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Advances in the field of signal processing, nonlinear dynamics, statistics, and optimization theory, combined with marked improvement in instrumentation and development of computer systems, have made it possible to apply the power of mathematics to the task of understanding the human brain. This veritable revolution has already resulted in widespread availability of high resolution neuroimaging devices in clinical as well as research settings. Breakthroughs in functional imaging are not far behind. Mathematical techniques developed for the study of complex nonlinear systems and chaos are already being used to explore the complex nonlinear dynamics of human brain physiology. Global optimization is being applied to data mining expeditions in an effort to find knowledge in the vast amount of information being generated by neuroimaging and neurophysiological investigations. These breakthroughs in the ability to obtain, store and analyze large datasets offer, for the first time, exciting opportunities to explore the mechanisms underlying normal brain function as well as the affects of diseases such as epilepsy, sleep disorders, movement disorders, and cognitive disorders that affect millions of people every year. Application of these powerful tools to the study of the human brain requires, by necessity, collaboration among scientists, engineers, neurobiologists and clinicians. Each discipline brings to the table unique knowledge, unique approaches to problem solving, and a unique language.
1 Applications of Global Optimization and Dynamical Systems to Prediction of Epileptic Seizures.- 1.1 Introduction.- 1.2 Background.- 1.3 Method and Results.- 1.4 Quadratic 0–1 Programming.- 1.5 Conclusions and Discussion.- 2 Nonlinear Neurodynamical Features in an Animal Model of Generalized Epilepsy.- 2.1 Introduction.- 2.2 Materials and Methods.- 2.3 Results.- 2.4 Discussion.- 3 Optimization Techniques for Independent Component Analysis with Applications to EEG Data.- 3.1 Introduction.- 3.2 Extraction via maximization of the absolute value of the cumulants.- 3.3 A generalization of the fixed point algorithm.- 3.4 Examples and remarks for practical implementation.- 3.5 Combining second and fourth order statistics.- 3.6 Experiment with EEG data for detection of eye movements.- 3.7 Results.- 3.8 Conclusion.- 4 On a New Quantization in Complex Systems.- 4.1 Introduction.- 4.2 A Nonlocal Correlation between Sequences.- 4.3 The Formations of Integer Relations: Processes with Integer Particles.- 4.4 Geometrization of the Integer Relations Formations.- 4.5 Quantization of States of a Complex System as the Formations of Integer Relations.- 4.6 Conclusions.- 5 The Seizure Prediction Characteristic.- 5.1 Introduction.- 5.2 The Seizure Prediction Characteristic.- 5.3 Conclusion.- 6 Seizure Prediction Methods.- 6.1 Introduction.- 6.2 EEG data and patients characteristics.- 6.3 Seizure Prediction Methods.- 6.4 Calculation of the seizure prediction characteristic.- 6.5 Results and Discussion.- 6.6 Conclusion.- 7 Controlling Neurological Disease at the Edge of Instability.- 7.1 Introduction.- 7.2 Mathematical background and outline.- 7.3 Changes in neural synchrony.- 7.4 Multistability in delayed recurrent loops.- 7.5 Noise-induced switching between attractors.- 7.6 On-off intermittency: Parametric, or state-dependent, noise.- 7.7 Self-organized criticality.- 7.8 Discussion.- 8 Anatomical Connectivity in the Central Nervous System Revealed by Diffusion Tensor Magnetic Resonance Imaging (DT-MRI).- 8.1 Fundamentals of Diffusion Weighted Magnetic Resonance Imaging.- 8.2 Diffusion Tensor Magnetic Resonance Imaging.- 8.3 Scalar Measures Derived From DT-MRI.- 8.4 Fiber-Tract Mapping in Neural Tissue.- 8.5 Problems of DT-MRI Based Fiber Tracking.- Appendix: Diffusion Tensor and Displacement Profile.- 9 Epileptic Seizure Detection Using Dynamical Preprocessing (STLmax).- 9.1 Introduction.- 9.2 Nonlinear Dynamic Preprocessing.- 9.3 Detection Methods.- 9.4 Results and Discussion.- 9.5 Conclusion and Directions for Future Work.- 10 Role Of The Dorsocentral Striatum In Contralateral Neglect And Recovery From Neglect In Rats.- 10.1. The Rodent Model of Neglect and Recovery.- 10.2 Role of the Dorsocentral Striatum in Neglect.- 10.3 Role of induced plasticity in DCS in Recovery from AGm-induced Neglect.- 10.4 Clinical Significance.- 11 Binary and Sparse Checkerboard Visual Stimuli in Multiple Sclerosis Patients.- 11.1 Methods.- 11.2 Data Analysis.- 11.3 Results.- 11.4 Discussion.- 12 Spatiotemporal Transitions in Temporal Lobe Epilepsy.- 12.1 Introduction.- 12.2 Nonlinear Dynamical and Statistical Measures.- 12.3 Results.- 12.4 Discussion.- 13 Nonlinear Dynamical and Statistical Approaches to Investigate Dynamical Transitions Before Epileptic Seizures.- 13.1 Introduction.- 13.2 Measures and Methods.- 13.3 Results.- 13.4 Discussions.- 14 Testing Whether a Prediction Scheme is Better than Guess.- 14.1 Introduction and Modeling.- 14.2 Naive Prediction Schemes.- 14.3 Testing the Hypothesis that the New Method is Better.- 14.4 An Example.- 14.5 Power Analysis.- 14.6 Concluding Remarks.- Appendix: Optimal Prediction Strategy for the Next Event.