Florent Domine
email: florent.domine@gmail.com

Kevin Fourteau

Philippe Choler

Exploration of Thermal Bridging through Shrub Branches in Alpine Snow

The winter ground temperature in a snow-covered alpine area during a cold spell is 1.3°C colder under alders than under nearby grass.

• Using finite element modeling, we demonstrate that this is due to thermal bridging through alder branches.

• Plane-parallel snow layer geometry cannot be used to model ground temperature, soil freezing, and snow metamorphism if shrubs are present.

Introduction

In snow-covered areas, heat exchanges between the atmosphere and the ground through the snowpack determine winter ground temperature [START_REF] Domine | Major Issues in Simulating Some Arctic Snowpack Properties Using Current Detailed Snow Physics Models: Consequences for the Thermal Regime and Water Budget of Permafrost[END_REF]. Physical snow models describe heat exchanges through snowpacks based on a plane-parallel snow layer geometry. In Arctic and Alpine areas, shrubs are expanding [START_REF] Anthelme | Environmental and spatial influences of shrub cover (Alnus viridis DC.) on vegetation diversity at the upper treeline in the inner western Alps[END_REF][START_REF] Wiedmer | Biomass and primary productivity of an Alnus viridis stand -a case study from the Schachental valley, Switzerland[END_REF][START_REF] Ju | The vegetation greenness trend in Canada and US Alaska from 1984-2012 Landsat data[END_REF], inducing changes in snow properties. Shrubs trap blowing snow, increasing snow height [START_REF] Sturm | Snow-shrub interactions in Arctic tundra: A hypothesis with climatic implications[END_REF][START_REF] Lamare | Investigating the Role of Shrub Height and Topography in Snow Accumulation on Low-Arctic Tundra using UAV-Borne Lidar[END_REF], reducing snow compaction, and favoring the formation of the most thermally-insulating snow type: depth hoar [START_REF] Domine | The growth of shrubs on high Arctic tundra at Bylot Island: impact on snow physical properties and permafrost thermal regime[END_REF].. Furthermore, shrub branches disrupt the plane-parallel geometry. Recently, thermal bridging through frozen shrub branches buried in snow has been detected in the high Arctic [START_REF] Domine | Permafrost cooled in winter by thermal bridging through snow-covered shrub branches[END_REF], enhancing ground cooling in winter, despite widespread depth hoar.

Cooling averaged over the November-February period was 1.21°C, but reached 4°C during cold spells. Accurate simulation of the ground temperature by snow models is crucial as it impacts the temperature gradient in the snowpack, which determines metamorphism and snow physical properties. Ground temperature is equally important to land surface models to simulate carbon and nutrient recycling in soils, which considerably slows down when the ground freezes [START_REF] Saccone | The effects of snowpack properties and plant strategies on litter decomposition during winter in subalpine meadows[END_REF].

In the Arctic, the thermal conductivity contrast between snow found in shrubs and frozen shrub branches is a factor of 30 to 70. Snow within shrubs is mostly depth hoar with a thermal conductivity of 0.03 to 0.07 Wm -1 K -1 in the Arctic [START_REF] Domine | The growth of shrubs on high Arctic tundra at Bylot Island: impact on snow physical properties and permafrost thermal regime[END_REF][START_REF] Domine | Permafrost cooled in winter by thermal bridging through snow-covered shrub branches[END_REF] while the thermal conductivity of frozen wood is around 2 Wm -1 K -1 [START_REF] Domine | Permafrost cooled in winter by thermal bridging through snow-covered shrub branches[END_REF]. Therefore, even though high Arctic shrub branches rarely exceed 3 cm in diameter [START_REF] Sturm | Snow-shrub interactions in Arctic tundra: A hypothesis with climatic implications[END_REF][START_REF] Domine | Permafrost cooled in winter by thermal bridging through snow-covered shrub branches[END_REF], the thermal conductivity contrast is sufficient for thermal bridging through frozen shrub branches to efficiently cool the ground.

Whether thermal bridging also operates in Alpine regions is uncertain because the thermal contrast between alpine snow and the presumably unfrozen alpine shrub branches is much less than in the Arctic. [START_REF] Morin | In-situ measurement of the effective thermal conductivity of snow[END_REF], [START_REF] Calonne | Numerical and experimental investigations of the effective thermal conductivity of snow[END_REF] and [START_REF] Riche | Thermal conductivity of snow measured by three independent methods and anisotropy considerations[END_REF] measured thermal conductivity values from 0.05 to 0.46 Wm -1 K -1 for Alpine snow. These values were however for snow without shrubs, whose effects are therefore not considered.

Snow temperature in Alpine regions where shrubs are found seldom drop durably below -10°C [START_REF] Morin | In-situ measurement of the effective thermal conductivity of snow[END_REF][START_REF] Morin | An 18-yr long (1993-2011) snow and meteorological dataset from a mid-altitude mountain site (Col de Porte, France, 1325 m alt.) for driving and evaluating snowpack models[END_REF]. At these temperatures, shrub branches probably do not freeze because of physiological freeze resistance [START_REF] Squeo | Cold resistance mechanisms in high desert Andean plants[END_REF][START_REF] Zhang | Freezing resistance in Patagonian woody shrubs: the role of cell wall elasticity and stem vessel size[END_REF]. The axial thermal conductivity of fresh wood is 0.5 to 1 W m -1 K -1 , with the higher values more frequent [START_REF] Turrell | Thermal Conductivity of Functional Citrus Tree Wood[END_REF][START_REF] Deliiski | Computation of the wood thermal conductivity during defrosting of the wood[END_REF][START_REF] Laguela | Thermal conductivity measurements on wood materials with transient plane source technique[END_REF]. The thermal contrast between snow and buried shrub branches in Alpine regions is therefore in the range 1 to 20, much less than in the Arctic. Alpine shrub branches are however much bigger than in the Arctic, e.g.green alders (Alnus alnobetula) have branches up to 10 cm in diameter [START_REF] Wiedmer | Biomass and primary productivity of an Alnus viridis stand -a case study from the Schachental valley, Switzerland[END_REF].

The cross-sectional surface area of shrub branches may therefore be at least a factor of 10 greater than in the Arctic, compensating the lower thermal conductivity contrast.

Thermal bridging through shrub branches buried in Alpine snow therefore deserves investigation. This preliminary study of the process explores whether a more complete study with extensive instrumental deployment is justified. We report data from the 2021-2022 winter near the Col du Lautaret, French Alps. Using finite element modeling, we conclude that thermal bridging through shrub branches is necessary to explain observations. More detailed studies of thermal bridging in Alpine conditions are therefore warranted.

Materials and Methods

Species and site description

The green alder, Alnus alnobetula , is a tall deciduous shrub growing up to 5-6 m. It exhibits clonal growth with many ascending branches, which sprout from a horizontal thickened stem. In the Alps, green alders form dense stands in many north-exposed slopes between 1000 m and 2000 m [START_REF] Richard | Écologie de L'aune Vert (Alnus viridis): Facteurs Climatiques et édaphiques[END_REF][START_REF] Wiedmer | Biomass and primary productivity of an Alnus viridis stand -a case study from the Schachental valley, Switzerland[END_REF]). The estimated above-ground biomass of dense Alnus stands amounts 100 t ha -1 [START_REF] Richard | Écologie de L'aune Vert (Alnus viridis): Facteurs Climatiques et édaphiques[END_REF], much more than in the Arctic, where values around 10 and 2 t ha -1 have been estimated in the low and high Arctic, respectively [START_REF] Greaves | High-resolution mapping of aboveground shrub biomass in Arctic tundra using airborne lidar and imagery[END_REF][START_REF] Paradis | Greater effect of increasing shrub height on winter versus summer soil temperature[END_REF]. Over the recent decades, the green alder has been expanding at the expense of formerly grazed grasslands (Figure S1 in supporting information), affecting biodiversity [START_REF] Anthelme | Environmental and spatial influences of shrub cover (Alnus viridis DC.) on vegetation diversity at the upper treeline in the inner western Alps[END_REF].

The study site (45.034750°N, 6.413630°E, 2034 m asl), called Alnus-Nivus, is a northfacing slope dominated by green alders up to 2.5 m tall (Figure 1 and Figure S2 in Supporting Information). It is located within a nature reserve so that placing visible instruments is not permitted. Only buried sensors could be deployed. The site was selected because of the presence of the nearby FR-Clt meteorological station (45.041278°N, 6.410611°E, 2046 m asl, [START_REF] Gupta | Impact of distributed meteorological forcing on simulated snow cover and hydrological fluxes over a mid-elevation alpine micro-scale catchment[END_REF]) about 750 m away on a 4° south-facing slope across the valley, where there is no alder, and which is outside the nature reserve. This station is part of the Integrated Carbon Observation System network (https://www.icos-cp.eu/observations/ecosystem/stations).

Instruments

We placed 5TM temperature and fractional volume water content sensors from Decagon in the ground at 5 and 15 cm depths at two spots: (i) under a green alder, near the emergence of ascending branches, hereafter called ALNUS; and (ii) in a nearby grass patch about 6 m away, hereafter called GRASS. Values of air temperature, wind speed and downwelling longwave radiation measured at FR-Clt are probably reasonably similar to those at Alnus-Nivus. However, downwelling shortwave radiation is significantly different in winter because the site is entirely shaded by mountains. Snow height is higher at Alnus-Nivus. On 31 March 2021, we measured over 1.5 m of snow at both ALNUS and GRASS vs only 0.5 m at the FR-Clt station.

Field measurements

On 19 September 2022, we measured the ground thermal conductivity at 5 and 15 cm depths at ALNUS and GRASS using a TP02 heated needle probe from Hukseflux, as described in [START_REF] Domine | Automatic monitoring of the effective thermal conductivity of snow in a low-Arctic shrub tundra[END_REF]. At ALNUS, we counted a total of 35 primary branches at 50 cm height, with a mean diameter of 6.5 cm. No winter measurements were performed because of the avalanche risk.

