Power Laws in Densities Functions
48(2)
3.3.2 Beyond Fitting: Variance and Correlation Scaling of BrainNoise
50(5)
3.3.2.1 Anomalous Scaling
51(1)
3.3.2.2 Correlation Length
52(3)
3.4 Beyond Averages: Spatiotemporal Brain Dynamics at Criticality
55(5)
3.4.1 fMRI as a Point Process
56(1)
3.4.2 A Phase Transition
57(2)
3.4.3 Variability and Criticality
59(1)
3.5 Consequences
60(3)
3.5.1 Connectivity versus Functional Collectivity
60(2)
3.5.2 Networks, Yet Another Circuit?
62(1)
3.5.3 River Beds, Floods, and Fuzzy Paths
62(1)
3.6 Summary and Outlook
63(1)
References
64(3)
4 The Dynamic Brain in Action: Coordinative Structures, Criticality, and Coordination Dynamics 67(38)
J.A. Scott Kelso
4.1 Introduction
67(1)
4.2 The Organization of Matter
68(4)
4.3 Setting the Context: A Window into Biological Coordination
72(2)
4.4 Beyond Analogy
74(1)
4.5 An Elementary Coordinative Structure: Bimanual Coordination
75(1)
4.6 Theoretical Modeling: Symmetry and Phase Transitions
76(4)
4.7 Predicted Signatures of Critical Phenomena in Biological Coordination
80(2)
4.7.1 Critical Slowing Down
80(1)
4.7.2 Enhancement of Fluctuations
81(1)
4.7.3 Critical Fluctuations
81(1)
4.8 Some Comments on Criticality, Timescales, and Related Aspects
82(2)
4.9 Symmetry Breaking and Metastability
84(3)
4.10 Nonequilibrium Phase Transitions in the Human Brain: MEG, EEG, and fMRI
87(1)
4.11 Neural Field Modeling of Multiple States and Phase Transitions in the Brain
88(1)
4.12 Transitions, Transients, Chimera, and Spatiotemporal Metastability
89(3)
4.13 The Middle Way: Mesoscopic Protectorates
92(2)
4.14 Concluding Remarks
94(1)
Acknowledgments
95(1)
References
96(9)
5 The Correlation of the Neuronal Long-Range Temporal Correlations, Avalanche Dynamics with the Behavioral Scaling Laws and Interindividual Variability 105(22)
Jaakko Matias Palva
Satu Palva
5.1 Introduction
105(1)
5.2 Criticality in the Nervous System: Behavioral and Physiological Evidence
106(3)
5.2.1 Human Task Performance Fluctuations Suggest Critical Dynamics
106(2)
5.2.2 Two Lines of Empirical Evidence for Critical-State Dynamics in Neuronal Systems
108(1)
5.3 Magneto- and Electroencephalography (M/EEG) as a Tool for Noninvasive Reconstruction of Human Cortical Dynamics
109(2)
5.4 Slow Neuronal Fluctuations: The Physiological Substrates of LRTC
111(4)
5.4.1 Infra-Slow Potential Fluctuations Reflect Endogenous Dynamics of Cortical Excitability
111(2)
5.4.2 Slow Fluctuations in Oscillation Amplitudes and Scalp Potentials are Correlated with Behavioral Dynamics
113(1)
5.4.3 Slow BOLD Signal Fluctuations in Resting-State Networks
114(1)
5.5 Neuronal Scaling Laws are Correlated with Interindividual Variability in Behavioral Dynamics
115(2)
