Neural source localization using particle filter with optimal proportional set resampling

ETRI JournalVolume 42, Issue 6 p. 932-942 ORIGINAL ARTICLEFree Access Neural source localization using particle filter with optimal proportional set resampling Santhosh Kumar Veeramalla, Corresponding Author Santhosh Kumar Veeramalla vsanthosh46@gmail.com Department of Electronics and Communication Engineering, National Institute of Technology, Warangal, Telangana, India Correspondence Santhosh Kumar Veeramalla, Department of Electronics and Communication Engineering, National Institute of Technology, Warangal, Telangana, India. Email vsanthosh46@gmail.comSearch for more papers by this authorV.K. Hanumantha Rao Talari, V.K. Hanumantha Rao Talari Department of Electronics and Communication Engineering, National Institute of Technology, Warangal, Telangana, IndiaSearch for more papers by this author Santhosh Kumar Veeramalla, Corresponding Author Santhosh Kumar Veeramalla vsanthosh46@gmail.com Department of Electronics and Communication Engineering, National Institute of Technology, Warangal, Telangana, India Correspondence Santhosh Kumar Veeramalla, Department of Electronics and Communication Engineering, National Institute of Technology, Warangal, Telangana, India. Email vsanthosh46@gmail.comSearch for more papers by this authorV.K. Hanumantha Rao Talari, V.K. Hanumantha Rao Talari Department of Electronics and Communication Engineering, National Institute of Technology, Warangal, Telangana, IndiaSearch for more papers by this author First published: 25 February 2020 https://doi.org/10.4218/etrij.2019-0020Citations: 2AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract To recover the neural activity from Magnetoencephalography (MEG) and Electroencephalography (EEG) measurements, we need to solve the inverse problem by utilizing the relation between dipole sources and the data generated by dipolar sources. In this study, we propose a new approach based on the implementation of a particle filter (PF) that uses minimum sampling variance resampling methodology to track the neural dipole sources of cerebral activity. We use this approach for the EEG data and demonstrate that it can naturally estimate the sources more precisely than the traditional systematic resampling scheme in PFs. 1 INTRODUCTION Magnetoencephalography (MEG) and Electroencephalography (EEG) provide estimations with exceptional time resolution, which enables us to monitor the temporal evolution of neural actions like no other cerebrum imaging procedure can do. They are also non-invasive, which permits us to take measurements for an extended period with no risk to the subject. Although these are considered as the most effective tools for accessing the brain, analyzing EEG or MEG data is a complicated process. Reproducing the hidden neural activations from MEG or EEG data permits localizing the dynamic cerebral areas with sensible spatial precision and empowers various applications, both in clinical and fundamental research [1]. Reconstruction of the EEG source is a challenging job and several techniques for the ill-posed inverse issue have been implemented. This can be categorized into two classes for the description of neural sources. The first modeling method includes the use of imaging models to explain the data. It uses a dense arrangement of current dipoles circulating in settled fields, such as Minimum Norm Estimation [2], and various types of beamformers [3]. A second method is a parametric approach that takes advantage of the fact that by using one equivalent current dipole, the dense sets of the current dipole could be supplanted [4]. While imaging-based methods may generate a thorough map of the neuronal activity of the cerebrum, the parametric methodology offers direct mapping of EEG measurements to a few parameters [5]. The present parametric method enables more intuitive interpretation of the equivalent models, which explains the electrical activity of the cerebrum and can be supported in emerging technologies, such as BCI systems. Although the parametric approach has the benefit of several automatic algorithms, manually assisted dipole fitting is the most commonly employed method because it is simple. The estimation of the dipole equivalent fields in cerebral 3-D volumes using EEG measurements collected from the scalp is a significant task for the parametric approach. Localization of the EEG source is based on two hypotheses: (a) the number of dipoles is established and (b) the sources are fixed at the time [6, 7]. In this article, we present an approach based on Bayesian filtering for the estimation of dynamical source models from MEG or EEG information. With this approach, we do not consider any assumptions, and the number of sources can vary with time. We apply multi-target Bayesian