Céline Charroud
email: celine.charroud@hotmail.fr

Nicolas Menjot

Nicolas Menjot De Champfleur

Emily Sanrey

Josef Pfeuffer

Jérémy Deverdun

Emmanuelle Le Bars

Philippe Coubes

Differential Effects of Hunger on Cerebral Blood Flow in Healthy Adolescents

Keywords: MeSH], Adolescent, Cerebrovascular Circulation, Mood, Healthy Volunteers, Hunger

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Food intake represents periodic behavior regulating body weight by obtaining the nutritional resources essential for survival. It involves complex homeostatic and nonhomeostatic mechanisms which leads transient glycemic changes by stimulation and inhibition feeding and appetite [START_REF] Liu | Homeostatic and non-homeostatic controls of feeding behavior: Distinct vs. common neural systems[END_REF] . Energy and weight regulation are thus directly link by the control of feeding in response to metabolic needs as well as by the experiences and hedonic aspects of food (psychological factors). Unfortunately, more than 300 million people worldwide are thought to suffer excessive food intake driving up obesity rates, especially in children and adolescents [START_REF] Strauss | Epidemic increase in childhood overweight, 1986-1998[END_REF][START_REF] Ogden | Prevalence of Overweight and Obesity in the United States, 1999-2004[END_REF] . Such observations are not surprising due to the fact that puberty is usually characterized by psychological changes (depression, anxiety, stress etc.) influencing food choice and intake [START_REF] Dietz | Health consequences of obesity in youth: childhood predictors of adult disease[END_REF][START_REF] Sahoo | Childhood obesity: causes and consequences[END_REF] . Indeed, it has been widely demonstrated that appetite is influenced by mood and emotional states such as depression, resulting in the consumption of palatable "comfort foods" (i.e. high-fat meals) as a means of alleviating negative feelings [START_REF] Singh | food, and obesity[END_REF] . Given the complexity of the notion of food intake, knowledge regarding the neural substrates of appetite is rather limited; hence the necessity of investigating the basis of hunger and satiety states in healthy adolescents.

Most studies have investigated neural substrates using functional MRI in adult populations. They focus on visual presentations by varying appetizing value by means of neutral, low-calorie and high-calorie food picture stimulation. They demonstrate that the food pictures activate two gustatory processing brain areas devoted to the reward value of taste: the orbitofrontal and temporo-insulo-opercular cortices [START_REF] Simmons | Pictures of appetizing foods activate gustatory cortices for taste and reward[END_REF][START_REF] Beaver | Individual Differences in Reward Drive Predict Neural Responses to Images of Food[END_REF][START_REF] Porubská | Subjective feeling of appetite modulates brain activity: an fMRI study[END_REF] . However, none of these trials recognize that specific types of food tend to be preferred under certain psychological conditions [START_REF] Singh | food, and obesity[END_REF] . Therefore, some authors have investigated the effect of food deprivation on brain networks by comparing two states: "hunger" meaning a great desire for food and "satiety" meaning fullness [START_REF] Tataranni | Neuroanatomical correlates of hunger and satiation in humans using positron emission tomography[END_REF][START_REF] Del Parigi | Neuroimaging and obesity: mapping the brain responses to hunger and satiation in humans using positron emission tomography[END_REF][START_REF] Frank | Fat intake modulates cerebral blood flow in homeostatic and gustatory brain areas in humans[END_REF][START_REF] Eldeghaidy | Prior Consumption of a Fat Meal in Healthy Adults Modulates the Brain's Response to Fat[END_REF] . Two studies using arterial spin labeling (ASL) imaging investigated mean CBF changes after consumption of either a high-fat meal or a water load [START_REF] Eldeghaidy | Prior Consumption of a Fat Meal in Healthy Adults Modulates the Brain's Response to Fat[END_REF] or high-fat yogurt [START_REF] Frank | Fat intake modulates cerebral blood flow in homeostatic and gustatory brain areas in humans[END_REF] . Although CBF was found to increase at 40 min post-meal after the water load in the thalamus and insula, it decreased in the hypothalamus and insula at 65min-120min post-meal after the high-fat meal or high-fat yogurt. Furthermore, positron emission tomography (PET) imaging of regional CBF demonstrated an increase in the vicinity of the hypothalamus, thalamus, putamen, insular cortex, orbitofrontal cortex and limbic areas in response to hunger [START_REF] Tataranni | Neuroanatomical correlates of hunger and satiation in humans using positron emission tomography[END_REF][START_REF] Del Parigi | Neuroimaging and obesity: mapping the brain responses to hunger and satiation in humans using positron emission tomography[END_REF] . Higher regional CBF has also been reported in the prefrontal cortex and inferior parietal lobule in response to satiety. However, all of these neuroimaging studies neglect the link between mood, emotional state, and feeding, as mentioned above. Some authors limit their investigation to exploring the "depression" effect found in individuals exhibiting symptoms of depression yet who fail to reach clinically significant levels that are common in the general population [START_REF] Ezzati | Hippocampal correlates of depression in healthy elderly adults[END_REF][START_REF] Spalletta | Hippocampal multimodal structural changes and subclinical depression in healthy individuals[END_REF][START_REF] Schweizer | Symptoms of depression in a large healthy population cohort are related to subjective memory complaints and memory performance in negative contexts[END_REF] . Their main aim was to investigate the effect of depression on structural changes in the hippocampus in healthy populations. They revealed bilateral hippocampal volume reduction which may predict subclinical depressive phenomenology in healthy adults. Unfortunately, none of the studies addressed the depressed mood effect in healthy populations in relation to the whole brain and even less so in the context of food. Furthermore, all of these studies are based on healthy adults and none focus on the adolescent population.

Therefore, the aim of the present research is to broaden knowledge of the brain metabolism involved in the food intake of healthy adolescents using arterial spin labeling. We expected cerebral blood flow in areas of the brain governing taste: (i) to be implicated in response to hunger compared to satiety and (ii) to be correlated to mood.

Materials and Methods

Population

Fifteen healthy right-handed adolescents (7 female; 8 male; age=14.1±1.3 years, mean±SD) with no history of eating disorders or other medical illnesses participated in this study. All of the participants were right-handed and weight was normal (body mass index -BMI=19.3±1.7 kg/m² mean±SD). Mood assessment was measured in all of the adolescents using the Multiscore Depression Inventory for Children (MDI-C), comprising anxiety, selfesteem, sad mood, instrumental helplessness, social introversion, low energy, pessimism and defiance subscales. The total depression score was defined as the sum of these subscales and was converted to standardized scores -T scores [START_REF] Berndt | Multiscore Depression Inventory for Children[END_REF] . Clinically significant depression was defined using a cut-off score of 56 or greater. This composite score is representative regardless of the sample size.

The study was conducted according to the Declaration of Helsinki and received ethical approval from the local ethics committee. Written informed consent was obtained from each participant (ClinicalTrials.gov Identifier: NCT02868619).

Study procedures

All participants underwent two MRI sessions: (i) one pre-lunch called hunger session and (ii) post-lunch called satiety session. They were prevented from eating at least 6h before the first MRI session, after which a standardized lunch was provided including bread, mini pasta shells (125g) with a portion of butter and gruyere, tomato sauce (300g), yogurt and compote. The second MRI session was performed one hour later when participants had reached a state of satiety [START_REF] Labar | Hunger selectively modulates corticolimbic activation to food stimuli in humans[END_REF] . Mood assessment was performed immediately after the meal.

MRI data acquisition

Neuroimaging data were collected using a 3T magnet (Skyra, Siemens, Germany) with a 32-channel head coil for each MRI session. Structural images (3DT1-MPRAGE) were acquired using the following parameters: echo time (TE) = 2.5ms; repetition time (TR) =1690ms; flip angle = 9°; inversion time = 922ms; voxel size = 0.98x0.98x1 mm 3 ; 176 slices. CBF data were acquired using a prototype sequence of 3D pseudo continuous arterial spin labeling (pCASL) according to parameters as follows: post-labeling delay (PLD) = 1800ms; bolus duration = 1800ms; TE/TR = 15.5/4600ms; 42 slices; voxel size = 1.64 x 1.64 x 3 mm with interpolation in acquisition plane.

2.4.MRI data pre-processing

Arterial spin labeling images were realigned to the baseline magnetization image acquired with a long repetition time.

The label images were then subtracted pair-wise from the time-matched control images using surround subtraction to acquire perfusion-weighted images.