5.6 Neuronal Avalanches, LRTC, and Oscillations: Enigmatic Coexistence?
117(2)
5.6.1 The Mechanistic Insights from Interindividual Variability in Scaling Laws
118(1)
5.7 Conclusions
119(1)
Acknowledgment,
120(1)
References
120(7)
6 The Turbulent Human Brain: An M HD Approach to the MEG 127(26)
Arnold J. Mandell
Stephen E. Robinson
Karen A. Selz
Constance Schrader
Torn Holroyd
Richard Coppola
6.1 Introduction
127(2)
6.2 Autonomous, Intermittent, Hierarchical Motions, from Brain Proteins Fluctuations to Emergent Magnetic Fields
129(1)
6.3 Magnetic Field Induction and Turbulence; Its Maintenance, Decay, and Modulation
130(9)
6.4 Localizing a Time-Varying Entropy Measure of Turbulence, Rank Vector Entropy (RVE) [ 35, 107], Using a Linearly Constrained Minimum Variance (LCMV) Beamformer Such as Synthetic Aperture Magnetometry (SAM) [ 25, 34], Yields State and Function-Related Localized Increases and Decreases in the RVE Estimate
139(3)
6.5 Potential Implications of the MHD Approach to MEG Magnetic Fields for Understanding the Mechanisms of Action and Clinical Applications of the Family of TMS (Transcranial Magnetic Stimulation) Human Brain Therapies
142(3)
6.6 Brief Summary of Findings
145(1)
References
145(8)
7 Thermodynamic Model of Criticality in the Cortex Based on EEG/ECoG Data 153(24)
Robert Kozma
Marko Puljic
Walter J. Freeman
7.1 Introduction
153(1)
7.2 Principles of Hierarchical Brain Models
154(4)
7.2.1 Freeman K-Models: Structure and Functions
154(1)
7.2.2 Basic Building Blocks of Neurodynamics
155(2)
7.2.3 Motivation of Neuropercolation Approach to Neurodynamics
157(1)
7.3 Mathematical Formulation of Neuropercolation
158(6)
7.3.1 Random Cellular Automata on a Lattice
158(1)
7.3.2 Update Rules
159(1)
7.3.3 Two-Dimensional Lattice with Rewiring
160(1)
7.3.4 Double-Layered Lattice
161(1)
7.3.5 Coupling Two Double-Layered Lattices
162(1)
7.3.6 Statistical Characterization of Critical Dynamics of Cellular Automata
163(1)
7.4 Critical Regimes of Coupled Hierarchical Lattices
164(3)
7.4.1 Dynamical Behavior of 2D Lattices with Rewiring
164(1)
7.4.2 Narrow Band Oscillations in Coupled Excitatory-Inhibitory Lattices
165(2)
7.5 BroadBand Chaotic Oscillations
167(6)
7.5.1 Dynamics of Two Double Arrays
167(3)
7.5.2 Intermittent Synchronization of Oscillations in Three Coupled Double Arrays
170(1)
7.5.3 Hebbian Learning Effects
170(3)
7.6 Conclusions
173(1)
References
174(3)
8 Neuronal Avalanches in the Human Brain 177(14)
Oren Shriki
Dietmar Plenz
8.1 Introduction
177(1)
8.2 Data and Cascade-Size Analysis
178(3)
8.3 Cascade-Size Distributions are Power Laws
181(1)
8.4 The Data are Captured by a Critical Branching Process
181(5)
8.5 Discussion
186(2)
8.6 Summary
188(1)
Acknowledgements
188(1)
References
188(3)
9 Critical Slowing and Perception 191(36)
Karl Friston
Michael Breakspear
Gustavo Deco
9.1 Introduction
191(2)
9.1.1 Perception and Neuronal Dynamics
191(1)
9.1.2 Overview
192(1)
9.2 Itinerant Dynamics
193(3)
9.2.1 Chaotic Itinerancy
193(1)
9.2.2 Heteroclinic Cycling
194(1)
9.2.3 Multistability and Switching
194(1)
9.2.4 Itinerancy, Stability, and Critical Slowing
195(1)
9.3 The Free Energy Principle
196(3)
9.3.1 Action and Perception
197(1)
9.3.2 The Maximum Entropy Principle and the Laplace Assumption
198(1)
9.3.3 Summary
199(1)
9.4 Neurobiological Implementation of Active Inference
