Random Convex Hulls and Extreme Value Statistics

In this paper we study the statistical properties of convex hulls of $N$ random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy's formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of $n$ independent random walks. In the continuum time limit this reduces to $n$ independent planar Brownian trajectories for which we compute exactly, for all $n$, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting

Neuropeptide Y phase shifts the circadian clock in vitro via a Y2 receptor

The suprachiasmatic nuclei (SCN) contain a circadian clock whose activity can be recorded in vitro for several days. This clock can be reset by the application of neuropeptide Y. In this study, we focused on determination of the receptor responsible for neuropeptide Y phase shifts of the hamster circadian clock in vitro. Coronal hypothalamic slices containing the SCN were prepared from Syrian hamsters housed under a 14 h:10 h light:dark cycle. Tissue was bathed in artificial cerebrospinal fluid (ACSF), and the firing rates of individual cells were sampled throughout a 12 h period. Control slices received either no application or application of 200 nl ACSF to the SCN at zeitgeber time 6 (ZT6; ZT12 was defined as the time of lights off). Application of 200 ng/200 nl of neuropeptide Y at ZT6 resulted in a phase advance of 3.4 h. Application of the Y2 receptor agonist, neuropeptide Y (3-36), induced a similar phase advance in the rhythm, while the Y1 receptor agonist, [Leu31, Pro34]-neuropeptide Y had no effect. Pancreatic polypeptide (rat or avian) also had no measurable phase-shifting effect. Neuropeptide Y applied at ZT20 or 22 had no detectable phase-shifting effect. These results suggest that the phase-shifting effects of neuropeptide Y are mediated through a Y2 receptor, similar to results found in vivo