Elsevier

NeuroImage

Volume 32, Issue 1, 1 August 2006, Pages 228-237
NeuroImage

Partial correlation for functional brain interactivity investigation in functional MRI

https://doi.org/10.1016/j.neuroimage.2005.12.057Get rights and content

Abstract

Examination of functional interactions through effective connectivity requires the determination of three distinct levels of information: (1) the regions involved in the process and forming the spatial support of the network, (2) the presence or absence of interactions between each pair of regions, and (3) the directionality of the existing interactions. While many methods exist to select regions (Step 1), very little is available to complete Step 2. The two main methods developed so far, structural equation modeling (SEM) and dynamical causal modeling (DCM), usually require precise prior information to be used, while such information is sometimes lacking. Assuming that Step 1 was successfully completed, we here propose a data-driven method to deal with Step 2 and extract functional interactions from fMRI datasets through partial correlations. Partial correlation is more closely related to effective connectivity than marginal correlation and provides a convenient graphical representation for functional interactions. As an instance of brain interactivity investigation, we consider how simple hand movements are processed by the bihemispheric cortical motor network. In the proposed framework, Bayesian analysis makes it possible to estimate and test the partial statistical dependencies between regions without any prior model on the underlying functional interactions. We demonstrate the interest of this approach on real data.

Introduction

Blood oxygen level-dependent (BOLD) functional magnetic resonance imaging (fMRI), which allows to dynamically follow metabolic and hemodynamic consequences of brain activity (Chen and Ogawa, 1999, Huettel et al., 2004), has deeply modified our knowledge about the brain (Frackowiak et al., 2004). Since its appearance, it has primarily been concerned with locating brain processes in determinate regions, thus proposing a topography of brain activity (Friston et al., 1994, Worsley and Friston, 1995). However, this approach conveys a rather static idea of brain processes. By contrast, it is now believed that processing of a functional task by the brain can only be performed through interaction of segregated regions within a complex network (Hebb, 1949, Tononi et al., 1998, Frackowiak et al., 2004, Sporns et al., 2004). Consequently, various methods have been proposed in fMRI data analysis to extract information of interaction from datasets, most of which rely on either functional or effective connectivity (for reviews and discussions, see, e.g., Stone and Kötter, 2002, Horwitz, 2003, Lee et al., 2003).

Effective connectivity aims at examining the influence that regions exert on each other (Friston et al., 1993a). Its investigation requires three distinct levels of information: (1) the regions involved in the process and forming the spatial support of the network, (2) the presence or absence of interactions between each pair of regions, and (3) the directionality of the existing interactions. Step 1, though far from being trivial, has already been examined in detail by the existing literature. As a result, many methods are available to this end, including activation maps (Huettel et al., 2004), as well as methods deriving from functional connectivity, such as principal or independent component analysis (Friston et al., 1993b, Arfanakis et al., 2000), or correlation maps (Biswal et al., 1995, Biswal et al., 1997, Xiong et al., 1999). In our study, we assume that this step has been successfully completed, providing us with a set of regions and corresponding signals.

Step 2, on which we will henceforth concentrate, then focuses on functional interactions per se. Structural equation modeling (SEM) is the most widespread way to model effective connectivity (Gonzalez-Lima and McIntosh, 1995, Bullmore et al., 2000), even though dynamical causal modeling (DCM) has recently been developed to the same goal (Friston et al., 2003, Penny et al., 2004b). Both methods rely on the definition of a model in the form of a directed graph prior to the analysis. The graph links are usually thought of as anatomical connections that are functionally relevant for the experiment under consideration. In this setting, there is hence an anatomical connection underlying each functional interaction. Unfortunately, several drawbacks still hamper the use of SEM and DCM, most of which originate from the difficulty to propose a prior model of effective connectivity.

