Ludovic Berthier

Patrick Charbonneau

Joyjit Kundu

Finite-dimensional vestige of spinodal criticality above the dynamical glass transition

Finite-dimensional signatures of spinodal criticality are notoriously difficult to come by. The dynamical transition of glass-forming liquids, first described by mode-coupling theory, is a spinodal instability preempted by thermally activated processes that also limit how close the instability can be approached. We combine numerical tools to directly observe vestiges of the spinodal criticality in finite-dimensional glass formers. We use the swap Monte Carlo algorithm to efficiently thermalise configurations beyond the mode-coupling crossover, and analyze their dynamics using a scheme to screen out activated processes, in spatial dimensions ranging from d = 3 to d = 9. We observe a strong softening of the mean-field square-root singularity in d = 3 that is progressively restored as d increases above d = 8, in surprisingly good agreement with perturbation theory.

I. INTRODUCTION

Spinodals predicted by mean-field theories do not exist in finite-dimensional systems because thermal (or other) fluctuations destabilize the precursor metastable state before the critical point can ever be reached [1][START_REF] Gunton | Intro to the Theory of Metastable and Unstable States[END_REF][START_REF] Ray | [END_REF][START_REF] Stillinger | Energy Landscapes, Inherent Structures, and Condensed-Matter Phenomena[END_REF]]. An Ising system prepared with the metastable magnetization, for instance, grows nuclei (instantons [START_REF] Langer | [END_REF]6]) of the opposite magnetization well ahead of the formal instability region. As the single-flip relaxation dynamics slows down critically, that of nucleation accelerates [7]; cluster moves only worsen the imbalance. Numerical studies have nevertheless observed convincing hints of the Ising spinodal pseudocriticality in high enough spatial dimension d. Nucleation kinetics being exponentially suppressed as d increases, the pseudocritical power-law scaling can then be made sufficiently extended [7].

More theoretically enticing than the Ising spinodal is that of models with disorder, which capture the essence of systems ranging from magnetic [8] to mesoporous [9,10] materials and also appear in social sciences and economics [11]. These models exhibit a rich set of activated processes, such as avalanches [8,12] and hopping [13,14], in addition to nucleation, which makes their criticality especially challenging to scrutinize. Hence, although the Ginzburg criterion for the corresponding cubic field theory without activation gives an upper critical dimension d u = 8 [15][16][17], it is unclear how relevant the associated pseudocriticality might be in any given system. Below d u , perturbative expansions relying on dimensional reduction [16,18] or direct expansion [19] further do not concur. Even if they did, one may expect non-perturbative fluctuations to also contribute [20]. Although these fluctuations should limit the relevance of perturbative approaches and question the very existence of an upper critical dimension, very little is known about their effects [21,[START_REF] Bhattacharyya | Proc. Nat. Acad. Sci. U.S.A[END_REF], emphasizing further the need for quantitative results in finite-dimensional systems.

This theoretical haze has not held back the use of mode-coupling theory (MCT) to describe the dynamics of supercooled liquids [START_REF] Götze | Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory[END_REF], which has been the subject of countless numerical and experimental tests [START_REF] Götze | Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory[END_REF][START_REF] Götze | [END_REF]. Although unclear in its initial derivations, it is now understood that MCT [25] as well as the mean-field d → ∞ description of liquids [26] indeed correspond to the limit of stability of the glass phase upon heating (or decompressing). The dynamical glass transition should thus be described as a spinodal (thermodynamic) instability in the presence of quenched disorder [20,27]. Unfortunately, the spinodal is found to be totally hidden by finite-dimensional effects in direct free energy calculations [28,29].

The associated critical scaling laws of the structural relaxation time τ α and of time correlation functions upon approaching the avoided MCT (dynamical) transition from the equilibrium liquid are instead much more frequently examined [START_REF] Götze | Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory[END_REF][START_REF] Götze | [END_REF]26]. These quantities are straightforwardly measured in both simulations and experiments, but they are also non-universal, i.e., model dependent. Theoretical predictions for the associated critical exponents are not only sensitive to the spatial dimension (even above d u ) and to activated processes, but also to fine details of the liquid structure and pair interactions. These predictions are thus typically of limited quantitative validity in finite d, but this limitation did not prevent MCT from making valid predictive statements regarding the glassy dynamics of a variety of materials [START_REF] Götze | Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory[END_REF][START_REF] Götze | [END_REF].

In addition to non-universal scaling laws, the spinodal criticality is associated to a few universal signatures. In particular, a square root singularity of the Edwards-Anderson parameter directly follows [START_REF] Götze | Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory[END_REF]30], which is dynamically accessible as the plateau height in time correlation functions or the typical cage size in particle displacements. Treating fluctuations beyond meanfield leads also to universal predictions regarding the behaviour of four-point susceptibilities [25,31,32]. Yet, because of the computational difficulty of equilibrating liquids beyond the avoided dynamical transition, and of the lack of experimental methods to screen out activated processes, it remains difficult to assess signatures of the spinodal, even in simple glass-forming liquids. As a result, the validity of the square root singularity remains a debated issue [33].

In this work, we exploit a recent implementation of swap Monte Carlo (SWAP) for continuously polydisperse systems, which bypasses the sluggishness associated with approaching the avoided dynamical transition [34][35][36], in order to probe the spinodal criticality beyond the MCT crossover in equilibrium conditions. By carefully controlling for activated processes, our analysis manages to extract a sufficiently long scaling regime of the typical cage size to estimate effective critical exponents and the putative presence of the square-root singularity across several space dimensions from the experimentally relevant d = 3, where we conclude that the singularity is considerably softened, up to d = 9 where a nearly perfect square-root scaling can be convincingly observed.

II. SIMULATION DETAILS

We consider a continuously polydisperse system of N hard spheres under periodic boundary conditions in a simulation box of constant volume V . A hypercubic box is used in d = 3, 4, 5, 6, and 8, while in d = 7 and in d = 9 we use the Wigner-Seitz cell of the checkerboard lattice that decreases the number of simulated particles while preserving the same effective box size. The particle-size distribution is P (σ) = K/σ 3 , with normalization constant K for diameters σ ∈ {σ max , σ min } where σ max , σ min are the maximum and minimum diameters for a given polydispersity. The average diameter σ sets the unit of length, the degree of polydispersity is defined by the standard deviation of the diameter distribution, and the packing fraction is ϕ = ρ Vd for a number density ρ = N/V and average volume of a d-dimensional hypersphere Vd . The degree of polydispersity is chosen to be the minimum needed for the SWAP efficiency to saturate, i.e., 23% for d = 3, 10% for d = 4 -8, and 8% for d = 9 [36], and suitably optimized SWAP sampling is used to equilibrate initial configurations. This approach ensures fast structural relaxation without crystallization or fractionation. Structural equilibration is notably validated by the complete decay of the self-part of the particle-scale overlap function

Q(t) = 1 N N i=1 Θ(a -|r i (t) -r i (0)|), ( 1
)
where Θ is the Heaviside function and a = 0.3σ is about the typical particle cage size [29,[35][36][37]. The associated structural relaxation time, τ α , is defined as Q(τ α ) = 1/e. From these initial equilibrium configurations, multiple simulations are then run with a purely local Monte Carlo dynamics. This computational scheme achieves equilibration at densities 3-8% above the estimated dynamical transition, depending on d [35,36]. This strategy opens a comfortable regime to study glassy dynamics approaching the mode-coupling crossover from the arrested phase, unavailable to previous computational work.

III. TYPICAL CAGE SIZE

We first consider the size of the typical cage, ∆, which in the MCT and the mean-field description of hard spheres is expected to scale as

∆(ϕ) = ∆d -A d (ϕ -ϕ d ) 1/δ , (2)
with 1/δ = 1/2 and ∆d = ∆(ϕ d ) for densities beyond the dynamical transition, ϕ > ϕ d . In the high-dimensional limit, ∆ could be extracted from the long-time plateau of the mean square displacement (MSD) of an individual particle, ∆ = ∆r 2 i (t) = r 2 G s (r, t)dr, where G s (r, t) is the self-part of the van Hove function. In finite dimensions, two difficulties arise. First, caging is heterogeneous, and hence the full distribution of cages must be considered. Second, a sharp plateau in the MSD can only be identified much beyond the (avoided) dynamical transition, i.e. too far beyond the regime of interest. We consider the second effect first. Because activated processes interfere with the formation of the MSD plateau, the size of the transient cage must be extracted over a finite time window. In order to identify the meanfield-like dynamical caging regime, we rely on the non-Gaussian parameter, α 2 (t) = d d+2 r 4 (t) r 2 (t) 2 -1, as illustrated in Fig. 1. More specifically, the upper bound of the time window is set at 20% of peak non-Gaussianity, 3) for different dimensions. The results for ϕ d are obtained by standard MCT dynamical scaling [26]. The direct evaluation of the typical cage size at the estimated ϕ d validates the value of the fitted quantity. Error bars on the fit parameters are determined from quality of the fit (R 2 ) to Eq. (2).

d
i.e., the maximum of α 2 (t), and the lower bound is set to the short-time plateau of α 2 (t). Because the lower bound is much smaller than the upper bound, results are insensitive to its precise value, while the upper bound only weakly affects the subsequent analysis as long as it is chosen consistently. Note that for packing fractions much above ϕ d (in d = 3, ϕ 0.635), the system does not relax within the simulation timescale (with standard dynamics), which results in a clear plateau in the MSD (Fig. 1). The upper bound is then even tighter. Note also that generalizing the MSD to further suppress the contribution of activated processes as in Ref. [13] markedly flattens the MSD, but does not quantitatively affect the subsequent analysis (see Appendix A.