199(6)
9.4.1 Perception and Predictive Coding
202(2)
9.4.2 Action
204(1)
9.4.3 Summary
204(1)
9.5 Self-Organized Instability
205(6)
9.5.1 Conditional Lyapunov Exponents and Generalized Synchrony
205(2)
9.5.2 Critical Slowing and Conditional Lyapunov Exponents
207(3)
9.5.3 Summary
210(1)
9.6 Birdsong, Attractors, and Critical Slowing
211(12)
9.6.1 A Synthetic Avian Brain
212(1)
9.6.2 Stimulus Generation and the Generative Model
213(1)
9.6.3 Perceptual Categorization
214(2)
9.6.4 Perceptual Instability and Switching
216(3)
9.6.5 Perception and Critical Slowing
219(2)
9.6.6 Summary
221(2)
9.7 Conclusion
223(1)
References
224(3)
10 Self-Organized Criticality in Neural Network Models 227(28)
Matthias Rybarsch
Stefan Bornholdt
10.1 Introduction
227(1)
10.2 Avalanche Dynamics in Neuronal Systems
228(3)
10.2.1 Experimental Results
228(1)
10.2.2 Existing Models
229(2)
10.3 Simple Models for Self-Organized Critical Adaptive Neural Networks
231(21)
10.3.1 A First Approach: Node Activity Locally Regulates Connectivity
231(4)
10.3.2 Correlation as a Criterion for Rewiring: Self-Organization on a Spin Lattice Neural Network Model
235(3)
10.3.3 Simplicity versus Biological Plausibility - and Possible Improvements
238(5)
10.3.3.1 Transition from Spins to Boolean Node States
238(1)
10.3.3.2 Model Definitions
239(1)
10.3.3.3 Exploration of Critical Properties - Activity-Dependent Criticality
240(2)
10.3.3.4 Extension of the Model: Thermal Noise
242(1)
10.3.4 Self-Organization on the Boolean State Model
243(6)
10.3.4.1 Model Definitions
244(1)
10.3.4.2 Rewiring Algorithm
245(2)
10.3.4.3 Observations
247(2)
10.3.5 Response to External Perturbations
249(3)
10.4 Conclusion
252(1)
Acknowledgments
252(1)
References
252(3)
11 Single Neuron Response Fluctuations: A Self-Organized Criticality Point of View 255(18)
Asaf Gal
Shimon Marom
11.1 Neuronal Excitability
255(2)
11.2 Experimental Observations on Excitability Dynamics
257(4)
11.3 Self-Organized Criticality Interpretation
261(2)
11.4 Adaptive Rates and Contact Processes
263(2)
11.5 Concluding Remarks
265(4)
References
269(4)
12 Activity Dependent Model for Neuronal Avalanches 273(20)
Lucilla de Arcangelis
Hans J. Herrmann
12.1 The Model
274(3)
12.1.1 Plastic Adaptation
276(1)
12.2 Neuronal Avalanches in Spontaneous Activity
277(3)
12.2.1 Power Spectra
278(2)
12.3 Learning
280(3)
12.4 Temporal Organization of Neuronal Avalanches
283(5)
12.5 Conclusions
288(1)
References
289(4)
13 The Neuronal Network Oscillation as a Critical Phenomenon 293(26)
Richard Hardstone
Huibert D. Mansvelder
Klaus Linkenkaer-Hansen
13.1 Introduction
293(1)
13.2 Properties of Scale-Free Time Series
294(8)
13.2.1 Self-Affinity
294(4)
13.2.2 Stationary and Nonstationary Processes
298(1)
13.2.3 Scaling of an Uncorrelated Stationary Process
298(2)
13.2.4 Scaling of Correlated and Anticorrelated Signals
300(2)
13.3 The Detrended Fluctuation Analysis (DFA)
302(2)
13.4 DFA Applied to Neuronal Oscillations
304(1)
13.4.1 Preprocessing of Signals
304(1)
13.4.2 Filter Design
305(1)
13.4.3 Extract the Amplitude Envelope and Perform DFA
305(1)
13.4.4 Determining the Temporal Integration Effect of the Filter
305(1)
13.5 Insights from the Application of DFA to Neuronal Oscillations
305(5)
13.5.1 DFA as a Biomarker of Neurophysiological Disorder
309(1)
