Dominique Gallez

Predictability of human EEG: a dynamical approach

The electroencephalogram recordings from human scalp are analysed in the framework of recent methods of nonlinear dynamics. Three stages of brain activity are considered: the alpha waves (eyes closed), the deep sleep (stage four) and the Creutzfeld-Jakob coma. Two dynamical parameters of the attractors are evaluated. These are the Lyapunov exponents, which measure the divergence or convergence of trajectories in phase space and the Kolmogorov or metric entropy, whose inverse gives the mean predicting time of a given EEG signal. In all the stages considered, the results reveal the presence of at least two positive Lyapunov exponents, which are the footprints of chaos. This number increases to three positive exponents in the case of alpha waves, indicating that although for very short episodes the alpha waves seem extremely coherent, the variability of the brain increases markedly over larger periods of activity. The degree of entropy/chaos increases from coma to deep sleep and then to alpha waves. The large predicting time observed for deep sleep suggests that these waves are related to a slow rate of information processing. The predicting time of the alpha waves is much smaller, indicating a rapid loss of information. Finally, with the help of the Lyapunov exponents, the attractors dimensions are evaluated using two different conjectures and compared to values obtained previously by the Grassberger-Procaccia algorithm.

Nonlinear evolution equations for thin liquid films with insoluble surfactants

The dynamics of a free‐liquid film with insoluble surfactants is followed until film rupture with a simple model based on three nonlinear evolution equations for the film thickness, the surfactants concentration and the tangential velocity of the fluid in the film. This model is derived asymptotically from the full Navier–Stokes equations for free films and incorporates the effect of van der Waals attraction, capillary forces and Marangoni forces due to gradients of surface tension. Different stability regimes are observed numerically for periodic and fixed boundary conditions and several initial conditions. Furthermore, the role of the relevant parameters (Hamaker constant, tension, Marangoni number) on the rupture time is assessed and comparison is made with the flow dynamics for a liquid film with insoluble surfactants on a solid substrate.

Nonlinear rupture of thin free liquid films

We study nonlinear effects on film rupture by investigating the stability of a viscous free film to finite amplitude disturbances. Until now, theoretical results were obtained through a linear stability analysis for infinitesimal perturbations, which is only valid for a short time since disturbances grow to finite size at rupture. The liquid film considered is uncharged, nondraining, and laterally unbounded. The dynamics of the film is described by the Navier–Stokes equations where attractive van der Waals forces are operative, with suitable boundary conditions at the two surfaces (in the particular case of tangentially immobile surfaces). A nonlinear evolution equation describing the variation of the film thickness along the lateral space dimension, is derived for the squeezing mode which is characterized by symmetrical surface waves. This highly nonlinear partial differential equation is solved numerically and rupture characteristics are predicted and compared to experiments. It is shown that the nonlin...

A Viscoelastic Approach to the Hydrodynamic Stability of Membranes

Abstract A linear stability analysis for two rheological behaviors of biological or model membranes is performed. The membrane considered is symmetrical, incompressible, and uncharged. No account is taken of mechanical anisotropy. The two fluids adjacent to the membrane are Newtonian viscous fluids. Two viscoelastic behaviors of the membrane phase are studied (1) the Kelvin—Voigt viscoelastic “solid” model and (2) the Maxwell viscoelastic “liquid” model. The mechanical boundary conditions on both faces of the membrane are the transversal momentum balance (Laplace condition) and the longitudinal momentum balance (Marangoni—Levich condition) . The van der Waals attraction forces between the two faces of the membrane are taken into account. For the symmetrical systems considered, the two modes of wavy perturbations of the membrane are uncoupled: the in-phase motion of both surfaces (stretching mode) and the 180° out-of-phase motion (squeezing mode) . The dispersion relation of both modes is solved analytically for the two models, in the limit of long-wavelength perturbations. Comparison with examples of biological membranes instabilities is performed.

On the dynamic stability of fluid dielectric films

Abstract A simplified model of a fluid dielectric (hydrocarbon) film is introduced suitable for investigating the influence of electrical forces on the dynamics and film stability. Electrical and long range van der Waals interactions are treated as body forces. Linear stability analysis is carried through for a mechanically symmetric film with (i) symmetric surface charge distribution and (ii) linear electric potential drop across the film. Results in the limit of negligible viscosities call for special attention to repulsive interaction mechanisms in dielectric films free of charge.

