Guijun Yang

Tingfeng Wu

Pan Wu

Jianrong Ma

Jian Zhou

Boqiang Qin

Céline Casenave

Xiaoxia Han

Prof Boqiang Qin

Effects of wind wave

Keywords: Wind waves, Turbulence, Phytoplankton community composition, Dominance, Cyanobacterial blooms

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

Introduction

The phytoplankton community is controlled not only by local environmental conditions but also by physical processes occurring on different temporal and spatial scales [START_REF] Donaghay | Toward a theory of biological-physical control of harmful algal bloom dynamics and impacts[END_REF]. Wind waves play an important role in water ecosystems, especially in large shallow lakes where some spatial and temporal phytoplankton patterns match the characteristic properties of wind-driven environments [START_REF] Estrada | Phytoplankton in a turbulent world[END_REF][START_REF] Cardoso | Hydrodynamics-driven plankton community in a shallow lake[END_REF]. Other studies have shown that wind waves can influence the phytoplankton settling velocity [START_REF] Ruiz | Turbulence increases the average settling velocity of phytoplankton cells[END_REF]), vertical and horizontal distribution [START_REF] Cao | Effects of wind and wind-induced waves on vertical phytoplankton distribution and surface blooms of Microcystis aeruginosa in Lake Taihu[END_REF][START_REF] Moreno-Ostos | The influence of wind-induced mixing on the vertical distribution of buoyant and sinking phytoplankton species[END_REF][START_REF] Wu | Horizontal distribution and transport processes of bloom-forming Microcystis in a large shallow lake (Taihu, China)[END_REF]) and formation of cyanobacterial blooms [START_REF] Wu | Dynamics of cyanobacterial bloom formation during short-term hydrodynamic fluctuation in a large shallow, eutrophic, and wind-exposed Lake Taihu, China[END_REF]. In fact, wind waves induce several physical processes in large, shallow lakes, the most important of which are turbulence, turbidity, and the release of nutrients [START_REF] Cardoso | Hydrodynamics-driven plankton community in a shallow lake[END_REF]. Turbulence has been shown to have some impacts on the phytoplankton community composition and can select for particular life-forms [START_REF] Arin | Combined effects of nutrients and small-scale turbulence in a microcosm experiment. I. Dynamics and size distribution of osmotrophic plankton[END_REF]. For example, intense turbulent mixing has caused species replacements from dinoflagellates in oceans or cyanobacteria in lakes/rivers towards diatoms/green algae dominated communities (reviewed by [START_REF] Huisman | Changes in turbulent mixing shift competition for light between phytoplankton species[END_REF]). In shallow lakes, turbulence can control and, in some situations, govern the phytoplankton community [START_REF] Cardoso | Hydrodynamics-driven plankton community in a shallow lake[END_REF]. This also explains why an increasing number of studies focusing on wind waves are being performed in large shallow lakes to further understand the spatial and/or temporal variations of phytoplankton communities [START_REF] Carrick | Wind influences phytoplankton biomass and composition in a shallow, productive lake[END_REF][START_REF] Cao | Effects of wind and wind-induced waves on vertical phytoplankton distribution and surface blooms of Microcystis aeruginosa in Lake Taihu[END_REF].

Lake Taihu (Taihu) is the third largest freshwater lake in China, which has an area of 2338 km 2 . It is a shallow, eutrophic lake with an average depth of 1.9 m and maximum depth of 2.6 m, which is strongly influenced by wind waves [START_REF] Qin | Environmental issues of Lake Taihu, China[END_REF]. Taihu is also a key drinking water source for 40 million people [START_REF] Xu | Nitrogen and phosphorus inputs control phytoplankton growth in eutrophic Lake Taihu, China[END_REF]. However, to the despair of the inhabitants around the lake, dense surface harmful cyanobacterial blooms (CyanoHABs) of the cyanobacterium Microcystis have appeared in the northern (Meiliang Bay) and western regions of the lake every summer since the mid-1980s [START_REF] Qin | A drinking water crisis in Lake Taihu, China: linkage to climatic variability and lake management[END_REF].

The effects of wind waves on the nutrient availability [START_REF] Qin | Environmental issues of Lake Taihu, China[END_REF][START_REF] Zhu | Effects of hydrodynamics on phosphorus concentrations in water of Lake Taihu, a large, shallow, eutrophic lake of China[END_REF]) and phytoplankton distributions [START_REF] Cao | Effects of wind and wind-induced waves on vertical phytoplankton distribution and surface blooms of Microcystis aeruginosa in Lake Taihu[END_REF][START_REF] Wu | Horizontal distribution and transport processes of bloom-forming Microcystis in a large shallow lake (Taihu, China)[END_REF]) have been well studied in Lake Taihu. However, our understanding of the relationships between wind waves and the phytoplankton community composition in the lake is still limited. [START_REF] Wu | Dynamics of cyanobacterial bloom formation during short-term hydrodynamic fluctuation in a large shallow, eutrophic, and wind-exposed Lake Taihu, China[END_REF] previously showed that cyanobacterial blooms in Taihu form and expand in the absence or presence of soft wind waves. However, the effects of intense wind waves on the phytoplankton community composition, and dynamics of harmful algal blooms (HABs) have not yet been explored.

In this paper, high-frequency observation of meteorological parameters and corresponding monthly phytoplankton and environmental data, coupled with turbulence bioassay experiments were conducted to characterise the impact of the wind wave turbulence on phytoplankton communities in Taihu. Specifically, this study aims: 1) to observe the monthly variation of phytoplankton community composition in Meiliang Bay and the central lake, which are two areas of the lake with different hydrodynamic conditions; 2) to identify the effects of wind wave turbulence on phytoplankton communities in bioassay experiments; and 3) to determine if turbulence can be used as a method to mitigate or prevent the formation of CyanoHABs in Taihu.

Materials and Methods

Field study

Study site

Two stations in Taihu were selected for sampling (Fig. 1): one in Meiliang Bay (Sta. 1) and the other in the central lake (Sta. 2). The two stations were chosen for their varying physico-chemical characteristics, extent of influence of wind waves and recorded occurrences of CyanoHABs [START_REF] Qin | Environmental issues of Lake Taihu, China[END_REF], providing good contrast for the study (Table 1). To understand potential shifts in the phytoplankton community and their underlying factors, monthly physical, chemical and biological data were collected in Station 1 and 2 for the entirety of 2013 by the Taihu Laboratory for Lake Ecosystem Research (TLLER), Nanjing Institute of Geography and Limnology, Chinese Academy of Sciences (Fig. 1). Additionally, a Sontek Argonaut XR (Sontek/YSI, San Diego, California, USA) was used to measure the wave height near Sta. 1 (Fig. 1), which was unavailable in the central lake. As a proxy, the wind speed was automatically measured in a monitoring station in Sta. 2. Then, the SWAN (simulating wave nearshore) model was used to transform the wind speed into a wave height [START_REF] Li | Numerical modeling of waves in Lake Taihu[END_REF]. Half-hourly datasets in the two stations were analysed to characterise the turbulence of wind waves in 2013.

Physical, chemical, and biological conditions

Physico-chemical and biological parameters were measured in the middle of each month for the entire-duration of the study. Physical parameters, including the water temperature (WT), dissolved oxygen (DO), pH, and electrical conductivity (EC), were measured with Yellow Springs Instruments (YSI, San Diego, California, USA) 6600 multi-sensor sonde.

Water samples were analysed for nutrients such as total nitrogen (TN), total dissolved nitrogen (TDN), dissolved inorganic nitrogen (DIN; ammonium (NH 4 + ) + nitrate (NO 3 -) + nitrite (NO 2 -

)), total phosphorus (TP), total dissolved phosphorus (TDP), and soluble reactive phosphorus (SRP), according to the methods described by [START_REF] Xu | Nitrogen and phosphorus inputs control phytoplankton growth in eutrophic Lake Taihu, China[END_REF]. The particulate nitrogen (PN) was obtained by subtracting the TDN from the TN and particulate phosphorus (PP) as the difference between the TDP from the TP.

Phytoplankton samples were fixed with Lugol's iodine solution (2% final conc.) and allowed to settle for 48 h prior to counting with a Sedgwick-Rafter counting chamber under microscopic magnifications of 200-400×. Phytoplankton species were identified following the taxonomic classification of [START_REF] Hu | Freshwater algae in China[END_REF]. Algal biovolumes were calculated from the cell numbers and cell size measurements. Conversion to biomass assumes that 1 mm 3 of the volume is equivalent to 1 mg of fresh-weight biomass [START_REF] Xu | Nitrogen and phosphorus inputs control phytoplankton growth in eutrophic Lake Taihu, China[END_REF].

Turbulence bioassay experiments Experimental setup

The experimental setup is consisted of a tank, energy dissipation plate, and frequency conversion wave-maker pump (Fig. 2). The tank and the energy dissipation plate consist of 8 mm thick Plexiglas; their structure and size are shown in Fig. 2. Each tank has a maximal capacity of 126 L. To reduce the rebound of waves, considering costs and feasibility, the tanks were designed with a sloping side with a slope being of 5:1 (Fig. 2a) where an energy dissipation plate was also fixed (Fig. 2b). The wave-maker pumps (Fig. 2c) are fixed on the opposite side of the tanks. A total of 6 units of this experimental setup were built.

Experimental design

To minimise the "bottle effects", the experiment was only conducted for a total of 15 days. The bioassay experiments were performed from the 7 th to the 22 nd of July 2014 at TLLER on the shores of Taihu, in Wuxi. Chemical contamination released from the material (Plexiglas) was avoided by immersing the tanks in water for 15 days before the start of the experiments. Each pre-cleaned tank was filled with 96 litres of lake water pumped from 0.2 m subsurface of the water column. Finally, all tanks were floated and fixed in an outside artificial pond (10 × 10 × 2 m) which was filled with lake water and dug 1 m into the ground. This mimics natural conditions as closely as possible. To prevent algal adhesion, the inside walls of each tank (except the bottom) were gently brushed at 6:30 PM every day.

