Ayham Tohmé

Stéphanie Labouille

Thierry Roisnel

Vincent Dorcet

Duncan Carmichael
email: duncan.carmichael@polytechnique.edu

Frédéric Paul
email: frederic.paul@univ-rennes1.fr

Redox-active phosphines: synthesis and crystal structures of palladium (II) complexes of a metallaphosphine in two different oxidation states

published or not. The documents may come

The redox-active metallaphosphine [Fe(dppe)(η η η η 5 -C 5 Me 5 ) (C≡ ≡ ≡ ≡C-PPh 2 )] reacts with [Pd(1,5-cod)Cl 2 ] give mono-and bis-phosphine complexes as a function of stoichiometry, and these complexes provide a stable redox-active platform which allows reversible one-electron {Fe(II)→ Fe + (III)} oxidations within the palladium coordination sphere.

Fine control of architecture and the ability to generate quite specific electronic environments 1 means that phosphines are exceptionally widely used in many metal-catalysed processes, [START_REF] Kamer | Phosphorus(III) Ligand Effects in Homogeneous Catalysis[END_REF] and their versatility has driven recent research into areas as varied as self-assembling phosphines for enantioselection, [START_REF] Wieland | [END_REF] supramolecular phosphines allowing unprecedented catalytic selectivities, 4 and hybrid chemical/biological phosphine-containing systems. 5 However, there has been comparatively little recent interest in the class of redox-active phosphines, 6 which should constitute an unusual class of "non-innocent" ligands 7,8 and have been invoked inter alia for the switching of catalytic processes. 9 We have recently demonstrated that phosphorus centres and redox-active organoiron endgroups can communicate strongly if linked through molecular wires such as acetylenes, and that oxidation of the Fe(II) endgroup in 1 provokes significant reactivity at the phosphorus centres (Scheme 1). 10 This unusual behaviour prompted a study of the coordination chemistry of this metallaphosphine towards Pd(II) centres. ). [START_REF] Dunne | arylphosphines to a monoelectronic oxidation of their coordinated metal is well known[END_REF] The oxidation has a significantly smaller effect upon the palladium coordination sphere: a slight shortening of the mean Pd-P distance, from 2.2274( 11) to 2.2191( 12) Å (-0.4%) and a modest (0.8%) shortening in the trans-Pd-Cl distance from 2.4447(10) to 2.4256(12)Å are observed, [START_REF]For comparison, values for Pd-Cl trans-to phosphine in trans-[Pd 2 Cl 4 (PR 3 ) 2 ] are[END_REF] whilst the sum of the angles made by the organic groups at the Pd-coordinated phosphorus also rises only modestly from 317.6 to 321.8°. The dimeric palladium core of the molecule can be expected to act as "spacer" between the two oxidised [Pd(1 + )] components and an Evans determination 17 of a room temperature magnetic moment of 2.21±0.1 BM at 293K is consistent with a ditopic diradical featuring two non-interacting spins. 18 Palladium dimers of type 3 are excellent sources of both homo-and heteroleptic [PdCl 2 L 2 ] complexes [START_REF] Smithies | [END_REF] and 5, which offers the possibility of two-electron redox chemistry within a Pd coordination sphere, was found to be accessible through either reaction of two equivalents of 1 with the palladium bridged dimer 3, or by simple reaction of two equivalents of 1 with [PdCl 2 (1,5-cod)] (Scheme 2). Single crystals of the purple, air-stable trans-configured 5 revealed metallaphosphine ligands that are very similar to those in 3 and Pd-P distances (2.338(1)Å) which are normal for mutually trans-phosphine ligands in a Pd(II) complex (Fig. 2). DFT data from a model for 5 [START_REF]In the models, H 2 PCH 2 CH 2 PH 2 ligands replace dppe, and Cp replaces Cp*. B3PW91 with SDD on Fe[END_REF] (Fig. 3) show a sharp increase in the NBO-calculated positive charge at the Pd-bound phosphorus (+0.62) and a significant rise at the ironcoordinated alkyne carbon (+0.10) upon coordination but only negligible changes (<0.04) elsewhere. Again, cyclic voltammetry showed closely-spaced reversible oxidation waves (E 1 1/2 = -0.02 V, E 2 1/2 = 0.11 V) and chemical oxidation using two equivalents of [FcH][PF 6 ] provided a black, air-sensitive product, 6. Crystals suitable for a diffraction study were not obtained, but comparison of the very similar 1 H NMR data for 4 and 6, 21 and an Evans determination showing a value of 2.42 ±0.1 BM (298 K) that is again classical for two unpaired electrons, [START_REF]Taking a statistical analysis wherein all the magnetic states are equally populated, an uncoupled biradical should exhibit 75% of the magnetisation that corresponds to its triplet state (3.04 BM[END_REF] imply a bis-Fe(III) complex showing little spin-spin coupling across the palladium centre. A DFT comparison of NBO charge distributions in models [START_REF] Campora | For a Pd(II) to Pd(IV) oxidation using [FcH[END_REF] for 5 and 6 confirms an iron based oxidation (δQ Fe(5→6) = +0.36) that causes significant changes in charge at the linker atoms (δ Q(Fe-C) = -0.13; δ Q(P-C) = +0.21) but significantly smaller effects at phosphorus and at palladium (δ Q(P, Pd) < 0.02). The individual redox steps that occur within the bound couples 3,4 and 5,6 appear to be wellbehaved, but the chemical reversibility of the system as a whole clearly also rests upon the integrity of the dissociated ligands. The stability of 1 is established but our previous study has shown that the iron (III) proligand 2' normally exists as a 1,1'-(biphosphine)dication 2 10 (Scheme 3); this raises the undesirable possibility of the transient dissociation of 4 with subsequent dimerisation of the Fe(III) ligand monomers to generate a 1,1'-(biphosphine)dication 2 reservoir and a ligand-deficient solution that deposits [PdCl 2 ]. Experiments conducted upon isolated 1,1'-(biphosphine)dication 2 and [PdCl 2 (1,5-cod)] indicate that this mixture cleanly generates complex 4 (with 2eq of Pd) or 6 (with 1eq of Pd), and therefore confirm that ligand dissociation from both systems is reversible. The most likely mechanism involves the transient dissociation of the P-P bond to generate the Fe(III) proligand 2' 10 prior to capture by the Pd centre (Scheme 3). To conclude, metallaphosphines bearing [Fe(κ 2 -dppe)(η 5 -C 5 Me 5 )] endgroups bind to Pd(II) in both their neutral and cationic forms and can therefore be used to generate a chemically stable, redox-switchable coordination sphere about PdCl 2 centres. The [Pd(II)-(1 + ) 2 ] ↔ [Pd(IV) 2+ -(1) 2 ] valence tautomer is firmly shifted to the left, so the metallaligand brings a potential for varying the overall electron count of the complex without excessively compromising the electronic identity of the palladium(II) centre. Reactivity studies of 3-6 are in progress.

Notes and references † Electronic Supplementary Information (ESI) available: Full experimental, spectroscopic and crystallographic data. See DOI: 10.1039/c4dt00455h.

We thank CNRS and the French ANR (to AT, under contract BLAN 719) for support; S L thanks MNERT and ENS (Cachan) for a scholarship.

13 1

 13 Fig 1. Left: Superimposed X-ray-structures showing one half of each of the dimers forming the 3, 4 redox couple, showing ligand superstructure for the neutral complex 3 (purple) and its oxidation product 4 (gold). (Pd in magenta, Fe in orange, Cl in green). See supplementary for individual depictions of these complexes. Right: X-ray structural changes upon passing from free ligand 1 (plain text) to neutral dimer 3 (italic) and dication 4 (bold). Values are averaged over chemically equivalent sites.

Fig. 2 :

 2 Fig. 2: X-ray structure of the neutral monomeric Pd complex 5.

Fig 3 :

 3 Fig 3: The B3PW91 calculated HOMO (left) and LUMO (right) for [PdCl 2 L 2 ], (L= FeCp{H 2 PCH 2 CH 2 PH 2 }C 2 PPh 2 ), a model for 5. Calculated HOMO-LUMO gap: 3.05eV.

and 5. Reagents

  [FcH][PF 6 ], 2 eq., CH 2 Cl 2 , 20ºC, 1h; ii: [CoCp 2 ] 3 eq., CH 2 Cl 2 , 20ºC, 1h.

