Axel Steiger
email: steiger@mpipsykl.mpg.de

Martin Dresler

Petra Schüssler

Michael Kluge

Ghrelin in mental health, sleep, memory

Keywords: ghrelin, sleep, depression, memory, mental health

Ghrelin acts as a neuropeptide. It participates in sleep-wake regulation. After systemic ghrelin treatment nonREM sleep is promoted in male humans and mice. This effect is influenced by gender, time of administration and depression. Ghrelin does not modulate sleep in healthy women and during the early morning in male subjects. In depressed women REM sleep is diminished after ghrelin. In elderly men and depressed men sleep promotion by ghrelin was preserved. In rats after central ghrelin feeding and wakefulness increased. The nocturnal secretion pattern of cortisol, GH, LH, FSH and hypothalamo-pituitary-thyroid hormones are influenced by ghrelin. Furthermore ghrelin appears to be related to memory and to be involved in the pathophysiology of CNS disorders, particularly depression.

A c c e p t e d M a n u s c r i p t

Introduction

The early observation that ghrelin, besides stimulating the release of growth hormone (GH) and other hormones [START_REF] Kojima | Ghrelin is a growth hormone-releasing acylated peptide from stomach[END_REF] enhances feeding in the rat [START_REF] Tschöp | Ghrelin induces adiposity in rodents[END_REF][START_REF] Wren | The novel hypothalamic peptide ghrelin stimulates food intake and growth hormone secretion[END_REF] suggests that it acts as a neuropeptide. In humans ghrelin stimulates appetite [START_REF] Schmid | Ghrelin stimulates appetite, imagination of food, GH, ACTH and cortisol, but does not affect leptin in normal controls[END_REF][START_REF] Wren | Ghrelin enhances appetite and increases food intake in humans[END_REF]. In addition to an increased appetite healthy volunteers reported visual imagination of their favourite meal after a single iv administration of ghrelin [START_REF] Schmid | Ghrelin stimulates appetite, imagination of food, GH, ACTH and cortisol, but does not affect leptin in normal controls[END_REF]. Obviously ghrelin affects higher CNS activity and is more than just a signal for hunger. This view is confirmed by various studies during the last decade showing effects of ghrelin on sleep-wake behaviour and memory. Furthermore a role of ghrelin in the pathophysiology of depression and other psychiatric disorders and epilepsy is discussed. The aim of this chapter is to summarize the state of the art in this field.

Ghrelin and sleep

Ghrelin is the only identified, perhaps the only existing natural ligand of the growth hormone (GH) secretagogue (GHS) receptor (GHSR). The GHSR was cloned before the identification of ghrelin, and synthetic ligands of the GHSR, called GHS were even known before this receptor was cloned. Studies in human volunteers showed effects of synthetic GH on sleep electroencephalogram (EEG). The sleep EEG of mammals contains periods of rapid eye movement (REM) and nonREM sleep. In healthy young subjects nonREM-and REM-sleep periods alternate in a cyclic fashion.

NonREM sleep consists of four stages according to the criteria by [START_REF] Rechtschaffen | A Manual of Standardized Terminology, Techniques and Scoring System for Sleep Stages of Human Subjects[END_REF]. A recent classification (American Academy of Sleep [START_REF] Medicine | The AASM manual for the scoring of sleep and associated events: Rules, terminology and technical specifications[END_REF] differentiates between three stages only. During sleep stage 1, a transition from drowsiness to light sleep, a slowing of EEG activity occurs. Sleep s pindles and K-complex wave forms are found during sleep stage 2. Synchronised slow waves are found during slow wave sleep (SWS), the sleep stages 3 and 4 [START_REF] Rechtschaffen | A Manual of Standardized Terminology, Techniques and Scoring System for Sleep Stages of Human Subjects[END_REF] or stage N3 (American Academy of Sleep [START_REF] Medicine | The AASM manual for the scoring of sleep and associated events: Rules, terminology and technical specifications[END_REF] respectively. During REM sleep a faster EEG activity, horizontal rapid eye movements and hypotonus of skeletal muscles are found. In contrast to humans sleep in animals is often polyphasic resulting in episodic sleep-wake patterns. In animals nonREM sleep is not divided into several stages, whereas its depth can be interpreted according to the power of SWS slow wave activity (SWA) [START_REF] Borbély | A two process model of sleep regulation[END_REF].

REM sleep is also found in animals. The analysis of eye movements in small animals A c c e p t e d M a n u s c r i p t 4 is complicated. Therefore a combination of wake-like EEG with flat electromyogram showing muscle ………. is used to define REM sleep. In humans and animals the microstructure of sleep is investigated by computerized quantitative EEG analysis including amplitudes and power in the EEG bands beta, sigma, alpha, theta and delta (synonymous: SWA). Various factors including monoamines cytokines, neuroactive steroids and peptides participate in sleep regulation [START_REF] Luppi | Neurochemistry of sleep: an overview of animal experimental work[END_REF][START_REF] Krueger | Cytokines in immune function and sleep regulation[END_REF][START_REF] Steiger | Neurochemical regulation of sleep[END_REF].

In young normal men slow wave sleep (SWS) increased and wakefulness decreased after one week of oral treatment with the GHS MK-677 [START_REF] Copinschi | Prolonged oral treatment with MK-677, a novel growth hormone secretagogue, improves sleep quality in man[END_REF]. This finding is highly interesting as it suggests that oral administration of a neuropeptide is capable of affecting CNS activity. The same authors studied the effects of MK-677 also in a small group of elderly subjects and did not find clear sleep-promoting effects in this age group [START_REF] Copinschi | Prolonged oral treatment with MK-677, a novel growth hormone secretagogue, improves sleep quality in man[END_REF]. GH-releasing peptide-6 (GHRP-6) was given intravenously during sleep via long catheter in a pulsatile fashion (4 bolus injections between 22.00 and 01.00 h) [START_REF] Frieboes | Growth hormone-releasing peptide-6 stimulates sleep, growth hormone, ACTH and cortisol release in normal man[END_REF]. This method appears to be crucial to detect sleep-EEG effects of neuropeptides as it mimics their physiological release [START_REF] Steiger | Neurochemical regulation of sleep[END_REF]. After intravenous GHRP-6 non-rapid eye movement (nonREM) sleep stage 2 increased and the nocturnal secretion of GH, corticotropin (ACTH) and cortisol was elevated. In studies with intranasal, oral and sublingual administration of GHRP-6 the effects on the sleep EEG and hormone secretion were less distinct and partly different from those after intravenous (iv) injection [START_REF] Frieboes | Effects of growth hormone-releasing peptide-6 on the nocturnal secretion of GH, ACTH and cortisol and on the sleep EEG in man: role of routes of administration[END_REF]. After hexarelin the amount of SWS decreased, whereas a similar pattern of endocrine effects occurred with a marked stimulation of GH and an increase of cortisol [START_REF] Frieboes | Hexarelin decreases slow-wave sleep and stimulates the sleep-related secretion of GH, ACTH, cortisol and prolactin during sleep in healthy volunteers[END_REF]. The decrease of SWS in this study may be related to negative feedback inhibition of endogenous GH-releasing hormone (GHRH). A single dose injection of GHRP-2 during the third period of rapid eye movement (REM) sleep did not affect sleep [START_REF] Moreno-Reyes | Evidence against a role for the growth hormone-releasing peptide axis in human slow-wave sleep regulation[END_REF]. This method may be insufficient to change sleep EEG.

