Spike autosolitons and pattern formation scenarios in the two-dimensional Gray-Scott model

We performed an extensive numerical study of pattern formation scenarios in the two-dimensional Gray-Scott reaction-diffusion model. We concentrated on the parameter region in which there exists a strong separation of length and/or time scales. We found that the static one-dimensional autosolitons (stripes) break up into two-dimensional radially-symmetric autosolitons (spots). The traveling one-dimensional autosolitons (wave fronts) can be stable or undergo breakup. The static two-dimensional radially-symmetric autosolitons may break up and self-replicate leading to the formation of space-filling patterns of spots, wave fronts, or spatio-temporal chaos due to the competition of self-replication and annihilation of spots upon collision. Received 6 November 2000 and Received in final form 27 February 2001     

Pattern formation in the thiourea-iodate-sulfite system: Spatial bistability, waves, and stationary patterns

We present a detailed study of the reaction-diffusion patterns observed in the thiourea-iodate-sulfite (TuIS) reaction, operated in open one-side-fed reactors. Besides spatial bistability and spatio-temporal oscillatory dynamics, this proton autoactivated reaction shows stationary patterns, as a result of two back-to-back Turing bifurcations, in the presence of a low-mobility proton binding agent (sodium polyacrylate). This is the third aqueous solution system to produce stationary patterns and the second to do this through a Turing bifurcation. The stationary pattern forming capacities of the reaction are explored through a systematic design method, which is applicable to other bistable and oscillatory reactions. The spatio-temporal dynamics of this reaction is compared with that of the previous ferrocyanide-iodate-sulfite mixed Landolt system.     

Spatio-temporal analysis of irregular vocal fold oscillations: biphonation due to desynchronization of spatial modes.

This report is on direct observation and modal analysis of irregular spatio-temporal vibration patterns of vocal fold pathologies in vivo. The observed oscillation patterns are described quantitatively with multiline kymograms, spectral analysis, and spatio-temporal plots. The complex spatio-temporal vibration patterns are decomposed by empirical orthogonal functions into independent vibratory modes. It is shown quantitatively that biphonation can be induced either by left-right asymmetry or by desynchronized anterior-posterior vibratory modes, and the term "AP (anterior-posterior) biphonation" is introduced. The presented phonation examples show that for normal phonation the first two modes sufficiently explain the glottal dynamics. The spatio-temporal oscillation pattern associated with biphonation due to left-right asymmetry can be explained by the first three modes. Higher-order modes are required to describe the pattern for biphonation induced by anterior-posterior vibrations. Spatial irregularity is quantified by an entropy measure, which is significantly higher for irregular phonation than for normal phonation. Two asymmetry measures are introduced: the left-right asymmetry and the anterior-posterior asymmetry, as the ratios of the fundamental frequencies of left and right vocal fold and of anterior-posterior modes, respectively. These quantities clearly differentiate between left-right biphonation and anterior-posterior biphonation. This paper proposes methods to analyze quantitatively irregular vocal fold contour patterns in vivo and complements previous findings of desynchronization of vibration modes in computer modes and in in vitro experiments.     

The effects of reaction delay in the Nagel-Schreckenberg traffic flow model

This paper studies a new model, which considers the effects of drivers reaction delay in the Nagel-Schreckenberg model. We studied the traffic flow properties in the new model under both periodic and open boundary conditions. The fundamental diagram, spatio-temporal patterns, density-density correlation functions, relaxation time, and distance headway distribution are investigated. Several interesting results are reported, for example, (i) the jam becomes less condensed when the delay effect is considered; (ii) the distance headway of the new model exhibits a multi-peak distribution when randomization p is small; (iii) for large p, the distribution of distance headway follows a power law in the new model; (iv) under open boundary conditions, the existence of a stationary jam near the left boundary will lower the flow rate.     

