Derek Skillings
email: derek.skillings@gmail.com

Peter Godfrey-Smith

Jessie Mccormack

Maureen O'malley

Keywords: holobiont, symbiosis, individuality, organism, bacteria, microbe

Introduction

Multicellular organisms have been engaged in symbiotic relationships with microorganisms throughout their evolutionary history [START_REF] Moran | Symbiosis[END_REF]. It was long thought that all macroorganisms are routinely colonized by a large number of microorganisms, but the details and extent of macrobe-microbe interactions remained difficult to uncover. As molecular sequencing technologies have advanced, the microbial world has been increasingly opened up to investigation by biologists. It is now widely accepted that microorganisms have always playedand still continue to play-many important roles in the lives of plants and animals [START_REF] Mcfall-Ngai | Animals in a bacterial world, a new imperative for the life sciences[END_REF][START_REF] Douglas | Holes in the hologenome: Why host-microbe symbioses are not holobionts[END_REF]. Symbiotic interactions between microbes and macrobes have been documented among diverse organisms, and many researchers maintain that all macrobes engage in symbiotic interactions with microbes in natural settings (Zilber- [START_REF] Zilber-Rosenberg | Role of microorganisms in the evolution of animals and plants: the hologenome theory of evolution[END_REF][START_REF] Bosch | Metaorganisms as the new frontier[END_REF][START_REF] Dupré | Varieties of living things: life at the intersection of lineage and metabolism[END_REF][START_REF] Gordon | Superorganisms and holobionts[END_REF][START_REF] Singh | Emerging importance of holobionts in evolution and in probiotics[END_REF][START_REF] Booth | Symbiosis, selection and individuality[END_REF]).

The term "holobiont" was coined by Lynn Margulis and used to refer to symbiotic associations that last throughout a significant portion of an organism's lifetime [START_REF] Margulis | Symbiogenesis and symbionticism[END_REF]. The term first found wide usage in coral biology where it was defined as a coral colony and its associated photosynthetic algal symbionts and bacterial communities [START_REF] Rohwer | Diversity and distribution of coralassociated bacteria[END_REF][START_REF] Knowlton | Multispecies microbial mutualisms on coral reefs: The host as a habitat[END_REF], Stat et al. 2012). The recent influx of interest in macrobe-microbe relations has led to a proliferation of the term "holobiont", now most often understood as a host macroorganism and all of its associated microbiota, including bacteria, archaea, viruses, protists, fungi, and microscopic multicellular animals such as nematodes (Zilber- [START_REF] Zilber-Rosenberg | Role of microorganisms in the evolution of animals and plants: the hologenome theory of evolution[END_REF][START_REF] Booth | Symbiosis, selection and individuality[END_REF][START_REF] Bordenstein | Host biology in light of the microbiome: ten principles of holobionts and hologenomes[END_REF][START_REF] Moran | The hologenome concept: helpful or hollow?[END_REF][START_REF] Douglas | Holes in the hologenome: Why host-microbe symbioses are not holobionts[END_REF][START_REF] Theis | Getting the hologenome concept right: an eco-evolutionary framework for hosts and their microbiomes[END_REF]. Because the holobiont includes all associated microbiota, the interactions between holobiont partners may be harmful, beneficial or of no consequence. Relationships between partners may be coevolved or opportunistic, competitive or cooperative.

Recognition that holobionts are common in nature has led many researchers to reassess their views about various processes and concepts that are foundational in biological thinking. [START_REF] Dupré | Varieties of living things: life at the intersection of lineage and metabolism[END_REF] was an important early step in this direction, suggesting that microbial biology has radical implications for the future development of many areas of philosophy of biology. Philosophers are not alone here. They are joined by biologists in calling for a transformation in our thinking. "Right now, for those of us who are not evolutionary biologists, it is enough to recognize that the very foundations of biology are being shaken by both the integration of microbiology into concepts of macroevolution and the recognition that hostmicrobe symbioses are a major theme in biological systems" [START_REF] Mcfall-Ngai | Introduction to the hologenome special series[END_REF]. One suggestion is that there is now a need to "upgrade fundamental theories" because holobiont systems "raise the discussion of individuality and organismality beyond its historical perspective to a level that challenges and extends current thinking" [START_REF] Bordenstein | Host biology in light of the microbiome: ten principles of holobionts and hologenomes[END_REF].

Criteria for individuating entities are of central importance in biology [START_REF] Hull | A matter of individuality[END_REF][START_REF] Hull | Individuality and selection[END_REF][START_REF] Hull | Individual[END_REF][START_REF] Buss | The evolution of individuality[END_REF][START_REF]Darwinian populations and natural selection[END_REF], 2013;Clark 2011;[START_REF] Pradeu | The limits of the self: immunology and biological identity[END_REF]. For example, population biologists and ecologists must be able to distinguish individuals in a population. Evolutionary biologists must be able to distinguish parents from their offspring, and one lineage from another. Immunologists and physiologists must be able to distinguish between an individual and its environment. Organisms have long been the paradigm of individuality-a horse, a tree, a human--both within and outside of biology (Aristotle 1984). Extensive experimentation and theoretical advances in biology, especially within the last century, have changed our understanding of how individuals can and did evolve [START_REF] Buss | The evolution of individuality[END_REF], Maynard Smith and Szathmáry 1995[START_REF] Michod | Darwinian dynamics. Evolutionary transitions in fitness and individuality[END_REF]). New conceptions of individuality have helped us to understand individuality across the biological hierarchy: genes, cells, multicellular organisms, superorganismal colonies, and multispecies symbiotic communities [START_REF] Dawkins | The Selfish Gene. Best Books Douglas AE (2008) Conflict, cheats and the persistence of symbioses[END_REF], Sober and Wilson 1989[START_REF] Pradeu | The limits of the self: immunology and biological identity[END_REF].

The question motivating this paper is: are holobionts biological individuals or communities? I start by introducing the biology of the coral holobiont as an example of the complexity and diversity of the interactions within a holobiont. I take the coral holobiont to be a good example of typical holobiont dynamics, and as such an appropriate test case for working through whether or not holobionts are biological individuals. In the following sections I expand on relevant accounts of biological individuality and claims made in favor of holobionts being biological individuals. I then consider whether holobionts meet some plausible and common criteria for either evolutionary individuality (reproduction and heritability) or organismality (functionally integrated interactors, metabolic collaboration, or cooperative low-conflict consortiums). I conclude that most holobionts share more affinities with communities than they do with individual wholes, and that, except for in rare and possibly unrealized cases, holobionts do not meet the criteria for being evolutionary individuals, units of selection, or organisms.

The Coral Holobiont

The evolutionary and ecological success of corals in the characteristically nutrient-poor environments of tropical and subtropical oceans is thought to be a direct consequence of their ability to form mutually beneficial symbioses with unicellular photoautotrophic dinoflagellates in the genus Symbiodinium, commonly referred to as zooxanthellae (Stat et al. 2012[START_REF] Lesser | The endosymbiotic dinoflagellates (Symbiodinium sp.) of corals are parasites and mutualists[END_REF]. The appearance of coral reefs in the Triassic has been attributed to the evolution of the symbiotic association between the coral host and Symbiodinium.

Corals are mainly colonized by free-living Symbiodinium from the environment, but a direct transfer of Symbiodinium from parent to offspring via inclusion in the egg also occurs in many brooding corals, and is occasionally seen in some spawning corals [START_REF] Thompson | Microbes in the coral holobiont: partners through evolution, development, and ecological interactions[END_REF]. The Symbiodinium are endosymbionts that reside within the cells of the coral host. They fix carbon through photosynthesis, which they provide, along with other nutrients, to their host in return for host waste metabolites and protection from grazing (Stat et al. 2012[START_REF] Roth | The engine of the reef: photobiology of the coral-algal symbiosis[END_REF]. This exchange of nutrients is mutually beneficial and helps the coral secrete the calcium carbonate skeletal structure that is shared by the coral colony and contributes to the formation of coral reefs. This endosymbiotic association is especially interesting because it involves two eukaryotic organisms and the genome of the symbiont is three times larger than the genome of its host [START_REF] Roth | The engine of the reef: photobiology of the coral-algal symbiosis[END_REF].

The total number of bacterial species associated with corals remain largely unavailable, but recent estimates put the number between 3000-6000 species (Stat et al. 2012). At least some of the coral-associated prokaryotes are beneficial to the coral host. Cyanobacteria provide nutrition through nitrogen fixation (Lesser et al., 2004), and bacteria residing in the coral's exterior mucous layer act as a first line of defense against pathogens by producing antimicrobial compounds and occupying space (Stat et al. 2012). Most coral-associated fungi and viruses are thought to be parasitic [START_REF] Golubic | Endolithic fungi in marine ecosystems[END_REF], but there may be exceptions. There is evidence that some endolithic fungi residing in the coral skeleton could be converting nitrate and nitrite to ammonia which could enable fixed nitrogen to cycle within the coral holobiont, and some of the bacteriophages might be helping to beneficially regulate associated bacterial communities [START_REF] Wegley | Metagenomic analysis of the microbial community associated with the coral Porites astreoides[END_REF]. With rare exceptions, the microorganismal component of the coral holobiont is obtained from the environment [START_REF] Apprill | The onset of microbial associations in the coral Pocillopora meandrina[END_REF].

There is a substantial amount of genetic diversity within Symbiodinium and evidence that some of it is reflected as functional diversity. This is important because corals can harbor more than one type of Symbiodinium at a time, and are known to shuffle and switch out their Symbiodinium types. The coral holobiont is very sensitive to changes in ocean temperature and lives close to their upper thermal tolerance limit. A prolonged temperature increase of as little as 1 ˚C causes stress and can lead to coral bleaching, the forceful expulsion of the Symbiodinium by the coral host. The coral holobiont is a dynamic system with members fluctuating on a daily basis depending on the environmental conditions and life-cycle requirements [START_REF] Thompson | Microbes in the coral holobiont: partners through evolution, development, and ecological interactions[END_REF].

