Unpinning of spiral waves by electrical forcing in excitable chemical media

Effect of excitability on partially pinned scroll waves in excitable chemical media

We present an investigation of excitability effects on the dynamics of scroll waves partially pinned to inert cylindrical obstacles in three-dimensional Belousov-Zhabotinsky excitable media. We also report on corresponding numerical simulations with the Oregonator model. The excitability varies according to the concentration of sulfuric acid [H_{2}SO_{4}] in the Belousov-Zhabotinsky (BZ) reaction and the parameter ɛ^{-1} in the Oregonator model. Initially, the freely rotating scroll segment rotates faster than the pinned one. The difference in the frequency of the two parts results in a transition from a straight pinned scroll wave to a twisted one, which helically wraps around the entire obstacle. The wave frequency in the whole volume is equal to that of the freely rotating scroll wave. When the excitability is increased, the time for the transition to the twisted wave structure decreases while the average speed s of the development increases. After the transition, the twisted wave remains stable. In media with higher excitability, the helical pitch is shorter but the twist rate ω is higher. Analysis presented in this study together with our previous findings of the effect of the cylindrical obstacle diameter on the wave dynamics results in common features: The average speed s and the twist rate ω of both studies fit well to functions of the difference in the initial frequency Δf of the freely rotating and untwisted pinned waves. We also demonstrate the robustness of the partially pinned scroll waves against perturbations from spontaneous waves emerging during the wave generation in the BZ medium with high [H_{2}SO_{4}]. Even though the scroll wave is partly disturbed at the beginning of the experiment, the spontaneous waves are gradually suppressed and the typical wave structure is finally developed.

Theory and experiments of spiral unpinning in the Belousov-Zhabotinsky reaction using a circularly polarized electric field

We present the first experimental study of unpinning an excitation wave using a circularly polarized electric field. The experiments are conducted using the excitable chemical medium, the Belousov-Zhabotinsky (BZ) reaction, which is modeled with the Oregenator model. The excitation wave in the chemical medium is charged so that it can directly interact with the electric field. This is a unique feature of the chemical excitation wave. The mechanism of wave unpinning in the BZ reaction with a circularly polarized electric field is investigated by varying the pacing ratio, the initial phase of the wave, and field strength. The chemical wave in the BZ reaction unpins when the electric force opposite the direction of the spiral is equal to or above a threshold. We developed an analytical relation of the unpinning phase with the initial phase, the pacing ratio, and the field strength. This is then verified in experiments and simulations.

Information Processing Using Networks of Chemical Oscillators

I believe the computing potential of systems with chemical reactions has not yet been fully explored. The most common approach to chemical computing is based on implementation of logic gates. However, it does not seem practical because the lifetime of such gates is short, and communication between gates requires precise adjustment. The maximum computational efficiency of a chemical medium is achieved if the information is processed in parallel by different parts of it. In this paper, I review the idea of computing with coupled chemical oscillators and give arguments for the efficiency of such an approach. I discuss how to input information and how to read out the result of network computation. I describe the idea of top-down optimization of computing networks. As an example, I consider a small network of three coupled chemical oscillators designed to differentiate the white from the red points of the Japanese flag. My results are based on computer simulations with the standard two-variable Oregonator model of the oscillatory Belousov−Zhabotinsky reaction. An optimized network of three interacting oscillators can recognize the color of a randomly selected point with >98% accuracy. The presented ideas can be helpful for the experimental realization of fully functional chemical computing networks.

The Concilium of Information Processing Networks of Chemical Oscillators for Determining Drug Response in Patients With Multiple Myeloma

It can be expected that medical treatments in the future will be individually tailored for each patient. Here we present a step towards personally addressed drug therapy. We consider multiple myeloma treatment with drugs: bortezomib and dexamethasone. It has been observed that these drugs are effective for some patients and do not help others. We describe a network of chemical oscillators that can help to differentiate between non-responsive and responsive patients. In our numerical simulations, we consider a network of 3 interacting oscillators described with the Oregonator model. The input information is the gene expression value for one of 15 genes measured for patients with multiple myeloma. The single-gene networks optimized on a training set containing outcomes of 239 therapies, 169 using bortezomib and 70 using dexamethasone, show up to 71% accuracy in differentiating between non-responsive and responsive patients. If the results of single-gene networks are combined into the concilium with the majority voting strategy, then the accuracy of predicting the patient’s response to the therapy increases to ∼ 85%.

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.

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.

Robustness of free and pinned spiral waves against breakup by electrical forcing in excitable chemical media

We present an investigation on the breakup of free and pinned spiral waves under an applied electrical current in the Belousov-Zhabotinsky reaction. Spiral fronts propagating towards the negative electrode are decelerated. A breakup of the spiral waves occurs when some segments of the fronts are stopped by a sufficiently strong electrical current. In the absence of obstacles (i.e., free spiral waves), the critical value of the electrical current for the wave breakup increases with the excitability of the medium. For spiral waves pinned to circular obstacles, the critical electrical current increases with the obstacle diameter. Analysis of spiral dynamics shows that the enhancement of the robustness against the breakup of both free and pinned spiral waves is originated by the increment of wave speed when either the excitability is strengthened or the obstacle size is enlarged. The experimental findings are reproduced by numerical simulations using the Oregonator model. In addition, the simulations reveal that the robustness against the forced breakup increases with the activator level in both cases of free and pinned spiral waves.

Propagation of spiral waves pinned to circular and rectangular obstacles

We present an investigation of spiral waves pinned to circular and rectangular obstacles with different circumferences in both thin layers of the Belousov-Zhabotinsky reaction and numerical simulations with the Oregonator model. For circular objects, the area always increases with the circumference. In contrast, we varied the circumference of rectangles with equal areas by adjusting their width w and height h. For both obstacle forms, the propagating parameters (i.e., wavelength, wave period, and velocity of pinned spiral waves) increase with the circumference, regardless of the obstacle area. Despite these common features of the parameters, the forms of pinned spiral waves depend on the obstacle shapes. The structures of spiral waves pinned to circles as well as rectangles with the ratio w/h∼1 are similar to Archimedean spirals. When w/h increases, deformations of the spiral shapes are observed. For extremely thin rectangles with w/h≫1, these shapes can be constructed by employing semicircles with different radii which relate to the obstacle width and the core diameter of free spirals.

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.