This book describes some advanced mathematical signal processing techniques applied to the estimation of the cortical connectivity in humans from non-invasive electroencephalographic (EEG) recordings. Some might think that mathematics is not the proper tool for a full comprehension of the brain functions, but often this is not the case. In the last ten years, many different brain-imaging devices have conveyed a lot of information about the brain functioning in different experimental conditions. In every case, the biomedical engineers, together with mathematicians, physicists and physicians are called to elaborate the signals related to the brain activity in order to extract meaningful and robust information to correlate with the external behavior of people. In such attempts, different signal processing tools used in telecommunications and other fields of engineering or even social sciences have been adapted and re-used in the neuroscience field. In particular, the science of complex networks has produced an increasing interest in the study of complex systems where interaction networks are crucial. Recently, the analysis of real networks led to a series of important results in various fields and to the identification of the basic principles common to all the networks that are being considered. Scientists have found that several systems can be represented as networks and that the study of the whole web of links connecting different parts rather than the analysis of single elements, could give a better comprehension of the system itself. In this sense, the analysis of the brain functional connectivity through a network-based approach is one of the most promising means by which to study the brain functioning during motor or cognitive tasks. The present book intends to offer a concise presentation of the theoretical aspects concerning i) the possibility to achieve the cortical functional connectivity of the human brain from standard EE
The present book illustrates the theoretical aspects of several methodologies related to the possibility of i) enhancing the poor spatial information of the electroencephalographic (EEG) activity on the scalp and giving a measure of the electrical activity on the cortical surface. ii) estimating the directional influences between any given pair of channels in a multivariate dataset. iii) modeling the brain networks as graphs. The possible applications are discussed in three different experimental designs regarding i) the study of pathological conditions during a motor task, ii) the study of memory processes during a cognitive task iii) the study of the instantaneous dynamics throughout the evolution of a motor task in physiological conditions. The main outcome from all those studies indicates clearly that the performance of cognitive and motor tasks as well as the presence of neural diseases can affect the brain network topology. This evidence gives the power of reflecting cerebral "states" or "traits" to the mathematical indexes derived from the graph theory. In particular, the observed structural changes could critically depend on patterns of synchronization and desynchronization - i.e. the dynamic binding of neural assemblies - as also suggested by a wide range of previous electrophysiological studies. Moreover, the fact that these patterns occur at multiple frequencies support the evidence that brain functional networks contain multiple frequency channels along which information is transmitted. The graph theoretical approach represents an effective means to evaluate the functional connectivity patterns obtained from scalp EEG signals. The possibility to describe the complex brain networks sub-serving different functions in humans by means of "numbers" is a promising tool toward the generation of a better understanding of the brain functions. Table of Contents: Introduction / Brain Functional Connectivity / Graph Theory / High-Resolution EEG / Cortical Networks in Spinal Cord Injured Patients / Cortical Networks During a Lifelike Memory Task / Application to Time-varying Cortical Networks / Conclusions
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