For example, corals associated with clade C Symbiodinium usually perform better than corals with clade D Symbiodinium in normal conditions with clade C-infected juveniles growing two to three times as fast as those infected with clade D [START_REF] Lesser | The endosymbiotic dinoflagellates (Symbiodinium sp.) of corals are parasites and mutualists[END_REF][START_REF] Hume | Ancestral genetic diversity associated with the rapid spread of stress-tolerant coral symbionts in response to Holocene climate change[END_REF]. But in hot environments where coral bleaching is common, corals with Clade D Symbiodinium do better. One reason is that clade D appears to be more heat-tolerant than Clade C, and may rapidly increase the heat stress tolerance of corals [START_REF] Lesser | The endosymbiotic dinoflagellates (Symbiodinium sp.) of corals are parasites and mutualists[END_REF][START_REF] Thompson | Microbes in the coral holobiont: partners through evolution, development, and ecological interactions[END_REF][START_REF] Hume | Ancestral genetic diversity associated with the rapid spread of stress-tolerant coral symbionts in response to Holocene climate change[END_REF]. This can lead to short-term benefits during periods of thermal stress, but corals dominated by clade D Symbiodinium show significantly decreased growth and reproduction in the long term. This suggests that the Symbiodinium switch between being mutualists and parasites of the host depending on current environmental conditions.

It has also been suggested that things are not what they seem for the Symbiodinium either. The coral host might be benefiting at the expense of their microbes by capturing and controlling its algal symbionts [START_REF] Garcia | The symbiont side of symbiosis: do microbes really benefit?[END_REF]. [START_REF] Wooldridge | Is the coral-algae symbiosis really 'mutually beneficial' for the partners?[END_REF] suggests that "the coral host exerts a 'controlled parasitism' over its algal symbionts that is akin to an enforced domestication arrangement" with the "…the coral host as an active 'farmer' of the energy-rich photoassimilates from its captive symbionts."

Which lineages come together to make up a coral holobiont is strongly contingent on environmental conditions and cooperation and conflict between different possible partners. This flexibility in lineage composition allows the coral to respond to the abiotic environment in ways that it wouldn't be able to if it was on its own, and was likely important for the success of corals. Understanding the mechanisms and relations that have evolved between the coral and its symbiotic partners is crucial for understanding coral biology.

Controversy about the Status of Holobionts

Several biologists and philosophers have claimed that holobionts, or similar multi-lineage assemblages of macrobes and microbes, constitute at least one level of organization at which natural selection acts. Biologists Zilber-Rosenberg and Rosenberg were the first to articulate what they call the ''hologenome theory of evolution,'' which they see as an alternative to ''currently accepted dogma,'' according to which the units of selection are individual organisms as traditionally conceived (2008,731). The hologenome is a collective unit made up of all of the host and microbial genomes of the holobiont. They write, "In the hologenome theory of evolution, we suggest that the holobiont… with its hologenome, acting in consortium, should be considered a unit of selection in evolution…'' (2008,723). They are not the only theorists to have made such claims:

-"Therefore the holobiont, i.e. the host including all symbionts, should be regarded as the unit of selection as the association between host and symbionts may affect the fitness of the holobiont depending on the environment" [START_REF] Feldhaar | Bacterial symbionts as mediators of ecologically important traits of insect hosts[END_REF]).

-"The hologenome theory of evolution considers the dynamic holobiont as a single unit for natural selection and provides a more accommodating view of evolution blending Darwinism and Lamarkism" [START_REF] Singh | Emerging importance of holobionts in evolution and in probiotics[END_REF] -"[A]n organism's genetics and fitness are inclusive of its microbiome" [START_REF] Brucker | Response to comment on "The hologenomic basis of speciation: gut bacteria cause hybrid lethality in the genus Nasonia[END_REF]).

-"Thus, the holobiont, with its integrated community of species, becomes a unit of natural selection whose evolutionary mechanisms are largely unexplored" [START_REF] Gilbert | Symbiosis as the way of eukaryotic life: the dependent co-origination of the body[END_REF]).

-"The hologenome concept is a holistic view of genetics in which animals and plants are polygenomic entities. Thus, variation in the hologenome can lead to variation in phenotypes upon which natural selection or genetic drift can operate" [START_REF] Bordenstein | Host biology in light of the microbiome: ten principles of holobionts and hologenomes[END_REF].

Without referring to holobionts explicitly, philosophers Dupré and O'Malley1 endorse a similar view about the fundamental entities that are operated on by natural selection: "…complex systems involving the collaboration of many highly diverse lineage forming entities. This sort of interactor, we also suggest, is the most fundamental unit of selection." [START_REF] Dupré | Varieties of living things: life at the intersection of lineage and metabolism[END_REF]. Ereshefsky andPedroso (2013, 2015) use biofilms as a case study to defend the position that multi-species consortiums can be units of selection.

Other authors have been critical of the claim that holobionts are important units of selection. [START_REF] Moran | The hologenome concept: helpful or hollow?[END_REF] state that "While biologists would agree that microorganisms have important roles in host evolution, this statement is a far cry from the claim that they are fused with hosts to form the primary units of selection, or that hosts and microorganisms provide different portions of a unified genome" and that "…some observations that superficially appear to support the concept of the hologenome have spawned confusion about real biological issues." [START_REF] Douglas | Holes in the hologenome: Why host-microbe symbioses are not holobionts[END_REF] state that "…it is highly unlikely that the entire microbiome will evolve as a "holobiont" with its host" and that "…the hologenome concept is unhelpful to the study of host interactions with resident microorganisms…"

Pushing back against recent criticism, Theis et al. ( 2016) have backed off a bit from claims that holobionts are always units of selection, but still consider the holobiont to be a level at which selection acts. They also reiterate that shifts in the microbial community are akin to changes in allele frequency in the host genome, suggesting that the hologenome is a single unit upon which selection acts.

Biological Individuality

I will start with some general remarks about biological individuality, before moving on to considerations about whether or not holobionts are biological individuals. Accounts of biological individuality tend to be clustered around a few different investigatory projects. These accounts are not exclusive, a particular biological entity may be an individual of more than one type.

Genealogical individuals are lineages such as species and phylogenetic taxa [START_REF] Hull | A matter of individuality[END_REF]. They are the units that can evolve. Evolutionary individuals are the individuals of natural selection [START_REF] Sober | Unto others: the evolution and psychology of unselfish behaviour[END_REF][START_REF] Ereshefsky | Rethinking evolutionary individuality[END_REF][START_REF] Clarke | Levels of selection in biofilms: multispecies biofilms are not evolutionary individuals[END_REF]. They are the units upon which natural selection operates and members of a population that has the capacity to evolve.

The traditional target of accounts of biological individuality is the organism, the phenomenologically discrete living entities inhabiting the world around us. I contrast organisms with evolutionary individuals in that the defining criteria of organismal individuality are not restricted to purely evolutionary considerations. As a first pass, organisms are bounded individuals that are functionally or metabolically integrated. They are systems with mutually dependent components that work together to maintain the system's structure or developmental trajectory (e.g., Pradeu 2010, Godfrey-Smith 2013). There are numerous accounts of organismality, many of which don't agree2 .

One other approach to organismality is one that focuses physiological individuality. This is a family of views that have developed somewhat independently of evolutionary views about individuality. Of these, the immunological account advocated by Pradeu ( 2012) is especially promising. On this account, the boundaries of physiological individuals (organisms) are established by the immune system of the host. Other nearby physiological views rest on the fact that symbionts are either critical for host development, or make something of the host work, that is, realize or help realize an important physiological function (See, for example, [START_REF] Bocci | The neglected organ: bacterial flora has a crucial immunostimulatory role[END_REF][START_REF] Berg | The indigenous gastrointestinal microflora[END_REF][START_REF] Xu | Honor thy symbionts[END_REF] 3 . This line of thinking has been important for understanding the boundaries, health and development of macroorganisms, especially large and complex vertebrates like us.

I will not discuss the relation between holobionts and physiological individuality for two reasons. First, I am concerned with starting from a more general analysis that doesn't privilege macroorganisms or index claims of individuality to the host. I take physiological individuality to be host-centric, subordinating microbe individuality to functional, immunological, and developmental considerations regarding the host. My focus is: how might higher level individuality emerge out of the general interactions between macrobes and microbes? Second, much of the controversy and debate surrounding holobionts has focused on whether holobionts ought to be understood as units of selection or communities, and the present paper is an attempt to take side in that controversy. Because of these reasons I will limit myself to evolutionary individuality and accounts of organismality that focus on functional integration of a collaborative or codependent nature.

Are Holobionts Evolutionary Individuals?

Evolutionary individuals are entities defined in terms of natural selection: they vary among each other, their variability causes variations in fitness, and that variation and fitness effect is heritable [START_REF] Lewontin | The units of selection[END_REF][START_REF]Darwinian populations and natural selection[END_REF][START_REF] Ereshefsky | Rethinking evolutionary individuality[END_REF][START_REF] Clarke | Levels of selection in biofilms: multispecies biofilms are not evolutionary individuals[END_REF]. The Darwinian population framework is one way to make evolutionary individuality more precise. This framework has its roots in the account of natural selection articulated in [START_REF] Lewontin | The units of selection[END_REF], and gets its name from Godfrey-Smith's ( 2009) extended update of that account. Building from Lewontin's three criteria of variation, heredity and differences in reproductive success, a Darwinian population is defined as "a collection of causally connected individual things in which there is variation in character, which leads to differences in reproductive output (differences in how much or how quickly individuals reproduce), and which is inherited to some extent" (2009,39). A member of such a population is a Darwinian individual (Godfrey- Smith 2009, 40). Darwinian individuals are units of selection, and as such are "the loci of causal action for the process of selection" (Booth 2014, 664). If holobionts are to be evolutionary individuals on an account like this, then they must (a) be able to reproduce at the level of the holobiont, and (b) there must be heritable differences at the level of the holobiont. I will argue in the next section that holobionts fulfill neither of these criteria.

Holobiont Reproduction

Godfrey-Smith (2015) makes further distinctions that are helpful for understanding the generation of new entities. He starts with the notion of recurring structures, which may either be reproducing or reconstructed. Reproducing things form parent-offspring lineages, whereas reconstructed ones do not. Reconstructed objects include organs and enzymes. Reproduction can be collective, simple, or scaffolded (Godfrey-Smith 2009). Collective reproducers include multicellular organisms. Simple reproducers, such as bacteria, can give rise to more things like themselves. Scaffolded reproducers rely on external machinery for their reproduction. Examples include genes and viruses.