Once the timescale for identifying the cage dynamics is set, we can measure the cage size for each particle, ∆ i , and average over samples and particles to define the cage size distribution P (∆ i ). The results in Fig. 2 show that cages generally tighten as density increases in all dimensions, but that fat tails at large displacements persist for all densities. These tails deviate significantly from the log-normal forms reported in some mean-field models [13]. Although relatively little theoretical guidance is available as to what the proper functional form for P (∆ i ) should be [38], our observations suggest that activated processes are not fully eliminated from the MSD analysis. To further sidestep this issue, we use as estima- tor of the typical cage size the mode of the distribution, ∆ = argmax ∆i P (∆ i ), which is much less sensitive to the activated processes that appear in the fat tail of the distribution than the mean cage size, but converges to the same quantity as d → ∞.

In order to assess the critical scaling of the typical cage size, ∆(ϕ) is fitted to Eq. ( 2) using δ, ∆d and A d as parameters, while ϕ d is obtained independently from the growth of the relaxation time τ α (ϕ) [36] (Fig. 3, Ta-bleI). Because of the uncertainty on ϕ d , the cage size at ϕ d , is not directly measured; but the fitted value is consistent with the direct estimate, which validates our approach. The values of the fit parameters and direct measurements are listed in Table I. Note that both the fit error on ∆ d at a given ϕ d and the propagated uncertainty from ϕ d are then included. In contrast to earlier (cruder) estimates [39], we find that for a given polydispersity, ∆d decreases monotonically with increasing dimension [30]. More significantly, we also find that 1/δ decreases monotonically, and nearly linearly, as dimension increases, from 0.74(9) in d = 3 down to values numerically indistinguishable from the mean-field prediction, 1/δ = 1/2, around d ≥ 8. Although the precise numerical estimates of δ are fairly robust to the details of the above analysis, they may still be fragile to the overall scheme. The detection of the upper critical dimension in the vicinity of d = 8, and the systematic softening of the square-root singularity below d u = 8 are nonetheless numerically robust (see Appendix A and B). Interestingly, the latter is in sharp contrast with the prediction from dimensional reduction [16,18] .

IV. CONCLUSION

The finite-dimensional vestige of the spinodal criticality associated with the dynamical transition of glass forming liquids has here been characterized by numerical simulations using the SWAP algorithm and a careful screening of activated processes across a broad range of spatial dimensions. Our simulations reveal that the square-root singularity, often used to describe experimental measurements in molecular glass formers, is dramatically softened in d = 3 for hard spheres, a canonical model for testing MCT predictions. The measured effective exponent, 1/δ ≈ 0.75, can still be considered as indirect evidence for an underlying avoided singularity, because 1/δ = 1 would be trivially expected for a featureless evolution of the cage size. Further, the slow variation of 1/δ towards 1/2 with spatial dimension d suggests that strong deviations from mean-field criticality exist even in large spatial dimensions. It takes simulations in dimensions d ≥ 8 to observe direct signatures of the squareroot scaling that underlies the mean-field dynamical glass transition. The agreement between our results with perturbative approaches is nevertheless surprising, given the expected role of nonperturbative physics in the avoidance of the dynamical glass transition [20], which can in prin- ciple persist even above d = 8. A possible explanation might be the relative insensitivity of our specific estimators to these effects.

Because this spinodal critical point is part of a broad universality class [20], we expect our results to apply to a variety of other systems, for which the interplay between activation and criticality might be harder to control. Most crucially, these results further motivate the use of the dynamical criticality and of the mean-field description in describing the behavior of finite-dimensional liquid glass formers. It was recently shown that deviations from the dynamic glass transition can be studied considering the degree of localization of unstable modes in the potential energy landscape [40]. Localized excitations are indeed expected to disappear as d increases, and our study thus also motivates verifying this prediction in larger dimensions. where S d-1 (r) = 2π d/2 r d-1 /Γ(d/2) is the surface area of a d-dimensional hypersphere, and G s (r, t) is the self-part of the van Hove function. Note, however, that a direct use of this expression is numerically impractical, because it requires perfect sampling around the maximum of the van Hove function. In practice, we thus consider the convergence of the analysis upon reducing n from the standard MSD definition with n = 1. Figure 4shows that reducing n markedly enhances plateau formation, as expected. Thus, for smaller n, the cage size distribution can be measured over a longer time window, which is particularly helpful as ϕ approaches ϕ d . For instance, with n = 0.07 the upper bound of the time window can be set at 25% (instead of 20%) of the peak position of α 2 (t). The critical exponent extracted from fitting results for different values of n to Eq. 2 is robust. As an illustration, results for d = 4 are reported in Fig. 5.

Appendix B: Finite-size Effects

To ensure the robustness of the critical scaling with system size, we consider the cage size distributions at three different densities above ϕ d for two system sizes in Fig. 6. The typical cage size or the mode of the distributions at a given density are here indistinguishable. The critical scaling of the typical cage is thus insensitive to the system size, in the regime considered here.

FIG. 2 .

 2 FIG. 2. Cage size distributions, P (∆i), in (a) d 3 for ϕ = 0.6032, 0.6111, 0.6208, 0.6309, and 0.6414 and (b) d = 6 for ϕ = 0.1810, 0.1829, 0.1842, 0.1865, 0.1892, and 0.1916. Note that ϕ d = 0.600(2) and 0.1808(8) in d = 3 and 6 respectively. Fat tails at large displacements persist over the whole density regime accessible in simulations. The estimator ∆ = argmax ∆ i P (∆i) (vertical dashed lines) nonetheless monotonically shifts to smaller values as ϕ increases.

FIG. 3 .

 3 FIG. 3. Critical scaling of the typical cage size in d = 3 . . 9 as well as (inset) fitted power-law exponents, 1/δ. For visual clarity, data is shifted by a factor of 3 d-3 . Error bars reflect the measurement uncertainty of ∆(ϕ) only with ∆ d and ϕ d here chosen to optimize the quality (R 2 ) of the fit. Inset: the critical exponent 1/δ versus d. A clear change in critical scaling is seen around the upper critical dimension, du = 8, predicted from perturbative approaches. For d > du, the results are consistent with the mean-field prediction (1/δ = 1/2, dashed line); for d < du the deviation from the mean-field result is approximately linear 1/δ-1/2 = B(du-d) with B = 0.049(3).

1 FIG. 5 .FIG. 6 .

 156 FIG. 5. Critical scaling of the typical cage size in d = 4 calculated from the generalized MSD with packing fraction for different values of n. Inset: the critical exponent δ is robust against the choice of n. All data are for N = 1000.

TABLE I .

 I Fit parameters A d and ∆d for Eq. (

		3	4	5	6	7	8	9
	ϕ d	0.600(2) 0.410(2) 0.277(1) 0.1808(8) 0.1147(6) 0.0716(3) 0.0426(2)
	∆d	0.015(2) 0.021(2) 0.020(2) 0.018(1) 0.016(1) 0.010(1) 0.013(1)
	∆(ϕ d ) 0.0148(5) 0.0208(5) 0.0196(5) 0.0175(6) 0.0161(6) 0.0101(4) 0.0129(5)
	1/δ	0.74(9) 0.68(9) 0.65(8) 0.61(9) 0.59(9) 0.53(9)	0.55(10)
	A d	0.11(2) 0.16(2) 0.15(2) 0.12(2) 0.11(2) 0.04(1)	0.09(3)

ACKNOWLEDGMENTS

We thank G. Biroli, G. Carra, S. Franz, G. Tarjus, and F. Zamponi for stimulating discussions, as well as Tom Milledge for technical support. We acknowledge support from the Simons Foundation grant (#454933, LB, # 454937, PC). Simulations were performed at Duke Compute Cluster (DCC), and at the Extreme Science and Engineering Discovery Environment (XSEDE), supported by National Science Foundation grant number ACI-1548562.

Appendix A: Generalized mean square displacement

In order to further suppress the contribution of activated processes, we can generalize the definition of the mean square displacement as in Ref. [13] ∆r 2 (t) = lim n→0 r 2n G s (r, t)S d-1 (r)dr


Jean-Jacques Lemaire
email: jjlemaire@chu-clermontferrand.fr.

