Two Remarks on Graph Norms

For a graph H, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions W in Lp, p≥e(H), denoted by t(H, W). One may then define corresponding functionals ∥W∥H:=|t(H,W)|1/e(H) and ∥W∥r(H):=t(H,|W|)1/e(H), and say that H is (semi-)norming if ∥⋅∥H is a (semi-)norm and that H is weakly norming if ∥⋅∥r(H) is a norm. We obtain two results that contribute to the theory of (weakly) norming graphs. Firstly, answering a question of Hatami, who estimated the modulus of convexity and smoothness of ∥⋅∥H, we prove that ∥⋅∥r(H) is neither uniformly convex nor uniformly smooth, provided that H is weakly norming. Secondly, we prove that every graph H without isolated vertices is (weakly) norming if and only if each component is an isomorphic copy of a (weakly) norming graph. This strong factorisation result allows us to assume connectivity of H when studying graph norms. In particular, we correct a negligence in the original statement of the aforementioned theorem by Hatami

High-Speed imaging reveals opposing effects of chronic stress and antidepressants on neuronal activity propagation through the hippocampal trisynaptic circuit

Antidepressants (ADs) are used as first-line treatment for most stress-related psychiatric disorders. The alterations in brain circuit dynamics that can arise from stress exposure and underlie therapeutic actions of ADs remain, however, poorly understood. Here, enabled by a recently developed voltage-sensitive dye imaging (VSDI) assay in mouse brain slices, we examined the impact of chronic stress and concentration-dependent effects of eight clinically used ADs (belonging to different chemical/functional classes) on evoked neuronal activity propagations through the hippocampal trisynaptic circuitry (HTC: perforant path -> dentate gyrus (DG) -> area CA3 -> area CA1). Exposure of mice to chronic social defeat stress led to markedly weakened activity propagations ("HTC-Waves"). In contrast, at concentrations in the low micromolar range, all ADs, which were bath applied to slices, caused an amplification of HTC-Waves in CA regions (invariably in area CA1). The fast-acting "antidepressant" ketamine, the mood stabilizer lithium, and brain derived neurotrophic factor (BDNF) exerted comparable enhancing effects, whereas the antipsychotic haloperidol and the anxiolytic diazepam attenuated HTC-Waves. Collectively, we provide direct experimental evidence that chronic stress can depress neuronal signal flow through the HTC and demonstrate shared opposing effects of ADs. Thus, our study points to a circuit level mechanism of ADs to counteract stress-induced impairment of hippocampal network function. However, the observed effects of ADs are impossible to depend on enhanced neurogenesis

Local Optogenetic Induction of Fast (20-40 Hz) Pyramidal-Interneuron Network Oscillations in the In Vitro and In Vivo CA1 Hippocampus: Modulation by CRF and Enforcement of Perirhinal Theta Activity

The neurophysiological processes that can cause theta-to-gamma frequency range (4-80 Hz) network oscillations in the rhinal cortical-hippocampal system and the potential connectivity-based interactions of such forebrain rhythms are a topic of intensive investigation. Here, using selective Channelrhodopsin-2 (ChR2) expression in mouse forebrain glutamatergic cells, we were able to locally, temporally precisely, and reliably induce fast (20-40 Hz) field potential oscillations in hippocampal area CA1 in vitro (at 25 degrees C) and in vivo (i.e., slightly anesthetized NEX-Cre-ChR2 mice). As revealed by pharmacological analyses and patch-clamp recordings from pyramidal cells and GABAergic interneurons in vitro, these light-triggered oscillations can exclusively arise from sustained suprathreshold depolarization (similar to 200 ms or longer) and feedback inhibition of CA1 pyramidal neurons, as being mandatory for prototypic pyramidal-interneuron network (P-I) oscillations. Consistently, the oscillations comprised rhythmically occurring population spikes (generated by pyramidal cells) and their frequency increased with increasing spectral power. We further demonstrate that the optogenetically driven CA1 oscillations, which remain stable over repeated evocations, are impaired by the stress hormone corticotropin-releasing factor (CRF, 125 nM) in vitro and, even more remarkably, found that they are accompanied by concurrent states of enforced theta activity in the memory associated perirhinal cortex (PrC) in vivo. The latter phenomenon most likely derives from neurotransmission via a known, but poorly studied excitatory CA1 -> PrC pathway. Collectively, our data provide evidence for the existence of a prototypic (CRF-sensitive) P-I gamma rhythm generator in area CA1 and suggest that CA1 P-I oscillations can rapidly up-regulate theta activity strength in hippocampus-innervated rhinal networks, at least in the PrC

