Mario Feingold

Passive scalars, three-dimensional volume-preserving maps, and chaos

The dynamics of a medium-sized particle (passive scalar) suspended in a general time-periodic incompressible fluid flow can be described by three-dimensional volume-preserving maps. In this paper, these maps are studied in limiting cases in which some of the variables change very little in each iteration and others change quite a lot. The former are called slow variables or actions, the latter fast variables or angles. The maps are classified by their number of actions. For maps with only one action we find strong evidence for the existence of invariant surfaces that survive the nonlinear perturbation in a KAM-like way. On the other hand, for the two-action case the motion is confined to invariant lines that break for arbitrary small size of the nonlinearity. Instead, we find that adiabatic invariant surfaces emerge and typically intersect the resonance sheet of the fast motion. At these intersections surfaces are locally broken and transitions from one to another can occur. We call this process, which is analogous to Arnold diffusion, singularity-induced diffusion. It is characteristic of two-action maps. In one-action maps, this diffusion is blocked by KAM-like surfaces.

Chaotic advection in three-dimensional unsteady incompressible laminar flow

We discuss chaotic advection in three-dimensional unsteady incompressible laminar flow, and analyse in detail the most important novel advection phenomenon in these flows : the global dispersion of passive scalars in flows with two slow and one fast velocity components. We make a comprehensive study of the first model of an experimentally realizable flow to exhibit this resonance-induced dispersion : biaxial unsteady spherical Couette flow is a three-dimensional incompressible laminar flow with periodic time dependence derived analytically from the Navier-Stokes equations in the low-Reynolds-number limit.

Stochastic resonance in the speed of memory retrieval.

Abstract. The stochastic resonance (SR) phenomenon in human cognition (memory retrieval speed for arithmetical multiplication rules) is addressed in a behavioral and neurocomputational study. The results of an experiment in which performance was monitored for various magnitudes of acoustic noise are presented. The average response time was found to be minimal for some optimal noise level. Moreover, it was shown that the optimal noise level and the magnitude of the SR effect depend on the difficulty of the task. A computational framework based on leaky accumulators that integrate noisy information and provide the output upon reaching a threshold criterion is used to illustrate the observed phenomena.

Timing the start of division in E. coli: a single-cell study

We monitor the shape dynamics of individual E. coli cells using time-lapse microscopy together with accurate image analysis. This allows measuring the dynamics of single-cell parameters throughout the cell cycle. In previous work, we have used this approach to characterize the main features of single-cell morphogenesis between successive divisions. Here, we focus on the behavior of the parameters that are related to cell division and study their variation over a population of 30 cells. In particular, we show that the single-cell data for the constriction width dynamics collapse onto a unique curve following appropriate rescaling of the corresponding variables. This suggests the presence of an underlying time scale that determines the rate at which the cell cycle advances in each individual cell. For the case of cell length dynamics a similar rescaling of variables emphasizes the presence of a breakpoint in the growth rate at the time when division starts, tau(c). We also find that the tau(c) of individual cells is correlated with their generation time, tau(g), and inversely correlated with the corresponding length at birth, L(0). Moreover, the extent of the T-period, tau(g) - tau(c), is apparently independent of tau(g). The relations between tau(c), tau(g) and L(0) indicate possible compensation mechanisms that maintain cell length variability at about 10%. Similar behavior was observed for both fast-growing cells in a rich medium (LB) and for slower growth in a minimal medium (M9-glucose). To reveal the molecular mechanisms that lead to the observed organization of the cell cycle, we should further extend our approach to monitor the formation of the divisome.

Rotation of single bacterial cells relative to the optical axis using optical tweezers

Using a single-beam, oscillating optical tweezers, we demonstrate trapping and rotation of rod-shaped bacterial cells with respect to the optical axis. The angle of rotation, θ, is determined by the amplitude of the oscillation. It is shown that θ can be measured from the longitudinal cell intensity profiles in the corresponding phase-contrast images. The technique allows viewing the cell from different perspectives and can provide a useful tool in fluorescence microscopy for the analysis of three-dimensional subcellular structures.

Timing of Z-ring localization in Escherichia coli.

Bacterial cell division takes place in three phases: Z-ring formation at midcell, followed by divisome assembly and building of the septum per se. Using time-lapse microscopy of live bacteria and a high-precision cell edge detection method, we have previously found the true time for the onset of septation, τ(c), and the time between consecutive divisions, τ(g). Here, we combine the above method with measuring the dynamics of the FtsZ-GFP distribution in individual Escherichia coli cells to determine the Z-ring positioning time, τ(z). To analyze the FtsZ-GFP distribution along the cell, we used the integral fluorescence profile (IFP), which was obtained by integrating the fluorescence intensity across the cell width. We showed that the IFP may be approximated by an exponential peak and followed the peak evolution throughout the cell cycle, to find a quantitative criterion for the positioning of the Z-ring and hence the value of τ(z). We defined τ(z) as the transition from oscillatory to stable behavior of the mean IFP position. This criterion was corroborated by comparison of the experimental results to a theoretical model for the FtsZ dynamics, driven by Min oscillations. We found that τ(z) < τ(c) for all the cells that were analyzed. Moreover, our data suggested that τ(z) is independent of τ(c), τ(g) and the cell length at birth, L(0). These results are consistent with the current understanding of the Z-ring positioning and cell septation processes.

An Introduction to Chaotic Advection

Understanding particle advection in incompressible laminar fluid flow, apart from being of theoretical interest, holds much relevance for technological applications. Properties of emulsions, dispersion of contaminants in the atmosphere and ocean, sedimentation, and mixing, are just a few examples. Chaotic advection is the complex behaviour a passive scalar— a fluid particle, or a passively advected quantity such as temperature or concentration of a second tracer fluid — can attain, driven by the Lagrangian dynamics of the flow. The surprise is that even laminar flow at low Reynolds number is capable of producing such behaviour. The importance of chaotic advection lies not least in the enhancement of transport it produces. In this review we provide an introduction to theoretical results, numerical simulations, and laboratory experiments on chaotic advection in two-dimensional unsteady, three-dimensional steady, and three-dimensional unsteady flow.

Global diffusion in a realistic three-dimensional time-dependent nonturbulent fluid flow.

We introduce and study the first model of an experimentally realizable three-dimensional time-dependent nonturbulent fluid flow to display the phenomenon of global diffusion of passive-scalar particles at arbitrarily small values of the nonintegrable perturbation. This type of chaotic advection, termed {\it resonance-induced diffusion\/}, is generic for a large class of flows.

Passive scalars and three-dimensional Liouvillian maps

Abstract Global aspects of the motion of passive scalars in time-dependent incompressible fluid flows are well described by volume-preserving (Liouvillian) three-dimensional maps. In this paper the possible invariant structures in Liouvillian maps and the two most interesting nearly-integrible cases are investigated. In addition, the fundamental role of invariant lines in organizing the dynamics of this type of system is exposed. Bifurcations involving the destruction of some invariant lines and tubes and the creation of new ones are described in detail.

Localization and spectral statistics in a banded random matrix ensemble

We investigate the localization properties of the eigenvectors of a banded random matrix ensemble, in which the diagonal matrix elements increase along the diagonal. We relate the results to a transition in the spectral statistics which is observed as a parameter is vaned, and discuss the relevance of this model to the quantum mechanics of chaotic Hamiltonian systems.