Modelling functional integration: a comparison of structural equation and dynamic causal models
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, X−T 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 by
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).
References (27)
- et al.
How good is good enough in path analysis of fMRI data?
NeuroImage
(2000) Bayesian estimation of dynamical systems: an application to fMRI
NeuroImage
(2002)- et al.
Analysis of fMRI time series revisited
NeuroImage
(1995) - et al.
Psychophysiological and modulatory interactions in neuroimaging
NeuroImage
(1997) - et al.
Dynamic causal modelling.
NeuroImage
(2003) - et al.
Modeling regional and psychophsyiologic interactions in fMRI: the importance of hemodynamic deconvolution
NeuroImage
(2003) - et al.
Connectivity analysis with structural equation modeling: an example of the effects of voxel selection
NeuroImage
(2003) - et al.
Attention to action: specific modulation of corticocortical interactions in humans
NeuroImage
(2002) - et al.
Temporal autocorrelation in univariate linear modelling of fMRI data
NeuroImage
(2001) Structural Equations with Latent Variables
(1989)
Modulation of connectivity in visual pathways by attention: cortical interactions evaluated with structural equation modelling and fMRI
Cerebral Cortex
Dynamics of blood flow and oxygenation changes during brain activation: The Balloon Model
Magnetic Resonance in Medicine
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