To ensure an accurate estimation of the cerebral blood flow (CBF), Asllani et al's partial volume effect (PVE) correction [START_REF] Asllani | Regression algorithm correcting for partial volume effects in arterial spin labeling MRI[END_REF] was implemented in MATLAB and applied in native space, using a 7 x 7 x 1 voxel regression kernel. Briefly, it considers that the measured signal is a ponderation of gray matter, white matter and cerebrospinal fluid contribution:

∆M = PGM * ∆MGM + PWM * ∆MWM + PCSF * ∆MCSF
Where PGM, PWM and PCSF are segmented posterior probabilities obtained from a high resolution T1 image, ∆M the control-label differences from the ASL sequence in a voxel and ∆MGM, ∆MWM and ∆MCFS the proper differences for each tissue in this voxel. Assuming a constant perfusion across a local 2D kernel (7x7), a system of equations can be defined to estimate ∆MGM, ∆MWM and ∆MCFS through a least square approach.

CBF computation was performed using a one compartment model 20 :

= 6000 . . ∆ . e , 2 . . . , . !1 -e $ % , & [()/100+/(,-]
where ∆ represents mean difference in signal intensity between label and control images, equals blood/tissue water partition coefficient, , /001 equals longitudinal relaxation time of blood, is inversion efficiency, 2 label duration, and blood magnetization.

We chose to use the estimation of gray matter ∆ and provided by PVE correction to compute gray matter CBF. Blood magnetization ( ) was estimated using local tissue perfusion [START_REF] Cavuşoğlu | Comparison of pulsed arterial spin labeling encoding schemes and absolute perfusion quantification[END_REF] .

The parameters used for quantification were: labeling efficiency = 0.85; longitudinal relaxation, , /001 = 1650 ms; and, as we focused on gray matter, 34 = 0.98.

Overall mean gray matter CBF was retrieved from the PVE-corrected CBF map in subject space, then normalized to the MNI (Montreal Neurological Institute) space.

MRI statistical analyses

The normalized CBF maps were entered into whole brain voxel-wise general linear model analysis using the SPM12 toolbox. Paired t-tests were conducted to examine CBF changes during both states (hunger and satiety). To limit the analysis to gray matter, a mean gray matter mask was generated by thresholding each subject's spatially normalized GM probability map to 0.8. Only voxels considered as cortical gray matter in all subjects were included in the mask. Furthermore, voxel-based morphometry (VBM) analysis was undertaken to assess the relationship between (i) CBF and total depression scores and (ii) CBF and BMI with respect to each state as well as to the comparison of two states. For this, the normalized CBF maps have been subtracted between hunger and satiety state, then resulting delta images have been entered in the VBM analysis. These VBM analyses aimed to identify differences in the local composition of brain tissue [START_REF] Mechelli | Voxel-Based Morphometry of the Human Brain: Methods and Applications[END_REF] .

All results were age-and sex-adjusted using two covariates of no interest in the general linear model (GLM) that is fit to each voxel's data. Results were reported with a statistical pvalue threshold corresponding to p<5.10 -4 voxel-based with a minimum cluster size of 30 voxels, uncorrected; and p<0.05 cluster-based, FDR corrected.

In addition, the estimated parameter values ('beta' images) were extracted from VBM analyses in each peak score T for each subject. Then, a correlation analyses were performed between beta values and total depression scores. They were carried out using the R software, version 3.1.0.

Being that the size of our sample is low (n=15), non-parametric tests were also performed for the paired-t-test using TFCE (Threshold-Free Cluster Enhancement) method, as described by Han et al. 2019. It consists to a generalized combination of cluster size and voxel intensity to increase statistical power [START_REF] Han | Evaluating Alternative Correction Methods for Multiple Comparison in Functional Neuroimaging Research[END_REF][START_REF] Spisák | Probabilistic TFCE: A generalized combination of cluster size and voxel intensity to increase statistical power[END_REF] . Our results were very similar than those by using the classic method, therefore they were not reported here for clarity (available on request).

Results

Participant characteristics

Among the participants only one individual was clinically diagnosed with depression (T>56) and was excluded from analysis. The mean total depression score (mean±SD) was 45.9±8.2. Furthermore, no correlation was found between CBF and BMI during hunger and satiety state separately as well as in the difference (hunger vs satiety).

CBF in hunger vs satiety states

Brain regions showing significant differences between hungry and satiated states are illustrated in Figure 1 and Table 1 with a statistical threshold of p<5.10 -4 voxel-based, uncorrected and p<0.05 cluster-based, FDR corrected.

By comparison with a state of satiety, during a state of hunger CBF increased significantly in the bilateral posterior insula.

By comparison with a state of hunger, a state of satiety showed significantly higher CBF within the right cuneus, left lingual gyrus and left precuneus.

CBF and mood

Results of whole-brain voxel-based morphometry in relation to CBF and total depression scores in each state are reported in Figure 2 and Table 2 with a statistical threshold of p<5.10 -4 voxel-based, uncorrected and p<0.05 cluster-based, FDR corrected.

During hunger, CBF increased proportionate to higher total depression scores in the right superior and inferior occipital gyrus, bilateral cuneus, left precuneus, left anterior cingulate gyrus, right anterior orbitofrontal gyrus and left cerebellum. No correlation was found for satiety and the comparison hunger vs satiety.

During hunger, a significant correlation was observed between the CBF and total depression score in the right superior occipital gyrus (R²=0.78, p=3.11e-05), right inferior occipital gyrus (R²=0.77, p=3.49e-05), left precuneus (R²=0.75, p=6.16e-05), left anterior cingulate gyrus (R²=0.74, p=7.16e-05), right anterior orbitofrontal cortex (R²=0.73, p=9.02e-05), left cerebellum (R²=0.71, p=1.42e-04), right cuneus (R²=0.64, p=5.97e-04) and left cuneus (R²=0.69, p=2.53e-04). The scatterplots were reported in the figure 3 in order to better represent the dispersion of data.

Discussion

The present study aims to explore the impact of hunger and mood on cerebral blood flow changes in healthy adolescents using arterial spin labeling.

Our results reveal that hunger is associated with increased CBF in the bilateral posterior insula, which is an area previously described as important in the anticipation and motivation of feeding. In contrast, satiation is associated with increased CBF in the left precuneus, left lingual gyrus and right cuneus which are involved in the aspects of response inhibition related to food intake. This comparison between states of hunger and satiety has enhanced our findings by using a most reproducible and robust ASL imaging sequence [START_REF] Alsop | Recommended implementation of arterial spin-labeled perfusion MRI for clinical applications: A consensus of the ISMRM perfusion study group and the European consortium for ASL in dementia[END_REF] . In the literature, certain authors focused on a pulsed ASL sequence [START_REF] Frank | Fat intake modulates cerebral blood flow in homeostatic and gustatory brain areas in humans[END_REF][START_REF] Eldeghaidy | Prior Consumption of a Fat Meal in Healthy Adults Modulates the Brain's Response to Fat[END_REF] while we used a 3D pseudo-continuous ASL sequence that is a typical approach in the context of clinical imaging, providing superior labeling efficiency and better compatibility of body coil RF transmission [START_REF] Alsop | Recommended implementation of arterial spin-labeled perfusion MRI for clinical applications: A consensus of the ISMRM perfusion study group and the European consortium for ASL in dementia[END_REF] .

Furthermore, we have shown that correlation between CBF and total depression scores was modulated by appetite in the left precuneus, left anterior cingulate gyrus, right anterior orbitofrontal gyrus, left cerebellum, bilateral cuneus and right inferior and superior occipital gyrus. However, these results have been reported only in the hunger state but not in the comparison between hunger vs satiety. Therefore, all of these confirm evidence of a moderate relationship between mood and appetite even in participants unaffected by depression, suggesting that adolescents could have subclinical symptoms of depression hence concurring with findings in the literature in healthy adult populations [START_REF] Ezzati | Hippocampal correlates of depression in healthy elderly adults[END_REF][START_REF] Spalletta | Hippocampal multimodal structural changes and subclinical depression in healthy individuals[END_REF][START_REF] Schweizer | Symptoms of depression in a large healthy population cohort are related to subjective memory complaints and memory performance in negative contexts[END_REF] . Indeed in our study, adolescents had total depression scores below the clinical threshold of depression but the subscale scores had a tendency to vary which may account for mood fluctuations in our participants and CBF variations.