13.6 Scaling Behavior of Oscillations: a Sign of Criticality?
310(6)
13.6.1 CRitical OScillations Model (CROS)
310(1)
13.6.2 CROS Produces Neuronal Avalanches with Balanced Ex/In Connectivity
311(2)
13.6.3 CROS Produces Oscillations with LRTC When there are Neuronal Avalanches
313(2)
13.6.4 Multilevel Criticality: A New Class of Dynamical Systems?
315(1)
Acknowledgment
316(1)
References
316(3)
14 Critical Exponents, Universality Class, and Thermodynamic "Temperature" of the Brain 319(16)
Shan Yu
Hongdian Yang
Oren Shriki
Dietmar Plenz
14.1 Introduction
319(1)
14.2 Thermodynamic Quantities at the Critical Point and Their Neuronal Interpretations
320(4)
14.3 Finite-Size Scaling
324(1)
14.4 Studying the Thermodynamics Properties of Neuronal Avalanches at Different Scales
325(5)
14.5 What Could be the "Temperature" for the Brain?
330(1)
Acknowledgment
331(1)
References
331(4)
15 Peak Variability and Optimal Performance in Cortical Networks at Criticality 335(12)
Hongdian Yang
Woodrow L. Shew
Rajarshy Roy
Dietmar Plenz
15.1 Introduction
335(1)
15.2 Fluctuations Are Highest Near Criticality
336(2)
15.3 Variability of Spatial Activity Patterns
338(1)
15.4 Variability of Phase Synchrony
339(3)
15.5 High Variability, but Not Random
342(1)
15.6 Functional Implications of High Entropy of Ongoing Cortex Dynamics
343(1)
References
344(3)
16 Criticality at Work: How Do Critical Networks Respond to Stimuli? 347(18)
Mauro Copelli
16.1 Introduction
347(4)
16.1.1 Phase Transition in a Simple Model
347(3)
16.1.2 What is the Connection with Neuronal Avalanches?
350(1)
16.1.3 What if Separation of Time Scales is Absent?
351(1)
16.2 Responding to Stimuli
351(8)
16.2.1 What Theory Predicts
352(4)
16.2.1.1 Self-Regulated Amplification via Excitable Waves
352(2)
16.2.1.2 Enhancement of Dynamic Range
354(1)
16.2.1.3 Nonlinear Collective Response and Maximal Dynamic Range at Criticality
355(1)
16.2.2 What Data Reveals
356(11)
16.2.2.1 Nonlinear Response Functions in Sensory Systems
356(1)
16.2.2.2 Enhanced Dynamic Range
356(1)
16.2.2.3 Scaling In Brain Dynamics
357(2)
16.3 Concluding Remarks
359(2)
Acknowledgements
361(1)
References
361(4)
17 Critical Dynamics in Complex Networks 365(28)
Daniel B. Larremore
Woodrow L. Shew
Juan G. Restrepo
17.1 Introduction: Critical Branching Processes
365(2)
17.2 Description and Properties of Networks
367(6)
17.2.1 Network Representation by an Adjacency Matrix
368(1)
17.2.2 Node Degrees
368(1)
17.2.3 Degree Distribution
369(1)
17.2.4 Degree Correlations
370(2)
17.2.5 Largest Eigenvalue and the Corresponding Eigenvector
372(1)
17.3 Branching Processes in Complex Networks
373(14)
17.3.1 Subcritical Regime
378(3)
17.3.2 Supercritical Regime
381(2)
17.3.3 Critical Regime
383(4)
17.4 Discussion
387(3)
References
390(3)
18 Mechanisms of Self-Organized Criticality in Adaptive Networks 393(10)
Thilo Gross
Anne-Ly Do
Felix Droste
Christian Meisel
18.1 Introduction
393(1)
18.2 Basic Considerations
393(2)
18.3 A Toy Model
395(2)
18.4 Mechanisms of Self-Organization
397(2)
18.5 Implications for Information Processing
399(1)
18.6 Discussion
400(1)
References
401(2)
