Analysis of a three-component model phase diagram by Catastrophe Theory: Potentials with two Order Parameters

In this work we classify the singularities obtained from the Gibbs potential of a lattice gas model with three components, two order parameters and five control parameters applying the general theorems provided by Catastrophe Theory. In particular, we clearly establish the existence of Landau potentials in two variables or, in other words, corank 2 canonical forms that are associated to the hyperbolic umbilic, D_{+4}, its dual the elliptic umbilic, D_{-4}, and the parabolic umbilic, D_5, catastrophes. The transversality of the potential with two order parameters is explicitely shown for each case. Thus we complete the Catastrophe Theory analysis of the three-component lattice model, initiated in a previous paper.Comment: 17 pages, 3 EPS figures, Latex file, continuation of Phys. Rev. B57, 13527 (1998) (cond-mat/9707015), submitted to Phys. Rev.

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

Wave competition in excitable modulated media.

The propagation of an initially planar front is studied within the framework of the photosensitive Belousov-Zhabotinsky reaction modulated by a smooth spatial variation of the local front velocity in the direction perpendicular to front propagation. Under this modulation, the wave front develops several fingers corresponding to the local maxima of the modulation function. After a transient, the wave front achieves a stationary shape that does not necessarily coincide with the one externally imposed by the modulation. Theoretical predictions for the selection criteria of fingers and steady-state velocity are experimentally validated

Asymptotic analysis and analytical solutions of a model of cardiac excitation

We 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