CBF in hunger vs satiety states

We found that CBF was significantly increased in the bilateral posterior insula when hungry compared to when satiated. This finding is in line with previous fMRI research reporting that the insula is activated when hungry relative to when satiated, and during visual presentation of high-calorie food pictures relative to neutral pictures [START_REF] Killgore | Cortical and limbic activation during viewing of high-versus low-calorie foods[END_REF][START_REF] Stoeckel | Widespread reward-system activation in obese women in response to pictures of high-calorie foods[END_REF] . Furthermore, ASL and PET imaging of regional CBF studies confirms these results by showing that CBF in the insula increases before liquid meal ingestion [START_REF] Tataranni | Neuroanatomical correlates of hunger and satiation in humans using positron emission tomography[END_REF] and decreases after administration of a liquid meal, high-fat yogurt or a high-fat meal [START_REF] Del Parigi | Neuroimaging and obesity: mapping the brain responses to hunger and satiation in humans using positron emission tomography[END_REF][START_REF] Frank | Fat intake modulates cerebral blood flow in homeostatic and gustatory brain areas in humans[END_REF][START_REF] Eldeghaidy | Prior Consumption of a Fat Meal in Healthy Adults Modulates the Brain's Response to Fat[END_REF] . Based on other studies highlighting a link between regional CBF changes in the insula and the motivational value of chocolate [START_REF] Porubská | Subjective feeling of appetite modulates brain activity: an fMRI study[END_REF][START_REF] Small | Changes in brain activity related to eating chocolate: from pleasure to aversion[END_REF] , certain authors have suggested that CBF changes in the insula are modulated by motivational factors such as feeling hungry. Another hypothesis is that such changes could also be linked to alteration in insulin concentration. Tataranni et al. have reported in extreme fast of 36h that an increase in plasma insulin concentration is correlated with a regional CBF decrease in the insula, supporting the hypothesis of the anorectic role of insulin [START_REF] Tataranni | Neuroanatomical correlates of hunger and satiation in humans using positron emission tomography[END_REF] . Therefore, the transition from hunger to satiety could result in changes in insulin levels in response to meals through a decrease in CBF in the insula which may be modulated by feelings of hunger, playing an essential role in food-related motivation. This result concurs with our findings which show an increase in CBF before food-intake corresponding to a decrease in CBF when satiated as described in the literature. In this way, other authors established that the posterior insula could dynamically contribute to conscious interoceptive using a posterior-to-anterior gradient within insular cortex [START_REF] Frank | Food related processes in the insular cortex[END_REF][START_REF] Gu | Anterior insular cortex is necessary for empathetic pain perception[END_REF][START_REF] Barrett | Interoceptive predictions in the brain[END_REF][START_REF] Simmons | Interoceptive contributions to healthy eating and obesity[END_REF] . Prediction-error information of interoception would be process in the posterior insula, whereas the cognitive and motivational information of interoception would be integrate in the anterior insula. Consequently, interoceptive awareness of variations from energy homeostasis can be a crucial part in the activity of insular cortex. Hunger feeling could lead to greater interoceptive prediction signals by anticipating of the food intake which could explain the increase of CBF observed during hunger compared to satiety.

By comparison with hunger, we have identified a significant increase in CBF in the left precuneus during satiety. This region, known to exhibit decreased activation during goaldirected action, has been shown to participate in abstract visuospatial mental imagery and self-processing operations [START_REF] Cavanna | The precuneus: a review of its functional anatomy and behavioural correlates[END_REF][START_REF] Tuulari | Neural Circuits for Cognitive Appetite Control in Healthy and Obese Individuals: An fMRI Study[END_REF] . Recent fMRI research has reported that meal ingestion increases activity in the precuneus suggesting that involvement of the precuneus would appear to be essential in the processing of satiety [START_REF] Cheah | Ageing diminishes the modulation of human brain responses to visual food cues by meal ingestion[END_REF] . The authors speculate that ingestion of a recent meal leads to changes in the imagined or anticipated taste of the food which may influence prospective food consumption by consciously suppressing of food cravings. In contrast, two PET studies were agreed to provide evidence that the hunger (before liquid meal ingestion) was associated with increased CBF in the bilateral precuneus [START_REF] Tataranni | Neuroanatomical correlates of hunger and satiation in humans using positron emission tomography[END_REF] or that the satiation was associated with decreased CBF bilaterally in the precuneus [START_REF] Gautier | Differential brain responses to satiation in obese and lean men[END_REF] . This change is interpreted as an increase in the attention and salience-encoding for the food when the appetite is high [START_REF] Bruce | A comparison of functional brain changes associated with surgical versus behavioral weight loss[END_REF][START_REF] Charbonnier | Functional MRI of Challenging Food Choices: Forced Choice between Equally Liked High-and Low-Calorie Foods in the Absence of Hunger[END_REF] . The discrepancies between all these studies could be explained by the differences between PET and fMRI imaging methodologies. We can also note that these studies have been performed with a shift of around 15 years which could explain the mismatches due to large advances in MRI technology and imaging analysis. Despite the lack of consensus, our results are in concordance with recent fMRI finding of Cheah [START_REF] Cheah | Ageing diminishes the modulation of human brain responses to visual food cues by meal ingestion[END_REF] and indicate that this functional increase in activity could be associated with the change in cerebral blood flow that was observed in our results. This could suggest the involvement of precuneus in the processing of satiety by consciously reducing food cravings. Furthermore, during satiety by comparison with hunger, we observed increased CBF in the left lingual gyrus and right cuneus, areas known to be involved in visual processing [START_REF] Gearhardt | Relation of obesity to neural activation in response to food commercials[END_REF] . Although rarely discussed, both of these regions have often been associated with the visual presentation of food especially in high-calorie activation [START_REF] Killgore | Cortical and limbic activation during viewing of high-versus low-calorie foods[END_REF][START_REF] Rothemund | Differential activation of the dorsal striatum by high-calorie visual food stimuli in obese individuals[END_REF][START_REF] Tang | Food and drug cues activate similar brain regions: A meta-analysis of functional MRI studies[END_REF][START_REF] Luo | Differential effects of fructose versus glucose on brain and appetitive responses to food cues and decisions for food rewards[END_REF] . This confirms the role of the left lingual gyrus and right cuneus in response to food cues. However, one fMRI study comparing satiety and hunger reported that the bilateral cuneus was only recruited during satiety [START_REF] Santel | Hunger and satiety in anorexia nervosa: fMRI during cognitive processing of food pictures[END_REF] , in discordance with our results which suggest a right-lateralization of the response to the satiety. Authors was also demonstrated that the bilateral lingual gyrus was involved during hunger compared to satiety, inconsistent with our findings. Two hypotheses could be raised to explain these discrepancies. First, the authors included only female participants whereas we recruited 7 female and 8 male adolescents. Given that CBF varies from women to men [START_REF] Rodriguez | Sex Differences in Regional Cerebral Blood Flow[END_REF] especially concerning responses to satiation [START_REF] Smeets | Effect of satiety on brain activation during chocolate tasting in men and women[END_REF] , these sex differences could account for the discrepancies between our results and those found in the literature, although this effect has been corrected in our analysis. A second explanation could implicate the methodology due to the fact that the authors used fMRI with visual presentations of food whereas we focused on ASL imaging at rest without visual stimuli. Moreover, other fMRI studies have demonstrated that the left lingual gyrus was positively coupled with appetite inhibition and satiety in anorexics [START_REF] Santel | Hunger and satiety in anorexia nervosa: fMRI during cognitive processing of food pictures[END_REF][START_REF] Ioannou | Preliminary fMRI findings concerning the influence of 5-HTP on food selection[END_REF] . Therefore, in our study, it may be suggested that the left lingual gyrus and right cuneus -like the left precuneus described above -were instrumental in consciously suppressing eating.

On the basis of such concordance, our results support the idea that the functional changes reported in fMRI studies conceivably reflect the blood flow changes identified in this work.

CBF and mood

We observed an influence of mood on CBF in terms of appetite only during hunger. Increased CBF was correlated with higher total depression scores in the precuneus, anterior cingulate gyrus, orbitofrontal cortex, cerebellum, cuneus and occipital gyrus.

Certain authors have reported that the left precuneus is metabolically overactive in treatment-resistant major depression [START_REF] Ho | Brain glucose metabolism during non-rapid eye movement sleep in major depression. A positron emission tomography study[END_REF][START_REF] Leech | The role of the posterior cingulate cortex in cognition and disease[END_REF] . Since the precuneus is involved in introspective processing, some authors have suggested that their findings may reflect a tendency towards rumination in depression, given that depressed individuals repetitively entertain these negative thoughts [START_REF] Cooney | Neural correlates of rumination in depression[END_REF] . Furthermore, other studies related to resting-state functional MRI describe greater connectivity in the precuneus among patients with major depressive disorder [START_REF] Greicius | Resting-State Functional Connectivity in Major Depression: Abnormally Increased Contributions from Subgenual Cingulate Cortex and Thalamus[END_REF][START_REF] Berman | Depression, rumination and the default network[END_REF] . This region constitutes a component of the default mode network (DMN) and underlies its participation in self-evaluation and emotional regulation [START_REF] Buckner | The serendipitous discovery of the brain's default network[END_REF] . Therefore, certain authors have interpreted their findings as changes in the DMN resulting in greater manipulation of negative emotional content, and they support the idea of a close relationship between increased CBF and excessive functional connectivity in depressed subjects [START_REF] Greicius | Resting-State Functional Connectivity in Major Depression: Abnormally Increased Contributions from Subgenual Cingulate Cortex and Thalamus[END_REF] . In our study, we also observed that increased CBF was correlated with higher total depression scores in the precuneus occurring only when hungry. This suggests that the more participants tend to feel depressed, the more they focus on eating and ruminative processes when they are deprived of food.

