Download or read book Analysing Mathematical Models of Intracellular Calcium Dynamics Using Geometric Singular Perturbation Techniques written by Emily Paige Harvey and published by . This book was released on 2011 with total page 151 pages. Available in PDF, EPUB and Kindle. Book excerpt: Oscillations in free intracellular calcium (Ca2+) concentration are known to act as signals in almost all cell types, transmitting messages which control cellular processes including muscle contraction, cellular secretion and neuronal firing. Due to the universal nature of calcium oscillations, understanding the physiological mechanisms that underlie them is of great importance. A key feature of intracellular calcium dynamics that has been found experimentally is that some physiological processes occur much faster than others. This leads to models with variables evolving on very different time scales. In this thesis we survey a range of representative models of intracellular calcium dynamics, using geometric singular perturbation techniques with the aim of determining the usefulness of these techniques and what their limitations are. We find that the number of distinct time scales and the number of variables evolving on each time scale varies between models, but that in all cases there are at least two time scales, with at least two slower variables. Using geometric singular perturbation techniques we identify parameter regimes in which relaxation oscillations are seen and those where canard induced mixed mode oscillations are present. We find that in some cases these techniques are very useful and explain the observed dynamics well, but that the theory is limited in its ability to explain the dynamics when there are three or more distinct time scales in a model. It has been proposed that a simple experiment, whereby a pulse of inositol (1,4,5)- trisphosphate (IP3) is applied to a cell, can be used to distinguish between two competing mechanisms which lead to calcium oscillations [53]. However, detailed mathematical investigation of models has identified an anomalous delay in the pulse responses of some models, making interpretation of the experimental data difficult [14]. In this thesis we find that the response of models to a pulse of IP3 can be understood in part by using geometric singular perturbation techniques. Using recently developed theory for systems with three or more slow variables, we find that the anomalous delay can be due to the presence of folded nodes and their corresponding canard solutions or due to the presence of a curve of folded saddles. This delay due to a curve of folded saddles is a novel delay mechanism that can occur in systems with three or more slow variables. Importantly, we find that in some models the response to a pulse of IP3 is contrary to predictions for all bifurcation parameter values, which invalidates the proposed experimental protocol.
Download or read book Models of Calcium Dynamics written by Nathan Pages and published by . This book was released on 2020 with total page 127 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis considers models of intracellular calcium ions (Ca2+). We aim to show how mathematical modelling can help us understand Ca2+ dynamics and how the investigation of Ca2+ dynamics models can motivate the development of new mathematical tools. The first part of the thesis presents a model of Ca2+ dynamics in parotid acinar cells. This model is simulated using a finite element method on an anatomically accurate reconstruction of a cluster of cells. Parotid acinar cells are exocrine cells; therefore, the Ca2+ model is coupled with a fluid flow model. From simulations, we gathered three main results. Firstly, the structure of the cell determines which of the possible mechanisms can create the observed Ca2+ concentration oscillations. Secondly, a wave propagation mechanism is needed to transport the Ca2+ oscillation from the apical to the basal region; we propose a mechanism based on calcium-induced calcium-release channels. Finally, there is a strong co-dependence between fluid secretion and Ca2+ dynamics; therefore, it is necessary to model fluid secretion alongside Ca2+ dynamics. Geometric singular perturbation theory (GSPT) in its classical form, which assumes that each variable is associated with a distinct timescale, has previously been used to study Ca2+ dynamics problems with multiple timescales. However, this association is not valid in general and particularly for models of Ca2+ dynamics; instead, a non-standard form of GSPT, which does not rely on the separation of variables by timescale, is more appropriately used for the analysis of Ca2+ models. We applied non-standard GSPT to a simplified canonical model of Ca2+ dynamics to explain the structure of its relaxation oscillations. We linked timescales to distinct physiological processes underlying different terms in the model, making possible a physiological interpretation of the analysis. Our approach overcomes problems that arise when using classical GSPT. Specifically, we were able to study models that exhibit more timescales than variables and in which a variable can be characterised as either fast or slow depending on the position in phase space. Our strategy of identifying timescales in a model based on careful consideration of the underlying physiology is quite general and is expected to be useful for other Ca2+ dynamics models or process-based models with multiple timescales.
