Dynamics of lattice spins as a model of arrhythmia

We consider evolution of initial disturbances in spatially extended systems with autonomous rhythmic activity, such as the heart. We consider the case when the activity is stable with respect to very smooth (changing little across the medium) disturbances and construct lattice models for description of not-so-smooth disturbances, in particular, topological defects; these models are modifications of the diffusive XY model. We find that when the activity on each lattice site is very rigid in maintaining its form, the topological defects - vortices or spirals - nucleate a transition to a disordered, turbulent state.Comment: 17 pages, revtex, 3 figure

The rate and predictors of healing of repaired lesser tuberosity osteotomy in reverse total shoulder arthroplasty

BACKGROUND: Evidence is building that a functional subscapularis improves function-specifically internal rotation tasks-following reverse total shoulder arthroplasty (rTSA). However, the optimal method for subscapularis repair during rTSA remains unknown with variable healing rates reported. This study aims to investigate the rate of and predictors for healing a lesser tuberosity osteotomy (LTO) following rTSA. METHODS: Following local institutional review board approval, patients with at least one-year follow-up for rTSA managed with an LTO and subsequent repair between March, 2017 and March, 2020 were retrospectively identified. Shoulders were selected for LTO repair based upon preoperative imaging and intraoperative assessment of subscapularis quality. All patients were implanted with a system consisting of a 150° or 155° (constrained) humeral neck-shaft angle and 2.5 to 4.5 millimeters (mm) of glenoid lateralization (Trabecular Metal Reverse Shoulder System; Zimmer Biomet, Warsaw, IN, USA). At a minimum of six months, radiographs were reviewed for an assessment of LTO healing by three independent reviewers. Healing was classified as displaced, fibrous union, or ossified union. For assessing predictors, the repair was considered intact if the LTO fragment was not displaced (fibrous union or ossified union). RESULTS: Sixty-five rTSA with LTO repair were performed in 64 patients. These patients had an average age of 67.2 years (range, 31-81) and 36 (55.4%; 36/65) were female. At an average follow-up of 15.2 months (range, 8-38), 50 cases (76.9%; 50/65) were classified as having an ossified union. The radiographic healing could not be assessed in a single case. Of the 14 cases without ossific union, 8 (12.3%; 8/65) were displaced and 6 (9.2%; 6/65) were classified as a fibrous union. In logistic regression, only combined humeral liner height predicted LTO displacement (odds ratio = 1.4 [95% confidence interval = 1.1-1.8]; CONCLUSION: This analysis demonstrates that radiographic healing of LTO repair is more favorable than published rates of healing after subscapularis tenotomy or peel in the setting of rTSA. Subscapularis management with LTO provides the ability to monitor repair integrity with plain radiographs and a predictable radiographic healing rate. The integrity of subscapularis repair may be influenced by the use of thicker humeral liners. Further investigation is needed to determine the functional impact of a healed subscapularis following rTSA

Limit theorems for weakly subcritical branching processes in random environment

For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the process may at the same time be subcritical and, conditioned on nonextinction, 'supercritical'. This so-called weakly subcritical case is considered in this paper. We study the asymptotic survival probability and the size of the population conditioned on non-extinction. Also a functional limit theorem is proven, which makes the conditional supercriticality manifest. A main tool is a new type of functional limit theorems for conditional random walks.Comment: 35 page

Doppler Effect of Nonlinear Waves and Superspirals in Oscillatory Media

Nonlinear waves emitted from a moving source are studied. A meandering spiral in a reaction-diffusion medium provides an example, where waves originate from a source exhibiting a back-and-forth movement in radial direction. The periodic motion of the source induces a Doppler effect that causes a modulation in wavelength and amplitude of the waves (``superspiral''). Using the complex Ginzburg-Landau equation, we show that waves subject to a convective Eckhaus instability can exhibit monotonous growth or decay as well as saturation of these modulations away from the source depending on the perturbation frequency. Our findings allow a consistent interpretation of recent experimental observations concerning superspirals and their decay to spatio-temporal chaos.Comment: 4 pages, 4 figure

