Kinoshita, T

Kochi, K

Koukkou, M

Lehmann, D

Pascual-Marqui, R D

ZORA

Zurich Open Repository and ArchiveUniversity of ZurichMain LibraryWinterthurerstr. 190CH-8057 Zurichwww.zora.uzh.chYear: 2011Functional independent components: revealingcortico-cortical, cross-frequency interactionsPascual-Marqui, R D; Kochi, K; Lehmann, D; Koukkou, M; Kinoshita, TPascual-Marqui, R D; Kochi, K; Lehmann, D; Koukkou, M; Kinoshita, T (2011). Functional independentcomponents: revealing cortico-cortical, cross-frequency interactions. Japanese Journal of Pharmaco-EEG,12:53-58.Postprint available at:http://www.zora.uzh.chPosted at the Zurich Open Repository and Archive, University of Zurich.http://www.zora.uzh.chOriginally published at:Pascual-Marqui, R D; Kochi, K; Lehmann, D; Koukkou, M; Kinoshita, T (2011). Functional independentcomponents: revealing cortico-cortical, cross-frequency interactions. Japanese Journal of Pharmaco-EEG,12:53-58.Pascual-Marqui, R D; Kochi, K; Lehmann, D; Koukkou, M; Kinoshita, T (2011). Functional independentcomponents: revealing cortico-cortical, cross-frequency interactions. Japanese Journal of Pharmaco-EEG,12:53-58.Postprint available at:http://www.zora.uzh.chPosted at the Zurich Open Repository and Archive, University of Zurich.http://www.zora.uzh.chOriginally published at:Pascual-Marqui, R D; Kochi, K; Lehmann, D; Koukkou, M; Kinoshita, T (2011). Functional independentcomponents: revealing cortico-cortical, cross-frequency interactions. Japanese Journal of Pharmaco-EEG,12:53-58. Functional independent components: revealing cortico-cortical, cross-frequency interactions  Roberto D. Pascual-Marqui1,2, Kieko Kochi1, Dietrich Lehmann1, Martha Koukkou1, Toshihiko Kinoshita2  1The KEY Institute for Brain-Mind Research, University Hospital of Psychiatry, Zurich, Switzerland 2Department of Neuropsychiatry, Kansai Medical University Hospital, Osaka, Japan   Abstract  In its most classical and simple setting, the statistical analysis of functional brain images consists of comparing one group of subjects under two different conditions, or comparing two independent groups of subjects, using voxel-by-voxel statistics with correction for multiple testing. This approach is widely used in almost all brain imaging literature, where the aim is the “localization of brain function”.  An alternative approach consists of abandoning the concept that brain function can be localized to a small number of hotspots, and to embrace the notion that brain function is more adequately characterized by the spatio-temporal/frequency distributions of activity. Recently, this concept is becoming popular with the use of independent component analysis (ICA) of fMRI data.  In this paper, using insights from a novel field in statistics called “functional data analysis”, a new formulation for the discovery of interaction patterns of brain activity across space, time and frequency is presented. The method is illustrated by comparing EEG data between schizophrenic patients and normal control subjects.   1. Introduccion  We will consider neuroimaging experiments that use techniques such as fMRI or use EEG tomographies such as LORETA [1-3]. These experiments typically produce data that consist of brain activity values, measured over time, from a large number of cortical voxels. For instance, the data for the i-th subject ( 1... Si N= ) is represented as a matrix V TN Ni ×∈X  , corresponding to brain activity sampled at TN  moments in time (number of columns), from VN  cortical voxels (number of rows). Alternatively, as is typical in high-time resolution EEG, frequency domain data is considered instead, where VN Ni Ω×∈X   may denote spectral density of the electric neuronal activity for NΩ  discrete frequencies.  In order to analyze jointly the data from all subjects, the collection of SN  matrices must be converted into a single object. For instance, it is very common to concatenate the matrices over time: Eq. 1 ( ) ( )1 2 ... V S Ts N N NN ×= ∈Y X X X   This procedure produces a new single larger matrix, where all the subjects are placed next to each other along the temporal dimension (increasing the number of columns), but without changing the spatial dimension (i.e. without changing the structure of the cortical voxels corresponding to the rows).  There are many recent publications in neuroimaging that try to discover cortical connectivity patterns from such experimental data. Two commonly used techniques are the singular value decomposition (SVD) as explained in [4], and independent component analysis (ICA) as reviewed in [5] and [6].  In the SVD approach [4], it is first necessary to standardize the group matrix Y  from Eq. 1 above, to ensure that the mean and standard deviations of the elements in each row are zero and one, respectively. Once the rows of the group matrix Y  are standardized, then the classical SVD is calculated: Eq. 2 TΛ= Y Y YY L R  where the columns of the matrix YL  are the left eigenvectors of Y. The brain image corresponding to the first (largest) left eigenvector, denoted as 1VN ×∈Y1L  , will detect regions of correlated voxels, as shown in [4].  In the ICA approach, one possible formulation out of many other possibilities consists of calculating: Eq. 3 = Y YY A S  with ×∈ VN KYA , ( )×∈ S TK N NYS , and where K denotes the number of components. Ideally, the K time series in the rows of the matrix YS  should be statistically independent in a strict sense, which can be approximately achieved in many different ways (see e.g. [7]). The columns of the matrix YA  contain the spatial components, where each one will detect interconnected brain regions.  