Ghrelin and GHRH also stimulate the release of GH from the pituitary. GHRH is the one neuropeptide, whose sleep-promoting effects are best documented so far (Obál & Krueger, 2004;[START_REF] Steiger | Effects of growth hormone-releasing hormone and somatostatin on sleep EEG and nocturnal hormone secretion in male controls[END_REF]. GHRH increased nonREM sleep when given by various routes of administration (iv, intracerebroventricular, intrapreoptic) to A c c e p t e d M a n u s c r i p t 5 laboratory animals [START_REF] Ehlers | Effects of corticotropin-releasing factor and growth hormone-releasing factor on sleep and activity in rats[END_REF][START_REF] Obál | Vasoactive intestinal polypeptide promotes sleep without effects on brain temperature in rats at night[END_REF]1996;[START_REF] Zhang | Intrapreoptic microinjection of GHRH or its antagonist alters sleep in rats[END_REF] and (iv, intranasal) to male humans subjects [START_REF] Steiger | Effects of growth hormone-releasing hormone and somatostatin on sleep EEG and nocturnal hormone secretion in male controls[END_REF][START_REF] Kerkhofs | Sleep-promoting effects of growth hormone-releasing hormone in normal men[END_REF][START_REF] Marshall | Greater efficacy of episodic than continuous growth hormone releasing hormone (GHRH) administration in promoting slow wave sleep (SWS)[END_REF][START_REF] Perras | Sleep and endocrine changes after intranasal administration of growth hormone-releasing hormone in young and aged humans[END_REF]. Various animal models showed that nonREM sleep is elevated, when the amount of GHRH is high and is decreased when GHRH is low (transgenic animals, GHRH antagonists, GHRH antibodies) (Obál & Krueger, 2004). GHRH appears to be involved in sleep promotion after sleep deprivation [START_REF] Obál | Growth hormone-releasing hormone antibodies suppress sleep and prevent enhancement of sleep after sleep deprivation[END_REF]. In young normal subjects after pulsatile iv administration of GHRH SWS and GH increase, whereas cortisol decreases [START_REF] Steiger | Effects of growth hormone-releasing hormone and somatostatin on sleep EEG and nocturnal hormone secretion in male controls[END_REF]. This effect is in contrast to the decrease of SWS and GH and the increase of cortisol after CRH in young male healthy volunteers [START_REF] Holsboer | Effects of intravenous corticotropin-releasing hormone upon sleep-related growth hormone surge and sleep EEG in man[END_REF]. These observations support the view of a reciprocal interaction of GHRH and CRH in sleep-endocrine regulation, which was first submitted by [START_REF] Ehlers | Slow-wave sleep: do young adult men and women age differently?[END_REF]. The effect of GHRH appears to depend on gender, age and the time of administration [START_REF] Steiger | Neurochemical regulation of sleep[END_REF]. In women, in contrast to men, sleep was impaired and ACTH and cortisol were elevated after GHRH (Antonijevic et al., 2000a;Antonijevic et al., 2000b). Only a moderate sleep-promoting effect of GHRH was found in healthy elderly subjects as the number of awakenings decreased after the peptide, whereas SWS remained unchanged [START_REF] Guldner | Reduced efficacy of growth hormone-releasing hormone in modulating sleep endocrine activity in the elderly[END_REF]. When GHRH was given during the early morning hours to young normal subjects sleep EEG remained unchanged [START_REF] Schier | Changes in sleependocrine activity after growth hormone-releasing hormone depend on time of administration[END_REF].

In addition to GHRH all components of the somatotrophic axis appear to be involved in sleep regulation (Obál & Krueger, 2004). Therefore it was of high interest to examine the effect of ghrelin on sleep after the substance became available. In a pioneering study [START_REF] Weikel | Ghrelin promotes slow-wave sleep in humans[END_REF] showed that during the whole night SWS and during the second half of the night its equivalent derived from quantitative EEG analysis, accumulated slow-wave activity (SWA) increased after pulsatile iv ghrelin administration (4 x 50 µg from 22.00 to 01.00) to healthy young men (see figure 1).

REM sleep was reduced during the second third of the night as during the whole night GH and prolactin were stimulated. Cortisol increased during the first half of the night in comparison to placebo (see figure 2). ACTH was elevated by trend only. The effects of ghrelin on sleep EEG and GH secretion resemble the changes seen after GHRH [START_REF] Steiger | Effects of growth hormone-releasing hormone and somatostatin on sleep EEG and nocturnal hormone secretion in male controls[END_REF], whereas the increase of cortisol after ghrelin is opposite to the decrease of this hormone after GHRH [START_REF] Steiger | Effects of growth hormone-releasing hormone and somatostatin on sleep EEG and nocturnal hormone secretion in male controls[END_REF] and resembles the A c c e p t e d M a n u s c r i p t 6 effects of GHRP-6 [START_REF] Frieboes | Growth hormone-releasing peptide-6 stimulates sleep, growth hormone, ACTH and cortisol release in normal man[END_REF] and hexarelin [START_REF] Frieboes | Hexarelin decreases slow-wave sleep and stimulates the sleep-related secretion of GH, ACTH, cortisol and prolactin during sleep in healthy volunteers[END_REF] suggesting that ghrelin may act as an interface between the hypothalamo-pituitary somatotrophic and the hypothalamo-pituitary adrenocortical systems. At the same time as the report by [START_REF] Weikel | Ghrelin promotes slow-wave sleep in humans[END_REF]), Obàl et al. (2003) found an increase of SWS after intraperitoneal ghrelin administration in mice. These authors showed that intact GHRH receptors are the prerequisite of the sleep-promoting action of ghrelin, since this effect was absent in mice with deficient GHRH receptors.

-Please insert figures 1 and 2 near to here -Beside of its effects on GH, the hypothalamo-pituitary adrenocorticoal hormones and prolactin, ghrelin influences the sleep related release of more endocrine systems.

Intracerebroventricular injection of ghrelin in rats suppressed the pulsatile secretion of hypothalamic luteinizing hormone (LH) releasing hormone and of LH [START_REF] Fernandez-Fernandez | Effects of ghrelin upon gonadotropinreleasing hormone and gonadotropin secretion in adult female rats: in vivo and in vitro studies[END_REF][START_REF] Furuta | Intracerebroventricular administration of ghrelin rapidly suppresses pulsatile luteinizing hormone secretion in ovariectomized rats[END_REF]. After pulsatile injection of ghrelin in healthy young male subjects lower mean plasma levels of both LH (23.40 until 02.00 h) and testosterone (00.40 to 03.00 h) occurred than after placebo. After the first administration of ghrelin LH peak levels were significantly smaller than after placebo (see figure 3). Furthermore the interval between this and the preceeding peak was significantly longer after ghrelin than after placebo. Significantly more LH pulses occurred after placebo than after ghrelin administration (Kluge et al., 2007a). In all, ghrelin caused both a delay and suppression of the amplitude of LH pulses.

