EEG Dynamic Source Localization using Marginalized Particle Filtering Bradley Ebinger * , Nidhal Bouaynaya * , Petia Georgieva ‡ and Lyudmila Mihaylova § * Department of Electrical and Computer Engineering Rowan University, USA Email: ebinge52@students.rowan.edu, bouaynaya@rowan.edu † Department of Electronics, Telecommunications, and Informatics University of Aveiro, Portugal Email: petia@ua.pt ‡ Department of Automatic Control and Systems Engineering University of Sheffield, United Kingdom Email: l.s.mihaylova@sheffield.ac.uk Abstract—Localization of the brain neural generators that create Electroencephalographs (EEGs) has been an important problem in clinical, research and technological applications re- lated to the brain. The active regions in the brain are modeled as equivalent current dipoles, and the positions and moments of these dipoles or brain sources are estimated. So far, the brain dipoles are assumed to be fixed or time-invariant. However, recent neurological studies are showing that brain sources are not static but vary (in terms of location and moment) depending on various internal and external stimuli. This paper presents a shift in the current paradigm of brain source localization by considering dynamic sources in the brain. We formulate the brain source estimation problem from EEG measurements as a (nonlinear) state-space model. We use the Particle Filter (PF), essentially a sequential Monte Carlo method, to track the trajectory of the moving dipoles in the brain. We further address the “curse of dimensionality,” issue of the PF by taking advantage of the structure of the EEG state-space model, and marginalizing out the linearly evolving states. A Kalman Filter is used to optimally estimate the linear elements, whereas the PF is used to track only the non-linear components. This technique reduces the dimension of the problem; thus exponentially reducing the computational cost. Our simulation results show that, where the PF fails, the Marginalized PF is able to successfully track two dipoles in the brain with only 500 particles. Keywords—Bayesian estimation, EEG inverse problem, Spatial- temporal brain source localization, Particle filtering, Kalman Fil- tering. I. I NTRODUCTION We formulate the brain source localization problem as a (nonlinear) state-space model, where the positions and mo- ments of the neural generators constitute the unknown or hidden state and the EEG measurements are the observations of the system. In a Bayesian context, inference of the hidden state given a realization of the observations relies upon the posterior density function (pdf) [1]. For systems with linear dynamics and Gaussian noise, the posterior distribution is Gaussian whose mean and covariance can be computed using the Kalman filter. For systems with non-linear dynamics, a Monte Carlo method, called the Particle Filter (PF) has emerged, which uses the concept of Sequential Importance Sampling (SIS) to estimate the posterior pdf using a finite number of weighted samples. In particular, the PF does not make any assumptions about the pdfs or the linearity of the system model. The power of the PF, however, comes at a computa- tional cost. In particular, the number of particles needed for the estimation increases super-exponentially with the dimension of the state [2]. This problem is commonly known as the “curse of dimensionality”, and makes it unreasonable to use the Particle filter for tracking problems in high dimensional spaces. In the context of EEG source localization, the dimension of the state space is six times the number of dipoles, causing the tracking of even two dipoles (12-dimensional problem) to be inaccurate unless a very large number of particles are used. To deal with the high-dimensionality issue, we propose to marginalize out the states in the system that are linear with respect to the measurements [3]. This allows the linear states in the state- space model to be estimated optimally using the Kalman Filter, whereas the non-linear states are estimated using the PF. By decreasing the dimensionality of the state, less particles can be used, allowing a decrease in computation time. Simulation results show that even a two dipole model cannot be localized using the traditional PF, but can be tracked accurately using the marginalized PF. II. EEG SOURCE LOCALIZATION MODEL Given M equivalent active dipoles in the brain, the mea- sured multichannel EEG signal z k from n z sensors at time k can be modeled as follows: z k = M ∑ m=1 L m (d k (m))s k (m)+ e k , (1) where M is the total number of dipoles, d k (m) is a 3×1 spatial position vector in the brain of dipole m at discrete time k. Each dipole m is defined as d k (m)=[x k (m),y k (m),z k (m)] t . L m (d k (m)) is the n z × 3-dimensional lead-field matrix for the m th dipole. s k (m) is a 3 × 1-dimensional moment of the m th dipole at time k. e k is a zero-mean white Gaussian noise with covariance R k . Most notably, the components of the leadfield matrix L m are non-linear functions of the dipole locations, electrodes’ positions and head position [4]. The EEG measurement equation described in (1) can be written