Source localization with focus on low resolution electromagnetic

“eLORETA is a genuine inverse solution and not merely a linear imaging method. We show that it has exact, zero error localization in the presence of measurement and structured biological noise.” Roberto D. Pascual-Marqui^2.2

As it is known, both conscious and unconscious mental activity induce continuous electromagnetic signals as a result of underlying neuronal dynamics and metabolic processes in the brain, either during task engagement, resting state or sleep (Hobson and McCarley, 2003, Horovitz et al., 2009). Thus, uncovering brain areas implicated in the production of different mental states requires the introduction of techniques able to locate the most likely brain sources that are consistently active during the experimental task (Vanoosterom, 1991, Jatoi et al., 2014). The systematic search (i.e., by following a mathematical algorithm) for the spatial localization of putative active sources inside the brain that are responsible (or at least play a moderating role) for the generation of EEG patterns recorded at the scalp level, is called brain source localization (Anderer et al., 2001, Silfverhuth et al., 2012). Source estimation based on EEG is theoretically grounded on the inverse problem of electromagnetism, which consists on calculating the distribution of current sources (the unknown variables) that produce a certain electromagnetic field, out of a finite sample of measurements (the known variables), and which can be proven to possess no unique solution (indeed, an infinite number, even if the number of sensors were infinite) (Pascual-Marqui, 2002).

The motivation to find source localization methods adapted to EEG data stems from the limitations of commonly used experimental techniques for functional imaging and localization of brain activity, such as fMRI or PET imaging (Pascual-Marqui et al., 1994, Nichols and Holmes, 2002). Both fMRI and PET constitute three-dimensional brain tomographies exhibiting excellent spatial resolution, however, their temporal resolution is quite limited (in the range of seconds in the case of fMRI) (Farthouat and Peigneux, 2015, Houck et al., 2017).

Therefore, these tomographies are not able to track the typical speed at

which most neuronal processes occur (Astolfi et al., 2007). For instance, in the case of fMRI, it can be shown that the temporal resolution of the BOLD signal is equivalent to a low-pass filter of ongoing neuronal activity, that is, only captures the lowest frequencies, while ignoring faster components in the frequency spectrum (Pascual-Marqui et al., 2002). On the other hand, EEG has a different (although somehow related) biophysical grounding, making it more adept for creating a high-temporal resolution tomography (Pascual-Marqui et al., 2011).

Briefly, small local clusters of synchronized cortical cells (whose typical dimensions are comprised between 40 to 200 mm²) superpose their electrical fields, and the spatial summation resulting from the overall activity (post synaptic potentials) at a given moment constitutes the signal registered by the EEG sensors (Cartwright, 2001). Hence, electrical signals recorded by EEG correspond to activity originating from cortical pyramidal cells (importantly, these neurons are oriented perpendicularly to the cortical surface), and can achieve a temporal resolution of milliseconds, thus, offering a great scientific potential for creating a suitable brain tomography (Jones, 2016). Based on the fact that pyramidal neurons producing the recorded potential fields are orthogonally oriented to the surface, modelled dipoles used for source reconstruction are constrained to follow the same orientation, thus, resulting in a modest simplification to the inverse solution problem (nonetheless, the strength of the dipoles remains unaffected by this geometrical constraint) (Pascual-Marqui et al., 2011). However, the achieved accuracy obtained through source localization reconstruction based on EEG is compromised by a number of limiting factors (Vanoosterom, 1991). The most important to mention are: volume conduction effects (making more ambiguous the position of the generators due to electrical propagation, possibly picked by several nearby sensors), EEG noise (either of instrumental or biological origin, as both contribute) and head-modelling errors (arising when indicating the spatial coordinates of the sensors, necessary for the calculation of the source generators) (Grech et al., 2008).

In general, source localization techniques can be divided into two main categories: parametric and non-parametric methods (Faes et al., 2012).

Parametric methods, aim to estimate the dipole parameters (including position) of a given number of dipoles (e.g., using a non-linear least square approach), whereas non-parametric methods calculate dipole magnitude and strength for a set of fixed positions distributed across brain volume (e.g., by iteration and regularization) (Grech et al., 2008).