19 Cortical Networks with Lognormal Synaptic Connectivity and Their Implications in Neuronal Avalanches 403(14)
Tomoki Fukai
Vladimir Klinshov
Jun-nosuke Teramae
19.1 Introduction
403(1)
19.2 Critical Dynamics in Neuronal Wiring Development
404(1)
19.3 Stochastic Resonance by Highly Inhomogeneous Synaptic Weights on Spike Neurons
405(4)
19.4 SSWD Recurrent Networks Generate Optimal Intrinsic Noise
409(1)
19.5 Incorporation of Local Clustering Structure
410(2)
19.6 Emergence of Bistable States in the Clustered Network
412(1)
19.7 Possible Implications of SSWD Networks for Neuronal Avalanches
413(1)
19.8 Summary
414(1)
Acknowledgment
414(1)
References
415(2)
20 Theoretical Neuroscience of Self-Organized Criticality: From Formal Approaches to Realistic Models 417(20)
Anna Levina
J. Michael Herrmann
Theo Geisel
20.1 Introduction
417(1)
20.2 The Eurich Model of Criticality in Neural Networks
417(3)
20.2.1 Model Description
418(1)
20.2.2 Simulations and Analysis
419(1)
20.3 LHG Model: Dynamic Synapses Control Criticality
420(9)
20.3.1 Model Description
420(3)
20.3.2 Mean-Field Approximation
423(1)
20.3.3 Toward a Realistic Model: Network Structure, Leakage, and Inhibition
424(3)
20.3.4 Synaptic Facilitation
427(2)
20.4 Criticality by Homeostatic Plasticity
429(4)
20.4.1 Branching Processes
429(1)
20.4.2 Self-Organization by Long-Term Plasticity
430(1)
20.4.3 Effects of Spike-Time-Dependent Plasticity and Network Structure
431(2)
20.5 Conclusion
433(1)
Acknowledgment
434(1)
References
434(3)
21 Nonconservative Neuronal Networks During Up-States Self-Organize Near Critical Points 437(28)
Stefan Mihalas
Daniel Millman
Ramakrishnan Iyer
Alfredo Kirkwood
Ernst Niebur
21.1 Introduction
437(2)
21.2 Model
439(5)
21.2.1 Analytical Solution
440(1)
21.2.2 Numerical Evolution of the Fokker-Planck Equation
441(1)
21.2.3 Fixed-Point Analysis
442(2)
21.3 Simulations
444(10)
21.3.1 Up- and Down-States
444(2)
21.3.1 Up-/Down-State Transitions
446(2)
21.3.2 Up-States are Critical; Down-States are Subcritical
448(1)
21.3.3 More Biologically Realistic Networks
449(3)
21.3.3.1 Small-World Connectivity
449(1)
21.3.3.2 NMDA and Inhibition
450(2)
21.3.4 Robustness of Results
452(2)
21.4 Heterogeneous Synapses
454(6)
21.4.1 Influence of Synaptic Weight Distributions
454(1)
21.4.2 Voltage Distributions for Heterogeneous Synaptic Input
455(1)
21.4.3 Results for Realistic Synaptic Distributions in the Absence of Recurrence and STSD
456(2)
21.4.4 Heterogeneous Synaptic Distributions in the Presence of Synaptic Depression
458(2)
21.5 Conclusion
460(1)
Acknowledgment
460(1)
References
460(5)
22 Self-Organized Criticality and Near-Criticality in Neural Networks 465(20)
Jack D. Cowan
Jeremy Neuman
Wim van Drongelen
22.1 Introduction
465(3)
22.1.1 Neural Network Dynamics
466(2)
22.1.2 Stochastic Effects Near a Critical Point
468(1)
22.2 A Neural Network Exhibiting Self-Organized Criticality
468(4)
22.2.1 A Simulation of the Combined Mean-Field Equations
470(1)
22.2.2 A Simulation of the Combined Markov Processes
471(1)
22.3 Excitatory and Inhibitory Neural Network Dynamics
472(3)
22.3.1 Equilibria of the Mean-Field Wilson-Cowan Equations
473(2)
