Concentration of the first eigenfunction for a second order elliptic operator

We study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold when the diffusion constant $\epsilon$ goes to zero. We assume that the first order term is given by a vector field $b$, whose recurrent components are either hyperbolic points or cycles or two dimensional torii. The limits of the normalized eigenfunctions concentrate on the recurrent sets of maximal dimension where the topological pressure \cite{Kifer90} is attained. On the cycles and torii, the limit measures are absolutely continuous with respect to the invariant probability measure on these sets. We have determined these limit measures, using a blow-up analysis.Comment: Note to appear in C.R.A.

Residence times of receptors in dendritic spines analyzed by simulations in empirical domains

Analysis of high-density superresolution imaging of receptors reveal the organization of dendrites at the nano-scale resolution. We present here simulations in empirical live cell images, which allows converting local information extracted from short range trajectories into simulations of long range trajectories. Based on these empirical simulations, we compute the residence time of an AMPA receptor (AMPAR) in dendritic spines that accounts for receptors local interactions and geometrical organization. We report here that depending on the type of the spine, the residence time varies from one to five minutes. Moreover, we show that there exists transient organized structures, previously described as potential wells that can regulate the trafficking of AMPARs to dendritic spines.Comment: 19 page