Dynamics in One-Dimension
In this chapter we discuss differential equations of one dependent variable. We treat fixed points and define rules to define the stability of these fixed points for linear and nonlinear equations, complemented by geometric reasoning. In the second part o
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ynamics of Neural Networks A Mathematical and Clinical Approach
Dynamics of Neural Networks
Michel J. A. M. van Putten
Dynamics of Neural Networks A Mathematical and Clinical Approach
123
Michel J. A. M. van Putten Clinical Neurophysiology Group University of Twente Enschede, The Netherlands Neurocenter, Dept of Neurophysiology Medisch Spectrum Twente Enschede, The Netherlands
ISBN 978-3-662-61182-1 ISBN 978-3-662-61184-5 https://doi.org/10.1007/978-3-662-61184-5
(eBook)
© Springer-Verlag GmbH Germany, part of Springer Nature 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer-Verlag GmbH, DE part of Springer Nature. The registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany
Preface
This book evolved from the course “Dynamics of Neural Networks in Health and Disease.” It treats essentials from neurophysiology (Hodgkin-Huxley equations, synaptic transmission, prototype networks of neurons) and related mathematical concepts (dimensionality reductions, equilibria, bifurcations, limit cycles, and phase plane analysis). This is subsequently applied in a clinical context, focusing on EEG generation, ischaemia, epilepsy, and neurostimulation. The book is based on a graduate course taught by clinicians and mathematicians at the Institute of Technical Medicine at the University of Twente. Throughout the text, we present examples of neurological disorders in relation to applied mathematics to assist in disclosing various fundamental properties of the clinical reality at hand. Exercises are provided at the end of each chapter; answers are included. Basic knowledge of calculus, linear algebra, differential equations, and familiarity with Matlab or Python is assumed. Also, students should have basic knowledge about essentials of (clinical) neurophysiology, although most con
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