uZ Te Te Te

The parameter matrices AF, AB ,AL encode forward, backward and lateral connections respectively. Within each subpopulation, the dynamics of neural states are determined by two operators. The first transforms the average density of presynaptic inputs into the average postsynaptic membrane potential. This is modeled by a linear transformation with excitatory (e) and inhibitory (i) kernels parameterized by He i and Te i. He i control the maximum postsynaptic potential and Te,i represent lumped rate constants (i.e. lumped across dendritic spines and the dendritic tree). The second operator S transforms the average potential of each subpopulation into an average firing rate. This is assumed to be instantaneous and is a sigmoid function. Intra-areal interactions among the subpopulations depend on constants 71...4 which control the strength of intrinsic connections and reflect the total number of synapses expressed by each subpopulation. In (16), the top line expresses the rate of change of voltage as a function of current. The second line specifies how current changes as a function of voltage, current and presynaptic input from extrinsic and intrinsic sources. For simplification, our description here has omitted the fact that in DCM for ERPs all intra- and inter-areal connections have conduction delays. This requires the use of delay differential equations (see David et al. 2006 for details).

For estimating the parameters from empirical data, a fully Bayesian approach is used that is analogous to that used in DCM for fMRI and is described in detail by David et al. (2006). The posterior distributions of the parameter estimates can be used to test hypotheses about the modeled processes, particularly differences in inter-areal connection strengths between different trial types. As in DCM for fMRI, Bayesian model selection can be used to optimize model structure or compare competing scientific hypotheses (Penny et al. 2004b).

8 Application of System Models in Functional Neuroimaging: Present and Future

Models of functional integration, which were originally developed for electro-physiological data from multi-unit recordings (Gerstein and Perkel 1968), are now taking an increasingly prominent role in functional neuroimaging. This is because the emphasis of the scientific questions in cognitive neuroscience is shifting from where particular processes are happening in the brain to how these processes are implemented. With increasing use, a word of caution may be appropriate here: Models of effective connectivity are not very useful without precise a priori hypotheses about specific mechanisms expressed at the level of inter-regional coupling. Simply describing patterns of connectivity that require post hoc interpretation does not lead to a mechanistic understanding of the system of interest. What is needed are parsimonious, well-motivated models that test precise hypotheses about mechanisms, either in terms of changes in particular connection strengths as a function of experimental condition, time (learning) or drug, or in terms of comparing alternative explanations by model selection (for examples, see Buchel & Friston 1997; Buchel et al. 1999; Honey et al. 2003; Mcintosh et al. 1994, 1998; Rowe et al. 2002; Stephan et al. 2003, 2005; Toni et al. 2002). Figure 6 shows an example of such a model (Friston et al. 2003) where the parameters are mechanistically meaningful.

This search for mechanisms seems particularly promising for pharmacological questions. Since many drugs used in psychiatry and neurology change synaptic transmission and thus functional coupling between neurons, their therapeutic effects cannot be fully understood without models of drug-induced connectivity changes in particular neural systems. So far, only relatively few studies have studied pharmacologically induced changes in connectivity, ranging from simple analyses of functional connectivity (e.g. Stephan et al. 2001a) to proper system models (e.g. Honey et al. 2003). As highlighted in a recent review by Honey and Bullmore (2004), an exciting possibility for the future is to use system models at the early stage of drug development to screen for substances that induce desired changes of connectivity in neural systems of interest with a reasonably well understood physiology. The success of this approach will partially depend on developing models that include additional levels of biological detail (e.g. effects of different neurotransmitters and receptor types) while being parsimonious enough to ensure mathematical identifiability and physiological interpretability; see Breakspear et al. (2003),

