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​Waves in Neural Media: From Single Neurons to Neural Fields surveys mathematical models of traveling waves in the brain, ranging from intracellular waves in single neurons to waves of activity in large-scale brain networks. The work provides a pedagogical account of analytical methods for finding traveling wave solutions of the variety of nonlinear differential equations that arise in such models. These include regular and singular perturbation methods, weakly nonlinear analysis, Evans functions and wave stability, homogenization theory and averaging, and stochastic processes. Also covered in the text are exact methods of solution where applicable. Historically speaking, the propagation of action potentials has inspired new mathematics, particularly with regard to the PDE theory of waves in excitable media. More recently, continuum neural field models of large-scale brain networks have generated a new set of interesting mathematical questions with regard to the solution of nonlocal integro-differential equations.

Advanced graduates, postdoctoral researchers and faculty working in mathematical biology, theoretical neuroscience, or applied nonlinear dynamics will find this book to be a valuable resource. The main prerequisites are an introductory graduate course on ordinary differential equations or partial differential equations, making this an accessible and unique contribution to the field of mathematical biology.

Inhaltsverzeichnis

Frontmatter

Neurons

Frontmatter

Chapter 1. Single Neuron Modeling

Abstract
Chapter 1 provides a detailed introduction to the working parts of a neuron, including conductance-based models of action potential generation, synaptic and dendritic processing, and ion channels. Two important mathematical topics are also introduced. First, the dynamics of a periodically forced neural oscillator is used to introduce phase-reduction and averaging methods, phase-resetting curves, and synchronization. These are later applied to the study of waves in oscillatory neural media. Second, a detailed account of stochastic ion channels and membrane voltage fluctuations is given, which also provides background material on stochastic processes. A major theme is how to model and analyze stochastic hybrid systems, in which a continuous variable (e.g., voltage) couples to a discrete jump Markov process (e.g., number of open ion channels). Spontaneous action potential generation is formulated as a first passage time problem, which is solved using perturbation methods such as WKB and matched asymptotics. These methods are later used to analyze related problems such as the generation of calcium sparks and bistability in populations of spiking neurons.
Paul C. Bressloff

Chapter 2. Traveling Waves in One-Dimensional Excitable Media

Abstract
Chapter 2 covers the classical problem of waves in one-dimensional excitable media, as exemplified by the FitzHugh–Nagumo model of action potential propagation along an axon. Standard methods for analyzing front and pulse solutions of PDEs are described, including phase-plane analysis, singular perturbation methods and slow–fast analysis, and Evans functions for wave stability. In addition, the problem of wave propagation failure in myelinated axons is considered, where an averaging method is used to determine the effects of spatial discreteness on wave speed. This method is later used to analyze wave propagation failure in inhomogeneous neural fields. Finally, stochastic traveling waves are considered, where formal perturbation methods are used to show how to separate out fast fluctuations of the wave profile from the slow diffusive-like wandering of the wave.
Paul C. Bressloff

Chapter 3. Wave Propagation Along Spiny Dendrites

Abstract
Chapter 3 presents two different models of traveling waves along spiny dendrites: a spike–diffuse–spike model of propagating voltage spikes mediated by active dendritic spines and a reaction–diffusion model of Ca2+–calmodulin-dependent protein kinase II (CaMKII) translocation waves. The former model introduces methods that are later used to analyze solitary waves propagating in spiking neural networks. The latter model turns out to be identical in form to the diffusive susceptible–infected (SI) model of the spread of epidemics, which is a generalization of the scalar Fisher–KPP equation of population genetics. One characteristic feature of such equations is that they support traveling fronts propagating into an unstable steady state, in which the wave speed and longtime asymptotics are determined by the dynamics in the leading edge of the wave—so-called pulled fronts. In particular, a sufficiently localized initial perturbation will asymptotically approach the traveling front solution that has the minimum possible wave speed. Hence, pulled fronts have very different properties from propagating action potentials. Homogenization methods are also presented, which allow one to approximate the discrete distribution of spines by a smooth distribution.
Paul C. Bressloff