Two treatments were tested, one (T) where a pump generated turbulence was applied and the other (C) was calm (no artificial turbulence was induced), which also served as the control. Both treatments were conducted in triplicate.

Turbulence generation

To achieve hydrodynamic turbulence that was as similar as possible to the one induced by natural wind waves, we used a submerged wave-maker pump (WP, Jebao, China), which was fixed under the water surface with some magnets and was also tested and demonstrated in previous studies [START_REF] Clarke | Effects of water motion and prey behavior on zooplankton capture by two coral reef fishes[END_REF][START_REF] Pekcan-Hekim | Predation by a visual planktivore perch (Perca fluviatilis) in a turbulent and turbid environment[END_REF][START_REF] Härkönen | Combined effects of turbulence and different predation regimes on zooplankton in highly colored water-implications for environmental change in Lakes[END_REF].

The pump frequency was set to 1 Hz and the turbulence generated was monitored and measured by an acoustic Doppler velocimeter (ADV) (10 MHz ADVField; Sontek/YSI, San Diego, California, USA). The turbulence intensity was measured from the middle of the tank with a 25 Hz measurement for a period of 2 min. Measurements were performed after the turbulent motion in the tank had reached a steady state (after 6-7 min).

To define the characteristic speed of the turbulence, the root mean square (RMS) velocity (cm s -1 ) was calculated using the following formula

𝑅𝑅𝑅 = �𝜇 𝑅𝑅𝑅 𝑥 2 +𝜇 𝑅𝑅𝑅 𝑦 2 + 𝜇 𝑅𝑅𝑅 𝑧 2 (1)
where

𝜇 𝑅𝑅𝑅 𝑥 = � ∑ 𝜇 𝑥 2 -(∑ 𝜇 𝑥 ) 2 /𝑛 𝑛-1 (2)
is the fluctuation of the flow for Cartesian vector x (which is similarly calculated for the y and z vectors) and n is the number of samples per measurement. The RMS velocities were expressed as averages for the whole tank. The energy dissipation rate (m 2 s -3 ), which describes the rate at which the turbulent energy decays over time, was deduced from the RMS velocity (m s -1 ) with the following formula [START_REF] Sanford | Turbulent mixing in experimental ecosystem studies[END_REF]):

ε = 𝐴 1 𝑅𝑅𝑅 3

𝑙

(3) where A 1 is an dimensional constant of order 1 (Kundu and Cohen 2010), and l is the water depth (m) describing the size of the largest vortices.

The Reynolds number (Re, the ratio of inertial forces to viscous forces) for a given turbulence level was calculated with the following formula [START_REF] Peters | Turbulence generation and measurement: application to studies on plankton[END_REF]:

𝑅𝑅 = 𝑅𝑅𝑅𝑙 𝑣
(4) where l is the water depth (m) and v is the kinematic viscosity for water (8.5 × 10 -7 m 2 s -1 ).

The RMS velocity used in the experiments was 2.532 cm s -1 , and the dissipation rate (ɛ) and Reynolds number (Re) corresponding to the turbulent velocity were 2.951 × 10 -5 m 2 s -3 and 16384, respectively. The hydrodynamic condition in Taihu was also measured in Sta. 1. The dissipation rate in Taihu varied between 6.014 × 10 -8 and 2.389 × 10 -4 m 2 s -3 , suggesting that the turbulence level used in our bioassay experiment was commonly found in Taihu.

Measurements

Physical and chemical parameters were measured every day from day 0 until day 9, and at days 12 and 15, between 7:00 and 8:00 in the morning. Biological parameters were measured starting at day 0 and every three days after that until day 15. First, physical parameters were measured at each tank. Then, chemical and biological samples were collected using a tube sampler, collecting 0.45 or 0.75 L water samples from each setup. The same methods of measurement and/or calculation of the physico-chemical and biological parameters were applied to both field observations and bioassay experiments.

The chlorophyll a (Chl a) concentrations were measured by the spectrophotometric method [START_REF] Pápista | Chlorophyll-a determination with ethanol-a critical test[END_REF]. Samples were first filtered through GF/F filters, frozen at -20 °C and pigments were extracted with 90% hot acetone.

The dominant species were determined by calculating the Berger-Parker Index (BP) which is simply the proportion of the most abundance species, BP = max (B i ), where B i is the relative biomass of the i th species [START_REF] Berger | Diversity of planktonic foraminifera in deep-sea sediments[END_REF].

Statistical analysis

Significant differences between Meiliang Bay and the central lake and between the turbulence treatment and control were determined by one-way analysis of variance (ANOVA). Post-hoc multiple comparisons of treatment means were performed by Tukey's least significant difference method and the standard deviation in the variations of the triplicates was calculated. Statistical analysis was performed in SPSS 22.0 statistical package, and the level of significance used was P < 0.05 for all tests.

Results

Dynamics of the phytoplankton community composition

The physical and chemical conditions (except for the wave height) in Meiliang Bay and the central lake generally did not differ significantly in 2013 (Table 1, P > 0.05). However, the wave height varied greatly in the two stations. For example, in Sta. 1, the wave height ranged from 0.2 to 32.4 cm in 2013 which varied seasonally; it was larger in winter and spring, and smaller in the summer and early autumn (Fig. 3a). In particular, summer had an average wave height of only 0.49 cm, which was mostly associated with waves below 1 cm that had a frequency of 94.91% (Fig. 4a). In Sta. 2, on the other hand, wind disturbance was always present resulting in waves with heights ranging from 6.34 to 37.44 cm (Fig. 3b). While in summer, the average wave height was 12.30 cm with most waves higher than 6 cm accounting for 87.01% of the total number of waves that occurred (Fig. 4b).

Relative biomass (B) of dominant phytoplankton taxa also varied with the season in both sampling stations. In Sta.1, the B of Cyanophyta (cyanobacteria) was above 0.7 in summer and autumn, while the B of Bacillariophyta (diatoms) and Chlorophyta (green algae) were below 0.2 (Fig. 3a). In Sta.2, the average P of cyanobacteria and diatoms/green algae in summer were 0.39 and 0.49, respectively. Interestingly, diatoms and green algae seem to be the dominant phyla during summer at the central lake (Fig. 3b), and there were significant differences in B between Sta.1 and Sta.2 (P < 0.05). In the succeeding months, species of Cyanophyta replaced the other groups especially from late autumn to winter in Sta. 2 (Fig. 3).

Physico-chemical and biological observations in bioassay experiments

The measured environmental correlates (physical, chemical and biological) from this study are presented in Table 2. The water temperature ranged from 25.68 to 30.21 °C with DO and pH higher in C than T during the experiments. Also, the average concentrations of the total and particulate nutrients (TN, PN, TP, and PP) varied between the treatments. For example, the average TN in C and T were 1.59 ± 0.22 and 1.85 ± 0.09 mg L -1 (P < 0.05), respectively. There was also a significant difference for TP with C (0.049 ± 0.022 mg L -1 ), which was lower than T (0.061 ± 0.016 mg L -1 ) (P < 0.05). The same was also true for the concentrations of different nitrogen (TDN and DIN) and phosphorus species (TDP and SRP). Moreover, Chl a concentration was significantly higher in T than C (P < 0.05).

Impacts of turbulence on phytoplankton communities

Initially, the phytoplankton community (relative biomass) in the tanks consisted of Cyanophyta (38.03%), Bacillariophyta (33.82%), Chlorophyta (15.50%), and other algae (12.66%). Cyanophyta increased to a maximum relative biomass of 84.97% at day 9 in C, while it only reached 78.58% at day 6 in T (Fig. 5) and these were significantly different (P < 0.05). In addition, Cyanophyta remained the most dominant phyla for the entire duration of the experiments in C, while it decreased quickly as Bacillariophyta increased in abundance after approximately 10 days in T (Fig. 5). At the end of the experiments, the proportions of Bacillariophyta in C (20.23%) were much lower than T (78.06%) (Fig. 5).

Fragilaria spp. and Microcystis spp. were the two dominant species during the experiments with B's of 0.02 and 0.35 at the start of the experiments. However, Microcystis spp. (Cyanophyta) increased only until day 6 in T and day 9 in C, and it decreased afterwards in both treatments (Fig. 6). A significant difference in the proportions of communities in between the treatments was observed just after 6 days (P < 0.05). In contrast, Fragilaria spp. rapidly increased in T after day 6 (Fig. 6b). At the end of the experiment, the B of Microcystis spp. and Fragilaria spp. were 0.67 and 0.16 in C and 0.19 and 0.77 in T. Microcystis spp. was always the dominant species in C (Fig. 6a), whereas in T, Fragilaria spp. became the dominant species and replace Microcystis spp. after approximately 10 days (Fig. 6b).

