Reducing crosstalk in optically-pumped magnetometer arrays

Gradient phase and amplitude errors in atomic magnetic gradiometers for biomagnetic imaging systems

The cross-axis projection error (CAPE) caused by residual magnetic fields has recently attracted widespread attention. In this study, we propose a more specific theoretical model and expand the CAPE in gradient measurements. We first report that differences in relaxation rate and residual magnetic field between optically pumped magnetometers (OPMs) introduce a significant error term in the output of OPM gradiometers, referred to as the gradient phase error. Furthermore, when the longitudinal field compensation is inadequate, the interaxial response interference of a single OPM is prominent, resulting in an amplitude distortion of the signal. This is further amplified in the gradiometer configuration, introducing the gradient amplitude error. Our experiments demonstrated that the efficacy of mitigating common-mode noise of OPM gradiometers was significantly impaired when existing the gradient errors. In addition, a simulation with a magnetoencephalography (MEG) system illustrated an induced source localization error of exceeding 2 cm, severely compromising the localization accuracy of OPM-MEG systems.

Adaptive multipole models of optically pumped magnetometer data

Multipole expansions have been used extensively in the Magnetoencephalography (MEG) literature for mitigating environmental interference and modelling brain signal. However, their application to Optically Pumped Magnetometer (OPM) data is challenging due to the wide variety of existing OPM sensor and array designs. We therefore explore how such multipole models can be adapted to provide stable models of brain signal and interference across OPM systems. Firstly, we demonstrate how prolate spheroidal (rather than spherical) harmonics can provide a compact representation of brain signal when sampling on the scalp surface with as few as 100 channels. We then introduce a type of orthogonal projection incorporating this basis set. The Adaptive Multipole Models (AMM), which provides robust interference rejection across systems, even in the presence of spatially structured nonlinearity errors (shielding factor is the reciprocal of the maximum fractional nonlinearity error). Furthermore, this projection is always stable, as it is an orthogonal projection, and will only ever decrease the white noise in the data. However, for array designs that are suboptimal for spatially separating brain signal and interference, this method can remove brain signal components. We contrast these properties with the more typically used multipole expansion, Signal Space Separation (SSS), which never reduces brain signal amplitude but is less robust to the effect of sensor nonlinearity errors on interference rejection and can increase noise in the data if the system is sub-optimally designed (as it is an oblique projection). We conclude with an empirical example utilizing AMM to maximize signal to noise ratio (SNR) for the stimulus locked neuronal response to a flickering visual checkerboard in a 128-channel OPM system and demonstrate up to 40 dB software shielding in real data.

Combining OPM and lesion mapping data for epilepsy surgery planning: a simulation study

When planning for epilepsy surgery, multiple potential sites for resection may be identified through anatomical imaging. Magnetoencephalography (MEG) using optically pumped sensors (OP-MEG) is a non-invasive functional neuroimaging technique which could be used to help identify the epileptogenic zone from these candidate regions. Here we test the utility of a-priori information from anatomical imaging for differentiating potential lesion sites with OP-MEG. We investigate a number of scenarios: whether to use rigid or flexible sensor arrays, with or without a-priori source information and with or without source modelling errors. We simulated OP-MEG recordings for 1309 potential lesion sites identified from anatomical images in the Multi-centre Epilepsy Lesion Detection (MELD) project. To localise the simulated data, we used three source inversion schemes: unconstrained, prior source locations at centre of the candidate sites, and prior source locations within a volume around the lesion location. We found that prior knowledge of the candidate lesion zones made the inversion robust to errors in sensor gain, orientation and even location. When the reconstruction was too highly restricted and the source assumptions were inaccurate, the utility of this a-priori information was undermined. Overall, we found that constraining the reconstruction to the region including and around the participant's potential lesion sites provided the best compromise of robustness against modelling or measurement error.

