Phase synchronization of a pair of spiral waves

Waves spontaneously generated by heterogeneity in oscillatory media

Wave propagation is an important characteristic for pattern formation and pattern dynamics. To date, various waves in homogeneous media have been investigated extensively and have been understood to a great extent. However, the wave behaviors in heterogeneous media have been studied and understood much less. In this work, we investigate waves that are spontaneously generated in one-dimensional heterogeneous oscillatory media governed by complex Ginzburg-Landau equations; the heterogeneity is modeled by multiple interacting homogeneous media with different system control parameters. Rich behaviors can be observed by varying the control parameters of the systems, whereas the behavior is incomparably simple in the homogeneous cases. These diverse behaviors can be fully understood and physically explained well based on three aspects: dispersion relation curves, driving-response relations, and wave competition rules in homogeneous systems. Possible applications of heterogeneity-generated waves are anticipated.

Wave competitions around interfaces of two oscillatory media

Wave competition is an important phenomenon for pattern formation and pattern control, which are required in many practical applications. In this work we consider the competition of wave trains around interfaces of two one-dimensional oscillatory media, modeled by complex Ginzburg-Landau equations with different system control parameters. The wave trains are generated by pacings at two boundaries of the media. In the presence of both normal wave trains and antiwave trains, extremely rich phenomena can be observed by varying the control parameters of the systems. These rich behaviors can be fully understood and physically well explained, based on the competitions of three kinds of wave trains: the two wave trains generated by external pacings and the wave train generated internally at the interface.

Pattern formation of coupled spiral waves in bilayer systems: rich dynamics and high-frequency dominance

The interaction of two spiral waves with independent frequencies in a bilayer oscillatory medium (one spiral in each layer) and with a symmetric coupling e is studied. If the spirals have different frequencies, the faster spiral is unaffected by the slower one, and the slower can show a variety of behaviors, which depend on e and include, in order of increasing e, phase drifting, amplitude modulation, amplitude domination, and phase synchronization. This high-frequency dominance, the asymmetric driving-response effect under the condition of a symmetric coupling, is generic and independent of whether the coupled spiral waves are outwardly rotating or inwardly rotating spirals. If the spirals have identical frequencies, they may even show complete synchronization, parallel drift, or circular drift, depending on the relative rotation direction of the two spirals and their initial separation distance. Comparisons with coupled spirals in monolayer media, previous works on coupled spirals in bilayer systems, and coupled phase oscillators are made.

Novel type of amplitude spiral wave in a two-layer system

Interaction of spiral waves in a two-layer system described by a model of coupled complex Ginzburg-Landau equations with negative-feedback couplings ε(1) and ε(2) is studied. Synchronization of two spiral waves can be broadly found if ε(1)+ε(2) is sufficiently large. Prior to the synchronization, under the condition of strongly asymmetric coupling (∣ε(1)-ε(2)∣≫0), a novel type of spiral wave, amplitude spiral wave, exists in the driven system. The pattern of amplitude spiral wave shows the spiral in the amplitude and without a singularity point (tip), compared to usual spiral waves known for phase with amplitude uniform far away from tips and rotating around tips.

Collision-based spiral acceleration in cardiac media: roles of wavefront curvature and excitable gap

We have previously shown in experimental cardiac cell monolayers that rapid point pacing can convert basic functional reentry (single spiral) into a stable multiwave spiral that activates the tissue at an accelerated rate. Here, our goal is to further elucidate the biophysical mechanisms of this rate acceleration without the potential confounding effects of microscopic tissue heterogeneities inherent to experimental preparations. We use computer simulations to show that, similar to experimental observations, single spirals can be converted by point stimuli into stable multiwave spirals. In multiwave spirals, individual waves collide, yielding regions with negative wavefront curvature. When a sufficient excitable gap is present and the negative-curvature regions are close to spiral tips, an electrotonic spread of excitatory currents from these regions propels each colliding spiral to rotate faster than the single spiral, causing an overall rate acceleration. As observed experimentally, the degree of rate acceleration increases with the number of colliding spiral waves. Conversely, if collision sites are far from spiral tips, excitatory currents have no effect on spiral rotation and multiple spirals rotate independently, without rate acceleration. Understanding the mechanisms of spiral rate acceleration may yield new strategies for preventing the transition from monomorphic tachycardia to polymorphic tachycardia and fibrillation.

Interface-selected waves and their influence on wave competition

In this paper we study nonlinear oscillatory systems consisting of two media, one supporting forward propagating waves and the other inwardly propagating waves, separated by an interface. We find that the interface can select the type of wave. Under certain well-defined parameter condition, these waves propagate in two different media with the same frequency and same wave number; the interface of the two media is transparent to these waves. The frequency and wave number of these interface-selected waves (ISWs) are predicted explicitly. When parameters are varied from this parameter set, the wave numbers of the two domains become different, and the difference increases from zero continuously as the distance between the given parameters and this parameter set increases from zero. It is found that ISWs can play crucial roles in practical problems of wave competition, e.g., ISWs can suppress spirals and antispirals.

Sinklike spiral waves in oscillatory media with a disk-shaped inhomogeneity

Spiral wave propagation in oscillatory media with a disk-shaped inhomogeneity is examined. Depending on the properties of the medium as well as the inhomogeneity (different frequencies in two regions), distinct spiral waves including sinklike spirals and dense-sparse spirals, are able to emerge. We find that, unlike the previously found outward group velocity for spiral waves (normal spirals or antispirals), the direction of the velocity of the sinklike spiral wave points inward. A qualitative analysis of the possible mechanism underlying their formation is discussed, considering the inhomogeneity as a wave sink or source. Numerical simulations performed on other typical reaction-diffusion models confirm this analysis and suggest that our findings are robust and could be observed in experiments.

Chirality effect on the global structure of spiral-domain patterns in the two-dimensional complex Ginzburg-Landau equation

It is well known that in the single-spiral-stable parameter regimes of the two-dimensional complex Ginzburg-Landau equation, spiral-domain patterns spontaneously appear. These patterns are disordered cells of frozen spiral waves well separated by thin walls (shocks), and to a good approximation, the walls are segments of hyperbolas. In this paper, we take a closer look at the global structure of spiral-domain patterns by using rigorous mathematical analysis and considering the unusual effect of the chirality (handedness) of spiral wave. An equation that determines the slope of the shock line is derived. We generalize this analytical method to study the interaction of a pair of spirals with different rotation frequencies, and obtain the geometrical structures of the shock line and the wave front of the invasion wave in transient processing.

Interaction and breakup of inwardly rotating spiral waves in an inhomogeneous oscillatory medium

We studied the effects of spatial inhomogeneities on inwardly rotating spiral waves in a typical type of oscillatory medium using the complex-Ginzburg-Landau equation. With a small degree of the inhomogeneity in the medium, the slower inward spiral always suppressed a faster spiral; when the inhomogeneity exceeded a critical value, however, a transition occurred to the coexistence of multiple inward spirals, insulated by regions of highly disordered wave break. The occurrence of this transition is examined theoretically and shown to be due to the Eckhaus instability.