The vector x (t) contains the time series for each region, the vector e (t) is a zero-mean uncorrelated noise process and the coefficient matrices A. can be found by solving the multivariate Yule-Walker equations. If transformed to the frequency domain, the MVAR equation takes on the more simple form x (f) = H (f) • E (f), where H(f) is the transfer function matrix (Kus, et al., 2004). From the frequency domain representation we can compute the Directed Transfer Function (DTF), a frequency dependent measure of the interaction between nodes in the network (Kaminski, et al., 2001). Similarly to phase-locking and coherence measures, the DTF is a measure of interaction at a single frequency, although these interactions are computed simultaneously across the entire frequency range of interest. An advantage of MVAR models is that they allow one to determine the direction of interaction by analogy to the concept of Granger Causality. Furthermore, unlike the SEM models, no a priori knowledge is required about connectivity in the network. The wide sense stationarity requirement does limit the utility of MVAR models in event related studies, but since they are parametric they can be fit using relatively few temporal samples (with the covariance computed by averaging across epochs) and nonstationarity then detected using a sliding window. Since the method is linear, crosstalk will have a significant effect on DTF. If the mixed signals are given by y = M • x, then instead of the 'true' coefficient matrices A one gets the mixed coefficient matrices AiM, but there is no way to identify the mixing matrix M from the estimated coefficient matrices.

Neural mass models (David et al., 2004, David and Friston, 2003, Jansen and Rit, 1995) are physiologically motivated models of cortical activity, which attempt a mathematical description of the network of neurons that generate the signals measured by MEG/EEG. The purpose of these models is to provide a realistic network simulation of cortical activity with controllable connectivity parameters, but they can also be used to estimate network parameters from real data (David and Friston, 2003). The basic elements of the model are excitatory and inhibitory columns of neurons (Jansen and Rit, 1995). The state of each element is described by an average membrane potential and a mean firing rate. The membrane potential and average firing rate are related by nonlinear input/output functions for each element and the parameters of these functions are thus the parameters of the element. A number of interaction constants between the ensembles can be used to describe the network structure. If no external input signal is used, the model is typically driven by random noise. David and Friston (2003) expanded the double column model proposed by Jansen and Rit (1995) to a meta-ensemble or -mass model, which instead of containing only two columns is comprised of N columns. These neuronal mass models can be thought of as cortical areas and by introduction of multiple areas and coupling between those, macroscopic behavior of the brain, as can be measured by MEG/EEG, can be simulated. Mass models can exhibit a broad range of frequencies and their non-linear nature and controllable macroscopic coupling make them good candidates for testing connectivity measures (David et al., 2004). Also by defining meta parameters of the model such as the number of areas and their connectivity, the models themselves can be used to fit real measurements and thus provide an estimate of functional connectivity (David and Friston, 2003).

References

Andres F, Gerloff C (1999) Coherence of Sequential Movements and Motor Learning.

Journal of Clinical Neurophysiology 16(6), 520-527 Astolfi L, Cincotti F, Babiloni C, Carducci F, Basilisco A, Rossini PM, Salinari S, Mattia D, Cerutti S, Ben Dayan D, Ding L, Ni Y, He B, Babiloni F (2005) Estimation of the Cortical Connectivity by High-Resolution EEG and Structural Equation Modeling: Simulations and Application to Finger Tapping Data. IEEE Transactions on Biomedical Engineering 52(5), 757-768 Baillet S, Mosher JC, Leahy RM (2001) Electromagnetic Brain Mapping. IEEE

Signal Processing Magazine 18 (6), 14-30 Baillet S, Riera JJ, Marin G, Mangin JF, Aubert J, Garnero L (2001) Evaluation of inverse methods and head models for EEG source localization using a human skull phantom. Physics in Medicine and Biology 46, 77-96 Berg P, Scherg M (1994) A fast method for forward computation of multiple-shell spherical head models. Electronencephalography and Clinical Neurophysiology 90, 58-64

Blair RC, Karninski W (1993) An alternative method for significance testing of waveform difference potentials. Psychophysiology 30(5), 518-524 Bollen KA (1989), Structural Equations with Latent Variables. Wiley. New York, Braun C, Kaiser S, Kincses WE, Elbert T (1997) Confidence Interval of Single Dipole

