Stable sets and mean Li-Yorke chaos in positive entropy systems

It is shown that in a topological dynamical system with positive entropy, there is a measure-theoretically "rather big" set such that a multivariant version of mean Li-Yorke chaos happens on the closure of the stable or unstable set of any point from the set. It is also proved that the intersections of the sets of asymptotic tuples and mean Li-Yorke tuples with the set of topological entropy tuples are dense in the set of topological entropy tuples respectively.Comment: The final version, reference updated, to appear in Journal of Functional Analysi

Multipotent vascular stem cells contribute to neurovascular regeneration of peripheral nerve.

BackgroundNeurovascular unit restoration is crucial for nerve regeneration, especially in critical gaps of injured peripheral nerve. Multipotent vascular stem cells (MVSCs) harvested from an adult blood vessel are involved in vascular remodeling; however, the therapeutic benefit for nerve regeneration is not clear.MethodsMVSCs were isolated from rats expressing green fluorescence protein (GFP), expanded, mixed with Matrigel matrix, and loaded into the nerve conduits. A nerve autograft or a nerve conduit (with acellular matrigel or MVSCs in matrigel) was used to bridge a transected sciatic nerve (10-mm critical gap) in rats. The functional motor recovery and cell fate in the regenerated nerve were investigated to understand the therapeutic benefit.ResultsMVSCs expressed markers such as Sox 17 and Sox10 and could differentiate into neural cells in vitro. One month following MVSC transplantation, the compound muscle action potential (CMAP) significantly increased as compared to the acellular group. MVSCs facilitated the recruitment of Schwann cell to regenerated axons. The transplanted cells, traced by GFP, differentiated into perineurial cells around the bundles of regenerated myelinated axons. In addition, MVSCs enhanced tight junction formation as a part of the blood-nerve barrier (BNB). Furthermore, MVSCs differentiated into perivascular cells and enhanced microvessel formation within regenerated neurovascular bundles.ConclusionsIn rats with peripheral nerve injuries, the transplantation of MVSCs into the nerve conduits improved the recovery of neuromuscular function; MVSCs differentiated into perineural cells and perivascular cells and enhanced the formation of tight junctions in perineural BNB. This study demonstrates the in vivo therapeutic benefit of adult MVSCs for peripheral nerve regeneration and provides insight into the role of MVSCs in BNB regeneration

Convergence to global equilibrium for Fokker-Planck equations on a graph and Talagrand-type inequalities

In recent work, Chow, Huang, Li and Zhou introduced the study of Fokker-Planck equations for a free energy function defined on a finite graph. When $N\ge 2$ is the number of vertices of the graph, they show that the corresponding Fokker-Planck equation is a system of $N$ nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. The different choices for inner products on the space of probability distributions result in different Fokker-Planck equations for the same process. Each of these Fokker-Planck equations has a unique global equilibrium, which is a Gibbs distribution. In this paper we study the {\em speed of convergence} towards global equilibrium for the solution of these Fokker-Planck equations on a graph, and prove that the convergence is indeed exponential. The rate as measured by the decay of the $L_2$ norm can be bound in terms of the spectral gap of the Laplacian of the graph, and as measured by the decay of (relative) entropy be bound using the modified logarithmic Sobolev constant of the graph. With the convergence result, we also prove two Talagrand-type inequalities relating relative entropy and Wasserstein metric, based on two different metrics introduced in [CHLZ] The first one is a local inequality, while the second is a global inequality with respect to the "lower bound metric" from [CHLZ]