Acquisition and storage of aversive memories is one of the basic principles of central nervous systems throughout the animal kingdom. In the absence of reinforcement, the resulting behavioural response will gradually diminish to be finally extinct. Despite the importance of extinction, its cellular mechanisms are largely unknown. The cannabinoid receptor 1 (CB1) and endocannabinoids are present in memory-related brain areas and modulate memory. Here we show that the endogenous cannabinoid system has a central function in extinction of aversive memories. CB1-deficient mice showed strongly impaired short-term and long-term extinction in auditory fear-conditioning tests, with unaffected memory acquisition and consolidation. Treatment of wild-type mice with the CB1 antagonist SR141716A mimicked the phenotype of CB1-deficient mice, revealing that CB1 is required at the moment of memory extinction. Consistently, tone presentation during extinction trials resulted in elevated levels of endocannabinoids in the basolateral amygdala complex, a region known to control extinction of aversive memories. In the basolateral amygdala, endocannabinoids and CB1 were crucially involved in long-term depression of GABA (gamma-aminobutyric acid)-mediated inhibitory currents. We propose that endocannabinoids facilitate extinction of aversive memories through their selective inhibitory effects on local inhibitory networks in the amygdala.

The endogenous cannabinoid system controls extinction of aversive memories

Sphere Packings, Lattices and Groups

Stochastic differential equations and diffusion processes

The uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of these subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane.

Scaling Limits of Loop-Erased Random Walks and Uniform Spanning Trees

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

The uniform spanning tree (UST) and the loop-erased random walk (LERW) are related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of the subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane. 
The scaling limits of these processes are conjectured to be conformally invariant in 2 dimensions. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the Loewner differential equation 
${\partial f\over\partial t} 
= z {\zeta(t)+z \over \zeta(t)-z} {\partial f\over\partial z}$ 
with boundary values $f(z,0)=z$, in the range $z\in\U=\{w\in\C\st |w|<1\}$, $t\le 0$. We choose $\zeta(t):= \B(-2t)$, where $\B(t)$ is Brownian motion on $\partial \U$ starting at a random-uniform point in $\partial \U$. Assuming the conformal invariance of the LERW scaling limit in the plane, we prove that the scaling limit of LERW from 0 to $\partial\U$ has the same law as that of the path $f(\zeta(t),t)$. We believe that a variation of this process gives the scaling limit of the boundary of macroscopic critical percolation clusters.

Scaling limits of loop-erased random walks and uniform spanning trees

This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain $D\mathop \subset \limits_ \ne \mathbb{C} $ is equal to the radial SLE2 path. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invarint. Assuming that ∂D is a C 1-simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper are A ⊂ ∂ D, is the chordal SLE8 path in $\overline D $ joining the endpoints of A. A by-product of this result is that SLE8 is almost surely generated by a continuous path. The result and proofs are not restricted to particular choice of Iattice.

https://projecteuclid.org/download/pdf_1/euclid.aop/1079021469

Conformal invariance of planar loop-erased random walks and uniform spanning trees

Suppose that $G_j$ is a sequence of finite connected planar graphs, and in each $G_j$ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit $G$ of such graphs. Assume that the vertex degrees of the vertices in $G_j$ are bounded, and the bound does not depend on $j$. Then after passing to a subsequence, the limit exists, and is a random rooted graph $G$. We prove that with probability one $G$ is recurrent. The proof involves the Circle Packing Theorem. The motivation for this work comes from the theory of random spherical triangulations.

/pdf/recurrence-of-distributional-limits-of-finite-planar-graphs-3aq1qwd7n1.pdf

Recurrence of Distributional Limits of Finite Planar Graphs

SLEκ is a random growth process based on Loewner’s equation with driving parameter a one-dimensional Brownian motion running with speed κ. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions.

/pdf/basic-properties-of-sle-1xte9z679t.pdf

Oded Schramm

Papers

Scaling Limits of Loop-Erased Random Walks and Uniform Spanning Trees

Scaling limits of loop-erased random walks and uniform spanning trees

Conformal invariance of planar loop-erased random walks and uniform spanning trees

Recurrence of Distributional Limits of Finite Planar Graphs

Basic properties of SLE