Bénédicte Pontier

Rémi Chaix

Youssef El Ouadih

Toufic Khalil

Denis Sinardet

Viorel Achim

Adina Postelnicu

Jerome Coste

Véronique Germain

Youssef El

Jérôme Coste

Catherine Sarret

Anna Sontheimer

Neural Correlates Of Consciousness And Related Disorders: From Phenotypic Descriptors Of Behavioral And Relative Consciousness To Cortico-Subcortical Circuitry

Keywords: consciousness, disorders of consciousness, fMRI, default mode network, behavior

published or not. The documents may come

RÉSUMÉ

Nous rapportons une revue des aspects médicaux de la conscience. La dimension comportementale, des descripteurs phénotypiques, la conscience relative et les corrélats neuronaux de la conscience et de ses troubles ont été abordés successivement selon une approche holistique et chronologique. La conscience est relative, spécifique à chacun, dans le temps et l'espace. Historiquement définie comme la perception de soi et de l'environnement, elle ne peut être séparée des comportements, ce qui implique la notion de comportements conscients avec des aspects métapraxiques et métagnosiques. L'observation de comportements spontanés ou provoqués permet de définir trois types de troubles de la conscience, le coma, l'état végétatif ou éveillé non-répondant, et l'état de conscience minimale ou pauci-relationnel. Des techniques modernes d'exploration fonctionnelle, comme l'imagerie, améliorent la compréhension de la conscience et de ses troubles. Le fait que la conscience soit une fonction supérieure et/ou une fonction instrumentale est discuté. Les corrélats neuronaux peuvent se subdiviser selon deux voies de l'éveil, la voie supérieure cholinergique thalamique et la voie inférieure extra-thalamique, et en une circuiterie cortico-sous-corticale. Les structures du cerveau profond sont celles décrites dans les boucles, bien connues, sensorimotrices, associatives et limbiques, comme illustré dans le modèle mésolimbique des troubles de la conscience. Les cortex peuvent être distingués selon leur appartenance à plusieurs réseaux se chevauchant : [START_REF] Graziano | Opinion | Are We Really Conscious? The New York Times[END_REF] un réseau global incluant des boucles thalamo-corticales ; (2) le réseau du mode par défaut et les réseaux intrinsèques liés, soit le central exécutif, le réseau par défaut médial et la saillance; (3) un triple model de réseau avec le système fronto-pariétal de contrôle et ses sous réseaux dorsal et ventral attentionnels, le réseau fronto-pariétal exécutif de contrôle et le réseau cingulo-operculaire de saillance ; (4) les cortex internes et externes, respectivement médiaux tournés vers le soi, et latéraux tournés vers l'environnement. La dynamique des réseaux est le reflet de la conscience, notamment les anticorrélations comme la baisse d'activité de la région cingulaire postérieure et de celle précunéale durant les tâches attentionnelles. Grâce aux avancées récentes sur la physiopathologie des troubles de la conscience, de futurs progrès thérapeutiques significatifs, adaptés au contexte sociétal, sont espérés. Ceci dépend notamment de la divulgation de la connaissance médicale et de son transfert au grand public.

INTRODUCTION

We assume that we are conscious, merely as a very rough attempt as the concept stems from humankind, and despite the fact that it raises questions [START_REF] Graziano | Opinion | Are We Really Conscious? The New York Times[END_REF]. Meta aspects of consciousness related to religion and advanced physics (e.g. brane theory) were not deemed. We also esteem that we are conscious of a part of the universe which includes our own internal universe or self. This implies the relativity of consciousness, being aware of or knowing, basic to cosmological dimensions, whether acquired or possibly innate. Basically, we focus on the medical implications of related disorders of consciousness (DOCs). Does it matter? It is an issue per se, which has not been developed. In any event, this objective is tackled by way of a condensed review starting from the behavioral dimension of consciousness, then phenotypic descriptors of consciousness followed by the relativistic dimension of consciousness, and finally the underlying cortico-subcortical circuitry. Our Ariane's thread was the acceptance that new treatments and practices will emerge from better knowledge, while being aware that this approach brings into play ethical, philosophical and societal dimensions, as recently illustrated by the case of Mr. Vincent Lambert [START_REF] Vincent | Lambert -Une mort exemplaire[END_REF]. Nevertheless, we do not discuss the quality of life in the context of severe DOCs, inferred by relatives and direct care givers, as well as by individuals and groups indirectly through the media, and possibly by the patient. This issue under debate is all the more complex as we currently have very limited understanding of what dysfunctional consciousness is, notably concerning the quality and quantity of content and access of and to consciousness, from the patient's point of view. Definitively, we have a certain will of rationality, "…a conviction, akin to religious feeling, of the rationality or intelligibility of the world lies behind all scientific work…." (Einstein, 1927, On the Scientific truth; in [START_REF] Kaiser | Science-Fostering Belief-Then and Now[END_REF]), for better patient care.

THE BEHAVIORAL DIMENSION OF CONSCIOUSNESS

There is no consciousness without behavior, and conversely, assuming that cognition is a prerequisite. This captures the behavioral dimension of the consciousness. De facto the consciousness was always implicitly related to behavior, as it was linked to experiencing and reportability of content (see hereafter; see also e.g. [START_REF] Dehaene | Experimental and theoretical approaches to conscious processing[END_REF]. We must assume that spontaneous unanticipated reflexive and instinctive aspects are part of non-cognitive, unconscious features, akin to unsupervised loops, which are out of our scope (figure 1). However, some of these features can reach the level of consciousness with the development of purposeful behaviors (see below). Consciousness, which has obviously not yet been fully elucidated, is processed by the cortices and deep brain structures, fed by internal and external stimuli, and generates overt and covert purposeful behaviors. Perception of the self and environment, which classically defines consciousness, particularly in medicine, is a behavior. On the other hand, we must accept that the distinction between cognitive behaviors and pure reflexes or unconscious actions sometimes leads to medical questions and that the existence of a continuum is an acceptable interim hypothesis.

Figure 1: Phenotypic descriptors of consciousness (shades of orange), from left (no consciousness; 0%) to right (full consciousness, 100%). Consciousness is cognitive and purposeful, and observable clinically (overt) or instrumentally (covert) through behaviors. The reflexes, such as stretching or instinct, do not rely on full consciousness, they use unsupervised (not involved in cognition) loops, however some can reach consciousness and then participate in purposeful behavior. Conscious behavior uses metagnosic and metapraxic resources linked by the same purpose.

Pragmatically, genuinely conscious behaviors are behaviors that are thought or minded. The gnosis and the praxis, as medical constructs, have been established since the beginning of the 20th century (e.g. [START_REF] De Ajuriaguerra | Le cortex cérébral: étude neuro-pshycho-pathologique[END_REF]), and enable specifying two dimensions of consciousness, recognition and action, respectively. Historically, they were formulated from the observation of disturbed motor activities resulting from impaired cognitive behaviors, the agnosia and the apraxia. Therefore, the integration of praxic and gnosic dimensions of cognitive, covert activities in consciousness were at the back but benefit increasingly from the development of advanced techniques such functional magnetic resonance imaging (fMRI, see hereafter). Although the perimeters of praxis and gnosis are still not fixed (see e.g. [START_REF] Cubelli | Definition: Apraxia[END_REF][START_REF] Kumar | Agnosia. StatPearls, Treasure Island[END_REF][START_REF] Petreska | Apraxia: a review[END_REF]), and although the distinction has been called into question [START_REF] De Ajuriaguerra | Le cortex cérébral: étude neuro-pshycho-pathologique[END_REF], we use them to specify dimensions of consciousness, while being aware of a certain vagueness. Gnosis refers to semantic recognition on the basis of the primary information received by the sensory and somesthetic cortices, and on internal sources of visceral origin, for example. The gnosis of thoughts should complete this description cogently, as the direct exploration of the neural correlates of thoughts has begun [START_REF] Karapanagiotidis | Tracking thoughts: Exploring the neural architecture of mental time travel during mind-wandering[END_REF][START_REF] Rosenbaum | Neural correlates of mindful emotion regulation in high and low ruminators[END_REF]. Praxis refers to goal-directed sequences of actions, acquired, overt, logically assembled, observable through spontaneous and provoked motor sequences. Each praxis covers a unique and limited goal, as an elementary unit or brick enabling, by assemblage, the achievement of complex tasks under full cognition. In line with the gnosis of thoughts, we assume that the praxis of thoughts will further complete this definition, as covert, non-clinically observable, elementary, logically assembled, purposedirected sequences of thoughts. Nevertheless, the place of gnosic and praxic processes in cognitive, covert, activity has not been determined. We propose that the metagnosic and metapraxic components of consciousness, in concomitance, are temporary concepts, working together purposefully and framing the behavioral dimension of consciousness. Metagnosis and metapraxis are the higher levels of recognition and action, belonging to conscious behaviors and hence cognitive. They superpose gnosis and praxis, which, conversely, taking into account clinical observations of agnosia and apraxia, are unconscious as instrumental means, and observable as pieces of conscious behaviors. Attention should be interpreted as a metagnosic behavior, which can be directed internally or externally. When attention is focused or goal-directed, it is part of a top-down process that awaits an item of information such as visual or auditive or visceral; it is a typical conscious behavior. When the mind is wandering internally (see the default mode network below) or externally (looking at clouds or listening to relaxing sounds) attention is not goal-directed, it is blurred. However, physiologically this wandering-mind state is voluntary and belongs to conscious behaviors. Any salient events can prompt conscious behaviors, such as sensorial triggers (see the attentional cortices below), and quite likely, thoughts. Consequently, physiologically, conscious behaviors are on the top, switching attention between focus and unfocused states. Language belongs to conscious behaviors and enables, among other things, spreading and manipulating knowledge, besides intra and inter individual interactions. From clinical studies we have learned that hemispheric language specialization or dominance is leftward lateralized, as is the well-known Gerstmann's syndrome observed following left parietal posterior lesions that mix agnosia and apraxia features. However, instead of a global hemispheric dominance suggested from historical studies, each hemisphere, cortices and basal ganglia, could play a specific role in different praxic aspects [START_REF] Petreska | Apraxia: a review[END_REF][START_REF] Gross | Update on Apraxia[END_REF], such as vestibular controls under the supervision of right parietal and thalamic regions [START_REF] Dieterich | Right-sided dominance of the bilateral vestibular system in the upper brainstem and thalamus[END_REF][START_REF] Mcassey | Different EEG brain activity in right and left handers during visually induced self-motion perception[END_REF].