filtering and the hypothesis of Random Finite Sets in a calculation that recovers the location and amplitudes of dipolar sources. The recreated dipoles are clustered in time and space to connect them with sources. We used this approach on synthetic data and demonstrate that it could naturally evaluate the source structure because this source reconstruction is an ill-posed inverse problem, and because the reconstructed sources are non-unique. We also used real EEG data to determine the unknown number of sources with their locations and amplitudes. Neural currents can be shown as a point source or a dipolar model. A current dipole represents the activity of a minor neural population; the whole brain activity can be viewed as the superposition of several such point currents [8]. The utilization of dipole models has been constrained up to this point by the difficulty of evaluating the number of dynamic sources from the information. These days, in any case, the expanded accessibility of computational resources has permitted the utilization of Monte Carlo strategies for approximating posterior distribution for an unidentified number of current dipoles. The fundamental idea of the Monte Carlo strategies is that of approximating an integral utilizing an arrangement of weighted particles drawn from some probability distribution. The Bayesian methodology is used to deal with inverse issues, and the goal is to compute and advantageously outline the posterior distribution [9], that is, the conditional distribution of the unknown number of sources, given the deliberate information. For the EEG inverse problem situation, specifically, Monte Carlo techniques have been utilized. The PF or sequential Monte Carlo methods are Monte Carlo algorithms that are used to estimate the states of dynamic systems. In PFs, they are utilized to yield an accurate filtering distribution, that is, the posterior distribution at a particular time, based upon the estimations recorded up to that particular time [10]. In the latest literature, a PF was used to locate the source of EEG measurements [6, 8, 11, 12]. The PF uses Sequential Importance Sampling to approximate the posterior distribution of the state at each time step. The use of particles is to sample the state-space of the framework. Particles are then weighted using the measurement model to give a measure of the state posterior density [13, 14]. In any PF algorithm, there are three steps namely particle generation, weight calculation, and resampling. The first stage is the generation of particles and in the second stage, the weights are assigned. In the last stage, resampling replaces both particles and weights with another set [15]. Resampling is a critical procedure, which is of both theoretical and practical significance, for efficient implementation of the PF. Without this step in the PF, particles with large weights could lead to undue influence over the particles with lower weighting particles [16]. This implies that the achieved evaluations may be incorrect and could have unsatisfactorily huge variances. With resampling, such decays are deflected, thus demonstrating the necessity of using a PF. Recently, several modifications have been proposed for PFs to address this problem. Rao-Blackwellised PFs [17], auxiliary PFs [18], adaptive PFs, unscented PFs [19], regularized PFs [20], and Gaussian PFs [21] are some of the modifications. However, these changes can be used for offline predictions only. Shorter execution time is preferred to prediction accuracy for online prediction. Several resampling algorithms have been added to the PF scheme to solve particle degeneration. The focus of the work reported in this paper is resampling, and its objective is to replicate the particles with higher weights. The particles with lower weights are replaced by those with higher weights so that the final particle cloud represents the posterior more accurately. This replacement would reweight the replaced particles so that the variance among the weights is reduced [22]. Traditional and traditional variation resampling are two types of resampling. In comparison with traditional resampling, traditional variation resampling requires additional computational resources for resampling. Because multiple computing frameworks are involved, a different resampling procedure with lower computational time for resampling in PFs is required. Some of the resampling algorithms classified as traditional resampling, include multinomial, systematic, stratified, and residual. A traditional variation resampling algorithm includes residual systematic resampling. The traditional systemic resampling technique improves the precision of neural sources and requires less time to compute. Although traditional algorithms are the most widely used and common methods, more advanced