Oscillatory clusters in a model of the photosensitive belousov-zhabotinsky reaction system with global feedback

Delayed global feedback in the genesis and stability of spatiotemporal excitation patterns in paced biological excitable media

Biological excitable media, such as cardiac or neural cells and tissue, exhibit memory in which a change in the present excitation may affect the behaviors in the next excitation. For example, a change in calcium (Ca²⁺) concentration in a cell in the present excitation may affect the Ca²⁺ dynamics in the next excitation via bi-directional coupling between voltage and Ca²⁺, forming a delayed feedback loop. Since the Ca²⁺ dynamics inside the excitable cells are spatiotemporal while the membrane voltage is a global signal, the feedback loop is then a delayed global feedback (DGF) loop. In this study, we investigate the roles of DGF in the genesis and stability of spatiotemporal excitation patterns in periodically-paced excitable media using mathematical models with different levels of complexity: a model composed of coupled FitzHugh-Nagumo units, a 3-dimensional physiologically-detailed ventricular myocyte model, and a coupled map lattice model. We investigate the dynamics of excitation patterns that are temporal period-2 (P2) and spatially concordant or discordant, such as subcellular concordant or discordant Ca²⁺alternans in cardiac myocytes or spatially concordant or discordant Ca²⁺ and repolarization alternans in cardiac tissue. Our modeling approach allows both computer simulations and rigorous analytical treatments, which lead to the following results and conclusions. When DGF is absent, concordant and discordant P2 patterns occur depending on initial conditions with the discordant P2 patterns being spatially random. When the DGF is negative, only concordant P2 patterns exist. When the DGF is positive, both concordant and discordant P2 patterns can occur. The discordant P2 patterns are still spatially random, but they satisfy that the global signal exhibits a temporal period-1 behavior. The theoretical analyses of the coupled map lattice model reveal the underlying instabilities and bifurcations for the genesis, selection, and stability of spatiotemporal excitation patterns.

Understanding the mechanisms of pattern formation in biological systems is of great importance. Here we investigate the dynamical mechanisms by which delayed global feedback affects excitation pattern formation and stability in periodically-paced biological excitable media, such as cardiac or neural cells and tissue. We focus on the formation and stability of the temporal period-2 and spatially in-phase and out-of-phase excitation patterns. Using models of different levels of complexity, we show that when the delayed global feedback is negative, only the spatially in-phase patterns are stable. When the feedback is positive, both spatially in-phase and out-of-phase patterns are stable, and the out-of-phase patterns are spatially random but satisfy that the global signals are temporal period-1 solutions.

Photochemically Excited, Pulsating Janus Colloidal Motors of Tunable Dynamics

Spontaneous periodicity is widely found in many biological and synthetic systems, and designing colloidal motors that mimic this feature may not only facilitate our understanding of how complexity emerges but also enable applications that benefit from a time-varying activity. However, there is so far no report on a colloidal motor system that shows controllable and spontaneous oscillation in speeds. Inspired by previous studies of oscillating silver microparticles, we report silver-poly(methyl methacrylate) microsphere Janus colloidal motors that moved, interacted with tracers, and exhibited negative gravitaxis all in an oscillatory fashion. Its dynamics, including pulsating speeds and magnitude, as well as whether moving forward in a pulsating or continuous mode, can be systematically modulated by varying chemical concentrations, light intensity, and the way light was applied. A qualitative mechanism is proposed to link the oscillation of Janus colloidal motors to ionic diffusiophoresis, while nonlinearity is suspected to arise from a sequence of autocatalytic decomposition of AgCl and its slow buildup in the presence of H₂O₂ and light. The generation of light-absorbing Ag nanoparticles is suspected to be the key. This study therefore establishes a robust model system of chemically driven, oscillatory colloidal motors with clear directionality, good tunability, and an improved mechanism, with which complex, emergent phenomena can be explored.

Cluster singularity: The unfolding of clustering behavior in globally coupled Stuart-Landau oscillators

The ubiquitous occurrence of cluster patterns in nature still lacks a comprehensive understanding. It is known that the dynamics of many such natural systems is captured by ensembles of Stuart-Landau oscillators. Here, we investigate clustering dynamics in a mean-coupled ensemble of such limit-cycle oscillators. In particular, we show how clustering occurs in minimal networks and elaborate how the observed 2-cluster states crowd when increasing the number of oscillators. Using persistence, we discuss how this crowding leads to a continuous transition from balanced cluster states to synchronized solutions via the intermediate unbalanced 2-cluster states. These cascade-like transitions emerge from what we call a cluster singularity. At this codimension-2 point, the bifurcations of all 2-cluster states collapse and the stable balanced cluster state bifurcates into the synchronized solution supercritically. We confirm our results using numerical simulations and discuss how our conclusions apply to spatially extended systems.

