É Thiébaut
email: thiebaut@obs.univ-lyon1.fr

Image Reconstruction in Optical Interferometry

Keywords:

Inverse problem approach is a suitable framework to analyze the challenging issues in image reconstruction from interferometric data. It can be exploited to describe and formally compare the new methods specifically developed for optical interferometry.

Optical multi-telescope interferometers provide sparse measurements of the Fourier transform Îλ (ν ) of I λ (θ ), the specific brightness distribution of the observed object in angular direction θ and at wavelength λ. Indeed, at instant t, the complex visibility of the fringe pattern of interferences between the telescopes j 1 and j 2 is:

V j 1 , j 2 (t) = g j 1 (t) g j 2 (t) Îλ ν j 1 , j 2 (t) with ν j 1 , j 2 (t) = r j 2 (t) -r j 1 (t) λ , (1)
with g j (t) and r j (t) the complex amplitude transmission and the projected position (on a plane perpendicular to the line of sight) of the j-th telescope. Averaging during an exposure (assumed short enough to neglect the Earth rotation) yields:

V j 1 , j 2 ,m = 〈V j 1 , j 2 (t)〉 m 〈g j 1 (t) g j 2 (t)〉 m Îλ (ν j 1 , j 2 ,m ) with ν j 1 , j 2 ,m = 〈ν j 1 , j 2 (t)〉 m , (2)
where 〈〉 m denotes time averaging during m-th exposure. At optical wavelengths, the optical path variations due to the turbulence involve that 〈g j 1 (t) g j 2 (t)〉 m 0. In other words, complex visibilities cannot be directly measured. To overcome this issue, other quantities such as the powerspectrum and the bispectrum are measured:

S j 1 , j 2 ,m = 〈|V j 1 , j 2 (t)| 2 〉 m ρ j 1 ,m ρ j 2 ,m | Îλ (ν j 1 , j 2 ,m )| 2 , (3)
B j 1 , j 2 , j 3 ,m = 〈V j 1 , j 2 (t) V j 2 , j 3 (t) V j 3 , j 1 (t)〉 m ρ j 1 ,m ρ j 2 ,m ρ j 3 ,m Îλ (ν j 1 , j 2 ,m ) Îλ (ν j 2 , j 3 ,m ) Îλ (ν j 3 , j 1 ,m ), (4)
where ρ j,m = 〈|g j (t)| 2 〉 m = 0 can be calibrated from photometric channels and a reference source. Of particular interest is the phase of the bispectrum, the so-called phase closure, since it is more robust to calibration errors.

Image Synthesis

The first problem in image reconstruction from interferometric data is to cope with voids in the sampled spatial frequencies (see Fig. 1a for a typical coverage of the measured spatial frequencies). At optical wavelengths, working with the powerspectrum and bispectrum data gives less Fourier phase information and introduces nonlinearities and degeneracies which make the image synthesis problem much more difficult. Nevertheless and without loss of generality, image synthesis from interferometric data is an inverse problem [START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF] that can be restated as a constrained minimization [START_REF] Thiébaut | Image reconstruction with optical interferometers[END_REF][START_REF] Thiébaut | Image Reconstruction in Optical Interferometry[END_REF] to find the best image parameters, e.g. the pixel intensities, x :

x + = argmax x ∈ f data (x , y) + f prior (x )
with, e.g., = x |x ≥ 0,

k x k = 1 (5)
with f data (x , y) a likelihood term (usually derived from the statistics of the noise) which enforces agreement with the measurements y, and f prior (x ) a regularization term which imposes some regularity, e.g. smoothness, for the 6); (e) reconstruction with edge-preserving regularization as in Eq. ( 7); (f) reconstruction with a quadratic regularization given by Eq. ( 8) and which imposes a compact field of view. All reconstructions by algorithm MiRA [START_REF] Thiébaut | MiRA: an effective imaging algorithm for optical interferometry[END_REF] and from the powerspectrum and the phase closures.

sought image. Other strict constraints such as the non-negativity and normalization are imposed by requiring that x belongs to a feasible set . In this general framework, an image reconstruction algorithm is specified by: (i) the image representation (e.g. pixels, building-blocks [START_REF] Hofmann | Iterative image reconstruction from the bispectrum[END_REF], . . . ) and its Fourier transform; (ii) the approximation of the likelihood (e.g. [START_REF] Meimon | Convex approximation to the likelihood criterion for aperture synthesis imaging[END_REF]); (iii) the imposed constraints (type of regularization, non-negativity, normalization, etc.); and (iv) the optimization strategy. Different choices for these ingredients have led to a variety of algorithms. For instance, Maximum Entropy Methods (MEM) [START_REF] Narayan | Maximum entropy image restoration in astronomy[END_REF] readily solve a problem like the one Eq. ( 5) with a regularization:

f prior (x ) = n x n log(x n /x n ) -x n + xn (6)
with x the default image (cf. Fig. 1d). BSMEM [START_REF] Buscher | Direct maximum-entropy image reconstruction from the bispectrum[END_REF] is a version of MEM applied to bispectral data. CLEAN [START_REF] Högbom | Aperture Synthesis with a Non-Regular Distribution of Interferometer Baselines[END_REF] uses a matching-pursuit algorithm to minimize the number of point-like sources that compose the image. This is very like the sparsity constraints used in Compressive Sensing [START_REF] Candes | Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information[END_REF] and expressed as: f prior (x ) = x 1 to minimize the 1 -norm of the parameters. The building-block method [START_REF] Hofmann | Iterative image reconstruction from the bispectrum[END_REF] can be seen as an adaptation of CLEAN to bispectral data. Imposing the sparsity of the spatial gradient of the image, a method known as Total Variation (TV) [START_REF] Strong | Edge-preserving and scale-dependent properties of total variation regularization[END_REF], or an edge preserving smoothness regularization is obtained with:

f prior (x ) = n ∇ n x 2 + ε 2 -ε , (7)
with = 0 for TV, > 0 otherwise and ∇ n x the image gradient at n-th pixel. When → ∞, Eq. ( 7) yields quadratic smoothness which is the regularization used by WIPE [START_REF] Lannes | Weak-phase imaging in optical interferometry[END_REF]. More recently, a simple separable quadratic regularization has proven to be successful for optical interferometry imaging [START_REF] Besnerais | Advanced Imaging Methods for Long-Baseline Optical Interferometry[END_REF]:

f prior (x ) = n w n x 2 n , (8)
with the weights w n ≥ 0 set so as to favor the concentration of the light near the center of the field of view and hence the smoothness of Îλ (ν ). This flexibility is exploited by versatile algorithms like MiRA [START_REF] Thiébaut | MiRA: an effective imaging algorithm for optical interferometry[END_REF] or WISARD [START_REF] Meimon | Reconstruction method for weak-phase optical interferometry[END_REF] which implement various kind of regularizations and can therefore emulate other algorithms (their however differ in their optimization strategies). Comparing the results obtained with different regularizations (cf. Fig. 1df) shows that the reconstructed images are very similar and close to the true object. This consistency is reassuring but deserves a quantitative comparison on the impact of the type of regularizations on a variety of cases.

An other advantage of the inverse approach is that the problem in Eq. ( 5) yields a solution whatever is the amount of missing data (though the less data the less information is gained with respect to the priors). For instance, apart for a 180 • ambiguity in the orientation, an image may be synthesized without any measured Fourier phase information (see Fig. 1c for an example of reconstruction).

For the moment, no algorithms really exploit the full potential of the multi-spectral data provided by optical interferometers. Either the object is considered as gray with the same distribution at all wavelengths, or a monochromatic image is synthesized for each spectral channel independently [START_REF] Le Bouquin | Pre-maximum spectro-imaging of the Mira star T Lep with AMBER/VLTI[END_REF]. The most exciting developments in image reconstruction algorithms for the years to come will be to use the spectral continuity of the sought brightness distribution to globally process all available data to produce a full 3-D distribution I λ (θ ).

Fig. 1 :

 1 Fig. 1: (a) Spatial frequency coverage. (b) Original object smoothed to the resolution of the interferometer (FWHM ∼ 15mas). (c) Image reconstruction without any Fourier phase information. Image reconstruction with various types of regularization: (d) reconstruction with maximum entropy regularization as in Eq. (6); (e) reconstruction with edge-preserving regularization as in Eq. (7); (f) reconstruction with a quadratic regularization given by Eq. (8) and which imposes a compact field of view. All reconstructions by algorithm MiRA[START_REF] Thiébaut | MiRA: an effective imaging algorithm for optical interferometry[END_REF] and from the powerspectrum and the phase closures.


Joanna Norton
email: joanna.norton@inserm.fr

PhD Marie-Laure Laure Ancelin

PhD Rob Stewart

PhD Claudine Berr

PhD Karen Ritchie

PhD Isabelle Carrière

Anxiety symptoms and disorder predict activity limitations in

Keywords: anxiety, disability, aged, longitudinal studies

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Anxiety is the most prevalent psychiatric disorder affecting one in four persons across their lifetime [START_REF] Kessler | Lifetime prevalence and age-of-onset distributions of DSM-IV disorders in the National Comorbidity Survey Replication[END_REF][START_REF] Ritchie | Prevalence of DSM-IV psychiatric disorder in the French elderly population[END_REF]. Prevalence estimates in later life range from 1.2% to 14.2% [START_REF] Bryant | The prevalence of anxiety in older adults: methodological issues and a review of the literature[END_REF][START_REF] Wolitzky-Taylor | Anxiety disorders in older adults: a comprehensive review[END_REF] depending on sampling procedure, diagnosis tools used, age cut-offs for classification as elderly and exclusion or not of anxiety cases due to general medical conditions.

As in younger adults, anxiety and depression are frequently comorbid in older people. [START_REF] Cairney | Comorbid depression and anxiety in later life: patterns of association, subjective wellbeing, and impairment[END_REF] found in a large community-based sample aged 55 and older, that 23% of those with anxiety disorders also met diagnosis criteria for major depressive disorder using the World Mental Health Composite International Diagnostic Interview [START_REF] Cairney | Comorbid depression and anxiety in later life: patterns of association, subjective wellbeing, and impairment[END_REF].