PHENOTYPIC DESCRIPTORS

In the general population, consciousness refers to the state of being aware or conscious, an intuitive or reflexive knowledge, the fact of being responsive, of perceiving, of itself and the world, and a higher level of mental life [14-16]. The Latin roots of the word [START_REF] Quicherat | Dictionnaire LATIN-FRANCAIS. 46ème[END_REF] point to a dual conception of (1) knowledge-awareness, of self, and sharing with others; and (2) of guilt and judgement. These aspects are mostly metagnosic, metapraxic, and cognitive. The perceptive dimension of known information either external or internal, also points to the notion of unknown information (see below). The judgment of behaviors, as illustrated by Hugo [START_REF] Hugo | La Légende des siècles: la conscience[END_REF] with the symbolic representation of guilty consciousness by an eye "L'oeil était dans la tombe et regardait Caïn" (The eye was in the tomb and looked at Cain), and mental life such as mind training [START_REF] Barinaga | Studying the well-trained mind[END_REF], is outside the scope of this article. Starting with Plum and Posner [START_REF] Plum | The diagnosis of stupor and coma[END_REF], the role of the cortex and subcortical activating systems, such as those of the brainstem and thalamus regions, was identified. Considering that the human brain is above all a behavior machine (see hereinabove), overt behaviors are manifest and detectable on the ground, whereas covert behaviors, if there is no movement or language, are objectively accessible only through functional explorations, such as fMRI protocols performed at rest or during tasks. At the bedside, consciousness was defined notably as the perception of the self and the environment [START_REF] Plum | The diagnosis of stupor and coma[END_REF]. The self includes the thoughts illustrated by "Cogito, ergo sum" (I think therefore I am) of Descartes [START_REF] Descartes | Discours de la méthode[END_REF], and the environment in its wide meaning which includes others. It is noticeable, as shown by the "Cogito, ergo sum", that cognitive, metagnosic and metapraxic aspects are present, through covert perception and recognition (of thought), and covert (cognitive; an inference) and overt (language; writing) behaviors.

Thus, the medical definition of consciousness is restrictive, almost exclusively perceptionmetagnosis oriented. Whatever the case, the medical perception of the self and the environment, supported by biological phenomena, is merely an extension of the general historical definitions. This is probably due to the fact that the focus then was essentially on wakefulness, with related behaviors remaining in the background. Yet, intuitively and objectively, as already mentioned, behaviors are systematically associated with awareness. Conscious behaviors are, however, not always easy to define at the bedside, particularly in the case of limited or inconsistent communication (see DOC hereinafter). Behaviors are often qualified as voluntary, meaning that the subject wants to display them purposefully, but the purpose is not always obvious to the observer. For instance, if the observed subjects cannot explain their behavior by any means, or if they suffer from confusion of whatever cause, the observer can miss the purpose. This is dramatic for locked-in patients if no diagnosis of preserved consciousness is made. The difference between voluntary and involuntary behaviors or actions is also more subtle when intention and necessity are taken into account in parallel with philosophical considerations [START_REF] Fulton | Physiology of the nervous system[END_REF]. Interestingly, adjectives such as conscious or unconscious or voluntary and involuntary, are used indifferently to describe behaviors and stimuli. So, the number of combinations of adjectiveword that describe the circumstances where consciousness is considered increases dramatically, sometimes leading to awkwardness. For instance, the perception of a spontaneous automatic neurovegetative involuntary acceleration of the heartbeat, e.g. following an involuntary arrhythmia or a voluntary movement or an instinctive fear, can generate a covert or overt conscious behavior, such as mental stress or alertness with reflexive neurovegetative modifications. This is particularly true in this case if the subject is aware of a personal cardiovascular disease. The second example comes from epilepsy, where spontaneous, involuntary, non-conscious neuronal discharges, trigger a loss of consciousness or consciousness-like behaviors qualified as automatic, with emotional dimensions, at least from the observer's viewpoint. The third example is the condition of subliminal stimuli and conscious behaviors. They have been described experimentally, and hence not truly spontaneous, but the authors [START_REF] Dehaene | Conscious, preconscious, and subliminal processing: a testable taxonomy[END_REF] showed that conscious behaviors can react to so-called subliminal or preconscious stimuli, under the modulation of attention. In the current perceptive clinical understanding of consciousness, the behaviors observed cover a wide range of situations, from simple, often qualified as reflex or automatic behaviors, to sophisticated, such as so-called executive or cognitive behaviors. The words, "reflex", "automatic" and "involuntary" are often used indifferently. That being so, roughly speaking, automatic behaviors should use little cognition, if any at all, and cognitive behaviors belong to superior or higher functions or high cognition, also referred to as intelligence, either overt or covert. The superior or higher brain functions cover a wide range of concepts in neurology, psychology and sociology [START_REF] Luria | Higher Cortical Functions in Man[END_REF]. In contemporary neurology and neuropsychology, studied at the university and in the hospital, the so-called superior functions cover language, praxis and gnosis, and cognitivebehavioral dimensions, such as the executive and intellectual functions and memory (see e.g. [START_REF] Eusop-Roussel | Évaluation des fonctions supérieures[END_REF]). Language, praxis and gnosis are, in this case, often called instrumental functions, considered as means enabling cognition. Interestingly, in this case, the related dysfunctions, aphasia, agnosia and apraxia, could be interpreted as consciousness disorders, as the patient can be unaware of certain aspects of the self and the environment, because of neglect or misunderstanding. Thus, whether consciousness should be understood as a global phenomenon, or specialized consciousnesses, such as knowledgeawareness of different parts of space or the body or mind resources, or as different functions such as language, dressing and walking, has not been settled. In the general population, cognition refers to the acquisition of knowledge and understanding through thought, experience, and the senses, and intellectual activity, which may be empirical [14][15][16]. The Latin roots illustrate the notions of knowledge and instruction (of something, an investigation) [START_REF] Quicherat | Dictionnaire LATIN-FRANCAIS. 46ème[END_REF]. Several theories describe different aspects of consciousness (see e.g. [START_REF] Gulick | Consciousness[END_REF]), including the neural approach whose aim is to decipher the neural correlates. In parallel, the cognitive nature of consciousness is always a subject of debate in psychology [START_REF] Brown | Consciousness doesn't overflow cognition[END_REF]. Nevertheless, the so-called "hard problem" [START_REF] Chalmers | The Puzzle of Conscious Experience[END_REF], i.e. how the brain generates a conscious experience, remains. From the above objectives the hard problem can be raised to the same level as cognition and related concepts. Experience refers to perception and thought, engaging cognition, which includes phenomenological consciousness related to the senses and immediate experience. A developmental and evolutionary approach inferred two aspects of consciousness [START_REF] Fabbro | Evolutionary aspects of selfand world consciousness in vertebrates[END_REF][START_REF] Vandekerckhove | The flow of anoetic to noetic and autonoetic consciousness: A vision of unknowing (anoetic) and knowing (noetic) consciousness in the remembrance of things past and imagined futures[END_REF]: anoetic consciousness, receptiveness or perceptual, primitive, or unknowing consciousness; the noetic and autonoetic, knowing consciousnesses, of a higher level, where noetic consciousness classifies information and autonoetic consciousness relates to the recall of autobiographical memories and one's projection of the self into the future. It is noticeable that humanconsciousness-like behaviors, which could be called experience by humans, are known in other species, such as emblematic ones in apes [START_REF] Martin-Ordas | Sequential tool use in great apes[END_REF], and little-known ones in birds [START_REF] Wimpenny | Cognitive processes associated with sequential tool use in New Caledonian crows[END_REF], and are most commonly encountered with pets. Each living system could have a relative consciousness (see hereinafter) according to its own experienced universe. Genuine conscious behaviors or features, mimicking conscious behaviors of humans, should not hide the difficulty of distancing ourselves from anthropocentric considerations. Whatever the case, hierarchical considerations on the consciousnesses of living systems are outside the scope of this article. Indeed, human consciousness could be specified as that which purposefully self-examines and takes into account others, over large time periods up to a lifespan and beyond. Beyond human consciousness could be, conceivably, the only one, if humankind is unique in the universe. On the basis of all the above-mentioned elements, we can reasonably infer that consciousness relies on specific stimuli and cognitive-motor processes that purposefully interact with the self and the environment. This definition partially overlaps with that of autopoiesis and the cognition of systems, which together define a living system [START_REF] Bourgine | Autopoiesis and cognition[END_REF]. Memory takes place in the consciousness process as an internal reference library. Experimental data show that the storage is distributed in the neocortices [START_REF] Wiltgen | Systems consolidation and the content of memory[END_REF] and that the prefrontal cortices are actively involved [START_REF] Tang | Minimally dependent activity subspaces for working memory and motor preparation in the lateral prefrontal cortex[END_REF], whereas memory elements and ensembles are processed (in and out) within the hippocampal formation [START_REF] Masse | Circuit mechanisms for the maintenance and manipulation of information in working memory[END_REF][START_REF] Matsuo | Irreplaceability of Neuronal Ensembles after Memory Allocation[END_REF]. Memory could be prefilled by innate references and further completed by following experiences, including education and training. However, there is no evidence of such an innate brain memory library. We observe only that the temporospatial development of the brain (neurons and myelin), at least genetically and epigenetically determined, enables the implementation of consciousness. The early development over the embryonic and fetal periods, could be the support of a transition from anoetic to noetic consciousness, but this raises and exacerbates the "hard problem". Other questions are raised by the irruption of memory recalls, whether passively or actively, voluntarily or involuntarily, following conscious or unconscious physiological or pathological events, and the vague field of knowledge of feelings of persons or situations. This is especially the case since it has been shown experimentally that false memory can be created in the hippocampus [START_REF] Ramirez | Creating a False Memory in the Hippocampus[END_REF]. This brings us to the frontiers of physiology, such as flashbacks and reviviscence, and of pathology, such as confusion, ghost limbs and hallucinations during neurologic and psychiatric diseases and syndromes, and of artificially altered states of consciousness using psychoactive drugs or cognitive approaches (e.g. [START_REF] Barinaga | Studying the well-trained mind[END_REF][START_REF] Winkelman | The Evolution of Consciousness? Transpersonal Theories in Light of Cultural Relativism[END_REF]. Although all these dimensions are linked with consciousness and rely on neural support, they are outside the scope of our topic.