An alternative analysis of reproduction, [START_REF] Griesemer | Development, culture and the units of inheritance[END_REF][START_REF] Griesemer | Reproduction and scaffolded developmental processes: an integrated evolutionary perspective[END_REF][START_REF] Griesemer | Reproduction in complex life cycles: toward a developmental reaction norms perspective[END_REF]) "reproducer" account, shares many similarities with the Darwinian individual account. It differs in that he emphasizes the necessity of material overlap between generations. "Reproducers are entities with the capacity to multiply because offspring bear relations of 'material overlap' with their parents" [START_REF] Griesemer | Reproduction in complex life cycles: toward a developmental reaction norms perspective[END_REF]. What makes reproduction different than recurrence or mere production is the conveyance of developmental cycles linked together in a lineage.

Griesemer and Godfrey-Smith agree that reproduction requires the formation of lineages. Their disagreement centers on the necessity of material transfer between generations and how to parse the divisions between scaffolded and collective reproduces [START_REF] Griesemer | Reproduction in complex life cycles: toward a developmental reaction norms perspective[END_REF]. For Griesemer, nearly all organisms are scaffolded in some way because they depend on some aspect of the environment for either their development or reproduction.

Questions about whether holobionts are evolutionary individuals-i.e., natural selection operates at the level of the holobiont-are intimately related to questions about holobiont recurrence. Specifically, is the generation of a new holobiont the result of reproduction or reconstruction, which is to ask: do holobionts reproduce as a whole such that that there holobiont-level lineages?

The answer depends primarily on how the microbial symbionts are transmitted.

The transmission routes by which microbial symbionts move from host to host vary considerably and are usually divided into two categories (Bright and Bulgherisi 2010). Vertical transmission is the direct transfer of symbionts from the host parent(s) to their offspring. Horizontally transmitted symbionts are acquired from other non-parental hosts or from free-living population in the environment. The majority of microbial associations of multicellular animals and plants are thought to be horizontally transmitted [START_REF] Moran | The hologenome concept: helpful or hollow?[END_REF]. Cases of vertical transmission are often not obligate, that is, even though the symbionts can be, or even often are, transmitted vertically, they can also be obtained horizontally. Obligate vertical transmission is thought to be rare, and may reliably occur only in cases of endosymbiosis. Many cases of vertical transmission are really part of a mixed-mode of transmission. Though it appears to be the case that rates of vertical transmission are exceedingly low relative to the rates of horizontal transmission, it is perhaps the case those microbes that are transmitted vertically play a relatively larger role in the lives of their hosts4 .

Few, if any, holobionts as individual units are reproducers because strict vertical inheritance is rare. If the set of lineages that make up the holobiont varies within and between host generations, then the holobiont cannot be a coherent unit of selection. More select partnerships between hosts and individual symbionts do in rare cases meet the criteria for reproduction, such as eukaryotes and their mitochondria or corals that pass along Symbiodinium in their eggs. High partner fidelity is fleeting without strict vertical transmission. If the component lineages can all go their separate ways between reproductive events and reassemble at a later time in at least a semi-random fashion, then there are no higher-level lineage connecting generations of holobionts.

The only way to draw something approximating parent-offspring relations between holobionts without vertical transmission of all the component lineages is to privilege one of the partners. This is nearly always the host, as it is the largest and likely the longest living part of a holobiont. Privileging the host is nearly inevitable, as holobionts are defined by picking out particular hosts with all of their associated microbes.

But what if we privileged one of the other collaborating lineages? Consider the human + gutmicrobiota holobiont with a different emphasis. A bacterium lives inside a doctor who hasn't been particularly careful about sanitation. The doctor goes into work and delivers an unrelated baby. The gut bacterium replicates and one of the offspring bacteria quickly colonizes the infant. A new holobiont is assembled in the collaboration of the human lineage and doctor's bacterial lineage. We can now pick out a new parent-offspring relation between the doctor holobiont and the baby holobiont. From a lineage-neutral perspective at the holobiont level, this is no stranger than saying that the parent-offspring relation is between the mother holobiont and the baby holobiont.

More sensibly, we can say that there is no fact of the matter about what constitutes the parentoffspring relation between host-microbe associations without vertical transmission. Perhaps even better: there are no parent-offspring relations between holobionts in these cases. The related concepts of parenthood and reproduction have simply been stretched too far, and most holobionts are marginal reproducers at best (Godfrey- [START_REF]Darwinian populations and natural selection[END_REF][START_REF]Agents and acacias: replies to Dennett[END_REF][START_REF]Agents and acacias: replies to Dennett[END_REF][START_REF] Booth | Symbiosis, selection and individuality[END_REF]. In cases of horizontal transmission, the host-microbe associations recur in each generation, but they do not reproduce as a unit and do not form lineages (Godfrey-Smith 2012, Booth 2014). A particular host-microbe association might even be obligatory for the reproduction of one the partners. But the reproductive events wouldn't be holobiont reproduction, rather they would be co-dependent scaffolded reproduction of partner lineages. That is to say, they are not evolutionary individuals on this account.

Holobiont Heritability and Holobiont Lineages

Holobiont reproduction with vertical transmission in itself is not sufficient for evolutionary individuality. Horizontal transmission or symbiont exchange during the host's lifetime can disrupt heritability. Even hypothetical holobionts where all of the microbial symbionts are vertically transmitted during host reproduction may not be evolutionary individuals. The second important consideration regarding selection at the holobiont level is whether or not there is partner fidelity: a stable association of host and symbionts across multiple host generations. High partner fidelity is a prerequisite for evolutionary individuality because the holobiont can only evolve as a unit if the host and its symbionts co-occur across multiple host generations. Only holobionts with both a high degree of vertical transmission and high partner fidelity will meet the criteria for evolutionary individuality.

Partner fidelity is expected to be highest when there is obligate vertical transmission, though high partner fidelity might also be possible in holobiont systems with horizontal transmission where the hosts provide their offspring with symbionts and where specificity is high [START_REF] Douglas | Holes in the hologenome: Why host-microbe symbioses are not holobionts[END_REF]. Partner fidelity is often imposed by vertical transmission because the microbial partners will have a strong selective interest in the reproductive fitness of the host when their fitness is tied to the reproductive success of the host. But partner fidelity and vertical transmission can come apart. Partner fidelity is expected to be lower when there are no obligate dependencies, or when the obligate dependencies can be supplied by many different symbiont partners. If partner lineages can jump ship, and horizontally transfer to other hosts, then the different parts of a holobiont aren't locked into a common fate. This leads to an expectation of increased conflict between the members of the holobiont as they "pursue their own goals"; namely, selection for increased replication of one's own lineage at the expense of the success of the multi-lineage holobiont. As conflicts of interests among partners increase (e.g., due to weak partner fidelity), then the holobiont is undermined as a higher-level unit of selection.

This is especially apparent if we consider a hypothetical coral holobiont where all partners have long life-spans. For example, coral A and dinoflagellate symbiont C are distinct lineages that interact to form coral holobiont α. Coral B and dinoflagellate symbiont D are distinct lineages that interact to form coral holobiont β. If the fates of those lineages are tied to the success of the holobiont that they help produce, then we have some notion of alignment of fitness. But if the collaborating lineages are independent of each other, it is possible that lineages C and D could switch partners. In that case A and D would interact to form a new holobiont, γ, and B and C would interact to form the new holobiont δ. Holobionts γ and δ survive, and lineages A, B, C and D eventually reproduce and make it into the next generation of holobionts.

The extinction of coral holobionts α and β did not prevent the proliferation of the lineages that produced them. The death of α and β might have fitness consequences for the individual lineages, but it needn't. In this example, the overall success of the holobiont(s) will have fitness consequences for the individual lineages that make them up, but the dissolution of any particular holobiont-because of either partner death or partner switching-needn't necessarily have fitness consequences for the individual lineages. Holobionts γ and δ could reproduce as whole units and still wouldn't be units of selection without high partner fidelity over the life of the partner lineages. A high degree of symbiont swapping will undermine selection at the level of the holobiont because horizontal swapping continually dissolves and creates individual holobionts over the course of the lifetimes of the individual partner lifetimes.

Yet again, it is difficult to pick out what counts as a case of a new or different holobiont without explicitly privileging one of the partners. I take this as a likely reason for why the concept of the holobiont rests on macrobe bias. Microbes are small. Macrobes are big. Microbes go where their macrobial associates go. Holobionts seem contiguous to us, in ways that symbiotic associations between macrobe-macrobe symbiotic associations like plants and pollinators don't. But this alone does not indicate that they are part of some larger whole. It is just an artifact of their size.

Indexing holobiont identity to the host is not without its benefits. Focusing on the larger and longer lived host makes it easier to demarcate holobionts, a necessity for tracking holobiont changes over time and in response to environmental changes. Indexing symbiont community identity to the host is appropriate when the host is of primary interest5 . But in the context of evolutionary individuality, all partners have equal weight, regardless of their size or longevity. Evolutionary individuals are only picked out by being entities that natural selection works on, not any physical or taxonomic features. Mistaking holobionts for units of selection appears to stem, in part, from host-centric thinking, macrobe bias and a reification of operational concepts like hologenome, microbiome, and metagenome6 .

Are Holobionts Organisms?

