Asymptotics of conduction velocity restitution in models of electrical excitation in the heart

We extend a non-Tikhonov asymptotic embedding, proposed earlier, for calculation of conduction velocity restitution curves in ionic models of cardiac excitability. Conduction velocity restitution is the simplest non-trivial spatially extended problem in excitable media, and in the case of cardiac tissue it is an important tool for prediction of cardiac arrhythmias and fibrillation. An idealized conduction velocity restitution curve requires solving a non-linear eigenvalue problem with periodic boundary conditions, which in the cardiac case is very stiff and calls for the use of asymptotic methods. We compare asymptotics of restitution curves in four examples, two generic excitable media models, and two ionic cardiac models. The generic models include the classical FitzHugh–Nagumo model and its variation by Barkley. They are treated with standard singular perturbation techniques. The ionic models include a simplified “caricature” of Noble (J. Physiol. Lond. 160:317–352, 1962) model and Beeler and Reuter (J. Physiol. Lond. 268:177–210, 1977) model, which lead to non-Tikhonov problems where known asymptotic results do not apply. The Caricature Noble model is considered with particular care to demonstrate the well-posedness of the corresponding boundary-value problem. The developed method for calculation of conduction velocity restitution is then applied to the Beeler–Reuter model. We discuss new mathematical features appearing in cardiac ionic models and possible applications of the developed method

Fast-slow asymptotics for a Markov chain model of fast sodium current

We explore the feasibility of using fast-slow asymptotic to eliminate the computational stiffness of the discrete-state, continuous-time deterministic Markov chain models of ionic channels underlying cardiac excitability. We focus on a Markov chain model of the fast sodium current, and investigate its asymptotic behaviour with respect to small parameters identified in different ways.Comment: 16 pages, 6 figures, as accepted to Chaos 2017/09/0

Dynamics of filaments of scroll waves

This has been written as a chapter for "Engineering Chemical Complexity II", and as such does not have an abstract.Comment: 18 pages, 10 figure

Exponential integrators for a Markov chain model of the fast sodium channel of cardiomyocytes

The modern Markov chain models of ionic channels in excitable membranes are numerically stiff. The popular numerical methods for these models require very small time steps to ensure stability. Our objective is to formulate and test two methods addressing this issue, so that the timestep can be chosen based on accuracy rather than stability. Both proposed methods extend Rush-Larsen technique, which was originally developed to Hogdkin-Huxley type gate models. One method, "Matrix Rush-Larsen" (MRL) uses a matrix reformulation of the Rush-Larsen scheme, where the matrix exponentials are calculated using precomputed tables of eigenvalues and eigenvectors. The other, "hybrid operator splitting" (HOS) method exploits asymptotic properties of a particular Markov chain model, allowing explicit analytical expressions for the substeps. We test both methods on the Clancy and Rudy (2002) INa Markov chain model. With precomputed tables for functions of the transmembrane voltage, both methods are comparable to the forward Euler method in accuracy and computational cost, but allow longer time steps without numerical instability. We conclude that both methods are of practical interest. MRL requires more computations than HOS, but is formulated in general terms which can be readily extended to other Markov Chain channel models, whereas the utility of HOS depends on the asymptotic properties of a particular model. The significance of the methods is that they allow a considerable speed-up of large-scale computations of cardiac excitation models by increasing the time step, while maintaining acceptable accuracy and preserving numerical stability.Comment: 9 pages, 5 figures main text + 14 pages, 1 figure appendix, as submitted in final form to IEEE TBME 2014/11/11. Copyright IEEE (2014

Quasisolitons in self-diffusive excitable systems, or Why asymmetric diffusivity does not violate the Second Law

Solitons, defined as nonlinear waves which can reflect from boundaries or transmit through each other, are found in conservative, fully integrable systems. Similar phenomena, dubbed quasi-solitons, have been observed also in dissipative, "excitable" systems, either at finely tuned parameters (near a bifurcation) or in systems with cross-diffusion. Here we demonstrate that quasi-solitons can be robustly observed in excitable systems with excitable kinetics and with self-diffusion only. This includes quasi-solitons of fixed shape (like KdV solitons) or envelope quasi-solitons (like NLS solitons). This can happen in systems with more than two components, and can be explained by effective cross-diffusion, which emerges via adiabatic elimination of a fast but diffusing component. We describe here a reduction procedure can be used for the search of complicated wave regimes in multi-component, stiff systems by studying simplified, soft systems.Comment: 11 pages, 2 figures, as accepted to Scientific Reports on 2016/07/0

