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EEG-fMRI Fusion: Adaptations of the Kalman Filter for Solving a High-Dimensional Spatio-Temporal Inverse Problem
[chapter]

Thomas Deneux

2011
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Adaptive Filtering
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Adaptive Filtering 234 dimensionality, due to the entanglement between the temporal and spatial dimensions, and high levels of non-linearity, due to the complexity of the physiological processes involved. Thus, we will present some new developments that we issued, in particular, the design of a variation of the Kalman filter and smoother which performs a bi-directional sweep, first backward and second forward. And we will show directions for the development of new algorithms. The results
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... The results presented in this chapter have already been published in (Deneux and Faugeras, 2010) . Therefore, we focus more here on explaining in detail our comprehension of the EEG-fMRI fusion problem, and of its solution through the design of new algorithms. In this introduction, we pose the problem, its specific difficulties, and advocate the use of adaptive filters to solve it. In a second part, we will tackle a simplified, linear, problem: we present our Kalman-based fusion algorithm, discuss its characteristics and prove that it is more suitable to estimate smooth activities, while the estimation of sparse activities would rather necessitate the development of new algorithms based on the minimization of a L 1 -norm. In a third part, we will address the problem of strong nonlinearities: we present a modification of the Kalman-based algorithm, and also call for the development of new, more flexible, methods based for example on particle filters. Figure 1 (A) briefly explains how the cerebral activity gives raise to the EEG/MEG and fMRI signals. EEG and MEG measure directly the electrical activity in the brain. In the case of EEG, a set of electrodes (up to 300 in the most advanced EEG helmets) are positioned on the head of the subject, in electric contact with the skin, and measure an electric potential. In the case of MEG, a set of coils are positioned around the head but without touching it, and measure the magnetic field generated by the currents circulating inside the head. These currents themselves are the consequence of the electric activity of a large number of neurons which are activated together. EEG and MEG have an excellent temporal resolution, since the propagation of currents is instantaneous at this temporal scale. They also provide some spatial information, since it is possible to model the current propagation and then solve an inverse problem to localize the activity which generated the specific pattern observed over the different sensors (Hämäläinen et al., 1993) . The spatial resolution of this localization however is poor (error range of ~1cm); even, this inverse problem is ill-posed since some sources configurations can generate no signal on the sensors. fMRI measures secondary effects of the electrical activity, called the hemodynamic response. Indeed, the increased energy consumption in an activated brain region leads to a chain of events, in particular a higher O 2 extraction from the blood, followed by an increase in the blood flow. This impacts the magnetic resonance signals recorded by the MRI scanner, because of the changes in the concentration of the deoxyhemoglobine molecule. Indeed, the magnetic properties of the hemoglobin molecule change after it delivered the oxygen molecule it was carrying, which induces higher decays of the magnetic resonance signals. All in one, a cerebral activity leads to a smooth increase in the MRI signal, also called blood-oxygen level dependent (BOLD) signal; this increase lasts for a few (3~4) seconds, and is usually followed by a small undershoot (Ogawa et al., 1993) . This BOLD signal is localized with a millimeter or sub-millimeter precision but, obviously, lacks temporal resolution. Physiological basis of EEG/MEG and fMRI www.intechopen.com EEG-fMRI Fusion: Adaptations of the Kalman Filter for Solving a High-Dimensional Spatio-Temporal Inverse Problem 235

doi:10.5772/16490
fatcat:tmybqhbtozdjdltcf6n25ynpiq