Real-time, model-based magnetic field correction for moving, wearable MEG

Most neuroimaging techniques require the participant to remain still for reliable recordings to be made. Optically pumped magnetometer (OPM) based magnetoencephalography (OP-MEG) however, is a neuroimaging technique which can be used to measure neural signals during large participant movement (approximately 1 m) within a magnetically shielded room (MSR) (Boto et al., 2018; Seymour et al., 2021). Nevertheless, environmental magnetic fields vary both spatially and temporally and OPMs can only operate within a limited magnetic field range, which constrains participant movement. Here we implement real-time updates to electromagnetic coils mounted on-board of the OPMs, to cancel out the changing background magnetic fields. The coil currents were chosen based on a continually updating harmonic model of the background magnetic field, effectively implementing homogeneous field correction (HFC) in real-time (Tierney et al., 2021). During a stationary, empty room recording, we show an improvement in very low frequency noise of 24 dB. In an auditory paradigm, during participant movement of up to 2 m within a magnetically shielded room, introduction of the real-time correction more than doubled the proportion of trials in which no sensor saturated recorded outside of a 50 cm radius from the optimally-shielded centre of the room. The main advantage of such model-based (rather than direct) feedback is that it could allow one to correct field components along unmeasured OPM axes, potentially mitigating sensor gain and calibration issues (Borna et al., 2022).

Light-noise suppression method for the single-beam optically-pumped magnetometer arrays

We propose a miniaturized single-beam optically pumped magnetometer (OPM) with a laser power differential structure, along with a dynamically-adjusted detection circuit. This design enables the suppression of optical fluctuation noise and the enhancement of magnetometer sensitivity. For a single-beam OPM, pump light fluctuation noise is a significant contributor to output noise. To address this, we propose an OPM with a laser differential structure that separates the pump light as a part of the reference signal before it enters the cell. The reference current and OPM output current are then subtracted to suppress the noise introduced by pump light fluctuations. To achieve optimal optical noise suppression, we implement balanced homodyne detection (BHD) with real-time current adjustment, which dynamically adjusts the reference ratio between the two currents according to their amplitude. Ultimately, we can reduce the noise introduced by pump light fluctuations by 47% of the original. The OPM with laser power differential achieves a sensitivity of 17.5 fT/Hz^1/2, with the optical fluctuation equivalent noise at 13 fT/Hz^1/2.

An integrated full-head OPM-MEG system based on 128 zero-field sensors

Compact optically-pumped magnetometers (OPMs) are now commercially available with noise floors reaching 10 fT/Hz^1/2. However, to be used effectively for magnetoencephalography (MEG), dense arrays of these sensors are required to operate as an integrated turn-key system. In this study, we present the HEDscan, a 128-sensor OPM MEG system by FieldLine Medical, and evaluate its sensor performance with regard to bandwidth, linearity, and crosstalk. We report results from cross-validation studies with conventional cryogenic MEG, the Magnes 3,600 WH Biomagnetometer by 4-D Neuroimaging. Our results show high signal amplitudes captured by the OPM-MEG system during a standard auditory paradigm, where short tones at 1000 Hz were presented to the left ear of six healthy adult volunteers. We validate these findings through an event-related beamformer analysis, which is in line with existing literature results.

Improved Biomagnetic Signal-To-Noise Ratio and Source Localization Using Optically Pumped Magnetometers with Synthetic Gradiometers

Optically pumped magnetometers (OPMs) can capture brain activity but are susceptible to magnetic noise. The objective of this study was to evaluate a novel methodology used to reduce magnetic noise in OPM measurements. A portable magnetoencephalography (MEG) prototype was developed with OPMs. The OPMs were divided into primary sensors and reference sensors. For each primary sensor, a synthetic gradiometer (SG) was constructed by computing a secondary sensor that simulated noise with signals from the reference sensors. MEG data from a phantom with known source signals and six human participants were used to assess the efficacy of the SGs. Magnetic noise in the OPM data appeared predominantly in a low frequency range (<4 Hz) and varied among OPMs. The SGs significantly reduced magnetic noise (p < 0.01), enhanced the signal-to-noise ratio (SNR) (p < 0.001) and improved the accuracy of source localization (p < 0.02). The SGs precisely revealed movement-evoked magnetic fields in MEG data recorded from human participants. SGs provided an effective method to enhance SNR and improve the accuracy of source localization by suppressing noise. Software-simulated SGs may provide new opportunities regarding the use of OPM measurements in various clinical and research applications, especially those in which movement is relevant.

