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Atomic spin sensors offer precision measurements using compact, microfabricated packages, placing them in a competitive position for both market and research applications. Performance of these sensors such as dynamic range may be enhanced through magnetic field control. In this work, we discuss the design of miniature coils for three-dimensional, localized field control by direct placement around the sensor, as a flexible and compact alternative to global approaches used previously. Coils are designed on biplanar surfaces using a stream-function approach and then fabricated using standard printed-circuit techniques. Application to a laboratory-scale optically pumped magnetometer of sensitivity approximately 20 fT/Hz$^{1/2}$ is shown. We also demonstrate the performance of a coil set measuring $7 \times 17 \times 17$ mm$^3$ that is optimized specifically for magnetoencephalography, where multiple sensors are operated in proximity to one another. Characterization of the field profile using $^{87}$Rb free-induction spectroscopy and other techniques show $>$96% field homogeneity over the target volume of a MEMS vapor cell and a compact stray field contour of approximately 1% at 20 mm from the center of the cell., Comment: 14 pages, 7 figures

In this paper, we propose a method to estimate the position, orientation and gain of a magnetic field sensor using a set of (large) electromagnetic coils. We apply the method for calibrating an array of optically pumped magnetometers (OPMs) for magnetoencephalography (MEG). We first measure the magnetic fields of the coils at multiple known positions using a well-calibrated triaxial magnetometer and model these discreetly sampled fields using vector-spherical harmonics (VSH) functions. We then localize and calibrate a sensor by minimizing the sum of squared errors between the model signals and the sensor responses to the coil fields. We show that by using homogeneous and first-order gradient fields, the sensor parameters (gain, position, and orientation) can be obtained from a set of linear equations with pseudo-inverses of two matrices. By determining the coil currents based on the VSH models, the coil fields can approximate these low-order field components, yielding computationally simple initial estimates of the sensor parameters. As a first test of the method, we placed a fluxgate magnetometer at multiple positions and estimated the RMS position, orientation, and gain errors of the method to be 1.0 mm, 0.2{\deg} and 0.8%, respectively. Last, we calibrated a 48-channel OPM array. The accuracy of the OPM calibration was tested by using the OPM array to localize magnetic dipoles in a phantom, which resulted in an average position error of 3.3 mm. The results demonstrate the feasibility of using electromagnetic coils to calibrate and localize OPMs for MEG., Comment: 21 pages, 6 figures

Thermal motion of charge carriers in a conducting object causes magnetic field noise that interferes with sensitive measurements nearby the conductor. In this paper, we describe a method to compute the spectral properties of the thermal magnetic noise from arbitrarily-shaped thin conducting objects. We model divergence-free currents on a conducting surface using a stream function and calculate the magnetically independent noise-current modes in the quasi-static regime. We obtain the power spectral density of the thermal magnetic noise as well as its spatial correlations and frequency dependence. We describe a numerical implementation of the method; we model the conducting surface using a triangle mesh and discretize the stream function. The numerical magnetic noise computation agrees with analytical formulas. We provide the implementation as a part of the free and open-source software package bfieldtools.

In this paper, we analyze spatial sampling of electro- (EEG) magnetoencephalography (MEG), where the electric or magnetic field is typically sampled on a curved surface such as the scalp. Using simulated measurements, we study the spatial-frequency content in EEG as well as in on- and off-scalp MEG. The analysis suggests that on-scalp MEG would generally benefit from three times more samples than EEG or off-scalp MEG. Based on the theory of Gaussian processes and experimental design, we suggest an approach to obtain sampling locations on surfaces that are optimal with respect to prior assumptions. Additionally, the approach allows to control, e.g., the uniformity of the sampling locations in the grid. By simulating the performance of grids constructed with different priors, we show that for a low number of spatial samples, model-informed non-uniform sampling can be beneficial. For a large number of samples, uniform sampling grids yield nearly the same total information as the model-informed grids., Comment: arXiv admin note: substantial text overlap with arXiv:1912.05401

