Man vet hvor en planet befinner seg om tolv år, fire måneder og ni dager. Men man vet ikke hvor en sommerfugl vil være fløyet hen om et minutt.

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Garey Snow
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1 Man vet hvor en planet befinner seg om tolv år, fire måneder og ni dager. Men man vet ikke hvor en sommerfugl vil være fløyet hen om et minutt. Jens Bjørneboe, Kruttårnet, 1969 i

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3 Preface This thesis is submitted to the Norwegian University of Science and Technology (NTNU) in partial fulfillment of the degree of Dr. ing. It contains a short summary and eight scientific papers. The work has partly been carried out in the group of theoretical physics at the Faculty of physics, informatics and mathematics at NTNU, and partly at the Department of Physiology at McGill University, Montreal. I wish to express my gratitude to the members of these groups, for giving many constructive comments regarding my work. It has not been easy for me to convince all the members of these groups, partly because some of the results are controversial, and partly because the results are interdisciplinary in nature. I wish to thank my supervisor Jan Myrheim for his patience, encouragement and kind assistance. I will also thank Johan S. Høye for much help regarding the controversial parts of the thesis. The help and comments from Kåre Olaussen has been of invaluable importance through all stages of the thesis work, from the interpretation of the energy equations to brilliant advise regarding politics of publishing. It has been a pleasure for me to work with Nils Skarland, his language and humor has been very important. Michael R. Guevara and Kevin Hall at McGill University deserves credit for constructive criticism and feedback. Financial support from NTNU is acknowledged. I have received much encouragement from my parents, Randi and Endre Endresen, and from my parents in law, Sigrunn and Sverre Stensby. Finally I would like to express my deepest gratitude to my wife Mariane Cecilie Stensby for encouragement and support. Lars Petter Endresen June 23, 2000 iii

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5 Contents Preface Contents iii v Summary 1 Papers: I. A Possible Resolution of the Gating Paradox II. Modeling the Electrical Activity of Pacemaker Cells III. A Theory for the Membrane Potential of Living Cells IV. Chaos in Weakly Coupled Pacemaker Cells V. Limit Cycle Oscillations in Pacemaker Cells VI. Runge-Kutta Formulas for Cardiac Oscillators VII. A Formula for Steplength Control in Numerical Integration VIII. An Efficient Step Size Selection for ODE Codes v

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7 1 Summary The subject of my thesis is mathematical models of the heartbeat and their basis in laws of physics and chemistry. In particular, it is concerned with models of the cells in the pacemaker region of the heart, the so called sinoatrial node. Even with this limitation there are many aspects of the problem [1], and I have chosen to concentrate on few rather specific topics. My work is presented here in eight research papers, of various length and importance, following this brief introduction. The main papers are I and III, both emphasizing the physical understanding of the topics under discussion. Paper I deals with a so called paradox concerning experimental data on the gating of ionic channels in the cell membrane. Paper III describes a complete model of an oscillating cell, and discusses how it is related to general principles like the conservation laws for electric charge and energy. The underlying assumption of all models in this field, going back to van der Pol in 1926 [2], is that biological oscillators can be described by differential equations. The van der Pol equation, v (α 3γv 2 ) v + ω 2 0v = 0, (1) describes an oscillatory electrical circuit. Here v is voltage, t is time, v and v are the first and second time derivatives of v, while α, γ, and ω 0 are parameters. A typical behavior resulting from this equation is a so called relaxation oscillation, characterized by a long rest period followed by a rapid rise and fall of the potential, called an action potential. Thus, the equation provides an electrical analogy capturing one essential feature of the biological system, but is not intended as a completely realistic model. A realistic model of cardiac electrical activity was first developed by Denis Noble in 1962 [3], based on the work of Hodgkin and Huxley [4]. Since then many new models have been proposed, largely as a result of a bi directional interaction between theory and experiment [5]. The models in this thesis are the result of a bi directional interaction between theory and simulation. This means that we have used simulation to determine what is possible using well known principles from physics, and only afterwards compared with experiments. This approach is creative plagiarism, as described by Krauss [6]: One sees the same concepts, the same formalism, the same techniques, the same pictures, being twisted and molded as far as possible to apply to a host of new situations, as long as they have worked before. In this connection it is important to realize that model building may serve different and sometimes conflicting purposes. A very simple model, like the van der Pol equation, may give a useful qualitative understanding, from which it may be possible to predict qualitatively how the system responds to changes in