RELATIVE CONSCIOUSNESS AND ITS SPAN

The relativistic aspects of consciousness englobe consciousness in line, in one way or another, with the relativistic approach of cosmology. This relative consciousness expands the acquired consciousness of the universe and the self, from individual to populational aspects, as well as beyond the experienced universe (figure 2). Practically, relative consciousness expresses both the relativity of its content and the relativity of the examination of its content according to the knowledge and cognition of the observer, who can be the self or the other. At the same time, on an individual basis, the span of relative consciousness increases from a limited, primary consciousness, assimilable with anoetic consciousness through raw processes (figure 2). This anoetic consciousness is described as unknowing consciousness [START_REF] Vandekerckhove | The flow of anoetic to noetic and autonoetic consciousness: A vision of unknowing (anoetic) and knowing (noetic) consciousness in the remembrance of things past and imagined futures[END_REF]. It implies that we should consider an intermediate state, a pre-consciousness, enabling primary pre-consciousness behaviors without knowing, which brings us back to the notion of necessity, such as feeding. Hence relative consciousness is above all an acquired consciousness, fully operational as it reaches cognition, and which can be considered as noetic. Relative consciousness involves all consciousness related meta dimensions, metapraxis and metagnosis up to metacognition. The span of relative consciousness increases by training in all types of language, such as spoken, written, drawing and body languages, and hypothetically by genetics, and through individual and populational perspectives.

Figure 2: The relative consciousness of an individual (beige) within the extended universe (cosmological space-time) expresses the relativity of its content at a certain location and at a certain moment, accounting for before (back through the past) and after (forecast, into the future), over the life span (from birth, possibly also prenatal, to death). The span of relative consciousness reaches a maximum noetic dimension during life. It starts from a limited, primary state, with raw processes some of which could be innate. Fully operational relative consciousness relies on cognitive, purposeful, behaviors. Relative consciousness also expresses the relativity of the examination by the observer, who can be the self or the other.