First, it is virtually impossible for current methods to perform extensive model comparison. Indeed, the complexity of interactivity analysis increases exponentially with the number of regions. For instance, working with D regions in SEM brings up to 4D(D−1)/2 potential graphs. For a mere D = 6 regions, this amounts to 415 ≈ 109 graphs, which is already humongous and computationally out of reach. The complexity of DCM is at least as high. Methods have been developed to cope with this issue (Bullmore et al., 2000, Mechelli et al., 2002, Penny et al., 2004a). Yet, they are only able to perform model comparison within a limited subset of potential models given a priori.

Accordingly, it lies on the expert's hands to sift through the set of possible graphs and provide only a few potentially relevant models on which the method will focus. This process requires information that usually comes from literature on brain anatomy. A large amount of information is available from previous studies regarding the underlying neuroanatomy of some networks. Such evidence may come from lesion studies, electrophysiological investigations in monkeys or imaging analyses. However, the results of those experiments are usually incommensurate with one another for various reasons. The relevance of each finding to a different or more general group of subjects cannot always easily be assessed. Information originating from monkey studies must be used with care to constrain human structural connections. In what measure patient case studies can be used for healthy subjects remains an open issue; conversely, results on healthy subjects cannot usually be applied to just any patient. Results from different modalities suffer from different kinds of artifacts that make them hard to compare. Last, and most importantly, only a very limited amount of information is available for many other networks (e.g., language, executive control).

Further hypotheses have hence to be made, such as concentrating on anatomical connections that are assumed to be functionally relevant to the task under investigation. As a matter of fact, even though the brain is strongly interconnected, the principle of segregation states that only a few regions should participate to a given functional task (Tononi et al., 1998). Consequently, it could be hypothesized that only few of the anatomical connections are actually used during the processing of a task, particularly if the given task is simple. Unfortunately, relating anatomical connections to functional interactions remains an open issue. A functional interaction requires, yet is not entailed by, the presence of an anatomical connection. The cortical motor network is a remarkable illustration of this fact. From a purely anatomical perspective, this network is very likely to be fully connected, preventing the use of SEM due to a lack of degrees of freedom. Nonetheless, it still seems meaningful to study how regions of this network interact when performing a simple hand movement.

These issues clearly show that existing methods of effective connectivity require much prior information relative to the structure of functional interactions. Since the goal of the study is precisely to investigate this structure, that information is lacking in most cases. One must then resort to information relative to anatomical connectivity, which is often diffuse and cannot be easily translated in terms of functional interactions.

Use of directed graphs as structural models finally raises conceptual questions inherent to directed graphs, such as observational equivalence or identifiability (Pearl, 2001), which have not fully been taken into account in fMRI data analysis yet. These problems, which are critical for directed acyclic graphs, are even more crucial for cyclic graphs, which are commonly used in effective connectivity modeling, while many theoretical questions remain unsolved (Spirtes, 1995, Pearl and Dechter, 1996, Neal, 2000). These points remain to be tackled before interpretation of SEM and DCM analyses can be fully trusted.

For all these reasons, many situations occur where existing model-based approaches of effective connectivity cannot be applied. The issue to be faced is then the following: assuming that we are provided with a network of brain regions and have no other information of any kind relative to their functional interactivity, how can the global interaction structure of the network be inferred from fMRI data? In other words, is it possible to propose data-driven measures of direct functional interactions?

Although computationally convenient, data-driven functional connectivity (e.g., temporal correlation) cannot be used to this end, for it is well known that it can emerge from various configurations that are not related to direct interactions. Unlike effective connectivity, functional connectivity hence does not ensure that each functional interaction will be supported by an anatomical connection. Indeed, we showed that one of the reasons why functional connectivity was unable to extract information that could be interpreted in terms of effective connectivity was that it could not quantify interaction mediation (Marrelec et al., 2005a, Marrelec et al., 2005b). We introduced conditional correlation as a way to circumvent this flaw and hypothesized that a mediated interaction should translate into a zero conditional correlation.