While depression has been consistently demonstrated to be an independent risk factor for functional loss in the elderly [START_REF] Carriere | Late life depression and incident activity limitations: influence of gender and symptom severity[END_REF][START_REF] Carriere | Modelling complex pathways between late-life depression and disability: evidence for mediating and moderating factors[END_REF][START_REF] Schillerstrom | Depression, disability and intermediate pathways: a review of longitudinal studies in elders[END_REF], little is currently known about the association between anxiety disorders and activity limitations: previous research being largely based on cross-sectional observations [START_REF] Brenes | Anxiety, depression and disability across the lifespan[END_REF][START_REF] Cairney | Comorbid depression and anxiety in later life: patterns of association, subjective wellbeing, and impairment[END_REF][START_REF] De Beurs | Consequences of anxiety in older persons: its effect on disability, wellbeing and use of health services[END_REF][START_REF] Porensky | The burden of late-life generalized anxiety disorder: effects on disability, health-related quality of life, and healthcare utilization[END_REF][START_REF] Sareen | Disability and poor quality of life associated with comorbid anxiety disorders and physical conditions[END_REF]. To date only two prospective studies have been undertaken. In a sample of functionally limited women aged 65 and above, anxiety symptoms predicted activities of daily living (ADL) limitations and mild housework disability over a 3-year period [START_REF] Brenes | The influence of anxiety on the progression of disability[END_REF], and in a one-year prospective study of adults aged 72 and above, anxiety symptoms remained associated with incident ADL limitations after adjustment for physical functioning [START_REF] Tinetti | Shared risk factors for falls, incontinence, and functional dependence. Unifying the approach to geriatric syndromes[END_REF]. Previous longitudinal research has been limited due to the absence of adequate anxiety disorder diagnosis, failure to take into account common cooccurence with depression, and the use of partial dependency measures incorporating different items from ADL or instrumental activities of daily living (IADL) scales over short follow-up periods.

Another point which has not always been considered in previous analyses concerns confounding by other mental health factors, especially depression and somatic comorbidity which are common in the elderly. Symptoms such as dizziness, chest pain, shortness of breath, tiredness, exhaustion, palpitations and nausea in anxiety disorders also occur in common pathologies such as heart disease, chronic obstructive pulmonary disease, postural hypotension, thyroid disease in conjunction with some medications taken and as part of the general frailty syndrome [START_REF] Blazer | Generalized anxiety disorder and panic disorder in the elderly: a review[END_REF]. This may thus render difficult the diagnosis of anxiety especially when considering that the DSM-IV diagnosis of generalised anxiety disorder (GAD) cannot be established if it is the consequence of medical conditions. However, ruling out an organic disorder or a drug effect is often difficult in the elderly.

Anxiety pharmacotherapy also presents challenges. Benzodiazepines are the most frequently prescribed anxiolytic medication in older adults, of which approximately half are long half-life agents [START_REF] Gray | Benzodiazepine use and physical disability in community-dwelling older adults[END_REF]. However, they are associated with a number of adverse risks, especially those with a long half-life, including cognitive impairment, psychomotor impairment, excessive daytime sedation, instability of gait, falls, and hip fractures [START_REF] Gray | Is the use of benzodiazepines associated with incident disability?[END_REF][START_REF] Gray | Benzodiazepine use and physical disability in community-dwelling older adults[END_REF][START_REF] Ried | Benzodiazepine exposure and functional status in older people[END_REF]. The effect of anxiety on dependency may thus be confounded by benzodiazepine side-effects.

The purpose of this study, using data from a large cohort of community-living elderly persons allowing adjustment for a number of confounders (e.g. sociodemographic characteristics, physical and mental health, medications), is firstly to evaluate the 7-year longitudinal association between anxiety measured both with a standardized psychiatric interview and a self-reported symptom scale and activity limitations, and secondly to evaluate if the association persists independently of depressive symptoms.

METHOD

Study sample

The design of the ESPRIT study and participant recruitment have been published in detail elsewhere [START_REF] Ritchie | Prevalence of DSM-IV psychiatric disorder in the French elderly population[END_REF]. Participants were recruited over a 2-year period from Of the 2189 dementia-free participants included in the study, 141 were missing all follow-up evaluations for activity limitations (of whom 71 died) and 467 had missing data for at least one baseline adjustment variable. The present analysis was thus conducted on 1581 subjects.

Compared to the analysed sample, those not included in this analysis were more likely to be older (p<0.0001), have a lower level of education (p<0.006), have a lower level of income (p=0.002), be depressed (p<0.0001), have activity limitations (p<0.0001), and have visual (p=0.0008), hearing (p<0.0001) and cognitive impairment (p<0.0001). There was no significant difference regarding chronic disease status, intensity of anxiety symptoms or frequency of diagnosed anxiety.

Disability measures

As outcomes, we chose three disability measures widely used in elderly populations, in order to reflect varying forms of disability.

Social restriction was measured on a four-point scale ranging, by decreasing level of severity, from confinement to bed, to home, to the neighbourhood, to no restriction.

According to the World Health Organisation International Classification of Functioning, Disability, and Health (ICF, 2001), this can be considered as participation restriction resulting from personal functional and activity limitations and environmental factors. Subjects were classified as socially restricted if they reported any type of confinement.

With regard to activity limitations, mobility was assessed using the Rosow and Breslau scale [START_REF] Rosow | A Guttman health scale for the aged[END_REF] which assesses ability to perform physical tasks requiring mobility and strength such as doing heavy housework, walking half a mile, and climbing stairs, with binary yes/no responses for each item. The Lawton-Brody IADL scale was used to evaluate skills necessary to live independently such as the ability to use the telephone, to manage medication and money, to use public or private transport, to shop, and for women only, to prepare meals and do housework and laundry, with limitation severity response scales varying from three to five points [START_REF] Lawton | Assessment of older people: self-maintaining and instrumental activities of daily living[END_REF]. Compared to the Rosow and Breslau scale, this scale investigates daily activities which implicate both physical and cognitive abilities. For both types of activity limitations, participants indicating inability to perform one or more activities without help were considered as having mobility or IADL limitations.

Anxiety disorders

At baseline, a standardized psychiatric examination the Mini International Neuropsychiatry Interview (MINI), was used to investigate lifetime and current psychiatric disorder [START_REF] Sheehan | The Mini-International Neuropsychiatric Interview (M.I.N.I.): the development and validation of a structured diagnostic psychiatric interview for DSM-IV and ICD-10[END_REF]. This interview has been previously validated in France and widely used in many clinical and research settings (Lecrubier et al., 1997). The interview was administered by nurses and psychologists trained using video recordings of interviews by the clinicians responsible for the development of the French version of the tool. The MINI provides an extensive symptomatological examination, with diagnostic algorithms applied to assess 'caseness'. With regard to anxiety disorders, it provides diagnoses of both current and past GAD, social phobia, agoraphobia, obsessive compulsive disorder (OCD), panic and posttraumatic disorder(PTSD), according to DSM-IV criteria. Cases detected by the MINI were reviewed by a panel of psychiatrists to validate the initial diagnosis [START_REF] Ritchie | Prevalence of DSM-IV psychiatric disorder in the French elderly population[END_REF]).

Spielberger's State-Trait Anxiety Inventory (STAI), a well-established and validated self-report measure of anxiety, was used to measure trait anxiety [START_REF] Spielberger | Manual for the Sate-Trait anxiety Inventory (Form Y)[END_REF]. The scale is composed of 20 items, each rated on a four-point intensity scale. Overall scores range from 20 to 80. Different cut-offs are available in the literature for the elderly but none of them have been validated [START_REF] Devier | The impact of anxiety on conversion from mild cognitive impairment to Alzheimer's disease[END_REF][START_REF] Himmelfarb | The prevalence and correlates of anxiety symptoms in older adults[END_REF][START_REF] Tinetti | Shared risk factors for falls, incontinence, and functional dependence. Unifying the approach to geriatric syndromes[END_REF]. Himmelfarb (1983) established a cut-off score of 44+ as best discriminating between a hospitalised clinical sample and a non-hospitalised community sample of older persons [START_REF] Himmelfarb | The prevalence and correlates of anxiety symptoms in older adults[END_REF]. [START_REF] Devier | The impact of anxiety on conversion from mild cognitive impairment to Alzheimer's disease[END_REF] used a previously reported threshold of 30+ in studying conversion in patients from MCI to Alzheimer's disease [START_REF] Devier | The impact of anxiety on conversion from mild cognitive impairment to Alzheimer's disease[END_REF] and [START_REF] Tinetti | Shared risk factors for falls, incontinence, and functional dependence. Unifying the approach to geriatric syndromes[END_REF]) used Spielberger's (1983) normative mean score value of 32 in 50-69 year old adults to dichotomise the score [START_REF] Tinetti | Shared risk factors for falls, incontinence, and functional dependence. Unifying the approach to geriatric syndromes[END_REF]. In the absence of a validated cut-off score, we decided to group STAI trait scores into tertiles and to compare the highest tertile to the lowest two.

Medication

Use of antidepressant and anxiolytic medication in the previous month, validated by presentation of the prescription or medication, was recorded at baseline and type of medication was noted according to the World Health Organization's Anatomical Therapeutic Chemical (ATC) classification (for antidepressant medication: N06A; anxiolytics: N05B, of which short half-life benzodiazepines: N05BA04, N05BA06, N05BA12, N05BA21, and long (>20h) half-life benzodiazepines; N05BA01, N05BA05, N05BA08, N05BA09, N05BA11, N05BA16, N05BA18).

Depressive symptomatology

The MINI was also used for the diagnosis of current and past major depressive episodes (MDE), according to DSM-IV criteria. Severity of depressive symptoms was assessed using the 20-item Center for Epidemiologic Studies-Depression scale (CES-D) [START_REF] Radloff | The CES-D scale: a self-report depression scale for research in the general population[END_REF]. For this analysis, participants were classified into one of three groups [START_REF] Ryan | Late-life depression and mortality: influence of gender and antidepressant use[END_REF]. 'Severe depressive symptomatology' included participants with a current MDE or a CES-D score of 23 or over (allowing for the fact that some participants with severe symptoms did not reach DSM classification criteria, principally because of the duration of symptoms). 'Mild depressive symptomatology' was defined as a CES-D score between 16 and 22 and 'No depressive symptomatology' included participants with a CES-D score lower than 16.

