## X8

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.

### Acknowledgements

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.

### References

Aertsen A, Preißl H (1991) Dynamics of activity and connectivity in physiological neuronal Networks. In: Schuster HG (ed.) Non Linear Dynamics and Neuronal Networks, VCH Publishers, New York, pp. 281-302 Ashby WR (1956) An introduction to cybernetics. Chapman & Hall, London Bar-Yam Y (1997) Dynamics of complex systems. Addison-Wesley, Reading Bollen KA (1989) Structural equations with latent variables. John Wiley, New York Bossel H (1992) Modellbildung und Simulation. Vieweg, Braunschweig Breakspear M, Terry JR, Friston KJ (2003) Modulation of excitatory synaptic coupling facilitates synchronization and complex dynamics in a biophysical model of neuronal dynamics. Network: Computation in Neural Systems 14, 703-732

Büchel C, Friston KJ (1997) Modulation of connectivity in visual pathways by attention: cortical interactions evaluated with structural equation modelling and fMRI. Cerebral Cortex 7, 768-778 Büchel C, Coull JT, Friston KJ (1999) The Predictive Value of Changes in Effective

Connectivity for Human Learning. Science 283, 1538-1541 Bullmore ET, Horwitz B, Honey G, Brammer M, Williams S, Sharma T (2000) How good is good enough in path analysis of fMRI data? Neurolmage 11, 289-301 Buxton RB, Wong EC, Frank LR (1998) Dynamics of blood flow and oxygenation changes during brain activation: the balloon model. Magnetic Resonance Medicine 39, 855-864 Chong L, Ray LB (2002) Whole-istic biology. Science 295, 1661 David O, Friston KJ (2003) A neural mass model for MEG/EEG: coupling and neuronal dynamics. NeuroImage 20, 1743-1755 David O, Kiebel SJ, Harrison LM, Mattout J, Kilner JM, Friston KJ (2006) Dynamic causal modeling of evoked responses in EEG and MEG. Neurolmage 30, 1255-1272

Felleman DJ, Van Essen DC (1991) Distributed hierarchical processing in the primate cerebral cortex. Cerebral Cortex 1, 1-47 Friston KJ (1994) Functional and effective connectivity in neuroimaging: a synthesis.

Human Brain Mapping 2, 56-78 Friston KJ, Buüchel C, Fink GR, Morris J, Rolls E, Dolan RJ (1997) Psychophysio-

logical and modulatory interactions in neuroimaging. Neurolmage 6, 218-229 Friston KJ (1998) The disconnection hypothesis. Schizophrenia Research 30, 115-125

Friston KJ, Mechelli A, Turner R, Price CJ (2000) Nonlinear responses in fMRI: the Balloon model, Volterra kernels, and other hemodynamics. NeuroImage 12, 466-477

Friston KJ (2002) Beyond phrenology: What can neuroimaging tell us abut distributed circuitry? Annual Reviews in Neuroscience 25, 221-250 Friston KJ, Harrison L, Penny W (2003) Dynamic causal modelling. NeuroImage 19, 1273-1302

Friston KJ, Buüchel C (2004) Functional connectivity: eigenimages and multivariate analyses. In: Frackowiack R et al. (ed.) Human Brain Function, 2nd edition, Elsevier, New York, pp. 999-1018 Gerstein GL, Perkel DH (1969) Simultaneously recorded trains of action potentials:

Analysis and functional interpretation. Science 164, 828-830 Goebel R, Roebroeck A, Kim DS, Formisano E (2003) Investigating directed cortical interactions in time-resolved fMRI data using vector autoregressive modeling and Granger causality mapping. Magnetic Resonance Imaging 21, 1251-1261 Gottesman II, Gould TD (2003) The endophenotype concept in psychiatry: etymology and strategic intentions. American Journal of Psychiatry 160, 636-645 Granger CWJ (1969) Investigating causal relations by econometric models and cross-

spectral methods. Econometrica 37, 424-438 Harrison LM, Penny W, Friston KJ (2003) Multivariate autoregressive modeling of fMRI time series. NeuroImage 19, 1477-1491 Harrison LM, David O, Friston KJ (2005) Stochastic models of neuronal dynamics. Philosophical Transactions of the Royal Society London B Biological Sciences 360, 1075-1091

Honey GD, Fu CHY, Kim J, Brammer MJ, Croudace TJ, Suckling J, Pich EM, Williams SCR, Bullmore ET (2002) Effects of verbal working memory load on corticocortical connectivity modeled by path analysis of functional magnetic resonance imaging data. Neurolmage 17, 573-582 Honey GD, Suckling J, Zelaya F, Long C, Routledge C, Jackson S, Ng V, Fletcher PC, Williams SCR, BrownJ, Bullmore ET (2003) Dopaminergic drug effects on physiological connectivity in a human cortico-striato-thalamic system. Brain 126, 1767-1281

