Elsevier

NeuroImage

Volume 23, Supplement 1, 2004, Pages S264-S274
NeuroImage

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

https://doi.org/10.1016/j.neuroimage.2004.07.041Get rights and content

The brain appears to adhere to two fundamental principles of functional organisation, functional integration and functional specialisation, where the integration within and among specialised areas is mediated by effective connectivity. In this paper, we review two different approaches to modelling effective connectivity from fMRI data, structural equation models (SEMs) and dynamic causal models (DCMs). In common to both approaches are model comparison frameworks in which inferences can be made about effective connectivity per se and about how that connectivity can be changed by perceptual or cognitive set. Underlying the two approaches, however, are two very different generative models. In DCM, a distinction is made between the ‘neuronal level’ and the ‘hemodynamic level’. Experimental inputs cause changes in effective connectivity expressed at the level of neurodynamics, which in turn cause changes in the observed hemodynamics. In SEM, changes in effective connectivity lead directly to changes in the covariance structure of the observed hemodynamics. Because changes in effective connectivity in the brain occur at a neuronal level DCM is the preferred model for fMRI data. This review focuses on the underlying assumptions and limitations of each model and demonstrates their application to data from a study of attention to visual motion.

Introduction

Human brain mapping has been used extensively to provide functional maps showing which regions are specialised for specific functions (Frackowiak et al., 2003). A classic example is the study by Zeki et al. (1991) who identified V4 and V5 as specialised for the processing of colour and motion, respectively. More recently, these analyses have been augmented by functional integration studies, which describe how functionally specialised areas interact and how these interactions depend on changes of context.

Early analyses of functional integration used principal component analysis (PCA) to decompose neuroimaging data into a set of modes that are mutually uncorrelated both spatially and temporally. The modes are ordered according to the amount of variance they explain. By comparing the temporal expression of the first few modes with the variation in experimental factors, a distributed functional system associated with various factors can be identified (Friston et al., 1993). A more sophisticated use of PCA occurs in the context of generalised eigenimage analysis (Friston et al., 1997), where the principal component is found which is maximally expressed in one experimental condition or population and minimally expressed in another (e.g., control versus patient groups). If there are more than two experimental factors, this approach can be extended using a canonical variates analysis (CVA) or partial least squares (PLS) (MacIntosh et al., 1996).

More recently, independent component analysis (ICA) has been used to identify modes describing activity in a sparsely distributed network (McKeown et al., 1998). Such PCA/ICA-based methods are called analyses of functional connectivity as they are data-driven transform methods, which make no assumptions about the underlying biology. They are therefore of greatest practical use when it is not clear which regions are involved in a given task.

In contrast, analyses of ‘effective connectivity’ (see the following sections) are based on statistical models that make anatomically motivated assumptions (e.g., knowledge of structural connectivity) and restrict their inferences to networks comprising a number of preselected regions. Effective connectivity analyses are hypothesis driven rather than data driven and are most applicable when one can specify the relevant functional areas (e.g., from analyses of functional specialisation). The presence of connections, in the model, can be inferred from data obtained by invasive tracing procedures in primates, assuming homology between certain areas in the human and monkey brain. New imaging methodologies such as diffusion tensor imaging also hold the promise of providing information about anatomical connections for the human brain directly (Ramnani et al., 2004).

Detailed discussions of functional versus effective connectivity approaches can be found in chapters 48–53 of (Frackowiak et al., 2003). In this paper, we review the most widely used method for making inferences about functional integration from fMRI, namely, structural equation modelling (SEM). We also review dynamic causal modelling (DCM), a new approach that has been designed specifically for the analysis of fMRI time series.

The paper is structured as follows. The sections Structural equation models and Dynamic causal models describe the theoretical foundations of SEM and DCM, and the Attention to visual motion section presents exemplar analyses on fMRI data. We conclude with a discussion of the relative merits of the models in the Discussion section.

We use uppercase letters to denote matrices and lowercase to denote vectors. IK denotes the K × K identity matrix, 1K is a 1 × K vector of 1's and 0K is a 1 × K vector of zeros. If X is a matrix, Tr(X) denotes its trace, |X| its determinant, Xij the i, jth element, XT the matrix transpose, X−1 the matrix inverse, XT the transpose of the matrix inverse, vec(X) returns a column vector comprising its columns and ⊗ denotes the Kronecker product. The operator diag(x) returns a diagonal matrix with leading diagonal elements given by the vector x. log x denotes the natural logarithm. If p(x) = N (x; μ,Σ) then the d-dimensional random vector x is drawn from a multivariate Gaussian distribution with mean μ and covariance Σ. This is given byN(x;μ,Σ)=(2π)d/2|Σ|1/2exp(12(xμ)TΣ1(xμ))

Section snippets

Structural equation models

Structural equation models (SEMs) were developed in the field of econometrics and first applied to imaging data by McIntosh and Gonzalez-Lima (MacIntosh and Gonzalez-Lima, 1991). They comprise a set of regions and a set of directed connections. Importantly, a causal semantics is ascribed to these connections where an arrow from A to B means that A causes B. Causal relationships are thus not inferred from the data but are assumed a priori (Pearl, 1998).

An SEM with particular connection strengths

Dynamic causal models

Whereas SEM was developed in econometrics, dynamic causal modelling (DCM) (Friston et al., 2003) has been specifically designed for the analysis of functional imaging time series. The term ‘causal’ in DCM arises because the brain is treated as a deterministic dynamical system (see, for example, Section 1.1 in (Friston et al. (2003))) in which external inputs cause changes in neuronal activity, which in turn cause changes in the resulting blood oxygen level-dependent (BOLD) signal that is

Attention to visual motion

In previous work, we have established that attention modulates connectivity in a distributed system of cortical regions mediating visual motion processing (Buchel and Friston, 1997, Friston and Buchel, 2000). These findings were based on data acquired using the following experimental paradigm. Subjects viewed a computer screen that displayed either a fixation point, stationary dots or dots moving radially outward at a fixed velocity. In some epochs of moving dots, they had to attend to changes

Discussion

In this paper, we have compared the use of SEM and DCM for making inferences about changes in effective connectivity from fMRI time series. On our fMRI attention to visual motion data, both SEM and DCM approaches led to the same conclusions (i) that reciprocal models are superior to feedforward models, (ii) that models with reciprocal connections provide a good fit to the data and (iii) that attention significantly modulates the connectivity from V1 to V5.

There are data sets, however, where DCM

Acknowledgement

This study was funded by the Wellcome Trust. A Mechelli is supported by grant MH64445 from the National Institutes of Health (USA).

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