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Chemical oscillations in the classic Belousov-Zhabotinsky (BZ) system typically have a period of a few minutes, which can be increased significantly by changing the organic substrate. Here we show that by changing the temperature and concentrations, an increase of 3-4 orders of magnitude in the frequency of BZ oscillations can be obtained. At elevated temperatures, in high concentration mixtures, the cerium-catalyzed reaction exhibits sinusoidal oscillations with frequencies of 10 Hz or greater. We report the effect of temperature on the frequency and shape of oscillations in experiments under batch conditions and in a four-variable model. We show that our simple model accurately captures the complex temporal behavior of the system and suggests paths toward even higher frequencies.

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Breathing spiral waves are observed in the oscillatory chlorine dioxide-iodine-malonic acid reaction-diffusion system. The breathing develops within established patterns of multiple spiral waves after the concentration of polyvinyl alcohol in the feeding chamber of a continuously fed, unstirred reactor is increased. The breathing period is determined by the period of bulk oscillations in the feeding chamber. Similar behavior is obtained in the Lengyel-Epstein model of this system, where small amplitude parametric forcing of spiral waves near the spiral wave frequency leads to the formation of breathing spiral waves in which the period of breathing is equal to the period of forcing.

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Several reaction-diffusion systems that exhibit temporal periodicity when well mixed also display spatio-temporal pattern formation in a spatially distributed, unstirred configuration. These patterns can be travelling (e.g. spirals, concentric circles, plane waves) or stationary in space (Turing structures, standing waves). The behaviour of coupled and forced temporal oscillators has been well studied, but much less is known about the phenomenology of forced and coupled patterns. We present experimental results focusing primarily on coupled patterns in two chemical systems, the chlorine dioxide-iodine-malonic acid reaction and the Belousov-Zhabotinsky reaction. The observed behaviour can be simulated with simple chemically plausible models.

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We study wave propagation, interaction, and transmission across the boundary between two chemical media in a model of an oscillatory reaction-diffusion medium subjected to local periodic forcing. The forced waves can be either outwardly (OP) or inwardly propagating (IP), depending on the dispersion of the medium. Competition among forced waves, spontaneous spiral waves, and bulk oscillations is studied for both cases. We demonstrate development of a negatively refracted wave train when forced waves traverse the boundary between the OP medium and the IP medium.

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We describe a new type of solitary waves, which propagate in such a manner that the pulse periodically disappears from its original position and reemerges at a fixed distance. We find such jumping waves as solutions to a reaction-diffusion system with a subcritical short-wavelength instability. We demonstrate closely related solitary wave solutions in the quintic complex Ginzburg-Landau equation. We study the characteristics of and interactions between these solitary waves and the dynamics of related wave trains and standing waves.

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The best known Turing patterns are composed of stripes or simple hexagonal arrangements of spots. Until recently, Turing patterns with other geometries have been observed only rarely. Here we present experimental studies and mathematical modeling of the formation and stability of hexagonal and square Turing superlattice patterns in a photosensitive reaction-diffusion system. The superlattices develop from initial conditions created by illuminating the system through a mask consisting of a simple hexagonal or square lattice with a wavelength close to a multiple of the intrinsic Turing pattern's wavelength. We show that interaction of the photochemical periodic forcing with the Turing instability generates multiple spatial harmonics of the forcing patterns. The harmonics situated within the Turing instability band survive after the illumination is switched off and form superlattices. The square superlattices are the first examples of time-independent square Turing patterns. We also demonstrate that in a system where the Turing band is slightly below criticality, spatially uniform internal or external oscillations can create oscillating square patterns.

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Synaptic plasticity in CA1 hippocampal neurons depends on Ca2+ elevation and the resulting activation of calmodulin-dependent enzymes. Induction of long-term depression (LTD) depends on calcineurin, whereas long-term potentiation (LTP) depends on Ca2+/calmodulin-dependent protein kinase II (CaMKII). The concentration of calmodulin in neurons is considerably less than the total concentration of the apocalmodulin-binding proteins neurogranin and GAP-43, resulting in a low level of free calmodulin in the resting state. Neurogranin is highly concentrated in dendritic spines. To elucidate the role of neurogranin in synaptic plasticity, we constructed a computational model with emphasis on the interaction of calmodulin with neurogranin, calcineurin, and CaMKII. The model shows how the Ca2+ transients that occur during LTD or LTP induction affect calmodulin and how the resulting activation of calcineurin and CaMKII affects AMPA receptor-mediated transmission. In the model, knockout of neurogranin strongly diminishes the LTP induced by a single 100 Hz, 1 s tetanus and slightly enhances LTD, in accord with experimental data. Our simulations show that exchange of calmodulin between a spine and its parent dendrite is limited. Therefore, inducing LTP with a short tetanus requires calmodulin stored in spines in the form of rapidly dissociating calmodulin-neurogranin complexes.