algorithms are still required for certain applications. To the best of the author's knowledge, in the literature, no systematic empirical research exists that addresses the use of a PF with minimum sampling variance resampling for localizing equivalent current dipoles in EEG. This article focuses on applying the minimum sampling variance resampling technique [23] to locate the neural sources. Minimum sampling variance (MSV) is also called optimal proportional set resampling, and with this method, the sampling variance will be minimal. In this case, the PF is applied both to synthetic and real EEG measurements and is used to compare traditional dipole modeling and a Minimum Norm Estimate. Therefore, the efficiency of the proposed algorithm can be immediately compared with that of the other two techniques. The remaining parts of the paper are as follows. In Section 2, we briefly describe the mathematical version of the EEG inverse problem. Section 3 introduces the Bayesian filtering method with PF. In Section 4, the minimum sampling variance resampling algorithm of the PF is discussed, and in Section 5, application of a PF with a specific resampling scheme using synthetic data and real EEG records is described. Conclusions are presented in Section 6. 2 EEG SOURCE LOCALIZATION MODEL To develop a solution for the source localization model in EEG, the head is considered as nested concentric spheres (scalp, skull, and brain) with the corresponding conductivities of 0.3300 s/m, 0.0042 s/m, and 0.3300 s/m [11]. Let us consider that cerebral development rises at a minor part in the cortex positioned at and the estimated position x is some part of the long distance from this area. The current density inside a head volume produces a magnetic field that can be described using the Biot-Savart equation [5, 24]: (1) Here, the magnetic permittivity of vacuum is denoted by , and is steady inside the volume as well. In (1), the current density "j" consists of two parts: the "primary" current is the source of neural activity, and the "volume" current is a result of the interaction of the primary current within a conductive medium . The , can be modeled as a point source given by an equivalent current dipole as with moment . Because EEG measures differential electric potentials on the scalp surface, it relies upon the influences of volume conduction to deliver the signs we measure. Thus, EEG is extremely sensitive to differences in conductivity, not only in the tissue close to the cerebral current sources, but also in those present in the skull and scalp. Some tissues are extremely conductive (cerebrum, cerebrospinal fluid (CSF), skin), and others are less conductive (skull). For the calculation of the electric potential conveyance on the surface of the head (EEG), it is important to characterize the conductivities ( ) of the spherically symmetric layers. A realistic head model is essential for coordinating their properties effectively. The boundary element model (BEM) solution requires a head model that portrays the geometry of the head and the conductivities of the diverse tissues present. The BEM relies only upon the head geometry and conductivities. The volume current can be approximated using forward calculation [25-27]. The present dipole is an augmentation of the model of the combined charges dipole in electrostatics. Note that cerebral action does not comprise discrete arrangements of physical current dipoles, but instead the dipole is a helpful portrayal for the intelligible interaction of many cells. The present dipole model is the core of EEG behavior because an essential current source of an arbitrary amount can only separate into small regions, where every area is summarized by a proportionate current dipole [12]. Computation of the scalp potential requires a particular source to demonstrate that must be understood numerically. In case that the model relies upon realistic head shapes, the computation might be a positive test. Such models are mechanically utilized as a part of most extreme clinical and study bundles aimed at EEG source localization. Consider that N dipoles exhibit electrical activity in the cerebrum. To evaluate such activity, the multi-channel EEG data can be used with from number of sensors at a given time t, so that the forward EEG model is given by (2) Here is a three dimensional localization vector, the leadfield matrix for dipole i is denoted by , and the three-dimensional (3-D) moment vector (the source signal) is denoted as . The EEG model has observation noise that is represented by . The combination of head geometry, electrode positions, and dipole localization is achieved using a non-linear function . Sensors will measure the electrical activity according to a dipole source. These dipole sources contain unity moments along x, y, and z