Emergence of localized patterns in globally coupled networks of relaxation oscillators with heterogeneous connectivity

Oscillations in far-from-equilibrium systems (e.g., chemical, biochemical, biological) are generated by the nonlinear interplay of positive and negative feedback effects operating at different time scales. Relaxation oscillations emerge when the time scales between the activators and the inhibitors are well separated. In addition to the large-amplitude oscillations (LAOs) or relaxation type, these systems exhibit small-amplitude oscillations (SAOs) as well as abrupt transitions between them (canard phenomenon). Localized cluster patterns in networks of relaxation oscillators consist of one cluster oscillating in the LAO regime or exhibiting mixed-mode oscillations (LAOs interspersed with SAOs), while the other oscillates in the SAO regime. Because the individual oscillators are monostable, localized patterns are a network phenomenon that involves the interplay of the connectivity and the intrinsic dynamic properties of the individual nodes. Motivated by experimental and theoretical results on the Belousov-Zhabotinsky reaction, we investigate the mechanisms underlying the generation of localized patterns in globally coupled networks of piecewise-linear relaxation oscillators where the global feedback term affects the rate of change of the activator (fast variable) and depends on the weighted sum of the inhibitor (slow variable) at any given time. We also investigate whether these patterns are affected by the presence of a diffusive type of coupling whose synchronizing effects compete with the symmetry-breaking global feedback effects.

Complex mixed-mode oscillatory patterns in a periodically forced excitable Belousov-Zhabotinsky reaction model

The Oregonator is the simplest chemically plausible model for the Belousov-Zhabotinsky reaction. We investigate the response of the Oregonator to sinusoidal inputs with amplitudes and frequencies within plausible ranges. We focus on a regime where the unforced Oregonator is excitable (with no sustained oscillations). We use numerical simulations and dynamical systems tools to both characterize the response patterns and explain the underlying dynamic mechanisms.

Clustering in globally coupled oscillators near a Hopf bifurcation: theory and experiments

A theoretical analysis is presented to show the general occurrence of phase clusters in weakly, globally coupled oscillators close to a Hopf bifurcation. Through a reductive perturbation method, we derive the amplitude equation with a higher-order correction term valid near a Hopf bifurcation point. This amplitude equation allows us to calculate analytically the phase coupling function from given limit-cycle oscillator models. Moreover, using the phase coupling function, the stability of phase clusters can be analyzed. We demonstrate our theory with the Brusselator model. Experiments are carried out to confirm the presence of phase clusters close to Hopf bifurcations with electrochemical oscillators.

Preferred frequency responses to oscillatory inputs in an electrochemical cell model: linear amplitude and phase resonance

We investigate the dynamic mechanisms of generation of amplitude and phase resonance in a phenomenological electrochemical cell model in response to sinusoidal inputs. We describe how the attributes of the impedance and phase profiles change as the participating physicochemical parameters vary within a range corresponding to the existence of stable nodes and foci in the corresponding autonomous system, thus extending previous work that considered systems close to limit cycle regimes. The method we use permits us to understand how changes in these parameters generate amplifications of the cell's response at the resonant frequency band and captures some important nonlinear effects.

Swing, release, and escape mechanisms contribute to the generation of phase-locked cluster patterns in a globally coupled FitzHugh-Nagumo model

We investigate the mechanism of generation of phase-locked cluster patterns in a globally coupled FitzhHugh-Nagumo model where the fast variable (activator) receives global feedback from the slow variable (inhibitor). We identify three qualitatively different mechanisms (swing-and-release, hold-and-release, and escape-and-release) that contribute to the generation of these patterns. We describe these mechanisms and use this framework to explain under what circumstances two initially out-of-phase oscillatory clusters reach steady phase-locked and in-phase synchronized solutions, and how the phase difference between these steady state cluster patterns depends on the clusters relative size, the global coupling intensity, and other model parameters.

