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A Semy–Analytic Bayesian Approach
for Multiple Static Dipoles Estimation
from a Time Series of MEG Data
Sara Sommariva
Dipartimento di Matematica,
Universita` degli Studi di Genova
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1 / 16
1
MEG Inverse Problem and Source Model
2
Statistical Model
3
A Novel Approach: Semi–Analytic SMC
4
Results
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2 / 16
MEG Inverse Problem and Source Model
MEG Inverse Problem
The aim is to reconstruct the neural
currents inside the brain, from the
magnetic field recorded outside the scalp.
From the mathematical point of view:
inversion of the Biot–Savart operator.
MEG device
MEG sensors helmet
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MEG Inverse Problem and Source Model
Data: the magnetic field recorded by the Ns sensors at Nt time–points
y = (y1 , · · · , yNt ) ∈ RNs
Nt
Butterfly plot:
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MEG Inverse Problem and Source Model
Static Multi–Dipolar Model
with Time–Varying Moments
Single dipole:
q δ (r − r (c ))
where
q dipole moment
r (c ) dipole position.
c
Points of a brain–grid {r (c )}N
c =1
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MEG Inverse Problem and Source Model
Static Multi–Dipolar Model
with Time–Varying Moments
Single dipole:
q δ (r − r (c ))
where
q dipole moment
r (c ) dipole position.
c
Points of a brain–grid {r (c )}N
c =1
At each time–point t the neural currents are approximated with
d
X
(k )
qt δ r − r (c (k ) )
k =1
where d ≤ Dmax is the number of active dipoles.
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MEG Inverse Problem and Source Model
Data: the magnetic field recorded by the Ns sensors at Nt time–points
y = (y1 , · · · , yNt ) ∈ RNs
Nt
Unknown: number of dipoles and their parameters
x = (d , c, q)
= d , c (1) , . . . , c (d ) , q1(1) , . . . q1(d ) , . . . , qN(1t ) , . . . , qN(dt )
From Biot-Savart equation (conditionally linear model):
y = G (d , c)q + e
where
G (d , c) is the leadfield matrix
e is the noise affecting the measurements.
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Statistical Model
Bayesian Approach to Inverse Problems
Data, unknown and noise are modeled with random variables.
The aim becomes to approximate the posterior pdf π(x|y).
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Statistical Model
Bayesian Approach to Inverse Problems
Data, unknown and noise are modeled with random variables.
The aim becomes to approximate the posterior pdf π(x|y).
From Bayes theorem:
π(x|y) =
π(x)π(y|x)
π(y)
where
π(x) is the prior pdf:
embodies all information about the unknowns available before the data
are recorded;
π(y|x) is the likelihood function:
embodies the forward problem and the noise model.
π(y) is a normalizing constant.
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Statistical Model
We assume:
Prior: π(x) = π(d )π(c|d )π(q|d , c)
1
number of dipoles:
π(d ) Poisson with small, fixed parameter;
2
dipole positions:
π(c|d ) = π(c (1) , . . . , c (d ) |d ) Uniform on the brain–grid;
3
dipole moments (time–varying):
(1)
(d )
π(q|d , c) = π(q1 , . . . , qNt |d , c) = N (q; 0, σq 2 I).
Likelihood: π(y|x) = πe (y − G (d , c)q) = N (y − G (d , c)q; 0, σe 2 I)
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A Novel Approach: Semi–Analytic SMC
Novel Approach: Analytic Results
Under the previous assumption we obtain an analytic expression for
The marginal likelihood:
π(y|d , c) = N (y; 0, Γ(d , c))
The conditional posterior over the dipole moments:
π(q|y, d , c) =
N (q; σ2q G (d , c)T Γ(d , c)−1 y, σ2q I − σ2q G (d , c)T Γ(d , c)−1 G (d , c) G (d , c))
where Γ(d , c) = σ2q G (d , c)G (d , c)T + σ2e I
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A Novel Approach: Semi–Analytic SMC
Novel Approach: saSMC Algorithm
We propose the following two–step algorithm
[Sommariva and Sorrentino, submitted].
π(x|y) = π(d , c|y)π(q|y, d , c)
First step: approximate π(d , c|y) through Sequential Monte Carlo (SMC)
samplers [Del Moral et al., 2006]:
π(d , c|y) '
I
X
W i δ(d − D i )δ(c − Ci ).
i =1
Second step: for all the particles {(D i , Ci )}Ii=1 obtained at the previous step
we analytically compute π(q|y, D i , Ci ).
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A Novel Approach: Semi–Analytic SMC
First step: approximate π(d , c|y) through Sequential Monte Carlo (SMC)
samplers [Del Moral et al., 2006].
Sample sequentially from
πn (d , c|y) ∝ π(d , c)π(y|d , c)αn , 0 = α1 < · · · < αN = 1
That is
Initialization: sample the first set of I particles {(D1i , Ci1 )}Ii=1 from π(d , c).
Assign uniform weights W1i = 1I .
Evolution: for each particle, sample (Dni +1 , Cin+1 ) from Kn+1 ((Dni , Cin ), · ),
Kn+1 (·, ·) being a Markov kernel πn+1 –invariant (Reversible Jump).
Assign weight Wni +1 = Wni π(y |Dni , Cin )αn+1 −αn .
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A Novel Approach: Semi–Analytic SMC
Connection with Previous Works
Our work improves the one in [Sorrentino et al., 2014] 1 in which the SMC procedure
is used to approximate the whole posterior π(x|y) (full SMC).
Main advantages:
we can give in input time–window without increasing the computational cost;
the number of particles I being equal, reduction of the Monte Carlo variance.
1
Talk by Alberto Sorrentino on Wednesday 9th July at h. 11.00 MS 8.
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Results
Simulated Experiment
Forward Problem:
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Results
Simulated Experiment
Forward Problem:
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Results
Simulated Experiment
Forward Problem:
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Results
Simulated Experiment
Forward Problem:
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Results
Simulated Experiment
Forward Problem:
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Results
Simulated Experiment
Forward Problem:
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Source Reconstruction:
13 / 16
Results
Simulated Experiment
Forward Problem:
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Source Reconstruction:
13 / 16
Results
Real Data: SEF Data
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Results
Real Data: SEF Data
SA SMC:
full SMC:
dSPM:
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Results
Real Data: SEF Data
SA SMC:
full SMC:
dSPM:
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Results
Future Work and Acknowledgements
More accurate/automatic choice of the parameters, especially σq .
Application to other real data (good results with EEG auditory data).
Application to other inverse problems with conditionally linear models.
Joint work with Alberto Sorrentino,
Dipartimento di Matematica, Universita` di Genova.
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15 / 16
Results
Reference
Del Moral, P., Doucet, A., and Jasra, A. (2006).
Sequential Monte Carlo samplers.
Journal of the Royal Statistical Society B, 68:411–436.
Sommariva, S. and Sorrentino, A. (submitted).
Sequential monte carlo samplers for semi–linear inverse problems and
application to magnetoencephalography.
Inverse Problems.
Sorrentino, A., Luria, G., and Aramini, R. (2014).
Bayesian multi-dipole modeling of a single topography in meg by adaptive
sequential monte-carlo samplers.
Inverse Problems, 30:045010.
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16 / 16
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