These results are corroborated by Greicius et al. [START_REF] Greicius | Resting-State Functional Connectivity in Major Depression: Abnormally Increased Contributions from Subgenual Cingulate Cortex and Thalamus[END_REF] who demonstrate that subjects affected by major depressive disorder show increased functional connectivity in the anterior cingulate gyrus. They support the hypothesis of hyper-connectivity in the DMN which may reflect constrained thinking as a ruminative process [START_REF] Greicius | Resting-State Functional Connectivity in Major Depression: Abnormally Increased Contributions from Subgenual Cingulate Cortex and Thalamus[END_REF][START_REF] Berman | Depression, rumination and the default network[END_REF] . However, studies in ASL imaging (pulsed ASL) show inhomogeneous findings, reporting an increase [START_REF] Duhameau | Chronic and treatment-resistant depression: a study using arterial spin labeling perfusion MRI at 3Tesla[END_REF] or decrease [START_REF] Ho | Altered cerebral perfusion in executive, affective, and motor networks during adolescent depression[END_REF][START_REF] Ota | Characteristic distributions of regional cerebral blood flow changes in major depressive disorder patients: A pseudo-continuous arterial spin labeling (pCASL) study[END_REF] in CBF in the anterior cingulate gyrus in cases of major depressive disorder. Our findings show a correlation between increased CBF and higher total depression scores in the left anterior cingulate gyrus among healthy adolescents in a state of hunger, in accordance with Duhameau et al. [START_REF] Duhameau | Chronic and treatment-resistant depression: a study using arterial spin labeling perfusion MRI at 3Tesla[END_REF] The anterior cingulate gyrus is known to be involved in mood regulation and the processing of social information such as rewards, motivation, error monitoring and decision making [START_REF] Apps | The Anterior Cingulate Gyrus and Social Cognition: Tracking the Motivation of Others[END_REF] . This is consistent with our preceding assumption that, in a state of hunger, adolescents with higher total depression scores tend to dwell on the same thought by focusing on the benefit of an upcoming meal as a reward. Thus, we widen the scope of these results by showing a correlation between increased CBF and higher total depression scores in the right anterior orbitofrontal gyrus during a state of hunger. A number of studies have identified that the anterior orbitofrontal gyrus is positively related to the severity and various symptoms of major depressive disorder [START_REF] Périco | Relationship between regional cerebral blood flow and separate symptom clusters of major depression: A single photon emission computed tomography study using statistical parametric mapping[END_REF][START_REF] Drevets | Orbitofrontal cortex function and structure in depression[END_REF][START_REF] Jin | Orbitofrontal Cortex Activity and Connectivity Predict Future Depression Symptoms in Adolescence[END_REF] , suggesting that this type of dysfunction could mediate the manifestation of depression. It has been reported that the right anterior orbitofrontal gyrus plays a critical role in abstract goaloriented rewards [START_REF] Peters | Neural representations of subjective reward value[END_REF] and in recognition memory [START_REF] Morris | Involvement of Human Amygdala and Orbitofrontal Cortex in Hunger-Enhanced Memory for Food Stimuli[END_REF] . Consequently, these findings confirm our hypothesis that participants affected by depressed mood and in fasting state are able to recall their past experience of food-motivated behavior in order to evaluate the reward of an upcoming meal. Furthermore, we have observed that increased CBF is found to correlate with higher total depression scores in the bilateral cuneus and right occipital gyrus during a state of hunger. This concords with the study conducted by Hwang et al. showing increased CBF in the right superior occipital gyrus and cuneus in depression [START_REF] Hwang | Subthreshold depression is associated with impaired resting-state functional connectivity of the cognitive control network[END_REF] . These regions are known to participate in the processing of visual information and the generation of mental imagery [START_REF] Gearhardt | Relation of obesity to neural activation in response to food commercials[END_REF][START_REF] Farah | The neural basis of mental imagery[END_REF][START_REF] Bértolo | Visual imagery without visual perception?[END_REF][START_REF] Ganis | Visual imagery[END_REF] . We therefore postulate that a state of hunger could lead to anticipation of food intake using mental representations of meals among participants with higher total depression scores. They may retrieve previous food-related experiences from long-term memory into working memory to construct visual representations of food.

Limitations

Nevertheless, most studies on regional CBF imaging by fMRI and PET have reported the involvement of the orbitofrontal cortex in food-related processes [START_REF] Porubská | Subjective feeling of appetite modulates brain activity: an fMRI study[END_REF][START_REF] Tataranni | Neuroanatomical correlates of hunger and satiation in humans using positron emission tomography[END_REF][START_REF] Del Parigi | Neuroimaging and obesity: mapping the brain responses to hunger and satiation in humans using positron emission tomography[END_REF][START_REF] Killgore | Cortical and limbic activation during viewing of high-versus low-calorie foods[END_REF][START_REF] Stoeckel | Widespread reward-system activation in obese women in response to pictures of high-calorie foods[END_REF] . Consequently, we would expect to find CBF changes in this region with regard to appetite, yet no changes were observed when comparing states of hunger and satiety. Likewise, ASL studies reported no CBF changes in this region following consumption of a high-fat meal or water load [START_REF] Eldeghaidy | Prior Consumption of a Fat Meal in Healthy Adults Modulates the Brain's Response to Fat[END_REF] or highfat yogurt [START_REF] Frank | Fat intake modulates cerebral blood flow in homeostatic and gustatory brain areas in humans[END_REF] . A major explanation for this lack of activity could reside in imaging difficulties, causing magnetic susceptibility artifacts in the ventral brain areas especially when using highfield magnets; however we used pseudo-continuous ASL imaging which is known to demonstrate higher quality compared to the pulsed ASL sequences generally described in the literature. A second possible limitation addresses mood assessment that may be influenced by the reliability of the means of assessment since the measure of mood was self-reported. Furthermore, it might be worthwhile to perform this measure before the first MRI and after the second MRI scan to assess whether appetite influences participant responses. For our part we evaluated mood immediately after the meal, possibly entailing diminished impact of the appetite effect due to the fact that satiety had not yet been reached metabolically (postprandial phase). From a psychological perspective, the state of the participants would appear to represent neither hunger pangs nor complete satiety but may reflect a basal state, thus supporting the notion that total depression scores may be applicable to both states (hunger and satiety).

Conclusion

To conclude, our study confirms, by means of more efficient ASL sequences, involvement of the posterior insula and precuneus in the modulation of hunger. We also highlight a hitherto unpublished observation indicating that feelings of hunger may moderately modulate the relationship between mood and brain activity in specific regions. These results suggest that participants with higher total depression scores may use ruminative processes and visual representations of food to evaluate the reward of an upcoming meal. Therefore, this study involving healthy adolescents opens the way to potential avenues of future research on eating disorders such as obesity. Further research could also incorporate connectivity analysis, to link the identified regions with CBF changes, into the functional networks involved in hunger and mood fluctuations, namely depression.

Acknowledgements

This project is supported by NCT02868619 from the National Institute of Mental Health.

Author Contribution statement

Céline Charroud has a substantial contribution to the analysis of data, interpretation of data and the writing of the article.

Nicolas Menjot de Champfleur is involved in the revision of the article.

Emily Sanrey participates in concept and design, planned the acquisition and provided a feedback of the article.

Josef Pfeuffer conceived the design of ASL prototype sequence.

Jérémy Deverdun participates in the analysis of data.

Emmanuelle Le Bars contributes to the concept and design of the study and performs the critical revision of the article for important intellectual content.

Philippe Coubes directed the project and provided a feedback of the article.

All authors discussed the results and commented on the manuscript.

Declaration of conflicting interests

All procedures performed in this study were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

Informed consent was obtained from all individual participants included in the study.

All co-authors have seen and agree with the content of the manuscript. The authors hereby certify that the submission is not under review with any other publication. The authors declare that they have no conflict of interest.

Footnotes

During hunger compared to satiety, CBF increases in the bilateral posterior insula (red). During satiety compared to hunger, CBF increases in the right cuneus, left lingual gyrus and left precuneus (blue).

A statistical p-value threshold was used corresponding to p<5.10 -4 voxel-based with a minimum cluster size of 30 voxels, uncorrected and p<0.05 cluster-based, FDR corrected. Results were age-and sex-adjusted.

Abbreviations: CBF, Cerebral Blood Flow; L, Left; R, Right.

File name

Figure 2

Title

Statistical parametric maps showing a positive correlation between blood flow and mood assessment when hungry.