22.4 An E-I Neural Network Exhibiting Self-Organized Near-Criticality
475(6)
22.4.1 Modifiable Synapses
475(2)
22.4.2 A Simulation of the Combined Mean-Field E/I equations
477(1)
22.4.3 Balanced Amplification in E/I Patches
477(2)
22.4.4 Analysis and Simulation of the Combined E/I Markov Processes
479(2)
22.5 Discussion
481(1)
Acknowledgements
482(1)
References
482(3)
23 Neural Dynamics: Criticality, Cooperation, Avalanches, and Entrainment between Complex Networks 485(24)
Paolo Grigolini
Marzieh Zare
Adam Svenkeson
Bruce J. West
23.1 Introduction
485(2)
23.2 Decision-Making Model (DMM) at Criticality
487(6)
23.2.1 Intermittency
489(3)
23.2.2 Response to Perturbation
492(1)
23.3 Neural Dynamics
493(8)
23.3.1 Mittag-Leffler Function Model Cooperation
494(2)
23.3.2 Cooperation Effort in a Fire-and-Integrate Neural Model
496(5)
23.4 Avalanches and Entrainment
501(3)
23.5 Concluding Remarks
504(1)
References
505(4)
24 Complex Networks: From Social Crises to Neuronal Avalanches 509(16)
Bruce J. West
Malgorzata Turalska
Paolo Grigolini
24.1 Introduction
509(1)
24.2 The Decision-Making Model (DMM)
510(4)
24.3 Topological Complexity
514(3)
24.4 Temporal Complexity
517(1)
24.5 Inflexible Minorities
518(3)
24.6 Conclusions
521(1)
References
522(3)
25 The Dynamics of Neuromodulation 525(14)
Gerhard Werner
Bernhard J. Mitterauer
25.1 Introduction
525(1)
25.2 Background
525(4)
25.2.1 Gap Junctions and Neuroglia
525(2)
25.2.2 Brain Cell Microenvironment (Extracellular Fluid)
527(1)
25.2.3 Neuromodulatory Processes
528(1)
25.3 Discussion and Conclusions
529(3)
25.4 A Final Thought
532(1)
25.5 Summary
532(1)
References
532(7)
Color Plates 539(20)
Index 559
DIETMAR PLENZ is Chief of the Section on Critical Brain Dynamics in the Intramural Research Program at the National Institute of Mental Health. He received his Ph.D. in 1993 at the Max-Planck Institute of Biological Cybernetics and the University Tuebingen. Dr. Plenz joined the NIMH as an Investigator in 1999. He pioneered the development of in vitro networks to study and identify the emergence of neuronal avalanches in the brain.

ERNST NIEBUR is Professor of Neuroscience and of Brain and Psychological Sciences at Johns Hopkins University in Baltimore, USA. He holds degrees in Physics from the Universities of Dortmund, Germany and Lausanne, Switzerland, and a postgraduate certificate in Artificial Intelligence from the Swiss Federal Institute of Technology (EPFL). Prof. Niebur has authored more than 100 scientific articles in physics and computational neuroscience.

HEINZ GEORG SCHUSTER is Professor (em.) of Theoretical Physics at the University of Kiel in Germany. At the beginning of his academic career, he was appointed Professor at the University of Frankfurt am Main in Germany. He was a visiting professor at the Weizmann-Institute of Science in Israel and at the California Institute of Technology in Pasadena, USA. He is author and editor of research monographs and topical handbooks on chaos theory, nonlinear dynamics and neural networks, but also on popular science books, and editor of a Wiley series on Nonlinear Physics and Complexity.
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