Igor Schreiber

Strange attractors in coupled reaction-diffusion cells

Abstract A numerical study of two identical reaction cells with diffusion coupling has shown that the structure of motion in the system in principle agrees with results reported for variety of dynamic systems. When the characteristic parameter is varied, alternating sub-intervals of stable periodic ( P k ) and stable aperiodic ( A l ) solutions appear. The sub-intervals are connected by intervals, where tangent bifurcations and infinite sequences of subharmonic bifurcations occur. Feigenbaum relation holds for the studied sequence of subharmonic bifurcations. Aperiodic (chaotic) states are characterized by a complete set of one-dimensional Lyapunov exponents, by power spectra, and by corresponding Poincare maps. The spectra of Lyapunov exponents are of the type (+.0.-.-), and show that the topological dimension of the chaotic attractor is two. The power spectra are of two different types (a) the spectra with sharp peaks located above the broad-band noise, showing statistical phase coherence of the attractor, (b) the flat spectra showing only broad-band noise, corresponding to phase incoherent attractor. The phase coherence is present close after every point of accumulation. Phase incoherence arises when the strange attractors contain unstable periodic orbits with different topology. The relations of bifurcated stable and unstable periodic solutions (computed by means of continuation techniques) to the structure of the strange attractor is discussed and Poincare maps are used to illustrate the dependence of the structure of the attractors on the value of the characteristic parameter.

Transition to chaos via two-torus in coupled reaction-diffusion cells

Abstract A numerical example of the transition to turbulence, where instead of the progression of the period-doubling bifurcations occurs a transition to turbulence via tori is discussed. When the intensity of interaction of two diffusionally coupled Brusselators decreases, the torus appears after one period-doubling bifurcation; the torus undergoes a number of changes until a chaotic attractor is formed.

Mixed-mode oscillations in a homogeneous pH-oscillatory chemical reaction system.

We examine experimentally a chemical system in a flow-through stirred reactor, which is known to provide large-amplitude oscillations of the pH value. By systematic variation of the flow rate, we find that the system displays hysteresis between a steady state and oscillations, and more interestingly, a transition to chaos involving mixed-mode oscillations. The basic pattern of the measured pH in the mixed-mode regime includes a large-scale peak followed by a series of oscillations on a much smaller scale, which are usually highly irregular and of variable duration. The bifurcation diagram shows that chaos sets in via a period-doubling route observed on the large-amplitude scale, but simultaneously small-amplitude oscillations are involved. Beyond the apparent accumulation of period doubling bifurcations, a mixed-mode regime with irregular oscillations on both scales is observed, occasionally interrupted by windows of periodicity. As the flow rate is further increased, chaos turns into quasiperiodicity and later to a simple small-amplitude periodic regime. Dynamics of selected typical regimes were examined with the tools of nonlinear time-series analysis, which include phase space reconstruction of an attractor and calculation of the maximal Lyapunov exponent. The analysis points to deterministic chaos, which appears via a period doubling route from below and via a route involving quasiperiodicity from above, when the flow rate is varied.

Reduction waves in the BZ reaction: circles, spirals and effects of electric field

Abstract Spatiotemporal patterns in the Belousov-Zhabotinsky reaction medium in a Petri dish at low concentrations of malonic acid are studied experimentally and theoretically. Depending on the concentration of malonic acid, the interaction of reduction and oxidation front waves may lead to: (i) a disappearing reduction pulse wave, (ii) complex target structures, (iii) a stably propagating reduction pulse wave. A mechanical perturbation can lead either to reduction spirals or to healing effects causing heart-shaped structures. Imposed electric field causes a symmetry breaking of target patterns and can stabilize or destabilize the waves. A formal reaction-diffusion model reproduces qualitatively most of the experimentally observed phenomena in the absence of electric field. A modified Oregonator model that involves ionic migration describes well some experiments including those in the presence of electric field.

Chaotic patterns in a coupled oscillator-excitator biochemical cell system.

In this paper we examine dynamical modes resulting from diffusion-like interaction of two model biochemical cells. Kinetics in each of the cells is given by the ICC model of calcium ions in the cytosol. Constraints for one of the cells are set so that it is excitable. One of the constraints in the other cell - a fraction of activated cell surface receptors-is varied so that the dynamics in the cell is either excitable or oscillatory or a stable focus. The cells are interacting via mass transfer and dynamics of the coupled system are studied as two parameters are varied-the fraction of activated receptors and the coupling strength. We find that (i) the excitator-excitator interaction does not lead to oscillatory patterns, (ii) the oscillator-excitator interaction leads to alternating phase-locked periodic and quasiperiodic regimes, well known from oscillator-oscillator interactions; torus breaking bifurcation generates chaos when the coupling strength is in an intermediate range, (iii) the focus-excitator interaction generates compound oscillations arranged as period adding sequences alternating with chaotic windows; the transition to chaos is accompanied by period doublings and folding of branches of periodic orbits and is associated with a Shilnikov homoclinic orbit. The nature of spontaneous self-organized oscillations in the focus-excitator range is discussed. (c) 1999 American Institute of Physics.

