Asymptotic properties of mathematical models of excitability

We analyse small parameters in selected models of biological excitability, including Hodgkin-Huxley (1952) model of nerve axon, Noble (1962) model of heart Purkinje fibres, and Courtemanche et al. (1998) model of human atrial cells. Some of the small parameters are responsible for differences in the characteristic timescales of dynamic variables, as in the traditional singular perturbation approaches. Others appear in a way which makes the standard approaches inapplicable. We apply this analysis to study the behaviour of fronts of excitation waves in spatially-extended cardiac models. Suppressing the excitability of the tissue leads to a decrease in the propagation speed, but only to a certain limit; further suppression blocks active propagation and leads to a passive diffusive spread of voltage. Such a dissipation may happen if a front propagates into a tissue recovering after a previous wave, e.g. re-entry. A dissipated front does not recover even when the excitability restores. This has no analogy in FitzHugh-Nagumo model and its variants, where fronts can stop and then start again. In two spatial dimensions, dissipation accounts for break-ups and self-termination of re-entrant waves in excitable media with Courtemanche et al. (1998) kinetics.Comment: 15 pages, 8 figures, to appear in Phil Trans Roy Soc London

Conditions for propagation and block of excitation in an asymptotic model of atrial tissue

Detailed ionic models of cardiac cells are difficult for numerical simulations because they consist of a large number of equations and contain small parameters. The presence of small parameters, however, may be used for asymptotic reduction of the models. Earlier results have shown that the asymptotics of cardiac equations are non-standard. Here we apply such a novel asymptotic method to an ionic model of human atrial tissue in order to obtain a reduced but accurate model for the description of excitation fronts. Numerical simulations of spiral waves in atrial tissue show that wave fronts of propagating action potentials break-up and self-terminate. Our model, in particular, yields a simple analytical criterion of propagation block, which is similar in purpose but completely different in nature to the `Maxwell rule' in the FitzHugh-Nagumo type models. Our new criterion agrees with direct numerical simulations of break-up of re-entrant waves.Comment: Revised manuscript submitted to Biophysical Journal (30 pages incl. 10 figures

Dispersion of cardiac action potential duration and the initiation of re-entry: A computational study

BACKGROUND: The initiation of re-entrant cardiac arrhythmias is associated with increased dispersion of repolarisation, but the details are difficult to investigate either experimentally or clinically. We used a computational model of cardiac tissue to study systematically the association between action potential duration (APD) dispersion and susceptibility to re-entry. METHODS: We simulated a 60 × 60 mm 2 D sheet of cardiac ventricular tissue using the Luo-Rudy phase 1 model, with maximal conductance of the K+ channel gKmax set to 0.004 mS mm-2. Within the central 40 × 40 mm region we introduced square regions with prolonged APD by reducing gKmax to between 0.001 and 0.003 mS mm-2. We varied (i) the spatial scale of these regions, (ii) the magnitude of gKmax in these regions, and (iii) cell-to-cell coupling. RESULTS: Changing spatial scale from 5 to 20 mm increased APD dispersion from 49 to 102 ms, and the susceptible window from 31 to 86 ms. Decreasing gKmax in regions with prolonged APD from 0.003 to 0.001 mS mm-2 increased APD dispersion from 22 to 70 ms, and the susceptible window from <1 to 56 ms. Decreasing cell-to-cell coupling by changing the diffusion coefficient from 0.2 to 0.05 mm2 ms-1 increased APD dispersion from 57 to 88 ms, and increased the susceptible window from 41 to 74 ms. CONCLUSION: We found a close association between increased APD dispersion and susceptibility to re-entrant arrhythmias, when APD dispersion is increased by larger spatial scale of heterogeneity, greater electrophysiological heterogeneity, and weaker cell-to-cell coupling

Asymptotic analysis and analytical solutions of a model of cardiac excitation.