Finite element model calculations

Heat transfer between snow, soil, and shrub branches was simulated with finite element modeling, using the open source ElmerFEM software [START_REF] Malinen | Elmer Finite Element Solver for Multiphysics and Multiscale Problems[END_REF], as detailed in [START_REF] Domine | Permafrost cooled in winter by thermal bridging through snow-covered shrub branches[END_REF]. The shrub geometry used, based on field observations, is shown in Figure 2. Branches were constructed only to a height of 1 m, greater than the 0.7m snow height. All branches were 6.5 cm in diameter, the average measured. Secondary branches were omitted. The wood density and specific heat values were 900 kg m -3 and 1200 J kg -1 K -1 [START_REF] Radmanovic | Specific Heat Capacity of Wood[END_REF]. Since measured soil thermal conductivities were different at ALNUS and GRASS (Table S1 in Supporting Information), we divided the ground between an ALNUS ground with a radius of 0.75 m around the shrub, and a GRASS ground beyond that. The soil comprised three layers: 0-8 cm, 8-30 cm, and 30 cm to 10 m (Figure 2 and Table S1). For frozen soil, measured values were multiplied by two (Table S1), following measurements in the Arctic [START_REF] Domine | Seasonal evolution of the effective thermal conductivity of the snow and the soil in high Arctic herb tundra at Bylot Island, Canada[END_REF][START_REF] Domine | Permafrost cooled in winter by thermal bridging through snow-covered shrub branches[END_REF]. The baseline soil volumetric thermal capacity used was 2.6

x 10 6 J m -3 K -1 , corresponding to a density of 1300 kg m -3 and a specific heat of 2000 J K -1 kg -1 .

To represent the effect of water phase change on temperature, a Soil Freezing Characteristic Curve (SFCC) was used under the form of an increased apparent thermal capacity [START_REF] Tubini | A method for solving heat transfer with phase change in ice or soil that allows for large time steps while guaranteeing energy conservation[END_REF]. In practice, the thermal capacity of the soil was increased near the freezing point, accounting for the extra-energy needed to change the soil temperature just below 0°C, as a fraction of the energy is converted to latent rather than sensible heat. For instance, warming the soil from -1°C to 0°C requires an extra 9.1x10 6 J m -3 of energy, compared to 2.6x10 6 J m -3 if no freezing occurs. The SFCC was derived from water content versus temperature data measured at a tundra site at Bylot Island [START_REF] Domine | Seasonal evolution of the effective thermal conductivity of the snow and the soil in high Arctic herb tundra at Bylot Island, Canada[END_REF][START_REF] Domine | Meteorological, snow and soil data (2013-2019) from a herb tundra permafrost site at Bylot Island, Canadian high Arctic, for driving and testing snow and land surface models[END_REF]. The soil there was silt loam, reasonably similar to that at ALNUS, based on observations on 19 September 2022.

A time-constant snow height was used for all simulations. The Gmsh software [START_REF] Geuzaine | Gmsh: A 3-D finite element mesh generator with built-in pre-and postprocessing facilities[END_REF] was used to generate the finite element mesh. The simulations were run with a 30 min time step and forced at the top with the snow surface temperature, obtained from the longwave upwelling radiation measured at FR-Clt, using an emissivity of 1, and with the mean annual air temperature at 10 m depth, 3°C. Simulations ouputs used were temperatures in the vicinity (representing ALNUS) and away (representing GRASS) from the alder recorded at 5 and 15 cm depths, as shown in the bottom panel of Figure 2.

Results and discussion

Detection of thermal bridging from observations

Permanent snow cover at FR-Clt started on 25 November. Snow height exceeded 10 cm on 2 December 2021 and was 25 cm around 8 December (Figure 3). Snow height then varied between 30 and 70 cm, while the air temperature was often above 0°C and seldom dropped below -10°C (Figure 3). The snow surface temperature dropped below -20°C on 3 December and was often below -15°C. Temperatures are shown in Figure 3 as 24 h running means for clarity.

Measured ground temperatures and ground liquid water contents at 5 and 15 cm depths are shown in Figure 4. These data inform us on sensible and latent heat fluxes between the ground and the atmosphere through the snow. Our analysis focuses on the period with permanent snow cover, starting on 1 December 2021. This is consistent with an extra heat transport process existing at ALNUS. We hypothesize that thermal bridging through frozen shrub branches is that process. We detail two selected events.

Striking features in

A cold spell started on 4 January 2022 (Figure 3), with the air and snow surface temperatures dropping below 0°C. The daily-averaged snow surface temperature then remained below -5°C during 30 days. Figure 4 shows that at 5 cm depth at GRASS, the liquid water content and the ground temperature remained essentially unchanged throughout January, indicating little sensible or latent heat changes, probably because the limited heat loss through the insulating snowpack was compensated by the upward ground heat flux. At ALNUS however, most of the ground water froze rapidly on 6-7 January, indicating significant latent heat loss.

Once most of the water was frozen, the ground temperature dropped below 0°C, and the sensible heat loss is clearly visible in Figures 4a and4b, starting on 8 January.

Warming events also show much faster heat exchanges at ALNUS. The warming spell that started on 28 December led to the melting of essentially all the frozen ground water at ALNUS (Figures 4c,d), while only a small amount of frozen water melted at GRASS. Latent heat exchanges are thus much greater at ALNUS, and these exchanges most likely took place by heat transfer through the shrub branches.

Changes are similar at 15 cm depth, albeit slower because of the greater thermal inertia in deeper ground. At GRASS, the ground temperature remained at 0°C, while most of the water remained liquid. At ALNUS on the contrary, most of the water froze during cold spells and the temperature dropped to -0.7°C.

The extra sensible and latent heat exchanges at ALNUS at both depths may therefore reasonably be assigned to thermal bridging through snow. We test this hypothesis using finite element calculations.

Confirmation of thermal bridging by simulations

Finite element simulations were performed starting on 10 December to avoid the low and variable snow height in early December. Simulations were first performed with 70 cm of snow because we observed thicker snow at Alnus-Nivus than at FR-Clt. We used a snow thermal conductivity value of 0.1 Wm -1 K -1 , so that the wood/snow thermal conductivity contrast is a factor of 10. We first performed simulations without shrubs to ensure that the different ground properties at ALNUS and GRASS were not responsible for observations. Ground thermal property differences do result in a cooler ground at ALNUS (Figure 5a), but by only 0.25°C and only for two short periods, insufficient to explain measurements. Figures 5b and5c reproduce the data of Figure 4 reasonably well, given the simplifications. The GRASS temperatures remain close to 0°C, while those at ALNUS drop during the cold spells, to values identical to those measured. Despite the use of a SFCC for the soil, the zero-curtain effect is not as marked in simulations as in measurements, probably because of differences in SFCCs at Alvus-Nivus and at Bylot. Figures 5a to 5c demonstrate that including thermal bridging through shrub branches adequately simulates results.

Since snow height was not measured at Alnus-Nivus, we performed simulations for snow heights of 35 and 140 cm. Results are shown in Figure S3. As expected, thinner snow leads to in general colder ground while thicker snow leads to reduced cooling. However, qualitatively, results are similar, with a better thermal coupling between snow and ground at ALNUS, due to thermal bridging.

A higher snow thermal conductivity reduces the relative impact of thermal bridging.

Figure S4 in Supplementary Information shows simulations with a snow thermal conductivity of 0.2 Wm -1 K -1 . Expectedly, the temperature difference between ALNUS and GRASS decreases, but the general trends are still reasonably reproduced.

Branch diameter and thermal conductivity also affect heat exchanges. Figure S5 shows results with 70 cm of snow of thermal conductivity 0.1 Wm -1 K -1 , and branches of 32, 65 and 130 mm diameter. Obviously, thicker branches increase the temperature difference between ALNUS and GRASS, confirming that thermal bridging strongly affects the temperature at ALNUS. Alternative scenarios to shrub thermal bridging were explored to explain measurements.

Reduced snow height at ALNUS would increase atmosphere-ground thermal coupling there. We performed simulations without shrubs, with only 35 cm of snow at ALNUS and snow height increasing to 70 cm towards the edge of the simulation domain (Figure S5). Figure 5d shows that results at ALNUS are in fact reasonably similar to those of Figure 5b. However, daily temperature variations are visible, while they do not appear in the data (Figure 4a) indicating that the thermal insulance of the snowpack is too low to realistically simulate the data. Furthermore, general observations indicate that the effect of shrub on snow height is rather to increase it, not to decrease it [START_REF] Sturm | Snow-shrub interactions in Arctic tundra: A hypothesis with climatic implications[END_REF][START_REF] Busseau | Analysis of snow-vegetation interactions in the low Arctic-Subarctic transition zone (northeastern Canada)[END_REF][START_REF] Lamare | Investigating the Role of Shrub Height and Topography in Snow Accumulation on Low-Arctic Tundra using UAV-Borne Lidar[END_REF]. Moreover, snow height was not lower at ALNUS than at GRASS on 31 March 2021. The scenario of Figure 5d therefore appears unrealistic. Thermal bridging through alder branches is therefore the most reasonable explanation for the temperature difference between ALNUS and GRASS, and for the close thermal coupling between ALNUS and the atmosphere.

Upscaled impact and implementation in models

We calculated the impact of heat exchange through shrubs at the km scale by assuming a shrub density of one alder per square of 2x2m. By calculating the spatially-averaged heat flux with and without shrubs, we find that the average thermal insulance decreases from 7 to 5.9 K m - 2 W -1 when shrubs are added. Alders represented in Figure 2, with a 70 cm snowpack, thus increase the heat flux through the snowpack by 15%. It is not possible to translate this directly into a temperature change because in most Alpine areas the soil temperature is usually near 0°C, meaning that latent heat exchanges are important, so that much of the flux will serve to freeze soil water. Soil freezing considerably reduces nutrient recycling and favors carbon accumulation [START_REF] Saccone | The effects of snowpack properties and plant strategies on litter decomposition during winter in subalpine meadows[END_REF]. Given that green alders are now quite widespread in the French Alps (Figure S7), the impact may be quite large and deserves evaluation.

In snow models, shrub thermal effects may be implemented simply by adjusting the thermal insulance of the snowpack to account for the increased heat flux. A more sophisticated approach would introduce shrubs as an extra domain that thermally couples the atmosphere and the soil. However, shrubs also affect radiation transfer in snowpacks [START_REF] Belke-Brea | On the influence of erect shrubs on the irradiance profile in snow[END_REF], affecting snow temperature and metamorphism. Fully accounting for shrub effects would therefore require significant model modifications. The efforts are probably worthwhile, because shrubs keep expanding and adequate simulations of soil temperature are required for carbon budgets and productivity modeling [START_REF] Loranty | Reviews and syntheses: Changing ecosystem influences on soil thermal regimes in northern high-latitude permafrost regions[END_REF].

Conclusion

Field measurements and finite-element simulations lead to the conclusion that ground temperature under green alders is affected by thermal bridging through branches. Relative to the high Arctic, the lower thermal conductivity contrast between snow and branches is compensated by the greater branch cross section; 1095 cm 2 per simulated shrub here vs 92 cm 2 in the Arctic [START_REF] Domine | Permafrost cooled in winter by thermal bridging through snow-covered shrub branches[END_REF]. Thermal bridging through shrub branches buried in snow is therefore probably a general process. Given the ever-greater shrub expansion in seasonally snow-covered areas, this process must be considered in snow physics models to accurately describe the thermal gradient in snowpacks, snow metamorphism and snow physical properties.

