Particle Filtering in MEG: from single dipole filtering to Random Finite Sets

1. Particle Filtering in MEG: from single dipole filtering to Random Finite Sets A. Sorrentino CNR-INFM LAMIA, Genova methods for image and data analysis www.dima.unige.it/~piana/mida/group.html sorrentino@fisica.unige.it

2. Co-workers Genova group: Cristina Campi (Math Dep.) Annalisa Pascarella (Comp. Sci. Dep.) Michele Piana (Math. Dep.) Long-time collaboration Lauri Parkkonen (Brain Research Unit, LTL, Helsinki) Recent collaboration Matti Hamalainen (MEG Core Lab, Martinos Center, Boston)

3. Basics of MEG modeling Neural current Biot-Savart Ohmic term Poisson Biot-Savart Accurate model of brain conductivity Biot-Savart

4. 2 approaches to MEG source modeling Imaging approach Parametric approach Continuous current distribution Focal current Model M small N large Unknown Method Non-linear optimization methods Regularization methods Result

5. Automatic current dipole estimate • Common approximations to solve this problem: • Number of sources constant • Source locations fixed • Common methods: • Manual dipole modeling • Automatic dipole modeling • Estimate the number of sources • Estimate the source locations • Least Squares for source strengths Manual dipole modeling still the main reference method for comparisons (Stenbacka et al. 2002, Liljestrom et al 2005) Bayesian filtering allows overcoming these limitations

6. Bayesian filtering in MEG - assumptions Two stochastic processes: J1 J2 … Jt … B1 B2 … Bt … Markov process Markovian assumptions: Instantaneous propagation No feedback Our actual model The final aim:

7. Bayesian filtering in MEG – key equations Likelihood function “Observation” ESTIMATES Transition kernel “Evolution” … … Linear-Gaussian model  Kalman filter Non-linear model  Particle filter

8. Particle filtering of current dipoles The key idea: sequential Monte Carlo sampling. (single dipole space) Draw random samples (“particles”) from the prior Update the particle weights Resample and let particles evolve

9. A 2D example – the data

10. A 2D example – the particles

11. The full 3D case – auditory stimuli S. et al., ICS 1300 (2007)

12. Comparison with beamformers and RAP-MUSIC Pascarella et al., ICS 1300 (2007); S. et al. , J. Phys. Conf. Ser. 135 (2008) Two quasi-correlated sources Beamformers: suppression of correlated sources

13. Comparison with beamformers and RAP-MUSIC Pascarella et al., ICS 1300 (2007); S. et al. , J. Phys. Conf. Ser. 135 (2008) Two orthogonal sources RAP-MUSIC: wrong source orientation, wrong source waveform

14. Rao-Blackwellization Campi et al. Inverse Problems (2008); S. et al. J. Phys. Conf. Ser. (2008) Can we exploit the linear substructure? Analytic solution (Kalman filter) Sampled (particle filter) Accurate results with much fewer particles Statistical efficiency increased (reduced variance of importance weights) Increased computational cost

15. Bayesian filtering with multiple dipoles A collection of spaces (single-dipole space D, double-dipole space,...) A collection of posterior densities (one on each space) Exploring with particles all spaces (up to...) One particle = one dipole One particle = two dipoles One particle = three dipoles Reversible Jumps (Green 1995) from one space to another one

16. Random Finite Sets – why Non uniquess of vector representations of multi-dipole states: (dipole_1,dipole_2) and (dipole_2,dipole_1) same physical state, different points in D XD Consequence: multi-modal posterior density non-unique maximum non-representative mean A random finite set X of dipoles is a measurable function For some realizations, Let (W,s,P) be a probability space Where is the set of all finite subsets of (single dipole space) equipped with the Mathéron topology

17. Random Finite Sets - how Probability measure of RFS: a conceptual definition • Belief measure instead of probability measure Multi-dipole belief measures can be derived from single-dipole probability measures • Probability Hypothesis Density (PHD): the RFS-analogous of the conditional mean The integral of the PHD in a volume = number of dipoles in that volume Peaks of the PHD = estimates of dipole parameters Model order selection: the number of sources estimated dynamically

18. RFS-based particle filter: Results S. et al., Human Brain Mapping (2009) • Monte Carlo simulations: • 1.000 data sets • Random locations (distance >2 cm) • Always same temporal waveforms • 2 time-correlated sources • peak-SNR between 1 and 20 • Results: • 75% sources recovered (<2 cm) • Average error 6 mm, independent on SNR • Temporal correlation affects the detectability very slightly

19. RFS-based particle filter: Results S. et al., Human Brain Mapping (2009) Comparison with manual dipole modeling Data: 10 sources mimicking complex visual activation The particle filter performed on average like manual dipole modeling performed by uninformed users (on average 6 out of 10 sources correctly recovered)

20. In progress Source space limited to the cortical surface Two simulated sources

21. In progress Two sources recovered with orientation constraint Only one source recovered without orientation constraint

22. References • Sorrentino A., Parkkonen L., Pascarella A., Campi C. and Piana M. Dynamical MEG source modeling with multi-target Bayesian filteringHuman Brain Mapping 30: 1911:1921 (2009) • Sorrentino A., Pascarella A., Campi C. and Piana M. A comparative analysis of algorithms for the magnetoencephalography inverse problemJournal of Physics: Conference Series 135 (2008) 012094. • Sorrentino A., Pascarella A., Campi C. and Piana M. Particle filters for the magnetoencephalography inverse problem: increasing the efficiency through a semi-analytic approach (Rao-Blackwellization)Journal of Physics: Conference Series 124 (2008) 012046. • Campi C., Pascarella A., Sorrentino A. and Piana M. A Rao-Blackwellized particle filter for magnetoencephalographyInverse Problems 24 (2008) 025023 • - Sorrentino A., Parkkonen L. and Piana M. Particle filters: a new method for reconstructing multiple current dipoles from MEG data International Congress Series 1300 (2007) 173-176