Socio-demographic and clinical variables

The standardized interview included questions on demographic characteristics, level of education (<5 versus 6+ years of education), marital status, mode of living (alone or not) and income as well as an inventory of all drugs (including anxiolytic and antidepressant drugs) used over the preceding month. Information was obtained on type and quantity of alcohol consumption (number of units of alcohol per day: 0-36 versus >36 g/day) and tobacco use (classified as past, present or never users). Detailed medical questionnaires included history of stroke, angina pectoris, myocardial infarction, and cardio-vascular surgery established according to standardized questions, history of cancer and fracture with hospitalisation during the last two years, asthma (attacks during the last year), chronic bronchitis, dyspnoea, hypertension (>160/95 mm Hg or treated), diabetes (fasting glycemia > 7mmol/l or treated) and thyroid disease. Blood pressure was measured during the interview using a digital electronic tensiometer OMRON M4. The number of chronic diseases was calculated including: hypertension, diabetes, cardiovascular diseases, respiratory diseases, dyspnoea, thyroid disease, and cancer.

Cognitive impairment was defined as a Mini Mental State Examination (MMSE) [START_REF] Folstein | Mini-mental state". A practical method for grading the cognitive state of patients for the clinician[END_REF] score lower than 24. Visual impairment was defined as having a corrected near visual acuity (Parinaud scale) of more than 4 or difficulties recognizing a familiar face at 4 meters. Hearing impairment was defined as deafness or only being able to hear a conversation when a single person speaks loudly.

Statistical analyses

The Chi² test was used to compare subjects in the study sample to those excluded. For each indicator of activity limitations longitudinal associations with anxiety disorders were established from subjects free of that activity limitation at baseline. Thus the sample size was 955 for mobility, 1470 for IADL and 1456 for social restriction.

In longitudinal studies, the within-subject responses (repeated evaluations of activity limitations) are correlated. This correlation was accounted for by using a mixed logistic model [START_REF] Carriere | Choosing marginal or random-effects models for longitudinal binary responses: application to self-reported disability among older persons[END_REF]. Briefly, this model has four basic characteristics: (i) the individual time evolutions of activity limitations are entirely taken into account including possible reversion to normal state (ii) subjects with incomplete responses across time are included in the analysis; (iii) subjects do not have to be evaluated at the same time points; (iv) the model allows within-subjects dependency to vary from one subject to another, via the random part of the covariable linear combination. The SAS procedure NLMIXED was used to estimate the parameters (version 9.2).

We used univariate and multivariate models to determine if baseline anxiety was associated with odds of activity limitations or social restriction during follow up. Odds ratios were adjusted for sex, age, time and interaction time*age (Model 0). Multivariate adjusted logistic regression included covariates that were associated with the follow-up responses (p<0.15 in univariate model). Model 1 was adjusted for age, time, time*age, income, educational level and living alone. Model 2 was further adjusted for alcohol, body mass index (BMI), smoking, chronic disease status (three or more chronic diseases versus two or less), cognitive, and visual impairment and hearing impairment. Model 3 was further adjusted for depression (none, mild, severe) and Model 4 for past MDE, antidepressant and anxiolytic medication consumption. Interactions between anxiety and sex, and anxiety and depression were tested for all anxiety-activity limitation combinations. A stratified analysis was also carried out, testing the association between anxiety and activity limitations among subjects with and without depressive symptoms. Finally, concerning specific anxiety diagnoses, models were run for phobic disorders only as the frequency of the other disorders was too low.

RESULTS

Within this sample, 14.2% of the subjects had a least one DSM-IV anxiety disorder at baseline, 10.8% any type of phobia, 4.1% GAD, and less than 1% PTSD, OCD or panic disorder; 25.9% had at least one past anxiety disorder (Table 1). Mean score on the STAI Trait at baseline was 40.1 (SD: 9.1). Scores were dichotomised according to the upper tertile value, which corresponds to a cut-off score of 44. Among patients in the upper tertile defined as having a high level of trait anxiety, 24.1% reached criteria for a DSM-IV anxiety disorder compared to 9.8% in the lower tertiles.

Anxiolytic consumption was reported by 10.3% of the sample -the majority being benzodiazepine drugs (9.4%) of which half (5.1%) with a long (>20h) half-life -and antidepressant consumption by 6.5%. The prevalence of mild depression was 15.5% and severe depression 13.1%. Of the 225 subjects with anxiety disorder at baseline, 17.8% also had mild depression and 27.6% severe depression. Of the 490 subjects with scores in the upper tertile for trait anxiety, 28.6% reached criteria for mild depression and 32.6% for severe depression.

Mobility limitation at baseline was reported by 36.8% of the subjects compared to 4.3% for social restriction and 3.2% for IADL limitations. Cross-sectional associations with baseline anxiety disorder and symptomatology are shown in Table 2.

The analysis of repeated activity limitations over time was undertaken for each scale on the subjects without limitation on the considered scale at baseline. Table 3 shows rates and odds ratios for incident activity limitations over seven years according to the presence or absence of anxiety at baseline, in terms of both trait symptomatology and DSM-IV anxiety disorder. The odds ratios for having incident activity limitations are adjusted for sex, age, time and the interaction between time and age (Model 0). With regard to anxiety symptoms, the probability of developing any type of activity limitations increased significantly for subjects with a high trait symptom level at baseline. Anxiety disorder diagnosis was associated with a significantly increased probability of social restriction and IADL limitation but not mobility limitation.

All mixed logistic models showed a significant effect of age. Time showed a significant effect for IADL limitations only and age-time interaction for social restriction and IADL limitations, reflecting an increased effect of time with age. There was no interaction between anxiety and sex on activity limitations in any of the six mixed logistic regression models.

Table 4 shows the odds ratios for activity limitations according to anxiety at baseline with different levels of adjustment for possible confounders. All of the preceding associations persisted when adjusting for socio-demographic (Model 1) and physical health and lifestyle variables (Model 2). For social restriction, only trait anxiety symptoms remained significant when adding baseline depression (Model 3), baseline anxiolytic medication use and past MDE to the model (Model 4) (OR (95% CI): 2.41 (1.42-4.09)). The probability of incident limitations in mobility according to Trait anxiety symptoms at baseline became non significant when adjusting for further confounders in Models 3 and 4. With regard to IADL limitations, the probability increased significantly for anxiety disorder diagnosis at baseline only with an odds ratio (95% CI) of 1.84 (1.01-3.39) (Model 4).

None of the interaction terms between anxiety and baseline depressive symptomatology, further adjusted for covariates in Model 2, were significant. In a stratified analysis according to the presence or absence of mild and severe depressive symptoms, the association between anxiety disorder and incident IADL limitation remained significant in non-depressed subjects (CES-D <16) at baseline with a higher odds ratio (OR (95% CI) = 3.21 (1.31-7.89), p=0.011) than that obtained for the whole sample (OR (95% CI) = 1.84

(1.01-3.39), p=0.048). The association between baseline trait anxiety and social restriction also remained significant in the sub-sample of non-depressed subjects (OR (95% CI) = 2.92

(1.41-6.05), p=0.004).

None of the interactions between chronic disease status and anxiety (symptoms and disorder) on the 3 activity limitation variables were significant. With regard to specific anxiety disorders, phobia was associated with incident IADL limitations in Model 1(OR (95% CI) = 1.9 (1.06-3.39), p=0.03); however the association became borderline significant when further adjusted on physical health and lifestyle variables (Model 2) (OR (95% CI) = 1.78 (0.99-3.20), p=0.053).

DISCUSSION

This study is one of the first to examine anxiety, both on a trait symptom level and as a diagnosed disorder, as an independent predictor of 7-year incident activity limitations in a representative sample of community-dwelling elderly persons. Our findings suggest both trait anxiety symptoms and disorders meeting strict diagnostic criteria are associated with incident activity limitations. Interestingly, trait anxiety was associated with social restriction, whereas anxiety disorders were associated with more severe IADL limitations. Associations remained significant when adjusting for both past and current depressive symptoms at baseline and in non-depressed subjects only, suggesting that despite the high level of comorbidity between anxiety and depression, anxiety is an independent risk factor for activity limitations.

Anxiety in the elderly

While the association between depression and onset of disability in the elderly has been extensively explored [START_REF] Carriere | Modelling complex pathways between late-life depression and disability: evidence for mediating and moderating factors[END_REF][START_REF] Schillerstrom | Depression, disability and intermediate pathways: a review of longitudinal studies in elders[END_REF] with depression being confirmed as an independent risk factor even when multiple confounding effects are taken into account [START_REF] Carriere | Late life depression and incident activity limitations: influence of gender and symptom severity[END_REF], less is known about the association between anxiety and disability even though the prevalence of anxiety disorder in the elderly appears to be as high and maybe even higher than that of depression. In the present sample, 2.3% of subjects aged 65 and above had a baseline major depressive episode and 13.2% were classified as having severe depressive symptoms, whereas 14.2% had at least one baseline anxiety disorder; there was a three-fold increase in the prevalence of MDE (6.7%) among subjects with at least one anxiety disorder. In the elderly, medical comorbidity is also very frequent and several physical conditions such as cardiac, respiratory and vestibular problems may both be a source of anxiety and be exacerbated by anxiety [START_REF] Wolitzky-Taylor | Anxiety disorders in older adults: a comprehensive review[END_REF].

Cognitive impairment may affect presentation of symptoms, experience of symptoms and the ability to communicate them, rendering difficult the recognition of anxiety. At last it is still unclear whether older anxious persons experience other symptoms (not taken into account in existing assessment tools) than younger persons. Anxiety among older adults has been suggested to take a more somatic form with complaints such as dizziness and shakiness [START_REF] Flint | Anxiety and its disorders in late life: moving the field forward[END_REF].

Anxiety as a risk factor for disability

Most studies of the association between anxiety and disability have been crosssectional, precluding inference about direction of causation. Only two studies have analysed the prospective association between anxiety symptoms (but not anxiety disorder) and disability or its progression in the elderly. In a sample of 1002 functionally limited women aged 65 and over, past week anxiety symptoms identified using the Hopkins Symptom Checklist predicted 3-year ADL and mild housework limitations, but not mobility and lifting disability [START_REF] Brenes | The influence of anxiety on the progression of disability[END_REF]. Models were adjusted for potential confounders (eg. number of chronic diseases and depressive symptoms) and mediators (eg. benzodiazepine and psychotropic medication use). In a one-year follow-up of adults aged 72 and over, [START_REF] Tinetti | Shared risk factors for falls, incontinence, and functional dependence. Unifying the approach to geriatric syndromes[END_REF] examined anxiety symptomatology measured by Spielberger's Trait Inventory applying the normative average score of 32 among 764 working adults aged 50-69 years as the cut-off [START_REF] Spielberger | Manual for the Sate-Trait anxiety Inventory (Form Y)[END_REF]. Anxiety remained a significant predictor of functional dependence after adjusting for other predisposing factors (lower extremity and upper extremity impairment and sensory impairment), incontinence and falling [START_REF] Tinetti | Shared risk factors for falls, incontinence, and functional dependence. Unifying the approach to geriatric syndromes[END_REF]. However, neither of these studies excluded from the analysis subjects with functional dependence or activity limitations at baseline precluding the distinction between a crosssectional and longitudinal effect. In the current study we not only examined incident disability over a longer time-period (7-years) than previous studies, but for each type of limitation investigated we excluded subjects with the condition at baseline thus giving accurate estimates of predictive risks.