Thermal bridging also affects the thermal regime of the ground and here causes extensive ground freezing. This strongly affects microbial metabolism and therefore the carbon budget of soils, nutrient recycling and primary productivity [START_REF] Saccone | The effects of snowpack properties and plant strategies on litter decomposition during winter in subalpine meadows[END_REF]. These widespread effects justify further detailed study where all relevant meteorological, snow and soil variables are measured. S1). Measurement spots at 5 and 15 cm depths, and 10 and 140 cm from the center are indicated. Snow height was 70 cm.

  Figure 4 include: (1) cooling stages at ALNUS in early January and March (Figure 4a,b), indicating large sensible heat losses at ALNUS, with no sensible heat losses at GRASS; (2) large rapid changes in liquid water content at ALNUS in early December, late December and early January (Figure 4c,d), indicating large latent heat exchanges at ALNUS, with much lower and/or much slower exchanges at GRASS. ALNUS is thus clearly closely coupled to atmospheric temperature changes, while this coupling is much weaker at GRASS.

Figure 1 .

 1 Figure 1. Map showing the location of the Alnus-Nivus site and of the FR-Clt meteorological station near Col du Lautaret, central French Alps. Maps from https://en-gb.topographicmap.com/.

Figure 2 .

 2 Figure 2. (Top) Perspective views of the alder mesh used for FEM simulations. Two concentric sets of 16 caudices 6.5 cm in diameter and 1 m high, with one extra central caudice, were used. The central trunk is 20 cm in diameter. (Bottom) Conditions for our baseline simulation. Three soil layers were used. The top two layers had different thermal properties under the alder and under the grass (TableS1). Measurement spots at 5 and 15 cm depths, and 10 and 140 cm from

Figure 3 .

 3 Figure 3. Meteorological variables measured at FR-Clt. (a) snow height; (b) 24h running-mean air and snow surface temperatures.

Figure 4 .

 4 Figure 4. Ground temperature and volume liquid water fraction measured under alders (ALNUS) and in the nearby grass patch (GRASS). (a) temperature at 5 cm depth; (b) at 15 cm depth; (c) volume liquid water fraction at 5 cm depth; (d) at 15 cm depth.

Figure 5 .

 5 Figure 5. Ground temperature simulated at ALNUS and GRASS. (a) at 5 cm depth, without shrubs; (b) at 5 cm depth, with shrubs; (c) at 15 cm depth, with shrubs; (d) at 5 cm depth without shrubs and with 35 cm of snow at ALNUS and 70 cm of snow at GRASS.

Acknowledgements

This work was funded by NSERC (discovery grant to FD). KF was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (IVORI grant 949516). Meteorological data are from the FR-Clt station, associated to the ICOS Ecosystem station network, and funded by labex OSUG@2020 and Jardin du Lautaret. We thank the staff of Jardin du Lautaret, in particular Pascal Salze and Lucie Liger, for logistical support.

Open Research

The meteorological and soil data shown in Figure 3 and4, as well as the simulation framework including the shrub mesh of Figure 2, are available on the Zenodo repository at https://doi.org/10.5281/zenodo.8057992, [START_REF] Domine | Soil and meteorological data, and finite element simulation framework for heat transfer through shrubs in winter near Lautaret pass, French Alps[END_REF].


Ashley J Thomas
email: ajthomas@mit.edu

Brandon Woo

Daniel Nettle

Elizabeth Spelke

Rebecca Saxe

Early concepts of intimacy: young humans use saliva sharing to infer close relationships

Across human societies, people form 'thick' relationships, characterized by strong attachments, obligations and mutual responsiveness. People in thick relationships engage in distinctive interactions, like sharing food utensils or kissing, that involve sharing saliva. Here we show that children (N=113), toddlers (N=190), and infants (N=81) infer that dyads who share saliva (compared to other positive social interactions) have a distinct relationship. Children expect saliva sharing to happen in nuclear families. Toddlers and infants expect that people who shared saliva will respond to one another in distress. Parents (N=129) confirm that saliva sharing is a valid cue of relationship thickness in their children's social environments. The ability to use distinctive interactions to infer categories of relationships thus emerges early in life, without explicit teaching, allowing young humans to rapidly identify close relationships, both within and beyond families.

One-Sentence Summary: Children, toddlers and infants infer that individuals who share saliva are in distinctive close social relationships.

Main Text:

In order to become a competent member of society, humans must learn how the people around them are related to each other [START_REF] Kaufmann | Wired for Society: Cognizing Pathways to Society and Culture[END_REF][START_REF] Thomsen | Navigating the Social World: What Infants, Children, and Other Species Can Teach Us[END_REF][START_REF] Hirschfeld | On a Folk Theory of Society: Children, Evolution, and Mental Representations of Social Groups[END_REF]. Across cultures, people distinguish a special category of relationships, which we will call 'thick' (4) [START_REF] Clark | Interpersonal attraction in exchange and communal relationships[END_REF][START_REF] Rai | Moral psychology is relationship regulation: moral motives for unity, hierarchy, equality, and proportionality[END_REF][START_REF] Sahlins | Stone Age Economics[END_REF], that feature strong levels of attachments, obligations, mutual responsiveness, and a feeling of oneness which is conceived in terms of shared bodily substance [START_REF] Clark | Interpersonal attraction in exchange and communal relationships[END_REF][START_REF] Rai | Moral psychology is relationship regulation: moral motives for unity, hierarchy, equality, and proportionality[END_REF][START_REF] Sahlins | Stone Age Economics[END_REF]. Thick relationships are often, but not always, between close genetic relatives [START_REF] Schneider | A Critique of the Study of Kinship[END_REF][START_REF] Sahlins | The Use and Abuse of Biology: An Anthropological Critique of Sociobiology[END_REF][START_REF] Jones | The generative psychology of kinship: Part 1. Cognitive universals and evolutionary psychology[END_REF][START_REF] Cronk | Kin terms and fitness interdependence[END_REF][START_REF] Carsten | the substance of kinship and the heat of the hearth: feeding, personhood, and relatedness among Malays in Pulau Langkawi[END_REF]. The fact that only some relationships are thick presents young humans with a problem: how do they identify which ones? For older children, distinct relationship categories can be explicitly verbally labelled [START_REF] Hirschfeld | Rethinking the Acquisition of Kinship Terms[END_REF]. Anthropologists have claimed that younger children and even infants must be sensitive to how relationships are embodied in distinctive interactions [START_REF] Cohen | The hard embodiment of culture[END_REF][START_REF] Fiske | Differently embodying different relationships[END_REF]. For example, interactions that involve deliberate consensual transfer of saliva, such as kissing or sharing food utensils, distinctively occur in thick relationships ( [START_REF] Sorokowska | Affective Interpersonal Touch in Close Relationships: A Cross-Cultural Perspective[END_REF][START_REF] Miller | Food sharing and feeding another person suggest intimacy; two studies of American college students[END_REF]; see Supplemental Materials [START_REF]An analysis of saliva sharing across cultures as described in ethnographies is available in the supplementary materials at the Science website[END_REF]). Here we apply experimental techniques from developmental science to test whether young children, toddlers, and infants do indeed infer that two individuals who share saliva are likely to be in a thick relationship.

In a first experiment, when presented with interactions between cartoon people, young children (Experiment 1, N=113, 5-7 years old, from an American urban environment) predicted that sharing utensils, or licking the same food item, would occur within nuclear families; whereas sharing toys and partitionable food would occur equally within friendships and families (χ2=72.74, p<.001, Figure 1; [START_REF] Olson | Foundations of Cooperation in Young Children[END_REF][START_REF] Spokes | Children's Expectations and Understanding of Kinship as a Social Category[END_REF][START_REF]Materials and methods are available as supplementary[END_REF]). Thus young children recognize that saliva-sharing interactions distinctively occur within nuclear families. In the next experiments, we tested whether toddlers and infants would predict that when two individuals have shared saliva, those individuals will be more emotionally responsive in future interactions (1). This experimental design was inspired by classic studies of vervet monkeys who heard a familiar juvenile in distress, and looked towards that juvenile's mother, as if expecting her to respond [START_REF] Seyfarth | The structure of social knowledge in monkeys[END_REF]. We used this design to test whether young humans use a brief observation of saliva sharing to infer a thick relationship between novel individuals, whose genetic relatedness is unknown.

Toddlers (Experiment 2A, N=27, 16.5-18.5 months old) and infants (Experiment 2B, N=20, 8.5-10 months old) saw a central puppet alternately eat from the same orange slice with one actress (implying saliva sharing), and play ball with another actress (Figure 2). Then they saw the puppet seated between the two actresses, expressing distress. We measured which actress participants looked toward first, and longer, as though expecting the actress to react to the puppet's distress. Both toddlers and infants looked first, and longer, toward the actress who had shared food and saliva with the puppet (First look: 2A Toddlers: [START_REF] Spokes | Children's Expectations and Understanding of Kinship as a Social Category[END_REF] Further experiments revealed that toddlers and infants looked toward the food-and-salivasharing actress (i) only when the central puppet expressed distress and (ii) only when the puppet in distress was the actress' own thick relation. When the central puppet was removed leaving only the two actresses, toddlers and infants looked at both actresses equally (Figure 2 and ( 21)).

When the central puppet was replaced by a new puppet, who then expressed distress, neither infants nor toddlers looked first or longer at the food sharer (First look: 2Ccontrol Toddlers: 12/28, BF01=3.287; 2Dcontrol Infants: 13/25; BF01=4.03; Proportion look: Toddlers: M=0.496, BF01=4.903; Infants: M=0.470, BF01=4.46). These results suggest that toddlers' and infants' expectations concern the relationship, not the individuals' traits.