-Please insert figure 3 near to here -Sleep-related secretion of profiles of follicle stimulating hormone (FSH) were investigated in young healthy male volunteers after 4 x 50 µg ghrelin or placebo given iv between 22.00 and 01.00 h. The FSH plasma levels were significantly lower after ghrelin than after placebo during the interval between 01.00 and 02.20 h (Kluge et al., 2009b). This study shows, that ghrelin moderately suppresses FSH secretion in young male subjects. The influence of ghrelin on the hypothalamo-pituitary-thyroid (HPT) axis was studied in animal models and in humans. Ghrelin or placebo was injected A c c e p t e d M a n u s c r i p t intracerebroventricularly every 24 hours for five days to rats. After ghrelin pituitary thyroid-stimulating hormone (TSH) cells were smaller and TSH plasma levels were blunted resulting in less active thyroid follicles and thyroxine (T 4 ) plasma levels [START_REF] Sosic-Jurjevic | Central ghrelin affects pituitary-thyroid axis: histomorphological and hormonal study in rats[END_REF]. The sleep-associated concentrations of the hormones of the HPT axis were studied in 10 young female and 10 young male healthy volunteers after 4 x 50 µg ghrelin or placebo given hourly between 22.00 and 01.00 h. Ghrelin induced a subtle increase of thyroxin (fT 4 ) followed by a marked decrease of TSH.

This decrease occurred after the physiological TSH surge, which was not affected.

The trijodothyronine (fT 3 ) and thyroid-binding globulin (TBG) were not affected (Kluge et al., 2010b). These findings support the hypothesis that ghrelin suppresses the HPT axis in humans, which counteracts its energy-saving effects.

The effects of ghrelin on sleep in humans are influenced by the time of administration, gender and age. When ghrelin was given in a pulsatile fashion between 04.00 and 07.00 h during the early morning sleep EEG remained unchanged, whereas GH and cortisol increased (Kluge et al., 2007c). This finding is similar to the lack of major sleep-EEG changes after GHRH, when given during the same interval to young healthy volunteers [START_REF] Schier | Changes in sleependocrine activity after growth hormone-releasing hormone depend on time of administration[END_REF]. Whereas after GHRH opposite effects on sleep EEG and nocturnal hormone secretion in women and men were found (Antonijevic et al., 2000a;2000b) sleep EEG remained unchanged in young healthy women after iv ghrelin (Kluge et al., 2007b) in contrast to the sleep-promoting effect in male subjects [START_REF] Weikel | Ghrelin promotes slow-wave sleep in humans[END_REF][START_REF] Kluge | Ghrelin alone or co-administered with GHRH or CRH increases non-REM sleep and decreases REM sleep in young males[END_REF]. The sexual dimorphism was also found in elderly healthy men and women. SWS, nonREM sleep, GH and cortisol increased and wakefulness decreased in healthy male volunteers, who were 60 years and older. In contrast in women of the same age range sleep EEG remained unchanged (Kluge et al., 2010a). The endocrine effects of ghrelin were similar to those in male subjects, and in young and elderly women as well (Kluge et al., 2007b;2010a). Since sleep-EEG effects were absent in premenopausal and postmenopausal women as well, it is unlikely that estrogens contribute to this sexual dimorphism.

The interaction of ghrelin, GHRH and CRH in sleep regulation was studied in young male healthy volunteers [START_REF] Kluge | Ghrelin alone or co-administered with GHRH or CRH increases non-REM sleep and decreases REM sleep in young males[END_REF]. In this study 10 young male healthy volunteers received in a randomised fashion four bolus injections of placebo (A), 4) but did not affect GH secretion. After GHRH the ghrelininduced GH secretion was elevated, whereas GHRH did not affect cortisol secretion.

In conclusion, ghrelin increased nonREM sleep and decreased REM sleep. Ghrelin and CRH also tended to enhance the effects of ghrelin on sleep. When CRH was coadministered with ghrelin it promoted sleep, in contrast to its sleep-impairing effect after having been administered alone [START_REF] Holsboer | Effects of intravenous corticotropin-releasing hormone upon sleep-related growth hormone surge and sleep EEG in man[END_REF]. Furthermore CRH decreased REM latency. This finding is in line with the REM sleep-promoting role of this peptide. For example REM sleep is enhanced in CRH-overexpressing mice compared to the wildtype [START_REF] Kimura | Conditional corticotropin-releasing hormone overexpression in the mouse forebrain enhances rapid eye movement sleep[END_REF].

-Please insert figure 4 near to here -In a single healthy volunteer we observed that after a bolus injection of 100 µg at 22.00 h the subject developed distinct hunger and claimed for an additional dinner and snacks. After additional injections of 3 x 100 µg ghrelin his sleep was impaired [START_REF] Weikel | Single case report: 100 µg ghrelin at 22:00 increases distinctly hunger, food intake and nocturnal plasma levels of GH, ACTH and cortisol in a young man -evidence for dose-dependent effects of ghrelin on appetite?[END_REF]. Night eating syndrome is characterized by repetitive awakening because of hunger and excessive eating. In a patient with night-eating syndrome, who was not obese, we observed higher nocturnal ghrelin levels than usually seen in healthy young women [START_REF] Rosenhagen | Elevated plasma ghrelin levels in night-eating syndrome[END_REF]. These case reports suggest a dose-dependent action of ghrelin during the night. Under physiological conditions sleep is a long period of fasting in healthy humans. Possibly elevated endogenous [START_REF] Rosenhagen | Elevated plasma ghrelin levels in night-eating syndrome[END_REF] or exogenous [START_REF] Weikel | Single case report: 100 µg ghrelin at 22:00 increases distinctly hunger, food intake and nocturnal plasma levels of GH, ACTH and cortisol in a young man -evidence for dose-dependent effects of ghrelin on appetite?[END_REF] ghrelin levels may disrupt sleep due to increased hunger, whereas after lower dosages (in our experiments 4 x 50 µg) sleep is promoted in male subjects or remained unchanged in female subjects. Hunger after ghrelin administration may explain the contrasting findings in a series of excellent studies by Szentirmai et al. After intracerebroventricular [START_REF] Szentirmai | Ghrelin-induced sleep responses in ad libitum fed and food-restricted rats[END_REF] and after intrahypothalamic [START_REF] Szentirmai | Ghrelin microinjection into forebrain sites induces wakefulness and feeding in rats[END_REF] ghrelin administration to the rat wakefulness and feeding increased. In detail intracerebroventricular ghrelin injections at light onset and at dark This decrease was followed by an increase in nonREM sleep during hours 3-12 after some dosages of ghrelin. In feeding-restricted rats ghrelin suppressed nonREM sleep during hours 1 and 2 and nonREM sleep during hours 3-12 [START_REF] Szentirmai | Ghrelin-induced sleep responses in ad libitum fed and food-restricted rats[END_REF].

Furthermore [START_REF] Szentirmai | Ghrelin microinjection into forebrain sites induces wakefulness and feeding in rats[END_REF] examined the sleep and feeding responses to the microinjections of three dosages of ghrelin into hypothalamic sites that are implicated in the release and regulation, such as the lateral hypothalamus, the medial preoptic area and the paraventricular nucleus at dark onset in rats. The microinjections were followed by an increase in wakefulness. At the same time food consumption increased. A decrease of the EEG SWA after injections into the medial preoptic area was followed by an increase. The authors discussed that sleep and feeding are mutually exclusive behaviours and an increase in feeding might result in shortened sleep time. Alternatively, hunger due to ghrelin injections may cause discomfort that could also interfere with sleep. However, the lowest injected dose of ghrelin into the paraventricular nucleus stimulated feeding as strongly as the higher dose, but did not affect sleep. The authors hypothesized that decreased wakefulness and increased feeding are two parallel outputs of the hypothalamic ghrelin-sensitive circuitry. Its activation appears to trigger the behavioural sequence during the first hours of the activity period in rats, the "dark onset syndrome". One may keep in mind a possible delayed sleep-promoting action of ghrelin following the feeding period in some of these experiments.

