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

Chapter 11 – Fish Schooling

https://princetonup.degruyter.com/downloadpdf/book/9780691212920/10.1515/9780691212920-007.pdf

Chapter 5 – Why Self-Organization?

Mitochondrial Ca2+ uptake is crucial for coupling receptor stimulation to cellular bioenergetics. Further, Ca2+ uptake by respiring mitochondria prevents Ca2+-dependent inactivation (CDI) of store-operated Ca2+ release-activated Ca2+ (CRAC) channels and inhibits Ca2+ extrusion to sustain cytosolic Ca2+ signaling. However, how Ca2+ uptake by the mitochondrial Ca2+ uniporter (MCU) shapes receptor-evoked interorganellar Ca2+ signaling is unknown. Here, we generated several cell lines with MCU-knockout (MCU-KO) as well as tissue-specific MCU-knockdown mice. We show that mitochondrial depolarization, but not MCU-KO, inhibits store-operated Ca2+ entry (SOCE). Paradoxically, despite enhancing Ca2+ extrusion and promoting CRAC channel CDI, MCU-KO increased cytosolic Ca2+ in response to store depletion. Further, physiological agonist stimulation in MCU-KO cells led to enhanced frequency of cytosolic Ca2+ oscillations, endoplasmic reticulum Ca2+ refilling, NFAT nuclear translocation and proliferation. However, MCU-KO did not affect inositol-1,4,5-trisphosphate receptor activity. Mathematical modeling supports that MCU-KO enhances cytosolic Ca2+, despite limiting CRAC channel activity.

https://biorxiv.org/content/10.1101/2021.04.07.438854v1.full.pdf

Regulation of Interorganellar Ca2+ Transfer and NFAT Activation by the Mitochondrial Ca2+ Uniporter

It is known that curvature relation plays a key role in the propagation of two-dimensional waves in an excitable model. Such a relation is believed to obey the eikonal equation for typical excitable models (e.g., the FitzHugh-Nagumo (FHN) model), which states that the relation between the normal velocity and the local curvature is approximately linear. In this paper, we show that for a simplified model of intracellular calcium dynamics, although its temporal dynamics can be investigated by analogy with the FHN model, the curvature relation does not obey the eikonal equation. Further, the inconsistency with the eikonal equation for the calcium model is because of the dispersion relation between wave speed $s$ and volume-ratio parameter $\gamma$ in the closed-cell version of the model, not because of the separation of the fast and the slow variables as in the FHN model. Hence this simplified calcium model may be an unexpected excitable system, whose wave propagation properties cannot be always understood by analogy with the FHN model.

Curvature dependence of propagating velocity for a simplified calcium model

Oscillations and travelling waves in the concentration of free cytosolic calcium are complex dynamical phenomena that play vital roles in cellular function, controlling such processes as contraction, secretion and differentiation. Although, nowadays, these oscillations and waves may be observed experimentally with relative ease, we still lack a rigorous understanding of, firstly, the mechanisms underlying these waves and oscillations in different cell types, and, secondly, the mathematical structures that underlie these complex dynamics. Thus, the study of calcium waves and oscillations is one area in which modellers have, over the years, played a major role. Here, we review our current understanding of the nonlinear dynamics of calcium waves and oscillations, restricting our attention almost wholly to deterministic models.

James Sneyd

Papers

Chapter 11 – Fish Schooling

Chapter 5 – Why Self-Organization?

Regulation of Interorganellar Ca2+ Transfer and NFAT Activation by the Mitochondrial Ca2+ Uniporter

Curvature dependence of propagating velocity for a simplified calcium model

The Nonlinear Dynamics of Calcium