For experiment 2.E, we recruited a larger, more economically, geographically and racially diverse sample of toddlers (N=118; age 14.5-19 months, see [START_REF]Materials and methods are available as supplementary[END_REF]). The familiarization events were the same. Then, the central puppet either expressed distress as before (2Etest) or asked for the ball (2Econtrol). When the central puppet expressed distress, toddlers looked first, and longer, toward the food-sharer rather than the ball-passer (2Etest: First look: 35/48; BF10=59.5; Proportion look: M =0.65; BF10=1002). By contrast, when the puppet requested the ball, toddlers looked first and longer at the ball-passer (First look: 2Econtrol :13/52; BF10 =267; Proportion look: M=0.37; BF10=54). These conditions differed decisively (First look: BF10 >1000; proportion of time: BF10 >1000). Thus, toddlers from a diverse range of households expect that two people who share food and saliva will respond to each other's distress, but not that they will be socially more responsive to one another in general. In Experiment 3, we isolated sharing saliva, without food, as the visible evidence of a thick relationship. Interacting with one puppet, a central actress put her finger in her own mouth, rotated it, put her finger in the puppet's mouth, rotated it, and finally returned her finger to her own mouth. When interacting with a second puppet, the actress performed the same rotating finger actions touching her own and the puppet's forehead. We then measured which puppet infants and toddlers looked toward, when the central actress expressed distress.

Toddlers (age 16.5-18.5 months) looked first, and longer, towards the puppet from the mouth-to-mouth interaction, when the actress expressed distress (Experiment 3A, N=27, First look: 20/26; BF10=10.796; Proportion look: M=0.746, BF10=477.6). These results were replicated in an independent sample of toddlers (3Ctest N=31, First Look: 25/31, BF10=91.15; Proportion look: M=0.748, BF10=14856). Infants' (age 8.5-10 months) first looks were distributed equally between both puppets (3B N=21, First look: 11/21; BF01=3.700; replication 3Dtest N=26; 14/26; BF01=3.886). Yet infants did look longer toward the puppet from the mouth-to-mouth interaction while the actress expressed distress (3B M=0.631; BF10=1.55; replication 3Dtest : M=0.716,

BF=26.812).

Further experiments revealed that toddlers and infants looked toward the mouth-to-mouth puppet (i) only when the actor in distress was the person in the initial interactions, and (ii) only when the central actress expressed distress. When the actress was removed and replaced by a new actor who expressed distress, infants and toddlers seemed to expect that the other puppet, who had the forehead-to-forehead interaction with the first actress, was more likely to react (Toddlers 3Ccontrol: First look: 6/23 BF10=3.462; Proportion look: M=0.364, BF10=1.023; Infants 3Dcontrol: First look: 8/23 BF01=1.403; Proportion look: M=0.252; BF10=88.72). An independent group of toddlers (Experiment 3E, ( 21)) looked toward the mouth-to-mouth puppet when the central actress expressed distress (3Etest First look: 40/52; BF10=823; Proportion look: M=0.690, BF10=71669), but not when the actress uttered a nonsense word (3Econtrol First look:18/45; BF01=2.43; Proportion look: M=0.426; BF01=0.84). These two conditions differed decisively (First look: BF10=418; Proportion look: BF10>1000). Thus, toddlers from a wide range of households expect saliva sharing to selectively predict responses to distress. The results of Experiments 2 and 3 suggest that when toddlers and infants observe two unfamiliar individuals share saliva, they infer that those people are in a thick relationship. A separate survey of parents (N=129, Experiment 4,( 21)) of infants and toddlers (age 8-19 months) from the same population suggested that this inference would be valid. The parents expressed comfort with their child having positive social interactions (e.g. playing, reading, hugging) with people in many different relationships; but they expressed comfort with saliva-sharing interactions (i.e. sharing a utensil, drinking from the same cup, kissing on the face) only in relationships the participants assessed as thick (BF>1000; Figure 4). In summary, saliva-sharing interactions provide externally observable cues of thick relationships, and young humans can use these cues to make predictions about subsequent social interactions. Substantial prior research has shown that infants have an "intuitive psychology," supporting inferences about individuals' traits (e.g. cooperative, fair) [START_REF] Hamlin | Moral Judgment and Action in Preverbal Infants and Toddlers: Evidence for an Innate Moral Core[END_REF][START_REF] Powell | Third-Party Preferences for Imitators in Preverbal Infants[END_REF][START_REF] Geraci | The developmental roots of fairness: infants' reactions to equal and unequal distributions of resources[END_REF], mental states (e.g. goals, perceptions) [START_REF] Woodward | Infants selectively encode the goal object of an actor's reach[END_REF][START_REF] Carey | The Origin of Concepts[END_REF][START_REF] Liu | Ten-month-old infants infer the value of goals from the costs of actions[END_REF], and group membership (e.g., ethnicity, language) [START_REF] Kinzler | The native language of social cognition[END_REF][START_REF] Liberman | Preverbal Infants Infer Third-Party Social Relationships Based on Language[END_REF][START_REF] Bar-Haim | Nature and nurture in own-race face processing[END_REF][START_REF] Powell | Preverbal infants expect members of social groups to act alike[END_REF][START_REF] Ting | Toddlers and infants expect individuals to refrain from helping an ingroup victim's aggressor[END_REF]. By contrast, representations of social relationships and social structures, which are a central concern of sociology and anthropology, have been relatively understudied using experimental techniques in infants [START_REF] Thomsen | Big and mighty: Preverbal infants mentally represent social dominance[END_REF][START_REF] Thomas | Infants Choose Those Who Defer in Conflicts[END_REF][START_REF] Liberman | Early emerging system for reasoning about the social nature of food[END_REF]. The key distinction revealed here, between thick and thin relationships, may be one foundation of infants' "intuitive sociology" (1, 2).

School-aged children's judgments about saliva sharing are likely reinforced by explicit prohibitions (particularly during a pandemic), but similar intuitions appear to originate earlier and to generalize beyond the content of verbal rules motivated by hygiene ((37)). We hypothesize that an early intuitive distinction between thick and thin relationships allows infants to rapidly learn the distinctive behaviors that occur in these relationships in their social environment [START_REF] Cohen | The hard embodiment of culture[END_REF][START_REF] Fiske | Differently embodying different relationships[END_REF]. These rapidly bootstrapped representations would be useful for parsing the small set of thick, intimate relationships from the larger set of thin, cooperative relationships in typical human social networks [START_REF] Cohen | The hard embodiment of culture[END_REF][START_REF] Fiske | Differently embodying different relationships[END_REF].

The current experiments have limitations. We have not established whether thick and thin relationships are conceived as qualitatively distinct categories (e.g. [START_REF] Clark | Interpersonal attraction in exchange and communal relationships[END_REF][START_REF] Sahlins | Stone Age Economics[END_REF][START_REF] Fiske | The four elementary forms of sociality: framework for a unified theory of social relations[END_REF], or as ends of a continuum, with close friends or confidants having a mixture of features [START_REF] Silk | Cooperation without counting. Genetic and cultural evolution of cooperation[END_REF][START_REF] Small | Someone To Talk To[END_REF]. Saliva sharing is likely only one example of the set of interactions that children and infants can use to distinguish thick relationships from other cooperative relationships. Interactions distinctively occurring in thick relationships include confiding and emotional comforting, consensual exchange of blood, and touching genitals [START_REF] Fiske | Differently embodying different relationships[END_REF]. Infants may also have expectations about the social categories of the people in thick relations (for example, older women may be more likely to be in thick relationships with infants than older men (41)). Conversely, some interactions involving transfer of saliva are aggressive and demeaning, like spitting on a person [START_REF]An analysis of saliva sharing across cultures as described in ethnographies is available in the supplementary materials at the Science website[END_REF]. We have not established whether toddlers and infants can distinguish between saliva sharing interactions that are consensual versus coercive.

Our evidence that young children, toddlers and infants make distinctive inferences about thick relationships has broad theoretical implications. Anthropologists have stressed that thick relationships, characterized by strong attachment obligation and mutual responsiveness, have highly variable mappings onto genetic relatedness [START_REF] Jones | The generative psychology of kinship: Part 1. Cognitive universals and evolutionary psychology[END_REF][START_REF] Murdock | Kin Term Patterns and Their Distribution[END_REF]. Here we identify a proximal psychological mechanism that might allow infants to parse this variability [START_REF] Kaufmann | Wired for Society: Cognizing Pathways to Society and Culture[END_REF][START_REF] Hirschfeld | On a Folk Theory of Society: Children, Evolution, and Mental Representations of Social Groups[END_REF]. A consistent inferential mechanism, operating in diverse social and ecological settings, could enable young humans to rapidly acculturate to their local and culturally-specific relationship and kinship structures. In sum, children, toddlers, and infants recognize saliva-sharing interactions as distinctive evidence of a thick relationship. The pattern of who does, and who does not, share saliva may help infants to distinguish among their many caregivers [START_REF] Hrdy | The emergence of emotionally modern humans: implications for language and learning[END_REF][START_REF] Hrdy | Mothers and Others[END_REF], those who are kin (e.g. parents, siblings, grandparents) versus non-kin (e.g. daycare teachers, nannies). Young humans may use observations of saliva sharing to inform their earliest understanding of the conceptual structure of family.

Fig. 1 .

 1 Fig. 1. Stimuli and Results for Experiment 1. (Top Panel): Sample images and text for one

Fig 2 .

 2 Fig 2. Stimuli and Results for Experiment 2. (A): Experimental design flowchart and stills

Fig 3 .

 3 Fig 3. Stimuli and Results for Experiment 3. (A) Experimental design flowchart and stills

Fig 4 .

 4 Fig 4. Results for Experiment 4. Parent-reported comfort (A,B) and frequency (C,D) of saliva-

Acknowledgments: The authors thank Claire Lu, Shruti Ravikumar, Margot Austin, Vanessa Kudrnova, Sara Alansaari, William Pepe, Anne Harris, and Misha Oraa Ali for assistance with data collection and video coding. Thank you to Emily Chen for assistance with trimming videos, recruitment, and editing drafts of this manuscript. Thank you to Laura Schulz, Tomer Ullman, Shari Liu, Halie Olson, Lindsey Powell, and Michelle Hung for comments on earlier drafts of the manuscript. Statistical support was provided by data science specialist Steven Worthington at the Institute for Quantitative Social Science, Harvard University. Funding: NIH National Research Service; Award 1F32HD096829 (AJT) Patrick J. McGovern Foundation (RS) Guggenheim Foundation (RS) Social Sciences and Humanities Research Council Doctoral Fellowship; Award 752-2020-0474 (BW) NSF Center for Brains Minds and Machines; Award Number CCF-1231216 (LS, AJT, BW, RS) Siegel Foundation; Award S4881 (LS)

Data and materials availability: All data are available on OSF (https://osf.io/c69d5). A subset of participant videos available on Databrary https://nyu.databrary.org/volume/1253.

