Explicit Results for the Distributions of Queue Lengths for a Non-Preemptive Two-Level Priority Queue

Explicit results are derived using simple and exact methods for the joint and marginal queue-length distributions for the M/M/c queue with two non-preemptive priority levels. Equal service rates are assumed. Two approaches are considered. One is based on numerically robust quadratic recurrence relations. The other is based on a complex contour-integral representation that yields exact closed-form analytical expressions, not hitherto available in the literature, that can also be evaluated numerically with very high accuracy

Exact Results for the Distribution of the Partial Busy Period for a Multi-Server Queue

Exact explicit results are derived for the distribution of the partial busy period of the M/M/c multi-server queue for a general number of servers. A rudimentary spectral method leads to a representation that is amenable to efficient numerical computation across the entire ergodic region. An alternative algebraic approach yields a representation as a finite sum of Marcum Q-functions depending on the roots of certain polynomials that are explicitly determined for an arbitrary number of servers. Asymptotic forms are derived in the limit of a large number of servers under two scaling regimes, and also for the large-time limit. Connections are made with previous work. The present work is the first to offer tangible exact results for the distribution when the number of servers is greater than two

A dynamic deep sleep stage in Drosophila

Howmight one determine whether simple animals such as flies sleep in stages? Sleep inmammalsis a dynamic process involving different stages of sleep intensity, and these are typically associated with measurable changes in brain activity (Blake and Gerard, 1937; Rechtschaffen and Kales, 1968; Webb and Agnew, 1971). Evidence for different sleep stages in invertebrates remains elusive, even though it has been well established that many invertebrate species require sleep (Campbell and Tobler, 1984; Hendricks et al., 2000; Shaw et al., 2000; Sauer et al., 2003). Here we used electrophysiology and arousal-testing paradigms to show that the fruit fly, Drosophila melanogaster, transitions between deeper and lighter sleep within extended bouts of inactivity, with deeper sleep intensities after15 and30 min of inactivity. As in mammals, the timing and intensity of these dynamic sleep processes in flies is homeostatically regulated and modulated by behavioral experience. Two molecules linked to synaptic plasticity regulate the intensity of the first deep sleep stage. Optogenetic upregulation of cyclic adenosine monophosphate during the day increases sleep intensity at night, whereas loss of function of a molecule involved in synaptic pruning, the fragile-X mental retardation protein, increases sleep intensity during the day. Our results show that sleep is not homogenous in insects, and suggest that waking behavior and the associated synaptic plasticity mechanisms determine the timing and intensity of deep sleep stages in Drosophila

A conserved role for sleep in supporting Spatial Learning in Drosophila

Sleep loss and aging impair hippocampus-dependent Spatial Learning in mammalian systems. Here we use the fly Drosophila melanogaster to investigate the relationship between sleep and Spatial Learning in healthy and impaired flies. The Spatial Learning assay is modeled after the Morris Water Maze. The assay uses a thermal maze consisting of a 5 × 5 grid of Peltier plates maintained at 36-37°C and a visual panorama. The first trial begins when a single tile that is associated with a specific visual cue is cooled to 25°C. For subsequent trials, the cold tile is heated, the visual panorama is rotated and the flies must find the new cold tile by remembering its association with the visual cue. Significant learning was observed with two different wild-type strains-Cs and 2U, validating our design. Sleep deprivation prior to training impaired Spatial Learning. Learning was also impaired in the classic learning mutant rutabaga (rut); enhancing sleep restored learning to rut mutants. Further, we found that flies exhibited a dramatic age-dependent cognitive decline in Spatial Learning starting at 20-24 days of age. These impairments could be reversed by enhancing sleep. Finally, we find that Spatial Learning requires dopaminergic signaling and that enhancing dopaminergic signaling in aged flies restored learning. Our results are consistent with the impairments seen in rodents and humans. These results thus demonstrate a critical conserved role for sleep in supporting Spatial Learning, and suggest potential avenues for therapeutic intervention during aging

