Weakly Connected Neural Networks

Hardcover | July 10, 1997

byFrank C. Hoppensteadt, Eugene M. Izhikevich

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Devoted to local and global analysis of weakly connected systems with applications to neurosciences, this book uses bifurcation theory and canonical models as the major tools of analysis. It presents a systematic and well motivated development of both weakly connected system theory and mathematical neuroscience, addressing bifurcations in neuron and brain dynamics, synaptic organisations of the brain, and the nature of neural codes. The authors present classical results together with the most recent developments in the field, making this a useful reference for researchers and graduate students in various branches of mathematical neuroscience.

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Devoted to local and global analysis of weakly connected systems with applications to neurosciences, this book uses bifurcation theory and canonical models as the major tools of analysis. It presents a systematic and well motivated development of both weakly connected system theory and mathematical neuroscience, addressing bifurcations...

Format:HardcoverDimensions:418 pages, 9.25 × 6.1 × 0.9 inPublished:July 10, 1997Publisher:Springer New York

The following ISBNs are associated with this title:

ISBN - 10:0387949488

ISBN - 13:9780387949482

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Table of Contents

Dedication.- Preface.- Acknowledgment.- Introduction.- Models in Mathematical Biology.- Ordinary Language Models.- Qualifications.- Dale's Principle.- Weakness of Synaptic Connections.- How small is E?- Size of PSP.- A Common Misinterpretation.- Spike Delays.- Changes of Synaptic Efficacy.- Empirical Models.- Activities of Neurons.

Editorial Reviews

From the reviews: "...After the introduction, written according to the authors in ordinary language, and well readable even for laymen, follows a nicely written Chapter 2 on bifurcations in neuron dynamics which must be read. Here also spiking and bursting phenomena are clearly described. Chapter 3 contains a short sketch of nonhyperbolic (when the Jacobian matrix of (1) has at least one eigenvalue with zero real part) neural networks. The remaining part of the book is mainly devoted to canonical models (Chapter 4), their derivation (Chapters 6--9), and their analysis (Chapters 10--12). The term canonical model is not precisely defined here. The authors say that a model is canonical if there is a continuous change of variables that transforms any other model from a given class into this one. As the method of deriving the canonical models, the authors exploit the normal form theory. Canonical models treated in the book have only restricted value: They provide information about local behavior of (1) when there is an exponentially stable limit cycle but they say nothing about global behavior of (1), including the transients. The last Chapter 13 describes the relationship between synaptic organizations and dynamical properties of networks of neural oscillators. In other words, the problem of learning and memorization of phase information in the weakly connected network of oscillators corresponding to multiple Andronov-Hopf bifurcation is treated analytically. Surprisingly the book ends without any conclusions. Also there are no appendices to the book. The references are representative and sufficiently cover the problematics treated in the book."  (Ladislav Andrey, Mathematical Reviews)