Metastable Patterns Research Articles

A circuit basis for morphogenesis

Morphogenesis is the generation of structural patterns through a dynamic process. The mathematical basis of morphogenesis has long been studied with the key initial work by Alan Turing. This paper explores the consequences of a circuit basis for reaction–diffusion systems on morphogenesis, including the reachability of patterns and the logical basis of pattern stability. We consider how morphological patterns can arise through iterative computation and produce robust forms. Through an exhaustive analysis of reaction–diffusion dynamics in a minimal model of morphogenesis, we show how the stability and reachability of morphologies are influenced by their circuit basis. We show that this model exhibits similar behavior to the recently experimentally observed dynamics not accounted for by Turing's original model. We conclude by presenting an additional class of metastable patterns exhibited in this model.

The frustrated brain: from dynamics on motifs to communities and networks.

Cognitive function depends on an adaptive balance between flexible dynamics and integrative processes in distributed cortical networks. Patterns of zero-lag synchrony likely underpin numerous perceptual and cognitive functions. Synchronization fulfils integration by reducing entropy, while adaptive function mandates that a broad variety of stable states be readily accessible. Here, we elucidate two complementary influences on patterns of zero-lag synchrony that derive from basic properties of brain networks. First, mutually coupled pairs of neuronal subsystems-resonance pairs-promote stable zero-lag synchrony among the small motifs in which they are embedded, and whose effects can propagate along connected chains. Second, frustrated closed-loop motifs disrupt synchronous dynamics, enabling metastable configurations of zero-lag synchrony to coexist. We document these two complementary influences in small motifs and illustrate how these effects underpin stable versus metastable phase-synchronization patterns in prototypical modular networks and in large-scale cortical networks of the macaque (CoCoMac). We find that the variability of synchronization patterns depends on the inter-node time delay, increases with the network size and is maximized for intermediate coupling strengths. We hypothesize that the dialectic influences of resonance versus frustration may form a dynamic substrate for flexible neuronal integration, an essential platform across diverse cognitive processes.

Metastable dynamical patterns and their stabilization in arrays of bidirectionally coupled sigmoidal neurons

Transient patterns in a bistable ring of bidirectionally coupled sigmoidal neurons were studied. When the system had a pair of spatially uniform steady solutions, the instability of unstable spatially nonuniform steady solutions decreased exponentially with the number of neurons because of the symmetry of the system. As a result, transient spatially nonuniform patterns showed dynamical metastability: Their duration increased exponentially with the number of neurons and the duration of randomly generated patterns obeyed a power-law distribution. However, these metastable dynamical patterns were easily stabilized in the presence of small variations in coupling strength. Metastable rotating waves and their pinning in the presence of asymmetry in the direction of coupling and the disappearance of metastable dynamical patterns due to asymmetry in the output function of a neuron were also examined. Further, in a two-dimensional array of neurons with nearest-neighbor coupling, intrinsically one-dimensional patterns were dominant in transients, and self-excitation in these neurons affected the metastable dynamical patterns.

Metastable periodic patterns in singularly perturbed state-dependent delayed equations

We consider the scalar delayed differential equation ϵẋ(t)=−x(t)+f(x(t−r)), where ϵ>0, r=r(x,ϵ)=1+η(ϵ)R(x), and f represents either a monotone positive feedback df/dx>0 or a monotone negative feedback df/dx<0, plus some other hypotheses. When the delay is a constant, i.e. r(x,ϵ)=1, this equation can support metastable rapidly oscillating solutions that are transients whose duration is of order exp(c/ϵ), for some c>0. In this paper, combining analytic and numerical techniques, we investigate whether this metastable behavior persists when the delay r(x,ϵ) depends non trivially on the state variable x. More precisely, we proceed in three steps. First we explore theoretically and numerically the existence of branches of rapid periodic solutions for these equations. Then, to predict conditions for metastability, we introduce transition layer equations that depict the asymptotic shape of transient oscillations when ϵ tends to zero and η(ϵ)∼ϵ. Finally, these predictions are validated by numerical simulations. Numerical explorations are also performed for other scalings of η(ϵ) with ϵ. Our conclusions are: (i) when η(0)≠0, there are no metastable transients; (ii) when η(0)=0 and η′(0) is finite, for monotone negative feedback, the metastable oscillations exist irrespective of choices of f and R; (iii) for monotone positive feedback, they require that the feedback f and the delay R satisfy extra conditions, such as f being an odd function and the delay R(x) an even function. One novel result is that state-dependent delays may lead to metastable dynamics in equations that cannot support such regimes when the delay is constant.

