Stationary Spatial Patterns Research Articles

Fourier analysis of Turing-like pattern formation in cellular automaton models

The novelty of this paper is the study of emergence of diffusion-induced (Turing-like) patterns from a microscopic point of view, namely, in terms of cellular automata. Formally, the cellular automaton model is described in lattice-gas terminology [H. Bussemaker, A. Deutsch, E. Geigant, Phys. Rev. Lett. 78 (1997) 5018–5021]. The automaton rules capture in abstract form the essential ideas of activator–inhibitor interactions of biological systems. In spite of the automaton’s simplicity, self-organised formation of stationary spatial patterns emerging from a randomly perturbed uniform state is observed. Fourier analysis of approximate mean-field kinetic difference equations [J.P. Boon, D. Dab, R. Kapral, A.T. Lawniczak, Phys. Rep. 273 (1996) 55–147] yields a critical wave length and a “Turing condition” for the onset of pattern formation.

Spatial instabilities in reaction random walks with direction-independent kinetics.

We study spatial instabilities in reacting and diffusing systems, where diffusion is modeled by a persistent random walk instead of the usual Brownian motion. Perturbations in these reaction walk systems propagate with finite speed, whereas in reaction-diffusion systems localized disturbances affect every part instantly, albeit with heavy damping. We present evolution equations for reaction random walks whose kinetics do not depend on the particles' direction of motion. The homogeneous steady state of such systems can undergo two types of transport-driven instabilities. One type of bifurcation gives rise to stationary spatial patterns and corresponds to the Turing instability in reaction-diffusion systems. The other type occurs in the ballistic regime and leads to oscillatory spatial patterns; it has no analog in reaction-diffusion systems. The conditions for these bifurcations are derived and applied to two model systems. We also analyze the stability properties of one-variable systems and find that small wavelength perturbations decay in an oscillatory manner.

Time scales of stratified turbulent flows and relations between second-order closure parameters and flow numbers

The description of turbulent mixing and chemical reactions by Lagrangian probability density function methods offers some significant advantages over other methods, mainly due to the simulation of mixing processes and the exact treatment of chemical transformations. A key problem of such methods is the information on the time scales of processes, because they determine the dynamics and intensity of mixing. This question is considered for stratified flow. Different models are presented for the development of these time scales in time and their stationary spatial patterns in dependence on shear and stratification. The model predictions are shown to be in agreement with large-eddy simulations of stratified homogeneous shear flow. Two further applications of these models are considered: the description of transitions between flow regimes (characterized by different scaling quantities) in the stationary atmospheric surface layer and, second, the simulation of buoyant plume rise. It is shown that the predictions of the stationary frequency model agree with measured data. The consideration of limit cases of this model leads to connections between second-order closure parameters and (critical) flow numbers that characterize these transitions. These relationships are shown to be very advantageous for the application of closure models. A new flow number that characterizes the transition to free convective flow under unstable stratification is introduced here in analogy to the critical gradient Richardson number, which characterizes the onset of turbulence in stably stratified flow. The second application provides a new theory for buoyant plume rise. Two parameters that describe the turbulent mixing in the entrainment and extrainment stages of plume rise are explained as ratios of the relevant time scales. The two-thirds power law of buoyant plume rise, which is observed for nonturbulent and neutrally stratified flow, is obtained without having to make ad hoc assumptions. For turbulent flow, the plume’s leveling-off is calculated in accord with measurements.

Predicting dynamics of spatial automata models using Hamiltonian equations

Grid-based models such as cellular automata are useful tools for investigating spatial dynamics in ecological systems. Quantitative measures have been proposed for characterizing static spatial patterns, but techniques for examining the dynamics of spatial pattern are less developed. I present an adaptation of Hamiltonian free energy equations to predict numbers of state changes resulting from different ecological mechanisms in spatial automata models. The three terms in the Hamiltonian estimate the number of grid cells changing state due to spontaneous state changes, to interactions among cells, and to heterogeneity in the behavior of different cells in the grid. PComp is a simple competition model in which each cell in the grid can be occupied by a single individual plant. The simulation calculates the Hamiltonian at each time step to indicate how many of the observed changes in cell occupancy are due to basic colonization and death rates, crowding effects, dispersal from adjacent cell, and differences in substrate suitability.