Footnotes

Mood assessment was measured using total depression scores. When hungry, a correlation between increased CBF and higher total depression scores was identified in the precuneus, anterior cingulate gyrus, anterior orbitofrontal cortex, cerebellum, cuneus and occipital gyrus. No correlation was found when satiated.

A statistical p-value threshold was used corresponding to p<5.10 -4 voxel-based with a minimum cluster size of 30 voxels, uncorrected and p<0.05 cluster-based, FDR corrected. Results were age-and sex-adjusted.

Abbreviations: CBF, Cerebral Blood Flow; L, Left; R, Right.

File name

10.1.3. Figure 3 Title Scatterplots of the correlation between blood flow and mood assessment when hungry. Abbreviations: a.u., arbitrary unity; CBF, Cerebral Blood Flow.

Footnotes

File name

Table 10.2.1. Table 1

Title

Brain regions showing increased cerebral blood flow in states of hunger and satiety.

Footnotes

Results were evaluated using paired t-test analysis adjusting for age and sex. A statistical pvalue threshold was used corresponding to p<5.10 -4 voxel-based with a minimum cluster size of 30 voxels, uncorrected and p<0.05 cluster-based, FDR corrected. Results were age-and sex-adjusted.

Abbreviations: sc, same cluster; L, Left; R, Right.

File name

Table1.docx

Table 2

Title

Brain regions showing correlation between cerebral blood flow and total depression scores when hungry.

Footnotes

Positive correlation was reported between cerebral blood flow and total depression scores during hunger. Results were evaluated using voxel-based morphometry analysis by adjusting for age and sex. A statistical p-value threshold was used corresponding to p<5.10 -4 voxelbased with a minimum cluster size of 30 voxels, uncorrected and p<0.05 cluster-based, FDR corrected.

Abbreviations: sc, same cluster; L, Left; R, Right

File name

Table2.docx


Pierre Maréchal
email: pr.marechal@gmail.com

Aude Rondepierre
email: aude.rondepierre@math.univ-toulouse.fr

A proximal approach to the inversion of ill-conditioned matrices

We propose a general proximal algorithm for the inversion of ill-conditioned matrices. This algorithm is based on a variational characterization of pseudo-inverses. We show that a particular instance of it (with constant regularization parameter) belongs to the class of fixed point methods. Convergence of the algorithm is also discussed.

Introduction

Inverting ill-conditioned large matrices is a challenging problem involved in a wide range of applications, including inverse problems (image reconstruction, signal analysis, etc.) and partial differential equations (computational fluid dynamics, mechanics, etc.). There are two classes of methods: the first one involves factorization of the matrix (SVD, QR, LU, LQUP); the second one involves iterative schemes (fixed point methods, projection onto increasing sequences of subspaces).

The main purpose of this note is to show that a particular instance of the Proximal Point Algorithm provides a fixed point method for the problem of matrix inversion. This fact is based on the observation that the pseudo-inverse M † of a matrix M ∈ R m×n satisfies the fixed point equation:

Φ = ϕ(Φ) := BΦ + C, with B := (I + µM ⊤ M) -1 and C := (M ⊤ M + µ -1 I) -1 M ⊤
where µ > 0. The corresponding fixed point iteration Φ k+1 = BΦ k + C is nothing but a proximal iteration. We see that ϕ is a contraction and that, if M ⊤ M is positive definite, then ϕ is a strict contraction. It is worth noticing that, in the proximal algorithm, µ may depend on k, allowing for large (but inaccurate) steps for early iteration and small (but accurate) steps when approaching the solution.

The Proximal Point Algorithm (PPA) was introduced in 1970 by Martinet [START_REF] Martinet | Régularisation d'inéquations variationelles par approximations successives[END_REF], in the context of the regularization of variational inequalities. A few years later, Rockafellar [START_REF] Rockafellar | Monotone operators and the proximal point algorithm[END_REF] generalized the PPA to the computation of zeros of a maximal monotone operator. Under suitable assumptions, it can be used to efficiently minimize a given function, by finding iteratively a zero in its Clarke subdifferential.

Throughout, we denote by M F the Frobenius norm of a matrix M ∈ R m×n and by M, N F the Frobenius inner product of M, N ∈ R m×n (which is given by M, N F = tr(MN ⊤ ) = tr(N ⊤ M)). In R m×n , we denote by dist F (M, S) the distance between a matrix M and a set S:

dist F (M, S) := inf{ M -M ′ F |M ′ ∈ S}.
The identity matrix will be denoted by I, its dimension being always clear from the context. The next theorem, whose proof may be found e.g. in [START_REF] Amodei | Analyse Numérique Matricielle[END_REF], provides a variational characterization of M † . Theorem 1.1 The pseudo-inverse of a matrix M ∈ R m×n is the solution of minimum Frobenius norm of the optimization problem

(P) Minimize f (Φ) := 1 2 MΦ -I 2 F over R n×m .

The proximal point algorithm

The proximal point algorithm is a general algorithm for computing zeros of maximal monotone operators. A well-known application is the minimization of a convex function f by finding a zero in its subdifferential. In our setting, it consists in the following steps:

1. Choose an initial matrix Φ 0 ∈ R m×n ;

2. Generate a sequence (Φ k ) k≥0 according to the formula

Φ k+1 = argmin Φ∈R m×n f (Φ) + 1 2µ k Φ -Φ k 2 F , (1)
in which (µ k ) k≥0 is a sequence of positive numbers, until some stopping criterion is satisfied.

Equation (1) will be subsequently referred to as the proximal iteration of Problem (P). The stopping criterion may combine, as usual, conditions such as

∇f (Φ k ) F ≤ ε 1 and Φ k -Φ k-1 F ≤ ε 2 ,
where the parameters ε 1 and ε 2 control the precision of the algorithm.

Clearly, the function f : Φ → MΦ -I 2 F /2 is convex and indefinitely differentiable. Therefore, solutions of the proximal iteration (1) are characterized by the relationship ∇f

(Φ k+1 ) + µ -1 k (Φ k+1 -Φ k ) = 0 i.e., (I + µ k M ⊤ M)Φ k+1 = Φ k + µ k M ⊤ . (2)
Since M ⊤ M is positive semi-definite and µ k is chosen to be positive for all k, the matrix (I + µ k M ⊤ M) is nonsingular and the proximal iteration also reads:

Φ k+1 = I + µ k M ⊤ M -1 Φ k + µ k M ⊤ . (3)
The iterates Φ k could be computed either exactly (in the ideal case), or approximately, using e.g. any efficient minimization algorithm to solve the proximal iteration [START_REF] Amodei | Analyse Numérique Matricielle[END_REF]. In that case, we need another stopping criterion and we here choose the following one suggested in [START_REF] Luque | Asymptotic convergence analysis of the proximal point algorithm[END_REF]:

Φ k+1 -A(Φ k + µ k M ⊤ ) F ≤ ǫ k min{1, Φ k+1 -Φ k r F }, r > 1 ( 4
)
where ǫ k > 0 and the series ǫ k is convergent. Notice that the larger r, the more accurate the computation of Φ k+1 . Notice also that, in the case where µ k = µ for all k, each proximal iteration involves the multiplication by the same matrix A := (I + µM ⊤ M) -1 , and that the latter inverse may be easy to compute numerically, if the matrix I + µM ⊤ M is well-conditioned.

We now turn to convergence issues. Recall that our objective function f is a quadratic function whose Hessian M ⊤ M is positive semi-definite. Nevertheless, unless M ⊤ M is positive definite, the matrix I -(I + µM ⊤ M) -1 is in general singular and the classical convergence theorem for iterative methods (see e.g. [START_REF] Ciarlet | Introduction to numerical linear algebra and optimisation[END_REF]) is not helpful here to prove the convergence of our proximal scheme. The following proposition is a consequence of Theorem 2.1 in [START_REF] Luque | Asymptotic convergence analysis of the proximal point algorithm[END_REF]. For clarity, we shall denote by M the linear mapping Φ → MΦ, by L the linear mapping Φ → M ⊤ MΦ and by A the linear mapping Φ → AΦ = (I + µM ⊤ M) -1 Φ. Proposition 2.1 Let α 1 be the smallest nonzero eigenvalue of L and let E 1 be the corresponding eigenspace. Assume that µ k = µ for all k and that Φ 0 is not •, • F -orthogonal to the eigenspace E 1 . Then,

A(Φ k+1 -Φ k ) F Φ k+1 -Φ k F → 1 1 + α 1 µ and Φ k+1 -Φ k Φ k+1 -Φ k F → Ψ 1 as k → ∞,
in which Ψ 1 is a unit eigenvector in E 1 . Moreover the sequence (Φ k ) generated by the proximal algorithm, either with infinite precision or using the stopping criterion (4) for the inner loop, converges linearly to the orthogonal projection of Φ 0 onto the solution set S := argmin f = M † + ker M.

Proof.

