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An important field of blood oxygen level dependent (BOLD) functional magnetic resonance imaging (fMRI) is the investigation of effective connectivity, that is, the actions that a given set of regions exert on one another. We recently proposed a data-driven method based on the partial correlation matrix that could provide some insight regarding the pattern of functional interaction between brain regions as represented by structural equation modeling (SEM). So far, the efficiency of this approach was mostly based on empirical evidence. In this paper, we provide theoretical fundaments explaining why and in what measure structural equation modeling and partial correlations are related. This gives better insight regarding what parts of SEM can be retrieved by partial correlation analysis and what remains inaccessible. We illustrate the different results with real data.

Blood oxygen level dependent (BOLD) functional magnetic resonance imaging (fMRI) is an imaging technique that allows to dynamically and noninvasively follow metabolic and hemodynamic consequences of brain activity [

Path analysis, or structural equation modeling (SEM), has been the major way to examine effective connectivity in fMRI [

We recently proposed a novel approach to gain insight on effective connectivity. We first showed that, unlike marginal (i.e., regular) correlation, conditional correlation could account for many patterns of interaction as modeled by SEM [

We here quickly recall the essentials of a previous study on which our investigation of partial correlation relies. For more detail, we refer the reader to Bullmore et al. [

Sample marginal correlation coefficients of the real data set examined in Bullmore et al. [

(1) | (2) | (3) | (4) | (5) | ||

VEC | PFC | SMA | IFG | IPL | ||

(1) | VEC | 1 | ||||

(2) | PFC | 0.661 | 1 | |||

(3) | SMA | 0.525 | 0.660 | 1 | ||

(4) | IFG | 0.486 | 0.507 | 0.437 | 1 | |

(5) | IPL | 0.731 | 0.630 | 0.558 | 0.517 | 1 |

A plausible structural model, henceforth referred to as the “theoretically preferred model” (or “TP”), was proposed and is represented in Figure

VEC

PFC

Structural models and path coefficients corresponding to the theoretically preferred (a) and best fit (b) models (from [

We now go back to a different perspective. Indeed, the structure of any SEM entails specific constraints on the covariance matrix, as well as other matrices characteristic of the process, such as the concentration matrix and the marginal and partial correlation matrices.

Generally speaking, a structural model can be defined in matrix form as

Classically, we further assume that the noise

Indeed,

In other words,

for the TP model:

for the BF model:

Partial correlation constraints in the TP and BF models (1/2). For each link between regions and each model, examination of whether (_{1}) and (_{2}) are satisfied.

Partial correlation constraints in the TP and BF models (2/2). For each link between regions and each model, examination of whether (_{1}) and (_{2}) are satisfied.

Structures that render either constraint

As correlation matrices are often easier to interpret than covariance matrices, we can decide to examine partial correlation matrices rather than concentration matrices. The partial correlation coefficient between two regions

As we saw, a structural model has unique implications in terms of the structural pattern of partial correlation that can be expected from the data. Since the partial correlation matrix is a quantity that can be inferred from the data, we can use partial correlation analysis as a way to validate a structural model by comparing what is expected and what is observed.

The approach consists of translating the structural hypotheses in terms of partial correlation. Indeed, according to Tables

Assessing the validity of the various hypotheses can be done by first estimating the partial correlation matrix. Inference of

Evidence

0 | |

3 | |

6 | |

10 | |

20 | |

30 | |

40 |

Since we here focus on the partial correlation constraints entailed by the structural models, (

Real data. Relevance of hypotheses related to the TP and the BF models, respectively. Log odd ratios above a threshold of 10 dB are represented in bold.

Structural model | Constituting hypotheses | Structural constraints | |
---|---|---|---|

TP | 1.6 dB | ||

9.7 dB | |||

BF | 1.6 dB | ||

6.4 dB | |||

9.7 dB |

In this paper, we further examined how partial correlation could be used to investigate effective connectivity in fMRI. We introduced theoretical fundaments explaining why and in what measure the structure of the partial correlation matrix can be related to a structural model. More precisely, we showed that, given a structural model, the partial correlation

When examining the global relevance of partial correlation analysis to the investigation of effective connectivity, we must jointly consider two complementary effects, namely, the theoretical relationship between structural models and partial correlation matrices on the one hand and, on the other hand, the quality of the inference process. From a purely theoretical standpoint, this result shows that partial correlation analysis comes up as a combination of two effects. First, constraints

Another theoretical issue that needs to be tackled is the fact that having a partial correlation that is not constrained to 0 (e.g.,

Another, more important issue deals with inference and how confident we can be in the partial correlation estimates and, critically, in the tests that their values are different from zero. The major difference between partial correlation and marginal correlation is that the former is obtained by removing the effect of

Altogether, these various factors, both theoretical and inferential, have different consequences on the relationship between the inferred pattern of partial correlation and the underlying structural model. Although we have observed a rather good agreement between expected and inferred patterns so far, in the lack of gold standard, these consequences must be further investigated.

Still, one of the main reasons why partial correlation analysis might become an important tool for the investigation of effective connectivity is that it is, to our knowledge, the only fully exploratory approach. Its key feature is its ability to retrieve local patterns of interaction. Indeed, while the method developed for the estimation of structural parameters, for example, (

In this paper, we determined whether certain coefficients could be considered as different from zero or not in a Bayesian framework. This led us to the use of the evidence

A last question is the possibility to apply partial correlation to other imaging modalities, such as electroencephalography (EEG) and magnetoencephalography (MEG). While the issue of removing the effect of other regions when considering the interactions between two regions remains relevant, whether partial correlation as defined here can provide a cogent solution remains to be investigated. One of the major properties of the fMRI signal is that, due to the convolution with the hemodynamic response, the temporal information that it conveys is usually considered as less relevant than in EEG or MEG. This is one of the major reasons why most EEG or MEG analyses are performed in the frequency domain. Of interest would therefore be to use partial correlation in this frequency domain. This analysis could be performed over time windows that are narrow enough to assume stationarity of the signal. How such an approach could be related to partial coherence [

Using standard Bayesian theory, it can be shown that the covariance matrix

sample

calculate

Once a large number