	Me	Fe Ph 2 P PPh 2 Me C C P Ph	Ph	Cl				Cl				Me	Me			Me	Fe Ph 2 P PPh 2 Me C C P Ph	Ph	Cl	Me	Me
	Me	Me Me		Cl	Pd	3	Cl	Ph Pd	P Ph	C	Fe Me PPh 2 Me Ph 2 P C	Me			Me	Me	Me	Me Me	Cl 5	Pd Ph	P Ph	C	Me Fe Ph 2 P C	Me PPh 2	Me
														i								Me	iii
						ii														PPh 2	C	C	Fe	Me	Me	iv
																						Ph 2 P	PPh 2	1
	Fe Me Me Ph 2 P PPh 2 Me Me Me C C P Ph Cl	Ph Pd	Cl Cl	Ph Pd	P Ph Cl	C	Fe Me Me PPh 2 Me Me Me Ph 2 P C	2+ 2PF 6	-	v	Fe Me Me Ph 2 P PPh 2 Me Me Me C C P Cl Ph	Ph Ph Pd	Ph P Cl	C	Ph 2 P C Me Fe Me	PPh 2 Me Me Me	2+ 2PF 6	-
									4														6
	Scheme 2: Coordination chemistry of 1 and redox reactivity of the resulting Pd(II)
	complexes 3
				Ph 2 P PPh 2
			Me	Me	Fe	C C PPh 2
			Me		Me
					1	Me				i				ii
						Ph		Ph				Me Me		Me
		Ph 2 P PPh 2				P		C		C		Fe	Me	Me
	Me	Me	Fe	C C P			Ph		Ph 2 P		PPh 2
	Me		Me			Ph
			Me

2 Scheme 1: Redox switching of metallaphosphine 1. Reagents: i: and conditions: i: [PdCl 2 (1,5-cod)] (1.0eq.), CH 2 Cl 2 , 15h, 97%; ii: [FcH][PF 6 ], CH 2 Cl 2 , 1h, > 98%; iii: [PdCl 2 (1,5-cod)] (0.5eq.), CH 2 Cl 2 , 15h, 96%; iv: [FcH][PF 6 ], CH 2 Cl 2 , 1h, > 98%; v: Compound 2 (1.0eq.), CD 2 Cl 2 , 15mn, > 95%.

For the effects of ligand bite angle on the electronics of metal centres, see: M.-N. Birkholz,


Ronald Phlypo
email: ronald.phlypo@inria.fr

Bertrand Thirion
email: bertrand.thirion@inria.fr

Gaël Varoquaux
email: gael.varoquaux@inria.fr

Deriving a multi-subject functional-connectivity atlas to inform connectome estimation

Keywords:

The estimation of functional connectivity structure from functional neuroimaging data is an important step toward understanding the mechanisms of various brain diseases and building relevant biomarkers. Yet, such inferences have to deal with the low signal-to-noise ratio and the paucity of the data. With at our disposal a steadily growing volume of publicly available neuroimaging data, it is however possible to improve the estimation procedures involved in connectome mapping. In this work, we propose a novel learning scheme for functional connectivity based on sparse Gaussian graphical models that aims at minimizing the bias induced by the regularization used in the estimation, by carefully separating the estimation of the model support from the coefficients. Moreover, our strategy makes it possible to include new data with a limited computational cost. We illustrate the physiological relevance of the learned prior, that can be identified as a functional connectivity atlas, based on an experiment on 46 subjects of the Human Connectome Dataset.

1 Functional Connectivity and rest fMRI Functional connectivity (FC) is a simple measure of the interactions between brain regions. First introduced for electro-physiological recordings, such as spiketrain recordings or electro-encephalography, it is now commonly used in spatiallyresolved neuro-imaging modalities despite their poor temporal resolution at the scale of a second or more. Reported first as an alternative tool for positron emission tomography analysis [START_REF] Friston | Functional connectivity: the principal-component analysis of large (PET) data sets[END_REF], it has become a prominent tool in functional magnetic resonance imaging (fMRI) analysis. Functional connectivity has initiated a paradigm shift in fMRI studies, since it makes possible to scan subjects without engaging them in a controlled task. This is especially important when dealing with non-cooperative subjects, such as children or patients with neuropathologic disorders [2, among others]. In addition, it creates an opportunity to study the brain as a whole, through its global interactions, rather than through local effects.

Although FC is an established framework in neuroimaging, there is still no consensus on how to measure it from the data [3]. The connectivity pattern can be seen as a graph with its nodes (vertices) and connections (edges). Nodes are associated with elementary brain regions, between which connection strengths are then estimated, and the graph structure as a whole is referred to as a functional connectome. Nodes may either be predefined regions of interest (ROIs) or voxels [START_REF] Hayasaka | Comparison of characteristics between region-and voxel-based network analyses in resting-state fMRI data[END_REF], or may consist of distributed structures estimated from the data [START_REF] Damoiseaux | Consistent resting-state networks across healthy subjects[END_REF][START_REF] Smith | Temporally-independent functional modes of spontaneous brain activity[END_REF]. Once the nodes have been defined, edge strengths are inferred either through Pearson's correlation coefficients [START_REF] Hayasaka | Comparison of characteristics between region-and voxel-based network analyses in resting-state fMRI data[END_REF], partial correlation coefficients [START_REF] Marrelec | Partial correlation for functional brain interactivity investigation in functional MRI[END_REF], or mutual information [START_REF] Ma | Dynamic changes of spatial functional network connectivity in healthy individuals and schizophrenia patients using independent vector analysis[END_REF]. Characteristic network properties are then derived from these graph structures based on graph theoretic measures such as small-worldness, modularity, etc [START_REF] Rubinov | Complex network measures of brain connectivity: Uses and interpretations[END_REF].

The central challenge in functional connectome estimation is that it is illposed, due to the small number of temporal samples in typical datasets. Regularization, e.g. with a sparsity assumption in the connections, is critical to recover connections from the data [3], yet at the expense of a bias on the estimated connectome. Ng et al [START_REF] Ng | A novel sparse graphical approach for multimodal brain connectivity inference[END_REF] have shown that introducing a non-uniform connectivity prior derived from anatomical connectivity greatly improves estimation. However, the matching between functional and anatomical connectivity is often very imperfect [START_REF] Ng | Implications of inconsistencies between fMRI and dMRI on multimodal connectivity estimation[END_REF]. For this reason, it is important to construct connectivity priors from functional connectivity information itself.

Here, we use a group of subjects to build such a prior in the form of an atlas that characterizes the stability of brain connections. Our estimation method is based on the precision matrix (inverse of correlation matrix, related to the partial correlation coefficients), since it has proven sensitive to correctly detect existing connections in a graph structure [3]. In contrast to existing stability studies, we do not focus on global properties of the graph, as in [START_REF] Rubinov | Complex network measures of brain connectivity: Uses and interpretations[END_REF][START_REF] Wang | Graph theoretical analysis of functional brain networks: Test-retest evaluation on shortand long-term resting-state functional MRI data[END_REF]. Rather, we aim at quantifying the probability of a connection being present between two pre-defined regions by means of a group study, and use this as a prior in the estimation of single subject FC. To alleviate the bias of sparsity-inducing penalization, we use a two-stage estimation procedure to obtain edge strengths.

Here, we focus on the qualitative properties of the model, which sketches a new generation of FC atlases, and leave quantitative validation for future work.

