Oscillatory dispersion and coexisting stable pulse trains in an excitable medium

Unusual spiral wave dynamics in the Kessler-Levine model of an excitable medium

The Kessler-Levine model is a two-component reaction-diffusion system that describes spatiotemporal dynamics of the messenger molecules in a cell-to-cell signaling process during the aggregation of social amoeba cells. An excitation wave arising in the model has a phase wave at the wave back, which simply follows the wave front after a fixed time interval with the same propagation velocity. Generally speaking, the medium excitability and the refractoriness are two important factors which determine the spiral wave dynamics in any excitable media. The model allows us to separate these two factors relatively easily since the medium refractoriness can be changed independently of the medium excitability. For rigidly rotating waves, the universal relationship has been established by using a modified free-boundary approach, which assumes that the front and the back of a propagating wave are thin in comparison to the wave plateau. By taking a finite thickness of the domain boundary into consideration, the validity of the proposed excitability measure has been essentially improved. A novel method of numerical simulation to suppress the spiral wave instabilities is introduced. The trajectories of the spiral tip observed for a long refractory period have been investigated under a systematic variation of the medium refractoriness.

Origin of finite pulse trains: Homoclinic snaking in excitable media

Many physical, chemical, and biological systems exhibit traveling waves as a result of either an oscillatory instability or excitability. In the latter case a large multiplicity of stable spatially localized wavetrains consisting of different numbers of traveling pulses may be present. The existence of these states is related here to the presence of homoclinic snaking in the vicinity of a subcritical, finite wavenumber Hopf bifurcation. The pulses are organized in a slanted snaking structure resulting from the presence of a heteroclinic cycle between small and large amplitude traveling waves. Connections of this type require a multivalued dispersion relation. This dispersion relation is computed numerically and used to interpret the profile of the pulse group. The different spatially localized pulse trains can be accessed by appropriately customized initial stimuli, thereby blurring the traditional distinction between oscillatory and excitable systems. The results reveal a new class of phenomena relevant to spatiotemporal dynamics of excitable media, particularly in chemical and biological systems with multiple activators and inhibitors.

Dynamics of localized structures in reaction-diffusion systems induced by delayed feedback

We are interested in stability properties of a single localized structure in a three-component reaction-diffusion system subjected to the time-delayed feedback. We shall show that variation in the product of the delay time and the feedback strength leads to complex dynamical behavior of the system, including formation of target patterns, spontaneous motion, and spontaneous breathing as well as various complex structures, arising from combination of different oscillatory instabilities. In the case of spontaneous motion, we provide a bifurcation analysis of the delayed system and derive an order parameter equation for the position of the localized structure, explicitly describing its temporal evolution in the vicinity of the bifurcation point. This equation is a subject to a nonlinear delay differential equation, which can be transformed to the normal form of the pitchfork drift bifurcation.

Towards nonlinear selection of reaction-diffusion patterns in presence of advection: a spatial dynamics approach

We present a theoretical study of nonlinear pattern selection mechanisms in a model of bounded reaction-diffusion-advection system. The model describes the activator-inhibitor type dynamics of a membrane reactor characterized by a differential advection and a single diffusion; the latter excludes any finite wave number instability in the absence of advection. The focus is on three types of different behaviors, and the respective sensitivity to boundary and initial conditions: traveling waves, stationary periodic states, and excitable pulses. The theoretical methodology centers on the spatial dynamics approach, i.e. bifurcation theory of nonuniform solutions. These solutions coexist in overlapping parameter regimes, and multiple solutions of each type may be simultaneously stable. The results provide an efficient understanding of the pattern selection mechanisms that operate under realistic boundary conditions, such as Danckwerts type. The applicability of the results to broader reaction-diffusion-advection contexts is also discussed.

Self-organized pacemakers and bistability of pulses in an excitable medium

Pattern formation in an excitable medium described by a three-component reaction-diffusion system is investigated. Our focus is on stable self-organized pacemakers which give rise to spatially extended target patterns. Bistability of pulse solutions in the excitable regime is also reported, and interactions of the different pulses with each other and the pacemaker are studied. Self-organized pacemakers are created by a suitable perturbation from the steady state or through interaction of pulses. Bound states of one-dimensional pacemakers and phase flips are also observed.

Generation of finite wave trains in excitable media

Spatiotemporal control of excitable media is of paramount importance in the development of new applications, ranging from biology to physics. To this end, we identify and describe a qualitative property of excitable media that enables us to generate a sequence of traveling pulses of any desired length, using a one-time initial stimulus. The wave trains are produced by a transient pacemaker generated by a one-time suitably tailored spatially localized finite amplitude stimulus, and belong to a family of fast pulse trains. A second family, of slow pulse trains, is also present. The latter are created through a clumping instability of a traveling wave state (in an excitable regime) and are inaccessible to single localized stimuli of the type we use. The results indicate that the presence of a large multiplicity of stable, accessible, multi-pulse states is a general property of simple models of excitable media.

Electric-field-induced wave groupings of spiral waves with oscillatory dispersion relation

The dynamic behavior of spiral-shaped excitation patterns with oscillatory dispersion is investigated under the influence of externally applied direct current or alternating current. For these two types of electric field, wave-grouping phenomena are generally observed. For the direct current field, the spiral wave drifts approximately along a straight line and wave groupings appear in certain ranges of spatial polar angles when the strength of the external field is larger than a threshold. In terms of the Doppler effect induced by the drift of the spiral tip and the oscillatory dispersion, we propose a theory model to predict the spatial distribution of wave grouping and the critical strength of the current. In contrast, for the alternating current field, the spiral wave may stay stationary and wave grouping may appear in the whole space with a different manner. This finding indicates that movement of the spiral tip is not necessary for the appearance of wave grouping.

