Biomedical subjects

I G Kevrekidis

Publications and source records attributed to I G Kevrekidis.

14 recordsLinked to original sources

PubMed2003

Coupling fields and underlying space curvature: an augmented Lagrangian approach.

We demonstrate a systematic implementation of coupling between a scalar field and the geometry of the space which carries the field. This naturally gives rise to a feedback mechanism between the field and the geometry. We develop a systematic model for the feedback in a general form, inspired by a specific implementation in the context of molecular dynamics (the so-called Rahman-Parrinello molecular dynamics, or RP-MD). We use a generalized Lagrangian that allows for the coupling of the space's metric tensor to the scalar field, and add terms motivated by RP-MD. We present two implementations of the scheme: one in which the metric is only time-dependent (which gives rise to an ordinary differential equation for its temporal evolution), and the other with spatiotemporal dependence (wherein the metric's evolution is governed by a partial differential equation). Numerical results are reported for the (1+1)-dimensional model with a nonlinearity of the sine-Gordon type.

Journal Article↗

PubMed2003

Wave initiation through spatiotemporally controllable perturbations.

We study the initiation of pulses and fronts in a two-dimensional catalytic reaction-diffusion system: CO oxidation on Pt(110). Using a computer-controlled mobile focused laser beam, we impart various patterns (in space and time) of localized temperature "kicks" to the surface. We explore, and also rationalize through modeling, the cooperativity of such individually subcritical perturbations in both the excitable and the bistable regime.

Journal Article↗

PubMed2003

Gentle dragging of reaction waves.

Using a recently realized "addressable catalyst surface" [Science 294, 134 (2001)]] we study the interaction of chemical reaction waves with prescribed spatiotemporal fields. In particular, we study how a traveling chemical pulse is "dragged" by a localized, moving temperature heterogeneity as a function of its intensity and speed. The acceleration and eventual "detachment" of the wave from the heterogeneity is also explored through simulation and stability analysis.

Journal Article↗

PubMed2002

Continuum approach to discreteness.

We study analytically and numerically continuum models derived on the basis of Padé approximations and their effectiveness in modeling spatially discrete systems. We not only analyze features of the temporal dynamics that can be captured through these continuum approaches (e.g., shape oscillations, radiation effects, and trapping) but also point out ones that cannot be captured (such as Peierls-Nabarro barriers and Bloch oscillations). We analyze the role of such methods in providing an effective "homogenization" of spatially discrete, as well as of heterogeneous continuum equations. Finally, we develop numerical methods for solving such equations and use them to establish the range of validity of these continuum approximations, as well as to compare them with other semicontinuum approximations.

Journal Article↗

PubMed2001

Heterogeneous versus discrete mapping problem.

We propose a method for mapping a spatially discrete problem, stemming from the spatial discretization of a parabolic or hyperbolic partial differential equation of gradient type, to a heterogeneous one with certain comparable dynamical features pertaining, in particular, to coherent structures. We focus the analysis on a (1+1)-dimensional phi(4) model and confirm the theoretical predictions numerically. We also discuss possible generalizations of the method and the ensuing qualitative analogies between heterogeneous and discrete systems and their dynamics.

Journal Article↗

PubMed2001

Spatiotemporal addressing of surface activity.

We have modified surface catalytic activity in real time and space by focusing an addressable laser beam to differentially heat a platinum (110) single-crystal surface. Ellipsomicroscopy imaging of local conditions (such as reactant and product local coverages) enabled us to close the loop between sensing and actuation (both spatiotemporally resolved). Pulses and fronts, the basic building blocks of patterns, could be formed, accelerated, modified, guided, and destroyed at will. Real-time image processing and feedback allow the design and implementation of new classes of nonlocal evolution rules.

Journal Article↗

PubMed2001

Pulse bifurcations and instabilities in an excitable medium: computations in finite ring domains.