Step 1. The error ∆ k+1 := Φ k+1 -Φ k (at iterate k + 1) satisfies: ∆ k+1 = (I + µM ⊤ M) -1 ∆ k . The latter iteration is that of the power method for the linear mapping A.

Clearly, A is symmetric and positive definite. Consequently, its eigenvalues are strictly positive and there exists an unique eigenvalue of largest modulus (not necessarily simple).

Notice now that the eigenspace E 1 associated to the eigenvalue α 1 is nothing but the eigenspace of A for its largest eigenvalue strictly smaller than 1, namely, 1/(1 + α 1 µ). We proceed as in [START_REF] Vige | Proximal-Point Algorithm for Minimizing Quadratic Functions[END_REF]Theorem 1] to obtain the desired convergence rate via that of the iterated power method.

Step 2. We now establish the linear convergence of the sequence (Φ k ). First, the solution set S is clearly nonempty since it contains M † . Moreover, let (Φ k ) be a sequence generated by the PPA algorithm using the stopping criterion (4). Let us prove that

∃a > 0, ∃δ > 0, ∀Φ ∈ R m×n , ∇f (Φ) F < δ ⇒ dist F (Φ, S) ≤ a ∇f (Φ) F , (5)
which is nothing but Condition (2.1) in [4, Theorem 2.1], in our context. Let Φ ∈ R m×n and let Φ be the orthogonal projection of φ over (ker M) ⊥ . It results from the classical theory of linear least squares that dist

F (Φ, S) = Φ-M † F . Since M ⊤ M Φ-M ⊤ = ∇f ( Φ) and M ⊤ MM † -M ⊤ = 0, we also have: ∇f (Φ) = M ⊤ M( Φ -M † ). Moreover, Φ -M † ∈ (ker M) ⊥ = (ker L) ⊥ , so that ∇f (Φ) F = M ⊤ M( Φ -M † ) F ≥ α 1 Φ -M † F .
It follows that ( 5) is satisfied with a = 1/α 1 . The conclusion then follows from [4, Theorem 2.1]: the sequence (Φ k ) converges linearly with a rate bounded by a/ a 2 + µ 2 = 1/ 1 + µ 2 α 2 1 < 1.

Step 3. By rewriting the proximal iteration in an orthonormal basis of eigenvectors of L, we finally prove that the limit of the sequence (Φ k ) is the orthogonal projection of Φ 0 onto argmin f . A complete numerical study, which goes beyond the scope of this paper, is currently in progress and will be presented in a forthcoming publication. Let us merely mention that our proximal approach makes it possible to combine features of factorization methods (in the proximal iteration) with features of iterative schemes. In particular, if M if invertible, it shares with iterative methods the absence of error propagation and amplification, since each iterate can be regarded as a new initial point of a sequence which converges to the desired solution.

3 Comments Tikhonov approximation. A standard approximation of the pseudo-inverse of an illconditioned matrix M is (M ⊤ M + εI) -1 M ⊤ , where ε is a small positive number. This approximation is nothing but the Tikhonov regularization of M † , with regularization parameter ε. It is worth noticing that the choice Φ 0 = 0 in the proximal algorithm yields the latter approximation for ε = 1/µ after one proximal iteration.

Trade-offs. At the k-th proximal iteration, the perturbation of the objective function f is, roughly speaking, proportional to the square of the distance between the current iterate and the solution set of (P), and inversely proportional to µ k . In order to speed up the algorithm, it seems reasonable to choose large µ k for early iterations, yielding large but inaccurate steps, and then smaller µ k for late iterations, where proximity with the solution set makes it suitable to perform small and accurate steps. This is especially true in the case where M is invertible, since the solution set then reduces to {M -1 }. Moreover, numerical accuracy in early proximal iteration may be irrelevant, since the limit of the proximal sequence is what really matters. A trade-off between a rough approximation of the searched proximal point and an accurate and costly solution must be found. As suggested in [START_REF] Correa | Convergence of some algorithms for convex minimization[END_REF], one may use the following stopping criterion for the proximal iteration:

f (Φ k+1 ) -f (Φ k ) ≤ δ ∇f (Φ k+1 ), Φ k+1 -Φ k F .
This criterion is an Armijo-like rule: the algorithm stops when the improvement of the objective function f is at least a given fraction δ ∈ (0, 1) of its ideal improvement.

Inversion versus linear systems. It is often unnecessary to compute the inverse of a matrix M, in particular when the linear system Mx = d must be solved for a few data vectors d only. In such cases, of course, the usual proximal strategy may be used to compute least squares solutions. It is important to realize that, although the regularization properties of the proximal algorithm are effective at every proximal iteration, perturbations of d may still have dramatic effects on the algorithm if M is ill-conditioned. In applications for which no perturbation of the data must be considered, accurate solutions may be reached by a proximal strategy. We emphasize that, in the minimization of Φ → MΦ -I F , the data I undergoes no perturbation whatsoever.


Vincent Caldeira

Julien Thiel

René Lacoste

Laetitia Dubau

Marian Chatenet

Improving Zinc Porous Electrode for Secondary Alkaline Batteries: Toward a Simple Design of Optimized 3D Conductive Network Current Collector

Keywords: Zinc-Nickel (Zn / Ni) batteries, Multilayer current collector, Migration control, Core-shell mechanism

The internal migration of zinc within the thickness of porous composite negative electrodes for rechargeable zinc batteries causes detrimental shape change of the zinc electrode. It depends on various parameters linked to the electron conduction in the electrode: electronic percolation, conductive additives, primary current collector. This study highlights a simple solution, that drastically improves the electrochemical performance of thick and porous zinc electrodes, by rethinking their architecture. Using a multiplicity of thin current collectors enables to create a 3D network of electronic percolation and a better repartition of the current lines within the electrode; as a result, the formation of multiple active zinc cores in the electrode volume is better controlled. This strategy allows to better manage zinc migration within the electrode structure, leading to minimal densification of each zinc core, and overall enables higher reversibility of zinc oxidation/reduction upon discharge/charge. It finally mitigates the initial drastic capacity fading observed on standard type calcium zincate negative electrode.

Introduction

Global warming, pollution, energy consumption, non-renewable sources, fossil fuels... These words are overwhelming the international scientific community and are levers that enable to improve scientific and technological aspects of "green" energy. In response, dedicated works concern electrochemical storage and conversion systems, which are widely used and developed for automotive and stationary applications. Among a wide portfolio of batteries, alkaline ones with zinc negative electrodes, either zinc-air or zinc-nickel systems, are interesting candidates, owing to their attractive specific capacity, high power density, safety and environmental friendliness [START_REF] Linden | Handbook of Batteries[END_REF]. They are essentially composed of abundant and non-toxic materials and can be manufactured through environmentalfriendly processes. However, these technologies have lost attention in the 1980s, because of the emergence of more energetic Li-ion batteries. Zinc batteries usually suffer from non-negligible capacity loss and low cycle life, caused by multiple phenomena that occur during their charge and discharge. For example, shape change of the zinc electrode, a phenomenon caused by a heterogeneous migration of zinc within the electrode, leads to drastic morphological changes and contributes to longterm cycle life failure. Many research works have aimed to mitigate shape change, notably by using additives in the electrodes [START_REF] Mcbreen | The electrochemistry of metal oxide additives in pasted zinc electrodes[END_REF][START_REF] Mcbreen | Bismuth oxide as an additive in pasted zinc electrodes[END_REF][START_REF] Bass | Methods for the reduction of shape change and dendritic growth in zinc-based secondary cells[END_REF][START_REF] Mclarnon | The Secondary Alkaline Zinc Electrode[END_REF][START_REF] Zheng | Effects of barium on the performance of secondary alkaline zinc electrode[END_REF][START_REF] Cheng | Effects of additives on performance of zinc electrode[END_REF]. Usually, zinc electrodes are made of a blend of zinc oxide with conductive additives, shape change stabilizer, fibrous polymer binder, etc. in an aqueous media, pasted onto the current collector, before being dried and rolled/laminated/pressed. In most studies, a single layer of current collector is used per zinc negative electrode, either as porous (foam [START_REF] Gay | Method for preparing electrode with porous current collector structures and solid reactants for secondary electrochemical cells[END_REF][START_REF] Yuan | Electrochemical performances of Bi based compound film-coated ZnO as anodic materials of Ni-Zn secondary batteries[END_REF][START_REF] Moser | In situ X-ray diffraction investigation of zinc based electrode in Ni-Zn secondary batteries[END_REF], expanded/mesh/gauze/ribbon metal [START_REF] Jain | Development of long-lived high-performance zinc-calcium/nickel oxide cells[END_REF][START_REF] Rivers | Multi-layer current collector[END_REF][START_REF] Othman | Zinc-Air Cell with KOH-Treated Agar Layer between Electrode and Electrolyte Containing Hydroponics Gel[END_REF][START_REF] Striebel | Laboratory-scale evaluation of secondary batteries for electric vehicles[END_REF][START_REF] Hampson | The Electrochemistry of Porous Zinc V. The cycling behaviour of plain and polymer-bonded porous electrodes in KOH solutions[END_REF][START_REF] Gan | Zinc electrode with anion conducting polyvinyl alcohol/poly(diallyldimethylammonium chloride) film coated ZnO for secondary zinc air batteries[END_REF][START_REF] Wen | Lead ion and tetrabutylammonium bromide as inhibitors of the growth of spongy zinc in single flow zinc/nickel batteries[END_REF][START_REF] Jindra | Sealed nickel-zinc cells[END_REF][START_REF] Nichols | Active material redistribution rates in zinc electrodes: effect of alkaline electrolyte compositions having reduced zinc oxide solubility[END_REF][START_REF] Li | Effect of electrolyte concentration and depth of discharge for zincair fuel cell[END_REF][START_REF] Wang | Morphology control of zinc regeneration for zinc-air fuel cell and battery[END_REF][START_REF] Gunther | Zinc Electrode Shape Change in Cells with Controlled Current Distribution NiOOH NiOOH[END_REF]) or as foil/sheet [START_REF] Muller | Optimized zinc electrode for the rechargeable zinc ± air battery[END_REF][START_REF] Phillips | High rate, thin film, bipolar, nickel zinc battery having oxygen recombination facility[END_REF][START_REF] Zhong | Electrode formation by lamination of particles onto a current collector[END_REF][START_REF] Turney | Development and testing of an economic grid-scale flow-assisted zinc/nickel-hydroxide alkaline battery[END_REF] structures. Recently, copper foams emerged as the most employed primary current collector materials.