The original publication is available at www.springerlink.com - http://link.springer.com/article/10.1007/s11538-007-9267-0Journal ArticleCopyright © SpringerWe describe an asymptotic approach to gated ionic models of single-cell cardiac excitability. It has a form essentially different from the Tikhonov fast-slow form assumed in standard asymptotic reductions of excitable systems. This is of interest since the standard approaches have been previously found inadequate to describe phenomena such as the dissipation of cardiac wave fronts and the shape of action potential at repolarization. The proposed asymptotic description overcomes these deficiencies by allowing, among other non-Tikhonov features, that a dynamical variable may change its character from fast to slow within a single solution. The general asymptotic approach is best demonstrated on an example which should be both simple and generic. The classical model of Purkinje fibers (Noble in J. Physiol. 160:317-352, 1962) has the simplest functional form of all cardiac models but according to the current understanding it assigns a physiologically incorrect role to the Na current. This leads us to suggest an "Archetypal Model" with the simplicity of the Noble model but with a structure more typical to contemporary cardiac models. We demonstrate that the Archetypal Model admits a complete asymptotic solution in quadratures. To validate our asymptotic approach, we proceed to consider an exactly solvable "caricature" of the Archetypal Model and demonstrate that the asymptotic of its exact solution coincides with the solutions obtained by substituting the "caricature" right-hand sides into the asymptotic solution of the generic Archetypal Model. This is necessary, because, unlike in standard asymptotic descriptions, no general results exist which can guarantee the proximity of the non-Tikhonov asymptotic solutions to the solutions of the corresponding detailed ionic model

Tissue engineering of the human atrium : approaching mechanisms of genesis and control of atrial fibrillation

Cardiovascular disease is prevalent across the western world and is a major cause of morbidity and mortality, accounting for approximately a third of all fatalities. Investigating the heart by simulating its electrophysiology via the aid of mathematical models has advanced significantly over the past 60 years and is now a well established field. While much of the research focus is placed on the ventricles, the study of the atria is in comparison neglected. Therefore this Thesis is focused on the genesis and maintenance of atrial fibrillation (AF). A series of case studies are performed whereby established biophysically detailed mathematical models are implemented and modified to incorporate electrophysical alterations of atrial cells resulting from a variety of external conditions. The opening section of this Thesis is dedicated to developing a background to the field, including a discussion into the clinical aspect of the diagnosis and management of AF. The suitability of two atrial cell models is discussed and the development of single cell, 1D, 2D, and 3D multi-scale simulation protocols are described in detail. In addition measurements taken to quantify the arrhythmogenic properties of the cells susceptibility to AF are outlined. The second section is focused on the incorporation of conditions thought to enhance atrial tissues ability to initiate and maintain the genesis of AF. Included is a case study into the missence S140G gene mutation, and elevated physiological levels of the hormone Homocystein. The third section investigates the effectiveness of well established and widely used pharmacological treatments such as Beta-Blockers. In addition possible avenues of investigations for the development of atrial specific drugs are explored. These include blocking of the ultra rapid potassium channel and a more novel target for therapy via the targeting of 5HT4 receptors; which is transcribed solely in the atria and alters the electrophysical properties of the L-type Calcium current. The final part of this Thesis is dedicated to the development of a 2D atrial sheet model which includes electrical and spatial heterogeneities via the inclusion of multiple cell types and basic fiber orientation respectively. This allows for an investigation into the role that heterogeneities play in role genesis and maintenance of AF. The main finding of this Thesis is that alterations to the electrophysiology of atrial cells, due to external factors, can be successfully simulated via the implementation of mathematically detailed atrial cell models. It is concluded that simulations of the KENQ1 mutation and elevated levels of Homocystein successfully reproduce conditions which increase the onset of AF. Established treatments such as Beta-Blockers are found to have limited effectiveness. Possible theoretical treatments, such as the blocking of IKur, are found to provide a small amount of therapeutic benefit. In contrast, investigations into the effects of Serotonin were inconclusive. The study into the 2D atria indicated the importance that heterogeneities play in atria. The conclusions show that models provide a powerful tool when investigating how changes to electrophysiology of cells are manifested at a multi-scale level. The models also have their limitations and require further advancement to improve their accuracy.EThOS - Electronic Theses Online ServiceEPSRCGBUnited Kingdo

Stories from different worlds in the universe of complex systems: A journey through microstructural dynamics and emergent behaviours in the human heart and financial markets