We investigate the instabilities and bifurcations of traveling pulses in a model excitable medium; in particular, we discuss three different scenarios involving either the loss of stability or disappearance of stable pulses. In numerical simulations beyond the instabilities we observe replication of pulses ("backfiring") resulting in complex periodic or spatiotemporally chaotic dynamics as well as modulated traveling pulses. We approximate the linear stability of traveling pulses through computations in a finite albeit large domain with periodic boundary conditions. The critical eigenmodes at the onset of the instabilities are related to the resulting spatiotemporal dynamics and "act" upon the back of the pulses. The first scenario has been analyzed earlier [M. G. Zimmermann et al., Physica D 110, 92 (1997)] for high excitability (low excitation threshold): it involves the collision of a stable pulse branch with an unstable pulse branch in a so-called T point. In the framework of traveling wave ordinary differential equations, pulses correspond to homoclinic orbits and the T point to a double heteroclinic loop. We investigate this transition for a pulse in a domain with finite length and periodic boundary conditions. Numerical evidence of the proximity of the infinite-domain T point in this setup appears in the form of two saddle node bifurcations. Alternatively, for intermediate excitation threshold, an entire cascade of saddle nodes causing a "spiraling" of the pulse branch appears near the parameter values corresponding to the infinite-domain T point. Backfiring appears at the first saddle-node bifurcation, which limits the existence region of stable pulses. The third case found in the model for large excitation threshold is an oscillatory instability giving rise to "breathing," traveling pulses that periodically vary in width and speed.

Journal Article↗

PubMed2001

Formation of two-dimensional concentration pulses on microdesigned composite catalyst surfaces.

We study the effect of microdesigned composite geometries on pattern formation during the catalytic oxidation of CO on Pt-Ti, Pt-Rh, and Pt-Pd composite catalysts. In particular, we find experimentally (and rationalize through modeling) that adsorbate surface transport through the second (active) component drastically affects the shapes and interactions of concentration patterns (traveling pulses) observed on pure Pt.

Journal Article↗

PubMed2000

"Coarse" stability and bifurcation analysis using time-steppers: a reaction-diffusion example.

Evolutionary, pattern forming partial differential equations (PDEs) are often derived as limiting descriptions of microscopic, kinetic theory-based models of molecular processes (e.g., reaction and diffusion). The PDE dynamic behavior can be probed through direct simulation (time integration) or, more systematically, through stability/bifurcation calculations; time-stepper-based approaches, like the Recursive Projection Method [Shroff, G. M. & Keller, H. B. (1993) SIAM J. Numer. Anal. 30, 1099-1120] provide an attractive framework for the latter. We demonstrate an adaptation of this approach that allows for a direct, effective ("coarse") bifurcation analysis of microscopic, kinetic-based models; this is illustrated through a comparative study of the FitzHugh-Nagumo PDE and of a corresponding Lattice-Boltzmann model.

Computer Simulation↗

PubMed1994

Microbial predation in coupled chemostats: a global study of two coupled nonlinear oscillators.

Predator-prey systems in continuously operated chemostats exhibit sustained oscillations over a wide range of operating conditions. When two such chemostats interact through flow exchange, the interplay of the oscillation frequencies gives rise to a wealth of dynamic behavior patterns. Using numerical bifurcation techniques, we perform a detailed computational study of these patterns and the transitions between them as the coupling strength and relative frequencies of the two chemostats vary. We concentrate on certain strong resonance phenomena between the two frequencies as well as their mutual extinction and provide a representative sampling of possible phase portraits for our model system. Our observations corroborate recent mathematical results and case studies of coupled nonlinear chemical oscillators in which regions of mutual extinction as well as the Arnol'd structure for two-parameter families of maps of the plane have been observed. We highlight certain unexpected features of the operating diagram discovered through our computational study and discuss their implication for the dynamic response of the chemostat system.

Animals↗

PubMed1993

Immune network behavior--I. From stationary states to limit cycle oscillations.

We develop a model for the idiotypic interaction between two B cell clones. This model takes into account B cell proliferation, B cell maturation, antibody production, the formation and subsequent elimination of antibody-antibody complexes and recirculation of antibodies between the spleen and the blood. Here we investigate, by means of stability and bifurcation analysis, how each of the processes influences the model's behavior. After appropriate nondimensionalization, the model consists of eight ordinary differential equations and a number of parameters. We estimate the parameters from experimental sources. Using a coordinate system that exploits the pairwise symmetry of the interactions between two clones, we analyse two simplified forms of the model and obtain bifurcation diagrams showing how their five equilibrium states are related. We show that the so-called immune states lose stability if B cell and antibody concentrations change on different time scales. Additionally, we derive the structure of stable and unstable manifolds of saddle-type equilibria, pinpoint their (global) bifurcations and show that these bifurcations play a crucial role in determining the parameter regimes in which the model exhibits oscillatory behavior.

