Oregonator Model Research Articles

Influence of excitability on unpinning and termination of spiral waves.

Application of electrical forcing to release pinned spiral waves from unexcitable obstacles and to terminate the rotation of free spiral waves at the boundary of excitable media has been investigated in thin layers of the Belousov-Zhabotinsky (BZ) reaction, prepared with different initial concentrations of H_{2}SO_{4}. Increasing [H_{2}SO_{4}] raises the excitability of the reaction and reduces the core diameter of free spiral waves as well as the wave period. An electric current with density stronger than a critical value Junpin causes a pinned spiral wave to drift away from the obstacle. For a given obstacle size, Junpin increases with [H_{2}SO_{4}]. Under an applied electrical current, the rotation center of a free spiral wave drifts along a straight path to the boundary. When the current density is stronger than a critical value Jterm, the spiral tip is forced to hit the boundary, where the spiral wave is terminated. Similar to Junpin for releasing a pinned spiral wave, Jterm also increases with [H_{2}SO_{4}]. These experimental findings were confirmed by numerical simulations using the Oregonator model, in which the excitability was adjusted via the ratio of the excitation rate to the recovery rate of the BZ reaction. Therefore, our investigation shows that decreasing the excitability can facilitate elimination of spiral waves by electrical forcing, either in the presence of obstacles or not.

Inhibitory effect of oxygen on excitation waves in the Belousov–Zhabotinsky reaction with different excitability

Oxygen diffuses into an open-air surface of a vertically oriented thin layer of the Belousov–Zhabotinsky reaction and creates a millimeter-order inhibited layer where excitation fronts cease. When the excitability of the reaction is weakened via a decrement of [H2SO4] or [NaBrO3], the velocity of spiral waves located far from the open-air surface decreases and the width of the inhibited layer increases. The experimental findings agree well with simulations using the Oregonator model where the excitability is adjusted via two parameters ɛ−1 and f which mimic the change in [H2SO4] or [NaBrO3] and the oxygen profile along the vertical direction, respectively.

Three-dimensional autonomous pacemaker in the photosensitive Belousov-Zhabotinsky medium

In experiments with the photosensitive Belousov-Zhabotinsky reaction (PBZR) we found a stable three-dimensional organizing center that periodically emits trigger waves of chemical concentration. The experiments are performed in a parameter regime with negative line tension using an open gel reactor to maintain stationary non-equilibrium conditions. The observed periodic wave source is formed by a scroll ring stabilized due to its interaction with a no-flux boundary. Sufficiently far from the boundary, the scroll ring expands and undergoes the negative line tension instability before it finally develops into scroll wave turbulence. Our experimental results are reproduced by numerical integration of the modified Oregonator model for the PBZR. Stationary and breathing self-organized pacemakers have been found in these numerical simulations. In the latter case, both the radius of the scroll ring and the distance of its filament plane to the no-flux boundary after some transient undergo undamped stable limit cycle oscillations. So far, contrary to their stationary counterpart, the numerically predicted breathing autonomous pacemaker has not been observed in the chemical experiment.

Entropy production in the Oregonator model perturbed in a calorimeter with a chemical pulse

The classical Oregonator model of the Belousov–Zhabotinsky reaction is used to examine the response of an oscillating reaction to a chemical perturbation in a batch calorimeter. To do this, a reaction step for the perturbing chemical substance and the energy balance equation are added to the model. Changes in the frequency and amplitude of the oscillations occur as a function of the concentration of the perturbing substance, and these changes are quantified by applying the thermodynamics of irreversible processes in order to determine the rate of entropy production. Various criteria are proposed to evaluate the time series and measurable quantities with linear correlations between the entropy production due to transfer and the concentration of the chemical perturbation are obtained, showing that it is possible to implement experimental protocols in the proposed methodology.

Stationary spots and stationary arcs induced by advection in a one-activator, two-inhibitor reactive system.