The literature usually suggests that zinc electrode shape change is a surface modification phenomenon, but a novel mechanism has been recently proposed for thick porous electrodes [START_REF] Caldeira | Controlling the shape change and dendritic growth in Zn negative electrodes for application in Zn/Ni batteries[END_REF]: the migration of zinc responsible for the shape change would occurs within the core of the electrode, in the vicinity of the primary current collector, leading to the formation of a partially-active zinc core. This internal phenomenon follows a working gradient within the thickness of the electrode, coming from the current collector to the surface of the electrode. In a desired configuration, the current collector is not in direct contact with the electrolyte to avoid dendrite formation. As thick zinc electrodes are not used at 100% of their theoretical capacity, but instead with a nominal capacity selected between a few 10% and 80% of the theoretical capacity, the surface of the electrode remains intact, after repeated charge/discharge cycles, which shall reduce the possibility of dendrite formation and maintain the active part of the electrode within the core. However, this new mechanism is accompanied by its own problem, which is the self-densification of the zinc core. During repetitive cycling, as the zinc core is only partially active, because of the local inaccessibility of zinc resulting from porosity plugging, and as the demand in energy is the same for each charge/discharge cycle, zinc must increasingly be utilized towards the electrode surface (further away from the primary current collector from cycle to cycle). This densification likely reduces the mass and charge transfer kinetics, at least in the vicinity of this zinc core, inducing cell overvoltage and practical capacity loss.

In order to prevent those eventual drawbacks, electrodes embedding a multiplicity of primary current collector were designed and patented [START_REF] Caldeira | Electrode with current collection multiple array[END_REF]. This study shows the potential of the invention and reports how controlling the formation of a multitude of small zinc cores (on each multilayered current collector), instead of a large one as usually observed in model (thick) zinc-based electrodes, improves the electrochemical performances. Such laboratory-scaled electrodes were prepared and cycled using a homemade designed Swagelok cell, and analyzed post-test using cross-section scanning electron microscopy, to follow the fate (possible redistribution) of zinc and of the electrode additives within the composite electrode.

Experimental

Preparation of Zn electrodes for 10 mAh theoretical laboratory Swagelok cell

Circular calcium zincate-based mono-facial electrodes were prepared by deposition of an aqueous ink on one layer of a copper current collector (Fig. 1). The ink was composed of calcium zincate synthesized by a hydro-micromechanical (EASYL supplier d50 = 10 ± 2 µm) [START_REF] Lacoste | Method for manufacturing calcium zincate crystals, and the uses thereof[END_REF][START_REF] Caldeira | Ultrafast Hydro-Micromechanical Synthesis of Calcium Zincate: Structural and Morphological Characterizations[END_REF] as active material, titanium nitride and bismuth oxide as conductive additives, and polyvinyl alcohol (Fluka 72 000 Mw) as binder, in which the corresponding mass ratio was 73:12:5:10. Depending on the number of current collectors used (1, 2 or 3), the amount of electrode material deposit was adapted: 1 layer = 30 mg, 2 layers = 2 × 15 mg and 3 layers = 3 × 10 mg of calcium zincate equivalent active material mass. The number of layers was defined by the amount of copper and the practical limit of processing, because of thinner layers of electrode matter, that affects the adhesion and robustness of the layers (at least using this ink process). After drying the deposit in an oven at T = 50°C, the layers were superposed, placed inside a PTFE ring and pressed (1.5 T). The PTFE ring aim to protect the edge of the electrodes, avoiding dendrite formation on the sides, and allowing to have the same electrode/electrolyte surface. A copper foam, 300 g/m², was used as model electrode for the "1 layer" electrode and copper mesh (100 g/m², with wire diameter = 70 µm) was used as multilayer current collector. The choice of these current collector is based on their economic and ecologic value: the manufacturing process to obtain copper foam needs heavy chemical processing, hence high economic and ecological price compared to copper grids. Pasted NiOOH/Ni(OH)2 on nickel foam was used as counter-electrode with an over-dimensioned capacity, to perform this study with the zinc electrode as the limiting electrode. The NiOOH/Ni(OH)2 electrodes were extracted from a commercial ZnNi cell.

The working and counter electrodes were separated with a nonwoven polyolefin separator (Viledon®) and a polypropylene membrane (Celgard®), assembled in a homemade designed cell, made from a PFA T-type tubing pipe from Swagelok®, and filled with an aqueous solution of 7 M KOH with 10 g/L LiOH [START_REF] Bass | Methods for the reduction of shape change and dendritic growth in zinc-based secondary cells[END_REF] and saturated in ZnO as electrolyte. In the present configuration of electrode, the current collector is not directly exposed to the electrolyte (it is considered as the core of the electrode); the surface of the electrode, considered as the shell, was directly exposed to the electrolyte and faces the separator soaked with electrolyte and then the counter electrode; the cell geometry is detailed in supplementary information Fig.Si. 1. The nominal capacity was based on 40% of the theoretical capacity, thus used between 40% of SOC and 100% of DOD. A potentiostat-galvanostat OGF500 temperature) to avoid adverse overvoltage and hydrogen formation. Before cycling, electrode formation was applied following 3 cycles at C / 10 in charge and C / 5 in discharge.

Fig. 1. Schematic representation of zinc-based electrodes with 1 layer copper foam (a), 2 layers of copper mesh (b) and 3 layers of copper mesh (c)

Physico-chemical characterizations

Post-test analyses were used to characterize the new/aged composite electrodes. These essentially consist of electron microscopy imaging on electrode cross-sections. To that goal, the electrodes were embedded inside an epoxy resin and polished until mid-height with a mirror-finish. A carbon coating was applied on the resin-embedded samples, using a carbon evaporator (BALTEC CED030, Bal-Tec Union Ltd., Liechtenstein), to avoid any surface charge effect under the electron beam. Morphological analyses were achieved using a field-emission gun scanning electron microscope (FEG-SEM Ultra 55, Carl Zeiss, Germany) working at UHT = 20 KeV as acceleration voltage within a minimum time laps after sample preparation to prevent surface modification or atmospheric corrosion. Back-Scattered electron imaging (BSD) and X-ray Energy dispersive spectroscopy mapping (X-EDS) were performed to provide colored elemental maps (each color being linked to an element) to visualize the elemental distribution within the electrode thickness. The color convention used in this paper was: zinc in red, calcium in green, titanium in dark blue and copper in bright blue. As bismuth oxide was present in small amounts in the electrode and in order to minimize the numbers of colors on the micrographs (for clarity), this element was not affiliated to any color (and appears white in the chemical maps).

Results and Discussions

Internal morphological evolution of calcium zincate-based electrodes and prediction

Our previous work [START_REF] Caldeira | Controlling the shape change and dendritic growth in Zn negative electrodes for application in Zn/Ni batteries[END_REF] revealed a mechanism of the formation of a zinc core in porous bifacial zincbased electrodes, made either with calcium zincate or a mix of zinc oxide and calcium hydroxide. The apparition of agglomerated zinc oxide in the vicinity of the copper foam upon the ageing of the electrode is driven by a working gradient, as observed using laboratory-scale mono-facial electrodes.

