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Generalized correlated states in a ring of coupled nonlinear oscillators with a local injection.

In this paper, we study the spatiotemporal dynamics of a ring of diffusely coupled nonlinear oscillators. Floquet theory is used to investigate the various dynamical states of the ring, as well as the Hopf bifurcations between them. A local injection scheme is applied to synchronize the ring with an external master oscillator. The shift-invariance symmetry is thereby broken, leading to the emergence of generalized correlated states. The transition boundaries from these correlated states to spatiotemporal chaos and complete synchronization are also derived. Numerical simulations are performed to support the analytic approach.

Journal Article↗

Excitable CO oxidation on Pt(110) under nonuniform coupling.

A new feedback scheme for guided spatiotemporal pattern formation in reaction-diffusion systems is introduced. In contrast to previously established control methods, we present a coupling protocol that is sensitive to the presence of coherent structures in the medium. Applying this feedback to the catalytic CO oxidation on Pt(110) in both experiments and numerical simulations, we show that temporal evolution and spatial extension of self-organizing objects can be efficiently controlled.

Journal Article↗

Dynamical systems and cognitive linguistics: toward an active morphodynamical semantics.

We propose a novel dynamical system approach to cognitive linguistics based on cellular automata and spiking neural networks. How can the same relationship 'in' apply to containers as different as 'box', 'tree' or 'bowl'? Our objective is to categorize the infinite diversity of schematic visual scenes into a small set of grammatical elements and elucidate the topology of language. Gestalt-inspired semantic studies have shown that spatial prepositions such as 'in' or 'above' are neutral toward the shape and size of objects. We suggest that this invariance can be explained by introducing morphodynamical transforms, which erase image details and create virtual structures or singularities (boundaries, skeleton), and call this paradigm 'active semantics'. Singularities arise from a large-scale lattice of coupled excitable units exhibiting spatiotemporal pattern formation, in particular traveling waves. This work addresses the crucial cognitive mechanisms of spatial schematization and categorization at the interface between vision and language and anchors them to expansion processes such as activity diffusion or wave propagation.

Algorithms↗

Synchronization in nonidentical complex Ginzburg-Landau equations.

A cross-correlation coefficient of complex fields has been investigated for diagnosing spatiotemporal synchronization behavior of coupled complex fields. We have also generalized the subsystem synchronization way established in low-dimensional systems to one- and two-dimensional Ginzburg-Landau equations. By applying the indicator to examine the synchronization behavior of coupled Ginzburg-Landau equations, it is shown that our subsystem approach may be of better synchronization performance than the linear feedback method. For the linear feedback Ginzburg-Landau equation, the nonidentical system exhibits generalized synchronization characteristics in both amplitude and phase. However, the nonidentical subsystem may exhibit complete-like synchronization properties. The difference between complex fields for driven and response systems gives a linear scaling with the change of their parameter difference.

Journal Article↗

Oscillatory and chaotic dynamics in compartmentalized geometries.

The effects of spatial compartmentalization of a multistep reaction mechanism (Willamowski-Rössler model) whose mass action rate law shows oscillations and chaotic dynamics are explored. The mechanism is decomposed into subsets of reactions that are then assumed to take place in distinct regularly or randomly distributed spatial domains in the system. The reactive domains are coupled by diffusion. The spatiotemporal system states are investigated as a function of the system size and geometrical arrangement of the domains. A compartmentalization is chosen where the isolated domain attractors are simple steady states. It is then shown that changes in the system size or domain geometry can produce bifurcations leading to simple or period-doubled oscillatory attractors as well as chaotic states. These bifurcations are analyzed by direct simulations of the compartmentalized reaction-diffusion equations and by an analysis in terms of integral equations.

Journal Article↗

A C. elegans Hox gene switches on, off, on and off again to regulate proliferation, differentiation and morphogenesis.