Surface currents provide a general way to model static magnetic fields in source-free volumes. To facilitate the use of surface currents in magneto-quasistatic problems, we have implemented a set of computational tools in a Python package named bfieldtools. In this work, we describe the physical and computational principles of this toolset. To be able to work with surface currents of arbitrary shape, we discretize the currents on triangle meshes using piecewise-linear stream functions. We apply analytical discretizations of integral equations to obtain the magnetic field and potentials associated with the discrete stream function. In addition, we describe the computation of the spherical multipole expansion and a novel surface-harmonic expansion for surface currents, both of which are useful for representing the magnetic field in source-free volumes with a small number of parameters. Last, we share examples related to magnetic shielding and surface-coil design using the presented tools., Comment: Updated document style, some references and wordings. 15 pages, 8 figures

We present a novel open-source Python software package, bfieldtools, for magneto-quasistatic calculations with current densities on surfaces of arbitrary shape. The core functionality of the software relies on a stream-function representation of surface-current density and its discretization on a triangle mesh. Although this stream-function technique is well-known in certain fields, to date the related software implementations have not been published or have been limited to specific applications. With bfieldtools, we aimed to produce a general, easy-to-use and well-documented open-source software. The software package is written purely in Python; instead of explicitly using lower-level languages, we address computational bottlenecks through extensive vectorization and use of the NumPy library. The package enables easy deployment, rapid code development and facilitates application of the software to practical problems. In this paper, we describe the software package and give an extensive demonstration of its use with an emphasis on one of its main applications -- coil design., Comment: Updated abstract, document style and reference format. 13 pages, 7 figures

We present a theoretical framework for analyzing spatial sampling of fields in three-dimensional space. The framework bridges Shannon's sampling and information theory to Bayesian probabilistic inference and experimental design. Based on the theory, we present an approach for optimal sampling on curved surfaces and analyze spatial sampling of EEG as well as that of on- and off-scalp MEG. Our spatial-frequency analysis of simulated measurements shows that the available spatial degrees of freedom in the electric potential are limited by the smoothing due to head tissues, while those in the magnetic field are limited by the measurement distance. On-scalp MEG would generally benefit from three times more samples than EEG or off-scalp MEG. Assuming uniform whole-head sampling and similar signal-to-noise ratios, on-scalp MEG has the highest total information content among these modalities. If the number of spatial samples is small, nonuniform sampling can be beneficial.

Tens of millions of people live blind, and their number is ever increasing. Visual-to-auditory sensory substitution (SS) encompasses a family of cheap, generic solutions to assist the visually impaired by conveying visual information through sound. The required SS training is lengthy: months of effort is necessary to reach a practical level of adaptation. There are two reasons for the tedious training process: the elongated substituting audio signal, and the disregard for the compressive characteristics of the human hearing system. To overcome these obstacles, we developed a novel class of SS methods, by training deep recurrent autoencoders for image-to-sound conversion. We successfully trained deep learning models on different datasets to execute visual-to-auditory stimulus conversion. By constraining the visual space, we demonstrated the viability of shortened substituting audio signals, while proposing mechanisms, such as the integration of computational hearing models, to optimally convey visual features in the substituting stimulus as perceptually discernible auditory components. We tested our approach in two separate cases. In the first experiment, the author went blindfolded for 5 days, while performing SS training on hand posture discrimination. The second experiment assessed the accuracy of reaching movements towards objects on a table. In both test cases, above-chance-level accuracy was attained after a few hours of training. Our novel SS architecture broadens the horizon of rehabilitation methods engineered for the visually impaired. Further improvements on the proposed model shall yield hastened rehabilitation of the blind and a wider adaptation of SS devices as a consequence.

Convolutional Neural Networks (CNN) outperform traditional classification methods in many domains. Recently these methods have gained attention in neuroscience and particularly in brain-computer interface (BCI) community. Here, we introduce a CNN optimized for classification of brain states from magnetoencephalographic (MEG) measurements. Our CNN design is based on a generative model of the electromagnetic (EEG and MEG) brain signals and is readily interpretable in neurophysiological terms. We show here that the proposed network is able to decode event-related responses as well as modulations of oscillatory brain activity and that it outperforms more complex neural networks and traditional classifiers used in the field. Importantly, the model is robust to inter-individual differences and can successfully generalize to new subjects in offline and online classification., Comment: 12 pages, 4 figures, 4 tables. keywords: MEG, BCI, real-time, convolutional neural networks