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1

References [START_REF] Hrdy | Mothers and Others[END_REF](45)(46)(47)(48)(49)(50)(51)(52)(53)(54)(55)(56)


PhD Thomas Guerrier

MD Myriam Labalette

PhD David Launay

PhD Catalina Lee-Chang

PhD Olivier Outteryck

PhD Guillaume Lefèvre

PhD Patrick Vermersch

PhD Sylvain Dubucquoi

MD Hélène Zéphir
email: helene.zephir@chru-lille.fr

Objective: To assess whether any alteration of B-cell subset distribution and/or the cytokine production capacities of B cells could be associated with any stage of MS and could be predictive of MS evolution.

Methods:

We prospectively enrolled radiologically isolated syndrome (RIS), clinically isolated syndrome (CIS), naive patients with relapsing remitting MS (RRMS) of any disease modifying drug, and healthy controls (HCs). Peripheral blood B-cell subset distributions and the interleukin (IL)-6/IL-10-producing B-cell ratio were assessed by flow cytometry to evaluate their proinflammatory and anti-inflammatory functional properties.

Results: Twelve RIS, 46 CIS, 31 RRMS patients, and 36 HCs were enrolled. We observed that a high IL-6/IL-10-producing B-cell ratio in patients with RIS/CIS was associated with the evolution of the disease in the short term (6 months). This imbalance in cytokine production was mainly explained by an alteration of the production of IL-10 by B cells, especially for the transitional B-cell subset. In addition, a significant increase in IgD 2 /CD27 2 B cells was detected in patients with CIS and RRMS compared with HCs (p 5 0.01). Apart from this increase in exhausted B cells, no other variation in B-cell subsets was observed.

Conclusions:

The association between a high IL-6/IL-10-producing B-cell ratio and the evolution of patients with RIS/CIS suggest a skew of B cells toward proinflammatory properties that might be implicated in the early phases of MS disease.

MS is a well-known T cell-dependent disease. Cumulative data revealed the involvement of B cells in the pathophysiology of the disease. [START_REF] Cross | Rituximab reduces B cells and T cells in cerebrospinal fluid of multiple sclerosis patients[END_REF] Although the efficacy of B cell-targeting drugs 2 implies a key role for B cells in MS, the exact molecular mechanisms of this role remain to be defined. [START_REF] Cross | Rituximab reduces B cells and T cells in cerebrospinal fluid of multiple sclerosis patients[END_REF] The lack of impact of B-cell depletion on CSF oligoclonal bands suggests that the role of B cells in the lesional processes is not restricted to antibody-dependent mechanisms but could involve their cellular functions. [START_REF] Cross | Rituximab reduces B cells and T cells in cerebrospinal fluid of multiple sclerosis patients[END_REF] Data from experimental autoimmune encephalomyelitis models showed that B cells can balance either a proinflammatory response by producing interleukin (IL)-6 [START_REF] Barr | B cell depletion therapy ameliorates autoimmune disease through ablation of IL-6producing B cells[END_REF] or a regulatory response through IL-10 production. [START_REF] Fillatreau | B cells regulate autoimmunity by provision of IL-10[END_REF] Some B-cell subsets have been associated with such pro-and anti-inflammatory profiles. Previous studies working on cytokine production by B cells in MS focused on anti-inflammatory cytokines (IL-10) and proinflammatory cytokines (IL-6, granulocyte-macrophage colony-stimulating factor [GM-CSF], lymphotoxin [LT], or tumor necrosis factor-alpha [TNF]a) in established MS disease including patients with disease modifying drugs. Most of those transversal studies showed a decrease of IL-10 and an increase of proinflammatory cytokines produced by B cells i.e., LT and TNFa, [START_REF] Bar-Or | Abnormal B-cell cytokine responses a trigger of T-cell-mediated disease in MS?[END_REF] IL-6, [START_REF] Barr | B cell depletion therapy ameliorates autoimmune disease through ablation of IL-6producing B cells[END_REF] or GM-CSF. [START_REF] Li | Proinflammatory GM-CSF-producing B cells in multiple sclerosis and B cell depletion therapy[END_REF] However, Duddy et al. [START_REF] Duddy | Distinct effector cytokine profiles of memory and naive human B cell subsets and implication in multiple sclerosis[END_REF] did not find any difference in LT and TNFa secretion by B cells. Michel et al. [START_REF] Michel | Unaltered regulatory B-cell frequency and function in patients with multiple sclerosis[END_REF] did not observe any alteration in IL-10 production by B cells in MS. Such discrepancy might be explained by the studied population, especially the treatment status of patients with MS, but also by heterogeneous methodological conditions mixing early and late MS, treated and naive patients, precluding any firm conclusion. We therefore conducted this prospective study focusing on the early phases of MS disease in patients naive of any disease modifying drug. We aimed to analyze whether from the initial phase of MS naive of any disease modifying drug, the evolution of the disease may be associated with any imbalance in cytokine production capacities by B cells.

METHODS Patients and healthy volunteers. Patients in the MS group were enrolled from the Department of Neurology in the University Hospital of Lille. In that group, patients with radiologically isolated syndrome (RIS) and clinically isolated syndrome (CIS) were included as well as those with relapsing remitting MS (RRMS) who were defined according to the 2010 McDonald criteria. [START_REF] Polman | Diagnostic criteria for multiple sclerosis: 2010 revisions to the McDonald criteria[END_REF] RIS was defined according to Okuda criteria. [START_REF] Okuda | Radiologically Isolated Syndrome Consortium (RISC); Club Francophone de la Sclérose en Plaques (CFSEP). Radiologically isolated syndrome: 5-year risk for an initial clinical event[END_REF] Patients with CIS did not fulfill temporal and spatial dissemination for MS at baseline. Exclusion criteria were any history of taking disease modifying drugs; previous corticosteroid use for management of relapses was accepted. However, all blood samples had to be collected at least 1 month after the last steroid intake. Healthy subjects were enrolled as a control.

All biological procedures and statistical analyses are described in e-Methods at http://links.lww.com/NXI/A15, figures e-1 to e-9 at http://links.lww.com/NXI/A19. For all subjects, phenotypic and functional studies of B cells were performed at the time of the inclusion, i.e., at baseline, to assess (1) any differences between healthy controls (HCs) and the subgroups of patients with MS and (2) to define a potential prognosis biomarker associated with the evolution of the disease. Functional studies were focused on the analysis of intracellular IL-6 and IL-10 production by B cells.

All patients were routinely clinically visited at baseline before blood sampling and every 6 months. In our clinical practice, usually brain and spinal MRIs were performed at diagnosis time and 3 months afterward for patients with CIS.

Standard protocol approvals, registrations, and patients consents. This study was approved by the local ethical commit- tee (CPP Nord Ouest IV no. IDRCB: 2014 A00248 39). Informed written consent was obtained from all participants.

RESULTS

Eighty-nine patients with MS and 36 HCs were enrolled between 2013 and 2016 (table).

There was no difference in age and sex between the different groups.

The frequency of IgD 2 /CD27 2 B cells is increased in CIS and RRMS. The phenotypical analysis of peripheral blood lymphocytes showed both similar counts of lymphocytes and distribution of CD4 1 , CD8 1 and total T cells, NK cells, and total B cells between HCs and patients with RIS, CIS, and RRMS (figure e-3).

We observed a similar distribution of the different B-cell subsets (transitional, mature naive, marginal zone, switched memory B cells, IgM-only, IgD-only B cells, and plasmablasts) (figures e-1 and 1A). Double-negative IgD 2 /CD27 2 B cells were the only subset that was significantly increased in patients with CIS and RRMS when compared with HCs (p 5 0.01).

No difference in IL-10 and IL-6 production by B cells in patients with MS compared with controls. Stimulating B cells with CD40 and CpG reliably induced IL-6 and IL-10 production. After 48 hours of polyclonal stimulation, no differences in IL-6 or IL-10expressing B cells or in IL-6/IL-10-producing B-cell ratio were observed between the different groups of patients and HCs (figure 1B). Total B cells did not present any alteration in IL-10 and IL-6 production at any phase of MS disease (RIS, CIS, or RRMS stage) compared with HCs.

Comparison of B-cell subset distribution according to the MS evolution over time. Among cytometric and functional B-cell characteristics obtained at baseline after 48-hour polyclonal stimulation, we searched for parameters with a predictive value of the disease evolution at the maximum follow-up of 11.2 months (66.7). In the following 2 groups, RIS/CIS and RRMS, we distinguished patients who evolved and patients who did not, at 6 months (short term) and after 6 months (long term). Evolution was defined as (1) conversion from RIS to CIS, (2) conversion from CIS to MS according to 2010 MacDonald criteria, and (3) another relapse for RRMS (table). We did not find any association between the different lymphocytes including B cells populations, at baseline, and the evolution at short term and long term in patients with RIS/CIS and in patients with RRMS at short term (figures e-4 and e-5). No statistical analysis was performed for patients with RRMS for the long-term evolution (n 5 2).

An increased IL-6/IL-10-producing B-cell ratio is associated with the evolution of MS. To assess the predictive value for MS outcome of IL-6 and IL-10 production by B cells at baseline, we compared IL-6/IL-10producing B-cell ratio at baseline between evolving and not-evolving patients with MS. We observed that the IL-6/IL-10-producing B-cell ratio was significantly increased in patients with RIS/CIS who evolved at 6 months (p 5 0.021, figure 2A). Next, we compared the IL-6-and IL-10-producing B cells separately and did not observe any variation in IL-6-producing B cells according to the outcome of patients with RIS/CIS. Conversely, we observed a significant decrease of IL-10-producing B cells (p 5 0.0016) at baseline in patients with RIS/CIS who evolved at 6 months (figure 2, B andC). We did not observe any association between the IL-6/IL-10 ratio and the evolution of the RIS/CIS group in the long term and of the patients with RRMS in the short term (figure e-6, A-F).

IL-6/IL-10 ratio was significantly correlated with IL-10 but not with IL-6 B-cell production (figure 2, D andE). The IL-6/IL-10-producing B-cell ratio was also inversely correlated with the delay of evolution of the disease (r spearman 5 20.42; p 5 2.10 22 ). Even not significant, the IL-6/IL-10-producing B-cell ratio is higher in patients with early evolution of disease (figure 3A). In addition, the percentage of IL-10-producing B cells was also correlated with the delay of evolution of the disease (r spearman 5 0.54; p 5 2.10 23 ). Even not significant, the percentage of IL-10-producing B-cell ratio is lower in patients with early evolution of disease (figure 3B).

