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Applications of 2-Riemannian Helmholtz-Hodge Decomposition to MEG Source Dynamics

Sheraz Khan1*, Julien Lefevre2, Manoj Raghavan1 and Sylvain Baillet1
  • 1 Medical College of Wisconsin, Athena project team, United States
  • 2 University of Mediterranee, LSIS - UMR CNRS 6168 , France

MEG sources computed using distributed models exhibit complex spatial and temporal dynamics. We have recently proposed that optical flow techniques could be adapted to the evaluation of basic kinematics of MEG source current on the cortical surface [1, 2]. Helmholtz-Hodge Decomposition (HHD) is a technique that decomposes a 2D (resp. 3D) continuous vector field into a sum of three parts: a non-rotational component deriving from the gradient of a scalar potential U, a non-diverging component deriving from the rotation of a scalar potential A (resp. vectorial potential), and a harmonic vector field whose Laplacian vanishes [3, 4]. Even when described on polyhedral surfaces [5] computations are typically performed locally in Euclidean space. Surface curvature has to be taken into account for a proper estimation of vector fields on tangent spaces [1], and convergence is altered by non-flatness properties. We redefined HHD in Riemannian space which enables us to detect features in motion-fields even on highly-curved surfaces, like the cortex.We demonstrate automatic extraction of salient dynamical features from MEG source activity during a somatosensory stimulation task, thereby facilitating reproducible automated analysis of experimental data. We further show how HHD can be used to characterize the dynamics of focal epileptic activity by finding critical points in the surface distribution of scalar potential U. Divergence provides a quantitative measure of cortical dynamics that appears to reveal areas of initiation of spike and seizure activity within epileptic networks.This methodology and early applications suggest that HHD-based measures can capture complex dynamical features from the time-evolving vector motion field of cortical current flow.

References

1. Lefévre, J. and Baillet, S., “Optical flow and advection on 2-Riemannian manifolds: a common framework,” IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 30, no. 6, pp. 1081–92, 2008.

2. Khan, S., Lefévre, J. and Baillet, S. “Feature Extraction from Time-Resolved Cortical Current Maps using the Helmholtz-Hodge Decomposition”, Proc. 15th Int. Conf. on Human Brain Map., San Francisco, 2009

3. Guo, Q., Mandal, M.K, and Li, M.Y, “Efficient Hodge–Helmholtz decomposition of motion fields,” Pattern Recognition Letters, vol. 26, no. 4, pp. 493–501, 2005.

4. Tong, Y., Lombeyda, S., Hirani, A.N., and Desbrun, M., “Discrete multiscale vector field decomposition,” ACM Transactions on Graphics, vol. 22, no. 3, pp. 445–452, 2003.

5. Polthier, K. and Preuss, E., “Identifying vector fields singularities using a discrete Hodge decomposition,” Visualization and Mathematics, vol. 3, pp. 113–134, 2003.

Conference: Biomag 2010 - 17th International Conference on Biomagnetism , Dubrovnik, Croatia, 28 Mar - 1 Apr, 2010.

Presentation Type: Poster Presentation

Topic: MEG Modeling

Citation: Khan S, Lefevre J, Raghavan M and Baillet S (2010). Applications of 2-Riemannian Helmholtz-Hodge Decomposition to MEG Source Dynamics. Front. Neurosci. Conference Abstract: Biomag 2010 - 17th International Conference on Biomagnetism . doi: 10.3389/conf.fnins.2010.06.00067

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Received: 22 Mar 2010; Published Online: 22 Mar 2010.

* Correspondence: Sheraz Khan, Medical College of Wisconsin, Athena project team, Milwaukee, United States, shkhan@mcw.edu