Oregonator Model Research Articles

How Does a Simple Network of Chemical Oscillators See the Japanese Flag?

Chemical computing is something we use every day (e.g., in the brain), but we can still not explore and master its potential in human-made experiments. It is expected that the maximum computational efficiency of a chemical medium can be achieved if information is processed in parallel by different parts of the medium. In this paper, we use computer simulations to explore the efficiency of chemical computing performed by a small network of three coupled chemical oscillators. We optimize the network to recognize the white and red regions of the Japanese flag. The input information is introduced as the inhibition times of individual oscillators, and the output information is coded in the number of activator maxima observed on a selected oscillator. We have used the Oregonator model to simulate the network time evolution and the evolutionary optimization to find the best network for the considered task. We have found that even a network of three interacting oscillators can recognize the color of a randomly selected point with 95% accuracy.

Численное моделирование неустойчивых режимов орегонатора

A system of partial differential equations is presented, which is a mathematical model of a chemically active system (oregonator) with a diffusive type of coupling between components. The relevance of the study of systems with diffusion is related to the problem of the origin and forming of spatial structures in chemical, biological, and ecological systems. Equations for the stationary state of the system are written. The transition from the original system of differential equations to the system of differential equations in perturbations is performed. Computational algorithms have been developed for calculating the parameters of the oregonator model. Numerical studies of the presented model are carried out in the MATLAB package. Stationary States of the oregonator are calculated for different values of the stoichiometric coefficient that correspond to the physical meaning of the process. The stoichiometric coefficient is a bifurcation parameter of the system, and each of its values corresponds to a single positive stationary solution. The dispersion equation is derived. The instability criterion is the positive values of both the growth rate and the frequency of disturbances in the oregonator. Numerical modeling of the stability of the stationary state with respect to perturbations is carried out. Two types of instability in the oregonator are identified: change of stability and oscillatory instability. The results of computational experiments have shown that the diffusion of components generates more unstable modes with wave numbers other than zero. This indicates additional diffusive instability, which is a mechanism for the formation of spatial structures.

Effect of straining on spiral wave dynamics in excitable media

The present article envisages to explore the dynamics of stable rotating spiral waves in excitable media under the effect of shear strain through the Oregonator model. We employ an existing unconditionally stable, implicit high order compact finite difference scheme to discretize the highly nonlinear system governing the spiral wave patterns. In the process, we also study the effect of reaction parameters on the dynamics of spiral waves in the absence of straining, which results in the transition from the stable (periodic) rotation to compound (non-periodic) rotation of spiral waves. Our study reveals the existence of three regimes with a new time range, which is different from the earlier ones reported in literature; they characterize the extremely complex behaviour of the patterns under the effect of straining. The aforesaid conclusion is arrived at by carrying out a comprehensive analysis of the power spectrum of the tip motion. Besides establishing the stability of simulated rotating waves, the accuracy of the computed solutions has been ascertained through a convergence analysis and mesh independence study.

Study on the Formation of Complex Chemical Waveforms by Different Computational Methods

Chemical wave is a special phenomenon that presents periodic patterns in space-time domain, and the Belousov–Zhabotinsky (B-Z) reaction is the first well-known reaction-diffusion system that exhibits organized patterns out of a homogeneous environment. In this paper, the B-Z reaction kinetics is described by the Oregonator model, and formation and evolution of chemical waves are simulated based on this model. Two different simulation methods, partial differential equations (PDEs) and cellular automata (CA) are implemented to simulate the formation of chemical waveform patterns, i.e., target wave and spiral wave on a two-dimensional plane. For the PDEs method, reaction caused changes of molecules at different location are considered, as well as diffusion driven by local concentration difference. Specifically, a PDE model of the B-Z reaction is first established based on the B-Z reaction kinetics and mass transfer theory, and it is solved by a nine-point finite difference (FD) method to simulate the formation of chemical waves. The CA method is based on system theory, and interaction relations with the cells nearest neighbors are mainly concerned. By comparing these two different simulation strategies, mechanisms that cause the formation of complex chemical waves are explored, which provides a reference for the subsequent research on complex systems.

Non-linear Chemical Reactions

In this article, we discuss the Belousov–Zhabotinsky (BZ) reaction and its relation with the Oregonator model of chemical reactions. The Oregonator model has a limit cycle, and the BZ reaction shows chemical oscillations. We compare the time series obtained from the Oregonator model, with the time series obtained froman experimental study of the BZ reaction and show that the qualitative behaviour is similar. We also construct the limit cycle from the experimental data using the Takens’ embedding theorem and compare it with the limit cycle from the Oregonator model. We conclude with a demonstration of the spatio-temporal patterns which can be seen in the two-dimensional version of this system.