Anxiety symptoms versus disorder

Our findings suggest that both anxiety symptoms and disorder are risk factors for disability, with anxiety symptoms associated with a less severe form of disability: a high level of trait anxiety was associated with incident social restriction, whereas anxiety disorder was associated with onset of IADL limitations. A tendency was also found for an association between trait anxiety symptoms and mobility limitations. Contrary to De Beurs' crosssectional study where current sub-syndromal and DSM-IV anxiety disorder were considered as two mutually-exclusive categories based on strict diagnostic criteria [START_REF] De Beurs | Consequences of anxiety in older persons: its effect on disability, wellbeing and use of health services[END_REF], we used two different approaches to measure anxiety which are likely to capture different forms of anxiety. In fact, only 52.4% of patients with anxiety disorder had a high level of trait anxiety (defined as the overall top tertile) indicating only partial overlap. There are many possible explanations for this. Firstly, trait anxiety measures long-lasting persistent symptoms referring to how the subject usually feels, with responses given on a four-point severity scale and no specific time-frame whereas anxiety disorders refer to symptoms experienced over the past month (or six months for GAD), with binary responses and filter questions. Secondly, Spielberger's STAI is a self-report instrument whereas the MINI is administered by a trainedlay interviewer, each approach introducing different types of bias. It can tentatively be argued that these different forms of anxiety will trigger different types of disability. Subjects with underlying anxiety symptoms may be more likely to become social restricted as a result of actual physical limitations or through fear of falling and going out. On the other hand, anxiety disorder would lead to dependency for specific daily living skills through a combination of reduced physical and cognitive abilities.

Our findings stress the importance of considering underlying trait anxiety in the elderly, anxiety symptoms not necessarily fulfilling diagnostic criteria also causing distress and activity restriction.

With regard to specific anxiety disorders, the association between phobia and incident IADL limitations became non significant when adjusting for physical and mental health disorders. Given that two-thirds of persons with anxiety disorder at baseline had phobic disorder, the lack of association probably results from a loss of statistical power.

Depressive symptoms as a potential confounder

The complex relationship between anxiety and depression, with shared symptoms and high levels of comorbidity especially at low symptom levels, warrants a study of the independent association of both anxiety symptoms and disorder with disability. In the two prospective studies of anxiety symptoms, [START_REF] Brenes | The influence of anxiety on the progression of disability[END_REF] adjusted their analysis on depression at baseline only, whereas [START_REF] Tinetti | Shared risk factors for falls, incontinence, and functional dependence. Unifying the approach to geriatric syndromes[END_REF] selected anxiety symptoms over depressive symptoms as a predisposing risk factor for three outcomes: falls, incontinence and functional dependence [START_REF] Brenes | The influence of anxiety on the progression of disability[END_REF][START_REF] Tinetti | Shared risk factors for falls, incontinence, and functional dependence. Unifying the approach to geriatric syndromes[END_REF]. In the latter study, the selection which aimed to reduce the number of covariates actually eliminated depressive symptoms in the final model based on a slightly higher risk for anxiety symptoms. Our findings remain significant when adjusting for both current and past depression at baseline and when restricting the analysis to subjects free of mild and severe depression at baseline, suggesting that anxiety is an independent predictor of disability.

Other confounding and mediating effects

Anxiolytic use, particularly benzodiazepines, is a known risk factor for disability and was entered as an adjustment variable in the final models. It remained a significant and independent predictor of mobility and IADL limitations, but not of social restriction.

Associations between benzodiazepines use and onset of mobility and ADL limitations have previously been found [START_REF] Gray | Benzodiazepine use and physical disability in community-dwelling older adults[END_REF] and benzodiazepine use in the past year has been shown to be associated with both ADL and IADL limitations [START_REF] Ried | Benzodiazepine exposure and functional status in older people[END_REF]. Although many possible confounding variables such as antidepressant use have been taken into account, the observed associations for mobility and IADL limitations are likely to be more strongly related to underlying symptoms associated with anxiolytic use rather than with use itself: an independent effect of anxiolytic use on activity limitations cannot be ruled out but the mechanisms involved are likely to be complex, especially over a 7-year follow-up period.

Taking into account both antidepressant and anxiolytic use in the final model, the relationships between trait anxiety and social restriction, and anxiety disorder and IADL limitations remain significant, suggesting medication use would only partially explain the associations.

Analyses were also adjusted for all other baseline covariables significantly associated with incident activity limitations. The associations of trait anxiety symptoms with social restriction and anxiety disorders with IADL remained significant after multiple adjustments indicating their robustness. In particular we adjusted for the main chronic diseases and health behaviours such as smoking and alcohol, which did not substantially change the estimation of the anxiety effect. We also adjusted for sensory impairments which have been demonstrated to be predictors of both functional disability and anxiety [START_REF] Crews | Vision impairment and hearing loss among community-dwelling older Americans: implications for health and functioning[END_REF][START_REF] Mehta | Prevalence and correlates of anxiety symptoms in well-functioning older adults: findings from the health aging and body composition study[END_REF].

Potential mechanisms for the relationship between anxiety and incident activity limitations

The mechanisms by which anxiety may lead to activity limitations are likely to be complex involving physical and psychiatric comorbidity, cognitive and sensory impairments, health behaviours and biological factors. Mechanisms have previously been reported for the association between depression and incident activity limitations [START_REF] Carriere | Late life depression and incident activity limitations: influence of gender and symptom severity[END_REF] but it is unclear to what extent they also apply to anxiety symptoms and disorder. Anxiety and depression have been shown to have different relationships to mortality [START_REF] Mykletun | Levels of anxiety and depression as predictors of mortality: the HUNT study[END_REF] which suggests the mechanisms may differ. Early signs of cognitive impairment are likely to induce anxiety. Yet the relationships between anxiety and activity limitations found in our study remained significant after adjusting for cognitive impairment at baseline with no significant interaction between cognitive impairment and anxiety. It is possible that the threshold for cognitive impairment was set too high (MMSE score <24) which identified as impaired only 3.3% of the sample and not those with early signs of impairment.

Brenes et al examined physical activity, benzodiazepine and psychotropic medication use and lack of emotional support, as possible mediators for the effect of anxiety on disability [START_REF] Brenes | The influence of anxiety on the progression of disability[END_REF]. Their findings failed to prove support for these three variables.

Although we did not examine the effect of mediating factors, our results suggest anxiolytic use in the month preceding baseline plays a limited role in explaining this relationship.

Another possible mechanism could be through the somatic symptoms of anxiety such as dizziness and shaking which affect functional capacity and thus the ability to carry out everyday functions [START_REF] Blazer | Generalized anxiety disorder and panic disorder in the elderly: a review[END_REF][START_REF] Flint | Anxiety and its disorders in late life: moving the field forward[END_REF]. This may hold for anxiety disorders but is unlikely to explain the relationship between underlying long-term trait anxiety and social restriction.

Comorbidity with physical diseases is a likely candidate for explaining the association between anxiety and incident activity limitations. For example, anxiety is often comorbid with hypertension [START_REF] Mehta | Prevalence and correlates of anxiety symptoms in well-functioning older adults: findings from the health aging and body composition study[END_REF] and heart disease [START_REF] Wolitzky-Taylor | Anxiety disorders in older adults: a comprehensive review[END_REF], both of which could result in increased activity limitations. This explanation is unlikely though, as we controlled for the main chronic diseases at baseline in our study. However, anxiety may precede these physical diseases which in turn cause disability. Finally, behavioural mechanisms could explain the association between anxiety and incident activity limitation with, for example, anxiety symptoms shaping the perception of risk and inducing fear which in turn would curtail activity levels especially social participation.

Strengths and Limitations

Our study is based on a large community sample of elderly subjects [START_REF] Ritchie | Prevalence of DSM-IV psychiatric disorder in the French elderly population[END_REF] with three follow-up examinations over 7 years, permitting a dynamic evaluation of activity limitation. We used validated scales to measure social restriction and activity limitations, corresponding to distinct levels of disability severity. We examined disability in relation to baseline anxiety, both on a symptom scale and as a diagnostic category. Our study is the first to include these complementary measures of anxiety, the former taking into account the severity of underlying and persisting symptoms and the latter, disorders meeting strict DSM-IV diagnostic criteria. Whereas symptom severity was self-reported, we used a structured diagnostic interview for anxiety disorders, thus minimizing misclassification. However, we were not able to fully rule out medical conditions or medication use as a cause of anxiety symptoms in the diagnosis of GAD. Nonetheless we controlled for a large number of potentially confounding factors particularly measures of physical and mental comorbidity and baseline use of antidepressant and anxiolytic medication. We used a statistical model particularly adapted to longitudinal analyses of reversible states; furthermore it did not require fixed time points at follow-up and included subjects with incomplete follow-up thus maximising sample size.

Concerning the limitations of our study, activity limitations were self-reported and may have been over-reported in subjects with anxiety. Although subjects with activity limitations at baseline were excluded, self-reported anxiety symptoms may have been overreported in subjects with early signs of social restriction and activity limitation. Bias could also be introduced by the non-random exclusion of subjects with missing data at baseline or lost to follow-up, who were more likely to have depressive symptoms, as well as activity limitations. However there was no difference regarding anxiety symptoms or disorders at baseline. Whereas the association between depression and incident disability was previously found to be gender-dependent [START_REF] Carriere | Late life depression and incident activity limitations: influence of gender and symptom severity[END_REF]), this appears not to be the case for trait anxiety symptoms or anxiety disorder. However this could be due to smaller sample size, the interaction term failing to reach significance. In spite of extensive adjustments for a large number of potential confounding factors, the possibility remains with observational data that unmeasured confounders such as individual behaviour or social environment may partly explain the associations. Lastly, despite the large overall sample size, we were not able to analyse ADL limitation due to too low incidence rate (0.7% to 2.0%).