Periodic and aperiodic regimes in coupled dissipative chemical oscillators

Dynamic behavior of two identical reaction cells with linear symmetric coupling is studied in detail. The standard model reaction scheme “Brusselator” is used as the description of the kinetics. The uncoupled cells can exhibit either a stable stationary state or stable periodic oscillations. A number of stationary and periodic oscillatory patterns arise as a result of the coupling. A non-homogeneous spatio-temporal organization includes homoclinic and heteroclinic oscillations as well as chaotic regimes. Numerical continuation algorithms are used to determine the dependence of stationary and periodic solutions on parameters. Stable stationary nonhomogeneous regimes exist typically at intermediate levels of coupling intensity. The nonhomogeneous periodic solutions arise either via Hopf bifurcatios from stationary solutions or via period-doubling bifurcations from the homogeneous periodic solutions. The results obtained may serve as a standard for the study of the behavior of other coupled systems in which either a stable stationary state or stable oscillations exist in the single cell.

Dynamical regimes of a pH-oscillator operated in two mass-coupled flow-through reactors

We present results of experiments focused on emergent and cooperative dynamics in a system of two coupled flow-through stirred reaction cells with diffusion-like mass exchange and a strongly nonlinear chemical reaction between hydrogen peroxide and thiosulphate catalysed by cupric ions in diluted solution of sulphuric acid. Due to complex mechanism, in which a crucial role is played by hydrogen and/or hydroxide ions, dynamics in a single cell entail multiple stationary states, excitability and oscillations conveniently indicated by measuring pH. When coupled, the system shows a plethora of dynamical regimes depending on the coupling strength and flow rate. Under certain conditions both cells display dynamics close to that in the absence of coupling, but majority of the regimes are emergent and cannot be deduced from dynamics of decoupled reactors. The most prominent is a stationary state maintaining highly acidic values of pH in one of the reactors and weakly acidic in the other. When each cell is set to display excitability and the coupled system is externally perturbed, the cells may cooperate and transmit excitations elicited by pulsed perturbations in one cell to the other. Periodic pulses induce firing patterns marked by a various degree of propagated excitations and by being periodic or irregular.

Control of Turing patterns and their usage as sensors, memory arrays, and logic gates

We study a model system of three diffusively coupled reaction cells arranged in a linear array that display Turing patterns with special focus on the case of equal coupling strength for all components. As a suitable model reaction we consider a two-variable core model of glycolysis. Using numerical continuation and bifurcation techniques we analyze the dependence of the systems steady states on varying rate coefficient of the recycling step while the coupling coefficients of the inhibitor and activator are fixed and set at the ratios 100:1, 1:1, and 4:5. We show that stable Turing patterns occur at all three ratios but, as expected, spontaneous transition from the spatially uniform steady state to the spatially nonuniform Turing patterns occurs only in the first case. The other two cases possess multiple Turing patterns, which are stabilized by secondary bifurcations and coexist with stable uniform periodic oscillations. For the 1:1 ratio we examine modular spatiotemporal perturbations, which allow for controllable switching between the uniform oscillations and various Turing patterns. Such modular perturbations are then used to construct chemical computing devices utilizing the multiple Turing patterns. By classifying various responses we propose: (a) a single-input resettable sensor capable of reading certain value of concentration, (b) two-input and three-input memory arrays capable of storing logic information, (c) three-input, three-output logic gates performing combinations of logical functions OR, XOR, AND, and NAND.

Electrodiffusion Kinetics of Ionic Transport in a Simple Membrane Channel

We employ numerical techniques for solving time-dependent full Poisson-Nernst-Planck (PNP) equations in 2D to analyze transient behavior of a simple ion channel subject to a sudden electric potential jump across the membrane (voltage clamp). Calculated spatiotemporal profiles of the ionic concentrations and electric potential show that two principal exponential processes can be distinguished in the electrodiffusion kinetics, in agreement with original Plancks predictions. The initial fast process corresponds to the dielectric relaxation, while the steady state is approached in a second slower exponential process attributed to the nonlinear ionic redistribution. Effects of the model parameters such as the channel length, height of the potential step, boundary concentrations, permittivity of the channel interior, and ionic mobilities on electrodiffusion kinetics are studied. Numerical solutions are used to determine spatiotemporal profiles of the electric field, ionic fluxes, and both the conductive and displacement currents. We demonstrate that the displacement current is a significant transient component of the total electric current through the channel. The presented results provide additional information about the classical voltage-clamp problem and offer further physical insights into the mechanism of electrodiffusion. The used numerical approach can be readily extended to multi-ionic models with a more structured domain geometry in 2D or 3D, and it is directly applicable to other systems, such as synthetic nanopores, nanofluidic channels, and nanopipettes.

On the origin of bistability in the Stage 2 of the Huang-Ferrell model of the MAPK signaling

Mitogen-activated protein kinases (MAPKs) are important signal transducing enzymes, unique to eukaryotes, that are involved in many pathways of cellular regulation. Successive phosphorylation cascades mediated by MAPKs serve as sensitive switches initiating various cellular processes. Apart from this basic feature, the underlying reaction network is capable of displaying other nonlinear phenomena including bistable steady states and hysteresis as well as periodic oscillations. We show that from the mechanistic point of view, bistability is a consequence of interaction between single and double phosphorylation/dephosphorylation pathways in a Stage 2 subsystem of the Huang-Ferrell model. Within this subsystem we uncover the core subnetwork obtained by systematic reduction relying on the methods of stoichiometric network analysis. For the core model we show that there is either one stable steady state or three steady states of which two are stable and point out the role of interplay between the single and double phosphorylation subnetworks in generating bistability.