Data Used in the Study

In this contribution we use resting state data as provided by the minimal preprocessed data [START_REF] Glasser | The minimal preprocessing pipelines for the human connectome project[END_REF] of the human connectome project (HCP) [START_REF] Smith | Resting-state fMRI in the human connectome project[END_REF] (release Q2). This comprises 46 subjects. All data were recorded in 3T MRI scanners with a repetition time of 720 ms. Data have been appropriately masked and motion corrected using the estimated motion parameters provided by HCP. In addition, we corrected for confounds using high variance confound regression [START_REF] Behzadi | A component based noise correction method (CompCor) for BOLD and perfusion based fMRI[END_REF] based on five principal components of 2 % highest variance voxel time series of whole brain image, and using five principal components of ventricle and white matter voxels (using standard masks provided in FSL, the mask of the latter eroded by 1 voxel to avoid overlap with gray matter).

Learning the Functional Connectivity Model

Graph Representation of Functional Connectivity

Given a set of p brain regions, the goal is to estimate an undirected graphical model G(V, E, Ω) as a connectome representation. The vertex set V = {v i , i ∈ [[p]]} represents weighted averages of time series with respect to each of the p given ROIs (here a smooth positive function over a bounded subset of R 3 ). Weights reflect 'probabilities' of a voxel to belong to the given ROI. In this work, we have chosen the Harvard-Oxford lateralised probabilistic cortical atlas (p = 96) [START_REF] Desikan | An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest[END_REF] (see, e.g., [START_REF] Wang | Graph theoretical analysis of functional brain networks: Test-retest evaluation on shortand long-term resting-state functional MRI data[END_REF] for ROI abbreviations and acronyms used in this contribution).

The edge set Negative strengths correspond to a phase inversion between the time series of interest. When modeling FC, it is often assumed that the graph is sparse, which corresponds to a cardinality of E being (largely) inferior to p(p -1)/2. This hypothesis is justified based on evidence provided by structural connectivity measurements [START_REF] Deligianni | A framework for inter-subject prediction of functional connectivity from structural networks[END_REF].

E = {e ij , (i, j) ∈ [[p]] 2 } contains e ij if

Estimation of the Graph Model Parameters

Estimating a sparse graphical-model is an NP-hard problem, and we choose graphical lasso [START_REF] Friedman | Sparse inverse covariance estimation with the graphical lasso[END_REF][START_REF] Banerjee | Model selection through sparse maximum likelihood estimation for multivariate Gaussian or binary data[END_REF] as an appropriate convex relaxation of our problem. We represent the graphical model through its adjacency matrix Θ def.

= (θ ij ) i,j , which under a Gaussian hypothesis corresponds to the rescaled precision matrix (inverse of the covariance matrix). Hence, θ ij = θ ji = 0 if and only if the two regions represented by v i and v j are independent, conditional on all other regions [START_REF] Lauritzen | Graphical Models[END_REF]. Graphical lasso combines the maximum likelihood estimate with a convex sparsity inducing penalty with an objective function given by1

Θ E = arg min Θ 0 ϕ E (Θ) def. = -log det (Θ) + trace (ΘS) + Λ • Θ 1 ( 1
)
where S is the sample correlation matrix of the time-series represented by the nodes in V . We minimize ϕ E (Θ) of Eq. ( 1) over the space of symmetric positive definite matrices using the algorithm of [START_REF] Mazumder | The graphical lasso: new insights and alternatives[END_REF], which guarantees Θ E 0 at any step of the algorithm, as such providing a procedure more robust against illconditioning of S than those originally proposed in [START_REF] Friedman | Sparse inverse covariance estimation with the graphical lasso[END_REF][START_REF] Banerjee | Model selection through sparse maximum likelihood estimation for multivariate Gaussian or binary data[END_REF].

The estimate Θ E is a biased estimate of Θ, and we only use it to infer the edge set E. We thus break our estimation scheme down into three steps:

(i) estimation of the hyperparameter Λ, (ii) estimation of the edge set E, and (iii) estimation of the edge weights or signed connectivity strengths Ω.

Step (i) will be discussed in the next section. Assuming for the moment that Λ is known, the last two steps are simply: (ii) The edges e ij in E correspond to those entries of Θ E for which θ ij > 0.

(iii) Given the estimated edge set, we estimate edge strengths Ω = (ω ij ) i,j as the maximum likelihood estimates restricted to the edge set E of the graph G, based on the sample correlation matrix S. Our estimator reads

Θ Ω = arg min Θ| G 0 ϕ Ω (Θ) def. = -log det (Θ) + trace (ΘS) (2)
which is over the space of positive-definite, symmetric matrices with a given support set E. The restriction to the graph prohibits the use of straightforward matrix inversion, and instead we use iterative proportional scaling (IPS), iteratively solving plain maximum likelihood models alternated over maximal cliques of the graph G [START_REF] Lauritzen | Graphical Models[END_REF]. The obtained solution is an unbiased maximum likelihood estimator conditional on the support set of the graph, hence the only bias in our final model is the pruning of the edges. Given new data (a subject, session, or condition), we can infer the corresponding functional connectome.

Choosing the Prior Atlas

The choice of the regularizer Λ is essential to optimize the bias/variance compromise. A first approach consists in choosing Λ = λ 11 T -Id p . As in [START_REF] Mazumder | The graphical lasso: new insights and alternatives[END_REF], this makes the problem low-dimensional. Hence, we use a repeated random jackknifing cross-validation scheme. As a score function, we opt for the negative log-likelihood, which is our main objective function. However, the extrapolation of that single parameter λ is an issue, since a given λ value corresponds to different cardinality of E for different datasets (even if ratios such as λ/λ max are considered, where λ max is the critical regularizer that yields a trivial solution). Instead we propose to accumulate the support estimates across subjects and sessions. The intuition behind this approach is that noise in the data can cause instabilities in the graph lasso estimate. However, these instabilities are not reproducible across subjects, and the population distribution of the presence of edges is reliable. This approach is related to the "stability selection" strategy, also developed for sparse Gaussian graphical models [START_REF] Meinshausen | Stability selection[END_REF]. Consider q subjects, each subject having 2 separate scan sessions2 .

Estimation of λ (i) s. For each subject i, we use a grid search to estimate λ (i) as

that λ in Λ = λ 11 T -Id p for which Θ (i)
Ω maximizes the likelihood generalization. The generalization capability of the likelihood can be approximated by using n samples from one session as a train session to estimate Θ (i)

E (i) ; E (i)
is then taken as the support in Eq. (1) to estimate Θ (i) Ω (i) from n samples. We then use the n complementary samples in the latter session to compute the likelihood of Θ (i) Ω (i) . This is repeated k times, and we report λ (i) , as that λ that maximizes the average likelihood. In this study, we have used n = 200, n = 600, k = 5, and for each repetition the session used for training has been chosen randomly. Estimation of Λ. Using the optimal λ (i) for subject i, we re-estimate E (i) from n = 400 samples, using Eq. ( 1) for each session, yielding E (i,1) and E (i,2) . Suppose that all subjects (and sessions) are drawn from a single population with P ij the marginal probability of observing edge e ij . We simply estimate these probabilities by using maximum likelihood under marginal Bernoulli distributions on the edges, based on the {E (i,1) , E (i,2) } estimates. We then define the penalization parameter Λ as 11 T -P, where P def.

= (P ij ) i,j . Note that, by construction, P is a symmetric matrix with ones on its diagonal.

Λ can then be used on a new subject as a penalty in the penalized maximum likelihood estimation [START_REF] Friston | Functional connectivity: the principal-component analysis of large (PET) data sets[END_REF]. Importantly, it appears as a penalty, not a constraint, and it is thus a "soft" prior. In other words, it does not force the existence or absence of a connections; it simply promotes these choices. Prior on the functional connectivity, noted P in the text: the coefficients in the matrix represent the frequencies of edge detections. ROI labels and their abbreviations, as well as a visualization of all networks can be found in an addendum to a preprint of this manuscript at http://hal.inria.fr/hal-00991124.

The probability estimates in P give a global view of the connectome in our training set of 46 subjects. It is important to note that the structure of P reflects the sparsity assumptions, which means that some connections have indeed never been 'observed' over the entire set of subjects, see Fig. 1. This connectivity prior matrix is very structured: it displays a checkerboard pattern: regions are most likely to be connected to a region in the same hemisphere than the corresponding one in the opposite hemisphere. In addition, visualized as a brain graph, it features well-known characteristics of brain connectivity, such as strong homologous inter-hemispheric or fronto-parietal connections.

