Metabolic Regulation of Neuronal Plasticity by the Energy Sensor AMPK

Long Term Potentiation (LTP) is a leading candidate mechanism for learning and memory and is also thought to play a role in the progression of seizures to intractable epilepsy. Maintenance of LTP requires RNA transcription, protein translation and signaling through the mammalian Target of Rapamycin (mTOR) pathway. In peripheral tissue, the energy sensor AMP-activated Protein Kinase (AMPK) negatively regulates the mTOR cascade upon glycolytic inhibition and cellular energy stress. We recently demonstrated that the glycolytic inhibitor 2-deoxy-D-glucose (2DG) alters plasticity to retard epileptogenesis in the kindling model of epilepsy. Reduced kindling progression was associated with increased recruitment of the nuclear metabolic sensor CtBP to NRSF at the BDNF promoter. Given that energy metabolism controls mTOR through AMPK in peripheral tissue and the role of mTOR in LTP in neurons, we asked whether energy metabolism and AMPK control LTP. Using a combination of biochemical approaches and field-recordings in mouse hippocampal slices, we show that the master regulator of energy homeostasis, AMPK couples energy metabolism to LTP expression. Administration of the glycolytic inhibitor 2-deoxy-D-glucose (2DG) or the mitochondrial toxin and anti-Type II Diabetes drug, metformin, or AMP mimetic AICAR results in activation of AMPK, repression of the mTOR pathway and prevents maintenance of Late-Phase LTP (L-LTP). Inhibition of AMPK by either compound-C or the ATP mimetic ara-A rescues the suppression of L-LTP by energy stress. We also show that enhanced LTP via AMPK inhibition requires mTOR signaling. These results directly link energy metabolism to plasticity in the mammalian brain and demonstrate that AMPK is a modulator of LTP. Our work opens up the possibility of using modulators of energy metabolism to control neuronal plasticity in diseases and conditions of aberrant plasticity such as epilepsy

A continuum limit for the PageRank algorithm

Semi-supervised and unsupervised machine learning methods often rely on graphs to model data, prompting research on how theoretical properties of operators on graphs are leveraged in learning problems. While most of the existing literature focuses on undirected graphs, directed graphs are very important in practice, giving models for physical, biological or transportation networks, among many other applications. In this paper, we propose a new framework for rigorously studying continuum limits of learning algorithms on directed graphs. We use the new framework to study the PageRank algorithm and show how it can be interpreted as a numerical scheme on a directed graph involving a type of normalised graph Laplacian. We show that the corresponding continuum limit problem, which is taken as the number of webpages grows to infinity, is a second-order, possibly degenerate, elliptic equation that contains reaction, diffusion and advection terms. We prove that the numerical scheme is consistent and stable and compute explicit rates of convergence of the discrete solution to the solution of the continuum limit partial differential equation. We give applications to proving stability and asymptotic regularity of the PageRank vector. Finally, we illustrate our results with numerical experiments and explore an application to data depth.</jats:p

Mean Curvature, Threshold Dynamics, and Phase Field Theory on Finite Graphs

In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study. We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow. We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as "freezing" or "pinning") and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation. © 2014 The Author(s)

Metformin and 2DG activate AMPK in hippocampal CA1 neurons.

<p>A) Schematic of the AMPK-mTOR pathway. B) AMPK is activated 30 min after exposure to 2DG (10 mM, p = 0.019, n = 9), metformin (5 µM, p = 0.005, n = 6), or phenformin (10 µM, p = 0.018, n = 6). Hippocampal slices were incubated in ACSF and drug for 30 minutes and subjected to western blot with anti-phospho-Thr172-AMPK antibody followed by βIII-tubulin as a loading control. Representative western blots of duplicate lanes are shown, together with their quantification from at least 10 samples per condition (C). D) ATP levels are reduced in the presence of 10 mM 2DG. Slices were incubated in ACSF+10 mM 2DG (n = 3) as above. Tissue was lysed and subjected to a CellTiter-Glo ATP assay (Promega). E) 2DG activates AMPK in cell bodies of the pyramidal layer (PL) and dendrites of the stratum radiatum (SR). Anti-phospho-Thr172-AMPK immunoreactivity is displayed in green. Neu-N is displayed in red. Scale bar: 10 µm.</p