Dynamic mechanisms of generation of oscillatory cluster patterns in a globally coupled chemical system

We use simulations and dynamical systems tools to investigate the mechanisms of generation of phase-locked and localized oscillatory cluster patterns in a globally coupled Oregonator model where the activator receives global feedback from the inhibitor, mimicking experimental results observed in the photosensitive Belousov-Zhabotinsky reaction. A homogeneous two-cluster system (two clusters with equal cluster size) displays antiphase patterns. Heterogenous two-cluster systems (two clusters with different sizes) display both phase-locked and localized patterns depending on the parameter values. In a localized pattern the oscillation amplitude of the largest cluster is roughly an order of magnitude smaller than the oscillation amplitude of the smaller cluster, reflecting the effect of self-inhibition exerted by the global feedback term. The transition from phase-locked to localized cluster patterns occurs as the intensity of global feedback increases. Three qualitatively different basic mechanisms, described previously for a globally coupled FitzHugh-Nagumo model, are involved in the generation of the observed patterns. The swing-and-release mechanism is related to the canard phenomenon (canard explosion of limit cycles) in relaxation oscillators. The hold-and-release and hold-and-escape mechanisms are related to the release and escape mechanisms in synaptically connected neural models. The methods we use can be extended to the investigation of oscillatory chemical reactions with other types of non-local coupling.

Phase and frequency entrainment in locally coupled phase oscillators with repulsive interactions

Recent experiments in one- and two-dimensional microfluidic arrays of droplets containing Belousov-Zhabotinsky reactants show a rich variety of spatial patterns [M. Toiya et al., J. Phys. Chem. Lett. 1, 1241 (2010)]. The dominant coupling between these droplets is inhibitory. Motivated by this experimental system, we study repulsively coupled Kuramoto oscillators with nearest-neighbor interactions, on a linear chain as well as a ring in one dimension, and on a triangular lattice in two dimensions. In one dimension, we show using linear stability analysis as well as numerical study that the stable phase patterns depend on the geometry of the lattice. We show that a transition to the ordered state does not exist in the thermodynamic limit. In two dimensions, we show that the geometry of the lattice constrains the phase difference between two neighboring oscillators to 2π/3. We report the existence of domains with either clockwise or anticlockwise helicity, leading to defects in the lattice. We study the time dependence of these domains and show that at large coupling strengths the domains freeze due to frequency synchronization. Signatures of the above phenomena can be seen in the spatial correlation functions.

Osmotically driven crystal morphogenesis: a general approach to the fabrication of micrometer-scale tubular architectures based on polyoxometalates

The process of osmotically driven crystal morphogenesis of polyoxometalate (POM)-based crystals is investigated, whereby the transformation results in the growth of micrometer-scale tubes 10-100 μm in diameter and many thousands of micrometers long. This process initiates when the crystals are immersed in aqueous solutions containing large cations and is governed by the solubility of the parent POM crystal. Evidence is presented that indicates the process is general to all types of POMs, with solubility of the parent crystal being the deciding parameter. A modular approach is adopted since different POM precursor crystals can form tubular architectures with a range of large cationic species, producing an ion-exchanged material that combines the large added cations and the large POM-based anions. It is also shown that the process of morphogenesis is electrostatically driven by the aggregation of anionic metal oxides with the dissolved cations. This leads to the formation of a semi-permeable membrane around the crystal. The osmotically driven ingress of water leads to an increase in pressure, and ultimately rupture of the membrane occurs, allowing a saturated solution of the POM to escape and leading to the formation of a "self-growing" microtube in the presence of the cation. It is demonstrated that the growth process is sustained by the osmotic pressure within the membrane surrounding the parent crystal, as tube growth ceases whenever this pressure is relieved. Not only is the potential of the modular approach revealed by the fact that the microtubes retain the properties of their component parts, but it is also possible to control the direction of growth and tube diameter. In addition, the solubility limits of tube growth are explored and translated into a predictive methodology for the fabrication of tubular architectures with predefined physical properties, opening the way for real applications.

Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: Propagation failure and control mechanisms

We study the evolution of fronts in a bistable equation with time-delayed global feedback in the fast reaction and slow diffusion regime. This equation generalizes the Hodgkin-Grafstein and Allen-Cahn equations. We derive a nonlinear equation governing the motion of fronts, which includes a term with delay. In the one-dimensional case this equation is linear. We study the motion of one- and two-dimensional fronts, finding a much richer dynamics than for the previously studied cases (without time-delayed global feedback). We explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a Hopf bifurcation as the delay time increases. We show that for certain delay times, the prevailing phase is different from that corresponding to the system in the absence of global coupling. Numerical simulations of the partial differential equation are in agreement with the analytical predictions.

Delay-induced spatial correlations in one-dimensional stochastic networks with nearest-neighbor coupling

We consider a network of deterministic and stochastic locally coupled oscillators with positive or negative dissipation and local time-delayed feedback. (i) For a deterministic system, we study propagation of waves through the network. We show that time delay leads to a coexistence of several neutral modes with different wave numbers and group velocities, which we compute analytically. (ii) For noisy system, we study the response of the network to external random forcing correlated in space and uncorrelated in time. Below the threshold of spatial instability, noise induces spatiotemporal fluctuations, which can be characterized by the structure function. We give an analytical expression for the structure function and demonstrate the effect of the time delay and of the correlation length of noise on the wave number of the most excited mode.