Conclusion

In conclusion, our findings highlight the importance of considering anxiety both on a dimensional symptom scale and as a disorder reaching diagnostic criteria. Firstly, both measures of anxiety are associated with incident disability with different levels of severity. These associations are independent of comorbid depressive symptomatology and persist in non-depressed elders. In addition, focusing on trait anxiety symptoms would allow the identification of a wide range of subjects who are missed if applying diagnostic criteria only but whose symptoms are likely to lead to dependency. Our study underlines the necessity to increase the recognition of anxiety in the elderly which to date remains largely underdiagnosed and undertreated.

  1999 to 2001, by random selection from the electoral rolls in Montpellier, France. Eligible persons, who were at least 65 years of age and non-institutionalised, were invited to participate and provided written informed consent. Participants were administered a number of standardised questionnaires by trained staff and underwent clinical examinations at baseline, 2 years, 4 years and 7 years in a dedicated centre or at home. Ethics approval for the ESPRIT study was granted by the Ethics Committee of the University Hospital of Kremlin-Bicêtre (France).

Table 1 .

 1 Sample description at baseline

	Total	Subjects with Trait	Subjects with DSM
		anxiety	IV anxiety disorder
	(N=1581)		(N=225)
	%	(N=490)	%
		%

Table 2 .

 2 Cross-sectional association between anxiety (baseline anxiety disorder and trait anxiety) and activity limitation (N=1581)

			Activity limitation
		N	(%)	OR (95% CI) #	p-value
	Social restriction: : home or neighbourhood confined
	Baseline Anxiety Disorder
	No	1356	3.83
	Yes	225	7.11	1.95 (1.05-3.63)	0.034
	Baseline Trait Anxiety
	Low-Moderate	1091	3.756
	High*	490	5.51	1.51 (0.89-2.55)	0.12
	Activity limitations: Mobility
	Baseline Anxiety Disorder
	No	1356	34.96
	Yes	225	48.0	1.61 (1.18-2.0)	0.003
	Baseline Trait Anxiety
	Low-Moderate	1091	32.26
	High*	490	46.94	1.83 (1.43-2.33)	<0.0001
	Activity limitations: IADL
	Baseline Anxiety Disorder
	No	1356	3.02
	Yes	225	4.44	1.56 (0.76-3.24)	0.23
	Baseline Trait Anxiety
	Low-Moderate	1091	3.02
	High*	490	3.67	1.26 (0.69-2.31)	0.44
	*high=top third tertile

# adjusted for age and sex

Anxiety and act lim 03-04-12

Anxiety and act lim 09-01-12


S F Eskildsen

V Fonov

P Coupé

D L Collins

Visualizing stages of cortical atrophy in progressive MCI from the ADNI cohort

Keywords: ADNI, Alzheimer's disease, MCI, MRI, cortical thickness, prediction

Learning objectives: To investigate the cortical pattern of atrophy at different stages in patients with mild cognitive impairment (MCI), which

Background: Amnestic mild cognitive impairment (MCI) is considered a condition where patients are at risk of developing clinically definite Alzheimer's disease (AD) with an annual conversion rate of approximately 15% [START_REF] Petersen | Current concepts in mild cognitive impairment[END_REF]. AD is characterized by progressive brain atrophy with major impact on the cerebral cortex and medial temporal lobe structures such as hippocampus. Understanding the structural pattern of cortical atrophy at different stages of MCI, before AD can be diagnosed, may help in patient monitoring and prognosis. In this study, we used data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) to calculate and visualize the cortical atrophy at different stages in patients who eventually converted to clinically definite AD.

Methods:

We selected patients with a diagnosis of MCI from the ADNI database who converted to AD during the follow--up period. T1--weighted MRI scans were collected at time of conversion(n=140) and at intervals prior to the date of conversion: 6 months(n=121), 12 months(n=128), 24 months(n=62) and 36 months(n=29). As control group we collected all baseline scans of healthy controls (HC)(n=226). Images were denoised [START_REF] Coupé | An Optimized Blockwise NonLocal Means Denoising Filter for 3--D Magnetic Resonance Images[END_REF], bias field corrected [START_REF] Sled | A nonparametric method for automatic correction of intensity nonuniformity in MRI data[END_REF], linearly co--registered [START_REF] Collins | Automatic 3D intersubject registration of MR volumetric data in standardized Talairach space[END_REF], skull--stripped [START_REF] Eskildsen | BEaST: Brain Extraction based on nonlocal Segmentation Technique[END_REF], and cortical thickness were measured using FACE [START_REF] Eskildsen | Active surface approach for extraction of the human cerebral cortex from MRI[END_REF] and mapped to an average cortical surface. Statistical maps of differences in cortical thickness between groups of MCI patients and HC were constructed and corrected for multiple comparisons.

Results: Three years prior to clinically definite AD, the MCI patients show signs of atrophy in the parahippocampal gyrus, the cingulate gyrus and temporal lobes (figures). At two years before conversion, atrophy has spread to large parts of the temporal and frontal lobes, much of the cingulate gyrus and has started in the orbitofrontal region. Twelve months before conversion, the atrophy is present all over the cortex except the occipital lobe and the sensory--motor areas. Six months later and at conversion the same pattern is seen with more severe atrophy.

Conclusions: Focal atrophy can be observed in patients with MCI three years before they are diagnosed with clinically definite AD. The atrophy accelerates during the MCI stage and affects the entire cortex except the occipital and sensory--motor cortex at the time of diagnosis. These patterns may be used to assess the disease progression in patients with MCI.

Figure 1 .

 1 Figure 1. Differences in mean cortical thickness for MCI patients compared to HC at different stages.

Figure 2 .

 2 Figure 2. Statistical maps of differences in cortical thickness for MCI patients compared to HC at different stages. FDR corrected p--values.


Jovan Radak

Nathalie Mitton

Transmission Power Adaptation Based Energy Efficient Neighborhood Discovery

Keywords: wireless sensor network, neighborhood discovery, HELLO message, transmission power adaptation, turnover;

Neighborhood discovery -detection of the devices within communication range using HELLO messages -is a fundamental mechanism in wireless sensor networks (WSN) which enables usage of many different topology control and routing algorithms. Even though it is very important, most of the algorithms does not take into account parameters of the neighborhood discovery. We present two algorithms that adapt power of transmission of the sensors in a mobile WSN by still adapting frequency of HELLO messages in order to save energy and get accurate neighborhood tables. First solution is based on the knowledge of turnover -change in the number of neighbors in consecutive iterations of the neighborhood discovery -used in conjunction with adaptation of frequency and transmission range, minimizing general cost of transmission of the HELLO messages. Second solution is based on computing of optimal range. Both algorithms are based on theoretical analysis. Results show that they are energy efficient and outperform solutions of the literature.

I. Introduction

Wireless sensor networks (WSN) can be defined as networks of small spatially distributed devices, called sensor nodes, which are working cooperatively -exchanging messages wirelessly -on the same application. Mobility of the sensor nodes is emerging problem and nowadays it becomes more and more under the attention of scientific community. Included in the WSN, mobility arises new problems and questions such as optimization of energy consumption, connectivity of the WSN, routing in mobile networks and many more. Nodes in WSN can have mobility included inherently, as it is case with nodes which are attached to the animals with the purpose to track animals' habit and their natural habitat 1 . In this case mobility pattern is very often hard to estimate and protocols built upon this problem must take into account possible losses of connectivity or delays in the transmission of messages. Other possibility would include mobile agents (robots) which may have given mobile pattern to follow or even use it to improve some of the parameters in the WSN [START_REF] Loscrì | Simulations of the impact of controlled mobility for routing protocols[END_REF]. In this case overall energy efficiency of transmission can be significantly augmented using controlled mobility smart placement of nodes related to the routing path. Due to the specific nature mobile networks some of the characteristic mechanisms used in static WSNs need to be redefined and adapted to the specific types of the mobility of the nodes. 1 http://wasp-project.org Neighborhood discovery is one of the most important protocols in functioning of WSN. Mechanism behind this protocol is rather simple, it includes periodical sending of specific type of the messages, called HELLO messages (also known as beacon messages ) and gathering the data from the received HELLO messages. Hello messages contain the data of the sender id, unique identification number for the node in the WSN -usually MAC address in practical applications. Each node, usually, acquires data from all HELLO messages that it has received and organizes them into the neighborhood table which can be further used for some kind of topology control [START_REF] Radak | Using Battery Level as Metric for Graph Planarization[END_REF] or proactive routing [START_REF] Jacquet | Optimized link state routing protocol for ad hoc networks[END_REF]. The main challenge, in the practical usage of the neighborhood discovery mechanism is the accuracy of the neighborhood table, size of HELLO messages, as well as their content and the frequency at which they are being sent in order not to spent network resource uselessly. Indeed if the Hello frequency is too low, nodes may not be detected by their neighbors, leading deprecated neighborhood tables, and protocol failures are likely to occur. But if the frequency is too high, neighborhood tables are up to date, but then energy and bandwidth are wasted to the detriment of data traffic. Similarly, nodes can adapt their range. The longer range, the more neighbors to be detected by the more energy is spent. It is shown [START_REF] Razafindralambo | Analysis of the impact of hello protocol parameters over a wireless network self-organization[END_REF] that varying of these parameters with other parameters of the protocol stack in WSN (MAC layer, clustering) can have significant impact on the performance of the nodes and whole network. In the presence of the mobility of the nodes we must observe that each of the nodes in successive periods of time ∆t is going to receive HELLO messages of the different nodes and also HELLO messages that the node itself is sending are going to be received by different sets of nodes. This means that neighborhood table is changing due to the mobility of the nodes. Question that arises is: how can we use mechanism of neighborhood discovery in the presence of the mobility and which parameters are the best for usage in these cases?

Our work, presented in this paper, relies on the Turnover based Adaptive Hello Protocol [START_REF] Ingelrest | A Turnover based Adaptive HELLO Protocol for Mobile Ad Hoc and Sensor Networks[END_REF] which exploits the idea of existence of optimal frequency of hello messages and using this result and the change of the neighborhood table, called turnover gives the optimal value for the turnover in the mobile WSNs and sets the basis for the future works which can include prediction of the mobility or some kind of topology control in the mobile WSNs. Coming from the fact that overall energy efficiency can be minimized when manipulating both power of transmission and the frequency of hello messages, we present two algorithms for the adaptation of the frequency of hello messages in parallel with the adaptation of the power of transmission of each node. Both algorithms rely on theoretical analysis. They periodically executed by each node (independently of the other nodes) in the mobile network, upon gathering the data from the hello messages when node receives them. To the traditional neighborhood table we have added also history table in which we are keeping successive neighborhood tables with the idea to track the changes between the neighborhood tables and according to those changes properly changes the factors of neighborhood discovery and transmission range of each node. Results show that our solutions maintain same accuracy as TAP protocol but saving energy.

