Sensitivity analysis for the EEG forward problem.
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Both functions were considered by some authors who compared their results for different dynamical systems (see Banks and Bihari, 2001; Kappel and Batzel, 2006; Banks et al., 2008).Afterward we compute the GSF for the same model.We perform some numerical experiments for both types of sensitivity functions and compare the results.
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Affiliation: Departamento de Matemática, Facultad de Ingeniería, Universidad de Buenos Aires Buenos Aires, Argentina.
ABSTRACT
Sensitivity analysis can provide useful information when one is interested in identifying the parameter θ of a system since it measures the variations of the output u when θ changes. In the literature two different sensitivity functions are frequently used: the traditional sensitivity functions (TSF) and the generalized sensitivity functions (GSF). They can help to determine the time instants where the output of a dynamical system has more information about the value of its parameters in order to carry on an estimation process. Both functions were considered by some authors who compared their results for different dynamical systems (see Banks and Bihari, 2001; Kappel and Batzel, 2006; Banks et al., 2008). In this work we apply the TSF and the GSF to analyze the sensitivity of the 3D Poisson-type equation with interfaces of the forward problem of electroencephalography. In a simple model where we consider the head as a volume consisting of nested homogeneous sets, we establish the differential equations that correspond to TSF with respect to the value of the conductivity of the different tissues and deduce the corresponding integral equations. Afterward we compute the GSF for the same model. We perform some numerical experiments for both types of sensitivity functions and compare the results. No MeSH data available. |
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Mentions: We calculate the TSF and the GSF of u with respect to σ1 for a dipole source Ji(x) = Mδ(x − q) for different locations q ∈ G1 in a spherical head model. This approximation for the head allows us to calculate the solution by the series formula appearing in Zhang (1995). Differentiating this series with respect to σ1 we obtain the sensitivity function s1 for the case of nested homogeneous spherical sets. We compare their values to find out the information that they provide about the variations on the scalp potential u when σ1 changes. The values of the sensitivities were simulated considering that the observations are measurements of the electric potential on the scalp collected by means of a set of electrodes with 10-10B configuration and that the “spike-instants” where the values of u are considered, were detected and marked on the data by experts. We consider the conductivity values in (2): σ1 = 0.33, σ2 = 0.0042, σ3 = 0.33(1/(Ω/m)) (see Geddes and Baker, 1967). We choose an order to enumerate the electrode positions and calculate the GSFinc and the TSF with respect to σ1 for different radial and tangential dipoles. The results obtained are shown in the figures below. Since no comparison between sensitivities for different parameters is presented, we have not normalized the values and have plotted the values of the absolute value of TSF directly. In Figure 1 we plot the values of GSFinc at the 20 electrode positions for the same dipole location rq = (0.3, 0.4, 0) and different dipoles moments, a radial one M = (3, 4, 0) and a tangential one M = (−4, 3, 0). |
View Article: PubMed Central - PubMed
Affiliation: Departamento de Matemática, Facultad de Ingeniería, Universidad de Buenos Aires Buenos Aires, Argentina.
No MeSH data available.