We then performed receiver operating characteristic (ROC) curve analysis to find a cutoff value for the IL-6/IL-10 ratio defining a higher risk of evolution. Areas under the curve were at 0.7 for the IL-6/IL-10-producing B-cell ratio and for the percentage of IL-10-producing B cells (figure 3C). Seven of 8 patients with RIS/CIS (87.5%) having a value above a ratio cutoff of 6.3 defined by an ROC curve analysis evolved at 6 months, with a good specificity of 97%. With a ratio below 6.3, 19 of 50 patients with RIS/ CIS (38%) evolved giving a sensitivity of 31%. The ROC curve analyses performed for the percentage of IL-10-producing B cells show the same performances with a cutoff at 4.5% (figure 3D). For the IL-10producing B-cell test, the graph representing its performance is similar (data not shown).

Even if no specific lymphocyte distributions at baseline seem to predict any disease evolution, an alteration of IL-10-expressing B cells and the IL-6/ IL-10-producing B-cell ratio, at the very early phase of the disease, is significantly associated with an evolution in the short term.

Potential B-cell subsets and mechanisms involved in the alteration of IL-10 production in early-phase MS.

Using short 5-hour stimulation instead of long 48hour stimulation in patients in the RIS/CIS group, we did not observe any further difference in IL-10producing B cells in evolving patients compared with nonevolving patients (figure e-7). We hypothesized that according to different stimulation methods, different B-cell subsets were involved in IL-10 production. We tested both stimulation methods in the different B-cell subsets from HCs. We observed that IL-10 was mainly produced by memory B cells with the short 5-hour stimulation, whereas IL-10 was mainly produced by marginal zone and transitional B cells with the long 48-hour one (figure e-8).

To decipher the mechanisms of IL-10 production by B cells after long stimulation, we explored the type I interferon pathway. Using peripheral blood DISCUSSION The main results of this study are as follows: (1) an increase of the IL-6/lL-10 ratio in patients with RIS/CIS at baseline is significantly associated with an activity of the disease at 6 months; (2) this proinflammatory imbalance is linked to a decrease of the IL-10-producing B cells, without alteration of the IL-6-producing B cells;

(3) this alteration of IL-10 production in patients with RIS/CIS who evolved concerns mainly transitional B cells and is independent of type I interferon secretion; (4) this feature was not observed when considering baseline samples from patients with RRMS, suggesting that impaired regulatory functions of B cells affect the early phase of MS disease.

Understanding the impact of the B-cell role in MS pathophysiology involves the analysis of peripheral blood B-cell subset distribution in patients, trying to associate any phenotypic characteristics with functional properties. Conflicting results are found in the literature. Immature B-cell phenotype (such as transitional B cells) has been associated with an antiinflammatory profile, [START_REF] Blair | CD19(1)CD24 (hi)CD38(hi) B cells exhibit regulatory capacity in healthy individuals but are functionally impaired in systemic Lupus Erythematosus patients[END_REF] whereas memory B-cell phenotype has been associated with a more proinflammatory profile. [START_REF] Duddy | Distinct effector cytokine profiles of memory and naive human B cell subsets and implication in multiple sclerosis[END_REF] Studies usually focused on the distribution of the following subsets: transitional, mature naive, and memory B cells. Although Haas et al. [START_REF] Haas | B cells undergo unique compartmentalized redistribution in multiple sclerosis[END_REF] described an increase of transitional B cells in CIS and RRMS patients with active disease, Lee-Chang et al. [START_REF] Lee-Chang | Primed status of transitional B cells associated with their presence in the cerebrospinal fluid in early phases of multiple sclerosis[END_REF] found a decrease of those same cells in the same patient groups. Memory B-cell decrease have also been described in RRMS, [START_REF] Haas | B cells undergo unique compartmentalized redistribution in multiple sclerosis[END_REF][START_REF] Niino | Memory and naive B-cell subsets in patients with multiple sclerosis[END_REF] whereas Michel et al. [START_REF] Michel | Unaltered regulatory B-cell frequency and function in patients with multiple sclerosis[END_REF] did not find any modification in the distribution of B-cell subsets in MS compared with HCs. In our study, we did not find any alteration in the distribution of those B-cell subsets at any phase of the disease. The only exception was that the double-negative (IgD 2 / CD27 2 ) B-cell subset is overrepresented in CIS and RRMS groups compared with HCs. Such doublenegative B cells have been reported as increased in other autoimmune conditions (systemic lupus and Sjögren syndrome) [START_REF] Wei | A new population of cells lacking expression of CD27 represents a notable component of the B cell memory compartment in systemic lupus erythematosus[END_REF][START_REF] Saadoun | Expansion of autoreactive unresponsive CD21(-/low) B cells in Sjögren's syndrome associated lymphoproliferation[END_REF] but have not been explored in MS except by Haas et al. [START_REF] Haas | B cells undergo unique compartmentalized redistribution in multiple sclerosis[END_REF] who also described their overrepresentation in CIS and RRMS. These cells are described as exhausted B cells in a chronic activation status, but their functional properties remain elusive. [START_REF] Seifert | Human memory B cells[END_REF][START_REF] Wu | The relationship between CD27 negative and positive B cell populations in human peripheral blood[END_REF] We observed that the distribution of some B-cell subsets like transitional B cells was more heterogeneous in patients with MS. [START_REF] Haas | B cells undergo unique compartmentalized redistribution in multiple sclerosis[END_REF][START_REF] Lee-Chang | Primed status of transitional B cells associated with their presence in the cerebrospinal fluid in early phases of multiple sclerosis[END_REF] This could mean that, despite an apparent homogeneity through the clinical clustering of patients in defined groups (RIS, CIS, and RRMS), every individual patient presents a peculiar heterogeneous disease history. Indeed Carr et al. [START_REF] Carr | The cellular composition of the human immune system is shaped by age and cohabitation[END_REF] recently reported how peripheral blood immune cell distribution could be heterogeneous and how that distribution was shaped by environmental conditions. Besides the lack of association between B-cell subsets and the stage of MS at baseline in our study, we did not find any association with MS evolution either. These data suggest that the phenotypic analysis of peripheral blood B cells could be insufficiently informative. We therefore assessed a more functional analysis of B cells through their cytokine production capacities. We hypothesized that initial phases of the disease could reflect a more homogenous pathologic process before any additional modifications brought by treatments and cumulative damage occurrence. In addition to patients with RRMS, we did select early MS disease (RIS and CIS) patients who are naive of any immunomodulating or immunosuppressive drugs. We did not see any difference in IL-6 and IL-10 production by B cells between HCs and RIS, CIS, or RRMS nor between the different subgroups of patients. Such discrepant results with literature might be linked to the different methodological approaches: cells studied and stimulation methods used. We chose to use stimulated fresh PBMCs to analyze specific cytokine production by B cells using a cytometric approach. Iwata et al. [START_REF] Iwata | Characterization of a rare IL-10-competent B-cell subset in humans that parallels mouse regulatory B10 cells[END_REF] demonstrated that this technique using PBMC is as convenient as purified B cells for the specific cytokine production analysis. Moreover, the stimulation pathway we used in our study has been demonstrated as optimal for the analysis of cytokine production by B cells. [START_REF] Menon | A regulatory feedback between plasmacytoid dendritic cells and regulatory B cells is aberrant in systemic lupus erythematosus[END_REF] In the longitudinal phase of our study, we concluded that, in the very early phases of MS (for RIS and CIS), the higher the IL-6/IL-10 B-cell ratio, the poorer the prognosis at 6 months.

The increase of this ratio was mainly explained by a decrease of IL-10 production. Indeed, although IL-6 production did not significantly vary, the decrease of the capacity of B cells to produce IL-10 could confer a higher proinflammatory property on those B cells. Moreover, in the group of patients who evolved at 6 months, we observed an inverse correlation between time to progression and IL-6/IL-10 ratio. A lower ratio was associated with a longer time to evolution.

Using an ROC curve approach, we delineated a cutoff to define a test with a high specificity. More than 87% of patients with RIS/CIS who presented less than 4.5% of B-cell-producing IL-10 have evolved at 6 months, corresponding to a specificity of 97%, but a low sensitivity of 31%. This approach points out that, under this percentage of IL-10producing B cells, the patients with RIS/CIS are at substantial risk of developing an active disease.

Those results were obtained with a long stimulation while they were not observed after short stimulation of B cells. We could hypothesize that the decrease in the level of IL-10 B-cell production may be restricted to transitional B cells. Moreover, previous studies have shown that IL-10 production by transitional B cells after long TLR9 stimulation could depend on type I interferon produced by PBMC. [START_REF] Carr | The cellular composition of the human immune system is shaped by age and cohabitation[END_REF] However, we did show that IL-10 B-cell production was independent of type I interferon synthesis in our population.

In the RRMS group, we did not observe that IL-6/ IL-10 ratios and IL-10 B cells were associated with any further disease activity during the follow-up. These data suggest that at the RRMS stage, peripheral blood no longer reflected the initial pathologic processes but, perhaps, a more heterogeneous process linked to accumulation, in a single patient, of lesions at various stages.

In conclusion, we suggested that B cells had a reduced capacity for producing the immunoregulatory cytokine IL-10 that may confer a proinflammatory profile to those cells. This phenomenon is objectively shown only at the early initial phases of the disease and could help at that step to identify patients who will need active treatment. A replication cohort would be helpful to substantiate that this test IL-6/IL-10 ratio in B cells is predictive of early disease activity in patients with RIS/CIS. acquisition of data and critical revision of the manuscript for intellectual content. Guillaume Lefèvre and David Launay: critical revision of the manuscript for intellectual content. Patrick Vermersch: acquisition of data and critical revision of the manuscript for intellectual content. Dubucquoi Sylvain: study concept and design, study supervision, analysis and interpretation of data, and critical revision of the manuscript for intellectual content. Hélène Zéphir: study concept and design, study supervision, acquisition of data, analysis and interpretation of data, and critical revision of the manuscript for intellectual content.

Figure 1

 1 Figure 1Phenotype and functional properties of B cells in MS and HCs at baseline

Figure 2

 2 Figure 2Functional properties of B cells according to disease evolution in patients with RIS/CIS

Figure 3

 3 Figure 3 Analysis of the prognostic value of IL-10-producing B cells or IL-6/IL-10-B-cell ratio measurements in patients with RIS/CIS MS

  Table Demographic and clinical features of patients with MS and HCs

			Age, y,		Disease duration,	Follow-up, mo,	Radiologic/clinical
	Diagnosis at inclusion	N	mean 6 SD	Sex ratio, F/M	mo, mean 6 SD	mean 6 SD	evolution
	HC	36	41.8 6 14.1	0.7	-	-	-
	RIS	12	37.0 6 10.7	2.0	-	12.0 6 6.3	2/12
	CIS	46	33.1 6 9.9	1.7	5.3 6 10.2	10.4 6 6.3	29/46
	RRMS	31	36.1 6 9.6	2.1	60.9 6 66.3	12.1 6 7.3	9/31

Abbreviations: CIS 5 clinically isolated syndrome; HC 5 healthy control; RIS 5 radiologically isolated syndrome; RRMS 5 relapsing remitting MS.