Exploring the limits of MEG spatial resolution with multipolar expansions

The advent of scalp magnetoencephalography (MEG) based on optically pumped magnetometers (OPMs) may represent a step change in the field of human electrophysiology. Compared to cryogenic MEG based on superconducting quantum interference devices (SQUIDs, placed 2-4 cm above scalp), scalp MEG promises significantly higher spatial resolution imaging but it also comes with numerous challenges regarding how to optimally design OPM arrays. In this context, we sought to provide a systematic description of MEG spatial resolution as a function of the number of sensors (allowing comparison of low- vs. high-density MEG), sensor-to-brain distance (cryogenic SQUIDs vs. scalp OPM), sensor type (magnetometers vs. gradiometers; single- vs. multi-component sensors), and signal-to-noise ratio. To that aim, we present an analytical theory based on MEG multipolar expansions that enables, once supplemented with experimental input and simulations, quantitative assessment of the limits of MEG spatial resolution in terms of two qualitatively distinct regimes. In the regime of asymptotically high-density MEG, we provide a mathematically rigorous description of how magnetic field smoothness constraints spatial resolution to a slow, logarithmic divergence. In the opposite regime of low-density MEG, it is sensor density that constraints spatial resolution to a faster increase following a square-root law. The transition between these two regimes controls how MEG spatial resolution saturates as sensors approach sources of neural activity. This two-regime model of MEG spatial resolution integrates known observations (e.g., the difficulty of improving spatial resolution by increasing sensor density, the gain brought by moving sensors on scalp, or the usefulness of multi-component sensors) and gathers them under a unifying theoretical framework that highlights the underlying physics and reveals properties inaccessible to simulations. We propose that this framework may find useful applications to benchmark the design of future OPM-based scalp MEG systems.

Design and Simulation of a Magnetization Drive Coil Based on the Helmholtz Principle and an Experimental Study

In order to realize the magnetization of hydrogel mixed with NdFeB powder, a magnetization drive coil based on a Helmholtz coil is designed in this paper. The 3D model of the magnetic field system is drawn by the Maxwell software 3D module, and the influence of different factors on the magnetic induction intensity is analyzed to obtain the optimized structure of the magnetization drive coil; then, the central magnetic induction intensity and magnetic induction line distribution density of the magnetization drive coil and Helmholtz coil are compared to verify the reliability of the structure optimization. The results show that the central magnetic induction intensity is the highest when the distance between the auxiliary coils is 70 mm, the central magnetic induction intensity of the magnetized drive coil is significantly higher than that of the Helmholtz coil when the number of turns is the same, and the central magnetic induction intensity of the optimized magnetized drive coil can reach 1.37 T with a more uniform and dense magnetic induction line distribution. After building the magnetization drive coil, the magnetic induction intensity of the center of the magnetization drive coil can reach 1.34 T by a handheld digital Gauss meter test, and the error is no more than 2% with the simulation result. This design approach provides a reference for the structural design and operating characteristics analysis of magnetized drive coils and shortens the design cost and development cycle of magnetized drive coils.

Cross-Axis Dynamic Field Compensation of Optically Pumped Magnetometer Arrays for MEG

We present dynamic field compensation (DFC), whereby three-axis field measurements from reference magnetometers are used to dynamically maintain null at the alkali vapor cells of an array of primary sensors that are proximal to a subject's scalp. Precision measurement of the magnetoencephalogram (MEG) by zero-field optically pumped magnetometer (OPM) sensors requires that sensor response is linear and sensor gain is constant over time. OPMs can be operated in open-loop mode, where the measured field is proportional to the output at the demodulated photodiode output, or in closed-loop, where on-board coils are dynamically driven to maintain the internal cell at zero field in the measurement direction. While OPMs can be operated in closed-loop mode along all three axes, this can increase sensor noise and poses engineering challenges. Uncompensated fluctuations in the ambient field along any statically nulled axes perturb the measured field by tipping the measurement axis and altering effective sensor gain - a phenomenon recently referred to as cross-axis projection error (CAPE). These errors are particularly problematic when OPMs are allowed to move in the remnant background field. Sensor gain-errors, if not mitigated, preclude precision measurements with OPMs operating in the presence of ambient field fluctuations within a typical MEG laboratory. In this manuscript, we present the cross-axis dynamic field compensation (DFC) method for maintaining zero field dynamically on all three axes of each sensor in an array of OPMs. Together, DFC and closed-loop operation strongly attenuate errors introduced by CAPE. This method was implemented by using three orthogonal reference sensors together with OPM electronics that permit driving each sensor's transverse field coils dynamically to maintain null field across its OPM measurement cell. These reference sensors can also be used for synthesizing 1st-gradient response to further reduce the effects of fluctuating ambient fields on measured brain activity and compensate for movement within a uniform field. We demonstrate that, using the DFC method, magnetic field measurement errors of less than 0.7% are easily achieved for an array of OPM sensors in the presence of ambient field perturbations of several nT.

