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

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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

Effect of Deep Inspirations on Airway Smooth Muscle Force.

Salivary fluid secretion involves an intricate choreography to result in the trans-epithelial movement of NaCl and water into the acinus lumen. Current models are based on experimental observations in enzymatically isolated cells where the Ca2+ signal invariably propagates globally and thus appears ideally suited to activate spatially separated Cl and K channels. We monitored Ca2+ signals and salivary secretion in live mice expressing GCamp6F, following stimulation of the nerves innervating the submandibular gland. Consistent with in vitro studies, Ca2+ signals were initiated in the apical endoplasmic reticulum. In marked contrast to in vitro data, highly localized trains of Ca2+ transients that failed to propagate from the apical region were observed. Following stimuli optimum for secretion, large apical-basal gradients were elicited. Given this incompatibility to the previous model, a new mathematical model was constructed to explain how salivary secretion can be efficiently stimulated by apically localized Ca2+ signals.

https://biorxiv.org/content/10.1101/2020.12.30.424864v1.full.pdf

The characteristics of intracellular Ca2+ signals in vivo necessitate a new model for salivary fluid secretion

We present a minimalist model to describe the interplay between burst firing and calcium dynamics in Gonadotropin-releasing hormone (GnRH) cells. This model attempts to give a qualitative representation of Duan's model [3], and it comprises two FithzHugh-Nagumo (FHN) coupled systems describing the dynamics of the membrane potential and calcium concentration in the GnRH cells. Within the framework of our minimalist model, we find that the calcium subsystem drives burst firing by making the voltage subsystem to undergo a Hopf bifurcation. Specifically, fast relaxation oscillations occur in a specific region of the c–z plane (c being the calcium concentration, and z a calcium-dependent gating variable). Slow calcium oscillations, instead, are carried by the voltage subsystem by successive shifts of the calcium steady state, and have the net effect of an external perturbation. The full comprehension of the phase-plane of the voltage subsystem and the 3-dimensional phase-space of the calcium subsystem ultimately allows us to study the behaviours of the entire model under the change of certain parameters. Those special parameters do not necessarily follow realistic assumptions, but merely intend to mimic some pharmacological tests which have been performed experimentally and also simulated by Duan's model under the corresponding physiological considerations.

https://iopscience.iop.org/article/10.1088/1742-6596/285/1/012040/pdf

A minimalist model of calcium-voltage coupling in GnRH cells

Calcium signaling data analysis has become increasing complex as the size of acquired datasets increases. In this paper we present a Ca2+ signaling data analysis method that employs custom written software scripts deployed in a collection of Jupyter-Lab “notebooks” which were designed to cope with this complexity. The notebook contents are organized to optimize data analysis workflow and efficiency. The method is demonstrated through application to several different Ca2+ signaling experiment types.

/pdf/tools-for-quantitative-analysis-of-calcium-signaling-data-ek5yrxhe.pdf

Tools for Quantitative Analysis of Calcium Signaling Data Using Jupyter-Lab Notebooks

We consider a general model for intracellular calcium dynamics. This model has solutions that evolve on multiple timescales, but cannot be written in the so-called standard form for geometric singular perturbation theory (GSPT). We propose a biologically motivated process-based approach for the identification of timescales and small parameters in the model. Using a coordinate-independent formulation of GSPT in combination with the blow-up method, we then prove the existence of relaxation oscillations with three distinct timescales in a certain parameter regime. Finally, we identify both singular and regular Andronov-Hopf bifurcations occurring as a possible mechanism for the onset of the relaxation oscillations under variations of important model parameters.

James Sneyd

Papers

Effect of Deep Inspirations on Airway Smooth Muscle Force.

The characteristics of intracellular Ca2+ signals in vivo necessitate a new model for salivary fluid secretion

A minimalist model of calcium-voltage coupling in GnRH cells

Tools for Quantitative Analysis of Calcium Signaling Data Using Jupyter-Lab Notebooks

Understanding Calcium Dynamics via Process-Oriented Geometric Singular Perturbation Theory