Nonlinear Chemical Dynamics: Oscillations, Patterns, and Chaos

Chemical reactions with nonlinear kinetic behavior can give rise to a remarkable set of spatiotemporal phenomena. These include periodic and chaotic changes in concentration, traveling waves of chemical reactivity, and stationary spatial (Turing) patterns. Although chemists were initially skeptical of the existence and the relevance of these phenomena, much progress has been made in the past two decades in characterizing, designing, modeling, and understanding them. Several nonlinear dynamical phenomena in chemical systems provide simpler analogues of behaviors found in biological systems.

Spatio-temporal self-organization of bone mineral metabolism and trabecular structure of primary bone.

A nonlinear two-variable reaction-diffusion model of bone mineral metabolism, built from an overall self-oscillatory compartmental model of calcium metabolism in vivo, has been studied for its ability to generate spatial and spatio-temporal self-organizations in a two-dimensional space. Analytical and numerical results confirm the theoretical properties previously described for this kind of model. In particular, it is shown that, for a given set of reactional parameter values and certain values of the ratio of the two diffusion coefficients, there exists a set of unstable wavenumbers leading spontaneously to the development, from the homogeneous steady state, of either different types of stationary spatial patterns (hexagonal, striped and re-entrant hexagonal patterns) or more or less complex spatio-temporal expressions. We discuss the relevance of analogies established between some spatial or spatio-temporal structures predicted by the model and some peculiar features of the primary bone trabecular architecture which appear during embryonic ossification.

Effects of local connectivity on striatal function: stimulation and analysis of a model.

Neuronal population activity was investigated by computer simulation of a network model based on the neostriatum. Three network topologies were studied, based on different assumptions about the synaptic connectivity among medium spiny neurons. In all networks neurons were interconnected by inhibitory synapses. The connectivity was either symmetric, in which case all connections between cells were reciprocal and equal in strength; or asymmetric. Simulations showed that networks with symmetric connectivity receiving randomly distributed afferent excitation produced stationary spatial activity patterns. In contrast, asymmetric connectivity in homogeneous networks produced slow travelling-wave activity across the network. We suggest that the shape of the medium spiny neurons is an important determinant of synaptic connectivity and that changes in the shape of these neurons caused by Huntington's disease would result in asymmetric connectivity. Slow travelling-wave activity produced by asymmetric connectivity in the neostriatum could explain some aspects of the choreic movement and some electromyographic features seen in Huntington's patients.

On the mechanical theory for biological pattern formation

We investigate the pattern-forming potential of mechanical models in embryology proposed by Oster, Murray and their coworkers. We show that the presence of source terms in the tissue extracellular matrix and cell density equations give rise to spatio-temporal oscillations. An extension of one such model to include ‘biologically realistic long range effects induces the formation of stationary spatial patterns. Previous attempts to solve the full system were in one dimension only. We obtain solutions in one dimension and extend our simulations to two dimensions. We show that a single mechanical model alone is capable of generating complex but regular spatial patterns rather than the requirement of model interaction as suggested by Nagorcka et al. and Shaw and Murray. We discuss some biological applications of the models among which are would healing and formation of dermatoglyphic (fingerprint) patterns.

A perturbation analysis of a mechanical model for stable spatial patterning in embryology

We investigate a mechanical cell-traction mechanism that generates stationary spatial patterns. A linear analysis highlights the model's potential for these heterogeneous solutions. We use multiple-scale perturbation techniques to study the evolution of these solutions and compare our solutions with numerical simulations of the model system. We discuss some potential biological applications among which are the formation of ridge patterns, dermatoglyphs, and wound healing.