Spherical harmonic based noise rejection and neuronal sampling with multi-axis OPMs

In this study we explore the interference rejection and spatial sampling properties of multi-axis Optically Pumped Magnetometer (OPM) data. We use both vector spherical harmonics and eigenspectra to quantify how well an array can separate neuronal signal from environmental interference while adequately sampling the entire cortex. We found that triaxial OPMs have superb noise rejection properties allowing for very high orders of interference (L=6) to be accounted for while minimally affecting the neural space (2dB attenuation for a 60-sensor triaxial system). We show that at least 11th order (143 spatial degrees of freedom) irregular solid harmonics or 95 eigenvectors of the lead field are needed to model the neural space for OPM data (regardless of number of axes measured). This can be adequately sampled with 75-100 equidistant triaxial sensors (225-300 channels) or 200 equidistant radial channels. In other words, ordering the same number of channels in triaxial (rather than purely radial) configuration may give significant advantages not only in terms of external noise rejection but also by minimizing cost, weight and cross-talk.

Quantum-assisted distortion-free audio signal sensing

Quantum sensors are known for their high sensitivity in sensing applications. However, this sensitivity often comes with severe restrictions on other parameters which are also important. Examples are that in measurements of arbitrary signals, limitation in linear dynamic range could introduce distortions in magnitude and phase of the signal. High frequency resolution is another important feature for reconstructing unknown signals. Here, we demonstrate a distortion-free quantum sensing protocol that combines a quantum phase-sensitive detection with heterodyne readout. We present theoretical and experimental investigations using nitrogen-vacancy centers in diamond, showing the capability of reconstructing audio frequency signals with an extended linear dynamic range and high frequency resolution. Melody and speech based signals are used for demonstrating the features. The methods could broaden the horizon for quantum sensors towards applications, e.g. telecommunication in challenging environment, where low-distortion measurements are required at multiple frequency bands within a limited volume.

High sensitivity in quantum sensing comes often at the expense of other figures of merit, usually resulting in distortion. Here, the authors propose a protocol with good sensitivity, readout linearity and high frequency resolution, and benchmark it through signal measurements at audio bands with NV centers.

On-scalp magnetocorticography with optically pumped magnetometers: Simulated performance in resolving simultaneous sources

Currently, the gold standard for high-resolution mapping of cortical electrophysiological activity is invasive electrocorticography (ECoG), a procedure that carries with it the risk of serious morbidity and mortality. Due to these risks, the use of ECoG is largely limited to pre-surgical mapping in intractable epilepsy. Nevertheless, many seminal studies in neuroscience have utilized ECoG to explore domains such as visual perception, attention, auditory processing, and sensorimotor behavior. Studies such as these, occurring in patients with epilepsy rather than healthy controls, may lack generalizability, and are limited by the placement of the electrode arrays over the presumed seizure focus. This manuscript explores the use of optically pumped magnetometers (OPMs) to create a non-invasive alternative to ECoG, which we refer to as magnetocorticography. Because prior ECoG studies reveal that most cognitive processes are driven by multiple, simultaneous independent neuronal assemblies, we characterize the ability of a theoretical 56-channel dense OPM array to resolve simultaneous independent sources, and compare it to currently available SQUID devices, as well as OPM arrays with inter-sensor spacings more typical of other systems in development. Our evaluation of this theoretical system assesses many potential sources of error, including errors of sensor calibration and position. In addition, we investigate the influence of geometrical and anatomical factors on array performance. Our simulations reveal the potential of high-density, on-scalp OPM MEG devices to localize electrophysiological brain responses at unprecedented resolution for a non-invasive device.