Anomalous pulse interaction in dissipative media

We review a number of phenomena occurring in one-dimensional excitable media due to modified decay behind propagating pulses. Those phenomena can be grouped in two categories depending on whether the wake of a solitary pulse is oscillatory or not. Oscillatory decay leads to nonannihilative head-on collision of pulses and oscillatory dispersion relation of periodic pulse trains. Stronger wake oscillations can even result in a bistable dispersion relation. Those effects are illustrated with the help of the Oregonator and FitzHugh-Nagumo models for excitable media. For a monotonic wake, we show that it is possible to induce bound states of solitary pulses and anomalous dispersion of periodic pulse trains by introducing nonlocal spatial coupling to the excitable medium.

Wave-pinned filaments of scroll waves

Scroll waves are three-dimensional excitation patterns that rotate around one-dimensional space curves. Typically these filaments are closed loops or end at the system boundary. However, in excitable media with anomalous dispersion, filaments can be pinned to the wake of traveling wave pulses. This pinning is studied in experiments with the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction and a three-variable reaction-diffusion model. We show that wave-pinned filaments are related to the coexistence of rotating and translating wave defects in two dimensions. Filament pinning causes a continuous expansion of the total filament length. It can be ended by annihilating the pinning pulse in a frontal wave collision. Following such an annihilation, the filament connects itself to the system boundary. Its postannihilation shape that is initially the exposed rim of the scroll wave unwinds continuously over numerous rotation periods.

Boundary-induced patterns in excitable systems: the structure of oscillatory domain

The present work extends our recently published study [Phys. Rev. E 73, 066224 (2006)] on a mechanism of pattern formation in excitable media due to inhomogeneous boundary conditions (BC). To that end, we analyze a pair of coupled excitable and oscillatory cells, a distributed FitzHugh Nagumo model, and a distributed five-variable model that describes CO catalytic oxidation. For the three systems we determine the structure of the oscillatory domains, composed of bands of complex firing solutions with period-adding bifurcations, and show the commonality of the structures. The obtained results account for the recently reported experimental observations of mixed-mode oscillations showing a period-adding bifurcation during CO oxidation on a disk-shaped catalytic cloth with imposed cold temperature BC.

Wave trains in an excitable FitzHugh-Nagumo model: bistable dispersion relation and formation of isolas

We investigate the dispersion relations of nonlinear periodic wave trains in excitable systems which describe the dependence of the propagation velocity on the wavelength. Pulse interaction by oscillating pulse tails within a wave train leads to bistable wavelength bands, in which two stable and one unstable wave train coexist for the same wavelength. The essential spectra of the unstable wave trains exhibit a circle of eigenvalues with positive real parts which is detached from the imaginary axis. We describe the destruction of the bistable dispersion curve and the formation of isolas of wave trains in a sequence of transcritical bifurcations unfolding into pairs of saddle-node bifurcations. It turns out that additional dispersion curves of unstable wave trains play an important role in the destruction of the bistable dispersion curve.

Propagation failures, breathing pulses, and backfiring in an excitable reaction-diffusion system

We report results from experiments with a pseudo-one-dimensional Belousov-Zhabotinsky reaction that employs 1,4-cyclohexanedione as its organic substrate. This excitable system shows traveling oxidation pulses and pulse trains that can undergo complex sequences of propagation failures. Moreover, we present examples for (i) breathing pulses that undergo periodic changes in speed and size and (ii) backfiring pulses that near their back repeatedly generate new pulses propagating in opposite direction.

Propagation failure dynamics of wave trains in excitable systems

We report experimental and numerical results on temporal patterns of propagation failures in reaction-diffusion systems. Experiments employ the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction. The propagation failures occur in the frontier region of the wave train and can profoundly affect its expansion speed. The specific rhythms observed vary from simple periodic to highly complex and possibly chaotic sequences. All but the period-1 sequences are found in the transition region between "merging" and "tracking" dynamics, which correspond to wave behavior caused by two qualitatively different types of anomalous dispersion relations.

Front aggregation in multiarmed excitation vortices

Using the Belousov-Zhabotinsky reaction, we study the pinning of multiarmed spiral waves to nonexcitable obstacles. With increasing obstacle size, the individual arms switch from a repulsive to an attractive state. This transition yields densely aggregated spiral arms and is caused by anomalous dispersion. A kinematic model reproduces the measurements quantitatively and identifies the transition as a supercritical pitchfork bifurcation.

Tracking waves and vortex nucleation in excitable systems with anomalous dispersion

We report experimental results obtained from a chemical reaction-diffusion system in which wave propagation is limited to a finite band of wavelengths and in which no solitary pulses exist. Wave patterns increase their size through repeated annihilation events of the frontier pulse that allow the succeeding pulses to advance farther. A related type of wave dynamics involves a stable but slow frontier pulse that annihilates subsequent waves in front-to-back collisions. These so-called merging dynamics give rise to an unexpected form of spiral wave nucleation. All of these phenomena are reproduced by a simple, three-species reaction-diffusion model that reveals the importance of the underlying anomalous dispersion relation.