Section II gives review of the literature related to our work. Section III presents our algorithms while Section V provides theoretical preliminaries and analysis. Simulation results are detailed in Section VI. Section VII gives conclusion and directions of future work.

II. Related work

Hello protocol is first introduced in OSPF [START_REF] Moy | OSPF -Open Shortest Path First[END_REF] and works as follows. Each node is sending HELLO messages at fixed frequency, noted f HELLO to allow other nodes knowledge of its presence, and at the same time each node collects data from HELLO messages that it receives from other nodes i.e. if node u for example receives HELLO message from the node v it puts node v in his neighborhood table, similarly if node v receives HELLO message from node u he will put it in his neighborhood table.

Neighborhood discovery mechanism as described in here assumes that f HELLO is fixed. In mobile networks this assumption in not adequate since it is more natural to assume that the nodes which are moving faster are also changing their neighbors faster thus they need to update neighborhood table more often but also since they are losing (and gaining) their neighbors more often it is logical that they send HELLO messages at a higher rate. In this case when we talk about mobility, we think of the relative mobility of the node itself referring to the other nodes. For example if we have a fleet of the nodes which are moving together at some fixed speed then they are in the relative sense static because each of the node always "sees" the same nodes around him. In [START_REF] Troel | Prise en compte de la mobilit dans les interactions de proximit entre terminaux profils htrognes[END_REF], assuming that there is a knowledge of relative speed between two nodes S and threshold distance in communication area aR such that a < 1, is given the optimal value for the frequency of HELLO messages:

f opt = 2S aR (1)
There exist several works which were trying to adapt frequency of HELLO messages using the information of position and speed thus assuming the presence of these devices on the sensor nodes. Turnover based adaptive hello protocol (TAP) [START_REF] Ingelrest | A Turnover based Adaptive HELLO Protocol for Mobile Ad Hoc and Sensor Networks[END_REF] is the first algorithm which adapts frequency of HELLO messages without using any additional hardware. In this paper authors are proposing usage of turnover value which can be calculated considering the differences of two neighborhood tables at fixed moments in time. The first neighborhood table is the current one and the other is the one obtained and saved at the previous moment. This protocol thus only requires periodical save of current neighborhood table in so called history table which will be then used for the calculation of the turnover.

In this case we come to the question how often we should update neighborhood table and which of the values should be considered as obsolete. Neighborhood lifetime algorithm (NLA) [START_REF] Kassem | Adapting dynamically neighbourhood table entry lifetime in wireless sensor networks[END_REF] works alongside the TAP and it adapts refreshing the entries of the neighborhood table using the information of the speed of the nodes and frequency of HELLO messages.

Knowledge of the relative speed of the nodes can also lead into the detection and estimation of the type of the mobility. In [START_REF] Li | Mobility prediction based neighborhood discovery in mobile ad hoc networks[END_REF] authors are presenting Autoregressive Hello protocol (ARH) and evaluate its performance alongside TAP. Different approach is presented in [START_REF] Kleisouris | Detecting intra-room mobility with signal strength descriptors[END_REF] where the authors are relying on the signal strength descriptors which are embedded in wireless radio (again without using any additional hardware) to detect mobility of the nodes in the closed spade (room). While giving excellent results in detection and prediction of mobility these works do not tackle the possibility of changing the transmission range of the nodes.

Distinction is made between initial and continuous neighborhood discovery [START_REF] Cohen | Continuous neighbor discovery in asynchronous sensor networks[END_REF]. Initial neighborhood discovery is applied when the sensor is unaware of its immediate environment, while the continuous neighborhood discovery is performed when the node is already aware of the neighborhood. Initial neighborhood discovery needs to be done by each sensor separately while continuous discovery can be applied as joint task of the segment of the neighborhood, and not each sensor. This joint task allows single node to spend more time in sleep mode and relies to the part of its neighborhood which is currently in active state. In this way they can lower energy consumption with high probability that new node is going to be properly discovered. The algorithm guarantees that new node is going to be discovered in given time slot with requested probability,

The best of our knowledge the algorithm that we are presenting is the first that tries to handle both frequency of the HELLO messages and the transmission range of the nodes when nodes are not aware of their position.

III. Algorithms for adaptation of frequency and transmission range

In this section we present our algorithms used to adapt both frequency of HELLO messages and node transmission range. The first algorithm is based on [START_REF] Ingelrest | A Turnover based Adaptive HELLO Protocol for Mobile Ad Hoc and Sensor Networks[END_REF] which relies on the calculation of turnover. The turnover of [START_REF] Ingelrest | A Turnover based Adaptive HELLO Protocol for Mobile Ad Hoc and Sensor Networks[END_REF] has been redefined to capture both unilateral and bilateral neighbors (that appear because of different transmission ranges of each node) along with optimal frequency of HELLO messages [START_REF] Troel | Prise en compte de la mobilit dans les interactions de proximit entre terminaux profils htrognes[END_REF]. This turnover helps in determining the HELLO frequency. Algorithm is then declined in two variants to define the transmission range. Second algorithm is combining results for optimal frequency and dependency between f and transmission range to adapt both frequency and transmission range without knowledge of turnover.

Both algorithms are based on the existence of an optimal frequency of HELLO messages [START_REF] Troel | Prise en compte de la mobilit dans les interactions de proximit entre terminaux profils htrognes[END_REF] given by Eq. ??.

A. Using the turnover

To be able to define this algorithm first step for us is to calculate turnover, r, in a real scenario. Number of new neighbors (theoretically calculated in previous section) is translated into the calculation of number the new neighbors with given weights.

• unilateral neighbors -the situation when the node finds new unilateral neighbor; it is quantified with θ uni multiplied by the total number of new unilateral neighbors in given period of time, • bilateral neighbors -the situation in which the node finds new bilateral neighbor; factor θ bi is used to quantify new bilateral neighbor which is found and it is multiplied by total number of new bilateral nodes y found in given period of time.

Current turnover, r, is calculated by each node independently according to their own track of the changes between the types of the links, using equation:

r = θ uni • x + θ bi • y x t + y t • T HELLO ∆t (2)
where

θ uni = 1, θ bi = 2,
d HELLO = d HELLO + d HELLO 4 • g(r) if r ≤ r opt d HELLO -d HELLO 4 • g(r) otherwise (3)
Function g(r) is retrieved using turnover:

g(r) = ( r-r opt r opt ) 2 if r < 2 • r opt , 1 otherwise (4)
Algorithm 1 Turnover based Power Transmission Adjustment 1: while 1 do

2: CalculateCurrentTurnover r = θ uni •x+θ bi •y x t +y t • T HELLO ∆t 3: if r ≤ r opt then 4:
Lower f with Eq. 3 5:

else if r > r opt then 6:

Augment f with Eq. 3 7:

end if

8:

Adapt Power o f T ransmission 9: end while Both variants differ in the call to Adapt Power of Transmission function in Line 8 of Algorithm 1. First variant uses Eq. 1 while second variant uses the analysis presented in previous section. This analysis as a result gives us R = α C 1 f -C as function for transmission range.

IV. Minimizing the energy consumption

This second algorithm is based on the idea that there exists a joint way to optimize the transmission range and the Hello frequency. Indeed, the energy spent in period ∆t of time by a node u can be expressed as the number of messages sent by u in ∆t multiplied by the cost of a message. The number of messages sent by u during ∆t is ∆t • f u (R u , t) where f u (R u , t) is the HELLO frequency of node u and R u is the range of node u at time t. We assume that the cost of a message follows energetic model given in [] with E(R u ) = R u (t) α + C, where α is a real constant (¿ 1) that represents the signal attenuation and c is the overhead due to signal processing. We assume that ∆t is such that u does not change its range nor its frequency during this period of time, so R u (t) = R. Thus, energy spent by node u with transmission range R during ∆t is:

cost ∆t (R) = ∆t • f (R) × (R α + C) (5)
Based on this equation and combining it to Eq. 1, we obtain a system with two equations and two unknowns: R and f that we compute in Section V-C. V. Theoretical analysis

Algorithm 2 Cost Based Transmission Power Adjustment

1: To get R, solve R α -aRC 1 2S + C = 0 based on α values. 2: f = 2S

A. Model and notations

Wireless networks are presented by a graph G = (V, E) in which V is set of nodes and E is set of edges such that E ⊆ V 2 . We suppose that nodes are randomly deployed using a Poisson Point Process [START_REF] Stoyan | Stochastic geometry and its applications[END_REF], with node positions which are independent and λ > 0, where λ represents the mean number of nodes per surface unit. Each node u has transmission range R u such that 0 < R u < R max .

In this paper, since nodes can adapt their range, they do not have necessarily the same range. So we differentiate two types of neighbors:

• bilateral neighbors -two nodes u and v with property

|uv| < min(R u , R v ) • unilateral neighbors -node u is unilateral neighbor of node v if min(R u , R v ) < |uv| < max(R u , R v )
Every node is moving at constant speed S in random direction. Position of node u at the moment t 0 is given as u 0 (respectively node v at t 0 has position v 0 ) and at the moment t 1 position is u 1 (respectively v 1 ). Distance covered by a node during the time ∆t is given as ∆d = V × ∆t. Angles α and β are given as

∠ ----→ u 1 , v 1 , ----→ u 1 , u 0 and ∠ ----→ u 1 , v 1 , ----→ v 1
, v 0 respectively, and they represent the directions from which nodes u and v come. In the worst case, node direction is random and thus α and β are uniformly distributed in [-π, π].