Discussion

Impact of wind waves on phytoplankton in Meiliang Bay and the central lake

Hydrodynamic conditions were significantly different between Meiliang Bay and the central lake (Table 1). It has been shown that turbulent mixing induced by wind, strongly affects the spatial distribution of the buoyant and potentially toxic Microcystis spp. [START_REF] Wu | Horizontal distribution and transport processes of bloom-forming Microcystis in a large shallow lake (Taihu, China)[END_REF][START_REF] Wu | Dynamics of cyanobacterial bloom formation during short-term hydrodynamic fluctuation in a large shallow, eutrophic, and wind-exposed Lake Taihu, China[END_REF]). Some efforts have been made to determine the threshold wind speed that prevented algal blooms [START_REF] Wong | A simple model for forecast of coastal algal blooms[END_REF][START_REF] Hunter | The spatial dynamics of vertical migration by Microcystis aeruginosa in a eutrophic shallow lake: a case study using high spatial resolution timeseries airborne remote sensing[END_REF]. In Lake Taihu (Taihu), to inhibit surface bloom formation, a minimum wind speed of 3.0 m s -1 and a wave height of approximately 6.2 cm are required [START_REF] Wu | Horizontal distribution and transport processes of bloom-forming Microcystis in a large shallow lake (Taihu, China)[END_REF][START_REF] Cao | Effects of wind and wind-induced waves on vertical phytoplankton distribution and surface blooms of Microcystis aeruginosa in Lake Taihu[END_REF]. When conditions are below these specific requirements, Microcystis mainly migrates horizontally which then facilitates the formation of surface blooms, suggesting that soft wind is beneficial to algal aggregation and HABs formation. On the contrary, the wind speed and wave height are larger than the threshold values, water mixing causes vertical transport of the phytoplankton and away from the euphotic zone inhibiting photosynthetic activities. [START_REF] Wu | Dynamics of cyanobacterial bloom formation during short-term hydrodynamic fluctuation in a large shallow, eutrophic, and wind-exposed Lake Taihu, China[END_REF] showed that when wind speed increases to higher than 6 m s -1 , the entire water column in Taihu is overturned (mixed from the bottom to the top).

In our study, the average wave height in Meiliang Bay was only 0.49 cm in summer. The low wind activity in the area during the said period allowed the buoyant and toxic cyanobacterium Microcystis to aggregate in the water surface, causing a shade-effect on the underlying phytoplankton communities (e.g., diatoms and green algae, [START_REF] Huisman | Changes in turbulent mixing shift competition for light between phytoplankton species[END_REF]). This advantage in the competition for light, coupled with low physical disturbance (i.e., wind mixing), caused Microcystis to become the most severe CyanoHABs in Meiliang Bay (Fig. 3a). By contrast, the impacts of strong wind mixing could have been more pronounced in the central lake, as indicated by the high frequency (87.01%) of waves with heights above 6 cm in the same period (Fig. 4b). As a consequence, mixing induced shifts in the phytoplankton community, favouring dominance of diatoms and green algae; therefore, there were no severe CyanoHABs at the central lake during summer.

Phytoplankton community composition in turbulent environments

Changes in the intensity and extent of turbulence in natural waters bodies, such as those driven by climate change, may induce major shifts in the species composition of phytoplankton communities [START_REF] Huisman | Changes in turbulent mixing shift competition for light between phytoplankton species[END_REF]). Furthermore, the turbulence from wind waves could select for particular life forms within the microbial communities such as phytoplankton [START_REF] Arin | Combined effects of nutrients and small-scale turbulence in a microcosm experiment. I. Dynamics and size distribution of osmotrophic plankton[END_REF]. Unfortunately, few studies have focused on this phenomenon in lakes. In the field, the impacts of turbulence cannot be separated from other physical processes, which makes the interpretation of the field datasets more complex. To evaluate the impact of wind wave turbulence on the biological communities, bioassay experiments were performed.

The dissipation rate (ɛ) used in these bioassay experiments is commonly found in Taihu and other large shallow lakes (Lake Blaton, Netherland, G. [START_REF] Tóth | Zooplankton community response to enhanced turbulence generated by water-level decrease in Lake Balaton, the largest shallow lake in Central Europe[END_REF].

Turbulence mixing can shift the competitive balance in phytoplankton species. Previous studies showed that the dominance of diatoms and green algae in turbulent environments is associated with a number of variables. First, turbulence affects the competition for nutrients. Enhancement of nutrient uptake in turbulent water is more important for larger cells (diatoms and green algae) than for small ones because of their volume:ratio advantage (Microcystis) [START_REF] Prairie | Biophysical interactions in the plankton: a cross-scale review[END_REF][START_REF] Barton | The impact of fine-scale turbulence on phytoplankton community structure[END_REF]. Second, turbulence affects the competition for light. Gas-vacuolated cyanobacteria can take advantage in calm waters by the increasing nutrient flux and light absorption through vertical migratory behaviour [START_REF] Oliver | Floating and sinking in gas-vacuolate cyanobacteria[END_REF]. On the other hand, non-motile diatoms and green algae are better adapted to fluctuating light conditions experienced in turbulent environment than buoyant cyanobacteria [START_REF] Huisman | Changes in turbulent mixing shift competition for light between phytoplankton species[END_REF]). Third, sedimentation losses of non-motile diatoms and green algae due to their size and weight is decreased by resuspension through mixing. Finally, turbulence impacts the feeding rate of heterotrophic or mixotrophic plankton (e.g. zooplankton, microzooplankton). Compared to cyanobacteria, chlorophytes and diatoms are generally considered as good-quality foods for herbivorous zooplankton [START_REF] Gulati | The role of food quality for zooplankton: remarks on the state-of-the-art, perspectives and priorities[END_REF][START_REF] Wilson | Effects of cyanobacterial toxicity and morphology on the population growth of freshwater zooplankton: meta-analyses of laboratory experiments[END_REF]). However, mixing in general increases the chances of encounter between the prey and predator, but also reduces capture rates (reviewed by Pécseli et al. 2014). Therefore, the low capture rates of diatoms and green algae by predators in turbulent environments contribute to factors that allow the dominance of diatoms and green algae. Consequently, hydrodynamics turbulence would favour the dominance of diatoms and green algae, which are, generally, better adapted to turbulent environments than buoyant cyanobacteria. In addition, depending on their habitat-dependent physiological characteristics, the coupling of hydrodynamics and nutrient concentrations could determine the dominant diatom species [START_REF] Romero | Dynamic forcing of coastal plankton by nutrient imbalances and match-mismatch between nutrients and turbulence[END_REF][START_REF] Wang | Can hydrodynamics change phosphorus strategies of diatoms? -nutrient levels and diatom blooms in lotic and lentic ecosystems[END_REF]. The results reported in this paper are consistent with previous field studies and theories proposed to explain shifts in the competitive balance between buoyant and sinking phytoplankton species in freshwater ecosystems [START_REF] Huisman | Changes in turbulent mixing shift competition for light between phytoplankton species[END_REF][START_REF] Cardoso | Hydrodynamics-driven plankton community in a shallow lake[END_REF].

Our work also aimed to examine the critical time for a shift in the phytoplankton community composition under the turbulent environment in Taihu. Previous studies have analysed the threshold wind influence in terms of the magnitude [START_REF] Wu | Horizontal distribution and transport processes of bloom-forming Microcystis in a large shallow lake (Taihu, China)[END_REF][START_REF] Cao | Effects of wind and wind-induced waves on vertical phytoplankton distribution and surface blooms of Microcystis aeruginosa in Lake Taihu[END_REF], but an understanding of the time length of the episodes is still lacking. To have significant and quick changes in the phytoplankton community, experiments are generally conducted under constant, relatively high turbulence conditions, lasting for several days [START_REF] Peters | Effects of turbulence on plankton: an overview of experimental evidence and some theoretical considerations[END_REF]. However, if the aim is to reproduce the turbulence found in natural conditions, such as that of the lake, time should be considered an important factor. Although it is true that relatively high turbulence levels, such as those used in many laboratory experiments are common in the lake, organisms may rarely experience them for a long time in nature [START_REF] Guadayol | Analysis of wind events in a coastal area: a tool for assessing turbulence variability for studies on plankton[END_REF]. In our study, the critical time for a shift in phytoplankton community composition was approximately 10 days (Fig. 5,6). However, short-term turbulence would be rather beneficial to Microcystis and harmful algae blooms, which is consistent with recent field studies that shortterm tropical cyclones (intense wind wave) stimulate buoyant Microcystis spp. algal blooms in Taihu [START_REF] Zhu | The role of tropical cyclones in stimulating cyanobacterial (Microcystis spp.) blooms in hypertrophic Lake Taihu, China[END_REF]. The relationship, however, between the magnitude and persistence of turbulence (time) in Taihu is not clear and still requires further study.

Finally, we observed that that the harmful Cyanophyta (almost Microcystis) was the dominant phylum in Taihu during late autumn and winter, even in the central area whereas the wind forcing hasn't change (Figure 3). During late autumn and winter, when the water is colder, the harmful Cyanophyta (mainly Microcystis) was the dominant taxa in Taihu, even in the central area where wind forcing was strong (Figure 3). Microcystis can persist better than other algae in cold waters, although their growth rate and photosynthetic activity remain low. This is supported by simulation experiments and field observations, all of which suggest that Microcystis in Taihu can to adapt to low temperatures and even form blooms in the winter (unpublished data). Persistent blooms of Microcystis aeruginosa has also been reported in Hartbeesport Dam, South Africa during the winter period [START_REF] Zohary | Hyperscums of the cyanobacterium Microcystis aeruginosa in a hypertrophic lake (Hartbeespoort Dam, South Africa)[END_REF].

Physico-chemical conditions and phytoplankton biomass in the bioassay experiments

Turbulence reduces loss due to sedimentation and enhances nutrient uptake which are good for phytoplankton growth [START_REF] Prairie | Biophysical interactions in the plankton: a cross-scale review[END_REF][START_REF] Barton | The impact of fine-scale turbulence on phytoplankton community structure[END_REF]. Higher primary productivity increased the concentrations of total and particulate nutrients (TN, PN, TP, and PP) including Chl a, which were higher in T than C (Table 2). Additionally, because turbulence enhances the nutrient uptake by the organisms and accelerates the incorporation of dissolved nutrients in water, TDN and DIN concentrations were reduced in T compared to C (Table 2). In addition, the synthesis and release of alkaline phosphatase (cleaves the phosphate group in compounds) were further enhanced by turbulence, which accelerated the biogeochemical cycle of phosphorus in the environment (unpublished data). Therefore, we observed that the TDP and SRP concentrations were higher in turbulent set-ups (Table 2).