From stripes to spots: prepatterns which can be produced in the skin by a reaction-diffusion system.

A key question in the area of spatial pattern formation in developmental biology is: how do groups of cells in a homogeneous tissue suddenly differentiate along entirely different developmental paths compared to neighbouring cells? Although experiments are now beginning to provide answers to this question, the mechanisms responsible for the development of repeated or periodic structures and spatial patterns, e.g., hair follicles and pigmentation patterns, are still unknown. Theoretical biologists and applied mathematicians have suggested various prepattern mechanisms as the primary cause of repeated or periodic spatial patterns. A class of biochemical reactions referred to here as reaction-diffusion (RD) systems, having the capacity to spontaneously generate stable stationary wavelike spatial patterns (Turing, 1952), has been suggested as a possible prepattern mechanisms, e.g., during hair follicle initiation and development (Nagorcka, 1989), and pigmentation patterns (Murray, 1989). Spatial patterns arising during development of the vertebrate skin are frequently complex. Spatial patterns in the skin can be seen to vary within an individual from one region of the skin to another. One pattern change commonly observed across the skin is from stripes to spots. An RD system is defined which is able to generate different spatial patterns depending on the value of a single parameter. The parameter varied controls the transport of the chemical components of the RD system across the basement membrane separating the epidermis and dermis. The patterns produced range from stripes to an irregular array of spots. Not only are different patterns produced, but a different time sequence of prepatterns is expected to arise in the different skin regions depending on whether the first prepattern in an array of spots or stripes. As a consequence it is possible to account for hair follicle initiation in the hair-bearing regions of the mammalian skin as well as the sequence of events required for the formation of dermatoglyphics in the volar regions.

Transition to chemical turbulence.

Experiments have been conducted on Turing-type chemical spatial patterns and their variants in a quasi-two-dimensional open spatial reactor with a chlorite-iodide-malonic acid reaction. A variety of stationary spatial structures-hexagons, stripes, and mixed states-were observed, and transitions to these states were studied. For conditions beyond those corresponding to the emergence of patterns, a transition was observed from stationary spatial patterns to chemical turbulence, which is marked by a continuous motion of the pattern within a domain and of the grain boundaries between domains. The transition to chemical turbulence was analyzed by measuring the correlation length, the average pattern speed, and the total length of the domain boundaries. The emergence of chemical turbulence is accompanied by a large increase in the defects in the pattern, which suggests that this is an example of defect-mediated turbulence.

Transverse laser patterns. I. Phase singularity crystals.

We analyze the interaction and the competition of a set of transverse cavity modes, which belong to a frequency-degenerate family. The laser turns out to be able to realize several different stationary spatial patterns, which differ in the transverse configuration of the intensity or of the field and are met by varying the values of the control parameters. A striking feature that emerges in almost all steady-state patterns is the presence of dark points, in which both the real and the imaginary part of the electric field vanish and such that, if one covers a closed loop around one of these points, the field phase changes by a multiple of 2\ensuremath{\pi}, which corresponds to the topological charge of the point. We show in detail the analogy of these phase singularities to the vortex structures well known in such fields as, for example, hydrodynamics, superconductivity, and superfluidity. In our case, at steady state, these singularities are arranged in the form of regular crystals, nd the equiphase lines of the field exhibit a notable similarity to the field lines of the electrostatic field generated by a corresponding set of point charges. We analyze in detail the patterns that emerge in the cases 2p+l=2 and 2p+l=3, where p and l are the radial and angular modal indices, respectively, and we compare the results with the experimental observations obtained from a ${\mathrm{Na}}_{2}$ laser. The observed patterns agree in detail with those found by theory; in particular, they exhibit the predicted phase singularities in each pattern. The transitions from one pattern to another, that one observes under variation of the control parameters, basically agree with those predicted by theory.