In another study Szentirmai et al. found that ghrelin knock out mice slept less than the wild type ( 2007). This study points to a sleep-promoting effect of ghrelin. In a similar vein [START_REF] Schüssler | Nocturnal ghrelin levels -relationship to sleep EEG, the levels of growth hormone, ACTH and cortisol -and gender differences[END_REF] reported an advanced increase of ghrelin during the recovery night after sleep deprivation (see figure 5). An endogenous substance which accumulates during sleep deprivation is a candidate for a sleep-promoting substance. A similar result derived from a study by [START_REF] Spiegel | Brief communication: Sleep curtailment in healthy young men is associated with decreased leptin levels, elevated ghrelin levels, and increased hunger and appetite[END_REF]. These authors compared intraindividually subjects with restricted (four hours per night) and prolonged (10 hours per night) sleep for two days during each condition. At daytime ghrelin levels were higher in the subjects with the restricted sleep condition when compared to the prolonged condition. Furthermore Taheri et al. [START_REF] Dzaja | Sleep enhances nocturnal plasma ghrelin levels in healthy subjects[END_REF].

-Please insert figure 5 near to here -The relationship between sleep, feeding, ghrelin and its antagonist in the energy balance, leptin, in the rat was investigated by [START_REF] Bodosi | Rhythms of ghrelin, leptin, and sleep in rats: effects of the normal diurnal cycle, restricted feeding, and sleep deprivation[END_REF]. Animals underwent three experimental conditions: Free feeding rats with normal diurnal rhythms, rats with feeding restricted to the 12 hours night period and rats subjected to five hours of sleep deprivation at the beginning of the light cycle. Plasma ghrelin and leptin levels showed diurnal rhythms. The ghrelin peak preceded and the leptin peak followed the major daily feeding peak in hour 1 after dark onset. Restricted feeding reversed this rhythm and the rhythm of REM sleep whereas the duration and intensity of nonREM sleep were responsive to restricted feeding. Sleep deprivation stimulated plasma ghrelin and induced feeding, but had no influence on leptin concentrations.

After sleep deprivation biphasic elevations in the hypothalamic ghrelin content occurred. The authors concluded that the variations in hypothalamic ghrelin contents might be associated with sleep-wake activity in rats (see figure 6).

-Please insert figure 6 near to here -Sleep and thermoregulatory responses to three days of cold exposure to the combined challenge of cold environment temperature and fasting were investigated in preproghrelin knockout mice (originally named ghrelin knockout mice) and in ghrelin receptor knockout mice. It was shown that mice lacking the preproghrelin gene, but not those lacking the ghrelin receptor, have impaired abilities to manifest and integrate normal sleep and thermoregulatory responses to metabolic challenges.

In response to fasting at 17° C the preproghrelin knockout mice entered hypothermic bouts associated with reduced sleep. Prior to treatment with obestatin, which is like a preproghrelin gene product, attenuated this hypothermic response of the preproghrelin knockout mice [START_REF] Szentirmai | The preproghrelin gene is required for the normal integration of thermoregulation and sleep in mice[END_REF]. The authors concluded, that products of the preproghrelin gene regulate physiologic sleep and body temperature responses to metabolic challenge in mice and may be involved in mechanisms of A c c e p t e d M a n u s c r i p t adaptation to harsh environmental conditions. The results may also help to better understand the physiology of sustaining energy stores in hibernating species.

Previous studies suggest the occurrence of peaks of ghrelin around sleep onset [START_REF] Cummings | A preprandial rise in plasma ghrelin levels suggests a role in meal initiation in humans[END_REF][START_REF] Dzaja | Sleep enhances nocturnal plasma ghrelin levels in healthy subjects[END_REF]. This issue was not supported in a study with a more frequent sampling of blood specimens [START_REF] Schüssler | Nocturnal ghrelin levels -relationship to sleep EEG, the levels of growth hormone, ACTH and cortisol -and gender differences[END_REF]. In the latter study a gender difference was found between the onset of the study at 20.00 and 23.00 h. During this period ghrelin levels increased in healthy male subjects. At 23.00 h a level was reached as found in women from 20.00 h. Until 07.00 h no major changes in ghrelin levels, particularly no ghrelin peak like the GH peak occurred (see figure 7). A trend suggesting a lower time spent in stage 1 sleep in subjects with high nocturnal ghrelin levels was found. Since sleep stage 1 means shallow sleep, also this finding suggests a sleep-promoting effect of ghrelin.

-Please insert figure 7 near to here -Furthermore in male patients with primary insomnia and in a control group ghrelin levels were measured three times during the night (at 23.00, 02.00 and 06.00 h). The insomnia patients showed less total sleep time, stage 2 and REM sleep were decreased, sleep efficiency and more stage 1 sleep than the controls. The ghrelin levels across the night were significantly lower in the insomnia patients, whereas leptin levels did not differ between groups [START_REF] Motivala | Nocturnal levels of ghrelin and leptin and sleep in chronic insomnia[END_REF]. Also this observation supports a sleep-promoting influence of ghrelin.

Ghrelin and memory

A growing number of studies in rodents suggests that ghrelin is involved in memory processes. Intracerebroventricular administration of ghrelin as well as injection of ghrelin in the hippocampus, amygdala and dorsal raphe nucleus dose dependently enhanced memory retention in an avoidance task in rats [START_REF] Carlini | Ghrelin increases anxiety-like behavior and memory retention in rats[END_REF][START_REF] Carlini | Differential role of the hippocampus, amygdala, and dorsal raphe nucleus in regulating feeding, memory, and anxiety-like behavioral responses to ghrelin[END_REF](Carlini et al., , 2007) ) and mice [START_REF] Diano | Ghrelin controls hippocampal spine synapse density and memory performance[END_REF]. Of note, this effect could be observed only if ghrelin was injected before training, but not before retrieval and only for long term memory, i.e. retention times of more than 3 hours (Carlini et al., 2010a). Also object recognition was enhanced by ghrelin administration to the hippocampus in rats (Carlini et al., 2007). Likewise in mice, decreases in object recognition performance A c c e p t e d M a n u s c r i p t 12 due to chronic food restriction were counteracted by ghrelin administration [START_REF] Carlini | Decreased memory for novel object recognition in chronically food-restricted mice is reversed by acute ghrelin administration[END_REF]. Subcutaneous injections of ghrelin or the ghrelin mimetic LY444711 led to a marked improvement in spatial memory retention in mice [START_REF] Diano | Ghrelin controls hippocampal spine synapse density and memory performance[END_REF].