So far I have only discussed accounts of biological individuality that are tied to reproduction at the level of the holobiont. Another approach that has been proposed is based on David Hull's interactor account of individuality [START_REF] Hull | Individuality and selection[END_REF][START_REF] Hull | Individual[END_REF], itself based on Richard Dawkins' replicator theory [START_REF] Dawkins | The Selfish Gene. Best Books Douglas AE (2008) Conflict, cheats and the persistence of symbioses[END_REF]). According to Hull, an interactor is "an entity that directly interacts as a cohesive whole with its environment in such a way that replication is differential" [START_REF] Hull | Individuality and selection[END_REF]). The replicators are entities which "pass on their structures largely intact from generation to generation" [START_REF] Hull | Individuality and selection[END_REF]. Replicators were originally conceived as an abstraction of the role of gene, while interactors are an abstraction of the role of organisms. The interactor is often identified as the primary unit of selection, but this role has also been extended to the replicator [START_REF] Lloyd | Units and levels of selection[END_REF]. The replicator-interactor framework has since been put to powerful use in the analysis of the complexities of inheritance and interaction in symbiotic consortiums [START_REF] Sterelny | Niche construction, developmental systems, and the extended replicator[END_REF][START_REF] Sterelny | Symbiosis, evolvability and modularity[END_REF][START_REF] Sterelny | Darwinian spaces: Peter Godfrey-Smith on selection and evolution[END_REF].

Criticism of the replicator-interactor framework has primarily focused on problems with replicator transmission and the reproduction of interactors (see for example, [START_REF] Griesemer | Development, culture and the units of inheritance[END_REF][START_REF]Darwinian populations and natural selection[END_REF]. This has led some to suggest a notion of biological individuality that maintains interactors while decoupling them from replicators or particular reproductive requirements [START_REF] Dupré | Varieties of living things: life at the intersection of lineage and metabolism[END_REF][START_REF] Dupré | The polygenomic organism[END_REF]Ereshefsky andPedroso 2013, 2015). For example, Dupré (2012) makes the following claim: "…the organisms that are parts of evolutionary lineages are not the same things as the organisms that interact functionally with their biological and non-biological surroundings." This is immediately followed by a much stronger claim: "The latter, which I take to be more fundamental, are composed of a variety of the former, which are the more traditionally conceived organisms" [START_REF] Dupré | The polygenomic organism[END_REF].

The move to an interactor-only, or "updated interactor", concept of individuality is meant to capture the fact that many multilineage symbiotic consortiums appear to function as organisms in their environments, while also recognizing that the many replicators coming together within these consortiums are not inextricably tied together. It is suggested that all that is needed for an entity to be an interactor is enough interaction between the member parts such that the success or failure of the interactor has a unitary effect on the success or failure of its members (Ereshefsky and Pedroso 2013). For example, a higher survivorship in the members then if the members were living independently from each other (Ereshefsky and Pedroso 2013).

In the case of holobionts, much more needs to be said about the necessary type and strength of interactions between holobiont partners such that the holobiont interacts as a functional whole (organisms) with the environment7 . I will examine two approaches to organismality that may be compatible with the view that holobionts are modified interactors and biological individuals. The first is an approach that focuses on functional integration through metabolic collaboration. The second is the cooperation and conflict framework outlined in [START_REF] Pradeu | The limits of the self: immunology and biological identity[END_REF].

Functional Integration and Metabolic Dependency

Metabolic dependencies are a hallmark of the close symbiotic relationships that inspired the adoption of the holobiont framework. A type of biological individuality that emerges from lineage-forming entities collaborating in metabolism is a position explored by [START_REF] Dupré | The polygenomic organism[END_REF] and [START_REF] Dupré | Varieties of living things: life at the intersection of lineage and metabolism[END_REF].

"My colleague Maureen O'Malley and I [START_REF] Dupré | Varieties of living things: life at the intersection of lineage and metabolism[END_REF] have suggested that the most fundamental way to think of living things is as the intersection of lineages and metabolism. The point we are making is that, contrary to the assumption that is fundamental to the one genome, one organism idea, the biological entities that form reproducing and evolving lineages are not the same as the entities that function as wholes in wider biological contexts. Functional biological wholes, the entities that we primarily think of as organisms, are in fact cooperating assemblies of a wide variety of lineageforming entities." [START_REF] Dupré | The polygenomic organism[END_REF] In their view of the natural world, "collaboration" among entities of fundamentally different types is essential to all living systems (Dupré and O'Malley 2009) 8 . Collaboration encompasses cooperation and competition, includes metabolic, structural, and fitness-affecting interactions, and involves entities at many levels of biological organization. Dupré and O'Malley do not discuss holobionts, and so it is unclear whether they would endorse the position that holobionts are organisms, but their collaboration criteria for organismality seems a promising option for those wishing to argue that multilineage systems like holobionts are biological individuals.

Individual holobionts will almost inevitably contain partnerships that vary across the full range of collaboration as described by [START_REF] Dupré | Varieties of living things: life at the intersection of lineage and metabolism[END_REF]. There are at least two reasons to be cautious about such a permissive approach to collaboration, if that collaboration is to be the glue that binds lower level individuals into a higher level individual.

First, recurring interactions, even ones with reciprocal benefits, needn't indicate that there is functional integration or active collaboration. Members of a particular host species will inevitably share similar physiologies, microbial defense mechanisms, and biochemistries compared to other species. The fact that selective microbial communities with the same composition are always associated with the same hosts may be due to differences in community assembly rather than cooperative behavior or a shared evolutionary history. Similar communities are to be expected across common similar environments.

Second, the evolution of metabolic dependencies or codependencies among host-microbe communities does not mean that the community, in this case a holobiont, is functionally integrated into a whole. When a nutrient is routinely provided by an organism's environment, selection for biochemical efficiency can lead to loss of genes in the particular biochemical pathway [START_REF] Morris | The Black Queen Hypothesis: evolution of dependencies through adaptive gene loss[END_REF]. The Black Queen Hypothesis states that this process is able to occur whenever there are microbial communities where "leaky products" are produced by different members that are routinely associated with each other [START_REF] Morris | The Black Queen Hypothesis: evolution of dependencies through adaptive gene loss[END_REF][START_REF] Douglas | Holes in the hologenome: Why host-microbe symbioses are not holobionts[END_REF]. Such processes can lead to interdependent communities without requiring selection or functional integration at the holobiont level [START_REF] Sachs | The origins of cooperative bacterial communities[END_REF][START_REF] Douglas | Holes in the hologenome: Why host-microbe symbioses are not holobionts[END_REF]. [START_REF] Mushegian | Rethinking "mutualism" in diverse host-symbiont communities[END_REF] give plausible examples that include protective symbioses based on secondary metabolic functions, such as detoxification of heavy metals or plant toxins, or production of defensive compounds against other microbes, which are likely to be beneficial regardless of whether the microbe is in a host or non-host environment. Perhaps a more compelling example of the independence of metabolic dependency and functional integration is the mammalian gut. Mammalian digestive tracks provide microorganisms with all sorts of beneficial "leaky products" such as warmth, moisture and nutrients. Commensalist bacteria utilize and may even depend on those resources, while the host is entirely unaffected by the bacterial presence, by definition.

Is recurrence with metabolic integration enough to infer that holobionts are whole entities in their own right rather than mere associations of individuals or ephemeral communities? No, because ecological communities can have reproducible dynamics and predictable outcomes for their members without being the result of selection, integration or coevolution at the level of the community [START_REF] Mushegian | Rethinking "mutualism" in diverse host-symbiont communities[END_REF]. Something further is needed to bind individuals into a whole. As argued in earlier sections, reproduction and cotransmission at the level of the whole are plausible options. Another possibility is cooperation.

Cooperation and Conflict

The second conception of organismality I will consider is the framework developed in Queller and Strassmann (2009, see also Queller and Strassmann, this special issue9 ), which takes a social behavior approach to defining organisms. This approach is probably the most amendable to treating multi-species symbiotic consortiums like holobionts as organisms. [START_REF] Pradeu | The limits of the self: immunology and biological identity[END_REF] define organisms as "the largest unit of near-unanimous design." They fill this out by saying that "the organism is simply a unit with high cooperation and very low conflict among its parts. That is, the organism has adaptations and it is not much disrupted by adaptations at lower levels" [START_REF] Pradeu | The limits of the self: immunology and biological identity[END_REF].

Conflicts of interest between the symbiont partners is a major obstacle to holobiont individuality. Within a single species, conflicts of interest can be suppressed by maintaining genetic homogeneity in the case of an individual multicellular organism, or high genetic relatedness in the case of a group of cooperating organisms. These routes are not available to multi-species holobionts.

Three factors that are important for conflict suppression in symbioses are: vertical transmission, specificity of symbiont relationships, and little or no dependency on a free-living state [START_REF] Herre | The evolution of mutualisms: exploring the paths between conflict and cooperation[END_REF][START_REF] Sachs | New paradigms for the evolution of beneficial infections[END_REF][START_REF] Lesser | The endosymbiotic dinoflagellates (Symbiodinium sp.) of corals are parasites and mutualists[END_REF]. Vertical transmission favors increased metabolic integration and can lead to symbiont genome reduction and obligate dependencies between partners, which in turn lowers conflict because the symbionts share a common fate [START_REF] Sachs | New paradigms for the evolution of beneficial infections[END_REF][START_REF] Lesser | The endosymbiotic dinoflagellates (Symbiodinium sp.) of corals are parasites and mutualists[END_REF]. A high specificity in symbiont relationships reduces the number of competitive phenotypes that a host has to contend with, and reduces the conflict between individual microbes associated with the host. Little or no dependency on a free-living state reduces the chance that a symbiont will have to contend with selective pressures from two environments, leading to specialized adaptation to symbiotic environments, and increasing the likelihood of coevolution and dependencies between partners.

These three factors can lead to the symbionts sharing common interests and an alignment of fitness. The difference between having mutually beneficial relationships and sharing common interests is a key difference between being just a group of interacting individuals and being a higher level individual. The open question is: how often does this happen?

We see again that there is a problem when we look at the holobiont as a whole unit. Some of the host-microbe relations might meet criteria for multi-species organismality, but it seems highly unlikely that all of the host-microbe relations will be cooperative. Even if there is cooperation without conflict between a host and all of its microbial symbionts, there will still inevitably be conflict in the holobiont between the microorganisms. For example, microbes will compete for resources within the host, including nutrients and space, as well as for access to the next host generation. The host will remain the site of a whole ecosystem of complex microorganismal interactions (see Mushegian and Ebert 2016 for a similar view). The idea that there is a single interaction between the host and its microbiome is an artifact of macrobe bias and the difficulty of gathering information about all the various host-microbe interactions. Epistemic limitations shouldn't tempt us to overly simplistic conclusions about biological individuality.

