In the last section, we presented a number of examples in which large-scale neural models were used to simulate PET and fMRI data. In both the simulated and experimental cases, the data of interest usually are the "activations", which are the differences between regional activities in the condition of interest compared to activities in a control condition. However, a second, and increasingly important, way to analyze functional neuroimaging data is to use the covariance paradigm (Horwitz et al. 1992), which asserts that a task is mediated by a network of interacting brain regions, and that different tasks utilize different functional networks (see also McIntosh 2000). The fundamental concept employed by the covariance paradigm is functional connectivity, which is used in the context of neuronal processes to allude to the functional interactions between different areas of the brain. The formal mathematical quantification of these functional relationships depends on the use of interregional covariance or correlation coefficients (e.g., Friston 1994). However, the actual definition used by different researchers varies widely, as do the computational algorithms employed to evaluate interregional functional connectivity (Horwitz, 2003).

Functional connectivity doesn't distinguish explicitly between the case where regions are directly influencing one another (e.g., along specific anatomical pathways) compared to simply indicating some kind of indirect interaction. To do the former, some type of computational top-down modeling needs to be employed (Friston 1994; Horwitz 1994; McIntosh & Gonzalez-Lima 1994) to calculate the interregional effective connectivity. This term has come to mean, at the systems level, the direct influence of one neural population on another (Friston 1994). We think of it as meaning the functional strength of a particular anatomical connection; for instance, during one task a specific anatomical link may have a stronger effect than during a second condition; likewise, such a link may be stronger in normal subjects than in a patient group. The evaluation of effective connectivity requires modeling because one needs to select a small group of brain regions to include in the network, and one needs to combine functional neuroimaging data with information about the anatomical linkages between these regions. That is, the validity of effective connectivity modeling builds on a combination of implied anatomical and functional connections between brain regions. A number of chapters in this volume address functional and effective connectivity; see, for example, the contributions by Stephan and Friston, by Bullmore, and by Strother.

As we have emphasized in several previous papers (Horwitz 2003; Horwitz et al. 2005), there are multiple measures of functional and effective connectivity in use, and there is no guarantee that the conclusions one draws using one measure will be the same as using another. Furthermore, the terms functional and effective connectivity are applied to quantities computed on types of functional imaging data (e.g., PET, fMRI, EEG) that vary in spatial, temporal, and other dimensions, using different definitions (even for data of the same modality) and employing different computational algorithms. Until it is understood what each definition means in terms of an underlying neural substrate, comparisons of functional and/or effective connectivity across studies may lead to inconsistent or misleading conclusions. Perhaps more important is the fact that since the neural substrates of each measure are unknown, it is unclear how well a particular way in which the functional or effective connectivity is computed accurately represents the underlying relationships between different brain regions.

To address these issues, we have started using our large-scale neural models to help determine the neurobiological substrates for the most widely used definitions and algorithms for evaluating interregional functional and effective connectivity. That is, we simulate fMRI time series with our models and compute a particular version of the functional or effective connectivity. Because in the model we know what each neuron and synapse are doing at every moment in time (unlike the situation for real brain data), we can determine how well the computed functional/effective connectivity reflects the actual underlying neural relationships in the models. Because our models are neurobiologically plausible, complex, contain both excitatory and inhibitory neurons, have feedforward and feedback connections and include a diversity of regions containing neurons that possess different response properties, they provide useful testing grounds for investigating various kinds of data analysis methods (Horwitz 2004).

One example of this approach can be found in Horwitz et al. (2005), where one of the simplest definitions of fMRI functional connectivity - the within-condition correlation between fMRI time series - was examined. The crucial aspect of simulating functional and effective connectivity is to be able to simulate in biologically plausible ways variability in neural activity, because the key to evaluating functional or effective connectivity is to assess interregional co-variability. There are multiple sources of the variability found in functional neuroimaging data. Although some of these originate from the scanning technique and some are non-neural in origin (e.g., changes in the vasculature may lead to changes in the fMRI hemodynamic response function), some of the variability observed in the functional neuroimaging signal can be utilized to provide the covariance needed to evaluate functional or effective connectivity. The main idea is that variability in the activity in one region of the neural network mediating the task under study is propagated to other regions, resulting in a larger covariance between the regions than would be the case if they were not interacting with one another. The various methods of evaluating functional connectivity try to tap one or more of these neurally-based sources of covariation. For fMRI, the three main sources of variation that can be employed to assess within-condition functional connectivity are subject-to-subject variability, block-to-block variability and item-to-item (or MR volume-to-volume) variability (see Horwitz et al. (2005) for a more extensive discussion of this topic).

In our modeling approach, these types of variability were incorporated as subject-to-subject differences in the strengths of anatomical connections, scan-to-scan changes in the level of attention, and trial-to-trial interactions between the neurons mediating the task and nonspecific neurons processing noise stimuli (Horwitz et al. 2005). Recall that in our modeling framework, simulated fMRI is computed by integrating the absolute value of the synaptic activities in each module every 50msec (we call this the integrated synaptic activity or ISA), convolving this with a function representing the hemodynamic response and then sampling the resulting time series every TR seconds. Because the hemodynamic convolution and sampling lead to a loss of temporal resolution, one can think of the functional connectivities calculated from the ISA as a kind of 'gold standard' in that they represent the most ideal evaluation of the neural interrelationships that one could get at the systems level. Indeed, it has been argued by Gitelman and colleagues (Gitelman et al. 2003) that deconvolving an experimental fMRI time series improves the evaluation of functional and effective connectivity.