Fig. 6. DCM analysis of a single subject fMRI data from a study of attention to visual motion in which subjects viewed identical stimuli (radially moving dots) under different levels of attention to the stimuli (Buchel & Friston 1997). The model was introduced and described in detail by Friston et al. (2003). The figure is reproduced (with permission from Elsevier Ltd.) from Stephan et al. (2004). Only those conditional estimates are shown alongside their connections for which there was at least 90% confidence that they corresponded to neural transients with a half life shorter than 4 seconds. The temporal structure of the inputs is shown by box-car plots. Dashed arrows connecting regions represent significant bilinear affects in the absence of a significant intrinsic coupling. Fitted responses based upon the conditional estimates and the adjusted data are shown in the panels connected to the areas by dotted lines. The important parameters here are the bilinear ones. Note that while the intrinsic connectivity between areas V1 and V5 is non-significant and basically zero, motion stimuli drastically increase the strength of this connection, "gating" V1 input to V5. Top-down effects of attention are represented by the modulation of backward connections from the inferior frontal gyrus (IFG) to the superior parietal cortex (SPC) and from SPC to V5. See Penny et al. (2004b) and Stephan (2004) for a discussion how different neurophysiological mechanisms can be modeled with DCM

Harrison et al. (2005), Jirsa (2004) and Robinson et al. (2001) for examples that move in this direction.

Another important goal is to explore the utility of models of effective connectivity as diagnostic tools (Stephan 2004). This seems particularly attractive for psychiatric diseases whose phenotypes are often very heterogeneous and where a lack of focal brain pathologies points to abnormal connectivity (dysconnectivity) as the cause of the illness. Given a pathophysiological theory of a specific disease, connectivity models might allow one to define an en-dophenotype of that disease, i.e. a biological marker at intermediate levels between genome and behaviour, which enables a more precise and physiologically motivated categorization of patients (Gottesman & Gould 2003). Such an approach has received particular attention in the field of schizophrenia research where a recent focus has been on abnormal synaptic plasticity leading to dysconnectivity in neural systems concerned with emotional and perceptual learning (Friston 1998; Stephan et al. 2006). A major challenge will be to establish neural systems models which are sensitive enough that their connectivity parameters can be used reliably for diagnostic classification and treatment response prediction of individual patients. Ideally, such models should be used in conjunction with paradigms that are minimally dependent on patient compliance and are not confounded by factors like attention or performance. Given established validity and sufficient sensitivity and specificity of such a model, one could use it in analogy to biochemical tests in internal medicine, i.e. to compare a particular model parameter (or combinations thereof) against a reference distribution derived from a healthy population (Stephan et al. 2006). Such procedures could help to decompose current psychiatric entities like schizophrenia into more well-defined subgroups characterized by common pathophysiological mechanisms and may facilitate the search for genetic underpinnings.

The authors are supported by the Wellcome Trust. We would like to thank our colleagues for stimulating discussions on modeling, particularly Lee Harrison, Will Penny, Olivier David, Stefan Kiebel, Randy Mcintosh and Michael Breakspear.

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Multichannel Data Analysis in Biomedical Research

Maciej Kaminski

Department of Biomedical Physics, Institute of Experimental Physics, Warsaw University, ul. HoZa 69, 00-681, Warszawa, Poland

Multivariate data can be encountered in many fields of science or engineering. Any experiment or measurement with several quantities simultaneously recorded delivers multivariate datasets. This is especially true in biomedical investigations where most of the recordings are nowadays multichannel. EEG equipment is able to record signals from still more and more electrodes, but the notion is not confined only to electrode recordings: even fMRI data can be treated as a multivariate set of voxels changing their state in time. Such multichannel recordings are intended to deliver more information about the investigated object. However, the amount of knowledge obtained from the data analysis depends on the analysis method used. An analytical tool which is improperly applied may not give the correct answers, in fact, it may deliver incomplete or false information unbeknownst to the researcher.

The aim of this chapter is to describe the main aspects that are important in processing multivariate data. In the first, theoretical part, the specific properties of that type of data will be described and related functions will be introduced. In the second part, examples of treating typical problems arising during the analysis will be presented from a practical point of view and solutions will be proposed. The linear modeling approach will be discussed. This chapter can be viewed as a guide explaining basic properties of multivariate datasets and problems specific for multivariate analysis presented with selected relevant examples. For the issues considered in the text references to the literature for further, more comprehensive reading will be given.