Animals↗

PubMed1993

Immune network behavior--II. From oscillations to chaos and stationary states.

Two types of behavior have been previously reported in models of immune networks. The typical behavior of simple models, which involve B cells only, is stationary behavior involving several steady states. Finite amplitude perturbations may cause the model to switch between different equilibria. The typical behavior of more realistic models, which involve both B cells and antibody, consists of autonomous oscillations and/or chaos. While stationary behavior leads to easy interpretations in terms of idiotypic memory, oscillatory behavior seems to be in better agreement with experimental data obtained in unimmunized animals. Here we study a series of models of the idiotypic interaction between two B cell clones. The models differ with respect to the incorporation of antibodies, B cell maturation and compartmentalization. The most complicated model in the series has two realistic parameter regimes in which the behavior is respectively stationary and chaotic. The stability of the equilibrium states and the structure and interactions of the stable and unstable manifolds of the saddle-type equilibria turn out to be factors influencing the model's behavior. Whether or not the model is able to attain any form of sustained oscillatory behavior, i.e. limit cycles or chaos, seems to be determined by (global) bifurcations involving the stable and unstable manifolds of the equilibrium states. We attempt to determine whether such behavior should be expected to be attained from reasonable initial conditions by incorporating an immune response to an antigen in the model. A comparison of the behavior of the model with experimental data from the literature provides suggestions for the parameter regime in which the immune system is operating.

Animals↗

PubMed1992

Microbial predation in a periodically operated chemostat: a global study of the interaction between natural and externally imposed frequencies.

Predator-prey systems in continuously operated chemostats exhibit sustained oscillations over a wide range of operating conditions. When the chemostat is operated periodically, the interaction of the natural oscillation frequency with the external forcing gives rise to a wealth of dynamic behavior patterns. Using numerical bifurcation techniques, we perform a detailed computational study of these patterns and the transitions (local and especially global) between them as the amplitude and frequency of the forcing vary. The transition from low-forcing-amplitude quasiperiodicity to entrainment of the chemostat behavior by strong forcing (involving the concerted closing of resonance horns) is analyzed. We concentrate on certain strong resonance phenomena between the two frequencies and provide an extensive atlas of computed phase portraits for our model system. Our observations corroborate recent mathematical results and case studies of periodically forced chemical oscillators. In particular, the existence and relative succession of several distinct types of global bifurcations resulting in chaotic transients and multistability are studied in detail. The location in the operating diagram of several key codimension 2 local bifurcations of periodic solutions is computed, and their interaction with an interesting feature we name "real-eigenvalues horns" is examined.

Animals↗

PubMed1992

Evidence for p55-p75 heterodimers in the absence of IL-2 from Scatchard plot analysis.

The high affinity receptor for IL-2 is composed of at least two chains, a p55 chain that binds IL-2 with low affinity and a p75 chain that binds with intermediate affinity. Two molecular mechanisms have been proposed for the formation of the high affinity receptor-ligand complex: The affinity conversion model proposes that the high affinity receptor is formed via stepwise binding in which IL-2 first binds to the p55 chain and the resulting complex then associates with the p75 chain to form a high affinity ternary complex. In the performed heterodimer model the p55 and p75 chains form a non-covalently linked high affinity heterodimer in the absence of IL-2. We show that these two models can be distinguished on the basis of equilibrium binding experiments using cell lines expressing different numbers of p55 chains. To make this distinction we develop a general model for the interaction of IL-2 with its various receptors. We than analyze the case in which heterodimers exist in the absence of IL-2 and the case in which no preformed heterodimers exist. For both cases we predict the shape of equilibrium Scatchard plots. We then show that published IL-2 binding studies are consistent with a model in which a large concentration of preformed heterodimers is present on the cell surface and inconsistent with a model in which preformed heterodimers are absent from the cell surface. The models that we develop should have general applicability to the entire class of receptor systems in which low and intermediate affinity chains interact to constitute a high affinity receptor.

Cell Membrane↗