Pattern formation in thin liquid films with insoluble surfactants

The problem of pattern formation in thin liquid films with insoluble surfactants under attractive and repulsive forces is addressed. A thin fluid film bounded by a wall is modeled by a set of two nonlinear evolution equations for the film thickness and surfactant concentration on the free interface. We perform a bifurcation analysis valid for the general case of apolar and polar forces and predict a supercritical bifurcation to new stationary and periodic structures. Numerical simulations for the particular case of a negative apolar spreading coefficient (attractive van der Waals forces) and a positive polar spreading coefficient (repulsive hydration pressure) are discussed in terms of the analytical predictions. Nonlinearities in the competition between attractive and repulsive forces can lead to formation of periodic patterns for the film thickness with homogeneously distributed surfactants. Due to diffusion and Marangoni effects, insoluble surfactants alter the time required for pattern formation but d...

Repulsive hydration forces between charged lipidic bilayers: A linear stability analysis

Abstract The stability of the aqueous film between two charged lipidic bilayers is investigated in order to express such phenomena as vesicle aggregation and fusion in model form. The three phases (the two external lipid phases and the internal electrolytic solution) are considered as viscous fluids and an intrinsic surface rheology is ascribed to the charged surface layers. The dynamic behaviour of the system, submitted to small fluctuations is analysed. A complete balance of forces is taken into account: at long approach distances the long-range repulsive electrical forces are due to the overlap of the double layers, while the attractive forces are the van der Waals forces, the range of which is larger than the film thickness. At short distances, strong repulsive hydration forces appear which decay exponentially with distance. The general dispersion equation displays two vibrational modes (bending and squeezing); the stability conditions of these are derived. A possible mechanism for the kinetics of vesicle aggregation and fusion is suggested.

Linear hydrodynamics of viscous thin films: II. Application to lipid films

Abstract While employing parameters taken from well-defined reproducible experiments the theory of Part I is exemplified for colored lipid films. Two electric profiles are considered: (i) applied electric potential, (ii) symmetric surface charge distribution. In both cases, instability of the film against periodic thickness fluctuations of long wavelengths is met. The maximum growth rate of the unstable mode is found to be rather small in agreement with experiment. The rate differs by seven orders of magnitude from previous theory and justifies thus a linear treatment. The growth rate is augmented strongly upon application of an electric field.

Linear hydrodynamics of viscous thin films: I. General theory

Abstract A hydrodynamic model suitable for investigating the dynamics of a prototypical symmetric thin film is introduced. The model displays two-dimensional surface phases with intrinsic rheology. Van der Waals and electrostatic interactions are treated as bulk forces. With interest being concentrated on colored films the theory is used to investigate the dynamic behavior of films submitted to periodic thickness fluctuations. The appropriate high-order characteristic value problem associated with the linearized equations of motion is solved analytically in the whole wavelength regime by employing the method of asymptotic expansion combined with a careful consideration of the orders of magnitude of the terms involved. Thus, a complete set of dispersion relations evolves together with a classification scheme for all solutions. Some solutions had never been found before; others assume a form more general than in previous work on symmetric thin films. The investigation culminates in establishing general stability criteria for liquid films undergoing squeezing-type deformations.

Ionic Strength Dependence of Localized Contact Formation Between Membranes: Nonlinear Theory and Experiment

Erythrocyte membrane surface or suspending phase properties can be experimentally modified to give either spatially periodic local contacts or continuous contact along the seams of interacting membranes. Here, for cells suspended in a solution of the uncharged polysaccharide dextran, the average lateral separation between localized contacts in spatially periodic seams at eight ionic strengths, decreasing from 0.15 to 0.065, increased from 0.65 to 3.4 micrometers. The interacting membranes and intermembrane aqueous layer were modeled as a fluid film, submitted to a disjoining pressure, responding to a displacement perturbation either through wave growth resulting in spatially periodic contacts or in perturbation decay, to give a plane continuous film. Measured changes of lateral contact separations with ionic strength change were quantitatively consistent with analytical predictions of linear theory for an instability mechanism dependent on the membrane bending modulus. Introduction of a nonlinear approach established the consequences of the changing interaction potential experienced by different parts of the membrane as the disturbance grew. Numerical solutions of the full nonlinear governing equations correctly identified the ionic strength at which the bifurcation from continuous seam to a stationary periodic contact pattern occurred and showed a decrease in lateral contact and wave crest separation with increasing ionic strength. The nonlinear approach has the potential to recognize the role of nonspecific interactions in initiating the localized approach of membranes, and then incorporate the contribution of specific molecular interactions, of too short a range to influence the beginning of perturbation growth. This new approach can be applied to other biological processes such as neural cell adhesion, phagocytosis, and the acrosome reaction.