In the summer of 2007, a massive cyanobacterial bloom made the drinking water plant stop working, leading to a highly publicised drinking water crisis in Wuxi (4 million inhabitants), which forced residents to resort to bottled drinking water [START_REF] Qin | A drinking water crisis in Lake Taihu, China: linkage to climatic variability and lake management[END_REF]. To control the blooms in Taihu, the government took many measures that mainly focused on controlling and reducing the nutrient (N and P) inputs [START_REF] Qin | A drinking water crisis in Lake Taihu, China: linkage to climatic variability and lake management[END_REF][START_REF] Xu | Nitrogen and phosphorus inputs control phytoplankton growth in eutrophic Lake Taihu, China[END_REF][START_REF] Paerl | Controlling harmful cyanobacterial blooms in a hyper-eutrophic lake (Lake Taihu, China): the need for a dual nutrient (N & P) management strategy[END_REF]. Artificial mixing by air bubbling was successfully used to prevent the cyanobacterial blooms in Lake Nieuwe Meer, a shallow, hypertrophic lake in the Netherlands [START_REF] Huisman | Changes in turbulent mixing shift competition for light between phytoplankton species[END_REF]). Our findings support the possible application of the same method in Taihu to prevent and mitigate the formation of surface blooms of harmful cyanobacteria. However, large-scale, field follow-up experiments may be needed to confirm the findings of the current study.

Conclusion

By combining field observation data with bioassay experiments, we investigated the impacts of wind wave turbulence on possible shifts in the phytoplankton community composition of Lake Taihu. Observations from the field showed that intense wind waves could be one of the factors preventing the occurrence of harmful cyanobacterial blooms as well as responsible for the shift of the competitive balance by favouring diatoms and green algae. This conclusion explains the low biomass of the cyanobacterium Microcystis at the central lake during summer. The turbulence bioassay experiments also confirmed this observation, which further showed that turbulence could cause shifts in the phytoplankton community composition. We also determined, through the mesocosm bioassay experiments, the critical time before the shift in the phytoplankton community becomes evident, which was approximately 10 days under turbulent conditions. In the long-term (> 10 days), turbulent mixing favoured the dominance of diatom Fragilaria spp., while short-term turbulence would be beneficial to Microcystis. This study further suggests that the duration of wind events and their associated hydrodynamics are key factors for understanding the temporal and spatial changes of phytoplankton communities and the occurrence of harmful cyanobacterial blooms.

Fig. 1

 1 Fig. 1 Lake Taihu: Sta.1 and Sta. 2 are the locations at which the phytoplankton samples, and high-frequency wave height for Sta.2, were routinely collected during 2013. XR shows the location of the wave height monitoring and TLLER (Laboratory for Lake Ecosystem Research) shows the location at which the turbulence bioassays were incubated.

Fig. 2

 2 Fig. 2 Diagram of a tank (a), an energy dissipation plate (b), and a wave-maker pump (c). All dimensions are given in cm.

Fig. 3

 3 Fig. 3 The half-hourly variation of the wave height and B of Cyanophyta and Bacillariophyta/Chlorophyta in Sta.1 (a) and Sta.2 (b) during 2013. Data were obtained from TLLER.

Fig. 4

 4 Fig. 4 Frequency histograms of the wave height for Sta.1 (a) and Sta.2 (b) in the summer of 2013. The line corresponds to the fitted Weibull frequency distributions that were obtained using the maximum likelihood method.

Fig. 5

 5 Fig. 5 The proportional contributions of Cyanophyta, Bacillariophyta, Chlorophyta, and others to the total biomass in control (a) and turbulence treatment (b) during the experiments.

Fig. 6

 6 Fig. 6 The B variation of Microcystis spp. and Fragilaria spp. with control (a) and turbulence treatment (b) during the experiments.

Table 1 Mean and standard deviation of the physical and chemical measurements in Meiliang Bay and the central lake during 2013. WT, water temperature; DO, dissolved oxygen; EC, electrical conductivity; WH, wave height; TN, total nitrogen; TDN, total dissolved nitrogen; NO 3

 1

	-, nitrate; NO 2 -, nitrite; NH 4 + , ammonium; TP, total

Table 2

 2 Mean and standard deviation of the physical, chemical and biological measurements with control (C) and turbulence treatment (T) during the experiments. WT, water temperature; DO, dissolved oxygen; TN, total nitrogen; PN, particulate nitrogen; TDN, total dissolved nitrogen; DIN, dissolved inorganic nitrogen; TP, total phosphorus; PP, particulate phosphorus; TDP, total dissolved phosphorus; and SRP, soluble reactive phosphorus.

		Parameter	C	T
	Physical	WT ( °C)	27.801 ± 1.412	27.791 ±1.411
		DO (mg L -1 )	8.593 ± 1.158	7.673 ± 0.414
		pH	8.737 ± 0.309	8.543 ± 0.241
	Chemical	TN (mg L -1 )	1.586 ± 0.217	1.849 ± 0.091
		PN (mg L -1 )	0.489 ± 0.068	0.822 ± 0.157
		TDN (mg L -1 )	1.097 ±0.190	1.027 ± 0.202
		DIN (mg L -1 )	0.684 ± 0.197	0.638 ± 0.224
		TP (mg L -1 )	0.049 ± 0.022	0.061 ± 0.016
		PP (mg L -1 )	0.038 ± 0.020	0.048 ± 0.016
		TDP (mg L -1 )	0.011 ± 0.007	0.013 ± 0.006
		SRP (mg L -1 )	0.0047 ± 0.0045	0.0053 ± 0.0048
	Biological	Chl a (µg L -1 )	19.425 ± 6.061	32.308 ± 8.362

Acknowledgements

We appreciate the very thorough and constructive reviews provided by two anonymous reviewers. This research was supported by the National Science Foundation of China (41230744, 41471021) and Water Pollution Control and Management Project (2012ZX07503-002). We thank the Taihu Laboratory for Lake Ecosystem Research (TLLER) for providing the physical, chemical and phytoplankton data.


Romain Cohendet
email: romain.cohendet@univ-nantes.fr

Matthieu Perreira
email: matthieu.perreiradasilva@univ-nantes.fr

Da Silva

Patrick Le Callet
email: patrick.lecallet@univ-nantes.fr

Emotional movie: A new art form designed at the heart of human-technology interaction

Keywords: EEG, Eye-tracking, Emotion, Interactive movie

Innovative art forms emerge along with the development of new materials for displaying multimedia content and the new possibilities for interaction with the spectator they offer. Interactive movie in one of them; It consists in providing a personalized content by interacting with the emotional state of the spectator. In this study, we define this new form of artwork and describe how it can be implemented by providing functional criteria that will determine how different parts of the movie will be connected to form a particular scenario. We measured neurophysiological and ocular activities of 60 subjects watching an experimental short-movie realized for the "emotional movie" project, composed of 12 different scenarios. We combined an electroencephalography headset which directly provides emotional data with an eye-tracker to investigate the simultaneous positions of gaze. From our results we propose that the emotional proximity between the scenario and the spectator could be a relevant selection criterion to customize the movie.

Introduction

In recent years, technology development has been accompanied by the emergence of new forms of interaction with users. The combination of art contents and new display materials, which open opportunities for innovative interactions, has the potential to lead to new forms of art which would no longer be limited to content. An artist controlling the consumption of his artwork by the user who consumes it could better transmit its artistic intent; he would have an additional way to play on the emotions of the individual.

Interactive film, as we understand it, is an art form that aims to provide a personalized scenario to each spectator. The movie is cut into several coherent sequences whose adjustment determines the particular scenario which will be viewed. The sequences articulation -and consequently the resulting specific scenario -is built based on interactivity with the spectator, depending on its reactions. At certain moments of the film, called junctions, a sequence is selected from among several possible to assign the film a specific scenario. In the "emotional" interactive movie, the particular scenario selected for a spectator is based on a measure of his emotional activity, regis-tered in a previous sequence. Consequently, the selection of a sequence at a junction requires: 1/ to have a means for directly measuring the emotional state of the spectator watching the movie, 2/ to have relevant criteria to define how this choice will be made (for instance: "The choice of the sequence is based on arousal: the chosen sequence is that with the potential for arousal closest to spectator status at a given time), and 3/ to determine on which sequence of the film the emotional state of the individual will be taken into account for this choice. For the latter, it is necessary to select sequences where the reactions of the spectators vary and not sequences that always trigger the same reaction, to use these differences to make the choice of a scenario.

In this study, we used an EEG headset to investigate emotional activity of spectators during the viewing of an interactive movie. These researches were conducted in order to provide relevant criteria on the basis of emotional activity, to choose which sequences will be connected at junctions to form the particular scenario that will be displayed. Electroencephalography is a cerebral exploration method that measures real-time neurophysiological activity through electrodes placed on the scalp. The analysis of the recorded signal tells us about brain processes at work [START_REF] Sanei | EEG signal processing[END_REF]. This technology is typically used in medical applications. However, EEG devices intended for a broad audience have recently become available, handy to use and relatively affordable [START_REF] Cernea | Detecting insight and emotion in visualization applications with a commercial EEG headset[END_REF]. This is the case of the Emotiv EEG headset used in our experiment, which directly provides emotional data (Excitement, Frustration, Meditation, and Engagement) [START_REF]Emotiv[END_REF]. Based on these data, we could characterize the interchangeable film sequences by their average score on each of these emotional dimensions. In addition, we could define a selection criterion for assembling sequences: sequence connected at a junction is chosen because it is the closest to the spectator emotional state (recorded on a previous film sequence).