The Disunity of the Holobiont

It is unlikely that there is any holobiont that is also an evolutionary individual or organism if the holobiont is defined as a macrobe host and all of its associated microorganisms. It is not impossible that a host and its symbionts could form a unit of selection, it is just that the conditions are unlikely to obtain. High partner fidelity and alignment of fitness are necessary. This is achieved by vertical inheritance or by strong mutual partner choice. Such high-fidelity associations are unlikely to occur across all of the partnerships within a holobiont. Where it does not, selective pressures at the level of the individual lineages will tend to put the partners into direct competition or active exploitation. Focusing on the processes, interactions, and relations that occur between holobiont partners like this opens up a suite of questions. Does vertical transmission lead to increased metabolic integration and alignment of fitness? Or are these necessary before vertical transmission becomes permanent? To what degree are holobiont partnerships species-specific coevolved consortiums vs. generalist assemblages taking advantage of leaky products or stable environments? What is the relationship between different biological parameters: mode of transmission vs. alignment of fitness vs. metabolic integration?

Many of these questions concern ecological relationships. As such, holobiont theory and research will be impoverished if it doesn't incorporate the powerful theoretical tools of community and ecosystem ecology. As we saw with the coral holobiont, holobionts are complex systems comprised of an array of lineages interacting in diverse ways. Holobionts are disunified in the sense that they share features of both individuals and communities. Some partner interactions are best considered as symbioses-ranging from mutualism to parasitism--where the partners mutually form a part of each other's environments. Other interactions long ago bound the individual lineages together into a higher-level lineage and evolutionary individual. I expect there are plenty of indeterminable cases on the road between ecological interaction and becoming a full-fledged individual. Holobionts are interesting because they share features of organisms and communities. Neither reducing the holobiont to a set of pairwise interactions between symbiont partners nor treating the entire community as a single biological individual is a universally appropriate approach.

O'Malley has since moved away from this view, recently stating that natural selection probably does not act at the collective level in multilineal systems, of which holobionts are one kind (O'Malley

2016).

See Clarke (2011) for a thorough survey of accounts of organismality and individuality.

I thank Thomas Pradeu for emphazing this point, as well as pointing out the gap in my treatment of physiological individuality in a previous draft.

I thank Thomas Pradeu for emphasizing this point.

See[START_REF] Sterelny | Local ecological communities[END_REF]) for a similar argument regarding the individuation of ecosystems.

See Huss (2014) for an extended discussion and warning about reifying categories such as metagenome, microbiome and enterotype.

One reviewer suggested that many of the claims about holobionts as units of selection found in the quotations presented in the section entitled "Controversy about the Status of Holobionts" be interpreted as claims about holobionts being interactors. I disagree that this is the correct interpretation of the presented views. At the very least, it is unclear exactly what the quoted authors mean when they say holobionts are a unit of selection.

Bouchard (2009) presents a similar view, arguing that "superindividuality" can emerge in persistent, functionallyintegrated, multispecies communities.

Queller and Strassman (this issue) argue that it is extremely unlikely that any holobionts qualify as organisms.


Maria J Esteban
email: esteban@ceremade.dauphine.fr

Michael Loss
email: loss@math.gatech.edu

SELF-ADJOINTNESS OF DIRAC OPERATORS VIA HARDY-DIRAC INEQUALITIES

Distinguished selfadjoint extension of Dirac operators are constructed for a class of potentials including Coulombic ones up to the critical case, -|x| -1 . The method uses Hardy-Dirac inequalities and quadratic form techniques.

Introduction.

In the units in which both the speed of light c and Planck's constant are equal to 1, the Dirac operator in the presence of an external electrostatic potential V is given by [START_REF] Dolbeault | Relativistic hydrogenic atoms in strong magnetic fields[END_REF] H 0 + V with H 0 := -i α • ∇ + β .

In (1), α 1 , α 2 , α 3 and β are 4 × 4 complex matrices, whose standard form (in 2 × 2 blocks) is

β = I 0 0 -I , α k = 0 σ k σ k 0 (k = 1, 2, 3) ,
where I = 1 0 0 1 and σ k are the Pauli matrices:

σ 1 = 0 1 1 0 , σ 2 = 0 -i i 0 , σ 3 = 1 0 0 -1 .
If V is a bounded function which tends to 0 at infinity, one can easily prove that the operator H 0 + V with domain H 1 (R 3 , C 4 ) is self-adjoint. If V has singularities, as it is the case for instance in many models for atoms, one is interested in defining self-adjoint extensions of T := H 0 +V |C ∞ 0 (R 3 ,C 4 ) . The method used to do this depends on the singularity. Let us for instance consider Coulomb potentials -ν/|x|, ν > 0. Then for ν ∈ (0, π/2] one can use the pseudo-Friedrich extension method to define an extension which satisfies [START_REF] Dolbeault | An analytical proof of Hardy-like inequalities related to the Dirac operator[END_REF] D(

H 0 + V ) ⊂ D(|H 0 | 1/2 ) = H 1/2 (R 3 , C 4 ) .
This result is obtained by using Kato's inequality :

|H 0 | ≥ 2 π|x| .
Actually one can prove that H 0 -ν |x| defined on

C ∞ 0 (R 3 , C 4 ) is essentially self-adjoint if ν < √ 3/2 ([8]).
When the singularities are stronger, that is, if ν ≥ √ 3/2, other methods need to be used. Various works have dealt with this issue, and it appears that for potentials V which have only a singularity at the origin, the condition

(3) sup x =0 |x| |V (x)| < 1 , is sufficient to define a distinguised selfadjoint extension of H 0 + V |C ∞ 0 (R 3 \{0},C 4 )
. This has been done by means of cut-off methods by Wüst [START_REF] Wüst | A convergence theorem for self-adjoint operators applicable to Dirac operators with cut-off potentials[END_REF][START_REF] Wüst | Distinguished self-adjoint extensions of Dirac operators constructed by means of cut-off potentials[END_REF][START_REF] Wüst | Dirac operators with strongly singular potentials[END_REF] in the case of semibounded potentials V and by Schmincke [START_REF] Schmincke | Distinguished self-adjoint extensions of Dirac operators[END_REF] without the assumption of semiboundedness. These extensions are characterized by the fact that the domain is contained in D(T * ) ∩ D(r -1/2 ). On the other hand, under the same assumption (3) Nenciu proved in [START_REF] Nenciu | Self-adjointness and invariance of the essential spectrum for Dirac operators defined as quadratic forms[END_REF] the existence of a unique selfadjoint extension T with domain contained in H 1/2 (R 3 , C 4 ). Finally, in the case of semibounded potentials satisfying (3), Klaus and Wüst proved in [START_REF] Klaus | Characterization and uniqueness of distinguished self-adjoint extensions of Dirac operators[END_REF] that all the aforementioned self-adjoint extensions coincide.

In this paper we address the same question from a different perspective, and we prove that the existence of a distinguished self-adjoint extension of Dirac operators

H 0 + V |C ∞ 0 (R 3 ,C 4
) can be achieved whenever there exists a Hardy-like inequality involving the operators H 0 and V . This is Date: June 9, 2007. somewhat surprising since the Dirac operator is not semibounded. This should be contrasted with the case -∆ -µ |x| 2 where the classical Hardy inequality defines the threshold of self-adjointness. In [START_REF] Dolbeault | On the eigenvalues of operators with gaps. Application to Dirac operators[END_REF] Hardy-like inequalities for Dirac operators were proved. They allow potentials which cover all admissible Coulomb like singularities, that is, all the potentials V = ν |x| , ν ∈ (0, 1]. As a result, our method allows to overcome the limitation contained in assumption (3) in a natural way by constructing distinguished selfadjoint extensions for potentials satisfying [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF] sup

x =0 |x| V (x) ≤ 1 ,
which is the maximal possible range.

One example of a Hardy-Dirac inequality is given by the following theorem.

Theorem 1 ([3]

). Let V be a function satisfying

(5) lim |x|→+∞ V (x) = 0 and - ν |x| -c 1 ≤ V ≤ Γ = sup(V ) , with ν ∈ (0, 1), c 1 , Γ ∈ R, c 1 , Γ ≥ 0, c 1 + Γ -1 < √ 1 -ν 2 . There exists a constant c(V ) ∈ (-1, 1) such that for all ϕ ∈ C ∞ c (R 3 , C 2 ), (6)
R 3 |σ • ∇ϕ| 2 1 + c(V ) -V + 1 -c(V ) + V |ϕ| 2 dx ≥ 0 .
Note that the argument in [START_REF] Dolbeault | On the eigenvalues of operators with gaps. Application to Dirac operators[END_REF] proceeded in an indirect way. Using spectral analysis of the Dirac operator, i.e., its selfadjointness, Theorem 1 was established for the case ν ∈ (0, 1). From this, a simple limiting argument yields the Hardy-Dirac inequality for the case ν = 1, (7)

R 3 |σ • ∇ϕ| 2 1 + 1 |x| + |ϕ| 2 dx ≥ R 3 |ϕ| 2 |x| dx .
This functional inequality was later derived in [START_REF] Dolbeault | An analytical proof of Hardy-like inequalities related to the Dirac operator[END_REF] directly, i.e. without any spectral analysis. Despite the fact that the eigenfunctions of the Dirac-Coulomb operator for ν = 1 can be explicitely constructed, nobody, to our knowledge, has ever succeeded in deducing selfadjointness of this operator from that fact.

The main result of our paper states that for any potential V for which an inequality like (6) (with c(V ) ∈ (-1, 1)) holds true, we can define a distinguished self-adjoint extension of the operator H 0 +V defined on C ∞ c (R 3 , C 4 ). In particular our analysis can be applied to the critical case ν = 1 of the Dirac-Coulomb Hamiltonian. This extension will not necessarily satisfy condition [START_REF] Dolbeault | An analytical proof of Hardy-like inequalities related to the Dirac operator[END_REF], that is, the domain of H 0 + V will not necessarily be a space contained in H 1/2 (R 3 ). For instance, this will be the case for the potential V = -1 |x| . The total energy, however, will still be finite for all functions in our domain.

