SEM has been an established statistical technique in the social sciences for several decades, but was only introduced to neuroimaging in the early 1990's by McIntosh & Gonzalez-Lima (1991). It is a multivariate, hypothesis-driven technique that is based on a structural model which represents the hypothesis about the causal relations between several variables (see McIntosh & Gonzalez-Lima 1994, Büchel & Friston 1997, Bullmore et al. 2000 and Penny et al. 2004a for methodological details). In the context of fMRI these variables are the measured BOLD (blood oxygen level dependent) time series y1 ...yn of n brain regions and the hypothetical causal relations are based on anatomically plausible connections between the regions. The strength of each connection yi ^ yj is specified by a so-called "path coefficient" which, by analogy to a partial regression coefficient, indicates how the variance of yj depends on the variance of yi if all other influences on yj are held constant.

The statistical model of standard SEM implementations for neuroimaging data can be summarized by the equation y = Ay + u (10)

where y is a n x s matrix of n area-specific time series with s scans each, A is a n x n matrix of path coefficients (with zeros for non-existent connections), and u is a n x s matrix of zero mean Gaussian error terms, which are driving the modeled system ("innovations", see (11)). Note that the model on which SEM rests is a special case of the general equation for non-autonomous linear systems (with the exception that SEM is a static model and the inputs to the modeled system are random noise; compare (11) with (6)). Parameter estimation is achieved by minimizing the difference between the observed and the modeled covariance matrix S of the areas (Bollen 1989). For any given set of parameters, S can be computed by transforming (10):

where I is the identity matrix and T denotes the transpose operator. The first line of 11 can be understood as a generative model of how system function results from the system's connectional structure: the measured time series y results by applying a function of the inter-regional connectivity matrix, i.e. (I - A)-1, to the Gaussian innovations u.

In the special case of fMRI, the path coefficients of a SEM (i.e. the parameters in A) describe the effective connectivity of the system across the entire experimental session. What one would often prefer to know, however, is how the coupling between certain regions changes as a function of experimentally controlled context, e.g. differences in coupling between two different tasks. Notably, SEM does not account for temporal order: if all regional time series were permuted in the same fashion, the estimated parameters would not change. In case of blocked designs, this makes it possible to proceed as if one were dealing with PET data, i.e. to partition the time series into condition-specific sub-series and fit separate SEMs to them. These SEMs can then be compared statistically to test for condition-specific differences in effective connectivity (for examples, see Buchel et al. 1999; Honey et al. 2002). An alternative approach is to augment the model with bilinear terms (cf. (9)) which represent the modulation of a given connection by experimentally controlled variables (e.g. Buchel & Friston 1997; Rowe et al. 2002). In this case, only a single SEM is fitted to the entire time series.

One limitation of SEM is that one is restricted to use structural models of relatively low complexity since models with reciprocal connections and loops often become non-identifiable (see Bollen 1989 for details). There are heuristics for dealing with complex models that use multiple fitting steps in which different parameters are held constant while changing others (see Mcintosh et al. 1994 for an example).

6 Multivariate Autoregressive Models (MAR)

in contrast to SEM, autoregressive models explicitly address the temporal aspect of causality in time series. They take into account the causal dependence of the present on the past: each data point of a regional time series is explained as a linear combination of past data points from the same region. MAR models extend this approach to n brain regions, modeling the n-vector of regional signals at time t (yt) as a linear combination of p past data vectors whose contributions are weighted by the parameter matrices Ai:

MAR models thus represent directed influences among a set of regions whose causal interactions are inferred via their mutual predictability from past time points. Although MAR is an established statistical technique, specific implementations for neuroimaging were suggested only relatively recently. Harrison et al. (2003) suggested a MAR implementation that allowed for the inclusion of bilinear variables representing modulatory effects of contextual variables on connections and used a Bayesian parameter estimation scheme specifically developed for MAR models (Penny & Roberts 2002). This Bayesian scheme also determined the optimal model order, i.e. the number of past time points (p in (12)) to be considered by the model. A complementary MAR approach, based on the idea of "Granger causality" (Granger 1969), was proposed by Goebel et al. (2003). In this framework, given two time-series yi and y2, yi is considered to be caused by y2 if its dynamics can be predicted better using past values from yi and y2 as opposed to using past values of yi alone.

