Analytical and Numerical Approaches to Initiation of Excitation Waves

This thesis studies the problem of initiation of propagation of excitation waves in one- dimensional spatially extended excitable media. In a study which set out to determine an analytical criteria for the threshold conditions, Idris and Biktashev [68] showed that the linear approximation of the (center-)stable manifold of a certain critical solution yields analytical approximation of the threshold curves, separating initial (or boundary) conditions leading to propagation wave solutions from those leading to decay solutions. The aim of this project is to extend this method to address a wider class of ex- citable 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). In the case of one-component excitable systems where the critical solution is the critical nucleus, we also extend the theory to a quadratic approximation for the purpose of improving the accuracy of the linear approximation. The applicability of the approach is tested through five test problems with either traveling front such as Biktashev model, a simplified cardiac excitation model or traveling pulse solutions including Beeler-Reuter model, near realistic cardiac excitation model. Apart from some exceptional cases, it is not always possible to obtain explicit solution for the essential ingredients of the theory due to the nonlinear nature of the problem. Thus, this thesis also covers a hybrid method, where these ingredients are found numerically. Another important finding of the research is the use of the perturbation theory to find the approximate solution of the essential ingredients of FitzHugh-Nagumo system by using the exact analytical solutions of its primitive ver- sion, Zeldovich-Frank-Kamenetsky equation

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

Fast-slow asymptotic for semi-analytical ignition criteria in FitzHugh-Nagumo system

We study the problem of initiation of excitation waves in the FitzHugh-Nagumo model. Our approach follows earlier works and is based on the idea of approximating the boundary between basins of attraction of propagating waves and of the resting state as the stable manifold of a critical solution. Here, we obtain analytical expressions for the essential ingredients of the theory by singular perturbation using two small parameters, the separation of time scales of the activator and inhibitor, and the threshold in the activator's kinetics. This results in a closed analytical expression for the strength-duration curve.Comment: 10 pages, 5 figures, as accepted to Chaos on 2017/06/2

Semi-analytical approach to criteria for ignition of excitation waves

We consider the problem of ignition of propagating waves in one-dimensional bistable or excitable systems by an instantaneous spatially extended stimulus. Earlier we proposed a method (Idris and Biktashev, PRL, vol 101, 2008, 244101) for analytical description of the threshold conditions based on an approximation of the (center-)stable manifold of a certain critical solution. Here we generalize this method to address a wider class of excitable systems, such as multicomponent reaction-diffusion systems and systems with non-self-adjoint linearized operators, including systems with moving critical fronts and pulses. We also explore an extension of this method from a linear to a quadratic approximation of the (center-)stable manifold, resulting in some cases in a significant increase in accuracy. The applicability of the approach is demonstrated on five test problems ranging from archetypal examples such as the Zeldovich--Frank-Kamenetsky equation to near realistic examples such as the Beeler-Reuter model of cardiac excitation. While the method is analytical in nature, it is recognised that essential ingredients of the theory can be calculated explicitly only in exceptional cases, so we also describe methods suitable for calculating these ingredients numerically.Comment: 31 page, 20 figures, as resubmitted to Phys Rev E on 2015/09/20 and accepted on 2015/09/2

Trigger versus Substrate: Multi-Dimensional Modulation of QT-Prolongation Associated Arrhythmic Dynamics by a hERG Channel Activator

Background: Prolongation of the QT interval of the electrocardiogram (ECG), underlain by prolongation of the action potential duration (APD) at the cellular level, is linked to increased vulnerability to cardiac arrhythmia. Pharmacological management of arrhythmia associated with QT prolongation is typically achieved through attempting to restore APD to control ranges, reversing the enhanced vulnerability to Ca²⁺-dependent afterdepolarisations (arrhythmia triggers) and increased transmural dispersion of repolarisation (arrhythmia substrate) associated with APD prolongation. However, such pharmacological modulation has been demonstrated to have limited effectiveness. Understanding the integrative functional impact of pharmacological modulation requires simultaneous investigation of both the trigger and substrate. Methods: We implemented a multi-scale (cell and tissue) in silico approach using a model of the human ventricular action potential, integrated with a model of stochastic 3D spatiotemporal Ca²⁺ dynamics, and parameter modification to mimic prolonged QT conditions. We used these models to examine the efficacy of the hERG activator MC-II-157c in restoring APD to control ranges, examined its effects on arrhythmia triggers and substrates, and the interaction of these arrhythmia triggers and substrates. Results: QT prolongation conditions promoted the development of spontaneous release events underlying afterdepolarisations during rapid pacing. MC-II-157c applied to prolonged QT conditions shortened the APD, inhibited the development of afterdepolarisations and reduced the probability of afterdepolarisations manifesting as triggered activity in single cells. In tissue, QT prolongation resulted in an increased transmural dispersion of repolarisation, which manifested as an increased vulnerable window for uni-directional conduction block. In some cases, MC-II-157c further increased the vulnerable window through its effects on INa. The combination of stochastic release event modulation and transmural dispersion of repolarisation modulation by MC-II-157c resulted in an integrative behavior wherein the arrhythmia trigger is reduced but the arrhythmia substrate is increased, leading to variable and non-linear overall vulnerability to arrhythmia. Conclusion: The relative balance of reduced trigger and increased substrate underlies a multi-dimensional role of MC-II-157c in modulation of cardiac arrhythmia vulnerability associated with prolonged QT interval