Excitation Fronts on a Periodically Modulated Curved Surface

Abstract

The evolution of an excitation front propagating on a nonuniformly curved surface is considered within the framework of a kinematical model of its motion. For the case of a surface with a periodically modulated curvature an exact solution of the front shape is obtained under the assumption of sufficiently small surface deformation. The results of the theoretical consideration are compared with the experimental data obtained with a modified Belousov-Zhabotinsky reaction in a thin nonuniformly curved layer.

A mathematical model of the unidirectional block caused by the pulmonary veins for anatomically induced atrial reentry

“…This agrees with the firstly predicted phenomenon from Eq. (33). However, the ς φφ anisotropy cannot block the propagation in the result.…”

“…The kinematic equation defines the relationship between the curvature of the wavefront and its arc length on curved surfaces to derive the conditions of the critical curvature, or the critical amount of line curvature of the wavefront, which results in breaking up the propagation. The kinematics approach has been successfully used to describe the geometrical effects on surfaces such as non-uniformly curved surfaces [32], periodically modulated curved surfaces [33] and toruses [34] as well as ring-shaped propagation on curved surfaces [35] and propagation on moving excitable media [36]. In addition, this approach was also used to analyze the role of anisotropy in the plane: the breakup of the propagation [37] and the propagation of curved fronts in anisotropic excitable media [38].…”

Critical properties of excitation waves on curved surfaces: Curvature-dependent loss of excitability

“…With decreasing parameter η we can expect that after some time its value will be so reduced that the condition (9) will be satisfied and we should be able to observe the damping. In the experiments described in [9] we used fresh reaction solution and thus no critical phenomena as considered in this work were observed.…”

“…The Gaussian curvature of the surface (5) can be written as [9] Γ(x, It is the same in the minima and the maxima and will induce periodic deformations of an initially flat front. These deformations were analytically studied by making use of the main kinematic equation (2) and the eikonal equation (3) and compared with experimental results [9]. The expression for the geodetic curvature of the front propagating in an arbitrary direction on the curved surface was in a very good agreement with the experimental results.…”

“…An important example is the heart, where the excitation waves move along the muscle in a ring-like fashion. Although during the last decade some properties of excitation fronts on curved surfaces were studied theoretically [4-6] and experimentally [7][8][9], there are still many gaps in our understanding of processes in non-planar systems. An important aspect, especially in biological systems, is the study of possible mechanisms for controlling these waves.…”