These methodologies are of proven value in the discovery of functional connectivity. When they are interpreted from the point of view of functional data analysis [8], new generalizations can be derived, giving detailed temporal information about the nature of the connectivity patterns. The aim of this study is to present a functional data analysis approach to functional connectivity that allows the discovery of brain interactions across space (cortical locations), time, and frequency.   2. Functional data analysis perspective  Typically, measures of connectivity are based on the “similarity” between the time series recorded at two different locations. A simple similarity index is, for instance, the cross-correlation coefficient. However, it is nearly impossible to analyze the massive number of similarities when one considers all possible pairs of voxels at all possible time lags.  A solution to this problem can be obtained by considering the basic data as a function of several variables: space (cortical voxels) and time. This is the approach used in functional data analysis [8]. The data from each subject, consisting of brain activity, is now represented as a vector: Eq. 4 ( )××∈ → ∈  1T VV T N NN N veci iX X  where the “vec” operator transforms a matrix into a vector by stacking the columns of the matrix one underneath the other [9]. Thus, the elements of the vector correspond to brain activity values sampled at points in the (space, time) hyperplane.  The new group functional data matrix is now defined as follows: Eq. 5 ( ) ( )×= ∈1 2 ... T V SS N N Nvec vec vecNZ X X X   This is the basic idea behind functional data analysis, and it may seem deceptively simple, but in fact it is radically different from any other published form of group analysis [5], [10], [11].   2.1. Functional singular value decomposition (fSVD)  The SVD method of Worsley et al (2005) [4] can now be applied to the standardized group functional data matrix Z in Eq. 5, giving: Eq. 6 TΛ= Z Z ZZ L R  As before, the columns of the matrix ZL  are the left eigenvectors of Z. However, the first (largest) left eigenvector ( ) 1T VN N ×∈∈Z1L   consists of time varying brain images, which will reveal the time course of interconnected brain regions.   2.2. Interpretation of the functional component  As an example, suppose that after appropriately thresholding of the elements of Z1L , brain region A at an early latency τ A  has high values, and is followed by high values in a different brain region B at a later latency τ B . Then the interpretation is that brain regions A and B are cross-correlated with the time lag τ τ−A B . Such cross-spatial and cross-temporal connections can be revealed without having to explore nor calculate and analyze explicitly all pairwise cross-correlations.   2.3. Functional independent component analysis (fICA)  The ICA method can now be applied to group functional data matrix Z in Eq. 5, giving: Eq. 7 = Z ZZ A S  with ( )T VN N K×∈ZA  , SK N×∈ZS  , and where K denotes the number of components. Ideally, the K rows of the matrix ZS  (corresponding to subject scores for each component) should be statistically independent in a strict sense, which can be approximately achieved in many different ways (see e.g. [7]). Each component column of the matrix ZA  (corresponding to a spatio-temporal vector in ( ) 1T VN N × ) conveys information on the time course of the correlated brain regions related to that component. This means that the interpretation explained above in Section 2.2 with respect to the functional SVD applies for each independent component here.   3. An example  The new fICA method will be illustrated with EEG data from a previous study [12] which compared 9 schizophrenic (Schz) patients with 36 normal control (NC) subjects.  The data, as previously analyzed in [12], consisted of eyes-closed, awake, resting EEG recordings on 19 scalp electrodes from the 10/20 electrode placement system, sampled at 128 Hz with 1-30 Hz band-pass. For each subject, 20 artifact free EEG epochs of 2 seconds duration each were used for power spectra computations. In the current re-analysis, we use standardized low resolution electromagnetic tomography (sLORETA) [1] to compute the spectral density of the electric neuronal generators on 6239 cortical voxels for the 7 classical EEG frequency bands [13] corresponding to delta (1.5-6 Hz), theta (6.5-8 Hz), alpha1 (8.5-10 Hz), alpha2 (10.5-12 Hz), beta1 (12.5-18 Hz), beta2 (18.5-21 Hz), beta3 (21.5-30 Hz).  In this way, the data for the i-th subject has the standard form VN Ni Ω×∈X  , with 1... Si N= , 45SN =  total number of subjects in the study. The matrix X contains spectral density values at each voxel ( 6239VN =  total number of cortical voxels), and for each frequency band ( 7NΩ = total number of frequency bands).  This classical form of the single subject data was then transformed into the “group functional data” form of Eq. 4 and Eq. 5 above, producing the matrix ( )V SN N NΩ ×∈Z   ( 7NΩ = , 6239VN = , 45SN = ). This was then analyzed with the method of functional ICA (fICA, Section 2.3), with 10K =  independent components. As a final validation step, the subject scores (10 scores per subject) contained in the matrix SK N×∈ZS   were used for comparing the schizophrenia and normal control groups. From the matrix ZA , the column ( ( ) 1VN NΩ × ) corresponding to the most significantly different component revealed the spatio-frequency pattern with highest loadings shown in Figures 1, 2, and 3.  