Given that modeled equations appearing in parametric methods are non-linear, non-parametric methods are much preferred due to their linear character, simplifying the inverse problem both at the theoretical and computational level (Faes et al., 2012). It can be shown mathematically that the linearity of non-parametric methods is a consequence of the huge simplification entailed by imposing from the beginning all dipole locations (usually forming three-dimensional spatially regular grids), hence, these degrees of freedom do not have to be estimated by any algorithm (Pascual-Marqui et al., 2002). Among non-parametric methods, we can mention: LORETA (low resolution electromagnetic tomography, (Pascual-Marqui et al., 1994)), WMN (weighted minimum norm, (Jihene et al., 2013)), SLF (shrinking LORETA focus, (Wu et al., 2005)), Backus-Gilbert (Wan et al., 2008) and S-MAP (sequential maximum a posteriori expectation maximization, (Paavolainen et al., 2014)); these non-parametric methods were computationally compared against a validated toy model (of previously known solution) by Grech et al., rendering LORETA as the method of choice, both in terms of accuracy (especially for deep sources) and noise robustness (Grech et al., 2008).

In the context of brain source localization based on EEG/MEG, Low Resolution Electromagnetic Tomography (LORETA, last version called eLORETA, for “exact”) is a computational tool built to solve the inverse problem in a biologically realistic way and estimate intracerebral current source densities (CSD) located in brain gray matter, thus, yielding three-dimensional CSD maps (Pascual-Marqui et al., 2002, Pascual-Marqui et al., 2011). CSD is a measure of energy density proportional to the total influx of electromagnetic current crossing an orthogonal plane of unit surface (thus, measured in units of μA/mm²), such as the sum of currents derived from the oscillation of electrical dipoles in cortical pyramidal neurons in a given local brain region (Babiloni et al., 2005, Astolfi et al., 2007). On the other hand,

three-dimensional CSD maps are based on a grid of several thousand volume elements or voxels (there are 6239 voxels of 5 millimeter resolution in LORETA), which are geometrically arranged according to an average adult brain neuroanatomical model, encompassing all Brodmann areas (following the digitized Talairach Human Brain Atlas of the Montreal Neurologic Institute) (Pascual-Marqui et al., 2002).

Solution space for CSD maps includes cortical and hippocampal gray matter, using a probability atlas (provided by the MNI brain atlas) to classify each voxel as belonging to either gray matter, white matter or cerebrospinal fluid (the latter being excluded from calculations, as there are no underlying neuronal oscillations associated with these voxels, e.g., in the ventricular system) (Lantz et al., 1997). The solution map is presented as a LORETA image for CSD, typically representing the squared magnitude (or power, measured in μA²/(mm⁴·Hz)) of all computed current density vectors (with one vector of three components assigned to voxel) (Anderer et al., 2001).

In addition, the linearity of LORETA allows to easily separate the different spectral characteristics of the generators, thus, providing CSD maps specific to each selected frequency band (Pascual-Marqui et al., 2011).

Given the existence of an infinite range of solutions to the inverse problem (i.e., an infinite set of different distributions for the current generators that can successfully reproduce the exact measured electrical signals), LORETA aims to constrain them until obtaining the single one with the property of maximal spatial smoothness (Pascual-Marqui et al., 2002). In technical terms (to be later simplified), a precise definition is that LORETA minimizes the squared norm of the Laplacian of the weighted three dimensional current density vector field (Del Felice et al., 2014). Thus, the LORETA solution is achieved by imposing a distribution for the current generators satisfying maximum similarity between nearby voxels (hence, aiming to reproduce a realistic neurophysiology of local clusters of synchronized cells), in terms of the module (strength) and spatial orientation of the current vectors representing neuronal activity (Lantz et al., 1997).

Therefore, LORETA computes current vector fields for the electrical activity originated by neuronal populations associated to each voxel, which the property that the difference between currents corresponding

to a small sphere of voxels centered on a particular voxel has minimum difference with the voxel located in the center (which is a more precise mathematical meaning of “smoothness” for the LORETA solution) (Pascual-Marqui et al., 1994). However, by incorporating the smoothness principle in the spatial distribution of sources, LORETA produces some blurring in the calculated solution, which is more obvious for deep sources due to their increased difficulty for precise localization (Pascual-Marqui et al., 2002, Grech et al., 2008).