To determine the parts of the film on which the emotional activity that would lead to the choice of scenario would be recorded, we combined EEG device with an eye tracker. The eye-tracker records with cameras eye movements of a spectator, from which can be deduced locations on the screen where he looked [START_REF] Duchowski | Eyetracking methodology: Theory and practice[END_REF]. The purpose in using this measure was to detect the film parts where the gaze activity was more varied among the spectators. Indeed, parts of the film where the spectators reactions are quite similar have only low power to discriminate and thus to determine which scenario assign them personally.

During the analysis of synchronous EEG and eye-tracking data we looked for times when a significant ocular variability occurred with significant emotional values. The corresponding film sequences were selected for recording emotional activity of the spectators, which determine the choice of scenario.

Materials and methods

In order to develop a first functional form of emotional movie, we conducted an experiment at the IVC laboratory of IRCCyN (Nantes, France). We combined EEG and eye-tracking so as to link emotional and ocular activities of the spectators of a prototype of emotional movie, which includes several scenarios. The objective was to establish clear criteria for choosing a particular scenario based on the spectator's emotional activity. In addition, we wanted to detect the most relevant movie parts for recording spectator emotional reactions -parts where the spectators' ocular reactions were the most different and the emotional reactions the most intense. The experiment will be explained in detail in the following section. The movie used, "Mademoiselle Paradis", is a prototype made for the emotional movie initiative by Marie-Laure Cazin [5]. It lasts about 20 minutes (depending on the possible scenarios). It includes several possible concatenations, built at two junctions, corresponding to a total of 12 different scenarios. Four different sequences may be connected for the first junction (at 07: 32), and three for the second junction (at about 11:00, depending of the sequence previously connected). Outside of these sequences, the rest of the film is the same for all spectators. In our experiment, we worked on four fixed versions of the film, A, B, C and D; all possible sequences were seen by at least one spectators group.

A total of 60 people (33 males and 26 females) compensated for their participation by the University of Nantes participated in the experiment. The age ranged from 19 to 61 years, with an average age of 24,1 years. All participants had normal or correctedwith glasses or contact lenses -acuity. They were divided into four groups corresponding to the four presented versions of the film.

The experiment was conducted in a lab specifically designed and equipped for audiovisual quality assessments according to recommendation ITU-R BT.500. The film was presented on a 40-inch monitor. The screen resolution was 1920×1080. The participants were seated at a distance of 150 cm from the screen. We used an Emotiv EPOC headset [START_REF]Emotiv[END_REF] which has 14 electrodes. The sampling rate of the Emotiv headset is 128Hz. The headset is associated with a data analysis software which provides realtime values on five emotional dimensions: Short-term excitement, Long-term excitement, Meditation, Frustration, and Engagement. We used an SMI eye tracker to record the gaze of the human observers. The system consists of an infrared camera and two infrared light sources, one on either side of the camera. The eye tracker records gaze points (GP) at 50 GP/sec for each eye. A calibration of the system for a particular person is done using 5-points calibration screen. [START_REF]Emotiv[END_REF] Results

In this section, three parts of the analysis are discussed: analysis for EEG emotional data, analysis for eye-tracking data and a cross-analysis of EEG and eye-tracking data. Analyses were performed on 59 subjects for the eye tracker and 52 subjects for EEG (due to data recording problems) and focused on the description of the film shots (chosen as coherent units). Less than or equal to 3 seconds shots were excluded from the analysis. Only parts of the movie before the second junction (from which the final scenario is in place) were considered. At first, we defined each shot in terms of mean emotional value from EEG recorded data for all participants. We considered "Short-term excitement", "Engagement" and "Frustration", but no "Long-term excitement" which is a simple smoothing from "Short-term excitement" nor "Meditation" which is the inverse of "Engagement". The results show differences for activation values of the three different emotional dimensions between all shots. Emotions conveyed by the shots are clearly different; so there is latitude for setting a quantitative selection criterion to determine on emotional base which scenario is chosen: we choose the scenario which is the closest to the emotional state of the spectator.

For this purpose, we described each unfixed sequences, corresponding to several shots, by the mean activation value for "Excitement" they generated. Then we created four categories by dividing into four groups of values our "excitement" data so that each possible sequence at a junction is within a different category. Category 1 includes mean activation values for "Excitement" between the minimum and -0.5; it corresponds to "Very low activation" category; Category 2 includes values between -0.5 and 0 and corresponds to "Low activation"; Category 3 includes values between 0 and 0.5 and corresponds to "Moderate activation"; Category 4 includes values above 0.5 and corresponds to "High activation". Each sequence has been classified into one of these categories according to the mean excitement activation they generated on spectators. The mean excitement values for all the possible sequences (S) (determining their category membership) are as follows: at junction 1, S1=-0.69; S2=-0.27; S3=0.49; S4=1.61 and at junction 2, S5=-1.31; S6=0.02; S7=1.05. From this division, we defined the following selection criterion for connecting an unfixed sequence at a junction: "The selected sequence is the one whose mean activation for excitement match with mean excitement of the individual spectator watching the movie calculated and categorized in the same way".

To determine on which parts of the movie the spectator's emotional activity would be recorded for comparison with the excitement conveyed by the unfixed sequences that may be proposed to individualize his scenario, we crossed EEG and eyetracking data. First, we determined the movie parts where inter-spectator reactions measured using eye-tracking were more variable. For this purpose, we characterized each film shot according to the corresponding dispersion of the gaze points (GP, calculated as the average between the two eyes). The mean dispersion of GP per frame was calculated as the sum of the distances of each GP observers from the centroid divided by the number of GP. Dispersion per shot was calculated as the sum of the mean dispersions of each frame in the shot divided by the number of frames in the shot. Then we crossed mean GP dispersion and mean activation values for excitement for all the shots (see Fig. 1.). The purpose was to determine the parts of the movie with both high excitement intensity and important GP dispersion. Selected shots for real-time emotional activity of spectators recording were those whose GP variability differed by more than two standard deviations of the mean (Sd=0, M=1, because the standardization of the data) and where the excitement corresponded to category 4 ("High activation"). Shots were as follows: for junction 1, the shot 7 (excitement = 1.06; dispersion = 2.57); for junction 2, shot 42 (E=1.37; D=1.83), shot 58 (E=1.29; D=2.11), shot 59 (E=2.98; D=1.53), shot 69 (E=1.43; D=1.41). Note that these shots are time-shifted for the other versions of the film (here they apply to version A) and therefore must be shifted accordingly. Thus, depending on the category wherein mean excitement of a spectator as measured by EEG in the shot 7 get into, the sequence connected at junction 1 will be the one belonging to the same category. Ditto for junction 2, for which the activity can be recorded on one of the following shots: 42, 58, 59 or 69.

Discussion

Our study allowed achieving a first functional adaptation of emotional film. We selected several film parts during which the real-time emotional reactions of a spectator is recorded. The artist may rely on emotional values obtained by a spectator on these parts of the film to choose to trigger a scenario rather than another to enforce his artistic intention. The choice of a scenario can also occur automatically, based on the criteria we have proposed: the sequence triggered at the junction will be the one closest to the excitement of the spectator. Of course it is possible to imagine other criteria for the emotional movie, and other types of sensor (e.g. Galvanic skin response, pupil dilatation). [START_REF] Wu | The analysis of emotion recognition from GSR based on PSO[END_REF][START_REF] Bradley | The pupil as a measure of emotional arousal and autonomic activation[END_REF] Apart from our interest in the emotional movie, this study is exploratory in nature. It helps us to better understand the interaction between spectators and multimedia contents. In particular, we are interested in the link between memory and multimedia content. Emotional movie can open us interesting doors on interactions between memory, emotion and visual attention, especially to study effects of emotional congruence. Eventually, we are interested in the consideration of the inter-observer variability in predictive models of visual attention and memory.

Fig. 1 .

 1 Fig. 1. Mean activation values for Emotiv EEG excitement dimension and mean gaze points dispersion along with shots according to their order of appearance in the movie (version A)


Romain Cohendet

Stéphane Rousset

Cohendet Romain

Rousset Stéphane

Role of memory re-evocation: Evolution of the what-where-when memory during long-term consolidation

published or not. The documents may come

Role of memory re-evocation: Evolution of the what-where-when memory during long-term consolidation

Introduction

Models of long-term consolidation suggest that memories, richly contextualized and fit for revival in their original form, become more generic over time, losing the particular occurrence of the event. However, this transition from an episodic nature to a semantic nature (i.e., semantization) spares certain memories which can be relived decades later as vividly as on the first day. The multi-trace model (Nadel & Moscovitch, 1997) assumes that the reevocation of a memory preserves it from semantization, contrary to the standard model (Alvarez & Squire, 1994) which postulates that the re-evocation leads to the semantization of memories.

• 24 participants (age range 18-26, mean age = 21.4, SD = 1.2, 5 males) • Re-evocation is manipulated during cued recall in an inter-subject design

Results

Effect of Delay (T1 vs. T3 within-subject): We observe a lapse of memory…

The context of memories is less well recalled after two days [F(1,23) = 13.66, p <.05]. The Remember response rate is also lower [F(1,23) = 7.10, p <.05].

Effect of re-evocation (inter-subject): However, if re-evocation has no effect on accuracy, it does have an effect on Remember response

The Remember response rate is higher for the memories re-evoked [F(1,22) = 6.52, p <.05]. There was no main effect of re-evocation on mean proportion of what-where-when correct recalls [F < 1], nor interaction between re-evocation and category of recall [F < 1].

