On the rate of convergence of the Hamiltonian particle-mesh method

The Hamiltonian Particle-Mesh (HPM) method is a particle-in-cell method for compressible fluid flow with Hamiltonian structure. We present a numer- ical short-time study of the rate of convergence of HPM in terms of its three main governing parameters. We find that the rate of convergence is much better than the best available theoretical estimates. Our results indicate that HPM performs best when the number of particles is on the order of the number of grid cells, the HPM global smoothing kernel has fast decay in Fourier space, and the HPM local interpolation kernel is a cubic spline

Early postnatal development of the visual cortex in mice with retinal degeneration

This study characterizes the early postnatal development of the visual neocortex in C3H/HeNRj mice. These mice are homozygous for the Pde6b mutation, which causes retinal degeneration starting from postnatal day 7 (P7). To monitor the development of the visual cortex between P3 and P28 we used eight antigens known to be expressed at different developmental stages (Nestin, tau3, β3- Tubulin, Calbindin, Doublecortin, MAP2, Parvalbumin and NeuN). Using semiquantitative analysis we traced the expression and localization of different developmental markers throughout the layers of the visual cortex. Cortical tissue sections corresponding to the first postnatal week (P3-P6) stained positively for Nestin, tau3, β3-Tubulin and Calbindin. These proteins are known to be involved in the migration of neural progenitor cells (NPCs) within the cortical plate. At the time of eye-opening (P14), Doublecortin, MAP2 and NeuN, markers for developing and maturing neurons involved in NPC differentiation are present. Between P9 and P21 Nestin and Calbindin disappear while NeuN and Parvalbumin expression increases in the course of visual neocortex development. The findings of this study provide a snapshot of the dynamic changes in cortex formation during early postnatal development. So far, it is the first investigation on the postnatal development of the mouse visual cortex. Our results indicate that in C3H/HeNRj mice retinal degeneration during these early stages may not influence the maturation of the visual cortex. Until P28 in this mouse strain, the development of the visual neocortex is in accordance with data from other mice (C57BL/6) without retinal degeneration. Whether in older individuals of the C3H/HeNRj strain the visual neocortex will show signs of functional impairment has to be shown by future work

On the wellposedness of some McKean models with moderated or singular diffusion coefficient

We investigate the well-posedness problem related to two models of nonlinear McKean Stochastic Differential Equations with some local interaction in the diffusion term. First, we revisit the case of the McKean-Vlasov dynamics with moderate interaction, previously studied by Meleard and Jourdain in [16], under slightly weaker assumptions, by showing the existence and uniqueness of a weak solution using a Sobolev regularity framework instead of a Holder one. Second, we study the construction of a Lagrangian Stochastic model endowed with a conditional McKean diffusion term in the velocity dynamics and a nondegenerate diffusion term in the position dynamics

Daphnias: from the individual based model to the large population equation

The class of deterministic 'Daphnia' models treated by Diekmann et al. (J Math Biol 61: 277-318, 2010) has a long history going back to Nisbet and Gurney (Theor Pop Biol 23: 114-135, 1983) and Diekmann et al. (Nieuw Archief voor Wiskunde 4: 82-109, 1984). In this note, we formulate the individual based models (IBM) supposedly underlying those deterministic models. The models treat the interaction between a general size-structured consumer population ('Daphnia') and an unstructured resource ('algae'). The discrete, size and age-structured Daphnia population changes through births and deaths of its individuals and throught their aging and growth. The birth and death rates depend on the sizes of the individuals and on the concentration of the algae. The latter is supposed to be a continuous variable with a deterministic dynamics that depends on the Daphnia population. In this model setting we prove that when the Daphnia population is large, the stochastic differential equation describing the IBM can be approximated by the delay equation featured in (Diekmann et al., l.c.)

Dynamical aspects of mean field plane rotators and the Kuramoto model

The Kuramoto model has been introduced in order to describe synchronization phenomena observed in groups of cells, individuals, circuits, etc... We look at the Kuramoto model with white noise forces: in mathematical terms it is a set of N oscillators, each driven by an independent Brownian motion with a constant drift, that is each oscillator has its own frequency, which, in general, changes from one oscillator to another (these frequencies are usually taken to be random and they may be viewed as a quenched disorder). The interactions between oscillators are of long range type (mean field). We review some results on the Kuramoto model from a statistical mechanics standpoint: we give in particular necessary and sufficient conditions for reversibility and we point out a formal analogy, in the N to infinity limit, with local mean field models with conservative dynamics (an analogy that is exploited to identify in particular a Lyapunov functional in the reversible set-up). We then focus on the reversible Kuramoto model with sinusoidal interactions in the N to infinity limit and analyze the stability of the non-trivial stationary profiles arising when the interaction parameter K is larger than its critical value K_c. We provide an analysis of the linear operator describing the time evolution in a neighborhood of the synchronized profile: we exhibit a Hilbert space in which this operator has a self-adjoint extension and we establish, as our main result, a spectral gap inequality for every K>K_c.Comment: 18 pages, 1 figur