BASIC CONCEPTS OF SELF-ORGANIZED PHENOMENA IN CHEMICAL SYSTEMS

We present the main concepts of nonlinear dynamics and thermodynamics of irreversible processes to introduce chemistry students to the topic of self-organized phenomena. This task is performed by theoretically describing the emergence of self-sustained oscillations, waves, and stationary patterns/Turing patterns in the Belousov-Zhabotinsky (BZ) reaction, through the Oregonator model. We carefully developed such a description, which resulted in long algebraic deductions and rich supplementary material. Considering that, we encourage the use of this material in undergraduate and graduate advanced physical chemistry classes.

Qualitative analysis of the Oregonator model

In this paper, we consider the properties of the solutions for the Oregonator system, which is the mathematical model of the celebrated Belousov–Zhabotinskiĭ reaction. We first investigate the dynamics of the model, and some fundamental analytic propertie

Self-organization of multiarmed spiral waves in excitable media.

We present an investigation of self-organized multiarmed spiral waves pinned to unexcitable circular obstacles in a thin layer of the excitable Belousov-Zhabotinsky reaction and in simulations using the Oregonator model. The multiarmed waves are initiated by a series of wave stimuli. In the proximity of the obstacle boundary, the wave rotation around the obstacle causes damped oscillations of the wave periods of all spiral arms. The dynamics of wave periods cause the wave velocities as well as the angular displacements between the adjacent arms to oscillate with decaying amplitudes. Eventually, all displacements approach approximately the same stable value so that all arms are distributed evenly around the obstacle. A further theoretical analysis reveals that the temporal dynamics of the angular displacements can be interpreted as underdamped harmonic oscillations. Far from the obstacles, the wave dynamics are less pronounced. The wave period becomes stable very soon after the initiation. When the number of spiral arms increases, the rotation of individual arms slows down but the wave period of the multiarmed spiral waves decreases. Due to their short period, multiarmed spiral waves emerging in the heart potentially result in severe pathological conditions.

Resonant Behavior in a Periodically Forced Nonisothermal Oregonator.

Nonisothermal chemical oscillators are poorly studied systems because chemical oscillations are conventionally studied under isothermal conditions. Coupling chemical reactions with heat generation and removal in a nonisothermal oscillatory system can lead to a highly nontrivial nonlinear dynamic behavior. For the current study, we considered the three-variable Oregonator model with the temperature incorporated as a variable (not a parameter), thus adding an energy balance to the set of equations. The effect of temperature on reaction rates is included through the temperature-dependent reaction rate coefficients (Arrhenius law). To model a continuous operation in a laboratory environment, the system was subjected to external forcing through the coolant temperature and infrared irradiation. By conducting numerical simulations and parametric studies, we found that the system is capable of a resonant behavior exhibiting induced oscillations. Our findings indicate that an external source of heat (e.g., via an infrared light emitting diode) can be used to induce a Hopf bifurcation under resonant conditions in an experimental Belousov-Zhabotinsky reactor.

Exploration of Light-Controlled Chemical Behavior and Mechanism in a Macrocyclic Copper Complex Catalyst–Acetone–Glucose–Bromate–Sulfuric Acid Oscillation System

In this paper, the effect of ultraviolet light on the [CuL](ClO4)2–glucose (Glu)–acetone (Act)–sodium bromate (NaBrO3)–sulfuric acid (H2SO4) oscillation system was studied. The reaction mechanism and Oregonator model were established to verify the mechanism. Comparison of the bromide ion electrode–platinum electrode correlation diagrams with and without ultraviolet light reveals a nontracking phenomenon in the bromide ion electrode–platinum electrode correlation diagram under illumination, indicating that the illumination will affect the changes in the bromide ion concentration in the system. During the process, as UV intensity increases, the concentration of bromide ions in the system increases, and bromide ions can inhibit chemical oscillations, resulting in a decrease in the amplitude of chemical oscillations, further verifying that the proposed mechanism is reasonable.

