Synopses & Reviews
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.
The strength of the book is that classes of interesting models are derived which may be used instead of complex models that have large numbers of unknown parameters and that are impossible to analyze. These models should inspire a host of theoretical analyses for years to come. The book is suitable for anyone who has an interest in dynamical systems. SIAM BOOK 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)
This book is devoted to an analysis of general weakly connected neural networks (WCNNs) that can be written in the form (0.1) m Here, each Xi E IR is a vector that summarizes all physiological attributes of the ith neuron, n is the number of neurons, Ii describes the dynam ics of the ith neuron, and gi describes the interactions between neurons. The small parameter indicates the strength of connections between the neurons. Weakly connected systems have attracted much attention since the sec ond half of seventeenth century, when Christian Huygens noticed that a pair of pendulum clocks synchronize when they are attached to a light weight beam instead of a wall. The pair of clocks is among the first weakly connected systems to have been studied. Systems of the form (0.1) arise in formal perturbation theories developed by Poincare, Liapunov and Malkin, and in averaging theories developed by Bogoliubov and Mitropolsky."
This book develops the bifurcation theory for weakly connected neural networks. The authors analyze the relationship between synaptic organizations (anatomy) and dynamical properties (function) of the brain. In particular the authors show that there are some synaptic organizations that have especially rich dynamic behavior.
Includes bibliographical references (p. -393) and index.
Table of Contents
On the Application of Kalman Filtering to Correct Errors due to Vertical Deflection in Inertial Navigation.
- Filtering and Detection Problems for Nonlinear Time Series.
- Spectral and Bispectral Methods for the Analysis of Nonlinear (Non-Gaussian) Time-Series Signals.
- Bilinear Time Series: Theory and Application.
- Bivariate Bilinear Models and Their Identification.
- Nonlinear Time Series Modelling in Population Biology.
- The Akaike Information Criterion in Threshold Modelling.
- Nonlinear Time Series Analysis for Dynamical Systems of Catastrophe Type.
- Nonlinear Processing with M-th Order Signals.
- Stochastic Circulatory Lymphocyte Models.