The blood-brain barrier, which is formed by the endothelial cells that line cerebral microvessels, has an important role in maintaining a precisely regulated microenvironment for reliable neuronal signalling. At present, there is great interest in the association of brain microvessels, astrocytes and neurons to form functional 'neurovascular units', and recent studies have highlighted the importance of brain endothelial cells in this modular organization. Here, we explore specific interactions between the brain endothelium, astrocytes and neurons that may regulate blood-brain barrier function. An understanding of how these interactions are disturbed in pathological conditions could lead to the development of new protective and restorative therapies.

Astrocyte–endothelial interactions at the blood–brain barrier

This book explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology "Dynamical Systems in Neuroscience" presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties The book introduces dynamical systems starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems Each chapter proceeds from the simple to the complex, and provides sample problems at the end The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum - or taught by math or physics department in a way that is suitable for students of biology This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience

/pdf/dynamical-systems-in-neuroscience-51dkewn57v.pdf

Dynamical Systems in Neuroscience

1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/596B270197FE27DE5FABB598B6210622/S0025557200150892a.pdf/div-class-title-span-class-italic-the-geometry-of-biological-time-span-by-a-t-winfree-pp-544-dm68-corrected-second-printing-1990-isbn-3-540-52528-9-springer-div.pdf

The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)

This book provides an introduction to the mathematics needed to model, analyze, and design feedback systems. It is an ideal textbook for undergraduate and graduate students, and is indispensable for researchers seeking a self-contained reference on control theory. Unlike most books on the subject, Feedback Systems develops transfer functions through the exponential response of a system, and is accessible across a range of disciplines that utilize feedback in physical, biological, information, and economic systems. Karl strm and Richard Murray use techniques from physics, computer science, and operations research to introduce control-oriented modeling. They begin with state space tools for analysis and design, including stability of solutions, Lyapunov functions, reachability, state feedback observability, and estimators. The matrix exponential plays a central role in the analysis of linear control systems, allowing a concise development of many of the key concepts for this class of models. strm and Murray then develop and explain tools in the frequency domain, including transfer functions, Nyquist analysis, PID control, frequency domain design, and robustness. They provide exercises at the end of every chapter, and an accompanying electronic solutions manual is available. Feedback Systems is a complete one-volume resource for students and researchers in mathematics, engineering, and the sciences.Covers the mathematics needed to model, analyze, and design feedback systems Serves as an introductory textbook for students and a self-contained resource for researchers Includes exercises at the end of every chapter Features an electronic solutions manual Offers techniques applicable across a range of disciplines

/pdf/feedback-systems-an-introduction-for-scientists-and-596yj6j5hm.pdf

Feedback Systems: An Introduction for Scientists and Engineers

http://mcb.berkeley.edu/courses/mcb137/exercises/Tyson%20(2003).pdf

Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell

Airway smooth muscle (ASM) is cyclically stretched during breathing, even in the active state, yet the factors determining its dynamic force-length behavior remain incompletely understood. We developed a model of the activated ASM strip and compared its behavior to that observed in strips of rat trachealis muscle stimulated with methacholine. The model consists of a nonlinear viscoelastic element (Kelvin body) in series with a force generator obeying the Hill force-velocity relationship. Isometric force in the model is proportional to the number of bound crossbridges, the attachment of which follows first-order kinetics. Crossbridges detach at a rate proportional to the rate of change of muscle length. The model accurately accounts for the experimentally observed transient and steady-state oscillatory force-length behavior of both passive and activated ASM. However, the model does not predict the sustained decrement in isometric force seen when activated strips of ASM are subjected briefly to large stretches. We speculate that this force decrement reflects some mechanism unrelated to the cycling of crossbridges, and which may be involved in the reversal of bronchoconstriction induced by a deep inflation of the lungs in vivo.

https://www.physiology.org/doi/pdf/10.1152/ajplung.00095.2009

Transient oscillatory force-length behavior of activated airway smooth muscle

We study wave propagation in the buffered bistable equation, i.e., the bistable equation where the diffusing species reacts with immobile buffers that restrict its diffusion. Such a model describes wave front propagation in excitable systems where the diffusing species is buffered; in particular, the study of the propagation of waves of increased calcium concentration in a variety of cell types depends directly upon the analysis of such buffered excitability. However, despite the biological importance of these types of equations, there have been almost no analytical studies of their properties. Here, we study the question of whether or not the inclusion of multiple buffers can eliminate propagated waves. First, we prove that a unique (up to translation) traveling wave front exists. Moreover, the wave speed is also unique. Then we prove that this traveling wave front is stable, i.e., that any initial condition which vaguely resembles a traveling wave front (in a way we make precise) evolves to the unique wave front. We thus prove that multiple stationary buffers cannot prevent the existence of a traveling wave front in the buffered bistable equation and may not eliminate stable wave fronts. This suggests (although we do not prove) that the same result is true for more complex and realistic models of calcium wave propagation, a result of direct physiological relevance.

Existence and stability of traveling waves in buffered systems

Tutorials in Mathematical Biosciences II

Intracellular calcium waves.

https://www.cell.com/biophysj/pdf/S0006-3495(97)78159-1.pdf

James Sneyd

Papers

Transient oscillatory force-length behavior of activated airway smooth muscle

Existence and stability of traveling waves in buffered systems

Tutorials in Mathematical Biosciences II

Intracellular calcium waves.

Generation of action potentials in a mathematical model of corticotrophs.