directions, and also possess zero moment in other directions. To compute the forward model " ," the prior knowledge of head geometry and electrode position is required. The representation of N number of dipoles is , where each single geometric position in 3-D is given as . Because there are N dipoles, the leadfield matrix can be written as . This leadfield matrix depends on the location of the dipole at time t. The vector of moments is , where each single moment or mind source signal in 3-D is given as . Now, (2) can be rewritten as (3) Then, convert the above (3) into matrix form. (4) Here (4) is considered a measurement model. For the state-space model, because the location is unknown and we need to learn how states can evolve, this can be considered as a random walk model in a source localization problem, (5) The above (4) and (5) can be considered as measurement (or observation) and state equation, respectively. From this, the EEG can be modeled as a state-space model. To evaluate the dynamic parameters (ie, location in x, y, and z directions) in the cerebrum, we use PF by considering the measured signal at time t [28]. 3 PARTICLE FILTER The PF is a sequential Monte Carlo technique, which is extremely suitable for non-linear and non-Gaussian models. By considering the measurements, the PF is used to assess the dynamic state variables by approximating the posterior likelihood density function of the unidentified state parameters at each time point [29]. For such a dynamic system, the state-space model can be described in terms of and as follows: (6)and (7)where consists of unidentified parameters at time t, contains the observations at time t, and are considered to be nonlinear functions for which is related to state-transition, and is related to the state vector along with the observation vector. The term is the state model error and is the observation noise. The PF can be used to estimate the joint posterior likelihood density function of at time t using N number of random particles along with weights as (8)where is a Dirac delta function. Using (8), the estimated state vector can be written as (9) To assign the weights for particles, the importance density plays a vital role. Based on the importance density, the sequential importance resampling (SIR) PF can be used to estimate the states. This PF has three phases. 3.1 Particle generation The first phase is related to the generation of particles denoted by . Particles are taken from , which is an importance density function, where . 3.2 Weight calculation The particle weights can be computed as (10) The sum of all the weights should equal 1, that is, , provided . Now, (10) can become . 3.3 Resampling Resampling is used to avoid the rapid degeneracy of particles. Specifically, most of the particles with higher weights will dominate those with smaller weights. The dominance of this particle degeneracy leads to a poor posterior likelihood density function. 3.3.1 Systematic resampling Systematic resampling is known as universal sampling. In this popular technique we draw only one random number, that is, one direction in the "wheel analogy," for one particle and other particles with directions being fixed at increments from the randomly picked direction [30]. Initially, is drawn from the regular distribution , and the remaining samples of the " " information are acquired conclusively, that is, (11) Because of the strong relationship between the random numbers, it is noted that the systematic resampling is computationally more efficient than other resampling methods. 4 PROPOSED NEW APPROACH FOR EEG SOURCE LOCALIZATION USING A PF WITH MINIMUM SAMPLING VARIANCE The particles with lower weights are displaced by those with higher weights. This substitution would reweight the deposed particles with the objective that change among the weights would be diminished [31, 32]. The resampling should be applied only when it is necessary and it is also important when resampling is done. Based on this, we divide the resampling methods into two types: deterministic and adaptive. In the deterministic type, the resampling has to be done at all time steps. This includes several resampling schemes such as multinomial resampling, stratified resampling, systematic resampling, and residual resampling [15]. In adaptive resampling, whenever the weight of a particle is below the threshold criteria, then resampling is triggered. This gives better performance by the PF than deterministic resampling gives. The model for actualizing resampling is generally used because of the variance of the weights, which replicates the level of weight degeneracy. The threshold criterion for the resampling is the effective sample size of ESS. It is given as (12)where is the normalized weight. This adaptive resampling scheme includes the minimum sampling variance