7 Dynamic Causal Modeling (DCM)

An important limitation of the models discussed so far is that they operate at the level of the measured signals. Taking the example of fMRI, the model parameters are fitted to BOLD series which result from a haemodynamic convolution of the underlying neural activity. Any inference about inter-regional connectivity obtained by PPI, SEM or MAR is only an indirect one because these models do not include the forward model linking neuronal activity to the measured haemodynamic data. In the case of EEG, this forward model means there is a big difference between signals measured at each electrode and the underlying neuronal activity: changes in neural activity in different brain regions lead to changes in electric potentials that superimpose linearly. The scalp electrodes therefore record a mixture, with unknown weightings, of potentials generated by a number of different sources.

The causal architecture of the system that we would like to identify is expressed at the level of neuronal dynamics. Therefore, to enable inferences about connectivity between neural units we need models that combine two things: (i) a parsimonious but neurobiologically plausible model of neural population dynamics, and (ii) a biophysically plausible forward model that describes the transformation from neural activity to the measured signal. Such models make it possible to fit jointly the parameters of the neural and of the forward model such that the predicted time series are optimally similar to the observed time series. In principle, any of the models described above could be combined with a modality-specific forward model, and indeed, MAR models have previously been combined with linear forward models to explain EEG data (Yamashita et al. 2004). So far, however, Dynamic Causal Modeling (DCM) is the only approach where the marriage between models of neural dynamics and biophysical forward models is a mandatory component. DCM has been implemented both for fMRI (Friston et al. 2003) and EEG/MEG data (David et al. 2006; Kiebel et al. 2006). These modality-specific implementations are briefly summarized in the remainder of this section (see Fig. 2 for a conceptual overview).

Hemodynamic forward model: neural activity^BOLD (nonlinear)

Hemodynamic forward model: neural activity^BOLD (nonlinear)

Electromagnetic forward model: neural activity ^EEG MEG (linear)

Electromagnetic forward model: neural activity ^EEG MEG (linear)

Neural state equation:

Neural model: 1 state variable per region bilinear state equation no propagation delays inputs inputs

Neural model: 8 state variables per region nonlinear state equation propagation delays

ERPs

Neural model: 8 state variables per region nonlinear state equation propagation delays

Fig. 2. A schematic overview that juxtaposes properties of DCM for fMRI and ERPs, respectively. It illustrates that DCM combines a model of neural population dynamics, following the generic form of (3), with a modality-specific biophysical forward model. Given appropriate formulations of the neural and the forward model, DCM can be applied to any kind of measurement modality

DCM for fMRI uses a simple model of neural dynamics in a system of n interacting brain regions. It models the change of a neural state vector x in time, with each region in the system being represented by a single state variable, using the following bilinear differential equation:

Note that this neural state equation follows the general form for deterministic system models introduced by (3), i.e. the modeled state changes are a function of the system state itself, the inputs u and some parameters 0n that define the functional architecture and interactions among brain regions at a neuronal level (n in 0n is not an exponent but a superscript that denotes "neural"). The neural state variables represent a summary index of neural population dynamics in the respective regions. The neural dynamics are driven by experimentally controlled external inputs that can enter the model in two different ways: they can elicit responses through direct influences on specific regions (e.g. evoked responses in early sensory cortices; the C matrix) or they can modulate the coupling among regions (e.g. during learning or attention; the B matrices).

Equation (13) is a bilinear extension of (6) that was introduced earlier as an example of linear dynamic systems. Given this bilinear form, the neural parameters 0n = {A, B, C} can be expressed as partial derivatives of F:

d2F dxduj dF du

The matrix A represents the effective connectivity among the regions in the absence of input, the matrices B(j) encode the change in effective connectivity induced by the jth input uj, and C embodies the strength of direct influences of inputs on neuronal activity (see Fig. 3 for a concrete example and compare it to Fig. 1).

DCM for fMRI combines this model of neural dynamics with an experimentally validated haemodynamic model that describes the transformation of neuronal activity into a BOLD response. This so-called "Balloon model" was initially formulated by Buxton et al. (1998) and later extended by

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