To visualize the latent network structure, we re-organized the matrix P using hierarchical clustering with complete linkage, choosing the number of clusters that minimizes the silhouette criterion [START_REF] Rousseeuw | Silhouettes: A graphical aid to the interpretation and validation of cluster analysis[END_REF]. Clustering on probabilities of edges in the support of a graph gives FC networks (see Fig. 2) similar to those found in group analyses based on independent component maps, see, e.g., [START_REF] Damoiseaux | Consistent resting-state networks across healthy subjects[END_REF]. This suggests that the FC atlas has indeed captured valuable information from our reference population.

Other than having an interpretable graph representation, it should be emphasized that our approach provides a readily-usable subject-specific estimator, alleviating the need of a search over the parameter space (λ) for a new subject. As any medical-imaging atlases, our model incorporates accumulated knowledge. In contrast to group analyses that need a complete re-computation of the graph or the components upon incorporation of a new subject [START_REF] Smith | Temporally-independent functional modes of spontaneous brain activity[END_REF][START_REF] Ng | A novel sparse group Gaussian graphical model for functional connectivity estimation[END_REF], we only need to update a single matrix, namely the estimator P. Moreover, based on the observations of [START_REF] Ng | A novel sparse graphical approach for multimodal brain connectivity inference[END_REF], we conjecture that the incorporation of the structured atlas-based prior will yield more accurate estimation of functional connectivity than traditional graph lasso estimators that enforce a uniform 1 shrinkage on all connection.

Conclusions

Sparsifying penalties are the most promising tools to learning functional connectomes from noisy and scarce fMRI data. Injecting structure in this prior, favoring connections that are known to be likely can greatly improve the ability of the estimation procedures to recover real connections. We have introduced a procedure to learn an atlas of brain functional connections by accumulating knowledge across subjects, associated with a subject-specific estimator of a sparse Gaussian graphical model that quantifies functional connectivity. Future work includes the validation of the atlas-based strategy for individual connectivity estimation and subjects connectivity comparison.

Fig. 1 .

 1 Fig.1. Prior on the functional connectivity, noted P in the text: the coefficients in the matrix represent the frequencies of edge detections. ROI labels and their abbreviations, as well as a visualization of all networks can be found in an addendum to a preprint of this manuscript at http://hal.inria.fr/hal-00991124.

Fig. 2 .

 2 Fig. 2. Lateral and top views of the four networks containing the most nodes. From left to right we distinguish a motor network, a visual network, as well as two lateralised TPJ-to-IFG networks (TPJ: temporo-parietal junction -IFG: Inferior Frontal Gyrus).

  and only if the ROIs represented by v i and v j are functionally connected, given all other v k , k / ∈ {i, j}. Edge weights Ω = {ω(e ij ), e ij ∈ E} represent signed connectivity strengths.

The original implementation of the penalization in graph lasso reads λ Θ 1, but since no penalization of the diagonal terms is required, we may write λ 11 T -Idp • Θ 1 , where 1 is a vector of ones in R p and • • • denotes the element-wise or Hadamard product. We use here an extension of this model, replacing λ 11 T -Idp by a symmetric matrix Λ = (λij)i,j, with 0 ≤ λij ≤ 1.

We consider the back-to-back recorded left-right and right-left phase encoding as a single session. To marginalize the effect of the phase encoding, we always sample both encodings evenly. Thus, if we state that n samples are jackknifed from a session, we actually mean that n/2 samples are jackknifed from each phase encoding associated with that session.

Acknowledgement Data were provided by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Uǧurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that


Birgitta Dresp

Osmund Keith Langley

Long-range spatial integration across contrast signs: a probabilistic mechanism ?

It is widely assumed that the integation of orientation contrast across spatial gaps within the long-range regime is not selective to the contrast sign of the individual stimuli.

Probabilistic models of perceptual integration, however, suggest that long-range spatial integration should be, if not selective, at least sensitive to local contrast signs. To clarify this issue, we tested predictions of a model based on conditional probabilistic weights of identical and opposite contrast signs in a simple spatial configuration of two co-linear lines. Contrast detection thresholds of the target line presented either by itself (control condition) or simultaneously with the co-linear inducer (test condition) were measured.

The contrast sign of targets and inducers was varied so that all four possible combinations of signs were produced in the test conditions: 1) dark target with dark inducer, 2) dark target with bright inducer, 3) bright target with bright inducer and 4) bright target with dark inducer. The contrast intensity (Weber ratio) of dark and bright inducers was identical. The coaxial distance between target and inducer was constant in each of two experiments, testing for two distances that corresponded to an angular separation within the long-range domain of spatial integration as defined previously. It is found that targets and inducers with identical contrast signs produce significantly stronger facilitating effects on detection than stimuli with opposite signs. The data closely match predictions consistent with those of a probabilistic model of line contrast integration across spatial gaps and contrast signs within the long-range regime.

INTRODUCTION

It is well established that the contrast detection of dots, line stimuli or Gabor patches (targets) is facilitated by the presence of a co-linear line or Gabor patch (inducers). Such facilitating interactions between co-linear stimuli in visual contrast integration have been found in psychophysical studies on humans (e.g. [START_REF] Polat | Lateral interactions between spatial channels: Suppression and facilitation revealed by lateral masking experiments[END_REF]1994[START_REF] Dresp | Bright lines and edges facilitate the detection of small light targets[END_REF][START_REF] Tzvetanov | Short-and long-range effects in line contrast integration[END_REF][START_REF] Li | Global contour saliency and local co-linear interactions[END_REF] and on the awake monkey [START_REF] Kapadia | Improvement in visual sensitivity by changes in local context: parallel studies in human observers and in V1 of alert monkeys[END_REF]. It is assumed that lateral interactions between neurons in primary visual cortex are the physiological substrate of these effects (e.g. [START_REF] Kapadia | Improvement in visual sensitivity by changes in local context: parallel studies in human observers and in V1 of alert monkeys[END_REF][START_REF] Gilbert | Adult cortical dynamics[END_REF], Polat, 1999), which appear to exhibit functional characteristics of visual learning (Polat & Sagi, 1994[START_REF] Dresp | The effect of practice on the visual detection of near-threshold lines[END_REF][START_REF] Adini | Context-enabled learning in the visual system[END_REF][START_REF] Li | Global contour saliency and local co-linear interactions[END_REF], possibly enabled by structurespecific (i.e. selective) cortical plasticity (e.g. [START_REF] Gilbert | Adult cortical dynamics[END_REF].

Contrast integration across spatial gaps and contrast signs

The spatial separation of co-linear stimuli and their contrast sign influence facilitating interactions. With coaxial gaps smaller than approximately 20 minutes of visual arc (arcmin) between a target and an inducer, stimuli with the same contrast sign produce facilitating interactions; with coaxial gaps larger than 20 arcmin, combinations of stimuli of either contrast sign also produce noticeable contrast detection facilitation [START_REF] Polat | Lateral interactions between spatial channels: Suppression and facilitation revealed by lateral masking experiments[END_REF][START_REF] Morgan | Contrast detection facilitation by spatially separated targets and inducers[END_REF][START_REF] Wehrhahn | Detection facilitation by collinear stimuli in humans: Dependence on strength and sign of contrast[END_REF]. Similar observations have been reported from orientation discrimination experiments with colinear lines and Gabor patches [START_REF] Brincat | Integration of foveal orientation signals : Distinct local and long-range spatial domains[END_REF]. These findings lend strong support to models which assume the existence of two stages, or functional regimes, of visual spatial integration: a short-range regime that is sensitive to local characteristics of visual stimuli such as their contrast signs, and a long-range regime that disregards local stimulus properties, such as the sign of contrast, and is sensitive only to the global configuration (e.g. [START_REF] Grossberg | Neural dynamics of perceptual grouping: textures, boundaries, and emergent segmentations[END_REF]. The long-range regime of visual spatial integration describes a stage of processing that is critical to form and object perception, since it generates the grouping of contour information across space, the emergent segmentation of features, and the figure-ground segregation of shapes.

