Approximate Entropy of Brain Network in the Study of Hemispheric Differences

Human brain, a dynamic complex system, can be studied with different approaches, including linear and nonlinear ones. One of the nonlinear approaches widely used in electroencephalographic (EEG) analyses is the entropy, the measurement of disorder in a system. The present study investigates brain networks applying approximate entropy (ApEn) measure for assessing the hemispheric EEG differences; reproducibility and stability of ApEn data across separate recording sessions were evaluated. Twenty healthy adult volunteers were submitted to eyes-closed resting EEG recordings, for 80 recordings. Significant differences in the occipital region, with higher values of entropy in the left hemisphere than in the right one, show that the hemispheres become active with different intensities according to the performed function. Besides, the present methodology proved to be reproducible and stable, when carried out on relatively brief EEG epochs but also at a 1-week distance in a group of 36 subjects. Nonlinear approaches represent an interesting probe to study the dynamics of brain networks. ApEn technique might provide more insight into the pathophysiological processes underlying age-related brain disconnection as well as for monitoring the impact of pharmacological and rehabilitation treatments.     

10.1007/s10955-006-9040-z

We derive hydrodynamic equations describing the evolution of a binary fluid segregated into two regions, each rich in one species,which are separated (on the macroscopic scale) by a sharp interface. Our starting point is a Vlasov-Boltzmann (VB) equation describing the evolution of the one particle position and velocity distributions, f_i (x, v, t), i = 1, 2. The solution of the VB equation is developed in a Hilbert expansion appropriate for this system. This yields incompressible Navier-Stokes equations for the velocity field u and a jump boundary condition for the pressure across the interface. The interface, in turn, moves with a velocity given by the normal component of u.     

Hydrodynamics of binary fluid phase segregation

Starting with the Vlasov-Boltzmann equation for a binary fluid mixture, we derive an equation for the velocity field u when the system is segregated into two phases (at low temperatures) with a sharp interface between them. u satisfies the incompressible Navier-Stokes equations together with a jump boundary condition for the pressure across the interface which, in turn, moves with a velocity given by the normal component of u. Numerical simulations of the Vlasov-Boltzmann equations for shear flows parallel and perpendicular to the interface in a phase segregated mixture support this analysis. We expect similar behavior in real fluid mixtures.     

A longitudinal study of MR diffusion changes in normal appearing white matter of patients with early multiple sclerosis

BACKGROUND AND PURPOSE: The stage at which normal appearing white matter (NAWM) abnormalities first appear in multiple sclerosis (MS) is not clear. The aim of our study was to monitor water diffusion changes over time in NAWM of patients with early MS.METHODS: Out of a consecutive series of patients enrolled in a MR study on clinically isolated syndrome (CIS), we selected 19 subjects who had completed a one year follow-up. The MR scans obtained at baseline and at 12 months were reviewed according to the new criteria on the diagnosis of MS. Lesion load on T2 and T1 weighted images and the trace of the apparent diffusion coefficient in NAWM were measured both at baseline and at 12 months in patients and in 12 healthy controls.RESULTS: In three patients the diagnosis of MS was done at baseline based on MR. Thirteen patients developed MS during the study and in three patients the diagnosis remained "possible MS." TADC in NAWM in patients was significantly higher than in controls at the 12 months' follow-up but not at baseline (controls mean tADC +/- sd = 0.745 +/- 0.02 mm(2)/sec x 10(-3); patients mean tADC(12) +/- sd = 0.767 +/- 0.02 mm(2)/sec x 10(-3); p < 0.02). TADC and T2 lesion load at 12 months were significantly correlated (p < 0.01). Patients exhibiting tADC(12) above a confidence interval had a significantly greater EDSS score at the same time period (EDSS(12) +/- sd = 1.9 +/- 0.5 and = 1.1 +/- 0.4 respectively; p < 0.01).CONCLUSIONS: This study suggests that diffusion MR cannot detect alterations in NAWM of patients with a CIS suggestive of MS. After one year, when most patients develop MS, diffusion MR abnormalities in NAWM become apparent. These abnormalities are correlated with T2 lesion load and may contribute to neurological impairment.     

Kinetic limits of the HPP cellular automaton

We study the Boltzmann-Grad limit in various versions of the two-dimensional HPP cellular automaton. In the completely deterministic case we prove convergence to an evolution that is not of kinetic type, a well-known phenomenon after Uchyiama's paper on the Broadwell gas, whereas the limiting equation becomes of kinetic type in the model with random collisions. The main part of the paper concerns the case where the collisions are deterministic and the randomness comes from inserting, between any two successive HPP updatings, ^- stirring updatings, <1 being any fixed positive number and a parameter which tends to 0. The initial measure is a product measure with average occupation numbers of the order of (low-density limit) and varying on distances of the order of ^–1. The limit as 0 of the system evolved for times of the order of ^-1- corresponds to the Boltzmann-Grad limit. We prove propagation of chaos and that the renormalized average occupation numbers (i.e., divided by) converge to the solution of the Broadwell equation. Convergence is proven at all times for which the solution of the Broadwell equation is bounded.     

On the derivation of the incompressible Mavier-Stokes equation for Hamiltonian particle systems

We consider a Hamiltonian paticle system interacting by means of a pair potetial. We look at the behavior of the system on a space scale of order ^-1, times of order ^-2 and mean velocities of order , with a scale parameter. Assuming that the phase space density of the particles is give by a series in (the analog of the Chapman-Enskog expansion), the behavior of the system under this rescaling is described, to the lowest order in , by the incompressible Navier-Stokes equations. The viscosity is given in terms of microscopic correlations, and its expression agrees with the Green-Kubo formula.