Using the visual model of Tagamets and Horwitz (1998), we (Horwitz et al. 2005) explored the functional connectivity, evaluated as the within-condition correlation between fMRI time series. Focusing on the link between IT and PFC (see Fig. 1 above), we found that time series correlations between ISAs between these two modules were larger during the DMS task than during a control task. These results were less clear when the integrated synaptic activities were hemodynamically convolved to generate simulated fMRI activities. In a second simulation using the auditory model of Husain et al. (2004), we found that as the strength of the model anatomical connectivity between temporal and frontal cortex was weakened, so too was the strength of the corresponding functional connectivity, although the relation was nonlinear. This latter result is important, since it demonstrates that the fMRI functional connectivity can appropriately reflect the strength of an anatomical link.

A final illustration of using large-scale neural modeling to investigate the neural substrates of functional/effective connectivity is a recent study by Lee et al. (in press). Friston and colleagues (Friston et al. 2003) developed a method called Dynamic Causal Modeling (DCM) for estimating and making inferences about the changes in the effective connectivities among small numbers of brain areas, and the influence of experimental manipulations on these couplings (see the article by Stephan and Friston in this volume for a thorough discussion of this method). Lee et al. used DCM to evaluate the change in effective connectivity using simulated data generated by our large-scale visual model implementing the DMS task. System-level models with hierarchical connectivity and reciprocal connections were examined using DCM and Bayesian model comparison (Penny et al. 2004), and revealed strong evidence for those models with correctly specified anatomical connectivity. An example of this is illustrated in Fig. 6. A simple model incorporating the first three regions of the large-scale visual model (V1/V2, V4 and IT) was analyzed. Three arrangements of the anatomical linkages between regions were compared: (1) the

Fig. 6. DCM analysis of visual model. (Top) Coupling parameters for the simple model, with interarea connectivity and modulatory input specified correctly with respect to the underlying large scale neural model. The posterior parameter estimates for the coupling parameters are shown in black and grey; the values in brackets are the confidence that these values exceed a threshold of ln2/4 Hz. Coupling parameters exceeding threshold with a confidence of greater than 90% are shown in black. The posterior parameter estimates for the coupling parameters for direct visual inputs are shown next to the solid grey arrows. The posterior parameter estimates for the coupling parameters for modulatory effect of task (shapes vs. degraded stimuli) are shown next to the dotted grey markers. The values in brackets are the percentage confidence that these values are greater than zero. (Middle) Coupling parameters for the simple model, specified as a hierarchy. Inputs are specified and displayed as in the top panel. (Bottom) Coupling parameters for the simple model, specified with full inter area connectivity. Inputs are specified and displayed as in the top panel. Modified from Lee et al. (in press), which should be consulted for further details

Fig. 6. DCM analysis of visual model. (Top) Coupling parameters for the simple model, with interarea connectivity and modulatory input specified correctly with respect to the underlying large scale neural model. The posterior parameter estimates for the coupling parameters are shown in black and grey; the values in brackets are the confidence that these values exceed a threshold of ln2/4 Hz. Coupling parameters exceeding threshold with a confidence of greater than 90% are shown in black. The posterior parameter estimates for the coupling parameters for direct visual inputs are shown next to the solid grey arrows. The posterior parameter estimates for the coupling parameters for modulatory effect of task (shapes vs. degraded stimuli) are shown next to the dotted grey markers. The values in brackets are the percentage confidence that these values are greater than zero. (Middle) Coupling parameters for the simple model, specified as a hierarchy. Inputs are specified and displayed as in the top panel. (Bottom) Coupling parameters for the simple model, specified with full inter area connectivity. Inputs are specified and displayed as in the top panel. Modified from Lee et al. (in press), which should be consulted for further details actual anatomical connectivity (top); (2) a hierarchical arrangement (middle) and (3) an arrangement in which every region was reciprocally connected to every other (bottom). It can be seen that all the coupling parameters in the model on the top of Fig. 6 (the correctly specified model) exceeded threshold with greater than 90% confidence. In Fig 6 middle, the posterior probability of correctly specified connections exceeded 90% confidence, whereas the incorrect connections did not. The middle model shown in Fig. 6 reduces to the correct model when the coupling parameters that do not exceed threshold are excluded. Bayesian model comparison suggested that there was positive evidence for the correct model (the top model) over the two alternatives. For the cases examined, Bayesian model comparison confirmed the validity of DCM in relation to our well established and comprehensive neuronal model.

It should also be mentioned that evaluation of interregional functional connectivity using EEG/MEG data is also performed by many groups, and computational neural modeling has been used to investigate these methods (e.g., David et al. 2004).

In summary, the use of large-scale modeling is starting to provide results that add support for some methods for computing functional and effective connectivity. However, the results so far also suggest that caution is needed in using fMRI-based calculations of functional and effective connectivity to infer the nature of interregional neural interactions from functional neuroimag-ing data.

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