On this premise, we here propose a novel procedure that extracts partial correlation from the data. Partial correlation has the interesting features of providing a convenient summary of conditional independences (Whittaker, 1990) and, hence, of being more closely related to effective connectivity than marginal correlation. Furthermore, its use provides a convenient framework for graphical representation. Methods using partial correlation have already been proposed (e.g., McIntosh et al., 1996, Sun et al., 2004). In these previous approaches, though, partial correlation was used to eliminate the effect of the experimental design. In our approach, its use is rather to subtract and remove mutual dependencies on common influences from other brain areas. By conditioning the dependencies between two areas on other areas, the ensuing functional connectivity (i.e., partial correlation) reflects interactions between the two areas in question. Therefore, the use of partial correlation, allowing access to a quantity that is more closely related to direct interaction, takes the analysis of functional connectivity closer to the characterization of functional interactions in terms of effective connectivity. It is data-driven in the sense that, unlike existing methods such as SEM and DCM, it does not require any prior information regarding functional interactions to proceed. The partial correlation approach is also unique in that it provides a first insight into the effective connectivity of the network. As such, it is not meant to replace SEM or DCM but may act as a preliminary step and possibly allow for a more efficient use of those methods.

In this paper, we examine a simple instance of brain interactivity investigation. More precisely, we concentrate on the bihemispheric cortical motor network involved in simple hand movements. Much is known about the regions involved in motor processing and their anatomical connections, mostly through investigation in primates and studies in patients (Kunzle, 1978, Leichnetz, 1986, Zigmond et al., 1999, Gazzaniga, 2000). Within the hierarchical organization of the motor system, the cortex stands as the highest level of motor control. Cortical regions known to be involved in the motor network include at least the two sensorimotor cortices, the two premotor areas, and the two supplementary motor areas. The main cortical inputs to the motor areas stem from the premotor and the supplementary motor cortices. Cortico-cortical inputs are also present, originating from the opposite hemisphere. Movement can elicit complex brain processes that are still under investigation (Kandel et al., 2000, Swinnen, 2002). How the available anatomical information can be translated in terms of models of effective connectivity therefore often remains unknown. Few structural models have been proposed so far (Calautti and Baron, 2003, Rogers et al., 2004), none of which is based on the six aforementioned regions and considers the interactions between hemispheres.

The objective of this article is therefore to investigate functional brain interactivity within the aforementioned network during simple hand movements. Since the regions defining the functional network are already known, correlation analysis (and, hence, functional connectivity) can only confirm that the regions selected are indeed relevant for the analysis. As to effective connectivity, it cannot be applied without the definition of a directed graph. Since the anatomical information available regarding this network does not put any constraint on the anatomical connectivity, no such graph can be proposed without strong hypothesis. We hence examine what kind of information can be extracted from the data using a partial correlation analysis.

The outline of this paper is the following. In the next section, we present the partial correlation model. A Bayesian scheme is then proposed to infer the interaction structure from the data. The following section is devoted to demonstrating the relevance of this approach through the analysis of real data from simple hand movements. Further issues are addressed in the discussion.

Section snippets

Partial correlations

Our objective is to investigate the functional interactions occurring between the D = 6 following cortical regions: the two supplementary motor areas, RSMA and LSMA (R standing for “Right”, L for “Left”), the two sensorimotor cortices, RSMC and LSMC, and the two premotor cortices, RPMC and LPMC. This set of regions is denoted by R, i.e.,R={RSMA,LSMA,RSMC,LSMC,RPMC,LPMC}.

Let y = (yt)t = 1,…,T be the BOLD fMRI time courses of these six regions. Each yt is further assumed to be a realization of a D

Bayesian inference

This model being set, it is now necessary to infer the true partial correlation matrix underlying the group data. If we knew exactly the model parameters μ and Σ, the exact partial correlation coefficients would be unambiguously determined by Eq. (2). However, since the true values of μ and Σ are unknown and only partly accessible through the data, so is the value of Π, which must hence be inferred.

Imaging

The MR protocol was carried out with a General Electric 1.5 T Sigma system. Functional MRI using BOLD contrast was performed. The protocol included (1) two runs comprising 42 T2*-weighted functional volumes each, each volume covering the whole frontal lobes (TR/TE/flip angle: 3000 ms/60 ms/90°, 20 contiguous slices per volume, 5 mm slice thickness, in-plane pixel size: 3.75 mm × 3.75 mm); and (2) one axial inversion recovery three-dimensional T1-weighted image for anatomical localization.