A physical system is said to be complex if it exhibits unpredictable structures, patterns or regularities emerging from microstructural dynamics involving a large number of components. The study of complex systems, known as complexity science, is maturing into an independent and multidisciplinary area of research seeking to understand microscopic interactions and macroscopic emergence across a broad spectrum systems, such as the human brain and the economy, by combining specific modelling techniques, data analytics, statistics and computer simulations. In this dissertation we examine two different complex systems, the human heart and financial markets, and present various research projects addressing specific problems in these areas. Cardiac fibrillation is a diffuse pathology in which the periodic planar electrical conduction across the cardiac tissue is disrupted and replaced by fast and disorganised electrical waves. In spite of a century-long history of research, numerous debates and disputes on the mechanisms of cardiac fibrillation are still unresolved while the outcomes of clinical treatments remain far from satisfactory. In this dissertation we use cellular automata and mean-field models to qualitatively replicate the onset and maintenance of cardiac fibrillation from the interactions among neighboring cells and the underlying topology of the cardiac tissue. We use these models to study the transition from paroxysmal to persistent atrial fibrillation, the mechanisms through which the gap-junction enhancer drug Rotigaptide terminates cardiac fibrillation and how focal and circuital drivers of fibrillation may co-exist as projections of transmural electrical activities. Financial markets are hubs in which heterogeneous participants, such as humans and algorithms, adopt different strategic behaviors to exchange financial assets. In recent decades the widespread adoption of algorithmic trading, the electronification of financial transactions, the increased competition among trading venues and the use of sophisticated financial instruments drove the transformation of financial markets into a global and interconnected complex system. In this thesis we introduce agent-based and state-space models to describe specific microstructural dynamics in the stock and foreign exchange markets. We use these models to replicate the emergence of cross-currency correlations from the interactions between heterogeneous participants in the currency market and to disentangle the relationships between price fluctuations, market liquidity and demand/supply imbalances in the stock market.Open Acces

HYSTERESIS IN REPOLARIZATION OF CARDIAC ACTION POTENTIALS: EFFECTS OF SPATIAL HETEROGENEITY AND SLOW REPOLARIZATION CURRENTS

Repolarization alternans, i.e. beat-to-beat variation of repolarization of action potential, is proposed as a predictor of life-threatening arrhythmias. Restitution relates repolarization duration with its previous relaxation time, i.e. diatstolic interval (DI), and is considered a dominant mechanism for alternans. Previously, we observed that different repolarization durations at the same DI during decelerating and accelerating pacing, i.e. restitution displays hysteresis, which is a measure of “cardiac memory”. Objective of the current study was to investigate in the pig 1) the mechanism for a previously observed hysteresis type phenomenon, where alternans, once started at higher heart rate, persists even when heart rate decreases below its initiating rate, 2) regional differences in expression of hysteresis, i.e. memory in restitution in the heart, and 3) changes in restitution and memory during manipulation of an important repolarization current, the slow delayed rectifier, IKs. Action potentials were recorded in pig ventricular tissues using microelectrodes. Regional differences were explored in endocardial and epicardial tissues from both ventricles. DIs were explicitly controlled in real time to separate restitution mechanism from non-restitution related effects. Stepwise protocols were used to explore the existence in hysteresis in alternans threshold, where DIs were held constant for each step and progressively decreased and then increased. Quantification of cardiac memory was achieved by sinusoidally changing DI protocols, which were used to investigate memory changes among myocytes from different regions of the heart and during IKs manipulation. Results show that during stepwise protocol, hysteresis in alternans still existed, which indicates that restitution is not the only mechanism underlying the hysteresis. When comparing hysteresis obtained from sinusoidally oscillatory DIs among different regions, results show memory is expressed differently with endocardium expressing the most and epicardium the least memory. This provides important implications about the location where arrhythmia would initiate. Results also show that measures for hysteresis loops obtained by sinusoidal DI protocols decreased (increased) after enhancement (attenuation) of IKs, suggesting decreased (increased) hysteresis, i.e. memory in restitution. This effect needs to be considered during drug development

Fundamentals of Multiphase Flow

The subject of multiphase flows encompasses a vast field, a host of different technological contexts, a wide spectrum of different scales, a broad range of engineering disciplines and a multitude of different analytical approaches. Not surprisingly, the number of books dealing with the subject is voluminous. For the student or researcher in the field of multiphase flow this broad spectrum presents a problem for the experimental or analytical methodologies that might be appropriate for his/her interests can be widely scattered and difficult to find. The aim of the present text is to try to bring much of this fundamental understanding together into one book and to present a unifying approach to the fundamental ideas of multiphase flows. Consequently the book summarizes those fundamental concepts with relevance to a broad spectrum of multiphase flows. It does not pretend to present a comprehensive review of the details of any one multiphase flow or technological context though reference to books providing such reviews is included where appropriate. This book is targeted at graduate students and researchers at the cutting edge of investigations into the fundamental nature of multiphase flows; it is intended as a reference book for the basic methods used in the treatment of multiphase flows