Hox genes establish body pattern throughout the animal kingdom, but the role these genes play at the cellular level to modify and shape parts of the body remains a mystery. We find that the C. elegans Antennapedia homolog, mab-5, sequentially programs many independent events within individual cell lineages. In one body region, mab-5 first switches ON in a lineage to stimulate proliferation, then OFF to specify epidermal structures, then ON in just one branch of the lineage to promote neuroblast formation, and finally OFF to permit proper sense organ morphology. In a neighboring lineage, continuous mab-5 expression leads to a different pattern of development. Thus, this Hox gene achieves much of its power to diversify the anteroposterior axis through fine spatiotemporal differences in expression coupled with a changing pattern of cellular response.

Alleles↗

[Spontaneous synchronous discharges in hippocampal slices. Simulation and experiment].

Chaotic oscillations of extracellular potential of field-type nerve tissues are simulated by a 2D coupled map lattice. These tissues, say, the fields of the hippocampus, are represented by neural mass sheets consisting of current sources. The relationship between the source-sink ensembles and the extracellular field potential at each discrete instant of time t = 1, 2, ... is described by a single-site map creating chaos. The 2D coupled map lattice is viewed as a network of diffusively coupled the maps creating spatiotemporal chaos. The conversion of chaotic oscillations into synchronous ones, which are typical for epileptiform discharges, is studied. The results obtained are in good agreement with those derived from hippocampal slices treated with picrotoxin.

Animals↗

Characterization of the calcium release domains during excitation-contraction coupling in skeletal muscle fibres.

The spatiotemporal properties of the Ca2+ release process in skeletal muscle fibres were determined using a confocal spot detection system. The low-affinity, fluorescent Ca2+ indicator Oregon Green 488 BAPTA-5N (OGB-5N) was used to record localized, action potential-induced fluorescence signals from consecutive locations separated by 200 nm within a single sarcomere. Three-dimensional reconstructions of the Ca2+ transients illustrated the existence of fluorescence domains around Ca2+ release sites, which are centred at the T-tubules. By constructing isochronal plots, it was estimated that the earliest detectable full width at half-maximum (FWHM) of the Ca2+ domains was 0.77+/-0.08 microm and increased rapidly with time to 1.4+/-0.04 microm at peak (17-18 degrees C). A delay of 0.64+/-0.1 ms was observed between the onset of the fluorescence transients at the Z- and M-lines. Deconvolution of fluorescence transients gave estimates of approximately 9 and 2 microM for the peak [Ca2+] changes at the Z and M-lines, respectively. Our results are compatible with the possibility that action potential stimulation elicits Ca2+ release from a region of the sarcoplasmic reticulum (SR) broader than the T-SR junction.

Action Potentials↗

Prolonged inactivation of nicotinic acid adenine dinucleotide phosphate-induced Ca2+ release mediates a spatiotemporal Ca2+ memory.

Although numerous extracellular stimuli are coupled to increases in intracellular Ca(2+), different stimuli are thought to achieve specificity by eliciting different spatiotemporal Ca(2+) increases. We investigated the effect of nicotinic acid adenine dinucleotide phosphate (NAADP) inactivation on spatiotemporal Ca(2+) signals in intact sea urchin eggs. The photorelease of NAADP but not inositol 1,4,5-trisphosphate or cyclic ADP-ribose resulted in self-inactivation. When NAADP was released first locally and subsequently globally, the spatial pattern of the first response shaped that of the second. Specifically, the local release of NAADP created a Ca(2+) gradient that was reversed during the subsequent global release of NAADP. Neither cyclic ADP-ribose nor inositol 1,4,5-trisphosphate showed a similar effect. In contrast to homogenates, NAADP inactivation was reversible in intact eggs with resensitization occurring in approximately 20 min. Because initial NAADP responses affect later responses, NAADP can serve as a mechanism for a Ca(2+) memory that has both spatial and temporal components. This NAADP-mediated Ca(2+) memory provides a novel mechanism for cells to control spatiotemporal Ca(2+) increases.

Animals↗

Nonintegrable dynamics of the triplet-triplet spatiotemporal interaction.

