Pattern formation in a four-ring reaction-diffusion network with heterogeneity

In networks of nonlinear oscillators, symmetries place hard constraints on the system that can be exploited to predict universal dynamical features and steady states, providing a rare generic organizing principle for far-from-equilibrium systems. However, the robustness of this class of theories to symmetry-disrupting imperfections is untested in free-running (i.e., non-computer-controlled) systems. Here, we develop a model experimental reaction-diffusion network of chemical oscillators to test applications of the theory of dynamical systems with symmeries in the context of self-organizing systems relevant to biology and soft robotics. The network is a ring of four microreactors containing the oscillatory Belousov-Zhabotinsky reaction coupled to nearest neighbors via diffusion. Assuming homogeneity across the oscillators, theory predicts four categories of stable spatiotemporal phase-locked periodic states and four categories of invariant manifolds that guide and structure transitions between phase-locked states. In our experiments, we observed that three of the four phase-locked states were displaced from their idealized positions and, in the ensemble of measurements, appeared as clusters of different shapes and sizes, and that one of the predicted states was absent. We also observed the predicted symmetry-derived synchronous clustered transients that occur when the dynamical trajectories coincide with invariant manifolds. Quantitative agreement between experiment and numerical simulations is found by accounting for the small amount of experimentally determined heterogeneity in intrinsic frequency. We further elucidate how different patterns of heterogeneity impact each attractor differently through a bifurcation analysis. We show that examining bifurcations along invariant manifolds provides a general framework for developing intuition about how chemical-specific dynamics interact with topology in the presence of heterogeneity that can be applied to other oscillators in other topologies.

Chemical micro-oscillators based on the Belousov–Zhabotinsky reaction

The results of studies on the development of micro-oscillators (MOs) based on the Belousov –Zhabotinsky (BZ) oscillatory chemical reaction are integrated and systematized. The mechanisms of the BZ reaction and the methods of immobilization of the catalyst of the BZ reaction in micro-volumes are briefly discussed. Methods for creating BZ MOs based on water microdroplets in the oil phase and organic and inorganic polymer microspheres are considered. Methods of control and management of the dynamics of BZ MO networks are described, including methods of MO synchronization. The prospects for the design of neural networks of MOs with intelligent-like behaviour are outlined. Such networks present a new area of nonlinear chemistry, including, in particular, the creation of a chemical ‘computer’.

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Impact of PDMS-Based Microfluidics on Belousov-Zhabotinsky Chemical Oscillators

Sub-nanoliter volumes of the Belousov-Zhabotinsky (BZ) reaction are sealed in microfluidic devices made from polydimethylsiloxane (PDMS). Bromine, which is a BZ reaction intermediate that participates in the inhibitory pathway of the reaction, is known to permeate into PDMS, and it has been suggested that PDMS and bromine can react ( J. Phys. Chem. A. 108, 2004, 1325-1332). We characterize the extent to which PDMS affects BZ oscillations by varying the volume of the PDMS surrounding the BZ reactors. We measure how the oscillation period varies with PDMS volume and compare with a theoretical reaction-diffusion model, concluding that bromine reacts with PDMS. We demonstrate that minimizing the amount of PDMS by making the samples as thin as possible maximizes the number of oscillations before the BZ reaction reaches equilibrium and ceases to oscillate. We also demonstrate that the deleterious effects of the PDMS-BZ interactions are somewhat mitigated by imposing constant chemical boundary conditions through using a light-sensitive catalyst, ruthenium, in combination with patterned illumination. Furthermore, we show that light can modulate the frequency and phase of the BZ oscillators contained in a PDMS matrix by 20-30%.

Membrane Structure Drives Synchronization Patterns in Arrays of Diffusively Coupled Self-Oscillating Droplets

Networks of diffusively coupled inorganic oscillators, confined in nano- and microcompartments, are effective for predicting and understanding the global dynamics of those systems where the diffusion of activatory or inhibitory signals regulates the communication among different individuals. By taking advantage of a microfluidic device, we study the dynamics of arrays of diffusively coupled Belousov-Zhabotinsky (BZ) oscillators encapsulated in water-in-oil single emulsions. New synchronization patterns are induced and controlled by modulating the structural and chemical properties of the phospholipid-based biomimetic membranes via the introduction of specific dopants. Doping molecules do not alter the membrane basic backbone, but modify the lamellarity (and, in turn, the permeability) or interact chemically with the reaction intermediates. A transition from two-period clusters showing 1:2 period-locking to one-period antiphase synchronization is observed by decreasing the membrane lamellarity. An unsynchronized scenario is found when the dopant is able to interfere with chemical communication by reacting with the chemical messengers.

Hierarchical network of pulse coupled chemical oscillators with adaptive behavior: Chemical neurocomputer

We consider theoretically a network of pulse coupled oscillators with time delays. Each oscillator is described by the Oregonator-like model for the Belousov-Zhabotinsky (BZ) reaction. Different groups of oscillators constitute five functional units: (1) a central pattern generator (CPG), (2) a "reader" unit that can identify dynamical modes of the CPG, (3) an antenna (A) unit that receives external signals and responds on them by generating different dynamical modes, (4) another reader unit for identification of the dynamical modes in the A unit, and (5) a decision making unit that switches the current dynamical mode of the CPG to the mode that is similar to the current mode in the A unit. We call this network a chemical neurocomputer, since chemical BZ reaction occurs in each micro-oscillator, while pulse connectivity of these cells is inspired by the brain.

Multivariate statistical analysis of chemical and electrochemical oscillators for an accurate frequency selection

The effect of experimental parameters on the frequency of chemical oscillators has been systematically studied since the first observations of clock reactions. The approach is mainly based on univariate changes in one specific parameter while others are kept constant. The frequency is then monitored and the effect of each parameter is discussed separately. This type of analysis, however, does not take into account the multiple interactions among the controllable parameters and the synergic responses on the oscillation frequency. We have carried out a multivariate statistical analysis of chemical (BZ-ferroin catalyzed reaction) and electrochemical (Cu/Cu₂O cathodic deposition) oscillators and identified the contributions of the experimental parameters on frequency variations. The BZ reaction presented a strong dependence on the initial concentration of sodium bromate and temperature, resulting in a frequency increase. The concentration of malonic acid, the organic substrate, affects the system but with lower intensity compared with the combination of sodium bromate and temperature. On the other hand, the Cu/Cu₂O electrochemical oscillator was shown to be less sensitive to changes in the temperature. The applied current density and pH were the two parameters which most perturbed the system. Interestingly, the frequency behaved nonmonotonically with a quadratic dependence. The multivariate analysis of both oscillators exhibited significant differences - while the homogenous oscillator displayed a linear dependence with the factors, the heterogeneous one revealed a more complex dependence with quadratic terms. Our results may contribute, for instance, in the synthesis of self-organized materials in which an accurate frequency selection is required and, depending on its value, different physicochemical properties are obtained.

Competition between global feedback and diffusion in coupled Belousov-Zhabotinsky oscillators

The Belousov-Zhabotinsky (BZ) reaction is a famous experimental model for chemical oscillatory reaction and pattern formation. We herein study a diffusive coupled system of two oscillators with global feedback using the photosensitive BZ reaction both experimentally and theoretically. The coupled oscillator showed in-phase and antiphase oscillations depending on the strength of diffusive coupling and light feedback. Moreover, we analyzed our model to locate the bifurcational origin and found the reconnection of the bifurcation branches for antiphase oscillation, which was induced by the competition between global feedback and the diffusion effect.