Interestingly, ghrelin receptor deficient mice express impairments in spatial learning, but not avoidance learning compared to wild types (Davis et al., in press). Since ghrelin receptor agonists enhanced memory retention in rats [START_REF] Atcha | Cognitive enhancing effects of ghrelin receptor agonists[END_REF], the central ghrelin receptor has been proposed as a drug target for therapeutic approaches to treat diseases affecting cognition (Thomas, 2006;[START_REF] Atcha | Cognitive enhancing effects of ghrelin receptor agonists[END_REF]Carlini et al., 2010a). Indeed, the administration of ghrelin ameliorated memory impairments in an Alzheimer's disease mouse model [START_REF] Moon | Ghrelin ameliorates cognitive dysfunction and neurodegeneration in intrahippocampal amyloid-β1-42 oligomer-injected mice[END_REF]. However, an impairment of memory retention in neonatal chicks after ghrelin administration has also been reported recently [START_REF] Carvajal | Central ghrelin increases anxiety in the Open Field test and impairs retention memory in a passive avoidance task in neonatal chicks[END_REF].

Data on the effects of ghrelin on human cognition is sparse. In contrast to its memory enhancing effects in rodents, serum ghrelin levels were recently shown to be negatively correlated with declarative memory in elderly adults [START_REF] Spitznagel | Serum ghrelin is inversely associated with cognitive function in a sample of non-demented elderly[END_REF]. Also on procedural memory consolidation the effects of ghrelin seem to be impairing rather than promoting: Nocturnal administration of ghrelin compared to placebo levelled gains in a sequential motor skill task normally seen after a night of sleep (see figure 8), however this difference slightly failed to be significant (Dresler et al., 2010).

-Please insert figure 8 near to here -

The neuroanatomical network integrating ghrelin into memory functions is not well understood (Olszewski et al., 2008). In mice, circulating ghrelin enters the hippocampus and binds to neurons of the hippocampal formation, promoting dendritic spine synapse formation and generation of longterm potentiation (LTP), while ghrelin-knockout mice have reduced spine density compared to controls [START_REF] Diano | Ghrelin controls hippocampal spine synapse density and memory performance[END_REF]. Ghrelin administration in rats reduced the threshold values to generate LTP in the dentate gyrus and increased the nitric oxide (NO) synthesis via NO synthase (NOS) activation in the hippocampus (Carlini et al., 2010b). Since NOS activity was also shown to be different in trained and untrained animals, it was interpreted as one of the endogenous factors in the hippocampus mediating effects A c c e p t e d M a n u s c r i p t 13 of ghrelin on memory (Carlini et al., 2010b). However, also striatal activity is modulated by ghrelin administration in humans [START_REF] Malik | Ghrelin modulates brain activity in areas that control appetitive behavior[END_REF], which might counteract possible beneficial effects of ghrelin on hippocampal contributions to sleep-associated sequential motor skill learning [START_REF] Albouy | Both the hippocampus and striatum are involved in consolidation of motor sequence memory[END_REF].

In conclusion, like its effects on sleep, administration of ghrelin elicits very diverse effects on memory. Before proposals about cognition enhancing drugs targeting the central ghrelin receptor are warrantable, much more research is needed to elucidate the effects of ghrelin and its receptor agonists on human memory.

Ghrelin and depression

A ghrelin gene polymorphism was found to be associated with depression [START_REF] Nakashima | Ghrelin gene polymorphism is associated with depression, but not panic disorder[END_REF]. The potential role of ghrelin in the development of depressive symptoms was investigated in a series of studies in mice [START_REF] Lutter | The orexigenic hormone ghrelin defends against depressive symptoms of chronic stress[END_REF]. Ghrelin levels were physiologically increased by restricting the food intake of mice with a diet containing 60 % of normal calories for 10 days. Ghrelin levels increased fourfold. In the elevated plus maze and forced swim test the calorierestricted wildtype mice showed robust anxiolytic-and antidepressant-like behaviour in comparison to wildtype mice fed ad libitum. After genetic blockade of ghrelin signalling in GHSR knockout (Ghr -/-) these anxiolytic-and antidepressant-like effects related to calorie restriction did not occur. It was clarified that differences between the two genotypes are not related to differences in sensory motor coordination, general locomotor activity or body weight. Furthermore mice received a subcutaneous dose of ghrelin that induces distinct feeding. Compared with placebo-injected controls these mice showed significantly less anxiety-and depression-like symptoms in the elevated plus maze and forced swim test. In addition it was tested whether ghrelin signalling regulates depressive symptoms in a mouse model of chronic stress. In the chronic social defeat stress procedure mice underwent repetitive bouts of social defeat by aggressive male mice. These mice showed lasting behavioural changes which were reversible by chronic antidepressant treatment. After a chronic social defeat stress mice had significantly elevated ghrelin levels for at least four weeks.

Ghsr -/-mice showed greater social avoidance after chronic social defeat stress.

Whereas body weight after chronic social defeat stress did not differ between Ghsr -/- mice and the wildtype, food intake was enhanced in the wildtype, but not in the A c c e p t e d M a n u s c r i p t 14 knockout mice. The authors concluded that activation of ghrelin signalling due to chronic stress may be a homeostatic adaptation that helps to cope with stress at the expense of increased calorie intake. The antidepressant actions of ghrelin may include activation of hypothalamic orexin-containing neurons. This view is supported by the finding that the antidepressant-like effects of ghrelin in the forced swim test were blocked in mice lacking orexin [START_REF] Lutter | The orexigenic hormone ghrelin defends against depressive symptoms of chronic stress[END_REF].

In contrast to the work by [START_REF] Lutter | The orexigenic hormone ghrelin defends against depressive symptoms of chronic stress[END_REF] other studies suggest an anxiogenic action of ghrelin. After intraventricular and intraperitoneal administration of ghrelin anxiogenic behaviour was induced in mice [START_REF] Asakawa | A role of ghrelin in neuroendocrine and behavioral responses to stress in mice[END_REF]. Similarly anxiogenesis by ghrelin in rats was reported [START_REF] Carlini | Ghrelin increases anxiety-like behavior and memory retention in rats[END_REF][START_REF] Carlini | Differential role of the hippocampus, amygdala, and dorsal raphe nucleus in regulating feeding, memory, and anxiety-like behavioral responses to ghrelin[END_REF].

Another series of studies suggested that ghrelin antisense oligonucleotides given into the lateral ventricle resulted in anxiolytic-like effects in rats as shown by elevated plus maze, black and white and conditioned fear tests. Also antidepressant-like effects in rats were reported as demonstrated by the forced swimming test [START_REF] Kanehisa | Administration of antisense DNA for ghrelin causes an antidepressant and anxiolytic response in rats[END_REF].

Major depression is a condition of prolonged stress. Nocturnal ghrelin levels however did not differ between patients with depression and normal controls (Kluge et al., 2009a). After four weeks of antidepressive therapy with mirtazapine nocturnal ghrelin levels decreased in comparison to baseline [START_REF] Schmid | Changes of sleep architecture, spectral composition of sleep EEG, the nocturnal secretion of cortisol, ACTH, GH, prolactin, melatonin, ghrelin and leptin and the DEX-CRH test in depressed patients during treatment with mirtazapine[END_REF]. It is likely that this effect is related to a moderate weight gain rather than to an improvement of psychopathology in this study. A subgroup of healthy volunteers who received after an overnight fast a single iv injection of ghrelin reported elevated mood [START_REF] Schmid | Ghrelin stimulates appetite, imagination of food, GH, ACTH and cortisol, but does not affect leptin in normal controls[END_REF]. After pulsatile ghrelin administration to male depressed patients an improvement of mood was found by trend as the score of the Befindlichkeits-Skala decreased (Kluge et al., in press). In contrast in women psychopathology remained unchanged. In the same study changes of sleep EEG and hormone secretion showed a sexual dimorphism. In male patients after ghrelin wakefulness decreased and nonREM sleep increased, whereas in female patients REM sleep was suppressed as the time spent in stage REM decreased and REM latency was prolonged. In both sexes after ghrelin the GH peak was augmented. Secretion patterns of cortisol differed between depressed men and women. In both sexes cortisol plasma levels were elevated after ghrelin being significantly higher than those A c c e p t e d M a n u s c r i p t 15 after placebo during the first half of the night. However only in men cortisol subsequently declined to significantly lower plasma levels in ghrelin-treated than in the placebo condition during the second half of the night. In contrast cortisol levels in women during this interval were similar with both treatments (see figure 9) (Kluge et al., in press).