Note that the functional component has two aspects: spatial localization and frequency band.  Figure 1 places emphasis on the frequency band information, without revealing localization information. The results indicate that slow oscillations (delta band) are inversely coupled to slightly faster oscillations (alpha1), i.e., when slow delta oscillations increase, the faster alpha1 oscillations decrease.  Figures 2 and 3 display the locations of these oscillations, showing that the slow delta oscillations are frontal, whereas the alpha1 oscillations are parietal and posterior cingulated.  Taken all together, the result shows that in the patients with schizophrenia as compared to the normal controls, an increase in frontal slow oscillations is coupled to a decrease of alpha1 oscillations in posterior regions.   4. Conclusion  According to these results, a tentative general conclusion is that the electrophysiological correlates of the pathology (schizophrenia) are not located to one brain area, nor are they occurring at a certain given frequency. Instead, they are distributed in the brain across several brain regions and across several oscillatory components, in the form of a spatial-frequency distribution. In other words, these results support the notion that brain states are better characterized by the functional relations between different regions operating at possibly different frequencies, i.e. the cortico-cortical, cross-frequency interactions.   5. References  1 Pascual-Marqui RD. Standardized low-resolution brain electromagnetic tomography (sloreta): Technical details. Methods Find Exp Clin Pharmacol. 2002; 24 Suppl D: 5-12. 2 Pascual-Marqui RD. Theory of the eeg inverse problem. In: Tong S, Thakor N, eds. Quantitative eeg analysis: Methods and clinical applications. Boston: Artech House 2009; 121-140. 3 Pascual-Marqui RD, Michel CM, Lehmann D. Low resolution electromagnetic tomography: A new method for localizing electrical activity in the brain. Int J Psychophysiol. 1994; 18: 49-65. 4 Worsley KJ, Chen JI, Lerch J, Evans AC. Comparing functional connectivity via thresholding correlations and singular value decomposition. Philos Trans R Soc Lond B Biol Sci. 2005; 360: 913-920. 5 Calhoun VD, Liu J, Adali T. A review of group ica for fmri data and ica for joint inference of imaging, genetic, and erp data. Neuroimage. 2009; 45: S163-172. 6 Eichele T, Calhoun VD, Debener S. Mining eeg-fmri using independent component analysis. Int J Psychophysiol. 2009; 73: 53-61. 7 Cichocki A, Amari S. Adaptive blind signal and image processing: Learning algorithms and applications. New York: John Wiley & Sons, 2002. 8 Ramsay JO, Silverman BW. Functional data analysis. 2nd edn. New York: Springer, 2005. 9 Magnus JR, Neudecker H. Matrix differential calculus with applications in statistics and econometrics. Rev. edn. New York: John Wiley, 1999. 10 Guo Y, Pagnoni G. A unified framework for group independent component analysis for multi-subject fmri data. Neuroimage. 2008; 42: 1078-1093. 11 Varoquaux G, Sadaghiani S, Pinel P, Kleinschmidt A, Poline JB, Thirion B. A group model for stable multi-subject ica on fmri datasets. Neuroimage. 2010; 51: 288-299. 12 Pascual-Marqui RD, Lehmann D, Koenig T, et al. Low resolution brain electromagnetic tomography (loreta) functional imaging in acute, neuroleptic-naive, first-episode, productive schizophrenia. Psychiatry Res. 1999; 90: 169-179. 13 Kubicki S, Herrmann WM, Fichte K, Freund G. Reflections on the topics: Eeg frequency bands and regulation of vigilance. Pharmakopsychiatr Neuropsychopharmakol. 1979; 12: 237-245.    delta theta       alpha beta  Figure 1: Frequency band profile for the functional independent component that has highest significant difference between patients with Schizophrenia and Normal Controls. For each frequency band, the maximum loading over all 6239 voxels is shown (relative scale). The highest loadings for this functional component occur for the delta and alpha1 bands (underlined).     Figure 2: Spatial (cortical) profile of the delta band corresponding to the functional independent component that has highest significant difference between patients with Schizophrenia and Normal Controls. Maximum (positive, color coded yellow) delta band location occurs in frontal regions. Notation. A: anterior; S: superior; P: posterior; L: left; H: hemisphere; V: view; R: right; B: bottom.     Figure 3: Spatial (cortical) profile of the alpha1a band corresponding to the functional independent component that has highest significant difference between patients with Schizophrenia and Normal Controls. Maximum (negative, color coded blue) alpha1 band location occurs in parietal and posterior cingulated cortex regions. Notation. A: anterior; S: superior; P: posterior; L: left; H: hemisphere; V: view; R: right; B: bottom.   

Functional independent components: revealing cortico-cortical, cross-frequency interactions

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Functional independent components: revealing cortico-cortical, cross-frequency interactions

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