Our method is not limited to Dirac operators with scalar potentials. For instance a Hardy-Dirac inequality for an hydrogenic atom in a constant magnetic field has been proved recently in [START_REF] Dolbeault | Relativistic hydrogenic atoms in strong magnetic fields[END_REF] provided the magnetic field is not too intense. As a consequence, a distinguished selfadjoint extension can be obtained also in this case.

Main result and proof.

In what follows we use the following assumption on the potential V .

Assumption (A) : V : R 3 → R is a function such that for some constant c(V ) ∈ (-1, 1), Γ := sup R 3 V < 1 + c(V ) and for every ϕ ∈ C ∞ c (R 3 , C 2 ) , (8)
R 3 |σ • ∇ϕ| 2 1 + c(V ) -V + 1 -c(V ) + V |ϕ| 2 dx ≥ 0 .
In what follows γ is any number in (Γ, 1 + c(V )). Consider the quadratic form

(9) b γ (ϕ, ϕ) := R 3 |σ • ∇ϕ| 2 γ -V + 2 -γ + V |ϕ| 2 dx defined on C ∞ c (R
b γ (ϕ, ϕ) = (S 1/2 γ ϕ, S 1/2 γ ϕ) , for all ϕ ∈ H γ +1 . We now show that the above construction of H γ +1 does not depend on γ. This follows from Proposition 2. Under assumption (A),

(12) b γ (ϕ, ϕ) ≤ b γ ′ (ϕ, ϕ) + [γ -γ ′ ] + 1 (γ ′ -Γ)(γ -Γ) + 1 ϕ 2 2 for all γ, γ ′ in (Γ, 1 + c(V )) where [γ -γ ′ ] + = max{γ -γ ′ , 0}.
As a consequence, the spaces H γ

±1

are independent of γ and we denote them by H ±1 .

It suffices to prove the inequality for spinors ϕ ∈ C ∞ c (R 3 , C 2 ). The proposition follows immediately from the elementary, pointwise inequality

(13) 1 γ -V (x) - 1 γ ′ -V (x) ≤ [γ -γ ′ ] + (γ -Γ)(γ ′ -Γ)
.

After this preparatory result we are ready to define an extension of the Dirac operator originally defined on

C ∞ c (R 3 , C 4 ). Definition 3. The domain D of the Dirac operator is the collection of all pairs ϕ ∈ H +1 , χ ∈ L 2 (R 3 , C 2 ) such that (14) (2 -γ + V )ϕ -iσ • ∇χ , -iσ • ∇ϕ + (V -γ)χ ∈ L 2 (R 3 , C 2 ) .
The meaning of these two expressions is in the weak sense, i.e., the linear functional (η, (Vγ)χ) + (-iσ • ∇η, ϕ) , which is defined for all test functions, extends uniquely to a bounded linear on L 2 (R 3 , C 2 ). Likewise the same for (η, (2 -γ + V )ϕ) + (-iσ • ∇η, χ) . From this definition it is clear that the domain does not depend on γ. Thus on this domain D, we define the Dirac operator as

(15) (H 0 + V ) ϕ χ = (V + 1)ϕ -iσ • ∇χ -iσ • ∇ϕ + (V -1)χ
Note that for all vectors (ϕ, χ) ∈ D the expected total energy is finite. Our main result is the following theorem.

Theorem 4. Under the assumption (A) on the potential V , the operator H, that is

H 0 + V defined on D is selfadjoint. It is the unique selfadjoint extension of H 0 + V on C ∞ c (R 3 , C 4 ) such that the domain is contained in H +1 × L 2 (R 3 , C 2 ).
If in addition to (A) we assume that (3) holds, it is natural to ask how our results relate to the ones in [START_REF] Wüst | A convergence theorem for self-adjoint operators applicable to Dirac operators with cut-off potentials[END_REF][START_REF] Wüst | Distinguished self-adjoint extensions of Dirac operators constructed by means of cut-off potentials[END_REF][START_REF] Wüst | Dirac operators with strongly singular potentials[END_REF] and [START_REF] Nenciu | Self-adjointness and invariance of the essential spectrum for Dirac operators defined as quadratic forms[END_REF]. For simplicity we shall assume that V (x) ≤ 0 and that (16) sup

R 3 \{0} |xV (x)| < 1 .
It was shown in [START_REF] Klaus | Characterization and uniqueness of distinguished self-adjoint extensions of Dirac operators[END_REF] that for such potentials V (x) all these various selfadjoint extensions, which we call H W , H N coincide, i.e, H W = H N =: H W,N and the domain is given by (17)

D W := D(T * ) ∩ D(r -1/2 ) .
Here T * is the adjoint of

H 0 + V restricted to C ∞ c (R 3 \ {0}, C 4 ). If we denote by S * the adjoint of H 0 + V restricted to C ∞ c (R 3 , C 4
), then S * ⊂ T * . Note that S * is actually the operator H 0 + V defined on the set of spinors (ϕ, χ) ∈ L 2 (R 3 , C 2 ) for which (14) holds. Because of Theorem 1 we find that (A) is satisfied for such potentials. Hence we have a selfadjoint operator H with domain D which satisfies (18)

T ⊂ S ⊂ H = H * ⊂ S * ⊂ T * , moreover we have Theorem 5. Assume that V (x) ≤ 0 and satisfies (16). Then H = H W,N .

The following two results are important in the proof of Theorem 4.

Proposition 6.

Under assumption (A) on the potential V ,

(19) H +1 ⊂ ϕ ∈ L 2 (R 3 , C 2 ) : -iσ • ∇ϕ γ -V ∈ L 2 (R 3 , C 2 ) ,
where ∇ϕ denotes the distributional gradient of ϕ. Therefore, we have the 'scale of spaces'

H +1 ⊂ L 2 (R 3 , C 2 ) ⊂ H -1 .
Proof. It suffices to prove the proposition for ϕ ∈ C ∞ c (R 3 , C 2 ). First we note that for all

x ∈ R 3 (20) 1 γ -V (x) - 1 1 + c(V ) -V (x) ≥ δ (γ -V (x)) 2 where (21) δ := (γ -Γ)(1 + c(V ) -γ) 1 + c(V ) -Γ .
This follows from the assumption that V (x) ≤ Γ for all x. Noting that 1 + c(V ) -γ > δ, we get

(22) b γ (ϕ, ϕ) ≥ δ R 3 |ϕ| 2 dx + δ R 3 |σ • ∇ϕ| 2 (γ -V ) 2 dx . Lemma 7. For any F in L 2 (R 3 , C 2 ) and for any γ ∈ (Γ, 1 + c(V )), (23)
-iσ • ∇ F γ -V ∈ H -1 ,
where, once again, the gradient is to be interpreted in the distributional sense.

Proof. By the definition of the distributional derivative, for every

η ∈ C ∞ c (R 3 , C 2 ), ( 24
) -iσ • ∇η, F γ -V = -iσ • ∇η γ -V , F ≤ η H+1 F 2 .
Hence, the linear functional

(25) η → -iσ • ∇η, F γ -V
extends uniquely to a bounded linear functional on H +1 .

Proof of Theorem 4. We shall prove Theorem 4 by showing that for γ ∈ (Γ, 1 + c(V )), H + 1 -γ is symmetric and a bijection from its domain D onto L 2 (R 3 , C 4 ). To prove the symmetry we have to show that for both pairs (ϕ, χ), ( φ, χ) in the domain,

(26) (H + 1 -γ) ϕ χ , φ χ = ((V -γ)χ -iσ • ∇ϕ, χ) + ((2 -γ + V )ϕ -iσ • ∇χ, φ) equals (27) (χ, (V -γ) χ -iσ • ∇ φ) + (ϕ, (2 -γ + V ) φ -iσ • ∇ χ) = ϕ χ , (H + 1 -γ) φ χ .
First, note that since (ϕ, χ) is in the domain,

(V -γ) χ + -iσ • ∇ϕ V -γ ∈ L 2 (R 3 , C 2 ) . (28)
We now claim that

(29) ((2 -γ + V )ϕ -iσ • ∇χ, φ) = (S γ ϕ, φ) + (V -γ) χ + -iσ • ∇ϕ V -γ , -iσ • ∇ φ V -γ
Note that each term makes sense. The one on the left, by definition of the domain and the first on the right, because both ϕ, φ are in H +1 . The second term on the right side makes sense because of (28) above and Proposition 6. Moreover both sides coincide for φ chosen to be a test function and both are continuous in φ with respect to the H +1 -norm. Hence the two expressions coincide on the domain. Thus we get that

(30) (H + 1 -γ) ϕ χ , φ χ equals (31) (S γ ϕ, φ) + (V -γ) χ + -iσ • ∇ϕ V -γ , χ + -iσ • ∇ φ V -γ
which is symmetric in (ϕ, χ) and ( φ, χ). To show that the operator is onto, pick any F 1 , F 2 in L 2 (R 3 , C 2 ). Since S γ is an isomorphism, there exists a unique ϕ in H +1 such that (32)

S γ ϕ = F 1 -iσ • ∇ F 2 γ -V . Indeed, F 1 is in L 2 (R
(η, (V +2-γ)ϕ)+(-iσ•∇η, χ) = (-iσ•∇η, -iσ • ∇ϕ γ -V )+(η, (V +2-γ)ϕ)+(-iσ•∇η, (χ+ -iσ • ∇ϕ V -γ )) which equals (35) (η, S γ ϕ) + (-iσ • ∇η, (χ + -iσ • ∇ϕ V -γ )) = (η, F 1 )
This holds for all test functions η, but since F 1 is in L 2 (R 3 , C 2 ), the functional η → (η, (V + 2γ)ϕ) + (-iσ • ∇η, χ) extends uniquely to a linear continuous functional on L 2 (R 3 , C 2 ) which implies that

(36) (V + 2 -γ)ϕ -iσ • ∇χ = F 1 .
Hence (ϕ, χ) is in the domain D and the operator (H + 1 -γ) applied to (ϕ, χ) yields (F 1 , F 2 ).

