Modern cognitive neuroscience has adopted an explicit system perspective. A commonly accepted view is that the brain regions that constitute a given system are computationally specialized, but that the exact nature of their individual computations depends on context, e.g. the inputs from other regions. The aggregate behavior of the system depends on this neural context, the context-dependent interactions between the system components (Mcintosh 2000; see also the chapter by Bressler & Mcintosh in this volume). An equivalent perspective is provided by the twin concepts of functional specialization and functional integration (Friston 2002). Functional specialization assumes a local specialization for certain aspects of information processing but allows for the possibility that this specialization is anatomically segregated across different cortical areas. The majority of current functional neuroimaging experiments have adopted this view and interpret the areas that are activated by a certain task component as the elements of a distributed system. However, this explanation is incomplete as long as no insight is provided into how the locally specialized computations are bound together by context-dependent interactions among these areas, i. e. the functional integration within the system. This functional integration within distributed neural systems can be characterized in two ways, functional connectivity and effective connectivity.
Functional connectivity has been defined as the temporal correlation between regional time series (Friston 1994). Analyses of functional connectivity therefore do not incorporate any knowledge or assumptions about the structure and the SFR of the system of interest. Depending on the amount of knowledge about the system under investigation, this can either be a strength or a weakness. If the system is largely unknown, functional connectivity approaches are useful because they can be used in an exploratory fashion, either by computing functional connectivity maps with reference to a particular seed region (Horwitz et al. 1998; McIntosh et al. 2003; Stephan et al. 2001a) or using a variety of multivariate techniques that find sets of voxels whose time series represent distinct (orthogonal or independent) components of the co-variance structure of the data (Friston & Biichel 2004; McIntosh & Lobaugh 2004). The information from these analyses can then be used to generate hypotheses about the system. Conversely, given sufficient information about the system structure and a specific hypothesis about the SFR of the system, models of effective connectivity are more powerful. Here, we only deal with models of effective connectivity. For analyses of functional connectivity, please see the chapters by Salvador et al., Bressler & McIntosh and Sporns & Tononi in this volume.
Effective connectivity has been defined by various authors, but in complementary ways. A general definition is that effective connectivity describes the causal influences that neural units exert over another (Friston 1994). Other authors have proposed that "effective connectivity should be understood as the experiment- and time-dependent, simplest possible circuit diagram that would replicate the observed timing relationships between the recorded neurons" (Aertsen & Preifil 1991). Both definitions emphasize that determining effective connectivity requires a causal model of the interactions between the elements of the neural system of interest. Such a causal model has to take into account the external inputs that perturb the system and the anatomical connections by which neural units influence each other. In other words, any such model is a special case of the general system model as described in Sect. 2 and formalized by (3).
The equations presented in Sect. 2 are extremely general. To illustrate how the concept of effective connectivity emerges naturally from system models, we discuss the special case of a linear dynamic system. Although most natural phenomena are of a nonlinear nature, linear models play an important role in systems science because (i) they are analytically tractable, and (ii) given sufficiently long observation periods and non-negligible external input, their dynamics are largely independent of the initial state (Bossel 1992). Therefore nonlinear systems are usually investigated in restricted sub-spaces of interest, using linear models as local approximations. The following model of n inter acting brain regions is a simple linear case of (3) which uses a single state variable per region and m external inputs:
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