-Please insert figure 9 near to here -

Ghrelin and other psychiatric and neurologic disorders

Plasma levels of ghrelin are elevated in comparison to healthy subjects in patients with anorexia nervosa [START_REF] Otto | Weight gain decreases elevated plasma ghrelin concentrations of patients with anorexia nervosa[END_REF]. In patients with Prader-Willi syndrome ghrelin levels were elevated in comparison to normal controls during 24 h [START_REF] Paik | Correlation between fasting plasma ghrelin levels and age, body mass index (BMI), BMI percentiles, and 24-hour plasma ghrelin profiles in Prader-Willi syndrome[END_REF]. These patients are obese. Elevated ghrelin levels in these patients are in contrast to low ghrelin levels observed regularly in obese patients. Also in a group of patients with night eating syndrome and obesity ghrelin levels during 24 h were blunted [START_REF] Allison | Neuroendocrine profiles associated with energy intake, sleep, and stress in the night eating syndrome[END_REF].

In a sample of patients with epilepsy serum ghrelin levels were measured in single blood samples taken in the morning. Mean ghrelin levels of the patients were significantly higher than in healthy volunteers [START_REF] Berilgen | Serum ghrelin levels are enhanced in patients with epilepsy[END_REF]. The authors discuss that the elevation of ghrelin levels could contribute to the lengthening of nonREM sleep in patients with epilepsy and by this way may play a role in the occurrence of seizures. Furthermore they propose that high levels of ghrelin, by affecting GH and prolactin secretion may facilitate the occurrence of seizures.

During treatment of patients with schizophrenia with the antipsychotic olanzapine plasma ghrelin levels increased [START_REF] Murashita | Olanzapine increases plasma ghrelin level in patients with schizophrenia[END_REF]. It is likely that the change of ghrelin is related to the weight gain which is frequently observed during treatment with olanzapine.

Ghrelin morning levels were determined in patients with obsessive compulsive disorder, accompanied by major depression or not and in normal controls. No differences between groups were found [START_REF] Emül | Ghrelin and leptin levels in patients with obsessive-compulsive disorder[END_REF]. Grey bar: significant changes after ghrelin administration. Weikel, J.C., Wichniak, A., Ising, M., Brunner, H., Friess, E., Held, K., Mathias, S., Schmid, D.A., Tests with contrasts following a Bonferroni-adjusted multivariate ANOVA were used for identifying statistical differences between treatment groups. Cortisol levels significantly differed between 22:20 and 04:00 h and at 06:20 h. At all these points in time, cortisol levels in the ghrelin+CRH condition were statistically significant different from those in the other conditions. In addition, cortisol levels were statistically significant different in the placebo condition compared to those in the ghrelin (22:20-02:20 h) and in the ghrelin+GHRH condition (22:20-02:40 h). %significant difference compared to ghrelin (po0.05).

Reprinted from Psychoneuroendocrinology, 33, Kluge, M., Schüssler, P. Bleninger, P., Kleyer, S., Uhr, M., Weikel, J.C., Yassouridis, A., Zuber, V., Steiger, A.,

  s c r i p t ghrelin (B), ghrelin + GHRH (C), or ghrelin + CRH (D). In all ghrelin conditions nonREM sleep increased significantly compared to placebo whereas REM sleep decreased. After CRH + ghrelin the time spent awake and the REM latency (interval between sleep onset and the first occurrence of sleep stage REM) increased compared to other treatment conditions. CRH enhanced the ghrelin-induced cortisol secretion (see figure

  A c c e p t e d M a n u s c r i p t 9 onset suppressed nonREM sleep and REM sleep for two hours in ad libitum-fed rats.

  (2004) reported a correlation between short sleep time and elevated ghrelin levels in a cohort of 1,024 volunteers from the Wisconsin Sleep Cohort Study. During sleep A c c e p t e d M a n u s c r i p t 10 deprivation nocturnal ghrelin levels were blunted around sleep onset in comparison to the sleep condition

  show that the action of ghrelin includes much more than promotion of feeding or appetite. Ghrelin is involved in sleep-wake regulation. It appears to influence memory and it may play a role in affective and other CNS disorders. Accordingly a better understanding of the effects of ghrelin in the CNS may help to develop new therapies for sleep and psychiatric disorders.

  Figure legendsFig.1Effects of ghrelin on delta power activity. Accumulated delta power (± SE) for time in bed (TIB) after injection of 4 x 50 µg of ghrelin compared with placebo (n ± 7).

  Uhr, M., Steiger, A.: Ghrelin promotes slow-wave sleep in humans. American Journal of Physiology, Endocrinology & Metabolism 284 (2003) E407-E415 © American Physiological Society, used with permission.

Fig. 2

 2 Fig. 2 Time course of nocturnal plasma hormone concentrations of growth hormone (GH) and cortisol (means ± SE) after iv injection of 4 x 50 µg of ghrelin compared with placebo (n = 7). Arrows: times of injections. Weikel, J.C., Wichniak, A., Ising, M., Brunner, H., Friess, E., Held, K., Mathias, S., Schmid, D.A., Uhr, M., Steiger, A.: Ghrelin promotes slow-wave sleep in humans. American Journal of Physiology, Endocrinology & Metabolism 284 (2003) E407-E415 © American Physiological Society, used with permission.

Fig. 3

 3 Fig. 3 Exemplary nocturnal LH secretion profiles of a healthy man receiving ghrelin or placebo. *, LH pulse.

Fig. 2

 2 Fig. 2 from Kluge M. et al. Ghrelin suppresses secretion of luteinizing hormone in humans. Journal of Clinical Endocrinology & Metabolism 92 (2007) 3202-3205. Copyright 2007, The Endocrine Society.

Fig. 4

 4 Fig. 4 Secretion profiles of cortisol in 10 healthy males receiving ghrelin, ghrelin+GHRH, ghrelin+CRH, or placebo (mean±SEM).

Fig. 5

 5 Fig. 5 Nocturnal ghrelin secretion (mean±standard error) at baseline and after sleep deprivation. Reprinted from Psychoneuroendocrinology, 31, Schüssler, P., Uhr, M., Ising, M., Weikel, J.C., Schmid, D.A., Held, K., Mathias, S., Steiger, A., Nocturnal ghrelin, ACTH, GH and cortisol secretion after sleep deprivation in humans, 915-923, 2006, with permission from Elsevier.