Discovery of New Metastable Patterns in Diblock Copolymers

AbstractThe ordered patterns formed by microphase-separated block copolymer systems demonstrate periodic symmetry, and all periodic structures belong to one of 230 space groups. Based on this fact, a strategy of estimating the initial values of self-consistent field theory to discover ordered patterns of block copolymers is developed. In particular, the initial period of the computational box is estimated by the Landau-Brazovskii model as well. By planting the strategy into the whole-space discrete method, several new metastable patterns are discovered in diblock copolymers.

Flux pattern transitions in the intermediate state of a type-I superconductor driven by an ac field

The intermediate state of a type-I superconductor Pb film is studied by a scanning Hall probe and scanning ac-susceptibility microscopies under static and oscillating applied magnetic fields. The structure of the typical flux patterns during magnetic field penetration/expulsion shows a strong hysteresis. Under the action of an ac field, the multiply quantized flux tubes in a type-I superconductor reveal a dynamical reordering similar to what is observed in the Campbell regime for vortices in a type-II superconductor. Most strikingly, after shaking, higher density flux tube patterns demonstrate a reorganization from a superheated metastable tubular pattern to a stable stripe pattern. We provide direct experimental evidence that the flux reorganization behavior is a dynamical transition. The local distribution of the potentials providing pinning to intermediate state patterns is mapped out, which is, as far as we know, the first direct visualization of confinement of intermediate state domains by pinning centers.

Compositional sorting dynamics in coexisting lipid bilayer phases with variations in underlying e-beam formed curvature pattern.

Nanometer-scale curvature patterns of an underlying substrate are imposed on lipid multibilayers with each pattern imparting distinctly different sorting dynamics to a metastable pixelation pattern of coexisting liquid ordered (Lo)-liquid disordered (Ld) lipid phases. Therefore, this work provides pathways toward mechanical energy-based separations for analysis of biomembrane-associate species. The central design concept of the patterned sections of the silica substrate is a square lattice pattern of 100 nm projected radius poly(methyl methacrylate) (PMMA) hemispherical features formed by electron beam lithography which pixelates the coexisting phases in order to balance membrane bending and line energy. In one variation, we surround this pattern with three PMMA walls/fences 100 nm in height which substantially slows the loss of the high line energy pixelated Lo phase by altering the balance of two competing mechanism (Ostwald ripening vs. vesiculation). In another walled variation, we form a gradient of the spacing of the 100 nm features which forces partitioning of the Lo phase toward the end of the gradient with the most open (400 nm spacing) lattice pattern where a single vesicle could grow from the Lo phase. We show that two other variations distinctly impact the dynamics, demonstrating locally slowed loss of the high line energy pixelated Lo phase and spontaneous switching of the pixel location on the unit cell, respectively. Moreover, we show that the pixelation patterns can be regenerated and sharpened by a heating and cooling cycle. We argue that localized variations in the underlying curvature pattern have rather complex consequences because of the coupling and/or competition of dynamic processes to optimize mechanical energy such as lipid diffusion, vesiculation and growth, and phase/compositional partitioning.

Sensorimotor coordination and metastability in a situated HKB model

Oscillatory phenomena are ubiquitous in nature and have become particularly relevant for the study of brain and behaviour. One of the simplest, yet explanatorily powerful, models of oscillatory Coordination Dynamics is the Haken–Kelso–Bunz (HKB) model. The metastable regime described by the HKB equation has been hypothesised to be the signature of brain oscillatory dynamics underlying sensorimotor coordination. Despite evidence supporting such a hypothesis, to our knowledge, there are still very few models (if any) where the HKB equation generates spatially situated behaviour and, at the same time, has its dynamics modulated by the behaviour it generates (by means of the sensory feedback resulting from body movement). This work presents a computational model where the HKB equation controls an agent performing a simple gradient climbing task and shows (i) how different metastable dynamical patterns in the HKB equation are generated and sustained by the continuous interaction between the agent and its environment; and (ii) how the emergence of functional metastable patterns in the HKB equation – i.e. patterns that generate gradient climbing behaviour – depends not only on the structure of the agent's sensory input but also on the coordinated coupling of the agent's motor–sensory dynamics. This work contributes to Kelso's theoretical framework and also to the understanding of neural oscillations and sensorimotor coordination.