Honey G, Bullmore E (2004) Human pharmacological MRI. Trends in Pharmacological Sciences 25, 366-374 Horwitz B, Rumsey JM & Donohue BC (1998) Functional connectivity of the angular gyrus in normal reading and dyslexia. Proceedings of the National Academy of Sciences USA 95, 8939-8944 Jirsa VK (2004) Connectivity and dynamics of neural information processing. Neu-

roinformatics 2, 183-204 Kiebel SJ, David O, Friston KJ (2006) Dynamic causal modelling of evoked responses in EEG/MEG with lead-field parameterization. Neurolmage 30, 1273-1284 Kotter R (2004) Online retrieval, processing, and visualization of primate connectivity data from the CoCoMac database. Neuroinformatics 2, 127-144 Magee JC, Johnston D (2005) Plasticity of dendritic function. Current Opinion in

Neurobiology 15, 334-342 Mcintosh AR, Gonzalez-Lima F (1991) Metabolic activation of the rat visual system by patterned light and footshock. Brain Research 547, 295-302 Mcintosh AR, Gonzalez-Lima F (1994) Structural equation modeling and its application to network analysis in functional brain imaging. Human Brain Mapping 2, 2-22

McIntosh AR, Grady CL, Ungerleider LG, Haxby JV, Rapoport SI, Horwitz B (1994) Network analysis of cortical visual pathways mapped with PET. Journal of Neuroscience 14, 655-666 Mcintosh AR, Cabeza RE, Lobaugh NJ (1998) Analysis of neural interactions explains the activation of occipital cortex by an auditory stimulus. Journal of Neurophysiology 80, 2790-2796 Mclntosh AR (2000) Towards a network theory of cognition. Neural Networks 13, 861-870

Mcintosh AR, Rajah MN, Lobaugh NJ (2003) Functional connectivity of the medial temporal lobe relates to learning and awareness. Journal of Neuroscience 23, 6520-6528

Mcintosh AR, Lobaugh NJ (2004) Partial least squares analysis of neuroimaging data: applications and advances. Neuroimage 23, S250-S263 Payne BR, Lomber SG (2001) Reconstructing functional systems after lesions of cerebral cortex. Nature Reviews Neuroscience 2, 911-919 Penny WD, Roberts SJ (2002) Bayesian multivariate autoregressive models with structured priors. iEE Proceedings of Vision and image Signal Processing 149, 33-41

Penny WD, Stephan KE, Mechelli A, Friston KJ (2004a) Modeling functional integration: a comparison of structural equation and dynamic causal models. Neuroimage 23, S264-274 Penny WD, Stephan KE, Mechelli A, Friston KJ (2004b) Comparing dynamic causal models. Neuroimage 22, 1157-1172

Robinson PA, Rennie CJ, Wright JJ, Bahramali H, Gordon E, Rowe DL (2001) Prediction of electroencephalographic spectra from neurophysiology. Physical Reviews E 63, 021903 Rowe JB, Stephan KE, Friston KJ, Frackowiak RJ, Lees A, Passingham RE (2002) Attention to action in Parkinson's disease. Impaired effective connectivity among frontal cortical regions. Brain 125, 276-289 Stephan KE, Magnotta VA, White TJ, Arndt S, Flaum M, O'Leary DS & Andreasen, NC (2001a) Effects of Olanzapine on cerebellar functional connectivity in schizophrenia measured by fMRI during a simple motor task. Psychological Medicine 31, 1065-1078 Stephan KE, Kamper L, Bozkurt A, Burns GAPC, Young MP, Kotter R (2001b) Advanced database methodology for the Collation of Connectivity data on the Macaque brain (CoCoMac). Philosophical Transactions of the Royal Society London B Biological Sciences 356, 1159-1186 Stephan KE, Marshall JC, Friston KJ, Rowe JB, Ritzl A, Zilles K, Fink GR (2003) Lateralized cognitive processes and lateralized task control in the human brain. Science 301, 384-386

Stephan KE (2004) On the role of general system theory for functional neuroimaging.

Journal of Anatomy 205, 443-470 Stephan KE, Harrison LM, Penny WD, Friston KJ (2004) Biophysical models of fMRI responses. Current Opinion in Neurobiology 14, 629-635 Stephan KE, Penny WD, Marshall JC, Fink GR, Friston KJ (2005) Investigating the functional role of callosal connections with dynamic causal models. Annals of the New York Academy of Sciences 1064:16-36 Stephan KE, Baldeweg T, Friston KJ (2006) Synaptic plasticity and dysconnection in schizophrenia. Biological Psychiatry 59, 929-939 Toni I, Rowe JB, Stephan KE & Passingham RE (2002) Changes of cortico-striatal effective connectivity during visuo-motor learning. Cerebral Cortex 12: 1040-1047

von Bertalanffy L (1969) General System Theory. George Braziller, New York Wiener N (1948) Cybernetics. Wiley, New York

Yamashita O, Galka A, Ozaki T, Biscay R, Valdes-Sosa P (2004) Recursive penalized least squares solution for dynamical inverse problems of EEG generation. Human Brain Mapping 21, 221-235

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

### 1 Introduction

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.

### 2 Terminology

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.

### 3 Multivariate Analysis

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

## Post a comment