Fig. 6

 6 Fig. 6 Diurnal rhythms (hourly mean ± SE) of time spent in non-rapid eye movement sleep (NREMS), and feeding activity were determined on 3 days and averaged for 24 h in 51 rats. Diurnal rhythms of plasma (PL) leptin (Lep) and ghrelin (Ghre) concentrations, and hypothalamic (HY) ghrelin contents were determined on day 4 of recording in groups of rats (n = 7-10) killed at 4-h intervals. The 24-h curves are double-plotted to promote visualization of the rhythms. Time 0, light onset. Gray columns, dark period. %Rec time, %recording time. B. Bodosi, J. Gardi, I. Hajdu, E. Szentirmai, F. Obal, Jr. and J. M. Krueger: Rhythms of ghrelin, leptin, and sleep in rats: effects of the normal diurnal cycle, restricted feeding, and sleep deprivation. American Journal of Physiology -Regulatory, Integrative & Comparative Physiology 287:1071-1079, 2004 © American Physiological Society, used with permission.

Fig. 7

 7 Fig. 7 Nocturnal ghrelin secretion in male and female subjects.Means and standard errors of the mean (SEM) are presented. ACTH levels were not determined in samples collected before lights were switched off (23:00 hours).Journal of Sleep Research, 14, Schüssler, P., Uhr, M., Ising, M.,Schmid, D., Weikel, J., Steiger A., Nocturnal ghrelin levels -relationship to sleep EEG, the levels of growth hormone, ACTH and cortisoland gender differences, 329-336, 2005, with permission from Wiley.

Fig. 8

 8 Fig. 8 Nocturnal administration of ghrelin does not promote memory consolidation in humans. Mean overnight increases in and standard errors of the mean (SEM) a sequential motor memory task are presented. Pharmacopsychiatry 43: 277-278, Dresler M, Kluge M, Genzel L, Schüssler P, Steiger A.: Nocturnal administration of ghrelin does not promote memory consolidation. 2010 © Georg Thieme Verlag, used with permission.

Fig. 9

 9 Fig. 9Cortisol secretion profiles in 7 women (A) and 7 men (B) with major depression receiving ghrelin or placebo (mean ± SEM). *Significant difference compared to placebo (p < 0.05).

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  Ghrelin alone or co-administered with GHRH or CRH increases non-REM sleep and decreases REM sleep in young males, 497-506, 2008, with permission from Elsevier.

Acknowledgement

Studies from the authors' laboratory were supported by grants from the Deutsche Forschungsgemeinschaft (DFG Ste 486/5-3 to 5-5).


David Coudert
email: david.coudert@inria.fr

Luc Hogie
email: luc.hogie@inria.fr

Aurélien Lancin
email: aurelien.lancin@inria.fr

Dimitri Papadimitriou
email: dimitri.papadimitriou@alcatel-lucent.com

Stéphane Perennes
email: stephane.perennes@inria.fr

Issam Tahiri
email: issam.tahiri@inria.fr

Feasibility study on distributed simulations of BGP

Keywords: distributed simulation, network, BGP, Internet

The Autonomous System (AS) topology of the Internet (up to 61k ASs) is growing at a rate of about 10% per year. The Border Gateway Protocol (BGP) starts to show its limits in terms of the number of routing table entries it can dynamically process and control. Due to the increasing routing information processing and storage, the same trend is observed for routing model simulators such as DRMSim specialized in large-scale simulations of routing models. Therefore, DRMSim needs enhancements to support the current size of the Internet topology and its evolution (up to 100k ASs). To this end, this paper proposes a feasibility study of the extension of DRM-Sim so as to support the Distributed Parallel Discrete Event paradigm. We first detail the possible distribution models and their associated communication overhead. Then, we analyze this overhead by executing BGP on a partitioned topology according to different scenarios. Finally, we conclude on the feasibility of such a simulator by computing the expected additional time required by a distributed simulation of BGP compared to its sequential simulation.

I. INTRODUCTION

The Internet evolution pushes its routing system to its limits in terms of i) memory cost due to the size of the routing tables (RT); ii) communication cost or complexity; iii) Border Gateway Protocol (BGP) complexity. Routing research has investigated new routing paradigms to address these issues. However, simulation of stateful distributed routing protocols operating asynchronously becomes a real issue at large-scale (order of 10k nodes) [START_REF] Yaun | Large scale network simulation techniques: Examples of TCP and OSPF models[END_REF]. No simulator provides the means to measure and to characterize the performance and behavior of such routing protocols on large networks and to compare them with BGP on the same simulation environment. For this purpose, we study the extension of DRMSim [START_REF]DRMSim simulator[END_REF] in order to support the distribution of the routing model by partitioning the topology with respect to its properties and by extending the communication model. This paper is organized as follows. After describing in Section II the state-of-the-art in routing model simulation and DRMSim, we detail two distributed models together with their associated communication overhead. In Section IV, we describe our simulation scenarios and execution environment followed by the simulation results as well as the impact of topology partition on the communication and their analysis. Finally, we conclude on the feasibility of such a simulator in Section V.

II. STATE OF THE ART

We can distinguish three classes of routing simulators: i) protocol simulators dedicated to the performance measurement and analysis of the routing protocol (procedures and format) at the microscopic level such as NS [START_REF]The Network Simulator[END_REF] and SSFNet [START_REF]Scalable Simulation Framework (SSF[END_REF]; ii) configuration simulators dedicated to the simulation of BGP protocol specifics like SimBGP [START_REF] Simbgp | a simple BGP simulator[END_REF] and C-BGP [START_REF] Quoitin | Modeling the routing of an autonomous system with C-BGP[END_REF]; iii) routing model simulators like DRMSim [START_REF]DRMSim simulator[END_REF] that do not execute the protocol low level procedures but their abstraction. Designed for the investigation of the performance of dynamic routing models on large-scale networks, these simulators allow execution of different routing models and enable comparison of their resulting performance.

Routing model simulators require specification of an abstract procedural model, data model, and state model sufficiently simple to be effective on large-scale networks but still representative of the actual protocol execution. However, incorporating (and maintaining up to date) routing state information is technically challenging because of the amount of memory required to store the data associated to each state.

DRMSim implements the Discrete-Event Simulation (DES) approach. It addresses the previous issue by means of efficient graph-based data structures. It allows us to defined simulation scenarios where topology dynamics like link/router failures are considered. Several models can be easly developed and integrated with their own metric model for performance measurement. The provided set of models includes BGP, the Routing Information Protocol (RIP), and compact routing schemes such as NSR [START_REF] Nisse | Distributed computing of efficient routing schemes in generalized chordal graphs[END_REF] and AGMNT [START_REF] Abraham | Compact name-independent routing with minimum stretch[END_REF].

III. DISTRIBUTION MODELS

Implementing DRMSim as a distributed parallel discrete event simulator implies to distribute routers with their data structures but also the events execution among the different computational resources. The simulation feasibility depends mainly on the number of events to be transmitted between each logical processes (LP) and the available bandwidth between them.

The main problems observed with the non-distributed version of DRMSim are:

Routers and BGP peering sessions are distributed among the different logical processes. We model the system by a graph G(V, E), where the set of vertices V is partitioned and every partition is managed by a logical process. The edges internal to a partition are known only by the corresponding LP. For edges with end-points in distinct partitions, there are two possible solutions.

• Solution A: update events including the corresponding entries are transmitted between LPs. • Solution B: boundary edge end-points are duplicated, implying that the original vertices (as an example v and w in Fig. 1 must synchronize their copies (v and w ). The advantage of Solution B compared to Solution A, is that only modified RT entries are sent, thus reducing the communication cost. However, duplication of vertices (routers) may be very harmful if the cuts resulting from the partitioning algorithm have many edges, reducing in turn the memory gain.

