Waves in the Skyrme--Faddeev model and integrable reductions

In the present article we show that the Skyrme--Faddeev model possesses nonlinear wave solutions, which can be expressed in terms of elliptic functions. The Whitham averaging method has been exploited in order to describe slow deformation of periodic wave states, leading to a quasi-linear system. The reduction to general hydrodynamic systems have been considered and it is compared with other integrable reductions of the system.Comment: 16 pages, 5 figure

A normal form for excitable media

We present a normal form for travelling waves in one-dimensional excitable media in form of a differential delay equation. The normal form is built around the well-known saddle-node bifurcation generically present in excitable media. Finite wavelength effects are captured by a delay. The normal form describes the behaviour of single pulses in a periodic domain and also the richer behaviour of wave trains. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with the saddle-node in a Bogdanov-Takens point, and a symmetry breaking spatially inhomogeneous pitchfork bifurcation. We verify the existence of these bifurcations in numerical simulations. The parameters of the normal form are determined and its predictions are tested against numerical simulations of partial differential equation models of excitable media with good agreement.Comment: 22 pages, accepted for publication in Chao

Wave Instabilities in Excitable Media with Fast Inhibitor Diffusion

An excitable activator-inhibitor system with relatively fast inhibitor diffusion is considered. Numerical simulations of wave propagation inside long channels show transitions from stable flat traveling waves to folded waves and further to spreading spiral turbulence as the inhibitor diffusivity is increased. For sufficiently narrow channels the suppression of turbulence and the development of regular steadily propagating patterns is observed. The curvature dependence of the wave propagation velocity is derived and used to interpret the observed phenomena

Controlling Spiral Waves in Confined Geometries by Global Feedback

The evolution of spiral waves on a circular domain and on a spherical surface is studied by numerical integration of a reaction-diffusion system with a global feedback. It is shown that depending on intensity, sign, and/or time delay in the feedback loop a global coupling can be effectively used either to stabilize the rigid rotation of a spiral wave or to completely destroy spiral waves and to suppress self-sustained activity in a confined domain of an excitable medium. An explanation of the numerically observed effects is produced by a kinematical model of spiral wave propagation

On the two-point boundary value problem for quadratic second-order differential equations and inclusions on manifolds

The two-point boundary value problem for second-order differential inclusions of the form (D/dt)m˙(t)∈F(t,m(t),m˙(t)) on complete Riemannian manifolds is investigated for a couple of points, nonconjugate along at least one geodesic of Levi-Civitá connection, where D/dt is the covariant derivative of Levi-Civitá connection and F(t,m,X) is a set-valued vector with quadratic or less than quadratic growth in the third argument. Some interrelations between certain geometric characteristics, the distance between points, and the norm of right-hand side are found that guarantee solvability of the above problem for F with quadratic growth in X. It is shown that this interrelation holds for all inclusions with F having less than quadratic growth in X, and so for them the problem is solvable

In silico optical control of pinned electrical vortices in an excitable biological medium

Vortices of excitation are generic to any complex excitable system. In the heart, they occur as rotors, spirals (2D) and scroll waves (3D) of electrical activity that are associated with rhythm disorders, known as arrhythmias. Lethal cardiac arrhythmias often result in sudden death, which is one of the leading causes of mortality in the industrialized world. Irrespective of the nature of the excitable medium, the rotation of a rotor is driven by its dynamics at the (vortex) core. In a recent study, Majumder et al (2018 eLife 7 e41076) demonstrated, using in silico and in vitro cardiac optogenetics, that light-guided manipulation of the core of free rotors can be used to establish real-time spatiotemporal control over the position, number and rotation of these rotors in cardiac tissue. Strategic application of this method, called 'Attract-Anchor-Drag' (AAD) can also be used to eliminate free rotors from the heart and stop cardiac arrhythmias. However, rotors in excitable systems, can pin (anchor) around local heterogeneities as well, thereby limiting their dynamics and possibility for spatial control. Here, we expand our results and numerically demonstrate, that AAD method can also detach anchored vortices from inhomogeneities and subsequently control their dynamics in excitable systems. Thus, overall we demonstrate that AAD control is one of the first universal methods that can be applied to both free and pinned vortices, to ensure their spatial control and removal from the heart and, possibly, other excitable systems