Particle Filtering in MEG from single dipole filtering

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

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

Basics of MEG modeling Neural current Biot-Savart Ohmic term Poisson Biot-Savart Accurate model of brain conductivity rt B ava S t io

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

Automatic current dipole estimate Common approximations to solve this problem: ü Number of sources constant ü Source locations fixed Common methods: Ø Manual dipole modeling Ø Automatic dipole modeling üEstimate the number of sources üEstimate the source locations üLeast 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

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

Bayesian filtering in MEG – key equations “Observation” Likelihood function Transition kernel “Evolution” Linear-Gaussian model Kalman filter Non-linear model Particle filter … … E S T I M A T E S

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

A 2 D example – the data

A 2 D example – the particles

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

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

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

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

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 Reversible Jumps (Green 1995) from one space to another one One particle = three dipoles

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 Let (W, s, P) be a probability space A random finite set X of dipoles is a measurable function Where is the set of all finite subsets of (single dipole space) equipped with the Mathéron topology For some realizations,

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

RFS-based particle filter: Results S. et al. , Human Brain Mapping (2009) Monte Carlo simulations: ü 1. 000 data sets üRandom locations (distance >2 cm) üAlways same temporal waveforms ü 2 time-correlated sources ü peak-SNR between 1 and 20 Results: ü 75% sources recovered (<2 cm) ü Average error 6 mm, independent on SNR ü Temporal correlation affects the detectability very slightly

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)

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

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

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) 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 magnetoencephalography Inverse 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