Few items gave rise to a complete binding performance (6,3%), Re-evocation had no effect on binding performance (F < 1).

Conclusion

The results show the existence of a lapse of memories in time, with a decrease of what-where-when context recuperation and a global transition from an autonoetic consciousness level to a noetic consciousness level.

Re-evocation from a contextual element enables to produce a greater proportion of Remember responses on another elements in subsequent testing. These results support the multi-trace model which suggests that re-evocation preserves overall the episodic nature of memories via the hippocampus. Unfortunately, we did not show any significant re-evocation effect on the where element.

Method 1 /

 1 Learning phase: Events = Solving tasks associated with memorization of a what-where-when context. « Find the object with the same orientation» 2/ Cued recall tasks: -3 sessions of recall each one day apart, with re-evocation in Session 2. -The factual information of the task was presented. Subjects answered on the what, the where or the when associated with the task. -A Remember-Know judgment is assessed for all responses (i.e., Remember = autonoetic consciousness; Know = noetic consciousness)

→

  We measured the correct response rate for the recalls and the Remember response rate.BibliographyAlvarez P., Squire L.R., 1994. Memory consolidation and the medial temporal lobe: a simple network model. Proceedings of the national academy of sciences, 91, 7041-7045. Nadel L., Mosovitch M., 1997. Memory consolidation, retrograde amnesia and the hippocampal complex. Neurobiology, 7, 217-227.. effect on what-where-when performance AIM The purpose of this study is to highlight the existence of a semantization process and to demonstrate that memories re-evocation preserves memories from semantization, in agreement with the Multi-Trace Model.


Douglas N Arnold
email: arnold@umn.edu

Guy David
email: guy.david@math.univ-psud.fr

David Jerison
email: jerison@math.mit.edu

Svitlana Mayboroda
email: svitlana@math.umn.edu

Marcel Filoche
email: marcel.filoche@polytechnique.edu

The effective confining potential of quantum states in disordered media

Keywords: numbers: 71.23.An, 72.15.Rn, 03.65.Ge

The amplitude of localized quantum states in random or disordered media may exhibit long range exponential decay. We present here a theory that unveils the existence of an effective potential which finely governs the confinement of these states. In this picture, the boundaries of the localization subregions for low energy eigenfunctions correspond to the barriers of this effective potential, and the long range exponential decay characteristic of Anderson localization is explained as the consequence of multiple tunneling in the dense network of barriers created by this effective potential. Finally, we show that the Weyl's formula based on this potential turns out to be a remarkable approximation of the density of states for a large variety of 1D systems, periodic or random.

Despite more than 50 years of research, many questions on the exact mechanism of Anderson localization still remain open [1][2][3][4]. One of the most puzzling aspects of this phenomenon is the strong spatial confinement of the one-particle quantum states, i.e., the exponential decay of the wave amplitude at long range in the absence of any confining potential [5,6]. In the interpretation due to Anderson, this decay comes from the destructive interferences between waves traveling from an initial source along different propagation pathways in the disordered potential [7]. At distances from the origin much larger than the correlation length ξ, the waves statistically almost cancel each other, leading to an exponential decay of the amplitude. However, this statistical result says nothing about the detailed nature of the decay. Does it occur smoothly, in a continuous way, when the distance goes to infinity? Or are there specific places where a transition between constructive and destructive interference can be observed?

A recent theory has shown that the precise spatial location of such quantum states in a potential V ( r) can be predicted using the solution u( r) of a simple associated Dirichlet problem, called the localization landscape [8]. While quantum states and their energies are respectively the eigenfunctions and the eigenvalues of the Hamiltonian of the system defined as Ĥ = -

2 2m
∆ + V , the landscape u is defined as the solution of Ĥu = -

2 2m ∆u + V u = 1 , (1)
the boundary conditions being either Dirichlet, Neumann, or periodic. In this theory, the localization subregions are delimited by the valley lines of the graph of u. This property directly derives from a fundamental inequality satisfied by any eigenfunction ψ of Ĥ with eigenvalue E, normalized so that its maximum amplitude is equal to 1:

|ψ( r)| ≤ E u( r) . (2)
In other words, the small values of u along its valley lines constrain the amplitude of ψ to be small along the same lines, and as a consequence, localize low energy eigenfunctions inside the regions enclosed by these lines [8].

We unveil here a different, and much more powerful role played by u, by showing that the function W ≡ 1/u can in fact be interpreted as a confining potential that is responsible for the exponential decay of the Anderson localized states even far from its main localization subregion. To that end, the original Schrödinger equation is transformed by introducing an auxiliary function ϕ such that ψ ≡ uϕ. Expressing that ψ is an eigenvector of the Hamiltonian writes

- 2 2m ∆ + V (uϕ) = E uϕ . (3)
Developing this equation and accounting for the definition of u in Eq. 1 gives

- 2 2m ∆ϕ -2 2 2m ∇u u • ∇ϕ + 1 u ϕ = Eϕ . (4)
The additional first order term proportional to ∇ϕ can be inserted into the second order term which finally yields:

- 2 2m 1 u 2 div u 2 ∇ϕ + W ϕ = Eϕ . (5)
One can see that the auxiliary function ϕ = ψ/u thus obeys a Schrödinger-type equation in which the original potential V ( r) has disappeared. Instead, a new function W ( r) now plays the role of "effective confining potential". One first notices that, since u is a solution of Eq. 1, W is indeed homogeneous to an energy. Moreover, the valleys of u which are the boundaries of the localization subregions [8] also correspond to the crest lines of this new potential. These crest lines act as barriers for the auxiliary function ϕ.

The demonstration that W plays the role of an effective potential derives from the following equality satisfied by any quantum state |ψ :

ψ| Ĥ|ψ = u p ψ u |u p ψ u + ψ| Ŵ |ψ . ( 6
)
The detailed proof of this equality is given in the appendix. It shows not only that, independently of its kinetic energy, the energy E of any quantum state |ψ can never be smaller than the one it would have in a potential W ( r), but also that, according to Agmon's inequality [START_REF] Agmon | Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators[END_REF][START_REF] Agmon | Lecture Notes in Mathematics[END_REF], the quantity (W -E) can be used to build an Agmon distance ρ E ( r 1 , r 2 ) that will control the decay of ψ( r) in the regions where E < W . This distance is defined as

ρ E ( r 1 , r 2 ) = min γ γ (W ( r) -E) + ds (7)
where the minimum is computed on all paths γ going from r 1 to r 2 . The control on the amplitude ψ( r) of an eigenfunction centered in r 0 of energy E is expressed through the inequality

|ψ ( r) | e -ρ E ( r 0 , r) . (8)
This exponential decay can be observed in Fig. 1 which displays the localization of quantum states in a random potential with periodic boundary conditions. The potential is piecewise constant on intervals of length 1, with values following a uniform law between 0 and V max = 8, where units of 2 /2m are considered (Fig. 1a). The domain length L = 64 thus corresponds to the total number of such intervals. Fig. 1b displays the effective confining potential W = 1/u computed from solving Eq. 1, and in Fig. 1c andd superimposed on W are the logarithmic plots of, respectively, the fundamental and second excited quantum states (ψ 0 and ψ 2 ) together with the effective confining potential W . In this log scale an exponential decay of the state amplitude translates into a linear drop. One can observe that this decay occurs exactly at the barriers of W and stops across the wells.

The confining properties of 1/u are even more interesting in two dimensions. In the case of a 2D random potential where no clear localization region can be outlined, it has already been observed that the localization landscape u exhibits marked valleys that determine the localization subregions [START_REF] Filoche | Contemporary Mathematics Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II[END_REF]. Here we show that 1/u exponentially controls the confinement of the quantum states in the entire domain through the heights and widths of its barriers. This is of particular interest in cases where the semi-classical approach fails, as for instance for a Boolean-type potential (a random potential that can take only two values). Figure 2a displays a realization of a 2D Boolean potential, i.e., a potential that can take only two values, V min = 0 or V max = 4. In the present computation, the domain is divided into 40 × 40 small unit squares, and the potential is piecewise constant on each of these unit squares, taking the value V min with probability p 0 = 0.6 and the value V max with probability p = 0.4 so that the V = 0 region has a large chance to percolate throughout the domain, as happens for the specific realization. Consequently, a classical particle would never be confined by such a potential, independently of its energy.

Both the effective potential W and the fundamental quantum state ψ 0 are obtained using 320,000 triangular Lagrange finite elements of degree 3, with a code based on the FEniCS finite element software environment [START_REF] Logg | Automated Solution of Differential Equations by the Finite Element Method[END_REF]. Figure 2b as the consequence of successive and cumulative decays across the dense network of barriers generated by W . In summary, the decay of the original eigenfunction ψ = u ϕ away from its maximum thus originates from two concurring contributions:

• First, u is small near the valleys surrounding the maximum of ψ, reducing accordingly the amplitude of ψ wherever E u ( r) < 1 (see [8]).

• Second, ϕ decays through the barriers of W , which can be interpreted as "quantum tunneling" of the auxiliary function when W is larger than E.

So, not only does the amplitude of the quantum state ψ decrease in the vicinity of the valleys of the landscape u, but also it decays when crossing the far away valleys as long as the value of W is larger than the energy E. This decay is directly governed by the exponential of the Agmon distance build from W . One can therefore conclude that the complicated interferential pattern of wave localization in V is translated into the picture of a classical confinement, observed through the glass of the wells and barriers of the new effective potential W . It is thus possible to graphically identify the localization subregions at energy E just by flooding the effective potential W up to the height E, and then observing the extent of the flooded basins. Each basin can be considered as local oscillator, the entire system then appearing as a set of oscillators, each independent at low energy and weakly coupled by quantum tunneling through the boundaries of the basins.