Locations Based on EEG Data. Human Brain Mapping 10(1), 31-39 Brovelli A, Ding M, Ledberg A, Chen Y, Nakamura R, Bressler SL (2004) Beta Oscillations in large-scale sensorimotor cortical network: Directional influences revealed by Granger causality. Proceedings of the National Academy of Sciences 101(26), 9849-9854

Dale AM, Serano MI (1993) Improved localization of cortical activity by combing EEG and MEG with MRI cortical surface reconstruction: a linear approach. Journal of Cognitive Neuroscience 5, 162-176 David O, Friston KJ (2003) A neural mass model for MEG/EEG:coupling and neuronal dynamics. Neuroimage 20, 1743-1755 David O, Cosmelli D, Friston KJ (2004) Evaluation of different measures of functional connectivity using a neural mass model. Neuroimage 21, 659-673 Darvas F, Pantazis D, Kucukaltun-Yildirim E, Leahy RM (2004) Mapping human brain function with MEG and EEG:methods and validation. Neuroimage 23S, 289-299

Darvas F, Rautiainen M, Pantazis D, Baillet S, Benali H, Mosher JC, Garnero L, Leahy RM (2005) Investigations of dipole localization accuracy in MEG using the bootstrap. Neuroimage 25, 355-368 Darvas F, Ermer JJ, Mosher JC, Leahy RM (2006) Generic Head Models for Atlas-

Based EEG Source Analysis. Human Brain Mapping 27, 129-143 David O, Garnero L, Cosmelli D, Varela FJ (2002) Estimation of Neural Dynamics from MEG/EEG Cortical Current Density Maps: Application to the Reconstruction of Large-Scale Cortical Synchrony. IEEE Transactions on Biomedical Engineering 49(9), 975-987 Fell J, Klaver P, Lehnertz K, Grunwald T, Schaller C, Elger CE, Fernandez G (2001) Human memory formation is accompanied by rhinal-hippocampal coupling and decoupling. Nature neuroscience 4(12), 1259-1264

Friston KJ (1994) Functional and effective connectivity in neuroimaging: a synthesis.

Human Brain Mapping 2, 256-278 Fuchs M, Wagner M, Kohler T, Wischmann HA (1999) Linear and Nonlinear Current

Density Reconstructions. Journal of Clinical Neurophysiology 163(3), 267-295 Fuchs M, Wagner M, Kastner J (2001) Boundary element method volume conductor models for EEG source reconstruction. Journal of Clinical Neurophysiology 112(8), 1400-1407

Fuchs M, Kastner J, Wagner M, Hawes S, Ebersole JS (2002) A standardized boundary element method volume conductor model. Journal of Clinical Neurophysiology 113(5), 702-712 Gevins AS, Doyle JC, Cutillo BA, Schaffer RE, Tannehill RS, Bressler SL (1985). Neurocognitive pattern analysis of a visuospatial task: rapidly-shifting foci of evoked correlations between electrodes. Psychophysiology 22, 32-43. Gevins AS, Bressler SL (1988) Functional topography of the human brain. In

Pfurtscheller G (Ed.) Functional Brain Imaging. Hans Huber, Bern, 99-116 Gevins AS, Cutillo BA (1993) Spatiotemporal dynamics of component processes in human working memory. Electroencephalography and clinical Neurophysiology 87, 128-143

Gross J, Tass PA, Salenius S, Hari R, Freund HJ, Schnitzler A (2000) Cortico-muscular synchronization during isometric muscle contraction in humans as revealed by Magnetoencephalography. Journal of Physiology 527(3), 623-631 Gross J, Kujala J, Hamaläinen M, Timmermann L, Schnitzler A, Salmelin R (2001) Dynamic imaging of coherent sources: Studying neural interactions in the human brain. Proceedings of the National Academy of Sciences 98(2), 694-699 Gross J, Timmermann L, Kujala J, Salmelin R, Schnitzler A (2003) Properties of MEG tomographic maps obtained with spatial filtering. Neuroimage 19, 1329-1336