Seven

Discussion

We introduced partial correlation analysis as a way to measure the statistical dependencies between two regions after removing the confounding effects of all other regions, hence providing data-driven measures that are closer to effective connectivity than marginal correlation. Given a set of regions and their corresponding time courses, we developed a Bayesian scheme that allowed to infer the underlying interaction structure from fMRI data. The proposed numerical sampling scheme allowed to

Conclusion

In this paper, we proposed to use partial correlations as data-driven measures of functional dependencies that are more closely related to effective connectivity than marginal correlations. In this framework, we measured the interaction strengths between six cortical regions of the motor network. Once the regions and the corresponding time courses were selected, the Bayesian scheme developed allowed for a fully data-driven procedure that led investigation of dependencies closer to effective

Acknowledgments

The authors are in debt to the two anonymous referees for significantly improving the quality of the original manuscript. They are also grateful to Vincent Perlbarg, Odile Jolivet, Saâd Jbabdi, and Pierre Bellec for insightful discussions. Guillaume Marrelec is supported partly by the Ministère de la Recherche, de la Science et de la Technologie du Québec and partly by the Fondation Fyssen (Paris, France). The work was supported by Grant PHRC AOR01109.

References (60)

  • L. Lee et al.

    A report of the functional connectivity workshop, Düsseldorf 2002

    NeuroImage

    (2003)
  • M.J. Lowe et al.

    Functional connectivity in single and multislice echoplanar imaging using resting-state fluctuations

    NeuroImage

    (1998)
  • A.R. McIntosh et al.

    Spatial pattern analysis of functional brain images using partial least squares

    NeuroImage

    (1996)
  • A. Mechelli et al.

    Effective connectivity and intersubject variability: using a multisubject network to test differences and commonalities

    NeuroImage

    (2002)
  • W.D. Penny et al.

    Comparing dynamic causal models

    NeuroImage

    (2004)
  • W.D. Penny et al.

    Modelling functional integration: a comparison of structural equation and dynamic causal models

    NeuroImage

    (2004)
  • M.A. Rocca et al.

    Cortical adaptation in patients with MS: a cross-sectional functional MRI study of disease phenotypes

    Lancet Neurol.

    (2005)
  • B.P. Rogers et al.

    Hemispheric asymmetry in supplementary motor area connectivity during unilateral finger movements

    NeuroImage

    (2004)
  • P.M. Rossini et al.

    Post-stroke plastic reorganisation in the adult brain

    Lancet Neurol.

    (2003)
  • O. Sporns et al.

    Organization, development and function of complex brain networks

    Trends Cogn. Sci.

    (2004)
  • J.V. Stone et al.

    Making connections about brain connectivity

    Trends Cogn. Sci.

    (2002)
  • F.T. Sun et al.

    Measuring interregional functional connectivity using coherence and partial coherence analyses of fMRI data

    NeuroImage

    (2004)
  • G. Tononi et al.

    Complexity and coherence: integrating information in the brain

    Trends Cogn. Sci.

    (1998)
  • N.S. Ward

    Plasticity and the functional reorganization of the human brain

    Int. J. Psychophysiol.

    (2005)
  • K.J. Worsley et al.

    Analysis of fMRI time-series revisited-again

    NeuroImage

    (1995)
  • B. Biswal et al.

    Functional connectivity in the motor cortex of resting human brain using echoplanar MRI

    Magn. Reson. Med.

    (1995)
  • B.B. Biswal et al.

    Simultaneous assessment of flow and BOLD signals in resting-state functional connectivity maps

    NMR Biomed.

    (1997)
  • C. Calautti et al.

    Functional neuroimaging studies of motor recovery after stroke in adults

    Stroke

    (2003)
  • W. Chen et al.

    Principles of BOLD functional MRI

  • D. Cordes et al.

    Mapping functionally related regions of brain with functional connectivity MR imaging

    Am. J. Neuroradiol.

    (2000)
  • Cited by (0)

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