Recent work by [START_REF] Tzvetanov | Short-and long-range effects in line contrast integration[END_REF] suggests that the spatial limit of the long-range regime of integration of two co-linear lines of a given length, presented on plain backgrounds, may extend to a co-axial target-inducer separation of 150 arcmin.

Moreover, data from psychophysical experiments by [START_REF] Li | Global contour saliency and local co-linear interactions[END_REF], with co-linear line targets and inducers presented on noisy backgrounds, suggest that the spatial limits of long-range integration are situated within a dynamic range of co-linear separations and may depend on training, or visual learning.

How "insensitive" is long-range spatial integration to the contrast sign of stimuli?

Assessing whether parts belong to one and the same or to different objects in the visual field is one of the major tasks the human perceptual system has to accomplish.

While two-stage integration models such as those referred to above use the working hypothesis that local information of contrast sign is irrelevant to visual grouping across larger gaps between stimuli activating the long-range regime of spatial integration, other models assume that whether a given information carried by a stimulus is relevant or not to this integration or grouping process is determined by probabilistic mechanisms. [START_REF] Feldman | Bayesian contour integration[END_REF] suggested that contour grouping relies on a probabilistic process by which spatial information from co-linear stimuli in the visual field is computed on the basis of likelihood estimates. In his model, spatial integration is defined as a process that makes decisions which, under conditions of uncertainty, are based on conditional probabilities associated with any event that may be critical to the perceptual interpretation of a visual scene or a stimulus. Feldman successfully applied Bayesian probability theory to perceptual judgements as to whether a given configuration of dots may form a smooth contour or not.

Extended to other situations, such as those where such a probabilistic process would have to decide whether two separate lines in the visual field form a contour or not, identical lines with constant spatial separation and the same contrast sign, for example, would be given equal probabilistic weights; objects with otherwise identical properties and constant separation, but opposite contrast signs, would be given different, conditional probabilistic weights. A probabilistic model would, in this case, imply that spatial contrast integration, or visual grouping, is sensitive to the contrast sign of stimuli.

As mentioned earlier, psychophysical data on facilitating interactions between colinear targets and inducers in contrast detection tend to support the view that the longrange regime of spatial integration, the one supposed to generate visual grouping and the formation of object contours, is not sensitive to the contrast sign of stimuli. However, phenomenal observation suggests that configurations consisting of stimuli with opposite contrast signs produce less salient visual groups compared with configurations consisting of stimuli with identical sign (Fig 1). Recently, [START_REF] Tzvetanov | Psychophysique et modélisation des traitements de bas niveau soustendant la vision des contours d'objets[END_REF] found results showing that co-linear targets and inducers with identical contrast signs systematically produce stronger facilitating interactions than stimuli with opposite contrast polarities in a contrast detection task activating the spatial regime of long-range line contrast integration (i.e. coaxial separations larger than 20 arcmin between target and inducer). The differences in thresholds found were small, but systematic for two out of three observers. How insensitive is the long-range regime of spatial integration to the contrast sign of stimuli ?

Given that the studies on facilitating long-range spatial interactions mentioned above concern data obtained with relatively small numbers of subjects, generally not more than three or four, it may be possible that a sensitivity to contrast sign exists, but could not be brought to the fore because a much larger sample of observations would have been required to do so.

Is long-range spatial integration across contrast signs influenced by a probabilistic mechanism ?

If long-range grouping of visual stimuli is sensitive to the contrast sign of stimuli, significant differences in contrast detection thresholds for targets and inducers of identical sign and thresholds for targets and inducers of opposite signs should be found with a sufficiently large number of subjects and observations. The next question to be addressed then, is which kind of mechanism or process would account for such a sensitivity to contrast sign. As mentioned above, probabilistic mechanisms of visual spatial integration (e.g. [START_REF] Feldman | Bayesian contour integration[END_REF] qualify as potential candidates.

To find out whether long-range spatial integration across contrast signs is consistent with a probabilistic model, it would have to be shown that facilitating longrange spatial interactions between co-linear targets and inducers can be "probabilistically" predicted. Here, we propose to test a simple model for visual grouping across co-linear space and contrast signs that associates mathematically and functionally plausible, conditional probabilistic weights with two stimuli of either identical or opposite contrast sign. Our study deals with contrast detection experiments designed to measure, or probe, contour integration or grouping. This is achieved by comparing the detection thresholds of line targets presented alone to thresholds of targets presented with a co-linear inducer. The relative facilitation magnitude generated by the presence of the inducer is computed as follows: a detection threshold ratio is obtained by dividing the threshold for the detection of the target presented with the co-linear inducer (d 1 ) by the threshold for the detection of the target presented without the inducer (d 2 ). The relative magnitude or amount of spatial facilitation (A) produced by the inducer on the detection of a co-linear target of either sign is obtained by subtracting the detection threshold ratio (R) from 1.

We consider a straightforwardly probabilistic model for a simple spatial configuration of one target and one co-linear inducer of identical length, with constant co-axial separation within the spatial limits previously defined as the long-range regime of integration (e.g. [START_REF] Wehrhahn | Detection facilitation by collinear stimuli in humans: Dependence on strength and sign of contrast[END_REF][START_REF] Brincat | Integration of foveal orientation signals : Distinct local and long-range spatial domains[END_REF][START_REF] Tzvetanov | Short-and long-range effects in line contrast integration[END_REF]. The wider theoretical background of the model is that of probabilistic theory in general, and that of likelihood ratios, or conditional probabilistic weights, associated with conditions of visual stimulation (e.g. Knill & Richards, 1996), in particular. To generate quantitative predictions, we start by defining a deterministic case of contour grouping where the likelihood that the local elements will produce spatial facilitation, i.e. will group, corresponds to a probability of p = 1. With two co-linear lines, such a case would be that of a target and an inducer with equal length and polarity, separated by a spatial gap that is empirically known to produce grouping or spatial facilitation. When no further variations occur in such a configuration, or in the conditions under which it is presented, its conditional probabilistic weight takes a theoretical, empirically justified, value of pw = 1. This conditional probabilistic weight changes when further variations are introduced in the initial spatial configuration, or in the conditions under which it is presented. For example, if half of the spatial configuration defined by the two co-linear lines is given the other sign, the conditional probabilistic weight of the new configuration is reduced to half that of the initial condition. This reduction is, in fact, the mathematical consequence of the assumption that contrast sign influences grouping or spatial facilitation through a probabilistic mechanism, with a likelihood ratio of 1/1 for the integration of a single contour across a single sign, and a likelihood ratio of 1/2 for the integration of a single contour across two signs, all other things, such as spatial separation or inducer contrasts, being equal. This way of reasoning is novel in the domain of contrast detection studies, where the usual models use summative or multiplicative mechanisms. In fact, if we simply consider the visual brain as a machine that has to make a decision whether to group two lines into a single one or not, it becomes clear that, under conditions of optimal spatial separation, it will readily group two lines of identical polarity into a single line by producing an integrated neural firing pattern coding for that polarity. When these two lines are given opposite polarity, the visual brain has to make a further decision. This consists of choosing, out of two possibilities, the polarity to be coded for by the grouped response given that an integrated firing pattern cannot code for two polarities at the same time. When only half of the configuration that is to be grouped signals for the polarity that will be chosen, then the resulting, integrated firing rate should only be half the initial rate. This would mean that the overall grouping power of the configuration is reduced to half that of the initial configuration with the single sign. This then leads to the hypothesis that the amount of psychophysical spatial facilitation produced by the configuration with two different contrast polarities would be only half that produced by the initial configuration.