Hamäläinen M, Hari R, Ilmoniemi RJ, Knuutila J, Lounasmaa OV (1993) Magne-toencephalography - theory, instrumentation, and applications to noninvasive studies of the working human brain. Reviews of Modern Physics 65(2), 413-497 Horwitz B (2003) The elusive concept of brain connectivity. Neuroimage 19, 466-470 Jamsek J, Stefanovska A, McClintock PVE, Khovanov IA (2003) Time-phase bis-

pectral analysis. Physical Review E 68, 016201 1-12 Jansen BH, Rit VG (1995) Electronencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns. Biological Cybernetics 73, 357-366

Jerbi K, Baillet S, Mosher JC, Nolter G, Garnero L, Leahy RM (2004) Localization of Realistic Cortical Activity in MEG using Current Multipoles. Neuroimage 22(2), 779-793

Johnson CR (1997) Computational and numerical methods for bioelectric field problems. Critical Reviews in Biomedical Engineering 25(1), 1-81 Kaminski M, Ding M, Truccolo WA, Bressler SL (2001) Evaluating causal relations in neural systems: Granger causality, directed transfer function and statistical assessment of significance. Biological Cybernectics 85(2), 145-157 Kim SG, Richter W, Ugurbil K (1997) Limitations of temporal resolution in functional MRI. Magnetic resonance in medicine 37(4), 631-636 Okada Y, Wu J, Kyuhou S (1997) Genesis of MEG signals in a mammalian CNS structure. Electroencephalography & Clinical Neurophysiology 103, 474-485

Kus R, Kaminski M, Blinowska KJ (2004) Determination of EEG activity propagation: pair-wise versus multi-channel estimate. IEEE Transactions on Biomedical Engineering 51(9), 1501-1510 Lachaux JP, Rodriguez E, Martinerie J, Varela FJ (1999) Measuring Phase Synchrony in Brain Signals. Human Brain Mapping 8, 194-208 Leahy RM, Mosher JC, Spencer ME, Huang MX, Lewine JD (1998) A study of dipole localization accuracy for MEG and EEG using a human skull phantom. Electroencephalography and Clinical Neurophysiology 107(2), 159-173 Lee L, Harrison LM, Mechelli A (2003) A report on functional connectivity workshop, Dusseldorf 2002. Neuroimage 19, 457-465 Le Van Quyen M, Foucher J, Lauchaux JP, Rodriguez E, Lutz A, Martinerie J, Varela FJ (2001) Comparison of Hilbert transform and wavelet methods for the analysis of neuronal synchrony. Journal of Neuroscience Methods 111, 83-89 Liu AK, Dale AM, Belliveau JW (2002) Monte Carlo Simulation Studies of EEG

and MEG Localization Accuracy. Human Brain Mapping 16, 47-62 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

Miltner W, Braun C, Arnold M, Witte H, Taub E (1999) Coherence of gamma-band

EEG activity as a basis for associative learning. Nature 397,434-436 Mima T, Oluwatimilehin T, Hiraoka T, Hallet M (2001) Transient Interhemispheric Neuronal Synchrony Correlates with Object Recognition. Journal of Neuroscience 21(11), 3942-3948 Mosher JC, Spencer ME, Leahy RM, Lewis P (1993) Error bounds for EEG and MEG dipole source localization. Electroencephalography and Clinical Neuro-physiology 86(5), 303-321 Mosher JC, Leahy RM, Lewis PS (1999a) EEG and MEG: forward Solutions for Inverse Methods. IEEE Transactions on Biomedical Engineering 46(3), 245-259 Mosher JC, Leahy RM (1999b) Source localization using recursively applied and projected (RAP) music. IEEE Transactions on Signal Processing 47(2), 332-340 Nikias CL, Mendel JM (1993) Signal Processing with Higher-Order Spectra. IEEE