Two-dimensional wave patterns of spreading depolarization: Retracting, re-entrant, and stationary waves

We present spatio-temporal characteristics of spreading depolarizations (SD) in two experimental systems: retracting SD wave segments observed with intrinsic optical signals in chicken retina, and spontaneously occurring re-entrant SD waves that repeatedly spread across gyrencephalic feline cortex observed by laser speckle flowmetry. A mathematical framework of reaction-diffusion systems with augmented transmission capabilities is developed to explain the emergence and transitions between these patterns. Our prediction is that the observed patterns are reaction-diffusion patterns controlled and modulated by weak nonlocal coupling such as long-range, time-delayed, and global coupling. The described spatio-temporal characteristics of SD are of important clinical relevance under conditions of migraine and stroke. In stroke, the emergence of re-entrant SD waves is believed to worsen outcome. In migraine, retracting SD wave segments cause neurological symptoms and transitions to stationary SD wave patterns may cause persistent symptoms without evidence from noninvasive imaging of infarction.     

Laboratory studies of spectral features of stimulated electromagnetic emission during the ionospheric heating experiments

We present the results of laboratory studies of the formation of a number of spectral components of stimulated electromagnetic emission, which are related to the excitation of small-scale irregularities in the heated ionosphere. In the laboratory experiment, the small-scale irregularity was formed as a result of thermal self-channeling of short-wavelength quasielectrostatic oscillations in a magnetoplasma. Using the method of probing waves, it is experimentally shown that the trapping and waveguide propagation in a small-scale plasma irregularity are exclusively due to Langmuir waves, whereas the upper-hybrid waves with anomalous dispersion are not trapped into the irregularity. It is found that satellites shifted by about 1–2 MHz from the carrier frequency (700 MHz under the experimental conditions) are formed in the Langmuir wave spectrum during the thermal self-channeling. Two mechanisms of generation of spectral satellites have been detected. The first (dynamic) mechanism is observed during the formation of a small-scale irregularity with rapidly increasing longitudinal size. In this case, one low-frequency satellite is excited in the trapped-wave spectrum. The mechanism of the formation of this satellite is apparently related to the Doppler shift of the frequency of the Langmuir waves trapped inside the irregularity. The second (stationary) mechanism is observed in the case of a developed irregularity where its shape is close to cylindrical. In this regime, the trapped-wave spectrum has two symmetric spectral satellites, namely, high- and low-frequency ones. It may be hypothesized that the generation of these satellites is due to scattering of trapped Langmuir waves from drift oscillations of the irregularity.     

Sequences in neural networks with temporal association

We solve the dynamics of the strongly diluted version of a model recently proposed by Herz et al. to store sequences of patterns with spatio-temporal retrieval properties. We analyze the spurious sequence solutions and we find the region in the (,T) plane where the only relevant attractors are the learnt cycles.     

Spiking in an activator-inhibitor model for elements with S-shaped negative differential conductivity

Self-organized spatio-temporal dynamics of electrical transport is described by a simple noncubic activator-inhibitor system derived for layered semiconductor structures. The model exhibits a novel mode of self-sustained oscillations due to current filaments switching on and off (spiking) which may be periodic or chaotic. Additionally, we obtain complex multifilamentary spatio-temporal patterns. As such phenomena have been observed in various devices exhibiting S-shaped negative differential conductivity, our model is suggested to describe a generic mechanism.     

Pattern formation in the ferrocyanide-iodate-sulfite reaction: the control of space scale separation

We revisit the conditions for the development of reaction-diffusion patterns in the ferrocyanide-iodate-sulfite bistable and oscillatory reaction. This hydrogen ion autoactivated reaction is the only example known to produce sustained stationary lamellar patterns and a wealth of other spatio-temporal phenomena including self-replication and localized oscillatory domain of spots, due to repulsive front interactions and to a parity-breaking front bifurcation (nonequilibrium Ising-Bloch bifurcation). We show experimentally that the space scale separation necessary for the observation of stationary patterns is mediated by the presence of low mobility weak acid functional groups. The presence of such groups was overlooked in the original observations made with hydrolyzable polyacrylamide gels. This missing information made the original observations difficult to reproduce and frustrated further experimental exploitation of the fantastic potentialities of this system. Using one-side-fed spatial reactors filled with agarose gel, we can reproduce all the previous pattern observations, in particular the stationary labyrinthine patterns, by introducing, above a critical concentration, well controlled amounts of polyacrylate chains in the gel network. We use two different geometries of spatial reactors (annular and disk shapes) to provide complementary information on the actual three-dimensional character of spatial patterns. We also reinvestigate the role of other feed parameters and show that the system exhibits both a domain of spatial bistability and of large-amplitude pH oscillations associated in a typical cross-shape diagram. The experimental method presented here can be adapted to produce patterns in the large number of oscillatory and bistable reactions, since the iodate-sulfite-ferrocynide reaction is a prototype of these systems.     