Universality classes and crossover behaviors in non-Abelian directed sandpiles

We study universality classes and crossover behaviors in non-Abelian directed sandpile models in terms of the metastable pattern analysis. The non-Abelian property induces spatially correlated metastable patterns, characterized by the algebraic decay of the grain density along the propagation direction of an avalanche. Crossover scaling behaviors are observed in the grain density due to the interplay between the toppling randomness and the parity of the threshold value. In the presence of such crossovers, we show that the broadness of the grain distribution plays a crucial role in resolving the ambiguity of the universality class. Finally, we claim that the metastable pattern analysis is important as much as the conventional analysis of avalanche dynamics.

Variable histone modifications at the Avy metastable epiallele

The ability of environmental factors to shape health and disease involves epigenetic mechanisms that mediate gene-environment interactions. Metastable epiallele genes are variably expressed in genetically identical individuals due to epigenetic modifications established during early development. DNA methylation within metastable epialleles is stochastic due to probabilistic reprogramming of epigenetic marks during embryogenesis. Maternal nutrition and environment have been shown to affect metastable epiallele methylation patterns and subsequent adult phenotype. Little is known, however, about the role of histone modifications in influencing metastable epiallele expression and phenotypic variation. Utilizing chromatin immunoprecipitation followed by qPCR, we observe variable histone patterns in the 5’ long terminal repeat (LTR) of the murine viable yellow agouti (Avy) metastable epiallele. This region contains 6 CpG sites, which are variably methylated in isogenic Avy/a offspring. Yellow mice, which are hypomethylated at the Avy LTR and exhibit constitutive ectopic expression of agouti (a), also display enrichment of H3 and H4 di-acetylation (p=0.08 and 0.09, respectively). Pseudoagouti mice, in which Avy hypermethylation is thought to silence ectopic expression, exhibit enrichment of H4K20 tri-methylation (p=0.01). No differences are observed for H3K4 tri-methylation (p=0.7), a modification often enriched in the promoter of active genes. These results show for the first time the presence of variable histone modifications at a metastable epiallele, indicating that DNA methylation acts in concert with histone modifications to affect inter-individual variation of metastable epiallele expression. Therefore, the potential for environmental factors to influence histone modifications, in addition to DNA methylation, should be addressed in environmental epigenomic studies.

Transitions and heteroclinic cycles in the general Gierer–Meinhardt equation and cardiovascular calcification model

The article is concerned with transitions and pattern selection analysis of the inhibitor–activator system proposed in connection with recent studies of cardiovascular calcification patterns. Explicit criteria are derived to enable us to distinguish between stable and metastable patterns. By deriving a reduced system of equations, the existence of certain complicated structures is discussed; in particular, heteroclinic cycles are identified and their properties are studied. It is also discussed that the change of boundary conditions can affect the transitions of the system. In this connection, we will also study asymptotic behavior of patterns after transitions and will compare the results with numerical simulations.

Metastable Periodic Patterns in Singularly Perturbed Delayed Equations

We consider the scalar delayed differential equation \({\epsilon\dot x(t)=-x(t)\,+f(x(t-1))}\), where \({\epsilon\,{>}\,0}\) and f verifies either df/dx > 0 or df/dx 0. We call it a metastable solution. During this transient a finite time span of the solution looks like that of a periodic function. It is remarkable that if df/dx > 0 then f must be odd or present some other very special symmetry in order to support metastable solutions, while this condition is absent in the case df/dx < 0. Explicit \({\epsilon}\)-asymptotics for the motion of zeroes of a solution and for the transient time regime are presented.

Stability of fronts and transient behaviour in KPP systems

We consider a system of two reaction diffusion equations with the Kolmogorov–Petrovsky–Piskunov (KPP) type nonlinearity which describes propagation of pressure-driven flames. It is known that the system admits a family of travelling wave solutions parameterized by their velocity. In this paper, we show that these travelling fronts are stable under the assumption that perturbations belong to an appropriate weighted L 2 space. We also discuss an interesting meta-stable pattern the system exhibits in certain cases.

Transient Moiré rotation patterns in thin colloidal crystals

A high resolution optical microscopy study on low salt charged sphere suspensions settling under gravity reveals the formation of AB-stacked triangular layers via different transient structures. In particular, metastable triangular Moire rotation patterns (formally corresponding to 1 × 1 Rα super structures) with rotation angles α = 27.8° and α = 38.2° are identified.

