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Colloquium on Neuroimaging of Human Brain Function
interactions among extrastriate areas. This reiterates the importance of cartography, in this instance, a cartography of connections. Another example of principles that derive from empirical observations can be taken from the work of complexity theorists, wherein certain degrees of “connectivity” at the level of gene-gene interactions give rise to complex dynamics (3). This principle of sparse connectivity has emerged again in relation to the complexity of neuronal interactions (4). What are the sorts of principles that one is looking for in imaging neuroscience? The “principle of functional specialization” is now well established and endorsed by human neuroimaging studies. If we define functional specialization in terms of anatomically specific responses that are sensitive to the context in which these responses are evoked, then, by analogy, “functional integration” can be thought of as anatomically specific interactions between neuronal populations that are similarly context-sensitive. In a sense, functional integration is a principle, but it is not very useful. Examples of more useful principles might include a principle of sparse connectivity (e.g., functional integration is mediated by sparse extrinsic connections that preserve specialization within systems that have dense intrinsic connections) or that, in relation to forward connections, backward connections are modulatory. In summary, a detailed empiricism is a prerequisite for the emergence of organizational principles. For some neuroscientists the principles themselves might be the ultimate goal, but these principles will only be derived from maps of the brain. For others, relating the maps to cognitive architectures and psychological models may be the ultimate goal, but this in itself requires a principled approach.
Models or Tools?
In this section we consider the status of various models that are used to analyze or characterize brain function and how they are likely to develop. The models one usually comes across in neuroscience are of three types. First, there are biologically plausible neural network or synthetic neural models (5). Second are the mathematical models employed in linear and nonlinear system identification, and third are the statistical models used to characterize empirical data (6). This section suggests that the increasing sophistication of statistical models will render them indistinguishable from those used to identify the underlying system. Similarly, synthetic neural models that are currently used to emulate brain systems and study their emergent properties will lend themselves to reformulations in terms of those required for system identification. The importance of this is that (i) the parameters of synthetic neural models, for example, the connection parameters and time constants, can be estimated directly from empirical observations and (ii) the validity of statistical models, in relation to what is being modeled, will increase. Another way of looking at the distinctions between the various sorts of models (and how these distinctions might be removed) is to consider that we use models either to emulate a system, or to define the nature or form of an observed system. When used in the latter context, the empirical data are used to determine the exact parameters of the specified model where, in statistical models, inferences can be made about the parameter estimates. In what follows we will review statistical models and how they may develop in the future and then turn to an example of how one can derive a statistical model from one normally associated with a nonlinear system identification. The importance of this example is that it shows how a model can be used, not only as a statistical tool, but as a device to emulate the behavior of the brain under a variety of circumstances.
Linear Models. The most prevalent model in imaging neuroscience is the general linear model. This simply expresses the response variable (e.g., hemodynamic response) in terms of a linear sum of explanatory variables or effects in a “design matrix.” Inferences about the contribution of these explanatory variables are made in terms of the parameter estimates, or coefficients, estimated by using the data. There are a number of ways in which one can see the general linear model being developed in neuroimaging; for example, the development of random- and mixed-effect models that allow one to generalize inferences beyond the particular group of subjects studied to the population from which the subjects came, or the increasingly sophisticated modeling of evoked responses in terms of wavelet decomposition. Here we will focus on two examples: (i) model selection and (ii) inferences about multiple effects using statistical parametric maps of the F statistic [SPM(F)].
Generally, when using statistical models, one has to choose from among a hierarchy of models that embody more and more effects. Some of these effects may or may not be present in the data, and the question is, “which is the most appropriate model?” One way to address this question is to see whether adding extra effects significantly reduces the error variance. When the fit is not significantly improved one can cease elaborating the model. This principled approached to model selection is well established in other fields and will probably prove useful in neuroimaging. One important application of model selection is in the context of parametric designs and characterizing evoked hemodynamic responses in fMRI. In parametric designs it is often the case that some high-order polynomial “expansion” of the interesting variable (7) (e.g., the rate or duration of stimulus presentation) is used to characterize a nonlinear relationship between the hemodynamic response and this variable. Similarly, in modeling evoked responses in fMRI, the use of expansions in terms of temporal basis functions has proved useful (8). These two examples have something in common. They both have an “order” that has to be specified. The order of the polynomial regression approach is the number of high-order terms employed, and the order of the temporal basis function expansion is the number of the basis functions used. Model selection has a role here in determining the most appropriate or best model order.
Models that use expansions bring us to the second example. Recall that in general the contribution of designed effects is reflected in the parameter estimates of the coefficients relating to these effects. In the case of polynomial expansions or temporal basis functions these are the set of coefficients of the high-order terms or basis functions. Unlike simple activations (or effects corresponding to a particular linear combination of the parameter estimates), inferences based on these high-order models must be a collective inference about all of these coefficients together. This inference is made with the F statistic and speaks to the usefulness of the SPM(F) as an inferential tool. The next section presents an example of the SPM(F) in action and introduces models used in nonlinear system identification.
Nonlinear Models. How can we best characterize the relationship between stimulus presentation and the evoked hemodynamic response in fMRI? Hitherto the normal approach has been to use a stimulus waveform that conforms to the presence or absence of a particular stimulus attribute, convolve (i.e., smooth) this with an estimate of the hemodynamic response function, and see if the result can predict what is observed. Our using an estimate of the hemodynamic response function assumes that we know the nature or form of this response and furthermore precludes nonlinear effects. A more comprehensive approach would be to use nonlinear system identification and pretend that the stimulus was the input and that the observed hemodynamic response was the output. This approach posits a very general form for the relationship and uses the observed inputs and outputs to determine the parameters of the model that optimize the match between the observed and predicted hemodynamic responses. The approach that we have adopted uses a Volterra series expansion (8). This