Theoretical and experimental evidence of level repulsion states and evanescent modes in sonic crystal stubbed waveguides

The complex band structures calculated using the Extended Plane Wave Expansion (EPWE) reveal the presence of evanescent modes in periodic systems, never predicted by the classical \omega(\vec{k}) methods, providing novel interpretations of several phenomena as well as a complete picture of the system. In this work we theoretically and experimentally observe that in the ranges of frequencies where a deaf band is traditionally predicted, an evanescent mode with the excitable symmetry appears changing drastically the interpretation of the transmission properties. On the other hand, the simplicity of the sonic crystals in which only the longitudinal polarization can be excited, is used to interpret, without loss of generality, the level repulsion between symmetric and antisymmetric bands in sonic crystals as the presence of an evanescent mode connecting both repelled bands. These evanescent modes, obtained using EPWE, explain both the attenuation produced in this range of frequencies and the transfer of symmetry from one band to the other in good agreement with both experimental results and multiple scattering predictions. Thus, the evanescent properties of the periodic system have been revealed necessary for the design of new acoustic and electromagnetic applications based on periodicity

First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds

We calculate the first and the second variation formula for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that can move the singular set of a C^2 surface and non-singular variation for C_H^2 surfaces. These formulas enable us to construct a stability operator for non-singular C^2 surfaces and another one for C2 (eventually singular) surfaces. Then we can obtain a necessary condition for the stability of a non-singular surface in a pseudo-hermitian 3-manifold in term of the pseudo-hermitian torsion and the Webster scalar curvature. Finally we classify complete stable surfaces in the roto-traslation group RT .Comment: 36 pages. Misprints corrected. Statement of Proposition 9.8 slightly changed and Remark 9.9 adde

Quantum state engineering assisted by entanglement

We suggest a general scheme for quantum state engineering based on conditional measurements carried out on entangled twin-beam of radiation. Realistic detection schemes such as {\sc on/off} photodetection, homodyne detection and joint measurement of two-mode quadratures are analyzed in details. Imperfections of the apparatuses, such as nonunit quantum efficiency and finite resolution, are taken into account. We show that conditional {\sc on/off} photodetection provides a reliable scheme to verify nonclassicality, whereas conditional homodyning represents a tunable and robust source of squeezed light. We also describe optical teleportation as a conditional measurement, and evaluate the degrading effects of finite amount of entanglement, decoherence due to losses, and nonunit quantum efficiency.Comment: Some pics with low resolution. Originals at http://www.qubit.i

Astrocytic Ion Dynamics: Implications for Potassium Buffering and Liquid Flow

We review modeling of astrocyte ion dynamics with a specific focus on the implications of so-called spatial potassium buffering, where excess potassium in the extracellular space (ECS) is transported away to prevent pathological neural spiking. The recently introduced Kirchoff-Nernst-Planck (KNP) scheme for modeling ion dynamics in astrocytes (and brain tissue in general) is outlined and used to study such spatial buffering. We next describe how the ion dynamics of astrocytes may regulate microscopic liquid flow by osmotic effects and how such microscopic flow can be linked to whole-brain macroscopic flow. We thus include the key elements in a putative multiscale theory with astrocytes linking neural activity on a microscopic scale to macroscopic fluid flow.Comment: 27 pages, 7 figure