method [23, 33]. In the literature, for source localization with a PF, most of the authors used the systematic [1, 11, 12, 34] resampling approach. Some of the authors mentioned metropolis resampling, but it requires only ratios between the weights, so there was no need to worry about weight normalization [35]. Hypothetically, better resampling relates to the minimal variance of the number of copies from a particle. Systematic resampling is done with the lowest resampling variance [36, 37]. Because of low sampling variance in systematic resampling, for this work, we chose systematic resampling to compare with minimum sampling variance (MSV) resampling for the localization of brain sources. We now consider concerns about streamlining the resampling calculation under certain sensible limitations. In the first place, to limit the variance of the weights (which to the greatest degree diminishes the degeneracy), the weights of the resampled particles might be set equivalent, as is usually done (ie, ). We call this the ideal weight condition, which completely expels the degeneracy. The new approach for the resampling technique is called minimum sampling variance (MSV) [33] resampling, which comprises two principle steps: Step 1: Each particle is first resampled times, leaving a weight residual . This step will yield, in total, L particles, where . Step 2: The particle with relatively larger weight residual, , will be further sampled one more time each. The MSV resampling ensures accomplishing the lowest sampling variance for a discretionary sample set while fulfilling the ideal weight condition and including precisely the predefined number of particles. The ideal identical distribution quality got by MSV resampling demonstrates a capacity to maximally protect the posterior distribution or, to state a capacity to maximally diminish data loss during the resampling, which is profoundly the best approach, in principle. 5 SIMULATION RESULTS The dynamical dipole model is considered for use with source (neural) triggering, which will be at every time step and in any active area described by the equivalent current dipole. All the parameters related to neural sources are assumed to be dynamical parameters. The estimation of several neural sources is done using data, and the estimation is considered a dynamical variable. For available source space, neural sources can jump among adjacent points; in this way, they are permitted to move in time. The process of estimation is as follows: Calculate the leadfield matrix and source space. The number of dipole sources are unknown and are time-variant. For this situation, we model these multiple dipole sources as a Random Finite Set. Start analyzing the data at time t = 1. Based on the probabilistic criteria initialize the particle. Assign the weights to each particle. Normalize the weights. Most of the particles with high weights will dominate the low-weight particles. The dominance of high weight particles leads to a poor posterior likelihood density function. Use resampling methods to discard the low-weight particles. Prepare dipole configuration for the next time step. Calculate the position and moment of the dipoles. These parameters do not recognize the true sources in the brain. Now, apply the clustering to the dipoles based on the position. When dipoles have been consigned to various clusters according to various neural sources, calculate the average position of the dipoles in each cluster. This average position can be considered the position of a neural source. The performance of the proposed approach was assessed by simulation experiments performed using both synthetic data and real EEG data. The resampling methods used for the simulation experiments include systematic and minimum sampling variance. 5.1 Synthetic Data Results To validate the potential of the proposed resampling schemes with PF applied for source localization, we took the synthetic data from [38] with dipoles located at Red (x = –1.37 cm, y = –5.43 cm, z = 7.34 cm) Blue (x = 3.74 cm, y = 4.54 cm, z = 5.66 cm), Green (x = –2.04 cm, y = 3.73 cm, z = 9.56 cm), and Magenta (x = 2.96 cm, y = 2.11 cm, z = 9.42 cm) with a brain volume of 10 cm radius. The experiment consisted of 20 000 particles for every resampling method. For the synthetic data, the true location of the sources and waveform of the sources are shown in Figure 1A and 1B. Figure 1Open in figure viewerPowerPoint (A) True Source locations of the synthetic data and (B) Source waveforms of the synthetic data For source localization, first we will compute the leadfield matrix (this can be done with the help of fieldtrip toolbox [39]). By applying the leadfield matrix and EEG data on the PF it is possible to estimate the positions of the dipoles and their amplitudes in relation to the time. For this