This paper studies the spatiotemporal dynamics of a reaction-diffusion-advection system corresponding to an extension of the Oregonator model, which includes two inhibitors instead of one. We show that when the reaction-diffusion, two-dimensional problem displays stationary patterns the addition of a plug flow can induce the emergence of new types of stationary structures. These patterns take the form of spots or arcs, the size and the spacing of which can be controlled by the flow.

A new model to imitate the foraging behavior of Physarum polycephalum on a nutrient-poor substrate

Abstract Researches on Physarum polycephalum show that methods inspired by the primitive unicellular organism can construct an efficient network and solve some complex problems in graph theory. Current models simulating the intelligent behavior of Physarum are mainly based on Hagen–Poiseuille Law and Kirchhoff Law, reaction–diffusion, Cellular Automaton and multi-agent approach. In this paper, based on an assumption that the plasmodium of Physarum forages for food along the gradient of chemo-attractants on a nutrient-poor substrate, a new model is proposed to imitate its intelligent foraging behavior. The key point of the model is that the growth of Physarum is determined by the simple particle concentration field relating the distance to food source and the shape of food source on a nutrient-poor substrate. To verify this model, numerical experiments are conducted according to Adamatzky׳s experiment. Results in spanning tree construction by this model are almost the same as those of Physarum and Oregonator model. The proposed model can also imitate Physarum to avoid repellents. Furthermore, the Euclidean Spanning tree built by this model is similar to its corresponding Minimal Euclidean Spanning tree.

Unpinning of spiral waves by electrical forcing in excitable chemical media.

We present experimental observations on the electrically forced release of spiral waves pinned to unexcitable circular obstacles in the Belosov-Zhabotinsky reaction. When the applied electric current density reaches the necessary current density J(unpin), the spiral tip is detached and subsequently drifts away from the obstacle. J(unpin) is found to increase with the obstacle diameter d. The growth rate ΔJ(unpin)/Δd is much higher for obstacles larger than the free spiral core compared to that for smaller obstacles. The experimental findings are confirmed by numerical simulations using the Oregonator model. The results imply that it is more difficult to release spiral waves pinned to larger obstacles, especially when the obstacle size exceeds that of the free spiral core.

Chiral selection and frequency response of spiral waves in reaction-diffusion systems under a chiral electric field

Chirality is one of the most fundamental properties of many physical, chemical, and biological systems. However, the mechanisms underlying the onset and control of chiral symmetry are largely understudied. We investigate possibility of chirality control in a chemical excitable system (the Belousov-Zhabotinsky reaction) by application of a chiral (rotating) electric field using the Oregonator model. We find that unlike previous findings, we can achieve the chirality control not only in the field rotation direction, but also opposite to it, depending on the field rotation frequency. To unravel the mechanism, we further develop a comprehensive theory of frequency synchronization based on the response function approach. We find that this problem can be described by the Adler equation and show phase-locking phenomena, known as the Arnold tongue. Our theoretical predictions are in good quantitative agreement with the numerical simulations and provide a solid basis for chirality control in excitable media.

Mechano-chemical coupling in Belousov-Zhabotinskii reactions

Mechano-chemical coupling has been recently recognised as an important effect in various systems as chemical reactivity can be controlled through an applied mechanical loading. Namely, Belousov-Zhabotinskii reactions in polymer gels exhibit self-sustained oscillations and have been identified to be reasonably controllable and definable to the extent that they can be harnessed to perform mechanical work at specific locations. In this paper, we use our theoretical work of nonlinear mechano-chemical coupling and investigate the possibility of providing an explanation of phenomena found in experimental research by means of this theory. We show that mechanotransduction occurs as a response to both static and dynamic mechanical stimulation, e.g., volume change and its rate, as observed experimentally and discuss the difference of their effects on oscillations. Plausible values of the quasi-stoichiometric parameter f of Oregonator model are estimated together with its dependence on mechanical stimulation. An increase in static loading, e.g., pressure, is predicted to have stimulatory effect whereas dynamic loading, e.g., rate of volume change, is predicted to be stimulatory only up to a certain threshold. Further, we offer a physically consistent explanation of the observed phenomena why some Belousov-Zhabotinskii gels require an additional mechanical stimulation to show emergence of oscillation or why "revival" of oscillations in Belousov-Zhabotinskii reactions is possible together with indications for further experimental setups.

