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Register a new userExcitable solitons: Annihilation, crossover, and nucleation of pulses in mass-conserving activator-inhibitor media
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Excitable pulses are among the most widespread dynamical patterns that occur in many different systems, ranging from biological cells to chemical reactions and ecological populations. Traditionally, the mutual annihilation of two colliding pulses is regarded as their prototypical signature. Here we show that colliding excitable pulses may exhibit soliton-like crossover and pulse nucleation if the system obeys a mass conservation constraint. In contrast to previous observations in systems without mass conservation, these alternative collision scenarios are robustly observed over a wide range of parameters. We demonstrate our findings using a model of intracellular actin waves since, on time scales of wave propagations over the cell scale, cells obey the conservation of actin monomers. The results provide a key concept to understand the ubiquitous occurrence of actin waves in cells, suggesting why they are so common, and why their dynamics is robust and long-lived.
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An understanding of the underlying mechanism of side--branching is paramount in controlling and/or therapeutically treating mammalian organs, such as lungs, kidneys, and glands. Motivated by an activator-inhibitor-substrate approach that is conjectured to dominate the initiation of side--branching in pulmonary vascular pattern, I demonstrate a distinct transverse front instability in which new fingers grow out of an oscillatory breakup dynamics at the front line, without any typical length scale. These two features are attributed to unstable peak solutions in 1D that subcritically emanate from the Turing bifurcation and that exhibit repulsive interactions. The results are based on a bifurcation analysis and numerical simulations, and provide a potential strategy toward developing a framework of side--branching also of other biological systems, such as plant roots and cellular protrusions.
Experimental studies of protein-pattern formation have stimulated new interest in the dynamics of reaction-diffusion systems. However, a comprehensive theoretical understanding of the dynamics of such highly nonlinear, spatially extended systems is still missing. Here we show how a description in phase space, which has proven invaluable in shaping our intuition about the dynamics of nonlinear ordinary differential equations, can be generalized to mass-conserving reaction-diffusion (McRD) systems. We present a comprehensive analysis of two-component McRD systems, which serve as paradigmatic minimal systems that encapsulate the core principles and concepts of the local equilibria theory introduced in the paper. The key insight underlying this theory is that shifting local (reactive) equilibria -- controlled by the local total density -- give rise to concentration gradients that drive diffusive redistribution of total density. We show how this dynamic interplay can be embedded in the phase plane of the reaction kinetics in terms of simple geometric objects: the reactive nullcline and the diffusive flux-balance subspace. On this phase-space level, physical insight can be gained from geometric criteria and graphical constructions. The effects of nonlinearities on the global dynamics are simply encoded in the curved shape of the reactive nullcline. In particular, we show that the pattern-forming `Turing instability in McRD systems is a mass-redistribution instability, and that the features and bifurcations of patterns can be characterized based on regional dispersion relations, associated to distinct spatial regions (plateaus and interfaces) of the patterns. In an extensive outlook section, we detail concrete approaches to generalize local equilibria theory in several directions, including systems with more than two-components, weakly-broken mass conservation, and active matter systems.
A thring is a recent addition to the zoo of spiral wave phenomena found in excitable media and consists of a scroll ring that is threaded by a pair of counter-rotating scroll waves. This arrangement behaves like a particle that swims through the medium. Here, we present the first results on the dynamics, interaction and collective behaviour of several thrings via numerical simulation of the reaction-diffusion equations that model thrings created in chemical experiments. We reveal an attraction between two thrings that leads to a stable bound pair that thwarts their individual locomotion. Furthermore, such a pair emits waves at a higher frequency than a single thring, which protects the pair from the advances of any other thring and rules out the formation of a triplet bound state. As a result, the long-term evolution of a colony of thrings ultimately yields an unusual frozen nonequilibrium state consisting of a collection of pairs accompanied by isolated thrings that are inhibited from further motion by the waves emanating from the pairs.
In this paper we present the results of parallel numerical computations of the long-term dynamics of linked vortex filaments in a three-dimensional FitzHugh-Nagumo excitable medium. In particular, we study all torus links with no more than 12 crossings and identify a timescale over which the dynamics is regular in the sense that each link is well-described by a spinning rigid conformation of fixed size that propagates at constant speed along the axis of rotation. We compute the properties of these links and demonstrate that they have a simple dependence on the crossing number of the link for a fixed number of link components. Furthermore, we find that instabilities that exist over longer timescales in the bulk can be removed by boundary interactions that yield stable torus links which settle snugly at the medium boundary. The Borromean rings are used as an example of a non-torus link to demonstrate both the irregular tumbling dynamics that arises in the bulk and its suppression by a tight confining medium. Finally, we investigate the collision of torus links and reveal that this produces a complicated wrestling motion where one torus link can eventually dominate over the other by pushing it into the boundary of the medium.
We study the dynamics and interaction of coaxial vortex rings in the FitzHugh-Nagumo excitable medium. We find that threading vortex rings with a vortex string results in significant qualitative differences in their evolution and interaction. In particular, threading prevents the annihilation of rings in a head-on collision, allows generic ring overtaking, and can even reverse the direction of motion of a ring. We identify that an important mechanism for producing this new behaviour is that threaded vortex rings interact indirectly via induced twisting of the threading vortex string.
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