NEURAL CORRELATES OF CONSCIOUSNESS AND DOC

We report neural correlates of human consciousness from stimulus processing to different cognitive processes, from the basic level, such as alertness, to a high level, such as intellectual processes. Data obtained from studies of DOCs are included, providing indirect insights into consciousness: coma, with no signs of consciousness and no eye-opening; vegetative state (VS), also named unresponsive wakeful syndrome (UWS) [START_REF] Laureys | Unresponsive wakefulness syndrome: a new name for the vegetative state or apallic syndrome[END_REF], with eye-opening and no consciousness behavior observed; minimally conscious state (MCS) [START_REF] Giacino | The minimally conscious state: definition and diagnostic criteria[END_REF], with inconstantly observed consciousness behaviors; MCS + and MCSare subcategories that depend on the complexity of observed behaviors [START_REF] Bruno | From unresponsive wakefulness to minimally conscious PLUS and functional locked-in syndromes: recent advances in our understanding of disorders of consciousness[END_REF][START_REF]Prolonged disorders of consciousness following sudden onset brain injury[END_REF]. These diagnoses are exclusively based on clinical observations, but we now know that a covert consciousness-like cerebral activity, revealed by fMRI, remains in the population of DOC patients (e.g. [START_REF] Laureys | Coma and consciousness: paradigms (re)framed by neuroimaging[END_REF]). The mismatch between two, covert and overt, consciousness-like activities was well exemplified in covertly aware vegetative patients [START_REF] Fernández-Espejo | A Thalamocortical Mechanism for the Absence of Overt Motor Behavior in Covertly Aware Patients[END_REF], further reported as cognitive motor dissociation [START_REF] Schiff | Cognitive Motor Dissociation Following Severe Brain Injuries[END_REF]. However this apparent mismatch is relative to the observer point of view and to the definition of DOC, and depends on the technique of brain function analysis. For instance emotion, which facilitates overt behaviors at bedside when relatives are present, can be used to reveal covert behaviors [START_REF] Sontheimer | fMRI study of graduated emotional charge for detection of covert activity using passive listening to narratives[END_REF]. Hence the concept of mismatch must be used very cautiously in clinics accounting the relativity of consciousness (see above), and the technical-knowledge dependence of the interpretation of clinical picture. One might suppose that historically locked-in patients could have been the very firsts with apparent cognitive motor dissociation. This raises numerous questions about the specificity and selectivity of the information collected at the bedside, and even for the information provided by older techniques such as electroencephalography (EEG) and evoked potentials. At the same time, this recent knowledge will very likely modify the classifications of DOCs, improving diagnoses, prognoses and treatments, including the cessation of treatments. Indeed, Naccache [START_REF] Naccache | Minimally conscious state or cortically mediated state?[END_REF] recently proposed to introduce new types of evidence from EEG, fMRI and Pet-Scan, in addition to clinical examination, to classify the impairments of consciousness. Hence, we see that we move slightly away from clinical to combined instrumental-clinical diagnosis. The notion of DOC, the pathologic dimension of consciousness in medicine, requires a brief introduction to the particularity of sleep and anesthesia. Indeed, although physiological with a circadian cycle, the example of sleep is often presented to the general public as an analogy of coma. This is also true of anesthesia, which is neither physiological nor a kind of sleep, though experimental studies of DOC have used anesthetic models as substitutes of true clinical pathophysiologic conditions, while exploring the mechanisms of anesthesia [START_REF] Macdonald | Anesthesia and neuroimaging: investigating the neural correlates of unconsciousness[END_REF]. The overt status of sleep and anesthesia illustrates the impact of observing the closing of eyes by relatives, who are often confused by the fact that the eyes are always closed in comatose patients, whereas they open, cyclically when waking up in normal life, or systematically after general anesthesia. Indeed, if a DOC patient opens their eyes it does not mean that the patient has woken up or recovered consciousness, but merely that certain brain functions restart, following a complex process of functional and structural reorganizations. The spontaneous dynamics of recovery from coma to VS and to MCS, and then the evolution toward disability or full recovery, dramatically complexifies the general public's understanding of DOC, especially since the average timeline of spontaneous progression is not always clearly discussed with the patient's relatives, and depends on the specific clinical context. Nowadays, the following terms should be recommended [START_REF]Prolonged disorders of consciousness following sudden onset brain injury[END_REF]: [START_REF] Graziano | Opinion | Are We Really Conscious? The New York Times[END_REF] prolonged -continuing DOC if the clinical state lasts more than one month; (2) chronic, after 3 months following anoxic or metabolic injury, and after 12 months following trauma for more severe patients, i.e. VS and MCS-, besides which 6 additional months must be added for less severe cases or MCS+; (3) permanent after 6 months of stable chronic conditions assessed regularly. Frequent DOC clinical pictures, following stroke and tumor-related (lesion and treatment) brain injury, are neglected and fall in-between anoxic-metabolic and traumatic conditions. They are also three other very common clinical pictures of DOC in addition to the DOCs described above, in the absence of psychiatric symptoms that could lead to biases: spatial or temporal disorientation; obnubilation (no reaction in this context, but "cloudy" or "obscure" in the literature); confusion, a dreamlike experience often associated with fluctuations of attention and vigilance (transitory spontaneous eye-closing). Obnubilation could be interpreted as a transitory lockdown of the attentional state often accompanied by an interruption of actions, notably mutism, such as that observed following concussion. As mentioned above, the modern medical dimension of consciousness was first considered through the prism of wakefulness, and at the bedside the absence of wakefulness was named coma (from Greek, "κώμα", meaning deep, prolonged sleep [START_REF] Bailly | Dictionnaire Grec -Français[END_REF]). Wakefulness is controlled by two main routes that originate within the brainstem, in the ascending reticular activation system, whose electric stimulation effects were described as early as 1949 by Moruzzi and Magoun [START_REF] Moruzzi | Brain stem reticular formation and activation of the EEG[END_REF]. The dorsal route or superior thalamic cholinergic pathway reaches the thalamus, and the ventral route or inferior serotoninergic-noradrenergic basal-forebrain pathway, or extra thalamic pathway, reaches the hypothalamus and forebrain (see [START_REF] Lemaire | The Brainstem and Cerebral Activation[END_REF] for a review). Acute lesions, either hemorrhagic or ischemic, of the upper brainstem are prone to induce DOC and particularly coma [START_REF] Plum | The diagnosis of stupor and coma[END_REF][START_REF] Duret | A rare and little known hemorrhagic syndrome][END_REF]. The electrical stimulation of the upper tegmental region, and possibly of the cuneiform nucleus in severe DOC patients has shown that frequencies of 50 Hz and 250 Hz can improve consciousness [START_REF] Mclardy | Attempted inset-electrodes-arousal from traumatic coma: neuropathological findings[END_REF][START_REF] Tsubokawa | Deep-brain stimulation in a persistent vegetative state: follow-up results and criteria for selection of candidates[END_REF]. It has be shown that vagus nerve stimulation that modulates tegmental and thalamic activities in particular, can also improve DOC signs (e.g. [START_REF] Corazzol | Restoring consciousness with vagus nerve stimulation[END_REF]). The thalamus is the emblematic deep-brain structure involved in consciousness. The nuclei have been placed in specific groups directly involved in sensory-motor activity, unspecific groups [START_REF] Llinas | Temporal binding via cortical coincidence detection of specific and nonspecific thalamocortical inputs: a voltage-dependent dye-imaging study in mouse brain slices[END_REF][START_REF] Van Der Werf | The intralaminar and midline nuclei of the thalamus. Anatomical and functional evidence for participation in processes of arousal and awareness[END_REF], or in principal or relay (=sensorimotor), limbic (anterior, midline, and medial intralaminar nuclei) and connection nuclei groups between the relay and the limbic groups [START_REF] Vertes | Limbic circuitry of the midline thalamus[END_REF]. According to the calbindin and parvalbumin bindings of cells, nuclei have also been segregated into matrixcalbindin and core-parvalbumin groups, which involve the main parts of the central lateral and the parafascicular nuclei, and the centromedian and the reticular nuclei respectively [START_REF] Jones | Viewpoint: the core and matrix of thalamic organization[END_REF][START_REF] Münkle | The distribution of calbindin, calretinin and parvalbumin immunoreactivity in the human thalamus[END_REF]. The notion of central thalamus was highlighted by Schiff [START_REF] Schiff | Central thalamic contributions to arousal regulation and neurological disorders of consciousness[END_REF] and covers the extended central region of the thalamus centered on the intralaminar nuclei and the close neighborhood. Generally, the activation of the cortices can be explained by three mechanisms: the coincidence of the activities of specific and unspecific groups [START_REF] Llinas | Temporal binding via cortical coincidence detection of specific and nonspecific thalamocortical inputs: a voltage-dependent dye-imaging study in mouse brain slices[END_REF]; the gateway to the cortex of stimuli by the control of the reticular thalamus [START_REF] Min | A thalamic reticular networking model of consciousness[END_REF][START_REF] Pinault | The thalamic reticular nucleus: structure, function and concept[END_REF]; the arousal of the cortices by the intralaminar nuclei [START_REF] Schiff | Central thalamic contributions to arousal regulation and neurological disorders of consciousness[END_REF][START_REF] Steriade | Projections of cholinergic and non-cholinergic neurons of the brainstem core to relay and associational thalamic nuclei in the cat and macaque monkey[END_REF][START_REF] Steriade | Cholinergic and non-cholinergic neurons of cat basal forebrain project to reticular and mediodorsal thalamic nuclei[END_REF]. The role of midline and intralaminar nuclei has also been highlighted in the cognitive and executive functions as well as in goal-directed behaviors [START_REF] Haber | The primate basal ganglia: parallel and integrative networks[END_REF][START_REF] Haber | The cortico-basal ganglia integrative network: the role of the thalamus[END_REF]. Nevertheless the precise mechanism of thalamo-cortical functionality is still unsettled [START_REF] Halassa | Thalamo-cortical circuit motifs: a general framework[END_REF]. Electric deep brain stimulation (DBS) of thalamic structures has shown clues and evidence of improved consciousness in severe DOC patients, either during acute low frequency [START_REF] Hassler | Behavioural and EEG arousal induced by stimulation of unspecific projection systems in a patient with post-traumatic apallic syndrome[END_REF] or chronic high and low frequency [START_REF] Tsubokawa | Deep-brain stimulation in a persistent vegetative state: follow-up results and criteria for selection of candidates[END_REF][START_REF] Cohadon | Deep Brain Stimulation in Cases of Prolonged Post-Traumatic Unconsciousness. Neurostimulation: An Overview[END_REF][START_REF] Cohadon | Deep cerebral stimulation in patients with post-traumatic vegetative state. 25 cases[END_REF][START_REF] Lemaire | Deep brain stimulation in five patients with severe disorders of consciousness[END_REF][START_REF] Schiff | Behavioural improvements with thalamic stimulation after severe traumatic brain injury[END_REF][START_REF] Yamamoto | Deep brain stimulation for the treatment of vegetative state[END_REF] conditions. The spontaneous restoration of thalamocortical metabolic connectivity also seems to be linked with recovery from UWS [START_REF] Laureys | Restoration of thalamocortical connectivity after recovery from persistent vegetative state[END_REF]. The role of deep gray nuclei in the consciousness process has been less explored. The striatum was suspected to participate in thalamo-tegmental controls of consciousness [START_REF] Llinas | Temporal binding via cortical coincidence detection of specific and nonspecific thalamocortical inputs: a voltage-dependent dye-imaging study in mouse brain slices[END_REF]. The claustrum and anterior insula should participate in the consciousness process through their interactions with cortices [START_REF] Crick | What is the function of the claustrum?[END_REF][START_REF] Critchley | Neural systems supporting interoceptive awareness[END_REF][START_REF] Koubeissi | Electrical stimulation of a small brain area reversibly disrupts consciousness[END_REF][START_REF] Taylor | Two systems of resting state connectivity between the insula and cingulate cortex[END_REF]; the anterior insula being nowadays included in the cortical structures that participate to the salience network [START_REF] Menon | Saliency, switching, attention and control: a network model of insula function[END_REF] (see below). Experimental studies have shown that the modulation of pallidum activity can provoke consciousness-like activity, which is suppressed by thalamic lesions [START_REF] Dieckmann | Cortical synchronized and desynchronized responses evoked by stimulation of the putamen and pallidum in cats[END_REF]. Acute 5-Hz electric stimulations of the pallidum, alone and coupled with 50-Hz reticular thalamic stimulation, provoked consciousness-like behaviors in a DOC patient [START_REF] Hassler | Behavioural and EEG arousal induced by stimulation of unspecific projection systems in a patient with post-traumatic apallic syndrome[END_REF]. 30-Hz chronic pallido-thalamic DBS provoked consciousness-like behaviors in DOC patients [START_REF] Lemaire | Deep brain stimulation in five patients with severe disorders of consciousness[END_REF].