Generative Force of Self-Oscillating Gel

We succeeded in measuring the generative force of a self-oscillating polymer gel in an aqueous solution comprising the three substrates of the Belousov-Zhabotinsky (BZ) reaction (malonic acid, sodium bromate, and nitric acid) under constant temperature. In this study, we developed an apparatus with a microforce sensor for measuring the generative force of small-sized gels (1 mm(3)). The self-oscillating polymer gel directly converts the chemical energy of the BZ reaction into mechanical work. It was determined that the generative force of the self-oscillating gel was 972 Pa, and the period of self-oscillation was 480 s at 18 °C. We demonstrated that the generative force of the gel was about a hundredth the generative force of a muscle in the body. We analyzed the time dependence of the color change in the self-oscillating polymer gel. The color of the gel changed periodically owing to the cyclic change in the redox state of the Ru moiety, induced by the BZ reaction. The peaks of the waveforms of the generative force and color change were almost identical. This result showed that the generative force was synchronized with the periodical change in the oxidation number of the Ru catalytic moiety in the gel. To understand a theoretical basis for the generative force of a self-oscillating gel, we considered a general theory that is based on the volume phase transition of gel and the two-parameter Oregonator model of the BZ reaction.

Chemical Wave Propagation Preserved on an Inhibitory Field in the Ruthenium-Catalyzed Belousov–Zhabotinsky Reaction

The propagation of a chemical wave on an inhibitory field, which was wedged between two excitable fields, was investigated for the photosensitive Belousov-Zhabotinsky (BZ) reaction. With an increase in the width of the inhibitory field between the excitable fields (W), the chemical wave divided into two waves at W = Wα. The divided chemical waves then coalesced at W = Wβ with a decrease in W. Wα was larger than Wβ, i.e., hysteresis on the width of the inhibitory field was observed between the division and coalescence of the chemical wave. The experimental results were qualitatively reproduced by a numerical calculation based on a three-variable Oregonator model modified for the photosensitive BZ reaction. These results suggest that the chemical wave may be preserved on the inhibitory field due to an activator supplied from a chemical wave on the excitable field.

Turing instability in reaction-diffusion systems with nonlinear diffusion

The Turing instability is studied in two-component reaction-diffusion systems with nonlinear diffusion terms, and the regions in parametric space where Turing patterns can form are determined. The boundaries between super- and subcritical bifurcations are found. Calculations are performed for one-dimensional brusselator and oregonator models.

Cross-diffusion in the two-variable Oregonator model

We explore the effect of cross-diffusion on pattern formation in the two-variable Oregonator model of the Belousov-Zhabotinsky reaction. For high negative cross-diffusion of the activator (the activator being attracted towards regions of increased inhibitor concentration) we find, depending on the values of the parameters, Turing patterns, standing waves, oscillatory Turing patterns, and quasi-standing waves. For the inhibitor, we find that positive cross-diffusion (the inhibitor being repelled by increasing concentrations of the activator) can induce Turing patterns, jumping waves and spatially modulated bulk oscillations. We qualitatively explain the formation of these patterns. With one model we can explain Turing patterns, standing waves and jumping waves, which previously was done with three different models.

Dynamic properties of the coupled Oregonator model with delay

This work explores a coupled Oregonator model. By analyzing the associated characteristic equation, linear stability is investigated and Hopf bifurcations are demonstrated, as well as the stability and direction of the Hopf bifurcation are determined by employing the normal form method and the center manifold reduction. We also discussed the Z2 equivariant property and the existence of multiple periodic solutions. Numerical simulations are presented to illustrate the results in Section 5.