Let us now prove the injectivity of H + 1 -γ. Assuming that

(37) H + 1 -γ ϕ χ = 0 0 ,
we find by (32) and (33),

χ = -iσ • ∇ϕ γ -V , S γ ϕ = 0 .
Since S γ is an isomorphism, this implies that ϕ = χ = 0. It remains to show the uniqueness part in our theorem. Assume that

C ∞ c (R 3 , C 4 ) ⊂ D ′ is another selfadjoint extension such that whenever (ϕ, χ) ∈ D ′ , the spinor ϕ ∈ H + and, of course, χ ∈ L 2 (R 3 , C 2 ). Since H is selfadjoint on this domain, (38) φ χ , H ϕ χ = H φ χ , ϕ χ for all ( φ, χ) ∈ C ∞ c (R 3 , C 4
). This, however, means that the expressions (14), defined in the distributional sense, belong to L 2 (R 3 , C 2 ). Thus, D ′ ⊂ D and hence D ′ = D.

Proof of Theorem 5

If we can show that D ⊂ D W it follows that H ⊂ H W,N and hence H = H W,N . This follows from the next proposition in which we show that D ⊂ D(r -1/2 ). Proposition 8. Assume that V (x) ≤ 0 and satisfies (16). Then in the above notation D ⊂ D(r -1/2 ).

Proof

We have by the assumption on V (x), for any ϕ ∈ C ∞ c (R 3 , C 2 ), (39)

-V |ϕ| 2 dx ≤ ν |ϕ| 2 |x| dx ≤ ν R 3 |σ • ∇ϕ| 2 1 + 1 |x| + |ϕ| 2 dx
where we have used [START_REF] Reed | Methods of modern mathematical physics[END_REF]. For γ small enough we have the elementary inequality

(40) ν 1 + 1 |x| ≤ ν 2 γ -V . Hence -ν |ϕ| 2 |x| dx ≤ ν 2 R 3 |σ • ∇ϕ| 2 γ -V + ν R 3 |ϕ| 2 dx (41) = ν 2 b γ (ϕ, ϕ) + (ν + ν 2 (γ -2)) |ϕ| 2 dx -ν 2 V
+ (V -γ) χ + -iσ • ∇ϕ V -γ , χ + -iσ • ∇ϕ V -γ which for (ϕ, χ) ∈ D implies that (46) 0 ≤ χ - -iσ • ∇ϕ γ -V , (γ -V ) χ - -iσ • ∇ϕ γ -V < ∞ .
This implies, via simple estimates, that (47)

R 3 (γ -V )|χ| 2 dx - R 3 |σ • ∇ϕ| 2 γ -V dx < ∞ .
Since ϕ ∈ H +1 , (ϕ, S γ ϕ) is finite and since |ϕ| 2 |x| dx < ∞, we also have that (48)

R 3 |σ • ∇ϕ| 2 γ -V dx < ∞ .
This implies the result.

  +1 . We denote by S γ the unique selfadjoint operator associated with b γ : for allϕ ∈ D(S γ ) ⊂ H γ +1 , (10) b γ (ϕ, ϕ) = (ϕ, S γ ϕ) .S γ is an isometric isomorphism from H γ +1 to its dual H γ -1 . Using the second representation theorem in[START_REF] Kato | Perturbation Theory for Linear Operators[END_REF] Theorem 2.23 we have that H γ +1 is the operator domain of S

	1/2 γ , and

3

, C 2 ). Note that by assumption

[START_REF] Schmincke | Essential self-adjointness of Dirac operators with strongly singular potential[END_REF]

this quadratic form is nonnegative and symmetric on C ∞ c (R 3 , C 2 ). Therefore it is closable and we denote its closure by b γ and its form domain by H γ

  3 , C 2 ) and therefore in H -1 . Moreover the second term is also in H -1 by

	Lemma 7.
	Now define χ by
	(33) which by Proposition 6 is in L 2 (R 3 , C 2 ). χ = F 2 +	-iσ • ∇ϕ γ -V
	Now for any test function η we have that
	(34)

  |x| dx ≤ ν 2 b γ (ϕ, ϕ) + (ν + ν 2 (γ -2)) |ϕ| 2 dx + ν 3 -ν 2 ) |ϕ| 2 |x| dx ≤ νb γ (ϕ, ϕ) + (1 + ν(γ -2)) |ϕ| 2 dxThis inequality extends to all ϕ ∈ H +1 . It remains to show a similar bound for χ.In the proof of the symmetry of H we derived the formula

	(42)				|ϕ| 2 dx
	Hence
	(43) which yields	ν	|ϕ| 2		|ϕ| 2 |x|	dx
	(44) (1 (45) H ϕ χ ,	ϕ χ	= (S

γ ϕ, ϕ)

Acknowledgments. M.J.E. would like to thank M. Lewin, E. Séré and J.-P. Solovej for various discussions on the self-adjointness of Dirac operators. M.J.E. and M.L. wish to express their gratitute to Georgia Tech and Ceremade for their hospitality. M.J.E. acknowledges support from ANR Accquarel project and European Program "Analysis and Quantum" HPRN-CT # 2002-00277. M.L. is partially supported by U.S. National Science Foundation grant DMS DMS 06-00037. c 2007 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.


Brahim Belaoucha

Théodore Papadopoulo

Spatial regularization based on dMRI to solve EEG/MEG inverse problem

In this paper, we present a new approach to reconstruct dipole magnitudes of a distributed source model for magnetoencephalographic (MEG) and electroencephalographic (EEG). This approach is based on the structural homogeneity of the cortical regions which are obtained using diffusion MRI (dMRI). First, we parcellate the cortical surface into functional regions using structural information. Then, we use a weighting matrix that relates the dipoles' magnitudes of sources inside these functional regions. The weights are based on the region's structural homogeneity. Results of the simulated and real MEG measurement are presented and compared to classical source reconstruction methods.

I. INTRODUCTION

EEG and MEG are two imaging modalities that provide information about brain activity with high temporal resolution. The sources of MEG/EEG are current sources that are modeled using dipoles which are distributed in the cortical surface. Obtaining the distributed sources activation from these measurements is an underdetermined problem due to the small number of measurements with respect to the number of distributed sources. To obtain a unique solution, different constraints can be used [START_REF] Grech | Review on solving the inverse problem in EEG source analysis[END_REF]. They can be divided into three different categories: spatial, temporal, and spatiotemporal constraints. In our work, we focus on spatial constraints. Minimum norm estimate (MNE) was introduced to obtain a unique source reconstruction [START_REF] Wang | Magnetic source images determined by a lead-field analysis: the unique minimumnorm least-squares estimation[END_REF][START_REF] Hämäläinen | Interpreting magnetic fields of the brain: minimum norm estimates[END_REF]. MNE gives a linear solution which makes it a simple and attractive approach. But it suffers from overestimation of the extents of the active regions. To address these limitations, many MNE variants have been proposed by introducing different weighting matrices [START_REF] Grech | Review on solving the inverse problem in EEG source analysis[END_REF]. But all neglect the structural information which has been shown to be in relation to the functional homogeneity of brain regions by several studies [START_REF] Brodmann | Vergleichende Lokalisationslehre der Grobhirnrinde[END_REF]- [START_REF] Sporns | Organization, development and function of complex brain networks[END_REF]. In Philippe et al. [START_REF] Philippe | A nested cortex parcellation combining analysis of MEG forward problem and diffusion MRI tractography[END_REF], the authors used a weighting matrix as a penalty term which favour constant activation per region (CP). In their work, they do not consider the possible structural inhomogeneity. dMRI is the only non-invasive imaging modality that allows the access to the anatomical connectivity. That is why it has been used to parcellate the cortical surface [START_REF] Rakic | Neurocreationism-making new cortical maps[END_REF][START_REF] Belaoucha | Cortical surface parcellation via dMRI using mutual nearest neighbor condition[END_REF]. In this work, we present another variant of the weighted minimum norm in which the elements of the weighting matrix depend on the structural homogeneity of the regions obtained from dMRI. We call it structural weighted minimum norm This work received funding from ERC under the European Union's Horizon 2020 research and innovation program (ERC Advanced Grant agreement No 694665 : CoBCoM) and French ANR contract Vibrations. Both authors are with the Université Côte d'Azur, Inria, Nice, France (email: brahim.belaoucha@inria.fr, theodore.papadopoulo@inria.fr). estimate (sWMNE). The proposed method is tested using synthetic data and is compared to some of the reconstruction algorithms that can be found in the literature. Also, real data was used to test the accuracy of the algorithm. The results were compared with the results of functional MRI (fMRI) group analysis study with the same subjects performing the same tasks [START_REF] Wakeman | A multi-subject, multi-modal human neuroimaging dataset[END_REF].

II. DATA ACQUISITION AND PROCESSING

Structural and diffusion MRI data were taken from Wakeman et al. [START_REF] Wakeman | A multi-subject, multi-modal human neuroimaging dataset[END_REF]. The 1 mm isotropic resolution T1 weighted images of size 256 × 256 × 192 were acquired by a Siemens 3T. The 2 mm isotropic resolution diffusion weighted images of size 96 × 96 × 68 were collected by the same scanner (64 gradient directions and b-value = 1000 s/mm 2 ), with one b0 image (an image without diffusion gradient). The cortical surface was extracted from T1, using Freesurfer [START_REF] Dale | Cortical surface-based analysis I: Segmentation and surface reconstruction[END_REF], and re-meshed to 10 4 vertices (sources). The transformation between the anatomical and diffusion space was obtained by using FSL [START_REF] Smith | Advances in functional and structural MR image analysis and implementation as FSL[END_REF].

MEG (102 magnetometers, 204 planar gradiometers) and EEG (70 electrodes) were measured simultaneously in a magnetically shielded room. The face stimuli contain three sets of 450 gray scale photographs, one third of unfamiliar people (unknown to the participants), one third of famous people and the remaining are of scrambled faces. The reader is referred to [START_REF] Wakeman | A multi-subject, multi-modal human neuroimaging dataset[END_REF] for more details.