In this paper we examine the coupling of two wave triplets sharing two common modes. The analysis is performed in the solitonic sector of the parameter space where uncoupled solutions departing from linearly unstable homogeneous initial conditions evolve into a collection of regularly interspersed, spatiotemporally localized spikes. The uncoupled system is integrable, but coupling destroys integrability. As coupling grows, energy transfer to smaller spatial scales does appear and becomes faster not only in linearly unstable, but also in linearly stable cases. Chaos in low-dimensional subsystems appears to be responsible for the transfer. We perform a series of numerical tests to verify this idea.

Journal Article↗

Spatiotemporal variation of metabolism in a plant circadian rhythm: the biological clock as an assembly of coupled individual oscillators.

The complex dynamic properties of biological timing in organisms remain a central enigma in biology despite the increasingly precise genetic characterization of oscillating units and their components. Although attempts to obtain the time constants from oscillations of gene activity and biochemical units have led to substantial progress, we are still far from a full molecular understanding of endogenous rhythmicity and the physiological manifestations of biological clocks. Applications of nonlinear dynamics have revolutionized thinking in physics and in biomedical and life sciences research, and spatiotemporal considerations are now advancing our understanding of development and rhythmicity. Here we show that the well known circadian rhythm of a metabolic cycle in a higher plant, namely the crassulacean acid metabolism mode of photosynthesis, is expressed as dynamic patterns of independently initiated variations in photosynthetic efficiency (phi(PSII)) over a single leaf. Noninvasive highly sensitive chlorophyll fluorescence imaging reveals randomly initiated patches of varying phi(PSII) that are propagated within minutes to hours in wave fronts, forming dynamically expanding and contracting clusters and clearly dephased regions of phi(PSII). Thus, this biological clock is a spatiotemporal product of many weakly coupled individual oscillators, defined by the metabolic constraints of crassulacean acid metabolism. The oscillators operate independently in space and time as a consequence of the dynamics of metabolic pools and limitations of CO(2) diffusion between tightly packed cells.

Biological Clocks↗

Motion selectivity and the contrast-response function of simple cells in the visual cortex.

The responses of simple cells were recorded from the visual cortex of cats, as a function of the position and contrast of counterphase and drifting grating patterns, to assess whether direction selectivity can be accounted for on the basis of linear summation. The expected responses to a counterphase grating, given a strictly linear model, would be the sum of the responses to the two drifting components. The measured responses were not consistent with the linear prediction. For example, nearly all cells showed two positions where the responses approached zero (i.e. two "null phase positions"); this was true, even for the most direction selective cells. However, the measured responses were consistent with the hypothesis that direction selectivity is a consequence of the linear spatiotemporal receptive-field structure, coupled with the nonlinearities revealed by the contrast-response function: contrast gain control, halfwave rectification, and expansive exponent. When arranged in a particular sequence, each of these linear and nonlinear mechanisms performs a useful function in a general model of simple cells. The linear spatiotemporal receptive field initiates stimulus selectivity (for direction, orientation, spatial frequency, etc.). The expansive response exponent enhances selectivity. The contrast-set gain control maintains selectivity (over a wide range of contrasts, in spite of the limited dynamic response range and steep slope of the contrast-response function). Rectification conserves metabolic energy.

Animals↗

Quasidiagonal approach to the estimation of lyapunov spectra for spatiotemporal systems from multivariate time series

We describe methods of estimating the entire Lyapunov spectrum of a spatially extended system from multivariate time-series observations. Provided that the coupling in the system is short range, the Jacobian has a banded structure and can be estimated using spatially localized reconstructions in low embedding dimensions. This circumvents the "curse of dimensionality" that prevents the accurate reconstruction of high-dimensional dynamics from observed time series. The technique is illustrated using coupled map lattices as prototype models for spatiotemporal chaos and is found to work even when the coupling is not strictly local but only exponentially decaying.

Journal Article↗

A fully coupled binary biochemical reactive-diffusion model with analytic solution.

