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

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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

www.dima.unige.it/~piana/mida/group.html

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)

Neural current

Biot-Savart

Ohmic term

Poisson

Biot-Savart

Accurate model of brain conductivity

Biot-Savart

Imaging approach

Parametric approach

Continuous current distribution

Focal current

Model

M small

N large

Unknown

Method

Non-linear optimization methods

Regularization methods

Result

- 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

Two stochastic processes:

J1 J2 … Jt …

B1 B2 … Bt …

Markov process

Markovian assumptions:

Instantaneous propagation

No feedback

Our actual model

The final aim:

Likelihood function

“Observation”

ESTIMATES

Transition kernel

“Evolution”

…

…

Linear-Gaussian model Kalman filter

Non-linear model Particle filter

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

S. et al., ICS 1300 (2007)

Pascarella et al., ICS 1300 (2007); S. et al. , J. Phys. Conf. Ser. 135 (2008)

Two quasi-correlated sources

Beamformers: suppression of correlated sources

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

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

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

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

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

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

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)

Source space limited to the cortical surface

Two simulated sources

Two sources recovered with orientation constraint

Only one source recovered without orientation constraint

- 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