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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 [email protected] 1 / 16 1 MEG Inverse Problem and Source Model 2 Statistical Model 3 A Novel Approach: Semi–Analytic SMC 4 Results [email protected] 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 [email protected] 3 / 16 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: [email protected] 4 / 16 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 [email protected] 5 / 16 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. [email protected] 5 / 16 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. [email protected] 6 / 16 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). [email protected] 7 / 16 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. [email protected] 7 / 16 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) [email protected] 8 / 16 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 [email protected] 9 / 16 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 ). [email protected] 10 / 16 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 . [email protected] 11 / 16 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. [email protected] 12 / 16 Results Simulated Experiment Forward Problem: [email protected] 13 / 16 Results Simulated Experiment Forward Problem: [email protected] 13 / 16 Results Simulated Experiment Forward Problem: [email protected] 13 / 16 Results Simulated Experiment Forward Problem: [email protected] 13 / 16 Results Simulated Experiment Forward Problem: [email protected] 13 / 16 Results Simulated Experiment Forward Problem: [email protected] Source Reconstruction: 13 / 16 Results Simulated Experiment Forward Problem: [email protected] Source Reconstruction: 13 / 16 Results Real Data: SEF Data [email protected] 14 / 16 Results Real Data: SEF Data SA SMC: full SMC: dSPM: [email protected] 14 / 16 Results Real Data: SEF Data SA SMC: full SMC: dSPM: [email protected] 14 / 16 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. [email protected] 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. [email protected] 16 / 16