Critical fronts in initiation of excitation waves

Copyright © 2007 The American Physical SocietyJournal ArticleWe consider the problem of initiation of propagating waves in a one-dimensional excitable fiber. In the FitzHugh-Nagumo theory, the key role is played by "critical nucleus" and "critical pulse" solutions whose (center-) stable manifold is the threshold surface separating initial conditions leading to propagation and those leading to decay. We present evidence that in cardiac excitation models, this role is played by "critical front" solutions

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

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

Using novel simplified models of excitation for analytic description of initiation propagation and blockage of excitation waves

We consider applications of a recently suggested new asymptotic approach to detailed ionic models of cardiac excitation. First, we describe a three-variable approximation for the excitation fronts in a detailed ionic model of human atrial kinetics. It predicts not only the speed of the fronts but also a condition for failure of propagation, i.e. gives an operational definition of absolute refractoriness. This prediction is confirmed by direct simulations of the full model. Next, we consider problem of initiation of excitation waves, using a piecewise linear caricature of the INa-driven excitation front. We identify the unstable propagating front solution (“critical front”) as the threshold event between successful initation and decay, which plays a role similar to the “critical nucleus” in the theory of initiation of waves in the FitzHugh-Nagumo system

Predicting critical ignition in slow-fast excitable models

This is the final version. Available from American Physical Society via the DOI in this record.Linearization around unstable travelling waves in excitable systems can be used to approximate strength-extent curves in the problem of initiation of excitation waves for a family of spatially confined perturbations to the rest state. This theory relies on the knowledge of the unstable travelling wave solution as well as the leading left and right eigenfunctions of its linearization. We investigate the asymptotics of these ingredients, and utility of the resulting approximations of the strength-extent curves, in the slow-fast limit in two-component excitable systems of FitzHugh-Nagumo type, and test those on four illustrative models. Of these, two are with degenerate dependence of the fast kinetic on the slow variable, a feature which is motivated by a particular model found in the literature. In both cases, the unstable travelling wave solution converges to a stationary ``critical nucleus'' of the corresponding one-component fast subsystem. We observe that in the full system, the asymptotics of the left and right eigenspaces are distinct. In particular, the slow component of the left eigenfunction corresponding to the translational symmetry does not become negligible in the asymptotic limit. This has a significant detrimental effect on the critical curve predictions. The theory as formulated previously uses an heuristic to address a difficulty related to the translational invariance. We describe two alternatives to that heuristic, which do not use the misbehaving eigenfunction component. These new heuristics show much better predictive properties, including in the asymptotic limit, in all four examples.Engineering and Physical Sciences Research Council (EPSRC)National Science Foundation (NSF)National Institute for Health Research (NIHR)Gordon and Betty Moore Foundatio

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

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