Let X denote a multivariate stochastic process containing k subprocesses. A value of the process at a time t can be expressed as (T denotes transposition)

In the further discussion we assume that every process Xi has a zero mean. In the data collecting practice, data acquisition requires sampling values of each process Xi at certain equally spaced time points. This results in a time domain representation of multiple time series with a certain sampling period At. The quantity fs = I/At is the sampling frequency. Although many characteristics of the measured signals can be evaluated directly from the data in the time domain, quite often spectral properties are of primary interest. The signal X(t) can be transformed into the frequency domain by application of the Fourier transform. The frequency representation of the signal, X(f), is a complex valued function describing amplitudes and phases of frequency components of the signal at the frequency f. Later in the text we will omit the tilde above symbols representing frequency representations of respective time domain quantities, remembering that both symbols (like X(t) and X(f)) signify different quantities. The power spectral density matrix of the signal X is defined as

where the superscript asterisk represents transposition and complex conjugate. The matrix S is often simply called a spectrum. Its diagonal elements are called the auto-spectra, the off-diagonal elements are called the cross-spectra.

Typically, when analyzing a univariate data record (a single time series), several quantities describing properties of the data are estimated. In the case of a multivariate dataset the same estimations can be repeated for each data channel separately. Information gathered from each channel can be very useful, for instance in mapping procedures. However, a multivariate dataset contains additional information about relations between channels. To evaluate these so called cross-relations, specific functions depending on two (or more) signals simultaneously were defined.

Covariance (or correlation in a normalized version) is a function describing common trends in behavior of two time series Xi and Xj. The function operates in the time domain. The simplest general formula can be written as:

Depending on the version used, the covariance may have different normalization terms t ; the set of indices m covers the range applicable in a particular case. The value s is called the time lag. Covariance applied to two different signals is known as cross-covariance (cross-correlation), and when Xi and Xj are the same signal the name auto-covariance (auto-correlation) is used.

In the frequency domain a function analogous to correlation is (ordinary) coherence. It compares common behavior of components of the signals at different frequencies. It is defined by means of elements of the spectral matrix S

and depends on the signals Xi and Xj and on the frequency f. For simplicity a notation will be introduced: subscript indices of the function variable (ordinary coherence K) will correspond to the indices of the signals:

The modulus of ordinary coherence takes values in the [0,1] range. It describes the amount of in-phase components in both signals at the given frequency f(0 indicates no relation).

When a multivariate dataset consists of more than two signals, relations between them can be of a more complicated structure. Let us consider three signals Xi, X2 and X3 constituting a multivariate set. If Xi influences X2 and X2 influences X3 then signals X1 and X3 will be related with each other as well. However, contrary to relations of X1 with X2 and X2 with X3, the relation of channels X1 and X3 will not be direct because of the presence of intermediate signal X2. Distinguishing between direct and indirect relations may play the crucial role in understanding the investigated system. To study relations within multi-(fc > 2)-variate datasets another functions were introduced.

Functions which help to decompose complex relations between signals and describe only direct ones are called partial functions. In multivariate systems partial functions identify only direct relations, with the influence of the rest of signals on that relations statistically removed. The partial coherence function Cij (f) describes the amount of in-phase components in signals i and j at frequency f while the part of the signals which can be explained by influence of a linear combination of the other signals is subtracted.

Its modulus takes values within the [0, 1] range similar to ordinary coherence, but it is nonzero only when the relation between channel i and j is direct. Mij is a minor (determinant) of S with i-th row and j-th column removed. After some algebraic manipulations Cij can be expressed by elements of the inverse of S: dij = [S_1]ij

We may notice that in multichannel sets a signal can be simultaneously related with more than one signal. Multiple coherence Gi(f) describes the amount of in-phase components in channel i common with any other channel of the set. It is given by the formula:

As in the case of other coherence functions, its modulus takes values within the [0, 1] range; its high value indicate the presence of a relation between the channel i and the rest of the set.

Partial coherences are especially useful to find a pattern of connections within sets of highly correlated signals. Example presented in Fig. 1 shows a result of coherence analysis of a 21-channels dataset of human scalp sleep EEG (see also Sect. 8). We see that each type of coherence forms a different pattern of connections. Multiple coherences (on the diagonal) are all high

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