Download or read book Waves in Mathematical Models of Intracellular Calcium and Other Excitable Systems written by Wenjun Zhang and published by . This book was released on 2011 with total page 121 pages. Available in PDF, EPUB and Kindle. Book excerpt: Oscillations in cytoplasmic calcium concentration are a crucial control mechanism in almost every cell type. Two important classes of oscillation are of particular interest: solitary and periodic waves. Both types of waves are commonly observed in physical experiments and found in mathematical models of calcium dynamics and other excitable systems. In this thesis, we try to understand these two classes of wave solutions. We first investigate wave solutions of the canonical excitable model, the FitzHugh-Nagumo (FHN) equations. We analyze the FHN equations using geometric singular perturbation theory and numerical integration, and find some new codimension-two organizing centres of the overall dynamics. Many analytical results about the FHN model in its classical form have already been established. We devise a transformation to change the form of the FHN equations we study into the classical form to make use of the results. This enables us to show how basic features of the bifurcation structure of the FHN equations arise from the singular limit. We then study waves of a representative calcium model. We analyze the dynamics of the calcium model in the singular limit, and show how homoclinic and Hopf bifurcations of the full system arise as perturbations of singular homoclinic and Hopf bifurcations. We compare the wave solutions in the FHN model and the calcium model, and show that the dynamics of the two models differ in some respects (most importantly, in the way in which diffusion enters the equations). We conclude that the FHN model should not uniformly be used as a prototypical model for calcium dynamics. Motivated by phenomena seen in the FHN and calcium models, we then investigate reduction techniques for excitable systems, including the quasi-steady state approximation and geometric singular perturbation theory, and show that criticality of Hopf bifurcations may be changed when applying these reduction methods to slow-fast biophysical systems. This suggests that great care should be taken when using reduction techniques such as these, to ensure that spurious conclusions about the dynamics of a model are not drawn from the dynamics of a reduced version of the model. Finally, we describe the class of numerical algorithms used to compute features of the detailed bifurcation sets for the FHN and calcium models, and show how these were used to locate a non-structurally stable heteroclinic connection between periodic orbits in a calcium model; this is the first time such a global bifurcation has been computed.
Download or read book Mathematical Analysis of Complex Cellular Activity written by Richard Bertram and published by Springer. This book was released on 2015-10-09 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains two review articles on mathematical physiology that deal with closely related topics but were written and can be read independently. The first article reviews the basic theory of calcium oscillations (common to almost all cell types), including spatio-temporal behaviors such as waves. The second article uses, and expands on, much of this basic theory to show how the interaction of cytosolic calcium oscillators with membrane ion channels can result in highly complex patterns of electrical spiking. Through these examples one can see clearly how multiple oscillatory processes interact within a cell, and how mathematical methods can be used to understand such interactions better. The two reviews provide excellent examples of how mathematics and physiology can learn from each other, and work jointly towards a better understanding of complex cellular processes. Review 1: Richard Bertram, Joel Tabak, Wondimu Teka, Theodore Vo, Martin Wechselberger: Geometric Singular Perturbation Analysis of Bursting Oscillations in Pituitary Cells Review 2: Vivien Kirk, James Sneyd: Nonlinear Dynamics of Calcium
Download or read book Models of Calcium Signalling written by Geneviève Dupont and published by Springer. This book was released on 2016-06-07 with total page 453 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book discusses the ways in which mathematical, computational, and modelling methods can be used to help understand the dynamics of intracellular calcium. The concentration of free intracellular calcium is vital for controlling a wide range of cellular processes, and is thus of great physiological importance. However, because of the complex ways in which the calcium concentration varies, it is also of great mathematical interest.This book presents the general modelling theory as well as a large number of specific case examples, to show how mathematical modelling can interact with experimental approaches, in an interdisciplinary and multifaceted approach to the study of an important physiological control mechanism. Geneviève Dupont is FNRS Research Director at the Unit of Theoretical Chronobiology of the Université Libre de Bruxelles; Martin Falcke is head of the Mathematical Cell Physiology group at the Max Delbrück Center for Molecular Medicine, Berlin; Vivien Kirk is an Associate Professor in the Department of Mathematics at the University of Auckland, New Zealand; James Sneyd is a Professor in the Department of Mathematics at The University of Auckland, New Zealand.
Download or read book Mathematical Modeling and Analysis of Intracellular Calcium Dynamics written by Alireza Atri and published by . This book was released on 1996 with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Modeling Calcium Signaling written by Ritu Agarwal and published by Springer Nature. This book was released on with total page 86 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Understanding Calcium Dynamics written by Martin Falcke and published by Springer. This book was released on 2014-03-12 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: Written as a set of tutorial reviews on both experimental facts and theoretical modelling, this volume is intended as an introduction and modern reference in the field for graduate students and researchers in biophysics, biochemistry and applied mathematics.
Download or read book A Mathematical Analysis of a Model of Drug Action on Intracellular Calcium Dynamics written by Marah Townzen Funk and published by . This book was released on 2019 with total page 50 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Nonautonomous Dynamical Systems in the Life Sciences written by Peter E. Kloeden and published by Springer. This book was released on 2014-01-22 with total page 326 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonautonomous dynamics describes the qualitative behavior of evolutionary differential and difference equations, whose right-hand side is explicitly time dependent. Over recent years, the theory of such systems has developed into a highly active field related to, yet recognizably distinct from that of classical autonomous dynamical systems. This development was motivated by problems of applied mathematics, in particular in the life sciences where genuinely nonautonomous systems abound. The purpose of this monograph is to indicate through selected, representative examples how often nonautonomous systems occur in the life sciences and to outline the new concepts and tools from the theory of nonautonomous dynamical systems that are now available for their investigation.