In parallel to the extensive knowledge obtained from the basal ganglia and other deep gray nuclei in neurodegenerative disorders, greater attention is being given to the implication of corticosubcortico-thalamo-cortical loops in consciousness. Whether the cortices or the deep brain are at the origin of the phenomena is still unknown, and could be vain as both are linked by the wellknown sensorimotor, limbic and associative loops (see reviews [START_REF] Haber | Corticostriatal circuitry[END_REF][START_REF] Lemaire | Related Circuitry and Synaptic Connectivity in Psychiatric Disorders[END_REF]). In this respect, the mesocircuit model was proposed to explain the different DOCs [START_REF] Giacino | Disorders of consciousness after acquired brain injury: the state of the science[END_REF][START_REF] Schiff | Recovery of consciousness after brain injury: a mesocircuit hypothesis[END_REF]: the downregulation of the cortex being the result of widespread lesions, either structural or functional, across corticosubcortical structures. A growing body of works has shown the implication of extra thalamic subcortical components in consciousness [START_REF] Crone | Testing Proposed Neuronal Models of Effective Connectivity Within the Cortico-basal Ganglia-thalamo-cortical Loop During Loss of Consciousness[END_REF][START_REF] Lutkenhoff | Thalamic and extrathalamic mechanisms of consciousness after severe brain injury[END_REF][START_REF] Moll | Waking up the brain: a case study of stimulation-induced wakeful unawareness during anaesthesia[END_REF][START_REF] Monti | Dynamic Change of Global and Local Information Processing in Propofol-Induced Loss and Recovery of Consciousness[END_REF] and DOCs [START_REF] Perri | Functional neuroanatomy of disorders of consciousness[END_REF]. However structural lesions of deep gray nuclei do not always provoke permanent DOC [START_REF] Duffau | Absence of movement disorders after surgical resection of glioma invading the right striatum[END_REF][START_REF] Fukuoka | Bilateral caudate nucleus infarction associated with a missing A1 segment[END_REF][START_REF] Jain | Bilateral large traumatic basal ganglia haemorrhage in a conscious adult: a rare case report[END_REF]. The figure 3 illustrates corticosubcortical circuits related to DOCs.

Figure 3:

Cortico-subcortical circuits related to disorders of consciousness: A, structural, B, functional aspects; c., cortex; n., nucleus; stn, subthalamic nucleus; GP, globus pallidum, e, extern, i, intern; SN, substantia nigra, c, compacta, r, reticulata; hyperdir., dir. and indir., hyperdirect (prepares action or overall inhibition), direct (selected action program, disinhibition) and indirect (stop action) cortico-subcortical pathways; black line, pathway; thal, thalamus; vta, ventral tegmental area; neg., inhibits, and pos., reinforces, behaviors; acetylcholine (orange), noradrenalin (light blue), glutamate (green, activation), serotonin (pink), dopamine (dark blue) Gaba (red, inhibition) neuromodulators.

The role of the cortices in consciousness was extensively studied using electrophysiological and neuroimaging works. These pioneering integrative works proposed a model called "global workspace" including thalamo-cortical loops [START_REF] Dehaene | Experimental and theoretical approaches to conscious processing[END_REF][START_REF] Baars | Global workspace theory of consciousness: toward a cognitive neuroscience of human experience[END_REF][START_REF] Baars | A Cognitive Theory of Consciousness[END_REF][START_REF] Dehaene | A neuronal model of a global workspace in effortful cognitive tasks[END_REF][START_REF] Newman | A Neural Global Workspace Model for Conscious Attention[END_REF]. They proposed that long distance cortical connections particularly of the pyramidal cells of the prefrontal, cingulate, and parietal regions would allow consciousness [START_REF] Dehaene | Experimental and theoretical approaches to conscious processing[END_REF][START_REF] Baars | Global workspace theory of consciousness: toward a cognitive neuroscience of human experience[END_REF][START_REF] Dehaene | A neuronal model of a global workspace in effortful cognitive tasks[END_REF], in line with numerous others works (see e.g. [START_REF] Llinas | Temporal binding via cortical coincidence detection of specific and nonspecific thalamocortical inputs: a voltage-dependent dye-imaging study in mouse brain slices[END_REF][START_REF] Aru | Coupling the State and Contents of Consciousness[END_REF][START_REF] Crick | A framework for consciousness[END_REF][START_REF] Raffone | A global workspace model for phenomenal and access consciousness[END_REF]). If undoubtedly cortices play an important role in consciousness, extreme pathological conditions such as hydranencephaly questions medicine, science and beyond [START_REF] Aleman | Consciousness without cortex: a hydranencephaly family survey[END_REF]. In parallel, the metabolic activity of the brain at rest, i.e. when a subject experiences covert wandering thoughts, reveals local specificities notably unique high activities of the medial cortices. This particular distribution was qualified as the Default Mode Network (DMN) defined from Pet-Scan studies [START_REF] Raichle | A default mode of brain function[END_REF][START_REF] Shulman | Common Blood Flow Changes across Visual Tasks: II. Decreases in Cerebral Cortex[END_REF] and further studied with fMRI (see e.g. [START_REF] Boveroux | Breakdown of within-and between-network Resting State Functional Magnetic Resonance Imaging Connectivity during Propofol-induced Loss of Consciousness[END_REF][START_REF] Buckner | The brain's default network: anatomy, function, and relevance to disease[END_REF][START_REF] Corbetta | Spatial neglect and attention networks[END_REF][START_REF] Corbetta | Control of goal-directed and stimulus-driven attention in the brain[END_REF][START_REF] Dang | Dopamine supports coupling of attention-related networks[END_REF][START_REF] Vanhaudenhuyse | Default network connectivity reflects the level of consciousness in noncommunicative braindamaged patients[END_REF]). The DMN is in fact the very first, robust, physiologic brain functional network, observed in imagery, and this, even if it relies on covert activity, which is fortunately at rest and quasi-reproduceable by any human being.

Moreover the highest metabolic rest activity of the brain is located in the posteromedial cortices, the precuneus and the posterior cingulate [START_REF] Gusnard | Searching for a baseline: functional imaging and the resting human brain[END_REF]. This special cortical region, particularly the posterior cingulate, seems to be a key structure in consciousness, DOC and cognition [START_REF] Leech | The role of the posterior cingulate cortex in cognition and disease[END_REF]. The characteristic of DMN is that the cortices deactivate when attention is focused during goaldirected behaviors, notably involving the external environment: the right and left, bilateral junction of the posterior cingulateprecuneus, the inferior lateral parietal lobule, and the medial prefrontal cortices; the left prefrontal cortices and the inferior temporal cortex; the right mesiotemporalamydagala region [START_REF] Shulman | Common Blood Flow Changes across Visual Tasks: II. Decreases in Cerebral Cortex[END_REF][START_REF] Fox | The human brain is intrinsically organized into dynamic, anticorrelated functional networks[END_REF]. The activity of the DMN is markedly reduced in DOC patients [START_REF] Hannawi | Resting brain activity in disorders of consciousness: a systematic review and meta-analysis[END_REF]. Interestingly, the potentiality of recovering impaired functional connectivity between the medial prefrontal cortex and the posterior cingulate observed in DOC, could be a prognostic marker of further clinical recovery [START_REF] Lemaire | Deep brain stimulation in five patients with severe disorders of consciousness[END_REF][START_REF] Silva | Disruption of posteromedial large-scale neural communication predicts recovery from coma[END_REF]. From the entire data of the total brain voxels constructed during an fMRI acquisition, notably at rest, the segregation of voxel activities according to temporal relationships, historically using independent component analysis, enabled to identify canonical functional networks, further named intrinsic connectivity networks (ICNs) [START_REF] Seeley | Dissociable intrinsic connectivity networks for salience processing and executive control[END_REF]. These ICNs are the maps of the functional networks permanently active in a living brain, revealed at rest and during tasks. Three main ICNs were segregated: the central executive network (CEN), lateral, with the dorsolateral prefrontal and parietal cortices; the DMN, medial, with the ventromedial prefrontal and the posterior cingulate cortices, which could be acronymized as DMNm; the salience network (SN), with the ventrolateral prefrontal, the anterior insula and the anterior cingulate cortices [START_REF] Seeley | Dissociable intrinsic connectivity networks for salience processing and executive control[END_REF][START_REF] Sridharan | A critical role for the right fronto-insular cortex in switching between central-executive and default-mode networks[END_REF]. The attentional cortices, explored during visual tasks, were segregated into dorsal and ventral networks (DAN, VAN), respectively related to spatial cues and non-spatial, salience or novelty, cues [START_REF] Corbetta | Spatial neglect and attention networks[END_REF][START_REF] Corbetta | Control of goal-directed and stimulus-driven attention in the brain[END_REF]. Salience reflects the distinctiveness and relevance of stimuli [START_REF] Corbetta | Spatial neglect and attention networks[END_REF]. Indeed, the DAN detects an element in space using voluntary attention, thus purposefully, and consequently was qualified as topdown attention. The VAN focuses on an item of information using "natural" spontaneous salience (e.g. brightness or size) during purposeless spatial exploration, and was qualified consequently as bottom-up attention. The DAN covers roughly the lateral pre frontal cortex (the frontal eye field, the intermediate and the inferior frontal gyri, and the inferior frontal junction), the lateral parietal cortex (including the inferior parietal lobule and the intraparietal sulcus), the lateral occipital cortex (including V3) and the posterior temporal region. The VAN covers roughly the supra marginal gyrus, the superior temporal gyrus, the inferior frontal gyrus, the insula and the inferior frontal junction; it predominates in the right hemisphere. Further fMRI studies have also enabled the identification of several other functional networks such as the largescale frontoparietal control system (FPCS; anterior insula, dorsolateral and ventrolateral prefrontal cortices, dorsal anterior cingulate, and inferior parietal lobule), the frontoparietal executive control network (FPE-CON; bilateral, lateral prefrontal cortex, dorsomedial prefrontal cortex, and inferior parietal lobule) and the cingulo-opercular salient network (CO-SAL; bilateral, anterior insula/frontal operculum and dorsal anterior cingulate cortex). The FPCS is coupled with both the DMN and the DAN [START_REF] Gao | Frontal parietal control network regulates the anti-correlated default and dorsal attention networks[END_REF][START_REF] Spreng | Intrinsic Architecture Underlying the Relations among the Default, Dorsal Attention, and Frontoparietal Control Networks of the Human Brain[END_REF] allowing the allocation of resources [START_REF] Dang | Dopamine supports coupling of attention-related networks[END_REF]. The FPE-CON is correlated with the visual cortex (the activities are considered as synchronous) linked with the lateral DAN, whereas the CO-SAL is correlated with the DMN [START_REF] Elton | Divergent task-dependent functional connectivity of executive control and salience networks[END_REF]. Effective connectivity analysis, prone to identify causalities, showed the major role of the right fronto-insular cortex (ventrolateral prefrontal cortex + anterior insular cortex) of the SN, which can switch between the DMNm and the CEN [START_REF] Sridharan | A critical role for the right fronto-insular cortex in switching between central-executive and default-mode networks[END_REF]. The Pet-Scan activity of deep thalamo-caudate structures and the fronto-parietal network is impaired in DOC patients [START_REF] Bruno | Functional neuroanatomy underlying the clinical subcategorization of minimally conscious state patients[END_REF][START_REF] Stender | Diagnostic precision of PET imaging and functional MRI in disorders of consciousness: a clinical validation study[END_REF], and the external stimulation of dorsolateral prefrontal cortices of patients increases signs of consciousness [START_REF] Hermann | Combined behavioral and electrophysiological evidence for a direct cortical effect of prefrontal tDCS on disorders of consciousness[END_REF][START_REF] Thibaut | Clinical Response to tDCS Depends on Residual Brain Metabolism and Grey Matter Integrity in Patients With Minimally Conscious State[END_REF]. The cortices were also segregated into external, i.e. turned to the perceptual awareness of the external environment, and the internal, i.e. turned to the self or stimulus independent thought ("wandering mind"), which are anticorrelated (when one works the other stops) [START_REF] Boly | Consciousness and cerebral baseline activity fluctuations[END_REF][START_REF] Thibaut | Metabolic activity in external and internal awareness networks in severely brain-damaged patients[END_REF][START_REF] Vanhaudenhuyse | Two distinct neuronal networks mediate the awareness of environment and of self[END_REF]. These cortical networks are summarized in figure 4.