Short term memory may be the depletion of the readily releasable pool of presynaptic neurotransmitter vesicles of a metastable long term memory trace pattern

The Tagging/Retagging model of short term memory was introduced earlier (Tarnow in Cogn Neurodyn 2(4):347-353, 2008) to explain the linear relationship between response time and correct response probability for word recall and recognition: At the initial stimulus presentation the words displayed tag the corresponding long term memory locations. The tagging process is linear in time and takes about one second to reach a tagging level of 100%. After stimulus presentation the tagging level decays logarithmically with time to 50% after 14 s and to 20% after 220 s. If a probe word is reintroduced the tagging level has to return to 100% for the word to be properly identified, which leads to a delay in response time. This delay is proportional to the tagging loss. The tagging level is directly related to the probability of correct word recall and recognition. Evidence presented suggests that the tagging level is the level of depletion of the Readily Releasable Pool (RRP) of neurotransmitter vesicles at presynaptic terminals. The evidence includes the initial linear relationship between tagging level and time as well as the subsequent logarithmic decay of the tagging level. The activation of a short term memory may thus be the depletion of RRP (exocytosis) and short term memory decay may be the ensuing recycling of the neurotransmitter vesicles (endocytosis). The pattern of depleted presynaptic terminals corresponds to the long term memory trace.

An unusual weak bonding mode of fluorine to single-walled carbon nanotubes

Single-walled carbon nanotubes (SWCNTs) show a new facet of their extremely diversified reactivity through the possibility of becoming spontaneously efficiently functionalized in the presence of gaseous fluorine at room temperature, in sharp contrast with any other carbon allotrope. The synthesis route employed here optimizes the formation of an unusual weak bonding mode of fluorine to such a carbon frame. This particular C/F interaction could then be studied independently using solid state 19F NMR spectroscopy and the abnormally high chemical shift measured shows a reduced covalence of the bond between carbon and fluorine. In compliance with its weakness, the C–F bond is elongated to about 0.140 nm. It is further suggested that a metastable endo-addition pattern may form in such experimental conditions.

Dynamic formation of metastable intermediate state patterns in type-I superconductors

Structure of the intermediate state in type-I superconducting lead (Pb) is shown to be very sensitive to the ramp rate of an applied magnetic field. The configurations of resulting static patterns depend sensitively on the shape of the specimen. In particular, geometric barrier, present in the samples with rectangular cross-section, plays an important role in determining the sharp boundary between the phases of different topology. We propose that seemingly laminar (stripe) pattern obtained as a result of the fast field ramp is simply an imprint left behind by the fast-moving flux tubes. Our results confirm that flux tube phase is topologically favorable.

Relevance of Abelian Symmetry and Stochasticity in Directed Sandpiles

We provide a comprehensive view of the role of Abelian symmetry and stochasticity in the universality class of directed sandpile models, in the context of the underlying spatial correlations of metastable patterns and scars. It is argued that the relevance of Abelian symmetry may depend on whether the dynamic rule is stochastic or deterministic, by means of the interaction of metastable patterns and avalanche flow. Based on the new scaling relations, we conjecture critical exponents for an avalanche, which is confirmed reasonably well in large-scale numerical simulations.

The Pervasiveness of 1/f Scaling in Speech Reflects the Metastable Basis of Cognition

Human neural and behavioral activities have been reported to exhibit fractal dynamics known as 1/f noise, which is more aptly named 1/f scaling. Some argue that 1/f scaling is a general and pervasive property of the dynamical substrate from which cognitive functions are formed. Others argue that it is an idiosyncratic property of domain-specific processes. An experiment was conducted to investigate whether 1/f scaling pervades the intrinsic fluctuations of a spoken word. Ten participants each repeated the word bucket over 1,000 times, and fluctuations in acoustic measurements across repetitions generally followed the 1/f scaling relation, including numerous parallel yet distinct series of 1/f fluctuations. On the basis of work showing that 1/f scaling is a universal earmark of metastability, it is proposed that the observed pervasiveness of 1/f fluctuations in speech reflects the fact that cognitive functions are formed as metastable patterns of activity in brain, body, and environment.

Proteins in a shear flow

The conformational dynamics of a single protein molecule in a shear flow is investigated using Brownian dynamics simulations. A structure-based coarse grained model of a protein is used. We consider two proteins, ubiquitin and integrin, and find that at moderate shear rates they unfold through a sequence of metastable states-a pattern which is distinct from a smooth unraveling found in homopolymers. Full unfolding occurs only at very large shear rates. Furthermore, the hydrodynamic interactions between the amino acids are shown to hinder the shear flow unfolding. The characteristics of the unfolding process depend on whether a protein is anchored or not, and if it is, on the choice of an anchoring point.