Coupled multicomponent biochemical reactive diffusion underlies a variety of biological signalling processes and pharmacokinetic applications, such as paracrine signalling involving "cocktails" comprised of growth promoter/inhibitor factors and proteases associated with tumor angiogenesis, invasion and metastasis, extravascular drug delivery, and polymeric controlled-release drug codelivery design. Here, we present a model and develop a new analytic solution to illustrate the spatiotemporal behavior associated with fully coupled binary biochemical reactive diffusion. The complete coupling renders the solution appreciably more complex in structure and behavior than solutions for unicomponent or partially coupled models. Concentration behavior is illustrated by the computational simulation of binary-species tumor angiogenesis factor reactive-diffusion in the extravascular tissue matrix. The computational results indicate that (a) steady-state concentration profiles are achieved within 1 h of a change in factor production; (b) in the steady state, the spatial profiles of the two components tend to be similar; (c) exceedingly steep concentration gradients, involving several orders-of-magnitude differences in concentration over a few tenths of a millimeter, can occur in the vicinity of boundary sources due to inter-species reaction; (d) the concentration profiles of the two species differ from unicomponent predictions due to the simultaneous mass interchange between the two species. The analytic solution predictions are also used to provide a first-ever validation of a time-dependent, binary-component Crank-Nicholson numerical solution. The ability to quantitatively model interacting and often strongly varying concentration levels as a function of time and position can serve as a powerful complementary tool to experimental analyses for assessing disease state and interventional pharmacological efficacy, especially when the spatial scales on which in vivo behavior occurs taxes the limits of imaging capabilities.

Angiogenesis Inducing Agents↗

Comparison of time-delayed feedback schemes for spatiotemporal control of chaos in a reaction-diffusion system with global coupling.

Time-delayed feedback control for stabilizing time periodic spatial patterns is investigated in a generic reaction-diffusion system with global coupling. We focus on the case of low-dimensional chaos where unstable patterns admit only a single unstable mode. Spatial degrees of freedom are taken into account to define different control schemes. The efficiency of these schemes is discussed, where control forces are motivated by physical requirements as well as by the possibility of obtaining analytically exact results. We find that control schemes that contain the full feedback of the inhibitor variable may finally destroy the control performance. Thus schemes that omit the inhibitor might be more efficient. Our numerical findings are explained in terms of Floquet spectra and compared with analytical solutions of particular coupling schemes.

Journal Article↗

Wave instability induced by nonlocal spatial coupling in a model of the light-sensitive Belousov-Zhabotinsky reaction.

We study spatiotemporal patterns resulting from instabilities induced by nonlocal spatial coupling in the Oregonator model of the light-sensitive Belousov-Zhabotinsky reaction. In this system, nonlocal coupling can be externally imposed by means of an optical feedback loop which links the intensity of locally applied illumination with the activity in a certain vicinity of a particular point weighted by a given coupling function. This effect is included in the three-variable Oregonator model by an additional integral term in the photochemically induced bromide flow. A linear stability analysis of this modified Oregonator model predicts that wave and Turing instabilities of the homogeneous steady state can be induced for experimentally realistic parameter values. In particular, we find that a long-range inhibition in the optical feedback leads to a Turing instability, while a long-range activation induces wave patterns. Using a weakly nonlinear analysis, we derive amplitude equations for the wave instability which are valid close to the instability threshold. Therein, we find that the wave instability occurs supercritically or subcritically and that traveling waves are preferred over standing waves. The results of the theoretical analysis are in good agreement with numerical simulations of the model near the wave instability threshold. For larger distances from threshold, a secondary breathing instability is found for traveling waves.

Journal Article↗

Measles metapopulation dynamics: a gravity model for epidemiological coupling and dynamics.

Infectious diseases provide a particularly clear illustration of the spatiotemporal underpinnings of consumer-resource dynamics. The paradigm is provided by extremely contagious, acute, immunizing childhood infections. Partially synchronized, unstable oscillations are punctuated by local extinctions. This, in turn, can result in spatial differentiation in the timing of epidemics and, depending on the nature of spatial contagion, may result in traveling waves. Measles epidemics are one of a few systems documented well enough to reveal all of these properties and how they are affected by spatiotemporal variations in population structure and demography. On the basis of a gravity coupling model and a time series susceptible-infected-recovered (TSIR) model for local dynamics, we propose a metapopulation model for regional measles dynamics. The model can capture all the major spatiotemporal properties in prevaccination epidemics of measles in England and Wales.

Demography↗