In this work, we are interested in localizing face recognition areas. For this reason, we use only the measurement acquired when using photos of famous people subtracted to the ones obtained when using scrambled faces. Low pass filter of cut-off frequency 45 Hz was used to smooth the data. The MEG/EEG forward problem, lead field matrix G, is obtained using OpenMEEG [START_REF] Gramfort | OpenMEEG: opensource software for quasistatic bioelectromagnetics[END_REF][START_REF] Kybic | A common formalism for the integral formulations of the forward EEG problem[END_REF].

III. CORTICAL PARCELLATION

The mutual nearest neighbor (MNN) parcellation algorithm divides the cortical surface of N points into regions with the highest homogeneity according to a similarity measure [START_REF] Belaoucha | Cortical surface parcellation via dMRI using mutual nearest neighbor condition[END_REF]. MNN uses connectivity profiles of sources to parcellate the cortex substrate. Connectivity profile of a source is a vector containing the probability of the existence of anatomical connections between the source and the other image voxels. This probability is obtained by running a probabilistic tractography using FSL [START_REF] Smith | Advances in functional and structural MR image analysis and implementation as FSL[END_REF]. MNN depends on one parameter, s, that controls the resulting number of regions. We use Tanimoto [START_REF] Bajusz | Why is tanimoto index an appropriate choice for fingerprint-based similarity calculations?[END_REF] measure to quantify the similarity between the connectivity profiles. The Tanimoto similarity measure between source i and i is:

d i,j = X i • X j X i • X j + X i -X j 2 2
where X i and X j is the vector form of the connectivity profile of source i and j respectively. The Tanimoto is symmetric and has values between 0 and 1. The MNN can result to small regions due to the structural inhomogeneity.

The small regions, regions with less than N s sources, were merged with valid regions in a way that give the highest mean similarity value.

IV. MEG/EEG INVERSE PROBLEM

In a distributed source model with m sources and n sensors, the EEG/MEG measurement is related to the sources' magnitudes by the following linear relation:

M = GS + (1)
where M ∈ R n×T is the measurement (T number of time samples), G ∈ R n×m is the lead field matrix that contains the contribution of sources to each sensor and S ∈ R m×T is a matrix that contain the sources' magnitudes. Finally, the measurement is corrupted by an additive Gaussian noise of a zero mean, . The minimum norm estimates sources' magnitudes by optimizing the following functional:

U (S) = M -GS 2 2 + λ S 2 2 (2)
where • is l 2 norm and λ (≥ 0) is the trade-off parameter between the prior and the data fit term. The solution of Eq (2) is:

S mne = (G t G + λI) -1 G t M (3)
where X t is the transpose of matrix X and I is m × m identity matrix. Let's define the following functional:

U (S) = M -GS 2 2 + µ WS 2 2 + λ S 2 2 (4)
where W is a weighting matrix and µ (≥ 0) is a regularization parameter. We propose to use a weighted matrix whose elements are defined by similarity values of the connectivity profile vectors (see section III). We consider W to be a block matrix in which each block corresponds to a cortical region defined by:

S i ∈ R p → W(i, j) =        1 -1 di if i = j - di,j √ didj if S j ∈ R p 0 if S j / ∈ R p (5)
where S i and R p are, respectively, the i th source and p th cortical region. d i,j is the similarity value between the connectivity profile vector of source i and j.

d i = |Rp| j=1 d i,j (6)
where |R p | is the number of sources in the p th region.

To have an idea about the effect of this weighting matrix, let's assume that we have five sources. P = [1, 1, 1, 2, 2] is a vector containing the label of each source. The weighting matrix, W, for this case is:

W =         1 -1 d1 -d1,2 √ d1d2 -d1,3 √ d1d3 0 0 -d1,2 √ d1d2 1 -1 d2 -d2,3 √ d2d3 0 0 -d1,3 √ d1d3 -d2,3 √ d2d3 1 -1 d3 0 0 0 0 0 1 -1 d4 -d4,5 √ d4d5 0 0 0 -d4,5 √ d4d5 1 -1 d5        
WS 2 is minimum when:

J 2 = d2 d1 J 1 , J 3 = d3 d1 J 1 , J 4 = d4 d5 J 5
If source i and j are in the same region, WS 2 constrains i and j to have activations related by the following relationship S j ≈ dj di S i . The solution of Eq (4) is:

S swmne = (G t G + µW t W + λI) -1 G t M (7)
Each block of W corresponds to a Laplacian weighted by the similarity measure values of each region. The method explained in Philippe et al. [START_REF] Philippe | A nested cortex parcellation combining analysis of MEG forward problem and diffusion MRI tractography[END_REF], that we call CP, can be obtained from our method by replacing all the d i,j 's by 1. The regularization parameters are obtained using crossvalidation [START_REF] Arlot | A survey of cross-validation procedures for model selection[END_REF].

V. RESULTS AND DISCUSSION

A. Simulation setup

For reasonable fast computation time and comparison between sWMNE, MNE and CP, we generated a simulation with 20 sensors and 500 points distributed equally into 50 regions i.e. 10 points in each region. For each region, R i , we generate randomly, from normal distribution N (100, 200), the mean structural connectivity vector (r i ) of length equal to 10 3 . We give to a source point j, that belongs to R i , the following connectivity profile vector:

r j = r i + j ( 8
)
where the elements of the vector j are obtained randomly from N (50, 100), values lower than 100 were set to zero. The elements of the weighting matrix are obtained by computing the Tanimoto similarity measure between the sources' connectivity vectors that belong to the same regions. The forward model G, gain matrix, is drawn from a N (0, 1). We consider two test configurations: Left panel: Results of the 1 st test configuration. Right panel: results of the 2 nd test configuration. We show in color, blue or/and green, the reconstructed magnitudes of the simulated region(s) and in black the remaining sources.

• One active region (see Figure 1 (a)).

• Two active regions (see Figure 1 (b)).

In the first test configuration, we activate 20 random regions.

For each active region, we run the reconstruction algorithms 100 times at three different noise levels (Signal-to-Noise ration (SNR) = {15, 10, 5} dB). In the second, we activate 20 random pairs of regions. For each pair of regions, we run the reconstruction algorithms 100 times at the same noise levels mentioned earlier. Figure 2 shows an average, one of the 20 randomly activated regions in both test configurations, the reconstructed sources using MNE, CP and sWMNE. The left and right panels correspond, respectively, to the results

where S gt and S r are, respectively, the simulated (ground truth) and reconstructed sources, see the upper part of Figure 3. Also, we test the focality of the three methods by counting the number of sources that have activations higher than 25% of the highest activation, see the lower part of Figure 3. A degree of activation of a source i is computed as the l 2 norm of the reconstructed time course of i.

The MNE smears the activation which makes it hard to localize spatially the active regions, see Figure 2. In the CP and sWMNE, we can distinguish from the reconstructions, see Figure 2, which regions are more active. sWMNE is more flexible than CP by allowing magnitude variation due to a difference of the structural connectivity between sources of the same region. sWMNE does not show significant improvement in terms of E r , see Figure 3, with compared to CP but the number of active sources obtained from sWMNE is closer to the ground truth.

B. Real data

We tested sWMNE using real MEG data obtained from [START_REF] Wakeman | A multi-subject, multi-modal human neuroimaging dataset[END_REF] which was presented briefly in section II. In this paper, we show only the results of one subject due to page limitation. In Figure 4, we show the normalized l 1 norm of the source reconstructed intensities using sWMNE (left) and MNE (right) between time t = 0, the onset, and t = 0.6s thresholded at 15%. In the contrary to MNE, sWMNE provides more focal reconstruction.

In the sWMNE, we can distinguish fusiform and occipital activation in both hemispheres. Also in all the eleven subjects Fig. 4: The l 1 -norm of the reconstructed sources, from MEG [START_REF] Wakeman | A multi-subject, multi-modal human neuroimaging dataset[END_REF], of Subject 1 displayed on the inflated surface. We show only values above 15% of the highest l 1 -norm. The s parameter of MNN was set to 800. in [START_REF] Wakeman | A multi-subject, multi-modal human neuroimaging dataset[END_REF], anterior temporal lobe and inferior pre-frontal were found to be active. Orbitofrontal cortex was found to be more active in subject 4, 9 and 11. The right temporal pole of subject 1 and 2 were found to be more active than the left one. These regions coincide with what was reported in Wakeman et al. [START_REF] Wakeman | A multi-subject, multi-modal human neuroimaging dataset[END_REF] in which they used the same paradigm and participants to acquire fMRI. The study whether some of these regions are dedicated to human faces [START_REF] Kanwisher | The functional architecture of the face system: Integrating evidence from fMRI and patient studies[END_REF] or intervene in other visual stimuli [START_REF] Haxby | Distributed and overlapping representations of faces and objects in ventral temporal cortex[END_REF] is outside the scope of this work. sWMNE shows better results, in terms of reconstruction error and focality, with compared to MNE. We observed no significant difference in the reconstruction error between CP and sWMNE but the latter gives a better approximation of the activation extent.

VI. CONCLUSION

This paper presented a method to use information from dMRI to solve the EEG/MEG inverse problem. A weighting matrix whose elements are obtained from computing the similarity between the connectivity vector pairs inside each region is used to regularize the inverse problem. This matrix allows sources to differ according to their structural homogeneity. Finally, the source reconstruction and cortical surface parcellation depend on the similarity measure used to quantify the structural homogeneity between the different connectivity profiles. Other similarity measures can be used and their effect on the reconstruction should be investigated.

Fig. 1 :

 1 Fig. 1: Simulated data used to activate one region (a) and two regions (b).

Fig. 2 :

 2 Fig. 2: Mean (over 100 runs) reconstruction using MNE, CP and sWMNE at different noise levels (SNR = 15, 10, 5 dB). Left panel: Results of the 1 st test configuration. Right panel: results of the 2 nd test configuration. We show in color, blue or/and green, the reconstructed magnitudes of the simulated region(s) and in black the remaining sources.

Fig. 3 :

 3 Fig. 3: Top raw: reconstruction error E r of the different algorithms. Bottom, the number of sources that have l 2 -norm ≥ 25% of the highest energy in the source space. The results are obtained for different noise levels. GT, in red, refers to ground truth.

VII. ACKNOWLEDGMENTS

We thank Dr. Daniel Wakeman and Prof. Richard Henson for providing the raw data.