Neurology: Neuroimmunology & Neuroinflammation

Neurology: Neuroimmunology & Neuroinflammation

ACKNOWLEDGMENT

The authors acknowledge the University of Lille 2 and the Fondation pour l'aide à la recherche sur la sclérose en plaques (ARSEP), which funded the study and Carine Hauspie for technical support.

STUDY FUNDING

This study was funded by Université de Lille 2 and the Fondation pour l'aide à la recherche sur la sclérose en plaques (ARSEP).

DISCLOSURE

T. Guerrier received research fellowships from Fondation pour la Recherche, Lille 2, and ARSEP; received research grants from Université Lille 2 and ARSEP; and received research support from GEN-ZYME. M. Labalette reports no disclosures. D. Launany received travel funding and/or speaker honoraria from Shire, GSK, CSL Behring, and Pfizer and is an associate editor of La Revue de Medecine Interne. C. Lee-Chang received lecture fees from Merck and research grants from Novartis Pharma AG. O. Outteryck received consulting fees and invitations for national and international congresses from Biogen, Merck, Teva, Sanofi Genzyme, Novartis, Bayer, Sanofi-Aventis, and Roche; received research support from Biogen, Bayer, and Novartis; and received academic research grants from the VISIO Foundation. G. Lefèvre received consulting fees and invitations for national and international congresses from LFB, Octapharma, and Shire, as well as received research supports from LFB, CSL Behring, Air Liquide, Octapharma, GRIFOLS, and GSK. P. Vermersch received honoraria and consulting fees from Biogen Idec, Sanofi Genzyme, Bayer, Novartis, Teva, Merck, Roche, Almirall, and Celgene; received research support from Biogen, Bayer, Novartis, Sanofi Genzyme, Roche, and Novartis-Merck; and served on the scientific advisory board of Sanofi, Biogen, Merck, Teva, Novartis, Roche, Serrvier, and Celgene. S. Dubucquoi: received consulting fees and invitations for national and international congresses from Biogen and Teva; received research support from Genzyme, Octapharma, Roche, CSL Behring, and Novartis; and received academic research grants from Lille University and the ARSEP Foundation. H. Zéphir received fees for consulting or lectures and invitations for national and international congresses from Biogen, Merck, Teva, Sanofi Genzyme, Novartis, and Bayer; received research support from Teva and Roche; and received academic research grants from Académie de Médecine, LFSEP, and ARSEP Foundation. Go to Neurology.org/nn for full disclosure forms.

AUTHOR CONTRIBUTIONS

Thomas Guerrier: study concept and design and acquisition, analysis, and interpretation of data. Myriam Labalette: study concept and critical revision of the manuscript for intellectual content. Caty Lee Chang: critical revision of the manuscript for intellectual content. Olivier Outteryck:

Proinflammatory B-cell profile in the early phases of MS predicts an active disease


Laurent Bonnasse-Gahot

Jean-Pierre Nadal
email: jean-pierre.nadal@phys.ens.fr

Information theoretic study of the neural geometry induced by category learning

Categorization is an important topic both for biological and artificial neural networks. Here, we take an information theoretic approach to assess the efficiency of the representations induced by category learning. We show that one can decompose the relevant Bayesian cost into two components, one for the coding part and one for the decoding part. Minimizing the coding cost implies maximizing the mutual information between the set of categories and the neural activities. We analytically show that this mutual information can be written as the sum of two terms that can be interpreted as (i) finding an appropriate representation space, and, (ii) building a representation with the appropriate metrics, based on the neural Fisher information on this space. One main consequence is that category learning induces an expansion of neural space near decision boundaries. Finally, we provide numerical illustrations that show how Fisher information of the coding neural population aligns with the boundaries between categories.

Introduction

The study of categorization is an important field of research both for biological and artificial neural networks. Here, we take an information theoretic approach to study the optimal properties of the neural representations induced by category learning. We extend the formalism introduced in [START_REF] Bonnasse | Neural coding of categories: Information efficiency and optimal population codes[END_REF][START_REF] Bonnasse | Perception of categories: from coding efficiency to reaction times[END_REF] to the case of multilayer networks, allowing to cast the approach and results within the machine learning framework. We consider multilayer feedforward networks whose goal is to learn a categorization task. We first introduce the mean Bayes risk adapted to a categorization task. We show that the minimization of this cost amounts to dealing with two issues: optimizing the decision stage in order to provide the best possible estimator of the category given the neural activities; and optimizing the stimulus encoding (through the multilayer processing) by maximizing the mutual information between categories and neural code.

We then characterize the mutual information between the discrete categories and the neural activities in a coding layer, in the limit of a high signal-to-noise ratio. This limit allows to reveal the neural metrics relevant for the categorization task. It shows that maximizing the mutual information leads to finding the feature space most relevant for the classification (and amenable to easy decoding), and to probe this space with a particular metric. The latter depends on a ratio between two Fisher information quantities: one that is specific to the neural coding, and one that quantifies the classification uncertainty. As a result of the optimization, the space will be expanded near a class boundary, and contracted far from a boundary. This implies a better ability to discriminate between nearby inputs in the vicinity of a class boundary, than far from such boundary. This effect, well studied in cognitive science, is called categorical perception [START_REF]Categorical Perception: The Groundwork of Cognition[END_REF]. Finally, we provide numerical experiments that illustrate how learning modifies the metrics defined by the neural activities coding for categories. In an Appendix we briefly discuss the links and differences with the information bottleneck approach [START_REF] Tishby | The information bottleneck method[END_REF][START_REF] Tishby | The information bottleneck method[END_REF].

Revealing the metric of internal representations

Cost function: Decoupling into coding and decoding parts. We assume we are given a discrete set of classes/categories, µ = 1, . . . , M . Each category is characterized by a density distribution P (s|µ) over the input (sensory) space. A sensory input s ∈ R N0 elicits a cascade of neural responses (multilayer feedforward processing), r (1) ∈ R N1 , . . ., r (L) ∈ R N L . Finally, an estimate µ of µ is extracted from the observation of the neural activity of the last coding layer r, possibly through a decoding cascade of processing. For a given stimulus s and a neural activity r, the relevant Bayesian quality criterion is given by the divergence C(s, r) between the true probabilities {P (µ|s), µ = 1, ..., M } and the estimator {g µ (r), µ = 1, ..., M }, defined as the Kullback-Leibler divergence (or relative entropy) [START_REF] Cover | Elements of Information Theory[END_REF], C(s, r) ≡ M µ=1 P (µ|s) ln P µ|s) gµ(r) . Averaging over r given s, and then over s, one can show that the resulting mean Bayesian cost C induced by the estimation can be written as:

C = C coding + C decoding (1)
with

C coding = I[µ, s] -I[µ, r],
where I[X, Y ] denotes the mutual information between the random variables X and Y , and C decoding = D KL (P µ|r ||g µ|r ) P (r) dr, is the average of the Kullback-Leibler divergence of P (µ|r) from the network output g µ (r).

The consequences of this decoupling are as follows.

(i) Optimal decoding. The decoding cost C decoding is the average relative entropy between the true probability of the category given the neural activity, and the output g. It is the only term depending on g, hence the function minimizing the cost given by Eq. ( 1) is (if it can be realized) g µ (r) = P (µ|r).

(ii) Optimal coding. The coding cost C coding is the difference between the information content of the signal and the mutual information between category membership and neural activity. Since processing cannot increase information [START_REF] Blahut | Principles and practice of information theory[END_REF], the information I[µ, r] conveyed by r about µ is at most equal to the one conveyed by the sensory input s:

I[µ, s] -I[µ, r] ≥ 0.
If it is possible to find parameters such that the optimal estimator is reached, then the full average cost function (1) reduces to this difference in mutual information quantities. In such case, the cost function is minimized by maximizing the mutual information I[µ, r] between neural activity and category membership. Hence the infomax principle is here an outcome of the global optimization problem. This result is somewhat related to the information bottleneck approach [START_REF] Tishby | The information bottleneck method[END_REF], see Appendix for details.

Geometry of internal representations. The whole chain of neural responses can be seen as first extracting a representation x ∈ R K over which N neurons have their activities r: the neurons form a population code covering a K-dimensional (K ≪ N ) feature space given by x. This is motivated by the many recent works that show how (both biological and artificial) neural activities can be understood as acting on a lower-dimensional manifold (for works in neuroscience, see e.g. 2, 21, 12, 14, 10, 16, and for the machine learning literature, see e.g. 18, 1, 20). Thus, one has the following Markov chain:

µ → s → x = X(s) → r → µ.
For the decoding part, the minimization of the cost C decoding implies that, in this asymptotic limit of large signal-to-noise ratio (large number N of coding cells), g µ (r) = P (µ|r) is an efficient estimator of P (µ|x): it is unbiased and saturates the associated Cramér-Rao bound [START_REF] Bonnasse | Perception of categories: from coding efficiency to reaction times[END_REF].

For the coding part, as we have seen, the minimization of the cost C coding leads to the maximization of the mutual information I[µ, r] between the categories and the neural representation provided by the network prior to decoding. From the data processing theorem, we have that

I[µ, r] ≤ I[µ, x] ≤ I[µ, s].
Thus, for a given projection space X, at best I[µ, r] = I[µ, x], and optimization with respect to the choice of the space X gives optimally I[µ, x] = I[µ, s]. The quality of the projection X is given by how much the probability of the category given the stimulus is well approximated by the probability of the category given the projection X(s). As for the mutual information between categories and neural code, in a regime of high signal-to-noise ratio, one can write [START_REF] Bonnasse | Neural coding of categories: Information efficiency and optimal population codes[END_REF]:

I[µ, r] = I[µ, x] - 1 2 tr F T cat (x) F -1 code (x) P (x) dx (2)
where F code (x) and F cat (x) are K × K Fisher information matrices:

F code (x) ij = -r ∂ 2 ln P (r|x) ∂xi∂xj P (r|x) dr, F cat (x) ij = - M µ=1
∂ 2 ln P (µ|x) ∂xi∂xj P (µ|x). In the K = 1-d case, it simply writes as:

I[µ, r] = I[µ, x] -1 2 Fcat(x)
Fcode(x) P (x) dx. The Fisher information F cat (x) characterizes the sensitivity of the category membership with respect to small variations of x.