Interference suppression techniques for OPM-based MEG: Opportunities and challenges

One of the primary technical challenges facing magnetoencephalography (MEG) is that the magnitude of neuromagnetic fields is several orders of magnitude lower than interfering signals. Recently, a new type of sensor has been developed - the optically pumped magnetometer (OPM). These sensors can be placed directly on the scalp and move with the head during participant movement, making them wearable. This opens up a range of exciting experimental and clinical opportunities for OPM-based MEG experiments, including paediatric studies, and the incorporation of naturalistic movements into neuroimaging paradigms. However, OPMs face some unique challenges in terms of interference suppression, especially in situations involving mobile participants, and when OPMs are integrated with electrical equipment required for naturalistic paradigms, such as motion capture systems. Here we briefly review various hardware solutions for OPM interference suppression. We then outline several signal processing strategies aimed at increasing the signal from neuromagnetic sources. These include regression-based strategies, temporal filtering and spatial filtering approaches. The focus is on the practical application of these signal processing algorithms to OPM data. In a similar vein, we include two worked-through experiments using OPM data collected from a whole-head sensor array. These tutorial-style examples illustrate how the steps for suppressing external interference can be implemented, including the associated data and code so that researchers can try the pipelines for themselves. With the popularity of OPM-based MEG rising, there will be an increasing need to deal with interference suppression. We hope this practical paper provides a resource for OPM-based MEG researchers to build upon.

Magnetic Field Mapping and Correction for Moving OP-MEG

BACKGROUND

Optically pumped magnetometers (OPMs) have made moving, wearable magnetoencephalography (MEG) possible. The OPMs typically used for MEG require a low background magnetic field to operate, which is achieved using both passive and active magnetic shielding. However, the background magnetic field is never truly zero Tesla, and so the field at each of the OPMs changes as the participant moves. This leads to position and orientation dependent changes in the measurements, which manifest as low frequency artefacts in MEG data.

OBJECTIVE

We model the spatial variation in the magnetic field and use the model to predict the movement artefact found in a dataset.

METHODS

We demonstrate a method for modelling this field with a triaxial magnetometer, then show that we can use the same technique to predict the movement artefact in a real OPM-based MEG (OP-MEG) dataset.

RESULTS

Using an 86-channel OP-MEG system, we found that this modelling method maximally reduced the power spectral density of the data by 27.8 ± 0.6 dB at 0 Hz, when applied over 5 s non-overlapping windows.

CONCLUSION

The magnetic field inside our state-of-the art magnetically shielded room can be well described by low-order spherical harmonic functions. We achieved a large reduction in movement noise when we applied this model to OP-MEG data.

SIGNIFICANCE

Real-time implementation of this method could reduce passive shielding requirements for OP-MEG recording and allow the measurement of low-frequency brain activity during natural participant movement.

Cross-Axis projection error in optically pumped magnetometers and its implication for magnetoencephalography systems

Optically pumped magnetometers (OPMs) developed for magnetoencephalography (MEG) typically operate in the spin-exchange-relaxation-free (SERF) regime and measure a magnetic field component perpendicular to the propagation axis of the optical-pumping photons. The most common type of OPM for MEG employs alkali atoms, e.g. ⁸⁷Rb, as the sensing element and one or more lasers for preparation and interrogation of the magnetically sensitive states of the alkali atoms ensemble. The sensitivity of the OPM can be greatly enhanced by operating it in the SERF regime, where the alkali atoms' spin exchange rate is much faster than the Larmor precession frequency. The SERF regime accommodates remnant static magnetic fields up to ±5 nT. However, in the presented work, through simulation and experiment, we demonstrate that multi-axis magnetic signals in the presence of small remnant static magnetic fields, not violating the SERF criteria, can introduce significant error terms in OPM's output signal. We call these deterministic errors cross-axis projection errors (CAPE), where magnetic field components of the MEG signal perpendicular to the nominal sensing axis contribute to the OPM signal giving rise to substantial amplitude and phase errors. Furthermore, through simulation, we have discovered that CAPE can degrade localization and calibration accuracy of OPM-based magnetoencephalography (OPM-MEG) systems.

