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Particle Filtering in MEG: from single dipole filtering to Random Finite Sets A. SorrentinoCNR-INFM LAMIA, Genova methods for image and data analysis

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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)

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

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2 approaches to MEG source modeling Continuous current distributionFocal current Imaging approachParametric approach Unknown Model Result N large M small Method Regularization methods Non-linear optimization methods

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Automatic current dipole estimate Common approximations to solve this problem: Number of sources constant Source locations fixed Bayesian filtering allows overcoming these limitations 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)

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Bayesian filtering in MEG - assumptions J 1 J 2 … J t … B 1 B 2 … B t … Two stochastic processes: Markovian assumptions: Our actual model The final aim: Markov process Instantaneous propagation No feedback

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

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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

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A 2D example – the data

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A 2D example – the particles

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

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

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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

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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

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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 dipoleOne particle = two dipolesOne particle = three dipoles Reversible Jumps (Green 1995) from one space to another one

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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 X D Consequence: multi-modal posterior density non-unique maximum non-representative mean Where is the set of all finite subsets of (single dipole space) equipped with the Mathéron topology A random finite set of dipoles is a measurable function Let ( , ,P) be a probability space For some realizations,

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Random Finite Sets - how Probability measure of RFS: a conceptual definition Belief measure instead of probability measure 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 Multi-dipole belief measures can be derived from single-dipole probability measures

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RFS-based particle filter: Results S. et al., Human Brain Mapping (2009) Monte Carlo simulations: 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

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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)

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

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

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References - Sorrentino A., Parkkonen L., Pascarella A., Campi C. and Piana M. Dynamical MEG source modeling with multi-target Bayesian filtering Human Brain Mapping 30: 1911:1921 (2009) -Sorrentino A., Pascarella A., Campi C. and Piana M. A comparative analysis of algorithms for the magnetoencephalography inverse problem Journal of Physics: Conference Series 135 (2008) 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) Campi C., Pascarella A., Sorrentino A. and Piana M. A Rao-Blackwellized particle filter for magnetoencephalography Inverse Problems 24 (2008) 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)

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