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.htmlsorrentino@fisica.unige.it

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)

Basics of MEG modeling

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2 approaches to MEG source modeling

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Method Regularization methods Non-linear optimization methods

Automatic current dipole estimate

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Number of sources constant Source locations fixed

Bayesian filtering allows overcoming these limitations

Common methods:

Manual dipole modeling Automatic dipole modelingEstimate the number of sourcesEstimate the source locationsLeast Squares for source strengths

Manual dipole modeling still the main reference method for comparisons (Stenbacka et al. 2002, Liljestrom et al 2005)

Bayesian filtering in MEG - assumptions

J1 J2 … Jt …

B1 B2 … Bt …

Two stochastic processes:

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Our actual model

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Markov process

Instantaneous propagation

No feedback

Bayesian filtering in MEG – key equations

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“Observation”

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ESTIMATES

Linear-Gaussian model Kalman filterNon-linear model Particle filter

Likelihood function

Transition kernel

Particle filtering of current dipoles

The key idea: sequential Monte Carlo sampling.

(single dipole space)

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Draw random samples (“particles”) from the prior

Update the particle weights

Resample and let particles evolve

A 2D example – the data

A 2D example – the particles

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

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

Comparison with beamformers and RAP-MUSICPascarella 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-BlackwellizationCampi et al. Inverse Problems (2008); S. et al. J. Phys. Conf. Ser. (2008)

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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 dipolesA 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...)

D:1 DD:2 DDD:3

One particle = one dipole One particle = two dipoles One particle = three dipoles

D

DD

DDD Reversible Jumps (Green 1995) from one space to another one

Random Finite Sets – why

DD:2

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 densitynon-unique maximumnon-representative mean ),(),( 122212 dddd

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

)(: DP)(DPD

For some realizations,

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Random Finite Sets - how

Probability measure of RFS: a conceptual definition

Belief measure instead of probability measure

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DCCC }))(|({P)(

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

RFS-based particle filter: ResultsS. et al., Human Brain Mapping (2009)

Monte Carlo simulations:1.000 data setsRandom 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

RFS-based particle filter: ResultsS. 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