Sequential Oscillations in the Ferroin‐Catalyzed Belousov‐Zhabotinsky Reaction: The Case of Oxalacetic Acid Substrate

AbstractA novel sequential, dual‐frequency oscillations were found in the Ferroin‐catalyzed Belousov‐Zhabotinsky (BZ) reaction when oxalacetic acid was employed as the substrate. The reactions were followed spectrophotometrically at a wavelength of 490 nm. The reaction revealed a “trumpet”‐shape oscillation pattern at relatively low oxalacetic acid (OAA) concentration, while at higher OAA concentration oscillation splits into a low and a high frequency region separated by a short second “induction period”. Furthermore, the effect of the initial reactant concentrations such as those of bromate, ferroin, sulfuric acid and oxalacetic acid were also investigated. It was observed that oscillation started to appear at shorter reaction time upon increasing the concentrations of bromate and ferroin, while it changed according to a minimum curve as a function of sulfuric acid concentration. However, in the case of OAA increasing its concentration resulted in increasing “induction periods”, i. e., the onset of oscillation shifted to longer reaction times. The origin of the “trumpet”‐shape oscillation patterns and the two‐phase oscillation was interpreted by means of the Oregonator model.

Twisted scroll wave dynamics: partially pinned waves in excitable chemical media

We present an investigation of the dynamics of scroll waves that are partially pinned to inert cylindrical obstacles of varying lengths and diameters in three-dimensional Belousov-Zhabotinsky excitable media. Experiments are carried out in which a scroll wave is initiated with a special orientation to be partially pinned to the obstacle. Numerical simulations with the Oregonator model are also carried out, where the obstacle is placed in the region of the core of a preexisting freely rotating scroll wave. In both cases, the effect of the obstacle on the wave dynamics is almost immediately observable, such that after the first revolution of the wave, the pinned region of the scroll wave has a longer period than that of the freely rotating scroll wave. The dependence of the scroll wave period on the obstacle position gives rise to a transition from a straight scroll wave to a twisted scroll wave in the pinned region, while the form of the freely rotating wave remains unchanged. The twisted scroll wave arises from the interaction of the freely rotating scroll wave with the obstacle, giving rise to a pinned twisted wave with the same period. The twisted scroll wave gradually advances, displacing the slower untwisted scroll wave until the scroll wave helically wraps around the entire obstacle. At this point, the period of the entire wave has a single value equal to that of the freely rotating scroll wave. The time for the transition to the twisted wave structure increases when either the obstacle length is increased or the obstacle diameter is decreased, while the average speed of the development increases with both the obstacle length and diameter. After the transition, the twisted wave remains stable, with its structure depending on the obstacle diameter - the larger the diameter, the shorter the helical pitch but the higher the twist rate.

Disruption and recovery of reaction–diffusion wavefronts colliding with obstacles

We study the damage to and restoration of planar reaction–diffusion wavefronts colliding with convex obstacles in narrow two-dimensional channels using finite-difference numerical integration of the Tyson–Fife reduction of the Oregonator model of the Belousov–Zhabotinsky reaction. We characterize the obstacles’ effects on the wavefront shape by plotting wavefront delay versus time. Due to the curvature dependent wavefront velocities, the initial planar wavefront (or iso-concentration line) is restored after a relaxation period that can be characterized by a power-law. We find that recovery times are insensitive to obstacle concatenation or to the upstream obstacle shape but are sensitive to the downstream shape, with a vertical back side causing the longest disruption. Delays vary cyclically with obstacle orientations. The relaxation power-laws confirm that larger obstacles produce larger wavefront delays and longer recovery times, and for a given area larger obstacle width-to-length ratios produce longer delays. Possible applications include elucidating the effect of inhomogeneities on wavefront recovery in cardiac tissue.

QUALITATIVE ANALYSIS OF DELAYED-CONCENTRATION IN OREGONATOR-BASED CHEMICAL OSCILLATORS

Beulosuv-Zhabotonksiy reaction is pure chemical reaction that exhibits sustained and relaxed oscillations, which are analgous to observed oscillations in many biological systems. The reaction is very complicated, however, its oscillatory mechanism is represented by a simple model, called the Oregonator. Since delays naturally appear in chemical reactions and they are often responsible for presence of complex behaviours, delay effects in Beulosuv-Zhabotonksiy reaction should be considered to better explain the mechanism of some biochemical oscillations. In this work, qualitative and asymptotic analysis of the Oregonator model by taking into account the delays arising due to the physical constraints will be considered. Some numerical results will given to present some certain properties of the considered model.

Study of the Mechanism of the Autowave Structure Formation at the Reaction Front

A qualitative mechanism of the formation of wave structures at the reaction front is proposed. It is assumed that the structures are formed as a result of the interaction of two subsystems, one of which is responsible for the front formation, and the other is responsible for the formation of structures themselves. Three models are considered; two-dimensional analogues of concentric and spiral waves are numerically demonstrated in each. Fitzhugh–Nagumo, Fisher–Kolmogorov–Petrovskii–Piscounov (Fisher–KPP), and Oregonator models were used as subsystems.