Universal Scaling of Wave Propagation Failure in Arrays of Coupled Nonlinear Cells

We study the onset of the propagation failure of wave fronts in systems of coupled cells. We introduce a new method to analyze the scaling of the critical external field at which fronts cease to propagate, as a function of intercellular coupling. We find the universal scaling of the field throughout the range of couplings, and show that the field becomes exponentially small for large couplings. Our method is generic and applicable to a wide class of cellular dynamics in chemical, biological, and engineering systems. We confirm our results by direct numerical simulations.Comment: 4 pages, 3 figures, RevTe

Helicoidal instability of a scroll vortex in three-dimensional reaction-diffusion systems

We study the dynamics of scroll vortices in excitable reaction-diffusion systems analytically and numerically. We demonstrate that intrinsic three-dimensional instability of a straight scroll leads to the formation of helicoidal structures. This behavior originates from the competition between the scroll curvature and unstable core dynamics. We show that the obtained instability persists even beyond the meander core instability of two-dimensional spiral wave.Comment: 4 pages, 5 figures, revte

Theory of spiral wave dynamics in weakly excitable media: asymptotic reduction to a kinematic model and applications

In a weakly excitable medium, characterized by a large threshold stimulus, the free end of an isolated broken plane wave (wave tip) can either rotate (steadily or unsteadily) around a large excitable core, thereby producing a spiral pattern, or retract causing the wave to vanish at boundaries. An asymptotic analysis of spiral motion and retraction is carried out in this weakly excitable large core regime starting from the free-boundary limit of the reaction-diffusion models, valid when the excited region is delimited by a thin interface. The wave description is shown to naturally split between the tip region and a far region that are smoothly matched on an intermediate scale. This separation allows us to rigorously derive an equation of motion for the wave tip, with the large scale motion of the spiral wavefront slaved to the tip. This kinematic description provides both a physical picture and exact predictions for a wide range of wave behavior, including: (i) steady rotation (frequency and core radius), (ii) exact treatment of the meandering instability in the free-boundary limit with the prediction that the frequency of unstable motion is half the primary steady frequency (iii) drift under external actions (external field with application to axisymmetric scroll ring motion in three-dimensions, and spatial or/and time-dependent variation of excitability), and (iv) the dynamics of multi-armed spiral waves with the new prediction that steadily rotating waves with two or more arms are linearly unstable. Numerical simulations of FitzHug-Nagumo kinetics are used to test several aspects of our results. In addition, we discuss the semi-quantitative extension of this theory to finite cores and pinpoint mathematical subtleties related to the thin interface limit of singly diffusive reaction-diffusion models

Renormalization Group Theory for a Perturbed KdV Equation

We show that renormalization group(RG) theory can be used to give an analytic description of the evolution of a perturbed KdV equation. The equations describing the deformation of its shape as the effect of perturbation are RG equations. The RG approach may be simpler than inverse scattering theory(IST) and another approaches, because it dose not rely on any knowledge of IST and it is very concise and easy to understand. To the best of our knowledge, this is the first time that RG has been used in this way for the perturbed soliton dynamics.Comment: 4 pages, no figure, revte

Finite-gap Solutions of the Vortex Filament Equation: Isoperiodic Deformations

We study the topology of quasiperiodic solutions of the vortex filament equation in a neighborhood of multiply covered circles. We construct these solutions by means of a sequence of isoperiodic deformations, at each step of which a real double point is "unpinched" to produce a new pair of branch points and therefore a solution of higher genus. We prove that every step in this process corresponds to a cabling operation on the previous curve, and we provide a labelling scheme that matches the deformation data with the knot type of the resulting filament.Comment: 33 pages, 5 figures; submitted to Journal of Nonlinear Scienc