Chapter 4. Calcium Waves and Sparks

Abstract
This chapter gives a comprehensive review of calcium wave modeling, with an emphasis on their role in neuronal calcium signaling. Two models of intracellular waves are considered in some detail: a reaction–diffusion model of calcium dynamics and the fire–diffuse–fire model of calcium release. The latter is formally very similar to the spike–diffuse–spike model of spiny dendrites and is analyzed accordingly. Stochastic models of spontaneous calcium release (calcium puffs and sparks) are then analyzed using the stochastic methods introduced in the first chapter. Finally, several models of intercellular calcium waves in astrocytes are presented. Traditionally, astrocytes were thought to be physiologically passive cells that only play a supporting role in the central nervous system by regulating and optimizing the environment within which neurons operate. However, there is an increasing amount of empirical data indicating that astrocytes play a more active role in modulating synaptic transmission and neuronal signal processing.
Paul C. Bressloff

Networks

Frontmatter

Chapter 5. Waves in Synaptically Coupled Spiking Networks

Abstract
In this chapter, we turn to neural network models of wave propagation in the cortex and other parts of the nervous system. There has been a rapid increase in the number of computational studies of network dynamics, which are based on biophysically detailed conductance-based models of synaptically (and possibly electrically) coupled neurons. These models provide considerable insights into the role of ionic currents, synaptic processing, and network structure on spatiotemporal dynamics, but they tend to be analytically intractable. This has motivated an alternative approach to network dynamics, involving simplified neuron models that hopefully capture important aspects of wave phenomena, while allowing a more concise mathematical treatment. In the case of oscillatory networks, such a simplification can be achieved by reducing a conductance-based neuron model to a phase model. Alternatively, one can use a simplified spiking neuron model such as integrate-and-fire in order to investigate waves in excitable and oscillatory neural media. Both of these approaches are considered in this chapter, which also provides a summary of various wave phenomena in cortical and subcortical structures.
Paul C. Bressloff

Chapter 6. Population Models and Neural Fields

Abstract
Chapter describes the construction of population-based rate models under the assumption that the spiking of individual neurons is unimportant. The issue of how stochasticity at the single-cell level manifests itself at the population level is discussed, introducing topics such as balanced networks, Poisson statistics, and asynchronous states. Stochastic methods are then used to analyze bistability in a stochastic population model. Finally, the transition from spatially structured neural networks to continuum neural fields is highlighted. The latter take the form of nonlocal integrodifferential equations, in which the integral kernel represents the distribution of synaptic connections.
Paul C. Bressloff

Chapter 7. Waves in Excitable Neural Fields

Abstract
Chapter 7 develops the theory of waves in excitable neural fields, where the fundamental network element is a local population of cells rather than a single neuron. It is shown how many of the PDE methods and results from the analysis of waves in reaction–diffusion equations considered in Chap. 2 can be extended to the nonlocal equations of neural field theory. First, the existence and stability of solitary traveling fronts and pulses in one-dimensional excitatory neural fields are considered. In the case of traveling pulses, it is necessary to include some form of local negative feedback mechanism such as synaptic depression or spike frequency adaptation. Two approaches to analyzing wave propagation failure in inhomogeneous neural media are then presented: one based on averaging methods and the other on interfacial dynamics. Finally, wave propagation in stochastic neural fields is analyzed, and oscillatory waves in two-dimensional neural media are briefly discussed.
Paul C. Bressloff

Chapter 8. Neural Field Model of Binocular Rivalry Waves

Abstract
In this chapter, neural field theory is used to model binocular rivalry waves. During binocular rivalry, visual perception switches back and forth between different images presented to the two eyes. The resulting fluctuations in perceptual dominance and suppression provide a basis for noninvasive studies of the human visual system and the identification of possible neural mechanisms underlying conscious visual awareness. Various psychophysical experiments have demonstrated that the switch between a dominant and suppressed visual percept propagates as a traveling front for each eye. In addition to considering the particular problem of binocular rivalry waves, the more general issue of how to develop neural field models of the functional architecture of primary visual cortex (V1) is discussed.
Paul C. Bressloff

Development and Disease

Frontmatter

Chapter 9. Waves in the Developing and the Diseased Brain

Abstract
Finally, in this chapter, a variety of topics regarding wavelike phenomena in the developing and diseased brain are presented. First, the possible role of calcium and retinal waves in early development is summarized. There is then a detailed description and analysis of cytoskeletal waves involved in neurite growth and cell polarization. This introduces another interesting phenomenon, namely, wave pinning. Three distinct examples of waves in the diseased brain are considered: spreading depression and migraine auras, epileptic waves, and the spread of neurodegenerative waves due to protein aggregation. In order to tackle the latter two phenomena, a review of complex network theory is included, covering topics such as small-world networks, scale-free networks, neuronal avalanches, branching processes, and epidemics on networks.
Paul C. Bressloff

Backmatter

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