control system (FPCS), which is composed notably of the dorsal attentional network (DAN) and the ventral attentional network (VAN); the frontoparietal executive control network (FPE-CON) and the cinguloopercular salient network (CO-SAL).

However, the distinction of all these functional networks, which overlap more or less depending on the methods and terminologies, still does not answer the "hard problem", sending us back to the roots of brain functioning, outside our scope here. Roughly, the dynamics of the cortical networks, notably anticorrelations, seems to be one of the most robust markers of consciousness [START_REF] Shulman | Common Blood Flow Changes across Visual Tasks: II. Decreases in Cerebral Cortex[END_REF][START_REF] Barttfeld | Signature of consciousness in the dynamics of resting-state brain activity[END_REF]. One should bear in mind that these cortices also play a role in several processes or functions, up to emotion [START_REF] Rosenbaum | Neural correlates of mindful emotion regulation in high and low ruminators[END_REF] and pure cognitive processes such as advanced mathematics [START_REF] Amalric | Origins of the brain networks for advanced mathematics in expert mathematicians[END_REF]. Topological approaches of functional connectivity notably according to resting-state conditions could help to decipher the complexity of brain networks, such as graph theory [START_REF] Achard | Hubs of brain functional networks are radically reorganized in comatose patients[END_REF][START_REF] Bullmore | Complex brain networks: graph theoretical analysis of structural and functional systems[END_REF][START_REF] Hagmann | Mapping the structural core of human cerebral cortex[END_REF], effective connectivity analysis [START_REF] Friston | Functional and Effective Connectivity: A Review[END_REF], dynamics of function-structure similarities [START_REF] Barttfeld | Signature of consciousness in the dynamics of resting-state brain activity[END_REF][START_REF] Uhrig | Resting-state Dynamics as a Cortical Signature of Anesthesia in Monkeys[END_REF], and contextmodulated functional connectivity [START_REF] Cisler | A Comparison of Statistical Methods for Detecting Context-Modulated Functional Connectivity in fMRI[END_REF]. Even if these methods coupled with structural analysis of brains (e.g. diffusion tensor tractography) will help to decipher the pathophysiology of DOC, the complexity of lesions (figure 5) and their dynamics over time, following acute and subacute mechanisms such as ischemia, hematoma, trauma, seizure and complications of intracranial hypertension, entangle substantially the analysis of mechanisms. We should also consider that DOC stricto sensu are also truly present following progressive diseases, even if they are not fully qualified or interpreted as such, in multifactorial neurologic and psychiatric diseases, e.g. neurodegenerative diseases such as dementia whatever the context. The neural correlates of consciousness, at least from the medical viewpoint, must also consider the biology of neurotransmitters as several drugs can alleviate signs of DOC (see e.g. for a review [START_REF] Lemaire | Electrical modulation of neuronal networks in brain-injured patients with disorders of consciousness: a systematic review[END_REF], but no cross-fertilization with the data collected by studies of structural and functional dysconnectivity has yet occurred, and we do not know how these drugs act.

CONCLUSION-PERSPECTIVE

The tremendous growth of knowledge collected on consciousness over recent decades contrasts with the still limited therapeutic breakthroughs achieved, and, consequently, still very limited benefits for patients. However, it is important to disseminate and to share knowledge (including the untold stories), even if it is often inconclusive. One can reasonably hypothesize that pertinent avenues of research will soon be proposed to improve the condition of DOC patients. Nonetheless, one of the most complex issues could very likely be our capacity of deciphering the segregation and the potential hierarchization of the so-called superior functions and their related behaviors, while accounting for the neural correlates. That being so, pragmatically, the "hard medical problem" could still be patients with a high lesion burden, thus the most fragile, for whom consciousness disorder is one aspect of brain dysfunction, and for whom the hope of qualitative recovery, yet to be defined, must be discussed by taking account of the societal context.

Figure 4 :

 4 Figure 4: Cortical networks related to consciousness (see text for details). A. the historical networks: the default mode network (DMN; blue; with the medial cortices of high metabolic activity, dark blue) and the cortices that decrease their activity during goal-directed tasks (DMNdeactivation; bilaterally, red; left hemisphere, yellow; right hemisphere, green); consciousness cortices of the global workspace (Global workspace). B. Recent functional networks, from left to right: the canonical models of the DMN, the central executive (CEN, light brown), the DMN medial (DMNm, yellow) and the salience (SN, pink) networks; the

Figure 5 :

 5 Figure 5: Image data sets and tractogram of a 32-year-old male in a vegetative state (tracheostomy, gastrostomy and ventriculoperitoneal shunt) since twelve years following a traumatic head injury, and showing complex severe subcortical lesions that predominate in the left insular and lenticular regions: A, Image data sets, (1) axial T1-weighted MRI slice, (2) 3D volumetric display of the brain (beige) and cerebrospinal fluid spaces (yellow), (3) axial (top) and coronal (bottom) inversion-recovery MRI slices, (4) axial FDG TEP-scan (top) and colorcoded diffusion tensor imaging (DTI, 20 directions bottom) slices [right row patient data; left row, top, average FDG TEPScan at rest (healthy subjects), bottom, healthy control, male, right handed, 60 directions], R, right hemisphere, L, lenticular nucleus, V, ventricle, T, thalamus, destructed left lenticular region*; B, Tractogram (DTI tractography, 20 directions), crosssections along MRI slices (top left, axial; top right, sagittal; bottom left, coronal) and posteromedial view of the 3D tractogram (bottom right) of the right hemisphere showing the complexity of the disorganization of fiber bundles, notably cortico-thalamic (CTf and FTf) around the lesioned thalamus (#), and of the cingulum; CC, corpus callosum, c, cingulum, Lsf, longitudinal superior fascicle, Lif, longitudinal inferior fascicle, IC, internal capsule, TOf, temporooccipital fascicle, CTf, cortico-thalamic fascicle, CSF, corticospinal fascicle, FTf, frontothalamic fascicle, cf, cerebellar fascicles.