Chiralities of spiral waves and their transitions

The chiralities of spiral waves usually refer to their rotation directions (the turning orientations of the spiral temporal movements as time elapses) and their curl directions (the winding orientations of the spiral spatial geometrical structures themselves). Traditionally, they are the same as each other. Namely, they are both clockwise or both counterclockwise. Moreover, the chiralities are determined by the topological charges of spiral waves, and thus they are conserved quantities. After the inwardly propagating spirals were experimentally observed, the relationship between the chiralities and the one between the chiralities and the topological charges are no longer preserved. The chiralities thus become more complex than ever before. As a result, there is now a desire to further study them. In this paper, the chiralities and their transition properties for all kinds of spiral waves are systemically studied in the framework of the complex Ginzburg-Landau equation, and the general relationships both between the chiralities and between the chiralities and the topological charges are obtained. The investigation of some other models, such as the FitzHugh-Nagumo model, the nonuniform Oregonator model, the modified standard model, etc., is also discussed for comparison.

Twists of opposite handedness on a scroll wave.

The dynamic interaction of scroll waves in the Belousov-Zhabotinsky reaction with a vertically orientated gradient of excitability is studied by optical tomography. This study focuses on scroll waves, whose filaments were oriented almost perpendicular to the gradient. Whereas scroll waves with filaments exactly perpendicular to the gradient remain unaffected, filaments with a component parallel to the gradient develop a twist. Scroll waves with U-shaped filaments exhibit twists starting from both of its ends, resulting in scroll waves whose filaments display a pair of twists of opposite handedness. These twists are separated by a nodal plane where the filament remains straight and untwisted. The experimental findings were reproduced by numerical simulations using the Oregonator model and a linear gradient of excitability almost perpendicular to the orientation of the filament.

Chemical patterns in translating vortices: Inter- and intra-cellular mixing effects

Stationary chemical patterns-flow distributed oscillations (FDOs)-are obtained when the Belousov-Zhabotinsky reaction is coupled with translating vortex flow in a Vortex Flow Reactor. For certain conditions, the FDOs are unstable with the observation of disappearing bands or complex patterns. The transitions between modes of pattern formation are reproduced in a modified Oregonator model consisting of two-zone cells connected in series. We show that increasing inter-cellular mixing of the outer zones results in a transition from FDO to absolute instabilities (AI) and increasing intra-cellular mixing between the core and outer zones can drive the reverse transition between modes (AI to FDO).

Chaotic Oscillation of Two-Coupled Chemical Oscillators under Time Delay with an Asymmetric Condition

The synchronized modes of two-coupled Oregonator models are investigated. The numerical results indicate that the combination of the time delay and asymmetry in the coupling between the independent oscillators generates the chaotic oscillations. This study also reveals the two types of routes to chaos of anti-phase mode → chaos → anti-phase mode and in-phase mode → chaos → anti-phase mode. Furthermore, we propose an equation for the one-dimensional map of the chaos observed in the two-coupled Oregonators.

Turing space in reaction-diffusion systems with density-dependent cross diffusion

Reaction-diffusion systems with cross-diffusion terms that depend linearly on density are studied via linear stability analysis and weakly nonlinear analysis. We obtain and analyze the conditions for the Turing instability and derive a universal form of these conditions. We discuss the features of the pattern-forming regions in parameter space for a cross activator-inhibitor system, the Brusselator model, and for a pure activator-inhibitor system, the two-variable Oregonator model. The supercritical or subcritical character of the Turing bifurcation for the Brusselator is determined by deriving an amplitude equation for patterns near the instability threshold.

Single-Hopf Bursting in Periodic Perturbed Belousov—Zhabotinsky Reaction with Two Time Scales

By introducing weak periodic perturbation to the Oregonator, we investigate a mathematical model with two time scales. The novel dynamical phenomena called the single-Hopf bursting with unusually quiescent state are observed. The related bifurcation mechanism is presented by means of the transition portrait and slow-fast analysis, which has rarely been reported in the previous works associated with the Oregonator model. Furthermore, the influence of forcing amplitude on bursting behaviors is studied. Further investigation finds that excitation amplitude may play an important role in a two-timescale system with the slow procedure dominated by periodic perturbation.