B. Theoretical analysis on probable number of new neighbors

In our analysis, we are interested in the mean number of new neighbors that node u is going to meet during the time interval ∆t. We focus on the typical node u. Let N bi (u) ∆t be the number of the bilateral neighbors of node u and N uni (u) ∆t the number of unilateral neighbors of node u that node u detects during period ∆t. Let v be a node at distance d (d < R max ) from node u at time t 1 = t 0 + ∆t. We note:

• P bi : the probability that node v is new bilateral neighbor of node u • P uni : the probability that node v is new unilateral neighbor of node u From this we can determine the average values of N bi (u) ∆t and N uni (u) ∆t : [START_REF] Troel | Prise en compte de la mobilit dans les interactions de proximit entre terminaux profils htrognes[END_REF] where P(R u ) is probability that u has radius R u and P(R v ) is probability that v has radius R v . Figure 1 illustrates our model in case when R u < R v . Circle C u,R u is centered at node u with radius R u . In this case, node v, with radius R v is a new neighbor if and only if v 0 does not lie in the area delimited by C u 0 ,R u and if v 1 lies in the area delimited by C u 1 ,R u . The blue dashed circle C u 1 ,∆d and the red dotted circle C v 1 ,∆d represent the possible positions of u 0 and v 0 .

E[N bi ] = R max R u =0 R max R v =0 R u d=0 P(R u )P(R v )λπd × P bi (d, R u , R v )dddR u dR v (6) E[N uni ] = R max R u =0 R max R v =0 R u d=0 P(R u )P(R v )λπd×P uni (d, R u , R v )dddR u dR v
1) Computing the number of new bilateral neighbors E[N bi ]: We are interested in the probability that at time t 0 , u and v were either not neighbors (|u

0 v 0 | > max(R u , R v )) or only unilateral neighbors (max(R u , R v ) > |u 0 v 0 | > min(R u , R v )).
We are interested in probability P bi that given R u , R v and d,

|u 0 v 0 | > min(R u , R v ) knowing that |u 1 v 1 | ≤ min(Ru, R v ).
We make the distinction between two cases:

• Case 1: R u ≤ R v We note P bi R u ≤R v the probability that v is a new bilateral neighbor of u if R u ≤ R v ; for this case P(R v ) = 1 R max and P(R u | R u < R v ) = R v R max , • Case 2: R u > R v We note P bi R u >R v the probability that v is a new bilateral neighbor of u if R u > R v ; for this case P(R v ) = 1 R max and P(R u | R u ≤ R v ) = R max -R v R max .
From these two cases we have:

E[N bi ] = R max R u =0 R max R v =0 R u d=0 P(R u )P(R v )λπd × P bi (d, R u , R v )dddR u dR v = R max R v =0 R v R u =0 R u d=0 R v R max × 1 R max × λπd × P bi R u ≤R v (d, R u , R v )dd(dR u )dR v + R max R v =0 R max R u =R v R u d=0 R -R v R max × 1 R max × λπd × P bi R u >R v (d, R u , R v )dd(dR u )dR v (8) a) Case 1: R u ≤ R v : We first note that if d ≥ R u , P bi R u ≤R v (d)
= 0 since v is not a neighbor of u at a time t 1 . Next, we find that there exists a value d min < R max such that, if d < d min , nodes u and v were already neighbors at the time t 0 regardless of α and β i.e. v cannot be a new neighbor of u. Justification and computation details are given in Appendix VIII (Eq. VIII-A). We have that d min = max(0, R u -2∆d) which leads to:

P bi (d, R u , R v ) =            π -π P bi R u ≤R v (d, R u , R v )dα i f d min < d < R u 0 otherwise (9) where P bi R u ≤R v (d, R u , R v ) is the probability that node v at distance d with radius R v is a new bilateral neighbor of node u with radius R u coming from direction α, assuming d min < d < R u . Now, we can compute P bi R u ≤R v (d, R u , R v ).
For this purpose, we can notice that, for a given value of d min < d < R u , there exists a value α min such that, for α < α min , node v was a bilateral neighbor of node u at t 0 regardless of its direction β, thus it cannot be a new neighbor. This is illustrated on figure ??. Indeed, for α < α min , the whole circle C v 1 ,∆d lies inside the circle C u 0 ,R u thus, for any direction that the node v may had it was already in the bilateral neighborhood of the node u. As a result we have: [START_REF] Kleisouris | Detecting intra-room mobility with signal strength descriptors[END_REF] where α min = arccos(

P bi (d, R u , R v ) = 0 i f α < α min
d 2 +2R u ∆d-R 2 u 2d∆d
). Computation of α min is given in Appendix VIII Eq. VIII-B.

For any α, such that α > α min , computing the probability

P bi R u ≤R v (d, R u , R v
) is equal to the computing the probability that node v is coming from the dotted blue angular sector on Fig. ??. Node v is a new neighbor of node u if and only if β is such that |u 0 v 1 | > R u i.e. such that v 0 is outside of the circle C u 0 ,R u . In this case node v is the new neighbor of u if and only if β -< β < β + , where β -and β + are the angles of the intersection points between C u 0 ,R u and C v 1 ,∆d , as illustrated on Fig. ??. As a result we have:

P bi R u ≤R v (d, R u , R v ) = β + β - dβ 2π = β + -β - 2π = 1 π arccos( R 2 u -∆d 2 -k 2 2d∆d ) if α > α min where k = |u 0 v 1 | = √ ∆d 2 + d 2 -2d∆d cos α.
The details of computation of β -, β + and k are given in Appendix VIII Eq. 23. Thus, when d > d min :

P bi R u ≤R v (d, R u , R v ) = 1 π arccos( R 2 u -∆d 2 -k 2 2d∆d ) i f α > α min 0 otherwise (11)
From Eq. 11, we derive:

P bi R u ≤R v (d, R u , R v ) = 2 × π 0 P bi R u ≤R v (d, R u , R v , α) π dα =            2 π 2 π α min ) arccos( R 2 u -∆d 2 -k 2 2d∆d )dα i f max(0, R u -2∆d) ≤ d ≤ R u 0 otherwise b) Case 2: R u > R v : Computing of P bi R u >R v (d, R u , R v ) is similar to the computing of the P bi R u ≤R v (d, R u , R v
). The differences are in the values of d min , β + , β -and α min as shown in Appendix.

P bi R u >R v (d, R u , R v ) =                2 π 2 π α min arccos( R 2 v -∆d 2 -k 2 2k∆d
)dα

i f max(0, R u -2∆d) ≤ d ≤ R u &R v -2∆d < R u 0 otherwise
2) Computing the number of new unilateral neighbors E[N uni ]: Value that is interesting for our analysis is the probability P uni that at time t 0 , when u and v were not neighbors with given R u , R v and d. We distinguish two cases:

• Case 1: R u ≤ R v , noted P uni R u ≤R v • Case 2: R u > R v , noted P uni R u >R v
From these two cases we have:

E[N bi ] = R R u =0 R R v =0 R u d=0 P(R u )P(R v )λπd × P uni (d, R u , R v )dddR u dR v = = R v R u =0 R R v =0 R u d=0 R v R × 1 R × λπd × P uni R u >R v (d, R u , R v )dddR u dR v + + R R u =R v R R v =0 R u d=0 R -R v R × 1 R × λπd × P uni R u >R v (d, R u , R v )dddR u dR v a) Case 1: R u ≤ R v :
In such a case, probability for node v to be a new unilateral neighbor of node u is null. Indeed, since R u ≤ R v , this is indeed u which is a bilateral neighbor of node v and not the opposite, thus we have:

P uni R u >R v (d, R u , R v ) = 0 (12)
b) Case 2: R u > R v : We compute the probability P uni R u >R v for the node v to be a new unilateral neighbor of node u knowing R u , R v and d. This study is similar to the one given for the bilateral neighbors with the differences in values for d min , β + , β -and α min as shown in Appendix.

P uni R u >R v (d, R u , R v ) =                  2 π 2 π arccos( d 2 +2Ru∆d-R 2 u 2d∆d ) arccos( R 2 u -∆d 2 -k 2 2k∆d
)dα

i f max(R v , R u -2∆d) ≤ d ≤ R u 0 otherwise (13)

C. Theoretical analysis on minimization of energy cost

In this section, we analyze jointly the optimal frequency and transmission range in order to define the f function used in Algo. 2.

Hello frequency is also depending of transmission range. The higher transmission range, the lower Hello frequency. In order to find appropriate transmission range we need to find the couple f, R that gives the minimum of the cost considered in Eq. [START_REF] Ingelrest | A Turnover based Adaptive HELLO Protocol for Mobile Ad Hoc and Sensor Networks[END_REF].

∂cost(R) ∂R = ∂ ∂R (∆t f (R)(R α + C)) (14)
Partial derivative given is applied in order to get the minimum of the cost function:

∂cost(R) ∂R = ∆tα f (R) × R α-1 + ∆t ∂ f (R) ∂R × (R α + C) (15)
minimum of this function is obtained when ∂cost ∂R = 0 (16) which gives us

∂ f (R) ∂R = -α f (R) R α-1 R α + C (17)
This differential equation gives as a solution:

f (R) = C 1 R α + C (18)
where C 1 is a constant defined by initial conditions. For parameters used later on in our simulations, e.g. α = 4, C = 10 8 given in [START_REF]Réseaux de capteurs[END_REF] gives C 1 = 2 × 10 8 . It becomes:

f (R) = 200000000 R 4 + 100000000 (19)
Optimum function f (R) for energy consumption is then depicted on Figure 2. Note that, as expected, the Hello frequency decreases when the range increases. By replacing f in Eq. 1, we then need to solve R α -aRC 1 2S + C = 0 for different α and C values.

VI. Simulation results

We obtained our results using WSNET 2 simulator. We use a model in which the range of transmission can be adapted according to the values that we retrieve using proposed algorithms. In our simulation setup 100 simulation nodes are randomly deployed in a rectangular area of 1000 × 1000m. Mobility model used for the nodes uniform linear velocity with perfect deflection i.e. billiard model of motion.

Important parameters for neighborhood discovery and nodes behavior are followed and their dependence with the respect of the nodes relative speed is given. In the following graphs we present values for the range of transmission, period of HELLO 2 http://wsnet.gforge.inria.fr/ messages, state of the nodes batteries, observed turnover and accuracy of obtained results are given as the functions of the nodes' speed. In these graphs we refer to the algorithms in the following way: Algorithm 2 is called NoTAP since it does not use turnover in calculations, Algorithm 1 with usage of optimal frequency is called Fopt, TAP algorithm [START_REF] Ingelrest | A Turnover based Adaptive HELLO Protocol for Mobile Ad Hoc and Sensor Networks[END_REF] is referred as TAP while Algorithm 1 with minimized cost is called Cost. We will present results retrieved from the simulations with the comment on the performance of the TAP, Fopt and NoTAP algorithm. Separate discussion will be given on the obtained results with the Cost algorithm for adaptation of the range using minimized energy consumption because results obtained using this algorithm are unrealistic (too good) and deserve additional explanation.