It is interesting to note that W can also be interpreted as a regularized version of the original potential V . This can be seen by rewriting Eq. 1 as

V -W = 2 2m ∆u u . ( 9
)
However, unlike in a classical smoothing procedure, the smoothing scale here is not constant but varies spatially depending on the value of u, hence of V . The smoothed effective potential W resulting from this non linear operation favors the emergence of well-formed wells surrounded by barriers, even in situations where none are visible in the original potential. In that sense, the behavior of W is much closer to that of a classical confining potential than V .

The emergence of localized states triggered by the quenched disorder may also strongly perturb the oneelectron density of states [START_REF] Xu | [END_REF]14]. In some cases, it may shift the energy of the fundamental state, leading to an enhancement of the already existing gap in crystalline semiconductors [15] or amorphous semiconductors [16] or the creation of a pseudogap in superconducting materials. Very generally, Weyl's laws states that the IDOS (integrated density of states) N (E) at a given energy E can be approximated by the volume (properly normalized) in the phase space x, k that can be explored by a classical particle of mechanical energy smaller than E:

N (E) ≈ (2π) -n H( x, k)≤E d n x d n k = (2π) -n 2 k 2 2m +V ( x)≤E d n x d n k . ( 10
)
where n is the spatial dimension. For example in the local band structure theory of semiconductors, the local density of states is obtained by assigning to V the value E c ( x), the bottom edge of the conduction band. However, Weyl's formula is only valid in the asymptotic limit E → +∞ and can be very inaccurate at low energies.

It follows immediately from Eq. 10 that Weyl's formula in one dimension writes N (E) ≈ N V (E) where

N V (E) = 1 2π 2 k 2 2m +V (x)≤E dx dk = 1 2π 2 k 2 2m ≤E-V (x) dk dx = √ 2m π V (x)<E [E -V (x)] 1/2 dx . (11)
Due to the analogy with classical mechanics that is implicitly behind Weyl's law, one can easily understand why this approximation is poor in quantum systems where wave interference plays a major role. This can be seen in Figure 3 where the density of states for three typical cases of one-dimensional potential are examined: random with uniform law (a), random boolean (c), and periodic (e). For each type of potential represented on the left, both the true counting function N (E) (stepwise solid line) and the approximation N V (E) (dotted red curve, obtained from Weyl's approximation) are represented. One can see that in all three cases, these two curves differ significantly as expected.

We now compute a new approximation of the counting function, this time based on the potential W deduced from u. The reason for doing so lies in the fact that W has been shown to behave as an effective potential energy (Eq. 6) with identifiable wells and barriers, in the spirit of a classical mechanical system. In all three frames (b), (d), and (f), the smooth curve (dotted red line) is the approximation N W (E) obtained by inserting the effective potential W into Eq. 11. One can immediately notice the remarkable agreement between the original counting function and N W (E), whereas the standard approximation N V using the original potential totally fails to predict the density of states. In particular, N W (E) detects precisely the shift of the lower edge of the conduction band induced either by disorder (Fig. 3b) or by the periodicity of the potential (Fig. 3f). This lower bound to all energy values corresponds to an added gap in the IDOS that is reminiscent of the missing density of states at lower energy in Anderson localization [17]. At higher energy, all three curves catch up, as they asymptotically follow the Weyl law without potential, which is proportional to E 1/2 . In summary, a new conceptual tool has been introduced for understanding the localization properties of quantum states. This object is the reciprocal of the localization landscape introduced in [8] and can be interpreted as an effective potential. We have shown that this effective potential not only determines the boundaries of the localization regions but also controls the long range decay of the quantum states through an Agmon metric. In particular, the exponential decay observed in Anderson localization has been shown not to occur uniformly, but rather to be concentrated across the barriers of this effective potential. Therefore the transport between adjacent basins of the effective potential involves an "effective quantum tunneling". In short, one can say that the effective potential captures the interference pattern created by waves in the original potential and converts it into a semi-classical picture of confining potential. Finally, this property enables us to build an approximation of the integrated density of states based on the Weyl law. Applied to various types of 1D potentials, either random or deterministic, this approximation showed a remarkable agreement with the actual IDOS, far more accurate than what can be obtained using Weyl's law, with or without the original potential.

Capturing the deep confining properties of a complex or random potential, the effective confining potential appears as a very promising tool for understanding the spatial properties of quantum waves in disordered or complex media, as well as the transition from localized to delocalized states.

FIG. 1 .

 1 FIG. 1. (a) Random piecewise constant potential. (b) Effective confining potential W computed by solving Eq. 1 then taking the reciprocal of u. The wells of W are outlined in grey. (c) and (d) Probability density of the fundamental state |ψ0 of energy E0 = 1.58 (resp., second excited state |ψ2 with E2 = 1.62) in logarithmic scale superimposed on the effective potential W (light gray line). One can see that the decay of both eigenfunctions occurs mostly outside these wells, i.e., in the barriers of W , and stops inside these wells.

  displays a color representation of W , and its crest lines computed using a watershed algorithm. Figure2cpresents the amplitude of ψ 0 overlapped with the same lines. One can notice how the decay of the amplitude closely follows the subregions delimited by these lines.In order to examine in detail the long range decay of this quantum state, the quantity |∇ log(|ψ|)| which can be interpreted as the inverse of the local length of decay of the wavefunction, is plotted in Figure2d. Superimposing the level sets of this quantity with the crest lines of 1/u (which are also the valleys of the landscape u) reveals a very strong correlation: the local length of decay increases substantially in the vicinity of the barriers of W . The long range exponential decay in Anderson localization does not occur uniformly but rather appears

FIG. 2 .

 2 FIG. 2. (Color online) (a) Boolean potential V . The black region corresponds to V = Vmax = 4. The white region, corresponding to V = 0, occupies about 60% of the domain and percolates from the center to the outer boundary of the domain. (b) Color representation of W ≡ 1/u, the effective potential as defined in Eq. 5, superimposed by the crest lines of this potential computed using a watershed algorithm. (c) Color representation of log 10 (ψ0( r)), ψ0 being the fundamental eigenfunction of energy E0 = 0.503, superimposed with the crest lines of W . (d) The logarithmic gradient of ψ0, thresholded so that values > 1 are in black, and superimposed with the crest lines of W . The clear match indicates that the exponential decay of ψ occurs exactly at the boundaries of the localization subregions determined by the effective potential W .

FIG. 3 .

 3 FIG. 3. (Left) Three different type of potentials: (a) random with uniform law on [0 1], (c) random Boolean (0 or 1), (e) periodic with 256 periods, V = 0 on the first half of the period, V = 1 on the second half. (Right) For each potential, the counting function N (solid black line) is represented, together with NV (blue dash-dotted), Weyl's approximation using the original potential V (as defined in Eq. 11), and NW (red dotted), the same with W . Notice in all cases the remarkable agreement between N and NW .

Doug Arnold is partially supported by the NSF grant DMS-1418805. Guy David is partially supported by an ANR Grant, programme blanc GEOMETRYA, ANR-12-BS01-0014. David Jerison is supported the NSF Grant DMS-1069225. Svitlana Mayboroda is partially supported by the Alfred P. Sloan Fellowship, the NSF CAREER Award DMS-1056004, the NSF MRSEC Seed Grant, and the NSF INSPIRE Grant. Marcel Filoche is partially supported by a PEPS-PTI Grant from CNRS.

Appendix: the effective potential 1/u Inspired from Agmon's work [START_REF] Agmon | Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators[END_REF][START_REF] Agmon | Lecture Notes in Mathematics[END_REF], any function V satisfying ψ| Ĥ|ψ ≥ ψ| V |ψ for any quantum state ψ can be used to define a distance whose exponential governs the decay of the eigenfunction in the places where the potential V is larger than the state energy E (the "barriers" of V ). We give here a sketch of the proof showing that W = 1/u, where u is as in Eq. 1, satisfies this inequality. To do so, we write ψ = uϕ, which yields:

This proves that 1/u can be used as an effective confining potential to build an Agmon distance governing the exponential decay of the wavefunctions.


Marine Azevedo Da Silva

Archana Singh-Manoux

Martin J Shipley

Jussi Vahtera

Eric J Brunner

Jane E Ferrie

Mika Kivimäki

Hermann Nabi
email: hermann.nabi@inserm.fr

Marine Azevedo

Da Silva

Sleep duration and sleep disturbances partly explain the association between depressive symptoms and cardiovascular mortality: the Whitehall

Keywords: Sleep disturbances, sleep duration, depression, mortality, epidemiology a Adjusted for sex, age, ethnicity, marital status and occupational grade

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

INTRODUCTION

There is consistent evidence for an increased risk of mortality, particularly cardiovascular deaths, among persons with depressive symptoms. [START_REF] Ariyo | Depressive symptoms and risks of coronary heart disease and mortality in elderly Americans. Cardiovascular Health Study Collaborative Research Group[END_REF][START_REF] Nicholson | Depressive symptoms and health-related quality of life in early pregnancy[END_REF][START_REF] Nabi | Effects of depressive symptoms and coronary heart disease and their interactive associations on mortality in middle-aged adults: the Whitehall II cohort study[END_REF][START_REF] Lefevre | Usefulness of a single-item measure of depression to predict mortality: the GAZEL prospective cohort study[END_REF] In parallel, an increasing number of studies have also found sleep duration and sleep disturbances to be associated with mortality risk [START_REF] Cappuccio | Sleep duration and all-cause mortality: a systematic review and meta-analysis of prospective studies[END_REF][START_REF] Rod | Sleep disturbances and cause-specific mortality: Results from the GAZEL cohort study[END_REF][START_REF] Ferrie | A prospective study of change in sleep duration: associations with mortality in the Whitehall II cohort[END_REF]. What is more, depression and sleep are associated with each other in a bidirectional fashion [START_REF] Mezick | Are sleep and depression independent or overlapping risk factors for cardiometabolic disease?[END_REF][START_REF] Breslau | Sleep disturbance and psychiatric disorders: a longitudinal epidemiological study of young adults[END_REF]. However, we are aware of no previous study that has examined the role of sleep in the association between depression and mortality. Accordingly, we explored the contribution of sleep duration and disturbances to the association between depressive symptoms, all-cause and cardiovascular mortality.