The relative magnitude (A) of spatial facilitation produced by a configuration of co-linear targets and inducers depending on its conditional probabilistic weight (pw) is then predicted by

A = (1 -R) x pw;
the probabilistic weight of a configuration of two stimuli with only one contrast sign is, as explained above, considered to be pw = 1. The probabilistic weight of a configuration with two stimuli carrying opposite contrast signs, given that the two stimuli possess equal length and constant separation, corresponds to the weight of the configuration with a single sign divided by two, pw = 1/2, as explained above. The relative amount of spatial facilitation generated by a configuration with a single contrast sign is then predicted by

A 1 = (1 -R) x 1
and the relative magnitude of spatial facilitation generated by configurations with the two opposite contrast signs by

A 2 = A 1 x 1/2
To test these predictions, we designed a contrast detection experiment where a line target was presented with and without a co-linear line inducer of identical length and with constant co-axial separation. The goal of the study was to clarify whether:

1) Long-range spatial integration of colinear lines is sensitive to the contrast sign of the individual stimuli. If such an assumption holds, co-linear lines of identical contrast sign would be expected to produce significantly stronger facilitation than co-linear lines of opposite sign.

2) The long-range regime of spatial integration generates constant facilitatory interactions between co-linear targets and inducers (e.g. [START_REF] Tzvetanov | Short-and long-range effects in line contrast integration[END_REF]. This would lead to the prediction that any effect of contrast sign should be of similar magnitude for any target-inducer separation that falls within the long-range spatial regime.

3) The magnitude of the effect of contrast sign on long-range spatial integration in a sufficiently simple spatial configuration is predicted by the average absolute magnitude of spatial facilitation multiplied by a probabilistic weight, as given above.

METHODS

We measured contrast detection thresholds of a line target presented foveally for a brief duration on a computer screen, either by itself (control condition) or simultaneously with a co-linear line, the so-called inducer (test condition). The contrast sign of target line and inducer was varied so that all four possible combinations of signs were produced in the test conditions: 1) dark target with dark inducer, 2) dark target with bright inducer, 3) bright target with bright inducer and 4) bright target with dark inducer. The contrast intensity (Weber ratio) of dark and bright inducers was identical. The coaxial distance between target and inducer was constant in each of the two experiments, each corresponding to an angular target-inducer separation within the long-range domain of spatial integration, as defined previously in the literature (e.g. [START_REF] Wehrhahn | Detection facilitation by collinear stimuli in humans: Dependence on strength and sign of contrast[END_REF][START_REF] Brincat | Integration of foveal orientation signals : Distinct local and long-range spatial domains[END_REF][START_REF] Tzvetanov | Short-and long-range effects in line contrast integration[END_REF]. In a first experiment, we tested all of the 20 observers for a coaxial target-inducer separation of 25 arcmin. To confirm that similar results are obtained at a larger target-inducer separation within the long-range spatial regime, we tested four of these 20 observers for a co-axial target-inducer separation of 80 arcmin.

Subjects

20 young volunteers with normal or corrected-to-normal vision participated in the first experiment. All subjects were asked to accomplish a given experiment in two successive afternoon sessions. The first afternoon session served as a training session.

Four of these 20 observers then participated in the second experiment with the larger target-inducer separation. This corresponded to two more, successive afternoon sessions for each of these four subjects.

Training

For training, each observer had to accomplish a full set of test and control conditions, presented in random order, on the first afternoon of a given experiment. This corresponded to 240 trials per experimental condition, with four test conditions and one control for each target polarity, representing a total of 1440 training trials per observer.

Individual data from the training session were analyzed to check whether a given observer was able to produce consistent data within a given experimental condition.

Observers who failed to produce acceptable psychometric functions, or who found it difficult to concentrate during the task, were not asked to participate in the experimental session the following afternoon.

Stimuli

The line stimuli were generated by an IBM compatible PC equipped with a graphic card (VGA Trident), and presented on a monochrome computer screen with a 60 Hz frame rate and a resolution of 640x480 pixels. For the measurement of contrast thresholds for the detection of the target line, six different luminance levels were generated by combinations of RGB signals, calibrated with a MINOLTA photometer. In the case of the bright target line, these luminance levels corresponded to 14.2, 14.5, 14.8, 15.1, 15.4 and 15.7 cd/m2; in the case of the dark target line, to 9.8, 9.5, 9.2, 8.9, 8.6 and 8.3 cd/m2. The luminance of the background on which the lines were presented was 12 cd/m2. The luminance of the co-linear inducer was 16 cd/m2 in the case of the dark inducer, and 8 cd/m2 in the case of the dark inducer. Target lines and co-linear inducers had an equal angular length of 20 arcmin, an equal width of one arcmin, and were separated by a coaxial gap of 25 arcmin in the first experiment, and by a coaxial gap of 80 arcmin in the second experiment.

Procedure

A stimulus presentation, or trial, corresponded to two successive stimulus intervals separated by a temporal delay of 200 milliseconds, with the target presented randomly during either the first or the second stimulus interval. The inducer was presented in both stimulus intervals in the test conditions. Targets and inducers were preceded by a brief sound, and presented for about 32 milliseconds in a given stimulus interval. They were presented simultaneously in stimulus intervals containing an inducer and a target. In a given trial block, the six different luminance levels of the target were presented 40 times in a random order according to the method of constant stimuli. A twoalternative temporal forced-choice procedure was employed, where observers had to press one of two possible keys on the computer keyboard to indicate whether they had seen the target in the first or the second of the two successive stimulus intervals. A new trial was initiated 800 milliseconds after a response had been given.

RESULTS

The individual data produced by observers in the experimental session on the second afternoon of a given experiment were used for further analysis. In a first step, thresholds for the detection of the target lines in the different conditions described above were determined. Each threshold estimate was based on a total number of 240 (6x40) trials. The percentage, or probability (p), of correct detection of the target line was calculated for each observer and each experimental. These probabilities were transformed On the basis of these individual thresholds, the detection threshold ratio (R), comparing the threshold measured with a colinear inducer against the threshold measured in the no-inducer situation in a given experimental condition, was computed. This ratio is obtained by dividing the threshold for the detection of the target presented with the colinear inducer (d 1 ) by the threshold for the detection of the target presented without the inducer (d 2 ). The relative amount or magnitude of spatial facilitation (A) produced by an inducer of a given sign on the detection of a co-linear target of a given sign was computed by subtracting the threshold ratios (R) from 1.

Target-inducer separation of 25 arcmin

Figure 2 shows relative magnitude of facilitation in the four experimental conditions for each of the 20 observers, identified by their initials. The data reveal that, occasionally, a specific configuration of opposite contrast signs may produce a relative amount of facilitation that is similar or identical to the magnitude produced by a configuration of identical sign. However, configurations with a single contrast sign generally tend to produce greater relative magnitudes of facilitation than configurations with the two opposite contrast signs.

The average A produced by a bright inducer on the detection of a bright target is 0.29 with a standard error of 0.013 (N=20), and the average A produced by a dark inducer on the detection of a dark target is 0.28 with a standard error of 0.020 (N=20). This gives an average relative amount or magnitude of facilitation (A 1 ) of 0.285 and a standard error of 0.0165 (N=40) for configurations consisting of stimuli with identical contrast signs.

The average A produced by a bright inducer on the detection of a dark target is 0.14 with a standard error of 0.016 (N=20), and the average A produced by a dark inducer on the detection of a bright target is 0.12 with a standard error of 0.014 (N=20). This gives an average magnitude of facilitation (A 2 ) of 0.13 and a standard error of 0.015 (N=40) for configurations consisting of targets and inducers with opposite contrast signs.

Figure 3 represents the average relative amount of facilitation plotted for each of the 20 observers and the two combinations of contrast signs in the stimulus configurations. The graph shows that the configurations with a single contrast sign systematically produce stronger facilitation than the configurations with the two opposite contrast signs. This effect of the contrast sign of the stimuli on the amount of facilitation was found to be statistically significant (T (1,78) = 9.7598; p<.001)).

Target-inducer separation of 80 arcmin

Figure 4 shows relative magnitudes of facilitation in the four experimental conditions for each of the four observers, identified by their initials. The data reveal that configurations with a single contrast sign generally tend to produce greater magnitudes of facilitation than configurations with the two opposite contrast signs. The effects are similar to those obtained with a target-inducer separation of 25 arcmin.