work, we used two resampling methods, the source positions estimated using systematic resampling and MSV resampling are shown in Figure 2A and 2B, respectively. The estimated source amplitude waveforms using systematic and MSV resampling are shown in Figure 3A and 3B, respectively, for the given time series. Figure 2Open in figure viewerPowerPoint Source locations estimated using (A) Systematic resampling and (B) MSV resampling Figure 3Open in figure viewerPowerPoint Source amplitude waveforms using (A) Systematic resampling and (B) MSV resampling In Figure 4 we gave the sampling variances of the particles before and after resampling for the MSV and the systematic resampling. This clearly shows that the variance in systematic resampling is very high. Figure 4Open in figure viewerPowerPoint Sampling variance of systematic and MSV resampling methods To validate the proposed approach, we calculate the RMSE (root mean square error) between the true positions of dipoles and the reconstructed dipole locations. (13) In (13) represents the 3-D location of the estimated source and represents the true source locations of the synthetic data. Comparison of the RMSE between the true source location and the estimated source location of the synthetic data is given in Table 1. From the comparison table, we can observe that the location RMSE of the dipoles and the computational time are less for the MSV resampling method. Table 1. Comparison of RMSE of the location and PC processing time Approach Systematic resampling MSV resampling Dipole 1 location RMSE (mm) 9.18 7.35 Dipole 2 location RMSE (mm) 8.09 6.28 Dipole 3 location RMSE (mm) 10.36 8.42 Dipole 4 location RMSE (mm) 10.03 8.32 \midruleAverage RMSE (mm) 9.415 7.59 PC processing time (s) 68.95 56.55 The simulations with synthetic data show the proficiency of the proposed strategy for confining and remaking exceptional cerebral sources from synthetic EEG information. In Table 2, we compared the performance of our proposed approach to the methods in [25]and [1]. It can be observed that the RMSE of the RBPF [25] is less than with the proposed approach, but it requires more particles, which also means more computational complexity. Moreover, we also need to know the number of dipoles priori. In [1], the number of dipoles to be estimated is higher but it took more particles and it also resulted in higher RMSE compared to the proposed approach. The system proposed obtains comparable estimated results with substantially reduced computational complexity when tracking an unknown number of dipoles. Table 2. Comparison of RMSE of the neural source location for the synthetic data Method No. of particles No. of dipoles Location RMSE (mm) RBPF [25] 50 000 2 (Known) 6.30 PF [1] 100 000 6 (Unknown) 9.10 PF (MSV resampling) 20 000 4 (Unknown) 7.59 5.2 Real EEG data In this section, we show the estimation precision of the proposed algorithm using real EEG data. The data were taken from the Brainstorm EEG/Epilepsy dataset [40]. This dataset was collected from a patient who has suffered from focal epilepsy since the age of 8 years. He is not facing any dangerous risks from his epilepsy. Focal epilepsy is defined by seizures originating from a particular portion of the brain. From the surgical perspective, the epileptogenic area is better defined by individual MRI, in addition to reducing additional postoperative risks. The value of source localization of EEG data in epilepsy patients has already been proven. This method can locate brain resources from extratemporal to deep temporal locations, as demonstrated in earlier research [41-45]. Knowing the exact location of focal epilepsy revolutionizes epileptic treatments both internally and surgically. The proposed methodology can be viewed as a supplement to this diagnostic procedure. The localization of EEG sources indicates focal epilepsy in the brain around the temporal lobe. The data were recorded at a frequency of 256 Hz using 29 channels. EEG signals were recorded from 29 terminals (FP1, FP2, F3, F4, C3, C4, P3, P4, O1, O2, F7, F8, T7, T8, P7, P8, Fz, Cz, Pz, T1, T2, FC1, FC2, FC5, FC6, CP1, CP2, CP5, CP6) as per the 10/20 International framework. The channel locations were displayed in Figure 5. The source of the reference coordinate system is situated at the focal point of the spheres with the x-axis indicating right-to-left, the y-axis pointing back to front, and the z-axis from the base upward [46]. The arrangement of the EEG source confinement problem requires a critical number of forward model assessments. The proposed calculation requires for its assessment a large number and variety of source areas. Furthermore, we set roughly grid points on the head model. The total dataset contained 921 600 samples, and we tested our algorithm by using samples 136 700 to 200 000. Figure 5Open in fi