Spectral methods for multiscale stochastic differential equations

This paper presents a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paper we introduce a new numerical methodology that is based on a spectral method. In particular, we use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients of the homogenized equation. Spectral convergence is proved under suitable assumptions. Numerical experiments corroborate the theory and illustrate the performance of the method. A comparison with the HMM and an application to singularly perturbed stochastic PDEs are also presented

Optimal importance sampling for overdamped Langevin dynamics

Calculating averages with respect to multimodal probability distributions is often necessary in applications. Markov chain Monte Carlo (MCMC) methods to this end, which are based on time averages along a realization of a Markov process ergodic with respect to the target probability distribution, are usually plagued by a large variance due to the metastability of the process. In this work, we mathematically analyze an importance sampling approach for MCMC methods that rely on the overdamped Langevin dynamics. Specifically, we study an estimator based on an ergodic average along a realization of an overdamped Langevin process for a modified potential. The estimator we consider incorporates a reweighting term in order to rectify the bias that would otherwise be introduced by this modification of the potential. We obtain an explicit expression in dimension 1 for the biasing potential that minimizes the asymptotic variance of the estimator for a given observable, and propose a general numerical approach for approximating the optimal potential in the multi-dimensional setting. We also investigate an alternative approach where, instead of the asymptotic variance for a given observable, a weighted average of the asymptotic variances corresponding to a class of observables is minimized. Finally, we demonstrate the capabilities of the proposed method by means of numerical experiments

The Potential of Stem Cell Therapy to Repair White Matter Injury in Preterm Infants: Lessons Learned From Experimental Models

Diffuse white matter injury (dWMI) is a major cause of morbidity in the extremely preterm born infant leading to life-long neurological impairments, including deficits in cognitive, motor, sensory, psychological, and behavioral functioning. At present, no treatment options are clinically available to combat dWMI and therefore exploration of novel strategies is urgently needed. In recent years, the pathophysiology underlying dWMI has slowly started to be unraveled, pointing towards the disturbed maturation of oligodendrocytes (OLs) as a key mechanism. Immature OL precursor cells in the developing brain are believed to be highly sensitive to perinatal inflammation and cerebral oxygen fluctuations, leading to impaired OL differentiation and eventually myelination failure. OL lineage development under normal and pathological circumstances and the process of (re)myelination have been studied extensively over the years, often in the context of other adult and pediatric white matter pathologies such as stroke and multiple sclerosis (MS). Various studies have proposed stem cell-based therapeutic strategies to boost white matter regeneration as a potential strategy against a wide range of neurological diseases. In this review we will discuss experimental studies focusing on mesenchymal stem cell (MSC) therapy to reduce white matter injury (WMI) in multiple adult and neonatal neurological diseases. What lessons have been learned from these previous studies and how can we translate this knowledge to application of MSCs for the injured white matter in the preterm infant? A perspective on the current state of stem cell therapy will be given and we will discuss different important considerations of MSCs including cellular sources, timing of treatment and administration routes. Furthermore, we reflect on optimization strategies that could potentially reinforce stem cell therapy, including preconditioning and genetic engineering of stem cells or using cell-free stem cell products, to optimize cell-based strategy for vulnerable preterm infants in the near future

Recruitment, augmentation and apoptosis of rat osteoclasts in 1,25-(OH)2D3 response to short-term treatment with 1,25-dihydroxyvitamin D3in vivo

Background Although much is known about the regulation of osteoclast (OC) formation and activity, little is known about OC senescence. In particular, the fate of of OC seen after 1,25-(OH)2D3 administration in vivo is unclear. There is evidence that the normal fate of OC is to undergo apoptosis (programmed cell death). We have investigated the effect of short-term application of high dose 1,25-dihydroxyvitamin D3 (1,25-(OH)2D3) on OC apoptosis in an experimental rat model. Methods OC recruitment, augmentation and apoptosis was visualised and quantitated by staining histochemically for tartrate resistant acid phosphatase (TRAP), double staining for TRAP/ED1 or TRAP/DAPI, in situ DNA fragmentation end labelling and histomorphometric analysis. Results Short-term treatment with high-dose 1,25-(OH)2D3 increased the recruitment of OC precursors in the bone marrow resulting in a short-lived increase in OC numbers. This was rapidly followed by an increase in the number of apoptotic OC and their subsequent removal. The response of OC to 1,25-(OH)2D3 treatment was dose and site dependent; higher doses producing stronger, more rapid responses and the response in the tibiae being consistently stronger and more rapid than in the vertebrae. Conclusions This study demonstrates that (1) after recruitment, OC are removed from the resorption site by apoptosis (2) the combined use of TRAP and ED1 can be used to identify OC and their precursors in vivo (3) double staining for TRAP and DAPI or in situ DNA fragmentation end labelling can be used to identify apoptotic OC in vivo