The average A produced by a bright inducer on the detection of a bright target is 0.25 with a standard error of 0.017 (N=4), and the average A produced by a dark inducer on the detection of a dark target is 0.30 with a standard error of 0.044 (N=4). This gives an average relative amount of facilitation (A 1 ) of 0.275 and a standard error of 0.03 (N=8) for configurations consisting of stimuli with identical contrast signs. The average A produced by a bright inducer on the detection of a dark target is 0.14 with a standard error of 0.036 (N=4), and the average A produced by a dark inducer on the detection of a bright target is 0.16 with a standard error of 0.035 (N=4). This gives an average magnitude of facilitation (A 2 ) of 0.15 and a standard error of 0.0355 (N=8) for configurations consisting of targets and inducers with opposite contrast signs.

Figure 5 shows the average relative magnitude of facilitation for each of the four observers and the two combinations of contrast signs in the stimulus configurations.

Again, the configurations with a single contrast sign systematically produce stronger facilitation than the configurations with the two opposite contrast signs. The effect of the contrast sign of the stimuli on the amount of facilitation at a target-inducer separation of 80 arcmin is statistically significant (T (1,14) = 4.190; p<.01)).

Probabilistic model

The psychophysical data were compared with the probabilistic model given above for a simple configuration of one line target and one co-linear line inducer of identical size with constant co-axial separation in the long-range regime of spatial integration. In the model, the probabilistic weight (pw) of a target-inducer configuration with a single contrast sign is considered to be maximal (pw = 1). This leads to the prediction that the probabilistic weight of a target-inducer configuration with the two, opposite contrast signs would be expected to be half that amount (pw = 1/2). The relative amount or magnitude of facilitation (A) multiplied by the conditional probabilistic weight (pw) of the configuration producing that magnitude predicts the effect of the sign of contrast of the stimuli. In the model as given above, the psychophysical result for a target-inducer configuration with the two, opposite contrast signs is predicted by

A 2 = A 1 x 1/2 which gives A 2 = 0.285 x 1/2
in the case of the target-inducer separation of 25 arcmin with thresholds from 20 observers. This gives a hypothetical relative magnitude of facilitation of 0.1425 for target-inducer configurations with opposite contrast signs. The theoretical value differs by only 0.0125 from the observed average value, given above, for this condition. Such a difference represents less than the standard error for the condition with targets and inducers with opposite contrast signs. The model, therefore, produces statistically reliable predictions of the psychophysical observations. The dotted horizontal line in the graph shown in Figure 3 indicates the predicted A 2 for targets and inducers of opposite contrast sign, the straight horizontal line indicates the average A 2 as observed in the experiment.

The prediction for a target-inducer configuration with opposite contrast signs and a target-inducer separation of 80 arcmin is

A 2 = 0.275 x 1/2
This gives a hypothetical relative magnitude of facilitation of 0.1375 for target-inducer configurations with opposite signs. This theoretical value differs by 0.0125 from the observed average value, given above, for this condition. The difference between prediction and data obtained with four observers for a target-inducer separation of 80 arcmin represents less than the standard error, given above, for the condition with targets and inducers with opposite contrast signs. The probabilistic model, again, is shown to produce statistically reliably predictions. The dotted horizontal line in the graph represented by Figure 5 indicates the predicted A 2 for targets and inducers of opposite contrast sign, the straight horizontal line indicates the average A 2 as observed in the experiment.

DISCUSSION

Our observations unambiguously show that long-range spatial integration of visual information across co-linear space and contrast signs is sensitive to the contrast sign of stimuli. They therefore clarify that, provided a sufficiently large number of observations is collected with a sufficiently large sample of trained observers, co-linear targets and inducers of identical contrast sign are found to generate significantly stronger long-range spatial facilitation in a line contrast detection task than stimuli with opposite contrast signs, although thresholds in the two conditions can be similar or identical for a given individual in a given experiment. In addition, our observations confirm that any combination of contrast signs produces exclusively facilitating interactions between colinear targets and inducers for target-inducer separations within the limits of the longrange spatial regime of integration. Therefore, they are basically consistent with earlier conclusions from similar experiments, conducted with considerably smaller numbers of observers, that long-range spatial integration is not selective to the contrast sign of colinear stimuli (e.g. [START_REF] Dresp | Contour integration across polarities and spatial gaps: From contrast filtering to bipole cooperation[END_REF][START_REF] Wehrhahn | Detection facilitation by collinear stimuli in humans: Dependence on strength and sign of contrast[END_REF][START_REF] Dresp | Dynamic characteristics of spatial mechanisms in contour processing by the human visual system[END_REF][START_REF] Brincat | Integration of foveal orientation signals : Distinct local and long-range spatial domains[END_REF]. However, our data clarify that "non-selectivity" to contrast sign does not imply "insensitivity". They thus reveal an important, new aspect of longrange spatial interactions which has theoretical implications with regard to certain twostage models which claim that the second stage of visual spatial integration, also referred to as the grouping stage or the long-range spatial regime, disregards the contrast signs of the individual stimuli of a spatial configuration (e.g. [START_REF] Prazdny | Illusory contours are not caused by simultaneous brightness contrast[END_REF][START_REF] Grossberg | Neural dynamics of perceptual grouping: textures, boundaries, and emergent segmentations[END_REF].

Implications for two-stage models of spatial grouping

Two-stage models of grouping, such as that suggested by [START_REF] Grossberg | Neural dynamics of perceptual grouping: textures, boundaries, and emergent segmentations[END_REF], claim that the short-range, or first stage of processing of visual spatial information, is selective to local stimulus properties such as their contrast sign, whereas the long-range, or second stage of integration, which operates across larger spatial gaps, would be selective to configurational properties only. This theoretical assumption seemed justified by the argument that, in order to achieve coherent global percepts, long-range grouping needs to eliminate local contrast signs via some kind of summative mechanism (e.g [START_REF] Prazdny | Illusory contours are not caused by simultaneous brightness contrast[END_REF][START_REF] Grossberg | Neural dynamics of perceptual grouping: textures, boundaries, and emergent segmentations[END_REF]Reid & Shapley, 1988). Such a rationale is supported by evidence that contour grouping operates efficiently across opposite contrast signs (e.g. [START_REF] Field | Contour integration by the human visual system: Evidence for a local "association field[END_REF], and that various phenomena of perceptual interpolation, including the one seen in figures with illusory contours, are as strong in spatial configurations of opposite contrast signs as in configurations of a single sign (e.g [START_REF] Prazdny | Illusory contours are not caused by simultaneous brightness contrast[END_REF][START_REF] Prazdny | On the nature of inducing forms generating perceptions of illusory contours[END_REF][START_REF] Dresp | Illusory form from inducers with opposite contrast polarity: Evidence for multi-stage integration[END_REF]. However, although the assumption of a non-selectivity of long-range grouping to local contrast signs does clearly hold, the claim that information of contrast polarity is completely irrelevant (e.g. [START_REF] Brincat | Integration of foveal orientation signals : Distinct local and long-range spatial domains[END_REF], or eliminated by summative mechanisms [START_REF] Grossberg | Neural dynamics of perceptual grouping: textures, boundaries, and emergent segmentations[END_REF] at that stage of processing, cannot be sustained in the light of our present findings. The fact that a simple spatial configuration of two identical colinear lines with opposite contrast signs is shown to have half the weight of that of two identical lines of the same sign in the genesis of facilitating interactions is consistent with a probabilistic mechanism of spatial integration across contrast signs. Such a mechanism would preserve the relative weight of a given contrast sign at the long-range stage of integration rather than completely eliminate differences in polarity. This is consistent with a probablistic theory of contour vision, like that suggested by [START_REF] Feldman | Bayesian contour integration[END_REF], which is based on Bayesian probability theory applied to visual perception (e.g. Knill & Richards, 1996).

