32 research outputs found
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 FitzHughâNagumo model and its variation by Barkley. They are treated with standard singular perturbation techniques. The ionic models include a simplified âcaricatureâ of Noble (J. Physiol. Lond. 160:317â352, 1962) model and Beeler and Reuter (J. Physiol. Lond. 268:177â210, 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 BeelerâReuter model. We discuss new mathematical features appearing in cardiac ionic models and possible applications of the developed method
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 (âcritical frontâ) as the threshold event between successful initation and decay, which plays a role similar to the âcritical nucleusâ in the theory of initiation of waves in the FitzHugh-Nagumo system
Excitable media in open and closed chaotic flows
We investigate the response of an excitable medium to a localized
perturbation in the presence of a two-dimensional smooth chaotic flow. Two
distinct types of flows are numerically considered: open and closed. For both
of them three distinct regimes are found, depending on the relative strengths
of the stirring and the rate of the excitable reaction. In order to clarify and
understand the role of the many competing mechanisms present, simplified models
of the process are introduced. They are one-dimensional baker-map models for
the flow and a one-dimensional approximation for the transverse profile of the
filaments.Comment: 14 pages, 16 figure
Analytically Solvable Asymptotic Model of Atrial Excitability
We report a three-variable simplified model of excitation fronts in human
atrial tissue. The model is derived by novel asymptotic techniques \new{from
the biophysically realistic model of Courtemanche et al (1998) in extension of
our previous similar models. An iterative analytical solution of the model is
presented which is in excellent quantitative agreement with the realistic
model. It opens new possibilities for analytical studies as well as for
efficient numerical simulation of this and other cardiac models of similar
structure
Vortex Glass and Vortex Liquid in Oscillatory Media
We study the disordered, multi-spiral solutions of two-dimensional
homogeneous oscillatory media for parameter values at which the single
spiral/vortex solution is fully stable. In the framework of the complex
Ginzburg-Landau (CGLE) equation, we show that these states, heretofore believed
to be static, actually evolve on ultra-slow timescales. This is achieved via a
reduction of the CGLE to the evolution of the sole vortex position and phase
coordinates. This true defect-mediated turbulence occurs in two distinct
phases, a vortex liquid characterized by normal diffusion of individual
spirals, and a slowly relaxing, intermittent, ``vortex glass''.Comment: 4 pages, 2 figures, submitted to Physical Review Letter
Scroll waves in isotropic excitable media : linear instabilities, bifurcations and restabilized states
Scroll waves are three-dimensional analogs of spiral waves. The linear
stability spectrum of untwisted and twisted scroll waves is computed for a
two-variable reaction-diffusion model of an excitable medium. Different bands
of modes are seen to be unstable in different regions of parameter space. The
corresponding bifurcations and bifurcated states are characterized by
performing direct numerical simulations. In addition, computations of the
adjoint linear stability operator eigenmodes are also performed and serve to
obtain a number of matrix elements characterizing the long-wavelength
deformations of scroll waves.Comment: 30 pages 16 figures, submitted to Phys. Rev.
A cross-diffusion model of forest boundary dynamics
A simple mathematical model of mono-species forest with two age classes which takes into account seed production and dispersal is presented in the paper. This reaction â diffusion type model is then reduced by means of an asymptotic procedure to a lower dimensional reaction â cross-diffusion model. The existence of standing and travelling wave front solutions corresponding to the forest boundary is shown for the later model. On the basis of the analysis, possible changes in forest boundary dynamics caused by antropogenic impacts are discussed
Dissipation of Excitation Fronts as a Mechanism of Conduction Block in Re-entrant Waves
Abstract. Numerical simulations of re-entrant waves in detailed ionic models reveal a phenomenon that is impossible in traditional simplified mathematical models of FitzHugh-Nagumo type: dissipation of the ex-citation front (DEF). We have analysed the structure of three selected ionic models, identified the small parameters that appear in non-standard ways, and developed an asymptotic approach based on those. Contrary to a common belief, the fast Na current inactivation gate h is not necessar-ily much slower than the transmembrane voltage E during the upstroke of the action potential. Interplay between E and h is responsible for the DEF. A new simplified model emerges from the asymptotic analysis and considers E and h as equally fast variables. This model reproduces DEF and admits analytical study. In particular, it yields conditions for the DEF. Predictions of the model agree with the results of direct numerical simulations of spiral wave break-up in a detailed model. 1 Introduction