Irregular subharmonic cluster patterns in an autonomous photoelectrochemical oscillator

Unusual subharmonic cluster patterns are observed during the oscillatory electro-oxidation of n-Si(111) under illumination. 2D in situ imaging of the electrode by means of an ellipsometric setup allows local variations in the oxide layer thickness to be monitored. The local oscillators exhibit an irregular distribution of the amplitude with the extrema locked to the constant base frequency of the total current. In addition, Ising 2-phase clustering occurs at half the base frequency. This intrinsic dynamics is described by means of a modified complex Ginzburg-Landau equation.

Mixed-mode oscillations and cluster patterns in an electrochemical relaxation oscillator under galvanostatic control

We studied the dynamics of a prototypical electrochemical model, the electro-oxidation of hydrogen in the presence of poisons, under galvanostatic conditions. The lumped system exhibits relaxation oscillations, which develop mixed-mode oscillations (MMOs) for low preset currents. A fast-slow analysis of the homogeneous dynamics reveals that the MMOs arise from a fast oscillating subsystem and a one-dimensional slow manifold. In the spatially extended system, the galvanostatic constraint imposes a synchronizing global coupling that drives the system into cluster patterns. The properties of the cluster patterns (CPs) result from an intricate interplay of the nature of the local oscillators, the global constraint, and a nonlocal coupling through the electrolyte. In particular, we find that the global constraint suppresses small-amplitude oscillations of MMOs and prevents domains oscillating out of phase from occupying equal regions in phase space. The nonlocal coupling causes each individual clustered region to oscillate on a different limit cycle. Typically multistability of CPs is found. Coexisting patterns possess different oscillation periods and a different total fraction in space that occupies the in-phase or out-of-phase state, respectively.

Localized patterns in reaction-diffusion systems

We discuss a variety of experimental and theoretical studies of localized stationary spots, oscillons, and localized oscillatory clusters, moving and breathing spots, and localized waves in reaction-diffusion systems. We also suggest some promising directions for future research in this area.

Localized structures in a nonlinear wave equation stabilized by negative global feedback: one-dimensional and quasi-two-dimensional kinks

We study the evolution of fronts in a nonlinear wave equation with global feedback. This equation generalizes the Klein-Gordon and sine-Gordon equations. Extending previous work, we describe the derivation of an equation governing the front motion, which is strongly nonlinear, and, for the two-dimensional case, generalizes the damped Born-Infeld equation. We study the motion of one- and two-dimensional fronts, finding a much richer dynamics than for the classical case (with no global feedback), leading in most cases to a localized solution; i.e., the stabilization of one phase inside the other. The nature of the localized solution depends on the strength of the global feedback as well as on other parameters of the model.

A hierarchy of global coupling induced cluster patterns during the oscillatory H2-electrooxidation reaction on a Pt ring-electrode

We report experimental results on spatiotemporal pattern formation during the oscillatory hydrogen electrooxidation reaction on a Pt ring-electrode under negative (desynchronizing) global coupling (GC). Spatially one-dimensional profiles of the interfacial potential drop along the angular direction of the ring electrode are recorded by means of a potential probe. The dynamics is investigated as a function of two control parameters, the applied voltage U and the strength of the global coupling. The latter is adjusted either by varying the distance between the working electrode (WE) and the reference electrode (RE) or by inserting a negative impedance device in series with the WE. In the absence of global coupling, uniform oscillations were destabilized by migration coupling, and electrochemical turbulence developed at large values of U (H. Varela, C. Beta, A. Bonnefont and K. Krischer, Phys. Rev. Lett., 2005, 94, 174104; ). Already low global coupling strengths sufficed to suppress turbulence. Instead, regular two-phase clusters formed. At higher coupling strength, a second type of two-phase cluster was observed as well as two types of irregular cluster patterns, which were connected with an irregular motion of the cluster boundaries and the emergence and disappearance of clusters through splitting and merging of the boundaries, respectively. Upon increasing the coupling strength even further, five-phase clusters were stabilized and at the highest coupling strength applied the cluster patterns transformed into strongly modulated pulses. The two types of two-phase clusters and the five-phase clusters are analyzed employing several signal processing techniques.