Complex dynamics of the biological rhythms: gallbladder and heart cases

A theoretical analysis of the mechanisms underlying the dynamics of gallbladder and heart pulsation could clarify the question regarding the classification as chaotic of the associated behaviour, eventually related to a normal and healthy beat; this analysis is particularly relevant in view of the control of dynamics bifurcations arising in situations of disease. In this work is presented a summary of the DFA method applied to gallbladder volume data for a modest number of healthy and ill patients: the presence of signal correlation is found in both cases, but the fit shapes differ from some critical values.Comment: 3 pages, 8 figures, to appear on Physica

Localization of response functions of spiral waves in the FitzHugh-Nagumo system

Dynamics of spiral waves in perturbed, e. g. slightly inhomogeneous or subject to a small periodic external force, two-dimensional autowave media can be described asymptotically in terms of Aristotelean dynamics, so that the velocities of the spiral wave drift in space and time are proportional to the forces caused by the perturbation. The forces are defined as a convolution of the perturbation with the spiral's Response Functions, which are eigenfunctions of the adjoint linearised problem. In this paper we find numerically the Response Functions of a spiral wave solution in the classic excitable FitzHugh-Nagumo model, and show that they are effectively localised in the vicinity of the spiral core.Comment: 11 pages, 2 figure

Evolution of spiral and scroll waves of excitation in a mathematical model of ischaemic border zone

Abnormal electrical activity from the boundaries of ischemic cardiac tissue is recognized as one of the major causes in generation of ischemia-reperfusion arrhythmias. Here we present theoretical analysis of the waves of electrical activity that can rise on the boundary of cardiac cell network upon its recovery from ischaemia-like conditions. The main factors included in our analysis are macroscopic gradients of the cell-to-cell coupling and cell excitability and microscopic heterogeneity of individual cells. The interplay between these factors allows one to explain how spirals form, drift together with the moving boundary, get transiently pinned to local inhomogeneities, and finally penetrate into the bulk of the well-coupled tissue where they reach macroscopic scale. The asymptotic theory of the drift of spiral and scroll waves based on response functions provides explanation of the drifts involved in this mechanism, with the exception of effects due to the discreteness of cardiac tissue. In particular, this asymptotic theory allows an extrapolation of 2D events into 3D, which has shown that cells within the border zone can give rise to 3D analogues of spirals, the scroll waves. When and if such scroll waves escape into a better coupled tissue, they are likely to collapse due to the positive filament tension. However, our simulations have shown that such collapse of newly generated scrolls is not inevitable and that under certain conditions filament tension becomes negative, leading to scroll filaments to expand and multiply leading to a fibrillation-like state within small areas of cardiac tissue.Comment: 26 pages, 13 figures, appendix and 2 movies, as accepted to PLoS ONE 2011/08/0

Strength-duration relationship in an excitable medium

This is the final version. Available on open access from Elsevier via the DOI in this recordWe consider the strength-duration relationship in one-dimensional spatially extended excitable media. In a previous study [36] set out to separate initial (or boundary) conditions leading to propagation wave solutions from those leading to decay solutions, an analytical criterion based on an approximation of the (center-)stable manifold of a certain critical solution was presented. The theoretical prediction in the case of strength-extent curve was later on extended to cover a wider class of excitable systems including multicomponent reaction-diffusion systems, systems with non-self-adjoint linearized operators and in particular, systems with moving critical solutions (critical fronts and critical pulses) [7]. In the present work, we consider extension of the theory to the case of strength-duration curve.Engineering and Physical Sciences Research Council (EPSRC)Ministry of National Education of the Republic of TurkeyNational Science FoundationNational Institutes of Health (NIH)Gordon and Betty Moore Foundatio