Measuring the cortical tracking of speech with optically-pumped magnetometers

During continuous speech listening, brain activity tracks speech rhythmicity at frequencies matching with the repetition rate of phrases (0.2-1.5 Hz), words (2-4 Hz) and syllables (4-8 Hz). Here, we evaluated the applicability of wearable MEG based on optically-pumped magnetometers (OPMs) to measure such cortical tracking of speech (CTS). Measuring CTS with OPMs is a priori challenging given the complications associated with OPM measurements at frequencies below 4 Hz, due to increased intrinsic interference and head movement artifacts. Still, this represents an important development as OPM-MEG provides lifespan compliance and substantially improved spatial resolution compared with classical MEG. In this study, four healthy right-handed adults listened to continuous speech for 9 min. The radial component of the magnetic field was recorded simultaneously with 45-46 OPMs evenly covering the scalp surface and fixed to an additively manufactured helmet which fitted all 4 participants. We estimated CTS with reconstruction accuracy and coherence, and determined the number of dominant principal components (PCs) to remove from the data (as a preprocessing step) for optimal estimation. We also identified the dominant source of CTS using a minimum norm estimate. CTS estimated with reconstruction accuracy and coherence was significant in all 4 participants at phrasal and word rates, and in 3 participants (reconstruction accuracy) or 2 (coherence) at syllabic rate. Overall, close-to-optimal CTS estimation was obtained when the 3 (reconstruction accuracy) or 10 (coherence) first PCs were removed from the data. Importantly, values of reconstruction accuracy (~0.4 for 0.2-1.5-Hz CTS and ~0.1 for 2-8-Hz CTS) were remarkably close to those previously reported in classical MEG studies. Finally, source reconstruction localized the main sources of CTS to bilateral auditory cortices. In conclusion, t his study demonstrates that OPMs can be used for the purpose of CTS assessment. This finding opens new research avenues to unravel the neural network involved in CTS across the lifespan and potential alterations in, e.g., language developmental disorders. Data also suggest that OPMs are generally suitable for recording neural activity at frequencies below 4 Hz provided PCA is used as a preprocessing step; 0.2-1.5-Hz being the lowest frequency range successfully investigated here.

Recent advances in the noninvasive detection of high-frequency oscillations in the human brain

In recent decades, a significant body of evidence based on invasive clinical research has showed that high-frequency oscillations (HFOs) are a promising biomarker for localization of the seizure onset zone (SOZ), and therefore, have the potential to improve postsurgical outcomes in patients with epilepsy. Emerging clinical literature has demonstrated that HFOs can be recorded noninvasively using methods such as scalp electroencephalography (EEG) and magnetoencephalography (MEG). Not only are HFOs considered to be a useful biomarker of the SOZ, they also have the potential to gauge disease severity, monitor treatment, and evaluate prognostic outcomes. In this article, we review recent clinical research on noninvasively detected HFOs in the human brain, with a focus on epilepsy. Noninvasively detected scalp HFOs have been investigated in various types of epilepsy. HFOs have also been studied noninvasively in other pathologic brain disorders, such as migraine and autism. Herein, we discuss the challenges reported in noninvasive HFO studies, including the scarcity of MEG and high-density EEG equipment in clinical settings, low signal-to-noise ratio, lack of clinically approved automated detection methods, and the difficulty in differentiating between physiologic and pathologic HFOs. Additional studies on noninvasive recording methods for HFOs are needed, especially prospective multicenter studies. Further research is fundamental, and extensive work is needed before HFOs can routinely be assessed in clinical settings; however, the future appears promising.

Pragmatic spatial sampling for wearable MEG arrays

Several new technologies have emerged promising new Magnetoencephalography (MEG) systems in which the sensors can be placed close to the scalp. One such technology, Optically Pumped MEG (OP-MEG) allows for a scalp mounted system that provides measurements within millimetres of the scalp surface. A question that arises in developing on-scalp systems is: how many sensors are necessary to achieve adequate performance/spatial discrimination? There are many factors to consider in answering this question such as the signal to noise ratio (SNR), the locations and depths of the sources, density of spatial sampling, sensor gain errors (due to interference, subject movement, cross-talk, etc.) and, of course, the desired spatial discrimination. In this paper, we provide simulations which show the impact these factors have on designing sensor arrays for wearable MEG. While OP-MEG has the potential to provide high information content at dense spatial samplings, we find that adequate spatial discrimination of sources (< 1 cm) can be achieved with relatively few sensors (< 100) at coarse spatial samplings (~ 30 mm) at high SNR. After this point approximately 50 more sensors are required for every 1 mm improvement in spatial discrimination. Comparable discrimination for traditional cryogenic systems require more channels by these same metrics. We also show that sensor gain errors have the greatest impact on discrimination between deep sources at high SNR. Finally, we also examine the limitation that aliasing due to undersampling has on the effective SNR of on-scalp sensors.