It is large at locations x near a boundary between categories, and small if x is well within a category. F code (x) is the 'usual' Fisher information considered in neuroscience, related to the discriminability measured in psychophysics [START_REF] Macmillan | Signal Detection Theory: A user's guide[END_REF][START_REF] Seung | Simple models for reading neuronal population codes[END_REF]. It characterizes the sensitivity of the neural activity r with respect to small variations of x. The expression (2) allows for a simple and intuitive interpretation.

• Finding a proper discriminant space. The first term, I[µ, x], characterizes the correlation between the categories and the underlying projection space X. Maximizing this term means finding a discriminant space, an appropriate space from the point of view of the categorization task.

• Finding a proper metric. The second term tells us what should be the metrics of the neural representation, how this space X should be probed: the Fisher information F code should be large where the categorical Fisher information F cat is large in order to minimize the second term. Thus for a given space X, minimization of the second term in the mutual information leads to a neural code such that F code (x) is some increasing function of F cat (x) (see Appendix A.2). Efficient coding in view of optimal classification is thus obtained by essentially matching the two metrics. Since F cat is larger near a class boundary, this should also be the case for F code (x). A larger F code (x) around a certain value of x means that the neural representation is stretched at that location (the neural representation tiles the space x more finely near than far from the class boundaries). Thus, category learning implies better cross-category than within-category discrimination, hence the so-called categorical perception.

3 Numerical experiments Two-dimensional example with Gaussian categories. We first consider a toy example involving a two-dimensional stimulus space with three overlapping Gaussian categories (see Fig. 1a). Given the small dimension of s, we work with x = s ∈ R 2 (hence this x is given, not found by the network, I[µ, x] and F cat (x) are here properties of the data). The neural network is a multilayer perceptron with one hidden layer of 32 cells. Each cell i has a noisy neural activity given by r i (x) = f i (x) + σ g i (x)z i , where f i is a sigmoidal activation function, z i is a unit normal random variable, and σ = 0.3. Here we take g i (x) = f i (x). Note that this multiplicative noise can be seen as a form of dropout, which in the original work [START_REF] Srivastava | Dropout: a simple way to prevent neural networks from overfitting[END_REF] consists in multiplicative noise in the form of Bernoulli or Gaussian noise (where g i (x) = f i (x) 2 ). The choice g i (x) = f i (x) yields a Poisson like noise, as commonly found in biological neural networks [START_REF] David J Tolhurst | The statistical reliability of signals in single neurons in cat and monkey visual cortex[END_REF][START_REF] William | The highly irregular firing of cortical cells is inconsistent with temporal integration of random epsps[END_REF]. We assume that the noise is not correlated between neurons given a stimulus x, so that we can write P (r|x) = P (r i |x), which in turn helps writing the Fisher information as F code (x) = i F code,i (x), where F code,i (x) is the Fisher information of neuron i.

Figure 1a shows that after learning the network has indeed learned to estimate the posterior probabilities P (µ|x), correctly partitioning the three categories into their respective regions. Figure 1b presents a representation of the Fisher information F code (x) on the x-plane after learning. Here, remember that the Fisher information is a 2 × 2 matrix. At each point, between classes, the eigenvector associated with the largest eigenvalue is orthogonal to the class boundary, and this eigenvalue is largest at the boundary between categories, illustrating the categorical perception phenomenon. Finally, we consider a 1d path in input space, depicted by the dark dots, interpolating between two items drawn from two different categories. We compute the scalar Fisher information of the neural code along this line. Figure 1c shows the results, together with the categorical prediction outputted by the network. As expected, the neural Fisher information is the greatest at the boundary between categories.

Images of handwritten digits. Here we consider the MNIST dataset [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF], a large dataset of 28 × 28 handwritten digits (hence, the stimulus s lives in a 784 dimensional space). The neural network is a multilayer perceptron with two hidden layers, each made of 256 cells with ReLU activation. Poisson like neuronal noise affects the last hidden layer, just as in the previous example, with σ = 0.1. A continuum between an item from the '4' category and an item from the '9' category (two categories that are among the most confusable ones) is created by interpolating between them in a latent space discovered by training an autoencoder to reconstruct digits from the MNIST training set [as done in 7]. Each image along the continuum lies in the relevant manifold of digits. The labels in the x-axis of Fig. 2a pictures a few samples from the continuum, which is made of 31 images. This continuum is considered as the 1d 'x' in the previous discussions. One can then compute the categorical predictions outputted by the neural network together with the scalar Fisher information of the last hidden layer of neurons. Once again, Fig. 2a shows that learning induces categorical perception, with larger Fisher information at the boundary between the two categories. In a previous work [START_REF] Bonnasse | Categorical perception: A groundwork for deep learning[END_REF], the cosine distance between the neural activities r(x) and r(x + δx) was used as a proxy for Fisher information F code (x), as it is much easier to compute. Fig. 2b shows that these two quantities indeed behave quite similarly.

Discussion and challenges

We have shown that minimizing the mean Bayes cost in a categorization task notably implies maximizing the mutual information between category membership and neural activity. This optimization leads to (i) finding an appropriate representation space, and, (ii) building a representation with the appropriate metrics on this space, leading to an expansion of neural space near decision boundaries.

The results presented here are based on previous works [START_REF] Bonnasse | Neural coding of categories: Information efficiency and optimal population codes[END_REF][START_REF] Bonnasse | Perception of categories: from coding efficiency to reaction times[END_REF][START_REF] Bonnasse | Categorical perception: A groundwork for deep learning[END_REF] and on a paper under preparation [START_REF] Bonnasse | Category learning in deep neural networks: Information content and geometry of internal representations[END_REF].

To conclude, we mention several challenging issues which should be addressed. (i) Our results are based on the use of the exact probability distributions of the data. They should be reconsidered in the context of learning with a finite set of examples. Note however that the numerical illustrations indicates that the main results hold in such a learning context. (ii) In the neuroscience context (but also in the machine learning context), one should study the effect of (possibly strong) noise at any stage of processing, also implying noise correlations in the subsequent layers -in particular one issue is how to estimate the Fisher information quantity, F code , as P (r|x) does not factorize in this case.

(iii) It would be interesting to find ways of estimating the categorical Fisher information, F cat . (iv) Finally an important issue is to understand the effect of noise correlations on the geometry of the neural space (in the spirit of [START_REF] Franke | Structures of neural correlation and how they favor coding[END_REF], but here for the case of category learning).

Figure 1 :Figure 2 :

 12 Figure 1: Two-dimensional example with three Gaussian categories. (a) Colored dots: training set, random samples from each of the categories. Background color: mix between the colors that correspond to each of three categories, proportionally to the posterior probabilities P (µ|x) as estimated by the neural network. Dark dots: a path interpolating between two samples from the blue and the red categories. (b) Visualization of the Fisher information matrix at each point on the (x 1 , x 2 ) plane, after learning. The small line represents the direction at this point of the eigenvector of the Fisher information matrix associated with the largest eigenvalue. The magnitude of this largest eigenvalue is represented by the color, the lighter the greater. (c) The dotted colored lines indicate the posterior probabilities, as found by the network, each color representing its respective category. The solid line is the (scalar) Fisher information along the 1d path shown in (a).a.

Appendix A Supplementary Material

A.1 Link with the information bottleneck approach

The information bottleneck (IB) approach [START_REF] Tishby | The information bottleneck method[END_REF][START_REF] Tishby | The information bottleneck method[END_REF] can be formulated as a rate distortion problem, the considered learning cost being a distortion function that measures how well the category µ is predicted from the compressed neural representation r compared to its prediction from the stimulus s. Tishby and collaborators developed this framework, theoretically and algorithmically, in the context of deep learning [START_REF] Tishby | Deep learning and the information bottleneck principle[END_REF][START_REF] Schwartz-Ziv | Opening the black box of deep neural networks via information[END_REF].

The qualitative idea of the IB approach is that the neural activity should convey as little information as possible about the stimulus provided the information about the category is preserved. Thus, with our notation, the goal is to minimize I[s, r] -βI[µ, r] where β is a Lagrange multiplier. Analyzing this optimization principle, Tishby et al [START_REF] Tishby | The information bottleneck method[END_REF] show that the Kullback-Leibler divergence D KL (P µ|s ||P µ|r ) emerges as the relevant effective distorsion measure. This divergence corresponds to our cost function once the decoding stage is optimized, that is g µ (r) = P (µ|r). Then one sees that the approach followed here is somewhat dual to the IB one. One starts from the K-L divergence, and the infomax criterion emerges from the cost function. There is however two important differences. First, the full cost function that we consider includes the decoding part, and second, the correspondence is with the IB cost in the β → ∞ limit. An alternative way to see this correspondence is to consider, from a distortion measure viewpoint, the IB cost associated to the Bayes cost:

Making use of the decomposition of C in coding and decoding parts (Eq. 1), we can write

is the usual information bottleneck cost function.

A.2 Minimization under constraints

We comment here on the minimization of the part of the coding cost which depends on the Fisher information quantities, F code and F cat , for a given projection space X -hence a given categorical information F cat . As explained in the main text, minimization of this term requires that F code essentially follows the categorical Fisher information F cat . The precise result will depend on the constraints on the neural system. The constraints may be on the neurons parameters, as in [START_REF] Berlemont | Confidence-Controlled Hebbian Learning Efficiently Extracts Category Membership From Stimuli Encoded in View of a Categorization Task[END_REF], or directly on the Fisher information considered as a function, as in [START_REF] Bonnasse | Neural coding of categories: Information efficiency and optimal population codes[END_REF]. In such case one minimizes the right hand side of equation ( 2) under the chosen constraint Ψ (for simplicity we consider the 1d case),

, which is meaningful for α > 1. The limit α → 0 corresponds to considering an information theoretic constraint, as we show now.

As presented Section A.1, adopting the viewpoint of the information bottleneck approach [START_REF] Tishby | The information bottleneck method[END_REF], we may minimize the mutual information I[x, r] under the constraint that the information conveyed by the neural code about the categories is large enough: E = I[x, r] -βI[µ, r] In the same asymptotic limit as the one considered here, I[x, r] behaves as 1 2 ln F code (x) P (x) dx (again here for K = 1) [START_REF] Brunel | Mutual information, Fisher information and population coding[END_REF] . Combining this result and the ones in [START_REF] Bonnasse | Neural coding of categories: Information efficiency and optimal population codes[END_REF], we can thus write

Up to the (here constant) term I[µ, x], this is equivalent to the cost (A.2), in the case Ψ(.) = ln(.), taking the dual approach -that is exchanging the roles of the cost and the constraint, β = 1/λ. The optimal function is here F code (x) ∝ F cat (x).