The radius and diameter are fundamental graph parameters, with several natural definitions for directed graphs. Each definition is well-motivated in a variety of applications. All versions of diameter and radius can be solved via solving all-pairs shortest paths (APSP), followed by a fast postprocessing step. However, solving APSP on n-node graphs requires Ω(n 2 ) time even in sparse graphs. We study the question: when can diameter and radius in sparse graphs be solved in truly subquadratic time, and when is such an algorithm unlikely? Motivated by our conditional lower bounds on computing these measures exactly in truly subquadratic time, we search for approximation and fixed parameter subquadratic algorithms, and alternatively, for reasons why they do not exist.We find that:• Most versions of Diameter and Radius can be solved in truly subquadratic time with optimal approximation guarantees, under plausible assumptions. For example, there is a 2-approximation algorithm for directed Radius with one-way distances that runs inÕ(m √ n) time, while a (2 − δ)-approximation algorithm in O(n 2−ε ) time is considered unlikely.• On graphs with treewidth k, we can solve all versions in 2 O(k log k) n 1+o(1) time. We show that these algorithms are near optimal since even a (3/2 − δ)-approximation algorithm that runs in time 2 o(k) n 2−ε would refute plausible assumptions.Two conceptual contributions of this work that we hope will incite future work are: the introduction of a Fixed Parameter Tractability in P framework, and the statement of a differently-quantified variant of the Orthogonal Vectors Conjecture, which we call the Hitting Set Conjecture. * The full version of the paper can be found at: http://arxiv.org/abs/1506. 01799. A.A and V
Summary The precise neural circuitry that mediates arousal during sleep apnea is not known. We previously found that glutamatergic neurons in the external lateral parabrachial nucleus (PBel) play a critical role in arousal to elevated CO2 or hypoxia. Because many of the PBel neurons that respond to CO2 express calcitonin gene-related peptide (CGRP), we hypothesized that CGRP may provide a molecular identifier of the CO2 arousal circuit. Here we report that selective chemogenetic and optogenetic activation of PBelCGRP neurons caused wakefulness, whereas optogenetic inhibition of PBelCGRP neurons prevented arousal to CO2, but not to an acoustic tone or shaking. Optogenetic inhibition of PBelCGRP terminals identified a network of forebrain sites under the control of a PBelCGRP switch that is necessary to arouse animals from hypercapnia. Our findings define a novel cellular target for interventions that may prevent sleep fragmentation and the attendant cardiovascular and cognitive consequences seen in obstructive sleep apnea.
Basic and clinical observations suggest that the caudal hypothalamus comprises a key node of the ascending arousal system, but the cell types underlying this are not fully understood. Here we report that glutamate-releasing neurons of the supramammillary region (SuMvglut2) produce sustained behavioral and EEG arousal when chemogenetically activated. This effect is nearly abolished following selective genetic disruption of glutamate release from SuMvglut2 neurons. Inhibition of SuMvglut2 neurons decreases and fragments wake, also suppressing theta and gamma frequency EEG activity. SuMvglut2 neurons include a subpopulation containing both glutamate and GABA (SuMvgat/vglut2) and another also expressing nitric oxide synthase (SuMNos1/Vglut2). Activation of SuMvgat/vglut2 neurons produces minimal wake and optogenetic stimulation of SuMvgat/vglut2 terminals elicits monosynaptic release of both glutamate and GABA onto dentate granule cells. Activation of SuMNos1/Vglut2 neurons potently drives wakefulness, whereas inhibition reduces REM sleep theta activity. These results identify SuMvglut2 neurons as a key node of the wake−sleep regulatory system.
The suprachiasmatic nucleus (SCN) of the hypothalamus, the master mammalian circadian pacemaker, synchronizes endogenous rhythms with the external day-night cycle. Older humans, particularly those with Alzheimer’s disease (AD), often have difficulty maintaining normal circadian rhythms compared to younger adults, but the basis of this change is unknown. We report that the circadian rhythm amplitude of motor activity in both AD subjects and age-matched controls is correlated with the number of vasoactive intestinal peptide-expressing SCN neurons. AD was additionally associated with delayed circadian phase compared to cognitively healthy subjects, suggesting distinct pathologies and strategies for treating aging- and AD-related circadian disturbances.
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