Figure 3(a) shows range adaptation as the function of speed of the nodes. We can notice that transmission range for the TAP algorithm is held on the same level, using the same approach (unit disk graph) as in [START_REF] Ingelrest | A Turnover based Adaptive HELLO Protocol for Mobile Ad Hoc and Sensor Networks[END_REF]. In the case for Fopt and NoTAP algorithm ranges are increasing as the speed is increasing with the difference that NoTAP algorithm increases range linearly. This behavior of these two algorithms is expected because when nodes are moving faster then they are also changing their neighbors faster so in order to maintain the number of new neighbors the algorithms are increasing the range. We have to note also that Fopt increases the range more than NoTAP since its range is not bounded to single value for a given speed as it is the case for NoTAP. On the other side smaller increase of range of NoTAP algorithm is compensated (as given in Eq.1) with the increase of the period of HELLO messages (Fig. 3(b)). Fopt is keeping the period on almost constant value because the adaptation to the higher speed is done with an increase of the range. TAP is decreasing HELLO period with the increase of speed because its range is fixed and the only way to adapt its behavior when nodes are moving faster is to lower the period (hence increase the frequency of HELLO messages). From Figure 3(c) we can see that all algorithms better balance energy than TAP which keeps transmission range on the same and due to this fact has worse results than others. We can also observe that energy loss is bigger as the speed of the nodes increases this is due to the adaptation of algorithms and their attempt to balance the values of turnover, frequency and range which compensates in higher energy loss.

In this case we have to point out that we used linear discharge model for the battery, decreasing certain amount energy from the battery multiplied by range of transmission each time when we transmit packet, constant value for each received packet and loss of energy in idle mode represented by constant value multiplied by the time spent in idle mode (the time between receptions or between reception and transmit and vice versa). This representation of battery is not the best one since it overstates the impact of transmission range which is in real case smaller and is given with the increase of power of transmission.

The results for turnover, shown on Figure 3(d), are more straightforward, we can see that for the algorithms that take turnover (Fopt and TAP) into account there is an increase of turnover with the increase of speed while NoTAP is having changes of turnover not depending on the speed. Fopt has slightly bigger change of turnover as a result of adaptation both frequency and range.

A. Comment on the Cost algorithm

Cost algorithm shows almost perfect results when taking into account state of the battery and accuracy of the neighborhood table, but looking into dependency of the turnover and range gave us the clue what is the reason for this. This algorithm tries to minimize the energy, and its doing it well, but at the cost of keeping the range of transmission on the lowest value thus gaining new neighbors occasionally and with big accuracy. Also since the transmission range is minimal it also preserves battery in the best way. What can we conclude from this is that the model that we imposed for this algorithm is too ideal and does not take into account the nuances this adaptation might have. Minimal transmission range also have another consequence, keeping transmission range on such a low value is ensuring low power consumption but at the same time threatens the correct neighborhood discovery since each node with such a small transmission range detects small number of neighbors and keep its neighborhood table almost empty.

One of the most important values to follow is the accuracy of the neighborhood tables, shown on the Figure 4. Accuracy is calculated checking the state of neighborhood table periodically and checking if all the nodes listed as the neighbors are still in the neighborhood of the node that is being checked. All algorithms are showing tendency of increasing accuracy with increase of the speed which is logical since the faster nodes spend less time in physical neighborhood of their neighboring node.

VII. Conclusion

In this paper we present neighborhood discovery algorithms that take into account both frequency of hello messages and the transmission range in order to save energy. We are using results obtained by the theoretical analysis on the neighborhood discovery to calculate turnover and its impact on the overall energy consumption. These results are combined with optimal frequency of hello messages for mobile nodes and to conceive two algorithms which are adapting both frequency of hello messages and transmission range. Simulations have proven that our algorithms outperform TAP (which only adapts frequency with fixed range) in terms of energy consumption while keeping the accuracy of the neighborhood table.

Future work would include further improvement of proposed algorithms, taking into account total number of neighbors and received messages thus ensuring both energy efficient and realistic neighborhood discovery (limiting the transmission range to a higher values). Also, since transmission range cannot be directly manipulated on the real sensor hardware connection between transmission range and power of transmission can be made allowing these algorithms to be run on the real wireless sensor networks.

VIII. Appendix

A. Computing d min

We need to compute the mean number of new neighbors node u detects for a ∆t time period. For it, we suppose a node v at a distance d from node u at time t

1 = t 0 + ∆t, such that 0 ≤ d ≤ R. 1) Case 1 -new bilateral neighbors: v is a new bilateral neighbor of node u which means that |u 0 v 0 | > min(R u , R v ) and |u 1 v 1 | < min(R u , R v ).
The value d min is such that, if d < d min , nodes u and v are neighbors at time t 0 whatever the directions α and β. As illustrated by the Fig. 5, the longest distance l between u 0 and v 0 at time t 0 is when both node directions are opposite (for example when α = 0 and β = π). We have l = 2∆d + d, where

2∆d + d > min(R u , R v ) ⇔ d > 2∆d -min(R u , R v ) ⇔ d min = 2∆d -min(R u , R v ) (20)
Since, depending of the node speed, we may have 2∆d < min(R u , R v ), we finally get:

d min = max(0, 2∆d -min(R u , R v )) (21)
2) Case 2 -new unilateral neighbors: By definition, for v to be a new unilateral neighbor of node u, d is such that d > R v . Then, similarly to the previous case, d min is such that if d > d min , nodes u and v were already neighbors at the time t 0 , whatever the values α and β.

Since, depending on the node speed, we may have 2∆d < min(R u , R v ), we finally get:

d min = max(R v , 2∆d -min(R u , R v )) (22)

B. Computing α min

In this section, we are interested in computing the value α min such that, given d > d min and α, if α < alpha min then for any β (no matter the value which it takes) v was neighbor of u at time t 0 .

For it, we introduce k such that k = |u 0 v 1 |, like illustrated in

k -∆d > min(R u , R v ) ⇔ k 2 > (min(R u , R v )) 2 + ∆d 2 + 2∆d • min(R u , R v ) ⇔ d 2 -2d∆d cos α > (min(R u , R v )) 2 + 2∆d • min(R u , R v ) ⇔ cos α < d 2 -(min(R u , R v )) 2 -2∆d • min(R u , R v ) 2d∆d ⇔ α > arccos( d 2 -(min(R u , R v )) 2 -2∆d • min(R u , R v ) 2d∆d ) ⇔ α min = arccos( d 2 -(min(R u , R v )) 2 -2∆d • min(R u , R v ) 2d∆d ) (24)
2) Case 2 -new unilateral neighbors: For v to be a new unilateral neighbor of node u we need that |u 0 v 0 | > max(R u , R v ) for any β i.e. that the disks delimited by C v 1 ,∆d and C u 0 ,max(R u ,R v overlap: We deduce that:

k -∆d > max(R u , R v ) ⇔ k 2 > (max(R u , R v )) 2 + ∆d 2 + 2∆d • max(R u , R v ) ⇔ d 2 -2d∆d cos α > (max(R u , R v )) 2 + 2∆d • max(R u , R v ) ⇔ cos α < d 2 -(max(R u , R v )) 2 -2∆d • max(R u , R v ) 2d∆d ⇔ α > arccos( d 2 -(max(R u , R v )) 2 -2∆d • max(R u , R v ) 2d∆d ) ⇔ α max = arccos( d 2 -(max(R u , R v )) 2 -2∆d • max(R u , R v ) 2d∆d ) (25
(β + -β -) = 2 arccos( min(R u , R v ) -∆d 2 -k 2 2k∆d ) (28)
2) Case 2 -new unilateral neighbors: In this case we consider â = arccos(

∆d 2 + k 2 -max(R u , R v ) 2k∆d ) (29)
We deduce that:

(β + -β -) = 2 arccos( max(R u , R v ) -∆d 2 -k 2 2k∆d ) (30)

aR 3 :Fig. 1 .

 31 Fig. 1. Global view. Circle C u,Ru is centered at the position of the node u with radius R u . In this case, node v, with radius R v is a new neighbor if and only if v 0 does not lie in the area delimited by C u 0 ,Ru and if v 1 lies in the area delimited by C u 1 ,Ru . The blue dashed circle C u 1 ,∆d and the red dotted circle C v 1 ,∆d represent the possible positions of the u 0 and v 0 .

Fig. 2 .

 2 Fig. 2. Solution of differential equation obtained for optimal energy cost.

Fig. 3 .

 3 Fig. 3. Different parameters of the WSN as the function of nodes speed

Fig. 5 .

 5 Fig. 5. Calculating d min

Fig. 6 .

 6 Fig. 6. Calculating α min

Fig 6 .Fig. 7 .

 67 Fig. 7. Zoom view

  ) C. Computing β + -β - For computing of the (β + -β -) we use notations from the Fig 7. We can notice that: 2π = (β + -β -) + 2â (26) 1) Case 1 -new bilateral neighbors: In that case we consider â = arccos( ∆d 2 + k 2min(R u , R v )

  Turnover based Power Transmission Adjustment. Algorithm 1 is executed by each node independently only based in the observation of is neighborhood. Algorithm 1 is TAP-fashion algorithm which adapt the hello frequency dynamically based on changes on node neighborhood. It aims to reach an optimal turnover previously computed thanks to computing of new neighbors provided in Section ??. Starting point for the algorithms are neighborhood table and history table, and all other values are calculated using these values, including the turnover. Neighborhood table is the standard neighborhood table, as explained in Section ??. History table is table in which we are preserving the values which we have obtained for a given time moment, number of new bilateral and unilateral neighbors and the changes between the type of the neighbors which allows us to calculate the turnover at that moment. Adjustment of f is calculated through the period between two HELLO messages d HELLO , where f = 1 d HELLO :

∆t is the time passed between updates of two tables that we are comparing, T HELLO is the period of HELLO messages, x t + y t presents total number of neighbors in neighborhood table and it is the sum of all unilateral and bilateral neighbors. These specific values for the θ parameters are used with the respect of the type of neighbor. For new unilateral neighbors we use smaller value of θ because we consider those links weaker and we want to give them less importance in calculation of turnover. Bilateral neighbors are considered stronger and they are multiplied by bigger θ.

First solution is detailed in Algorithm 1