Control of transversal instabilities in reaction-diffusion systems

In two-dimensional reaction-diffusion systems, local curvature perturbations on traveling waves are typically damped out and vanish. However, if the inhibitor diffuses much faster than the activator, transversal instabilities can arise, leading from flat to folded, spatio-temporally modulated waves and to spreading spiral turbulence. Here, we propose a scheme to induce or inhibit these instabilities via a spatio-temporal feedback loop. In a piecewise-linear version of the FitzHugh–Nagumo model, transversal instabilities and spiral turbulence in the uncontrolled system are shown to be suppressed in the presence of control, thereby stabilizing plane wave propagation. Conversely, in numerical simulations with the modified Oregonator model for the photosensitive Belousov–Zhabotinsky reaction, which does not exhibit transversal instabilities on its own, we demonstrate the feasibility of inducing transversal instabilities and study the emerging wave patterns in a well-controlled manner.

System identification of propagating wave segments in excitable media and its application to advanced control.

The present paper proposes a scheme for controlling wave segments in excitable media. This scheme consists of two phases: in the first phase, a simple mathematical model for wave segments is derived using only the time series data of input and output signals for the media; in the second phase, the model derived in the first phase is used in an advanced control technique. We demonstrate with numerical simulations of the Oregonator model that this scheme performs better than a conventional control scheme.

Spirals in a reaction-diffusion system: Dependence of wave dynamics on excitability

A detailed study of the effects of excitability of the Belousov-Zhabotinsky (BZ) reaction on spiral wave properties has been carried out. Using the Oregonator model, we explore the various regimes of wave activity, from sustained oscillations to wave damping, as the system undergoes a Hopf bifurcation, that is achieved by varying the excitability parameter, ε. We also discover a short range of parameter values where random oscillations are observed. With an increase in the value of ε, the frequency of the wave decreases exponentially, as the dimension of the spiral core expands. These numerical results are confirmed by carrying out experiments in thin layers of the BZ system, where the excitability is changed by varying the concentrations of the reactant species. Effect of reactant concentrations on wave properties like time period and wavelength are also explored in detail. Drifting and meandering spirals are found in the parameter space under investigation, with the excitability affecting the tip trajectory in a way predicted by the numerical studies. This study acts as a quantitative evidence of the relationship between the excitability parameter, ε, and the substrate concentrations.

Constrained stoichiometric network analysis.

Stoichiometric network analysis (SNA) is a method for studying the stability of steady states of stoichiometric systems by decomposing the corresponding network into elementary subnetworks (also known as extreme currents) and identifying those that may cause loss of a network's stability via interplay of positive and negative feedback. Experimentally studied complex (bio)chemical reactions often display dynamical instabilities leading to oscillations or bistable switches. When modelling such systems, a frequently met case is that an assumed detailed mechanism in terms of power law kinetics is available, but some of the rate coefficients are unknown and obtaining them by traditional kinetic methods based on a least-square fit is cumbersome or unfeasible. We propose a method combining the SNA and experimental data at the point of instability, which provides an estimate of the unknown rate coefficients along with unknown steady state concentrations. The core of the method rests in using constrained linear optimization to find a combination of the elementary subnetworks such that the dominant instability-causing subnetwork is just counter-balanced by stabilizing effects of all other subnetworks to obtain the instability threshold, and at the same time, the experimentally available data (inflow constraints, measured steady state concentrations of some species, frequency of emerging oscillations, etc.) are exactly matched. We illustrate this approach by examining two classical chemical oscillators: the Brusselator chosen as the simplest model for illustration of our methods and the Belousov-Zhabotinsky reaction and its mechanism represented by the Oregonator model as a more advanced example.

Cellular Automaton Belousov–Zhabotinsky Model for Binary Full Adder

The continuous increment in the performance of classical computers has been driven to its limit. New ways are studied to avoid this oncoming bottleneck and many answers can be found. An example is the Belousov–Zhabotinsky (BZ) reaction which includes some fundamental and essential characteristics that attract chemists, biologists, and computer scientists. Interaction of excitation wave-fronts in BZ system, can be interpreted in terms of logical gates and applied in the design of unconventional hardware components. Logic gates and other more complicated components have been already proposed using different topologies and particular characteristics. In this study, the inherent parallelism and simplicity of Cellular Automata (CAs) modeling is combined with an Oregonator model of light-sensitive version of BZ reaction. The resulting parallel and computationally-inexpensive model has the ability to simulate a topology that can be considered as a one-bit full adder digital component towards the design of an Arithmetic Logic Unit (ALU).