Alternatively, LORETA can also find a solution that, instead of calculating current vector fields, fixates from the beginning the orientation of electrical currents according to a template of the cortical surface of the brain, and with this anatomical constraint, calculates the solution that maximizes smoothness only for the modules of the vectors (i.e., spatial correlation between nearby voxels) (Seeck et al., 1996). LORETA solutions are also equally adept to be applied either in the time or frequency spaces (via Fourier transformation) (Pascual-Marqui, 2002).

In addition of computing a solution for the current generators explicitly aiming to be compatible with the known neurophysiological basis, LORETA is superior to the rest of inverse solution methods because it achieves true zero localization error (Pascual-Marqui, 2002, Jatoi et al., 2014). Indeed, since the purpose itself to create a source localization method is to be as precise as possible pinpointing brain structures that are active during a task, minimizing the localization error is arguably the most important criterion to adopt one particular neuroimaging method as compared to alternative approaches (Pascual-Marqui et al., 1994). Localization error and its minimization refers to the problem of producing an algorithm aiming to calculate a solution for the putative localization and activity levels of the generators fulfilling the condition of having the smallest possible difference when compared to the actual arrangement and activity levels of the real generators (Pascual-Marqui et al., 2002). It has been proven (using diverse computational models for comparison) that LORETA has the virtue of outcompeting all additional linear algorithms (e.g., weighted minimum norm method), and that its superiority becomes even clearer in the localization of deep

sources in the brain, generally doing so with virtually zero localization error (margin of error within only one voxel on average) (Lantz et al., 1997, Pascual-Marqui et al., 2002, Grech et al., 2008, Park et al., 2015).

Besides a robust computational and theoretical grounding, there exist a considerable number of studies that have validated the excellent accuracy of the LORETA method at the experimental level (Pascual-Marqui et al., 1994, Seeck et al., 1996, Jatoi et al., 2014). Empirical comparisons build upon the fact that the most critical brain sources linked to a given task are already known (in many cases, with an outstanding precision) from previous research (typically, by using fMRI), therefore, making them suitable for direct testing (i.e., comparing the relevant brain sources involved in an activity with the ones yielded by LORETA, completely on its own) (Lantz et al., 1997).

These studies comprise the correct localization of: auditory stimuli (including identification of mismatch negativity for grammatical errors in spoken language and activity of Wernicke’s area in semantic tasks, (Waberski et al., 2001)), visual and visuo-motor tasks (including facial recognition in the fusiform gyrus, (Pizzagalli et al., 2000)) and motor tasks (located in motor cortices, (Gomez et al., 2003)) during event-related potential (ERP) experiments (Pascual-Marqui et al., 2002, Jatoi et al., 2014). Furthermore, in patients with known epileptic foci, LORETA could correctly confirm the source of seizures that was diagnosed by trained neurologists (using either structural MRI or intracranial recordings) (Pascual-Marqui, 2002). Psychiatric diseases showing abnormal brain activity in previously recognized areas have also been used for testing, e.g., depression (exhibiting heightened activity in rostral anterior cingulate cortex), ADHD (pinpointing to the prefrontal cortex, which regulates attention and impulse control) and Alzheimer’s disease (with LORETA generators indicating the same affected structures than PET scans) (Marqui, 2002, Pascual-Marqui et al., 2011).

Finally, regarding biomedical applications derived from studies of source localization which underscore its importance, besides theoretical motivations, we can mention: diagnoses of mental diseases (e.g., schizophrenia, impulsivity, anxiety, etc.), localization of brain

tumors or foci of epileptic seizures, assessment of damage to brain tissue (e.g., after stroke or concussion) and other pathologies producing aberrant EEG patterns (e.g., excessive delta activity during wakefulness) (Esslen et al., 2008b, Gianotti et al., 2009, Ikeda et al., 2015).