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Molecular switches have been implicated in the storage of information in biological systems. For small structures such as synapses, these switches are composed of only a few molecules and stochastic fluctuations are therefore of importance. Such fluctuations could potentially lead to spontaneous switch reset that would limit the lifetime of information storage. We have analyzed a model of the calcium/calmodulin-dependent protein kinase II (CaMKII) switch implicated in long-term memory in the nervous system. The bistability of this switch arises from autocatalytic autophosphorylation of CaMKII, a reaction that is countered by a saturable phosphatase-1-mediated dephosphorylation. We sought to understand the factors that control switch stability and to determine the functional relationship between stability and the number of molecules involved. Using Monte Carlo simulations, we found that the lifetime of states of the switch increase exponentially with the number of CaMKII holoenzymes. Switch stability requires a balance between the kinase and phosphatase rates, and the kinase rate must remain high relative to the rate of protein turnover. Thus, a critical limit on switch stability is set by the observed turnover rate (one per 30 h on average). Our computational results show that, depending on the timescale of fluctuations in enzyme numbers, for a switch composed of about 15 CaMKII holoenzymes, the stable persistent activation can span from a few years to a human lifetime.

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We study the mechanism of development of superlattice Turing structures from photochemically generated hexagonal patterns of spots with wavelengths several times larger than the characteristic wavelength of the Turing patterns that spontaneously develop in the nonilluminated system. Comparison of the experiment with numerical simulations shows that interaction of the photochemical periodic forcing with the Turing instability results in generation of multiple resonant triplets of wave vectors, which are harmonics of the external forcing. Some of these harmonics are situated within the Turing instability band and are therefore able to maintain their amplitude as the system evolves and after illumination ceases, while photochemically generated harmonics outside the Turing band tend to decay.

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Turing patterns in the chlorine dioxide-iodine-malonic acid reaction are studied in a system consisting of two coupled gel layers. Patterns with two wavelengths are observed. Changing the strength of the interlayer coupling causes a transition between a superposition of Turing patterns and a superlattice pattern. The effects of the reactant concentrations on the pattern wavelengths are delineated.

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We study the Brusselator reaction-diffusion model under conditions where the Hopf mode is supercritical and the Turing band is subcritical. Oscillating Turing patterns arise in the system when bulk oscillations lose their stability to spatial perturbations. Spatially uniform external periodic forcing can generate oscillating Turing patterns when both the Turing and Hopf modes are subcritical in the autonomous system. Most of the symmetric patterns show period doubling in both space and time. Patterns observed include squares, rhombi, stripes, and hexagons.

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Families of complex superlattice structures, consisting of combinations of basic hexagonal or square patterns, are found in a photosensitive reaction-diffusion system. The structures are induced by simple illumination patterns whose wavelengths are appropriately related to that of the system's intrinsic Turing pattern. Computer simulations agree with the structures and their stability. The technique offers a general approach to generating superlattices for use in information storage and other applications.

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Spatial resonances leading to superlattice hexagonal patterns, known as "black-eyes," and superposition patterns combining stripes and/or spots are studied in a reaction-diffusion model of two interacting Turing modes with different wavelengths. A three-phase oscillatory interlacing hexagonal lattice pattern is also found, and its appearance is attributed to resonance between a Turing mode and its subharmonic.

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We study the evolution of fronts in a bistable reaction-diffusion system when the nonlinear reaction term is spatially inhomogeneous. This equation has been used to model wave propagation in various biological systems. Extending previous works on homogeneous reaction terms, we derive asymptotically an equation governing the front motion, which is strongly nonlinear and, for the two-dimensional case, generalizes the classical mean curvature flow equation. We study the motion of one- and two-dimensional fronts, finding that the inhomogeneity acts as a "potential function" for the motion of the front; i.e., there is wave propagation failure and the steady state solution depends on the structure of the function describing the inhomogeneity. (c) 2001 American Institute of Physics.

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The history of the discovery and study of chemical oscillations and waves is presented from the very first accidental observations up to the systematic design of chemical oscillators. Special emphasis is devoted to the long-term debate over the possibility of pure chemical oscillations, i.e., concentration oscillations in homogeneous closed systems.