Probabilistic contour integration

Feldman ( 2001) argued that a probabilistic theory of contour integration requires an explicit model, in terms of appropriate likelihood functions or probabilistic weights, of the alternative grouping hypotheses that are possible for a given configuration. He proposed conditional likelihood functions that successfully predict explicit grouping of individual dots into a smooth contour, or which grouping interpretation is likely to be chosen by a human observer when a dot configuration is ambiguous, and can be grouped into either a single or two distinct virtual lines. The model we propose here is consistent with such a probabilistic approach, but applies to a different kind of perceptual task. In spatial contrast integration studies, which are designed to study visual grouping across co-linear space (e.g. [START_REF] Polat | Lateral interactions between spatial channels: Suppression and facilitation revealed by lateral masking experiments[END_REF]1994[START_REF] Dresp | Bright lines and edges facilitate the detection of small light targets[END_REF][START_REF] Tzvetanov | Short-and long-range effects in line contrast integration[END_REF], observers are not asked to judge how co-linear elements would group, but their grouping potential is inferred from the magnitudes of facilitating spatial interactions between co-linear elements. Here, for the simplest possible configuration of two co-linear lines, we apply a simple probabilistic combination rule with mathematically coherent probabilistic weights attached to configurations of identical and opposite contrast signs.

These probabilistic weights successfully predict the magnitude of facilitating spatial interactions between co-linear lines with opposite contrast signs in comparison with facilitating interactions between lines of identical sign. Such local spatial interactions have been identified as probes for the neurophysiological substrates of contour grouping (e.g. [START_REF] Kapadia | Improvement in visual sensitivity by changes in local context: parallel studies in human observers and in V1 of alert monkeys[END_REF]. They involve visual cortical plasticity, as shown recently in the studies by [START_REF] Adini | Context-enabled learning in the visual system[END_REF] and [START_REF] Li | Global contour saliency and local co-linear interactions[END_REF].

The spatial boundaries of long-range contour integration and the influence of visual plasticity

The present findings, by revealing similar magnitudes of spatial facilitation for coaxial target inducer separations of 25 and 80 arcmin, suggest that the effect of contrast sign on local contour interactions is likely to be constant across the spatial regime of long-range integration. Findings from earlier studies (e.g. [START_REF] Brincat | Integration of foveal orientation signals : Distinct local and long-range spatial domains[END_REF][START_REF] Tzvetanov | Short-and long-range effects in line contrast integration[END_REF] suggested that the spatial limits of the long-range regime of integration of two co-linear lines extend between coaxial separations of 20 and 150 arcmin. Recently, [START_REF] Li | Global contour saliency and local co-linear interactions[END_REF] presented local contour configurations consisting of co-linear line segments embedded in a context of randomly oriented lines. Their results

show that the detectability of a contour configuration presented in global noise is determined by interactions between the relative number of co-linear lines and their coaxial separation. These findings further reveal the complexity of contextual effects on contour integration, which not only depends on the local characteristics of the contour elements as such, but also on the visual information by which they are surrounded.

Moreover, the absolute critical spacing between co-linear lines was shown to increase with decreasing surround noise density. This important finding highlights the role of topdown influences on early mechanisms of visual integration. Top-down influences are suitable candidates to account for apparently controversial psychophysical data related to the spatial limits of long-range facilitating interactions.

There is general agreement on the fact that co-linear facilitatory interactions drop off as the spatial separation between target and inducer increases, but it has remained unclear whether they drop off abruptly, or decrease gradually, and whether the long-range spatial regime has clear upper and lower boundaries. While data by Polat & Sagi (1994) or [START_REF] Kapadia | Improvement in visual sensitivity by changes in local context: parallel studies in human observers and in V1 of alert monkeys[END_REF] show gradually decreasing detection facilitation with increasing spatial separation between co-linear stimuli, [START_REF] Tzvetanov | Short-and long-range effects in line contrast integration[END_REF] have shown that spatial interactions between a low contrast inducing line and a co-linear target line show gradually decreasing detection facilitation between co-linear separations of 5 and 25 arcminutes and a plateau effect of constant facilitation with an abrupt drop back to control detection levels between co-linear separations of 140 and 160 arcminutes. The four subjects in that condition were all widely experienced psychophysical observers and, in addition, highly trained in that specific study task, i.e. had spent a whole year in the lab working on the project, which meant that they often spent several hours per day repeating measurements over relatively long periods. Results from another condition with three unexperienced observers, who were minimally trained for the study task, showed that a high-contrast inducer produced gradually increasing facilitation between co-linear separations of 5 and 25 arcminutes and a plateau effect of constant facilitation with a much more gradual return to control detection levels between co-linear separations of 80 and 170 arcminutes. These observations not only show that the contrast intensity of an inducer critically influences co-linear facilitation/masking, as previously demonstrated also by Polat & Sagi (1994 a), but they furthermore suggest an influence of individual visual experience, or training levels, on long-range interactions. The abrupt change from constant long-range facilitation to control threshold levels seen in the four highly trained and experienced observers in [START_REF] Tzvetanov | Short-and long-range effects in line contrast integration[END_REF] study could be explained by a specific training effect where the long-range regime of facilitating interactions is pushed to its upper limit, and the mechanism then responds in an "all or nothing" manner to stimuli beyond that limit. This interpretation is supported by further data from [START_REF] Tzvetanov | Psychophysique et modélisation des traitements de bas niveau soustendant la vision des contours d'objets[END_REF], but requires additional control experiments to be fully validated. If such an explanation holds, however, this would mean that training not only reinforces contextual effects on grouping at a given, constant spatial separation (e.g. [START_REF] Dresp | The effect of practice on the visual detection of near-threshold lines[END_REF], but also modifies the spatial boundaries of long-range integration.

Visual learning has been reported to influence the upper spatial limit of longrange integration across co-linear space (e.g. Polat & Sagi, 1994 b). It specifically seems to enable local contour detection across increasingly larger co-axial gaps, up to a limit defined by a co-axial separation of about two degrees of visual angle between co-linear lines [START_REF] Li | Global contour saliency and local co-linear interactions[END_REF], or possibly more, depending on noise density, or the number of distracting context elements in a display. This would be consistent with functional properties of cortical interactions in V1 (e.g. [START_REF] Grossberg | Contrast-sensitive perceptual grouping and object-based attention in the laminar circuits of the primary visual cortex[END_REF], which exhibit plasticity. Cortical plasticity, by enabling the visual system to adapt quickly to new stimulations, would fulfill an important ecological role in the development of specific visual skills and visual experience in general.

We conclude that multiple stages of processing influence spatial mechanisms of visual integration across gaps and contrast polarities. Such an integration preserves the relative weight of a local contrast sign, possibly at any separation between individual stimuli within the long-range spatial domain at a given, individual level of training. The process or mechanism by which this is achieved appears to obey, as the results of our study would suggest, a probabilistic principle when the simplest possible local configuration of two co-linear lines is presented to human observers on a plain background. [START_REF] Feldman | Bayesian contour integration[END_REF] pointed out that probabilistic mechanisms represent a "rational strategy for perception". The present data encourage us to suggest that probabilistic processing, as a functional property of the brain, may influence any stage of neuro-cognitive processing where critical decisions about a visual stimulus are to be made.

FIGURE CAPTIONS

Figure 1

The human perceptual system groups local contour signals across spatial gaps and contrast signs to achieve figure-ground segregation of objects and shapes. However, shapes with contours of a single contrast sign (upper half) may appear more salient than shapes with contours of alternating contrast signs (lower half).

Figure 2

Relative magnitude or amount of facilitation produced by target-inducer configurations of two co-linear lines with a coaxial separation of 25 arcmin for each of the 20 observers tested. Even though, for a given observer, a configuration with the two opposite contrast signs (bright vs dark) may produce a facilitating effect that is similar to that produced by a configuration with a single sign (see data of observers AF, AM, and EL), the latter are shown to produce generally stronger facilitating effects.

(

  ln (p/1p)) to produce linear psychometric functions (y = axb) of the absolute difference between the luminance of the target line and the luminance of the background. Detection thresholds (x) were calculated on the basis of the parameters of the individual psychometric functions obtained for each experimental condition (x = (y + b)/a). The detection threshold here corresponds to p = 0.75, which gives ln = 1.09 on the ordinate (y) of the psychometric function.

Figure 3

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Figure 5

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Figure 1 Figure 2 SFigure 5 S

 125 Figure 1