Acute control of the sleep switch in Drosophila reveals a role for gap junctions in regulating behavioral responsiveness

Sleep is a dynamic process in most animals, involving distinct stages that probably perform multiple functions for the brain. Before sleep functions can be initiated, it is likely that behavioral responsiveness to the outside world needs to be reduced, even while the animal is still awake. Recent work in Drosophila has uncovered a sleep switch in the dorsal fan-shaped body (dFB) of the fly’s central brain, but it is not known whether these sleep-promoting neurons also govern the acute need to ignore salient stimuli in the environment during sleep transitions. We found that optogenetic activation of the sleep switch suppressed behavioral responsiveness to mechanical stimuli, even in awake flies, indicating a broader role for these neurons in regulating arousal. The dFB-mediated suppression mechanism and its associated neural correlates requires innexin6 expression, suggesting that the acute need to reduce sensory perception when flies fall asleep is mediated in part by electrical synapses. We thank Leonie Kirszenblat for help and comments on the manuscript. We thank Eleni Notaras for help with behavioral experiments. We also thank Chia-Lin Wu for the INX6 antibody. This work was supported by an NIH grant RO1 NS076980-01 to PJS and BVS, and by an NHMRC grant GNT1065713 to BVS. The authors declare no conflicts of interest

Neurexin-1 regulates sleep and synaptic plasticity in Drosophila melanogaster

Neurexins are cell adhesion molecules that are important for synaptic plasticity and homeostasis, although links to sleep have not yet been investigated. We examined the effects of neurexin-1 perturbation on sleep in Drosophila, showing that neurexin-1 nulls displayed fragmented sleep and altered circadian rhythm. Conversely, the over-expression of neurexin-1 could increase and consolidate night-time sleep. This was not solely due to developmental effects as it could be induced acutely in adulthood, and was coupled with evidence of synaptic growth. The timing of over-expression could differentially impact sleep patterns, with specific night-time effects. These results show that neurexin-1 was dynamically involved in synaptic plasticity and sleep in Drosophila. Neurexin-1 and a number of its binding partners have been repeatedly associated with mental health disorders, including autism spectrum disorders, schizophrenia and Tourette syndrome, all of which are also linked to altered sleep patterns. How and when plasticity-related proteins such as neurexin-1 function during sleep can provide vital information on the interaction between synaptic homeostasis and sleep, paving the way for more informed treatments of human disorders. Neurexins are cell adhesion molecules important for synaptic plasticity and homeostasis. Drosophila neurexin-1 knockouts display fragmented sleep and altered circadian rhythm, as well as a reduction of active zones; conversely, over-expression of neurexin-1 can increase and consolidate night-time sleep. This effect can be induced acutely in adulthood, and is coupled with evidence for synaptic growth. Neurexin-1 thus appears to be dynamically involved in synaptic plasticity and sleep in Drosophila

The Caenorhabditis elegans microtubule minus-end binding homolog PTRN-1 stabilizes synapses and neurites

Microtubule dynamics facilitate neurite growth and establish morphology, but the role of minus-end binding proteins in these processes is largely unexplored. CAMSAP homologs associate with microtubule minus-ends, and are important for the stability of epithelial cell adhesions. In this study, we report morphological defects in neurons and neuromuscular defects in mutants of the C. elegans CAMSAP, ptrn-1. Mechanosensory neurons initially extend wild-type neurites, and subsequently remodel by overextending neurites and retracting synaptic branches and presynaptic varicosities. This neuronal remodeling can be activated by mutations known to alter microtubules, and depends on a functioning DLK-1 MAP kinase pathway. We found that PTRN-1 localizes to both neurites and synapses, and our results suggest that alterations of microtubule structures caused by loss of PTRN-1 function activates a remodeling program leading to changes in neurite morphology. We propose a model whereby minus-end microtubule stabilization mediated by a functional PTRN-1 is necessary for morphological maintenance of neurons. DOI: http://dx.doi.org/10.7554/eLife.01637.00