Download or read book A Bifurcation Analysis of a Mathematical Model of Intracellular Calcium Waves written by David Simpson (MSc.) and published by . This book was released on 2003 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Modelling Intracellular Calcium Dynamics written by Gita Caroline Chopra and published by . This book was released on 1998 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Mathematical Foundations of Neuroscience written by G. Bard Ermentrout and published by Springer Science & Business Media. This book was released on 2010-07-01 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book applies methods from nonlinear dynamics to problems in neuroscience. It uses modern mathematical approaches to understand patterns of neuronal activity seen in experiments and models of neuronal behavior. The intended audience is researchers interested in applying mathematics to important problems in neuroscience, and neuroscientists who would like to understand how to create models, as well as the mathematical and computational methods for analyzing them. The authors take a very broad approach and use many different methods to solve and understand complex models of neurons and circuits. They explain and combine numerical, analytical, dynamical systems and perturbation methods to produce a modern approach to the types of model equations that arise in neuroscience. There are extensive chapters on the role of noise, multiple time scales and spatial interactions in generating complex activity patterns found in experiments. The early chapters require little more than basic calculus and some elementary differential equations and can form the core of a computational neuroscience course. Later chapters can be used as a basis for a graduate class and as a source for current research in mathematical neuroscience. The book contains a large number of illustrations, chapter summaries and hundreds of exercises which are motivated by issues that arise in biology, and involve both computation and analysis. Bard Ermentrout is Professor of Computational Biology and Professor of Mathematics at the University of Pittsburgh. David Terman is Professor of Mathematics at the Ohio State University.
Download or read book Applied Mechanics Reviews written by and published by . This book was released on 1992 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Mathematics for Neuroscientists written by Fabrizio Gabbiani and published by Academic Press. This book was released on 2017-02-04 with total page 630 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics for Neuroscientists, Second Edition, presents a comprehensive introduction to mathematical and computational methods used in neuroscience to describe and model neural components of the brain from ion channels to single neurons, neural networks and their relation to behavior. The book contains more than 200 figures generated using Matlab code available to the student and scholar. Mathematical concepts are introduced hand in hand with neuroscience, emphasizing the connection between experimental results and theory. - Fully revised material and corrected text - Additional chapters on extracellular potentials, motion detection and neurovascular coupling - Revised selection of exercises with solutions - More than 200 Matlab scripts reproducing the figures as well as a selection of equivalent Python scripts
Download or read book Mathematical Modeling in Systems Biology written by Brian P. Ingalls and published by MIT Press. This book was released on 2022-06-07 with total page 423 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to the mathematical concepts and techniques needed for the construction and analysis of models in molecular systems biology. Systems techniques are integral to current research in molecular cell biology, and system-level investigations are often accompanied by mathematical models. These models serve as working hypotheses: they help us to understand and predict the behavior of complex systems. This book offers an introduction to mathematical concepts and techniques needed for the construction and interpretation of models in molecular systems biology. It is accessible to upper-level undergraduate or graduate students in life science or engineering who have some familiarity with calculus, and will be a useful reference for researchers at all levels. The first four chapters cover the basics of mathematical modeling in molecular systems biology. The last four chapters address specific biological domains, treating modeling of metabolic networks, of signal transduction pathways, of gene regulatory networks, and of electrophysiology and neuronal action potentials. Chapters 3–8 end with optional sections that address more specialized modeling topics. Exercises, solvable with pen-and-paper calculations, appear throughout the text to encourage interaction with the mathematical techniques. More involved end-of-chapter problem sets require computational software. Appendixes provide a review of basic concepts of molecular biology, additional mathematical background material, and tutorials for two computational software packages (XPPAUT and MATLAB) that can be used for model simulation and analysis.
Download or read book Biomolecular Feedback Systems written by Domitilla Del Vecchio and published by Princeton University Press. This book was released on 2014-10-26 with total page 287 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an accessible introduction to the principles and tools for modeling, analyzing, and synthesizing biomolecular systems. It begins with modeling tools such as reaction-rate equations, reduced-order models, stochastic models, and specific models of important core processes. It then describes in detail the control and dynamical systems tools used to analyze these models. These include tools for analyzing stability of equilibria, limit cycles, robustness, and parameter uncertainty. Modeling and analysis techniques are then applied to design examples from both natural systems and synthetic biomolecular circuits. In addition, this comprehensive book addresses the problem of modular composition of synthetic circuits, the tools for analyzing the extent of modularity, and the design techniques for ensuring modular behavior. It also looks at design trade-offs, focusing on perturbations due to noise and competition for shared cellular resources. Featuring numerous exercises and illustrations throughout, Biomolecular Feedback Systems is the ideal textbook for advanced undergraduates and graduate students. For researchers, it can also serve as a self-contained reference on the feedback control techniques that can be applied to biomolecular systems. Provides a user-friendly introduction to essential concepts, tools, and applications Covers the most commonly used modeling methods Addresses the modular design problem for biomolecular systems Uses design examples from both natural systems and synthetic circuits Solutions manual (available only to professors at press.princeton.edu) An online illustration package is available to professors at press.princeton.edu