Dynamic instabilities in scalar neural field equations with space-dependent delays

In this paper we consider a class of scalar integral equations with a form of space-dependent delay. These nonlocal models arise naturally when modelling neural tissue with active axons and passive dendrites. Such systems are known to support a dynamic (oscillatory) Turing instability of the homogeneous steady state. In this paper we develop a weakly nonlinear analysis of the travelling and standing waves that form beyond the point of instability. The appropriate amplitude equations are found to be the coupled mean-field Ginzburg-Landau equations describing a Turing-Hopf bifurcation with modulation group velocity of O(1). Importantly we are able to obtain the coefficients of terms in the amplitude equations in terms of integral transforms of the spatio-temporal kernels defining the neural field equation of interest. Indeed our results cover not only models with axonal or dendritic delays but also those which are described by a more general distribution of delayed spatio-temporal interactions. We illustrate the predictive power of this form of analysis with comparison against direct numerical simulations, paying particular attention to the competition between standing and travelling waves and the onset of Benjamin-Feir instabilities.     

Projecting low and extensive dimensional chaos from spatio-temporal dynamics

We review the spatio-temporal dynamical features of the Ananthakrishna model for the Portevin-Le Chatelier effect, a kind of plastic instability observed under constant strain rate deformation conditions. We then establish a qualitative correspondence between the spatio-temporal structures that evolve continuously in the instability domain and the nature of the irregularity of the scalar stress signal. Rest of the study is on quantifying the dynamical information contained in the stress signals about the spatio-temporal dynamics of the model. We show that at low applied strain rates, there is a one-to-one correspondence with the randomly nucleated isolated bursts of mobile dislocation density and the stress drops. We then show that the model equations are spatio-temporally chaotic by demonstrating the number of positive Lyapunov exponents and Lyapunov dimension scale with the system size at low and high strain rates. Using a modified algorithm for calculating correlation dimension density, we show that the stress-strain signals at low applied strain rates corresponding to spatially uncorrelated dislocation bands exhibit features of low dimensional chaos. This is made quantitative by demonstrating that the model equations can be approximately reduced to space independent model equations for the average dislocation densities, which is known to be low-dimensionally chaotic. However, the scaling regime for the correlation dimension shrinks with increasing applied strain rate due to increasing propensity for propagation of the dislocation bands. The stress signals in the partially propagating to fully propagating bands turn to have features of extensive chaos.     

Experimental evidence for a zigzag bifurcation in bulk lamellar eutectic growth

We present real-time observations of the directional-solidification patterns of a transparent nonfaceted eutectic alloy (CBr4-C2Cl6) in bulk samples. The growth front of the two-phase solid is observed from the top through the liquid and the glass wall of the container with a long-distance microscope. We show that, in near-eutectic CBr4-C2Cl6 alloys, the upper stability limit of the stationary lamellar patterns is due to a zigzag bifurcation, which occurs at an interlamellar spacing of about 0.85 lambda(m), where lambda(m) is the minimum-undercooling spacing. The zigzag patterns undergo a lamella breakup instability leading to the creation of new lamellae at about 1.1 lambda(m). On the other hand, the lower stability limit of the stationary patterns is due to the same instability as in thin samples, namely, a lamella termination instability that occurs at about 0.7 lambda(m).     