8 2 its environment. But if one needs more quantitative predictions, for example in order to understand arrhythmia and other pathological conditions of the heart, or the effect of drugs, one needs models that are much more detailed and realistic. The drawback of such elaborate models is that they tend to lose predictive power, because they become very complicated, and include a lot of parameters that have to be adjusted so that the model fits experimental observations [7]. All the models presented in the included papers are compromises between the two extremes. They are not realistic in every detail, but are sufficiently realistic to reproduce perfectly experimentally observed action potentials. At the same time, they do so at a rather small cost in terms of the number of parameters to be fitted. The simplest of these models is the one used in paper IV, with only two first order ordinary differential equations describing one cell. Paper II is a more general discussion of the principles of model reduction. Note that the gain by the reduction in complexity of a model may be at least twofold. There is an obvious gain in computer time, which is sometimes of crucial importance if the model is to be useful in numerical simulations of large systems with many coupled cells. But no less important is the general principle that a complicated model with a large number of parameters may fit observations within the range where it has been tested, without actually representing the physical reality. It may simply be flexible enough to fit anything. A simple model, on the other hand, is not so easily made to fit the observed data, unless it contains some grain of truth. Thus, if there is a choice between different models that all reproduce correctly the same set of data, and one needs to extrapolate into a region where no data exist, the simplest model is a reasonable choice. Simplicity is not the only criterion for choosing between competing models. An even more basic criterion is that a good model should be based on, or at least compatible with, the known laws of physics. The physicist, or scientist in general, should not be content with a phenomenological description of oscillating cells, no matter how successful, unless it has a sound basis in physics. The ambition should be to understand the behavior of the cells in terms of general laws, such as conservation laws for energy, electric charge and other quantities, and the laws of electromagnetism and statistical physics. This point of view is one main motivation for my work, and in particular for the papers I and III. Fortunately, these two ideals, that a model should be simple and have a sound physical basis, seem to be not only compatible, but often so closely related that they can be regarded as the two sides of the same coin. In fact, it is a rather general experience in physics that an improved understanding of some phenomenon also leads to a simpler description. I would like to point out two examples from paper III of increased understanding leading to simpler models. The first example is the simple and well known relation between the charge q, capacitance C and voltage v on a capacitor, q = Cv, (2)

9 which is used in this paper to determine the membrane voltage for a living cell. It has been customary to use the time derivative of this equation, giving a relation between current and voltage change, instead of using the equation directly. This is not difficult to understand, since the membrane currents are more easily observable than the corresponding fluctuations in membrane charge q. Second we found that the potential energy associated with transmembrane transport of ions can be written P = 1 2 Cv2 T s V π. (3) Here C is capacitance, v voltage, T temperature, s entropy relative to equilibrium, V cell volume and π osmotic pressure. The meaning of this equation is that whenever ions move across the membrane of a cell, one will notice a change in: 1. the transmembrane voltage difference, v; 2. the cells entropy relative to equilibrium, s; and 3. the transmembrane osmotic pressure difference, π. I have mostly considered models of one single cell. The exception is paper IV, in which a system of two coupled cells is investigated, from the point of view of the general theory of dynamical systems. In general, when two or more oscillators of different frequencies are coupled together, one may observe synchronization, also called phase locking, as in the normal state in the sinoatrial node. Another theoretical possibility is chaotic oscillations, as demonstrated in this paper for certain ranges of parameter values. In paper V it is shown, using the model of paper III, how one single cell can reach the same stable limit cycle from different initial conditions. As already mentioned, in computer simulations the computational efficiency of the model used is sometimes one of the most important factors. One way to improve the efficiency is to choose an efficient method for numerical integration. In the papers VI, VII and VIII this problem is investigated. From a medical point of view it would have been important and interesting to also focus on the effect of drugs. I regret that this topic is not discussed here, since it is most relevant for patients suffering from heart disease. For more extensive reviews for the topics discussed in this thesis I refer to Noble [1] and Wilders [7]. 3

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11 Bibliography [1] D. Noble. The initiation of the heartbeat. Oxford University Press (1979). [2] Van der Pol, Balthasar, et al. On Relaxation - Oscillations. 1922, London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, pp [3] D. Noble. A modification of the Hodgkin Huxley equations applicable to Purkinje fibre action and pacemaker potentials, J. Physiol 160: (1962). [4] Hodgkin, A. L. and Huxley, A. F. A quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve. 1952, J Physiol, pp [5] D. Noble. The Development of Mathematical Models of the Heart, Chaos solitons & fractals, 5: (1995). [6] L. M. Krauss. Fear of Physics: A Guide for the Perplexed. Basic Books (1994). [7] R. Wilders. From single channel kinetics to regular beating. A model study of cardiac pacemaking activity. PhD Thesis. pp Universiteit van Amsterdam. ISBN (1993). 5

A note on discontinuous rate functions for the gate variables in mathematical models of cardiac cells

Procedia Computer Science (2) (22) 6 945 95 Procedia Computer Science www.elsevier.com/locate/procedia International Conference on Computational Science ICCS 2 A note on discontinuous rate functions for