METHODS

Participants

The Whitehall II study, established in 1985, is a longitudinal study based on 10,308 civil servants (6,895 men and 3,413 women) [START_REF] Marmot | Cohort Profile: the Whitehall II study[END_REF]. Baseline examination (phase 1) took place between 1985 and 1988 and involved a clinical examination and a selfadministered questionnaire. Subsequent data collection phases have alternated between a questionnaire (even-numbered phases), and questionnaire plus clinical examination (oddnumbered phases). University College London Medical School Committee on the Ethics of Human Research approved the protocol and informed consent was gained from all participants.

Measures

Depressive symptoms assessed at phase 7 (2003)(2004) used the Center for Epidemiologic Studies Depression Scale (CES-D, Cronbach's alpha = 0.83) for the first time in the Whitehall II study. Scores ≥ 16 from a total possible score of 60 were used to distinguish depressed from non-depressed participants [START_REF] Radloff | The CES-D scale A self-report depression scale for research in the general population[END_REF].

Sleep duration was assessed at phase 7 by asking participants "how many hours of sleep do you have on an average week night?" Responses choices were: 5 hours or less, 6 hours, 7 hours, 8 hours, and 9 hours or more. Those reporting ≤5 hours per night were assigned to the "short sleep duration" category and those reporting ≥6 hours per night to the "long or normal sleep duration" category (the reference group) [START_REF] Gallicchio | Sleep duration and mortality: a systematic review and meta-analysis[END_REF]Kalesan, 2009, Groeger et al., 2004).

Sleep disturbances were assessed at phase 7 using the 4-item Jenkins Scale [START_REF] Jenkins | A scale for the estimation of sleep problems in clinical research[END_REF]. This scale includes 4 questions on "having trouble falling asleep", "waking up several times per night," "having trouble staying asleep," "waking up after the usual amount of sleep feeling tired and worn out" (i.e., waking without feeling refreshed); all items have a 6-point response scale (1 = never; 2 = 1-3 days; 3 = 4-7 days; 4 = 8-14 days; 5 = 15-21 days; 6 = 22-31 days). In the absence of defined cut-off score for the Jenkins scale, we categorized participants into two groups: No sleep disturbances (any sleep problem ≤14 days and no hypnotics use during the last month, the reference group). Those who reported experiencing any of the four items ≥15 days or reported the use of hypnotics were considered to have sleep disturbances.

Mortality: Mortality follow-up was available through the National Health Services Central Registry until August 2012. Death certificates were coded using the 10th revision of the International Classification of Disease (ICD). All-cause mortality and death from cardiovascular diseases (CVD; ICD-10 codes I20-I25 and I60-I67) were the outcomes of interest.

Covariates: Sociodemographic measures included age, sex, ethnicity, marital status and occupational grade assessed by British civil service grade of employment taken from the phase 7 questionnaire. Behavioural risk factors were assessed using responses to the phase 7 questionnaire: smoking status (none, former or current), recommended physical activity (yes or no), high alcohol consumption (yes or no). The following biological CVD risk factors at phase 7 clinical examination were considered: hypertension (systolic/diastolic blood pressure >140/90 mmHg or antihypertensive medication), body mass index (BMI <20, 20-24.9, 25-29.9, or ≥30 kg/m 2 ) and diabetes (yes or no).

Statistical analyses

We examined the associations of depressive symptoms with mortality outcomes using serially adjusted Cox regression models. The contribution of sleep to these associations was assessed by including the two sleep variables into the Cox regression model separately and simultaneously and calculating the percentage of reduction in the hazard ratios. We combined men and women in the analyses (p >0.05 for interaction with sex) and verified that the assumptions for proportional hazards were not violated (all p>0.05).

RESULTS

A total 5813 participants (4220 men and 1593 women) aged 50 to 74 had complete data on the variables of interest in 2002-04 (the baseline of the present study) and were included in the analysis. Of these participants, 14.6% were above our cut-point for depression, 8.0% were considered "short sleepers" and 31.0% as with "sleep disturbances". As expected the depressive symptoms score and sleep scores were correlated, with coefficients of -0.22 for sleep duration and 0.27-0.50 for sleep quality items.

During a mean follow-up of 8.8 years, 338 deaths from all causes occurred, including 98 deaths from cardiovascular diseases. There was some evidence suggesting an association between "short sleep" duration (fully adjusted HR=1.41; 95% 0.75-2.68) and "sleep disturbances" (Fully adjusted HR=1.26; 95% 0.83-1.90) and cardiovascular mortality, although the evidence was weak, probably due to lack of power. For all-cause mortality, HRs varied between 0.99 and 1.14 and there was no evidence of an association (data not shown).

Table 1 shows the associations between depressive symptoms and mortality and the contribution of sleep to these associations. After adjustment for sociodemographic variables, depressive symptoms were associated with an increased risk of all-cause mortality (HR= 1.57; CI 95%, 1.19-2.06). Additional adjustment for sleep variables did not materially changed the magnitude and the significance of this association.

Depressive symptoms were associated with increased cardiovascular mortality (HR=1.67; 95% CI, 1.02-2.75) in model adjusted for sociodemographic variables. Additional adjustment for sleep variables reduced the magnitude of the association by 21% (HR=1.53; 95% CI, 0.91-2.57) and rendered it statistically non-significant.

DISCUSSION

We sought to examine the role of sleep in explaining the association of depressive symptoms with all-cause and cardiovascular mortality. We found that depressive symptoms predicted both mortality outcomes even after adjustment for sociodemographic characteristics. However, the association between depressive symptoms and cardiovascular mortality was reduced and no longer significant after additional adjustment for sleep duration and disturbances.

Our finding is consistent with the notion that sleep in part mediates the association between depressive symptoms and cardiovascular mortality. However, given that depressive symptoms and sleep variables were measured concurrently, we cannot exclude the possibility that sleep may confound (rather than mediate) the observed association with cardiovascular mortality. For instance, sleep problems may independently influence the onset of depressive symptoms [START_REF] Mezick | Are sleep and depression independent or overlapping risk factors for cardiometabolic disease?[END_REF] and increase mortality risk, supporting their role as a confounding factor.

We found sleep to play a more important role in the association between depressive symptoms and cardiovascular mortality than in the association with all-cause mortality. This finding is biologically plausible because sleep problems have been found to be associated with major CVD risk factors, including obesity, hypertension, diabetes, and inflammation [START_REF] Tasali | Obstructive sleep apnea and metabolic syndrome: alterations in glucose metabolism and inflammation[END_REF][START_REF] Spiegel | Sleep loss: a novel risk factor for insulin resistance and Type 2 diabetes[END_REF][START_REF] Williams | Sleep duration and snoring in relation to biomarkers of cardiovascular disease risk among women with type 2 diabetes[END_REF].

To our knowledge, this is the first study to explore the role of sleep in explaining the association between depressive symptoms and mortality. We were able to adjust for a wide range of factors which can potentially confound the associations of interest. The present findings should be interpreted in light of some limitations. First, our study is based on government employees and is not representative of the general population, limiting the generalizability of these findings. Second, we examined depressive symptoms rather than clinical depression. Thus, the relationship between depression and mortality may have been underestimated. Moreover, sleep duration and sleep disturbances were self-reported.

Reporting bias and measurement errors due to how sleep duration was recorded (responses were recorded as whole numbers of hours) may have biased the contribution of sleep to the relationship between depression and cardiovascular mortality. Third, the mean follow-up period was 8.8 years which explains the low number of deaths, particularly CVD deaths. Larger studies with longer follow-up and with repeated measures of depressive symptoms and sleep are needed to test the role of sleep. Despite these limitations, the results represent a unique contribution to the literature.

These results suggest that sleep may partially explain the association between depressive symptoms and cardiovascular mortality. Although sleep may also act as a confounder, these findings underscore the importance of considering sleep in studies aimed at examining increased cardiovascular mortality in depressive individuals. If the mediating role of sleep is confirmed, the implication is that preventive efforts for persons with depression should take into account aspects related to the quality of their sleep.

Acknowledgements

We thank all participating civil service departments and their welfare personnel, and establishment officers; the Occupational Health and Safety Agency; the Council of Civil Service Unions; all participating civil servants in the Whitehall II study; all members of the Whitehall II study team. The Whitehall II Study team comprises research scientists, statisticians, study coordinators, nurses, data managers, administrative assistants and data entry staff, who make the study possible. HN is supported by a grant from Institut de Recherche en Santé Publique (IReSP 2011 A11228LS). MK is supported by the Medical Research Council, the EU OSH ERA Research Programme, the Academy of Finland, the US National Institutes of Health (R01HL036310, R01AG034454) and an ESRC professorship. MJS is supported by a grant from the British Heart Foundation. AS-M is supported by a "European Young Investigator Award" from the European Science Foundation and the National Institute on Aging, NIH (R01AG013196, R01AG034454).

Author contributorship

HN designed the study. MADS and HN managed the literature searches. MADS undertook the Statistical analysis and wrote the first draft of the manuscript with HN. All authors contributed to the interpretation of the results and have approved the final manuscript.