Signal Processing Magazine 10(3), 10-37 Nolte G, Bai O, Wheaton L, Mari Z, Vorbach S, Hallet M (2004) Identifying true brain interaction from EEG data using the imaginary part of coherency. Clinical Neurophysiology 115, 2292-2307 Nunez PL, Srinivasan R, Westdorp AF, Wijesinghe RS, Tucker DM, Silberstein RB, Cadusch PJ (1997) EEG coherency I:statistics, reference electrode, volume conduction, Laplacians, cortical imaging, and interpretation at multiple scales. Elec-tronencephalography and clinical Neurophysiology 103, 499-515 Pantazis D, Nichols TE, Baillet S, Leahy RM (2005) A comparison of random field theory and permutation methods for the statistical analysis of MEG data, Neuroimage 25, 383-394

Pascual-Marqui RD, Michel CM, Lehmann D (1994) Low resolution electromagnetic tomography: a new method for localizing electrical activity in the brain. International Journal of Psychophysiology 18(1), 49-65 Pfurtscheller G, Lopes da Silva FH (1999) Event-related EEG/MEG synchronization and desynchronization: basic principles. Journal of Clinical Neurophysiology 110, 1842-1857

Ramnani N, Behrens TEJ, Penney W, Matthews PM (2004) New Approaches for Exploring Anatomical and Functional Connectivity in the human Brain. Biological Psychiatry 56, 613-619 Riera JJ, Fuentes ME, Valdes PA, Oharriz Y (1998) EEG-distributed inverse solutions for a spherical head model. Inverse Problems 14, 1009-1019 Robinson SE, Vrba J.(1999). Functional neuroimaging by synthetic aperture magne-tometry (SAM). Recent Advances in Biomagnetism. Tohoku Univ. Press, Sendai, 302-305.

Rodriguez E, George N, Lauchaux JP, Matinerie J, Renault B, Varela FJ (1999) Perception's shadow: long-distance synchronization of human brain activity. Nature 397, 430-433

Sarvas J (1987) Basic mathematical and electromagnetic concepts of the biomagnetic inverse problems. Physics in Medicine and Biology 32, 11-22 Schack B, Vath N, Petsche H, Geissler HG, Moller E (2002) Phase-coupling of theta-gamma EEG rythms during short-term memory processing. International Journal of Psychophysiology 44, 143-163 Schanze T, Eckhorn R (1997) Phase correlation among rythms present at different frequencies: spectral methods, application to microelectrode recordings from visual cortex and functional implications. International Journal of Psychophys-iology 26, 171-189

Scherg M (1990) Fundamentals of dipole source potential analysis. In Grandori F,

Hoke M, Romani GL (Ed.) Advances of Audiology. Karger, Basel, 40-69 Simoes C, Jensen O, Parkkonen L, Hari R (2003) Phase locking between human primary and secondary somatosensory cortices. Proceedings of the National Academy of Sciences 100(5), 2691-2694 Singh KD, Barnes GR, Hillebrand A (2003) Group imaging of task-related changes in cortical synchronization using nonparametric permutation testing. Neuroimage 19, 1589-1601

Smith WE (1992) Estimation of the Spatio-Temporal Correlations of Biological Electrical Sources from Magnetic Their Fields. IEEE Transactions on biomedical engineering 39(10), 997-1004 Tallon-Baudry C, Bertrand O, Delpuech C, Pernier J (1997) Oscillatory y -Band (30-70 Hz) Activity Induced by a Visual Search Task in Humans. The Journal of Neuroscience 17(2), 722-734 Tass P, Rosenblum MG, Weule J, Kurths J, Pikovsky A, Volkmann J, Schnitzler A, Freund HJ (1998) Detection of n:m Phase Locking from Noisy Data: Application to Magnetoencephalography. Physical Review Letters 81(15), 3291-3294 Worsley KJ, Marret S, Neelin P, Vandal AC, Friston KJ, Evans AC (1996) A unified statistical approach for determining significant signals in images of cerebral activation. Human Brain Mapping 4, 58-73 Van Veen BD, van Drongelen W, Yuchtman M, Suzuki A (1997). Localization of brain electrical activity via linearly constrained minimum variance spatial filtering. IEEE Transactions on Biomedical Engineering 44 (9), 867-880 Zhang Z (1995) A fast method to compute surface potentials generated by dipoles within multilayer anisotropic spheres. Physics in Medicine and Biology 40, 335-349

Was this article helpful?

## Post a comment