Pattern formation for NO+NH3 on Pt(100): two-dimensional numerical results

The Lombardo-Fink-Imbihl model of the NO+NH3 reaction on a Pt(100) surface consists of seven coupled ordinary differential equations (ODE) and shows stable relaxation oscillations with sharp transitions in the relevant temperature range. Here we study numerically the effect of coupling of these oscillators by surface diffusion in two dimensions. We find different types of patterns, in particular phase clusters and standing waves. In models of related surface reactions such clustered solutions are known to exist only under a global coupling through the gas phase. This global coupling is replaced here by relatively fast diffusion of two variables which are kinetically slaved in the ODE. We also compare our simulations with experimental results and discuss some shortcomings of the model.

Control of Turing pattern formation by delayed feedback

The effect of the global delayed feedback technique on Turing pattern formation is investigated in the modified Lengyel-Epstein two-variable model. Feedback intensity, delay time, and feedback-imposing time (the period of time that feedback is present in the system) are all found to be of significant influence on Turing pattern formation time. Under appropriate parameter settings, delayed feedback could suppress or induce the Turing pattern if the feedback intensity is strong enough.

Complex oscillations in a simple model for the Briggs-Rauscher reaction

Complex oscillations in a simple model of the Briggs-Rauscher reaction mechanism in a continuously stirred tank reactor proposed by Kim et al. [J. Chem. Phys. 117, 2710 (2002)] are investigated numerically. The k(0)-[CH(2)(COOH)(2)](0) phase diagram is constructed first where k(0) is the flow rate and [...](0) is the input concentration. Within the region surrounded by the Hopf bifurcation curve, we find complex oscillation regions which are again separated from the regular oscillation region by the secondary Hopf bifurcation curves. Mixed mode oscillations with an incomplete Farey sequence, periodic-chaotic (or nonperiodic) sequence, and various types of burst oscillations are observed in complex oscillation regions. Also, chaotic burst oscillations, which are due to the transition from one kind of burst to another kind, are reported.

Pattern formation on the edge of chaos: mathematical modeling of CO oxidation on a Pt(110) surface under global delayed feedback

Effects of global delayed feedback on diffusion-induced turbulence are studied in a realistic model of catalytic oxidation of carbon monoxide on Pt(110). Spatiotemporal patterns resulting from numerical simulations of this model are identified and analyzed using a transformation into the phase and the amplitude of local oscillations. We find that chemical turbulence can be efficiently controlled by varying the feedback intensity and the delay time in the feedback loop. Near the transition from turbulence to uniform oscillations, various chaotic and regular spatiotemporal patterns-intermittent turbulence, two-phase clusters, cells of hexagonal symmetry, and phase turbulence-are found.

Pattern formation on the edge of chaos: experiments with CO oxidation on a Pt(110) surface under global delayed feedback

Experiments with catalytic oxidation of carbon monoxide on Pt(110) show that chemical turbulence in this system can be suppressed by application of appropriate global delayed feedback. Different spatiotemporal patterns, seen near the transition from turbulence to uniform oscillations, are investigated. Such patterns include intermittent turbulence, oscillatory standing waves, cellular structures, and phase clusters. Using a method based on the Hilbert transform, spatial distributions of local phase and amplitude in these patterns are reconstructed from the experimental data.

Pattern formation in a surface chemical reaction with global delayed feedback

We consider effects of global delayed feedback on anharmonic oscillations in the reaction-diffusion model of the CO oxidation reaction on a Pt(110) single-crystal surface. Depending on the feedback intensity and the delay time, we find that various spatiotemporal patterns can be induced. These patterns are characterized using a transformation to phase and amplitude variables designed for anharmonic oscillations. Typical feedback-induced patterns represent traveling phase flips, asynchronous oscillations, and dynamical clustering. Three different types of cluster patterns are identified: amplitude clusters, phase clusters, and cluster turbulence. For phase clusters, two different front instabilities are possible. A pitchfork bifurcation leads to propagation of cluster fronts. An instability of the state of phase balance results in spatial front oscillations.

Oscillatory clusters in the periodically illuminated, spatially extended Belousov-Zhabotinsky reaction

Cluster-cluster transitions in the periodically illuminated photosensitive Belousov-Zhabotinsky (BZ) reaction-diffusion system exhibit the same scenario as in the autonomous BZ system with negative global feedback: two-phase clusters <--> three-phase clusters <--> irregular clusters <--> localized clusters. Transitions induced by changing the dark ( TD) or light ( TL) phases of the periodic external square wave illumination are dependent not only on the frequency of illumination at constant TD/TL, but also on the ratio TD/TL at constant frequency (when TD+TL = const).