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Waves in neural media. From single neurons to neural fields. (English) Zbl 1296.92005
Lecture Notes on Mathematical Modelling in the Life Sciences. New York, NY: Springer (ISBN 978-1-4614-8865-1/pbk; 978-1-4614-8866-8/ebook). xix, 436 p. (2014).
This book deals with spatiotemporal phenomena (namely the time-dependent propagation of some quantity on space, whether it is chemical concentration, a protein aggregate, voltage, or spiking activity of a population of neurons) in mathematical neuroscience.
Its content is structurally divided into three parts dealing, respectively, with neurons, networks, and development and disease.
It begins by describing the kinetics of a single neuron via the Hodgkin-Huxley model. The kinetics of a single synapse and in general the stochastic process of ion channels are discussed. In view of the above discussions, the membrane voltage is stochastically described by a linear noise approximation of the stochastic differential equation (or Langevin equation).
However, the phase-plane analysis of wave propagation along the one-dimensional axon of a neuron is developed by considering the simpler FitzHugh-Nagumo (FN) model (chapter “Traveling waves in one-dimensional excitable media”). In particular, stability is illustrated for the so-called scalar bistable equation by avoiding the recovery variable in the FN equations. The wave propagation along spiny dendrites is also discussed. The chapter “Calcium waves and sparks” presents the dynamics of Ca\(^{2+}\) concentration in the endoplasmic reticulum and cytoplasm as two well-mixed homogeneous compartments. Stationary and traveling front solutions of the one-dimensional model for Ca\(^{2+}\) diffusion and release are addressed. Stochastic versions of the Li-Rinzel model that describes the binding of IP\(_3\) and the activating Ca\(^{2+}\) are stated. A stochastic Ca\(^{2+}\)-release model of Hinch in a single compartment is developed. The rate of escape from a metastable state is calculated by applying quasi-stationary approximation and perturbation methods.
The part on networks begins by summarizing the basic structure of the brain. The network of a finite number of synaptically coupled cortical neurons is studied as a coupled system of ODEs by the phase reduction method. The network integrate-and-fire (IF) neurons distributed along an infinite 1D lattice is analyzed, as a continuum, by assuming that each neuron fires only once during the passage of a traveling pulse. Alternatively, the network of synaptically coupled spiking neurons is partitioned into a finite number of homogeneous populations. In particular, the existence and stability of an asynchronous state in a large globally coupled network of IF neurons are presented. The spike trains are statistically given by the Poisson process. The author considers existence and stability of traveling front and pulse solutions to nonlocal integro-differential equations describing the neural field in a one-dimensional infinite medium. Two approaches of analyzing wave propagation failure in neural media with periodically modeled weight distribution are then presented: one based on averaging methods, and the other on interfacial dynamics. Brief discussions are provided for the stochastic interpretation as well as two-dimensional media. Models of binocular rivalry waves and models of visual cortex appear as illustrative applications of the above neural field theory.
In the final part of the book, a variety of topics regarding wave-like phenomena in the developing and diseased brain are presented.

MSC:
92-02 Research exposition (monographs, survey articles) pertaining to biology
92C20 Neural biology
92B20 Neural networks for/in biological studies, artificial life and related topics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
37N25 Dynamical systems in biology
37A60 Dynamical aspects of statistical mechanics
37C75 Stability theory for smooth dynamical systems
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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