Field patterns in periodically modulated optical parametric amplifiers and oscillators

Spatially localized and periodic field patterns in periodically modulated optical parametric amplifiers and oscillators are studied. In the degenerate case (equal signal and idler beams) we elaborate on the systematic method of construction of the stationary localized modes in the amplifiers, and study their properties and stability. We describe a method of constructing periodic solutions in optical parametric oscillators, by adjusting the form of the external driven field to the given form of either signal or pump beams.     

Order parameter equation and model equation for high Prandtl number. Rayleigh-Bénard convection in a rotating large aspect ratio system

We derive the order parameter equation which describes the evolution of spatio-temporal patterns close to the Bénard instability in a rotating large aspect ratio system for high Prandtl number fluids. Since this order parameter equation contains rather complicated nonlinear terms we present a model equation which can be obtained from the order parameter equation by suitable simplification of the nonlinearity. For this model equation we calculate the family of roll solutions and investigate their stability with respect to long scale instabilities and examine the onset of the Küppers-Lortz instability. Then we present spatiotemporal patterns which are obtained from a numerical evaluation of the model equation.     

Disordered heteropolymers: models for biomimetic polymers and polymers with frustrating quenched disorder

The ability to design and synthesize polymers that can perform functions with great specificity would impact advanced technologies in important ways. Biological macromolecules can self-assemble into motifs that allow them to perform very specific functions. Thus, in recent years, attention has been directed toward elucidating strategies that would allow synthetic polymers to perform biomimetic functions. In this article, we review recent research efforts exploring the possibility that heteropolymers with disordered sequence distributions (disordered heteropolymers) can mimic the ability of biological macromolecules to recognize patterns. Results of this body of work suggests that frustration due to competing interactions and quenched disorder may be the essential physics that can enable such biomimetic behavior. These results also show that recognition between disordered heteropolymers and multifunctional surfaces due to statistical pattern matching may be a good model to study kinetics in frustrated systems with quenched disorder. We also review work which demonstrates that disordered heteropolymers with branched architectures are good model systems to study the effects of quenched sequence disorder on microphase ordering of molten copolymers. The results we describe show that frustrating quenched disorder affects the way in which these materials form ordered nanostructures in ways which might be profitably exploited in applications. Although the focus of this review is on theoretical and computational research, we discuss connections with existing experimental work and suggest future experiments that are expected to yield further insights.     

Statistical Mechanics of Nucleosomes Constrained by Higher-Order Chromatin Structure

Eukaryotic DNA is packaged into chromatin: one-dimensional arrays of nucleosomes separated by stretches of linker DNA are folded into 30-nm chromatin fibers which in turn form higher-order structures (Felsenfeld and Groudine in Nature 421:448, 2003). Each nucleosome, the fundamental unit of chromatin, has 147 base pairs (bp) of DNA wrapped around a histone octamer (Richmond and Davey in Nature 423:145, 2003). In order to describe how chromatin fiber formation affects nucleosome positioning and energetics, we have developed a thermodynamic model of finite-size particles with effective nearest-neighbor interactions and arbitrary DNA-binding energies. We show that both one- and two-body interactions can be extracted from one-particle density profiles based on high-throughput maps of in vitro or in vivo nucleosome positions. Although a simpler approach that neglects two-body interactions (even if they are in fact present in the system) can be used to predict sequence determinants of nucleosome positions, the full theory is required to disentangle one- and two-body effects. Finally, we construct a minimal model in which nucleosomes are positioned primarily by steric exclusion and two-body interactions rather than intrinsic histone-DNA sequence preferences. The model reproduces nucleosome occupancy patterns observed over transcribed regions in living cells.     

Chaotic itinerancy, temporal segmentation and spatio-temporal combinatorial codes

We study a deterministic dynamics with two time scales in a continuous state attractor network. To the usual (fast) relaxation dynamics towards point attractors (“patterns”) we add a slow coupling dynamics that makes the visited patterns lose stability, leading to an itinerant behavior in the form of punctuated equilibria. One finds that the transition frequency matrix for transitions between patterns shows non-trivial statistical properties in the chaotic itinerant regime. We show that mixture input patterns can be temporally segmented by the itinerant dynamics. The viability of a combinatorial spatio-temporal neural code is also demonstrated.