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

Models of calcium dynamics

The Circulatory System

As we have already pointed out (Section 1.3), the basic building block of a Ca2+ response is a stochastic event at the level of an individual Ca2+ channel, whether an IP3 receptor, a ryanodine receptor, or a voltage-gated Ca2+ channel. The behaviours that we call oscillations, or spiking, or waves, or microdomain transients, are merely emergent properties of the interaction of channel-level stochastic processes.

Hierarchical and Stochastic Modelling

Oscillations in the concentration of free cytoplasmic calcium ([Ca2+] i ) play a vital role in the generation and maintenance of force by airway smooth muscle (ASM) cells. Mathematical models have an important role to play in the study of such complex dynamic phenomena, and can be used to construct and test hypotheses for how such oscillations might occur, and how properties such as the oscillation period might be controlled. We briefly discuss the underlying principles of the construction of mathematical models of calcium dynamics, and show how our current model can be used to understand how oscillations of [Ca2+] i in ASM are the result of a complex interplay between inositol trisphosphate receptors (IP3R) and ryanodine receptors (RyRs). Agonist-stimulated production of inositol trisphosphate (IP3) opens IP3R, resulting in the release of Ca2+ from the endoplasmic reticulum (ER). This released Ca2+ stimulates the release of additional Ca2+ from both IP3R and RyR, leading to cycles of Ca2+ release and reuptake from the ER. In the absence of IP3 (no agonist), when the ER is overloaded with Ca2+ these cycles of release and reuptake are mediated primarily by the RyR. Conversely, in the presence of IP3 (with agonist), when the ER is partially depleted of Ca2+, these cycles are mediated primarily by the IP3R. Thus, an understanding of both IP3R and RyR is required for an understanding of how [Ca2+] i oscillations are controlled in ASM.

James Sneyd

Papers

Models of calcium dynamics

The Circulatory System

Hierarchical and Stochastic Modelling

Mathematical Modeling of Calcium Dynamics in Airway Smooth Muscle Cells

Chapter 3 – Characteristics of Self-Organizing Systems