Herein, investigations were oriented towards newly designed monofacial electrodes (see Fig. 1). Fig. The delimited area in dashed red line represents the morphologically-modified material. After the formation cycle (Fig. 2.a.), zinc clusters are present close to the copper foam (bright blue) and around agglomerated titanium (dark blue), which confirms zinc preferential migration towards highlyelectron-conductive zones. Using ImageJ software, area measurement shows that ca. 36% of the equivalent total surface of the electrode thickness was used, value in agreement with the nominal capacity (CN = 40% of the theoretical capacity); this suggests that the volume of used active material initially corresponds to the chosen nominal capacity. In the present case, one notes that the working thickness repartition is heterogeneous and more important in the middle: a higher concentration of zinc (red) is observed in the middle.

After 50 charge/discharge cycles at C/3 (Fig. 2.b.), zinc cluster are displaced and electrodeposited onto the copper foam (red clusters on the bottom of the micrograph), creating zinc-depleted volumes along the electrode and consequently rich calcium regions (green). The surface of the electrode is still composed of the pristine calcium zincate material and additives, thereby confirming the core-shell mechanism. Zinc densification on itself upon cycling reduces its active surface area, and, as the charged capacity was kept constant, this leads to a necessary reduction of ZnO/zincate towards the outer volume of the electrode (further from the Cu-foam current collector) upon charge. Using the same technique of morphologically-modified volume determination by ImageJ on Fig. 2.b., one calculates that 65% of the whole electrode volume has been used at that stage.

Fig. 2 Compilation of multiple FEG-SEM X-EDS elemental mapping micrographs representative of a complete model negative electrode in cross-section, after a formation cycle (a) and after 50 cycles at C/3 (b). The red dashed line delimits the morphologically modified zones.

To prevent this gradual increase of the volume of used material, a simple and industrially scalable solution is proposed. If zinc clusters appear upon charge (which initiates in the formation cycles), the volume of used material must be controlled to prevent major segregation. From this statement, the idea of using more than one (primary) current collector appears obvious. In order to not increase the weight of copper current collector per electrode (300 g/m² for a Cu-foam collector), it has been chosen to use two or three layers of lighter copper grid (100 g/m²).

V3 = V1 + V2 Eq.1.
If the correlation between the active volume obtained from cross-section microscopy and the nominal used capacity, noted CN above, the working volume in such electrode could be predicted using:

V1 = CN x V3 Eq.2.
Introducing the idea of multiplying the number "n" of current collector within the thickness of the electrode, it could be possible to predict the active volume surrounded each layer of current collector "Vn" using Eq.3.

Vn = V1 / n Eq.3.
This simple idea, using basic elementary grade scholar equations, represents for the example in Fig. 3, 1 x 40% of active volume in model electrode (a) versus 13.3% per current collector in the threelayered electrode (b), for the same nominal capacity CN = 40%. Besides, electrodes made using this concept, Fig. 3.c. represents an example of "n = 3" layered conductive network calcium zincate electrode after 3 cycles of formation. X-EDS elemental mapping confirms the presence of active zones "A" constituted by a sponge-mix including active calcium zincate, zinc oxide and calcium hydroxide and alternatively superposed to inactive zones "B" constituted by lamellar calcium zincate. This last composed the surface of the electrode, named "shell" in this technology.

Fig. 3. Schematic representation of active core during cycling in model (a) versus "n = 3" layered conductive network porous zinc-based electrode (b) and FEG-SEM X-EDS elemental mapping of a "n = 3" layered conductive network calcium zincate based electrode in cross-section at initial state (c) and after 3 cycles of formation at C/10 (d).

Electrochemical and morphological investigation of calcium zincate-based electrode with "n" layers of current collector

In the aim to investigate the electrochemical performances stability of electrodes with 1 ≤ n ≤ 3, circular calcium zincate-based mono-facial electrodes of 10 mAh theoretical capacity were elaborated and cycled 50 times in charge / discharge at 4 mAh of nominal capacity (CN = 40% of 10 mAh). Fig. 4. presents morphological and electrochemical performances of model electrodes with "n = 1", "n = 2"

and "n = 3" layer of copper current collector. In agreement with our previous work [START_REF] Caldeira | Controlling the shape change and dendritic growth in Zn negative electrodes for application in Zn/Ni batteries[END_REF], cross-section X-EDS mapping shows the core-shell mechanism of operation of the porous electrode: zinc migration is observed in the vicinity of the copper foam (primary current collector). Electrodes in this study are made with calcium zincate as active material (discharge state), which means that zinc is chemically linked to calcium, at least initially (before the first charge). The zinc migration observed necessary means that calcium and zinc segregation from initial zincate has occurred, leading to the formation of a calcium-rich layer located between each zinc-core and still unused calcium zincate regions. Calcium hydroxide being electro-inactive and electronic-insulant, the formation of a separate layer between the zinc core and active calcium zincate means a loss of electrical contact, and only the core of zinc can be easily reached to perform the electrochemical oxidation-reduction reactions. This phenomenon is responsible for additional overvoltage contributions starting at cycle 10 below 100 min of charging, that reach the cut-off voltage of 1.93 V, and below 50 min at the 20 th charge, yielding a drastic decrease of the practical capacity. The output average median voltage was 1.66 V. The calculated average capacity of this electrode for the first 50 cycles is 1.27 mAh, which means that only 32% of the initial nominal capacity has been retained.

In order to reduce/avoid this phenomenon, by reducing the ohmic resistance, electrodes with "n > 1" layers of (primary) current collector were elaborated with the same formulation; they were charged and discharged 50 times and post-tested. The electrode with "n = 2" shows less pronounced overvoltage for cycle 10 and 20 during the charge and an output average median voltage equivalent to 1.72 V, but zinc segregation is still observed in X-EDS mapping. Even if zinc migration cannot be completely avoided, it is at least enough reduced to prevent the formation of a detrimentally-large and continuous layer of resistive calcium hydroxide, which is believed to be responsible for the appearance of the cell overvoltage. For the electrode with "n = 2", the average capacity is 2.3 mAh for the first 50 cycles, corresponding to 58% of nominal capacity retention.

Finally, the "n = 3" layered calcium zincate-based electrode confirms the predicted behavior: zinccalcium segregation leads to the formation of multiple small zinc clusters (in red) always close to the individual copper grids (bright blue) and surrounded by calcium-rich areas (green). The surface of the electrode, at the right of the image, presents initial calcium zincate crystals (yellow), that have not been used and still protect the electrode surface from dendrite growth. As expected from section 2.1., the active material seems efficiently used along the thickness of the electrode and with gradual performances related to "n" (the performances evolve as follows: "n = 3" > "n = 2" > "n = 1"). As a result, the "n = 3" calcium zincate-based electrodes show an average nominal capacity retention of 92% after 50 cycles and 1.71 V as output average median voltage. So not only the multiple current collectors enable larger capacity retention after 50 charge/discharge cycles, but also larger median voltage, hence practical larger energy. Note that for all the cases, the outer surface of electrodes remains unused -it is considered as a shell -and thus protects the core of the electrode from any dendrite growth.

Conclusion

As zinc-based electrode technology seems to be at a standstill of technological development, a new look has been made on how its current collection could be improved. This study highlights a simple and industrially-scalable way to improve thick zinc-porous electrodes by using an optimized conductive network of layered primary current collector. By reducing zinc segregation along the thickness of the electrodes, the multiple layers of current collector lead to improved electrochemical performances of calcium zincate-based electrodes and overcome the initial drastic capacity fading observed on standard configuration. This study shows the potential of such strategy at small electrode scale and opens the gate to prototype studies in order to confirm the enhancement in electrochemical performances for larger electrodes. Moreover, this invention could be easily applied to well-defined and mature manufacturing bases to create ultra-thick electrodes for high capacity systems such as Zinc-Air stationary application.

2 .

 2 compiles multiple FEG-SEM micrographs taken along negatives electrodes after only one formation cycle (a) and after 50 charge/discharge cycles (b); 5 micrographs per electrodes were assembled to show the full electrodes in cross-section. The specific methodology to obtain this result is detailed in supplementary information (Fig.Si. 2).

Fig. 3 .

 3 Fig. 3. is a schematic representation of the volume of active electrode during repeated

Fig. 4 .

 4 Fig. 4. FEG-SEM X-EDS elemental mapping of cross-sectioned electrodes, charge and discharge voltage curves and capacity retention curves of electrodes composed with 1 copper foam current collector compared to "n = 2" and "n = 3" layered copper current collector.

Acknowledgments

The author would like to thanks La Fondation de France (Grant n°0049328) and CNRS (Grant n°113815) for the financial support.

Conflict of interest

The authors declare no conflict of interest regarding the publication of this paper.