A bifurcation sequence to stationary spatial patterns in a nonuniform chemical model system with equal diffusion coefficients

Numerical studies of the Gray–Scott model with equal diffusion coefficients show that a simple kinetic scheme can produce a transition sequence from a simple spatial state to stable stationary structures in an open spatial reactor with external gradients. The system displays multistability between periodic states, diffusion-induced aperiodic states and different steady patterns. The main upshot of the investigation is that external gradients greatly facilitate the self-organization of chemical structures.

Nonlinear dynamics in the CO-oxidation on Pt single crystal surfaces

The types of dynamical behaviour that can be found in nonlinear systems far from equilibrium are briefly described. Then experimental manifestations of these phenomena in the CO oxidation on Pt are presented which include bistability, oscillations and chaos as well as propagating and stationary spatial patterns. It is demonstrated how a simple model can be refined to account for the observed complex behaviour.

Stationary spatial patterns in passive optical systems: Two-level atoms.

We discuss a novel type of optical instability, which leads to the spontaneous formation of a stationary spatial structure, instead of to the onset of oscillations or pulsations. A passive system is contained in an optical cavity, to which we add two lateral mirrors. Under appropriate conditions, diffraction of radiation can induce the onset of a transverse-stripe structure in an initially uniform plane-wave field configuration. In this paper, we consider the case of a homogeneously broadened two-level system. The model on which our analysis is based is derived from the Maxwell-Bloch equations in the paraxial approximation, using the mean-field limit and assuming a large Fresnel number. We show that, with respect to the bistability parameter C, the threshold of the spatial instability coincides with the bistability threshold for absorptive, resonant optical bistability. The spatial instability requires, however, a detuned configuration and can arise both in the absence and presence of bistability. In the latter case, it can occur both in the lower and in the upper branch of the steady-state curve. The instability is caused by competition between different transverse modes, which realize a situation of spatial coexistence, different from the state of temporal coexistence which characterizes the spontaneous oscillations that arise from the previously known multimode instabilities. Finally, we discuss some points relevant for an experimental observation of this phenomenon.

Surface-induced chemical oscillations and their influence on space- and time-periodic patterns

Within the framework of a reaction-diffusion model we study the influence of active or passive surfaces (e.g., membranes) on extended chemical systems. The surface is described by boundary conditions. Using linear stability analysis we find time-independent boundary conditions which induce oscillatory instabilities of the homogeneous state leading to boundary oscillations. Two cases are of special interest:(i) autocatalysis takes place only in the bulk, whereas coupling occurs only at the surface (or vice versa); (ii) all reactions take place at the surface but bulk diffusion is still essential for the oscillations. Phase diffusion theory and numerical simulation show that for oscillating and excitable (or bistable) bulk systems these boundary conditions can act as a pacemaker leading to phase-locked and quasiperiodic oscillations. The stability of the arising phase waves is discussed. For systems with stationary spatial bulk patterns boundary oscillations can constitute an efficient wavelength selection mechanism.

Large Scale Spatially Organized Activity in Neural Nets

A model for the spatiotemporal activity of neuronal nets is proposed. Stationary periodic spatial patterns are discussed from the point of view of bifurcation theory. Existence of spatial patterns on the whole line is established by the implicit function theorem. Singularity theory is used to study the local structure of the bifurcation equations. A Poincaré–Lindstedt series is developed to establish the form of the periodic stationary states and their stability. The biological relevance of these patterns is briefly discussed.

Self-organization in biological systems with multiple cellular contacts.

The self-organizing properties of an ensemble of interconnected units are studied by linear stability analyses. Small perturbations of a uniform steady-state may result in bifurcations to other solutions that exhibit spatial or temporal order. We show that increasing the number of connections that a unit makes with its neighbors changes the nature of these solutions and tends to destroy spatiotemporal patterns. If an unconnected system is originally stable, the formation of multiple interconnections can never induce temporal periodicity but may, under certain circumstances, allow the emergence of stationary spatial patterns. We have verified the predictions of the linear stability analysis on a model system and comment on the implications of these results for multicellular ensembles.