A. Partitioning algorithms and complexity

We assume that BGP routing updates constitute the main problem to solve, other type of events lead to negligible effects. For convenience, we use the following notations:

• G(V, E) the graph representing the network topology;

• The vertices of the graph are partitioned:

V = ∪ K i=1 V i ; • N V (v) the neighbors set of a vertex v on the subgraph induced by V ; • M E(v)
the set of modified RT entries at the router modeled by v ; We use mixed integer programs (MIP) to compute optimal bipartitions of the topology by minimizing the communication Fig. 1. Two solutions to allocate edges with end-points on different LPs overhead. In Solution A, we assign to every edge uv ∈ E(G) a weight W uv that approximates the number of entries exchanged on this edge. The sum of weights over all edges having end points in different subsets is minimized. In Solution B, we assign to every vertex v ∈ V (G) a weight W v that approximates |M E(v)|. In this case, the sum of weights over all vertices having at least one neighbor in the foreign subset is minimized.

• e(v, V ) = 1 if v is not in V and

IV. EXECUTION

Our objective is to determine the communication overhead of BGP caused by a distributed parallel implementation of DRMSim. We count the number of BGP update messages exchanged at the boundary edges of the topology partitions. The communication time overhead is derived by estimating the transmission and the propagation time for a single BGP update message between two LPs.

A. Simulation scenarios

We first consider MinRouteAdvertisementInterval (MRAI) = 0s in our experiments. It represents an upper bound on the amount of communication between BGP routers. Indeed, in the Internet, the default MRAI value is set to 30s in order to limit the message rate between BGP peers.

The BGP peering session establishment delay together with the update propagation delay between routers play a major role in the amount of transmitted updates. Three scenarios have been elaborated:

• Scenario 1: considers BGP peering sessions establishment before the start of updates exchanges. Once sessions are established, received updates are executed in their scheduled order. • Scenario 2: considers Scenario 1. However, upon reception, updates are executed in a random order simulating highly random communication delay. • Scenario 3: after one peering session establishment, the resulting updates are executed. Then, the next scheduled BGP peering session is established.

These scenarios are executed once on topologies of 2.5k, 3k, 3.5k, 4k, 4.5k and 5k nodes. These topologies are generated according to the Generalized Linear Preferential (GLP) model parameterized as specified in [START_REF] Bu | On distinguishing between Internet power law topology generators[END_REF].

B. Execution environment

Sequential simulations of BGP to derive inter-partitions communication have been executed with DRMSim on an Intel Xeon 3.20Ghz with 64GB of RAM. Execution of Scenario 1 took 52 minutes (min), Scenario 2 48min and Scenario 3 16min on topologies of 5k nodes.

C. Simulation results

We first measure to obtain reference values, the number of BGP updates with their respective number of entries on nonpartitioned topologies according to the different scenarios. 1) BGP reference results on non partitioned topologies: From Table I on non partitioned topologies, the number of BGP update entries increases drastically. If the communication overhead between LPs behaves similarly, then the time to perform simulations becomes excessive. This observation shows that setting MRAI value to 0s leads to detrimental effects in terms of routing convergence time. We also observe that the increase of BGP update entries over the topology size is linear in their root square, allowing us to extrapolate this number for the topology size of interest (100k nodes).

2) BGP results on bipartitioned topologies: We compute optimal bipartitions of the topologies according to the methodology presented in Section III-A with Solution A and B. As optimal bipartitions, we know that the measured communication give us a lower bound on the overhead.

In Solution A, for the three scenarios and a topology of 5k routers, around 10% of the total number of updated entries transit between partitions. In Solution B, we observe for the three scenarios that the number of transmitted entries improves compared to Solution A. Only 5.6% of the reference updated entries have to be transmitted with Scenario 1, 5% for Scenario 2 and 8% for Scenario 3.

3) Communication overhead: We measure the total size of updated entries transmitted for the different scenarios and topologies. The average size of each entry is 1 for Scenario 1 and between 2.8 and 4.5 for Scenarios 2 and Scenario 3. We also need to determine the time to transmit an updated entry between two LPs. Considering 1Gbps link using TCP/IP, we obtain an average of 0.26ms for each packet (over 1000 packets sent one after the other) to reach its destination.

With Solution A applied to Scenario 1 on a topology of 5k routers, we can expect 45min to transmit all updated entries between the LPs. By extrapolation, we can expect around 91min of overhead for a topology of 10k routers and 15hours for 100k routers. For the two other scenarios executed on a topology of 5k routers, Scenario 2 cumulates an overhead of 110min and Scenario 3 remains in the same order of time as Scenario 1. By extrapolating Scenario 2 overhead, we can expect about 4h of overhead for a topology of 10k routers and about 36hours for 100k routers.

V. CONCLUSION

The expansion of the Internet and its dynamics require to handle the evolutions of its routing system for the next decade. Moving DRMSim to a Distributed Parallel Discrete Event simulator seems to provide a promising technique to make abstraction of the topology size although it induces communication overhead between logical processes. Therefore, this paper quantifies the expected additional time needed to distribute the simulation of BGP on topologies representative of this growth, i.e., 10 to 100k. For this purpose, we have identified BGP updates as the main cause of communication between nodes. We computed the number of updates and their respective number of entries for different BGP execution scenarios. Then, we have computed the best bipartition of each topology -by using the previous measured number of updates -to derive the amount of communication (the overhead) between partitions. It appears that for Scenario 1, distributing the simulation between partitions requires an additional time ranging from 91min (10k nodes) to 15hours (100k nodes). As shown by the Scenario 2 and Scenario 3, this additional time clearly depends on the execution scenario and can reach up to 36hours for topologies of 100k nodes. Decreasing the number of updates can be achieved by increasing the MRAI time to a value higher than 0. As the MRAI increases, the number of BGP updates decreases significantly (in the order of factor 10), until the convergence time reaches an average minimum. The critical issue becomes thus to determine the optimal MRAI value, for which the decrease in update rate would not increase the convergence time while limiting the amount of additional required memory. In order to further reduce the overhead, we have proposed to synchronize boundary nodes between partitions. By applying this method, we were able to save half of the overtime when considering Scenario 1 and 2. Consequently, such a distributed parallel simulator seems thus feasible, leading in turn to the next step of our work being the extension of our routing model simulator DRMSim according to the proposed distribution models.

  has a neighbor in V , and 0 otherwise; • Esize the average size of a RT entry. Solution A implies no internal communication overhead, but leads to k i=1 v∈Vi (|N V \Vi (v)|.|M E(v)|) communication between LPs. In Solution B, as boundary nodes are duplicated, there are k i=1 v∈Vi [( k j=1 e(v, V j )).|M E(v)|] internal updates, but the communication between LPs is negligible since we only need to send an update event containing the identifiers of both routers. There is no memory overhead in Solution A compared to the |V |.

  v, V j ))] bits that are needed in Solution B to store the duplicated boundary nodes.

• Routing tables (RT): let n be the number of routers in the network and k the size of a routing entry. The memory needed for storing all the routing tables is O(k.n 2 ). • Entries updates: event with the highest number of occurrences. We always start from empty routing tables.