Numerical methods for the detection of phase defect structures in excitable media

Electrical waves that rotate in the heart organize dangerous cardiac arrhythmias. Finding the region around which such rotation occurs is one of the most important practical questions for arrhythmia management. For many years, the main method for finding such regions was so-called phase mapping, in which a continuous phase was assigned to points in the heart based on their excitation status and defining the rotation region as a point of phase singularity. Recent analysis, however, showed that in many rotation regimes there exist phase discontinuities and the region of rotation must be defined not as a point of phase singularity, but as a phase defect line. In this paper we use this novel methodology and perform comparative study of three different phase definitions applied to in-silico data and to experimental data obtained from optical voltage mapping experiments on monolayers of human atrial myocytes. We introduce new phase defect detection algorithms and compare them with those that appeared in literature already. We find that the phase definition is more important than the algorithm to identify sudden spatial phase variations. Sharp phase defect lines can be obtained from a phase derived from local activation times observed during one cycle of arrhythmia. Alternatively, similar quality can be obtained from a reparameterization of the classical phase obtained from observation of a single timeframe of transmembrane potential. We found that the phase defect line length was 35.9(62)mm in the Fenton-Karma model and 4.01(55)mm in cardiac human atrial myocyte monolayers. As local activation times are obtained during standard clinical cardiac mapping, the methods are also suitable to be applied to clinical datasets. All studied methods are publicly available and can be downloaded from an institutional web-server.

Particle-like behavior of defects near a defect line in 2D Ising model: Defect–antidefect pair production and interaction

The aim of this work is to investigate Casimir effect in a system comprising of a defect line along with isolated defects (vacancies) in 2D Ising model. We have found out that the interaction energy has a decaying exponent with distance between defects. We are interested in an analogy between Casimir behavior of this defect structure and quantum field theory. The simplest deformation of a defect line (a defect’s position change) can be treated as defect–“antidefect” pair creation. Single defect is attracted to a defect line. By means of Monte Carlo simulation, the energy of pair creation and Casimir interaction potential are calculated. The interaction turned out that a Yukawa potential turns to the Coulomb’s one at phase transition point.

Complex-Periodic Spiral Waves Induced by Linearly Polarized Electric Field in the Excitable Medium

Complex-periodic spiral waves are investigated extensively in the oscillatory medium. In this paper, the linearly polarized electric field (LPEF) is employed to induce complex-periodic spiral waves in the excitable medium with abnormal dispersion. As the amplitude of LPEF is increased beyond a threshold, the simple-periodic spiral wave converts into an irregularly complex-periodic one, in which, the local dynamics exhibit several regular spikes followed by one missed spiking period. Furthermore, with the increase of the LPEF amplitude, the missed spiking period follows different numbers of regular spikes [so-called period-1 (P-1), period-2 (P-2), etc.], even a mix of different periods. Meanwhile, the wavelength of the spiral wave transits from a short to a longer one. The pure-periodic (from P-6 to P-2) spirals generally contain defect lines, across which the phase of local oscillation changes by [Formula: see text]. In contrast, there is no defect line in the mixed-periodic spiral waves. This finding indicates that the defect line is not a necessary feature for complex-periodic spiral waves. Moreover, three types of tip trajectories of pure-periodic spiral waves are identified depending on the periods. That is, the outward-petal meandering, the outward-petal meandering with slow modulation, and drifting tip motion, and the tip trajectories could be used to distinguish them from the complex-oscillatory spiral waves.

PINNING ON A DEFECT LINE: CHARACTERIZATION OF MARGINAL DISORDER RELEVANCE AND SHARP ASYMPTOTICS FOR THE CRITICAL POINT SHIFT

The effect of disorder for pinning models is a subject which has attracted much attention in theoretical physics and rigorous mathematical physics. A peculiar point of interest is the question of coincidence of the quenched and annealed critical point for a small amount of disorder. The question has been mathematically settled in most cases in the last few years, giving in particular a rigorous validation of the Harris criterion on disorder relevance. However, the marginal case, where the return probability exponent is equal to $1/2$, that is, where the interarrival law of the renewal process is given by $\text{K}(n)=n^{-3/2}\unicode[STIX]{x1D719}(n)$ where $\unicode[STIX]{x1D719}$ is a slowly varying function, has been left partially open. In this paper, we give a complete answer to the question by proving a simple necessary and sufficient criterion on the return probability for disorder relevance, which confirms earlier predictions from the literature. Moreover, we also provide sharp asymptotics on the critical point shift: in the case of the pinning of a one-dimensional simple random walk, the shift of the critical point satisfies the following high temperature asymptotics $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D6FD}\rightarrow 0}\unicode[STIX]{x1D6FD}^{2}\log h_{c}(\unicode[STIX]{x1D6FD})=-\frac{\unicode[STIX]{x1D70B}}{2}.\end{eqnarray}$$ This gives a rigorous proof to a claim of Derrida, Hakim and Vannimenus (J. Stat. Phys. 66 (1992), 1189–1213).