2.1 Complementary information from extracellular neural fields

A neuron can be conceptually divided into intracellular, membrane, and extracellular regions (Fig 1a). Assuming nonlinear sources are confined to the membrane, the intra- and extracellular fields obey the Poisson equation, enabling use of Green’s theorem [28]. Previous work has applied this to express the extracellular potential in terms of both current densities and either the transmembrane potential or the inner and outer membrane potentials [29–31]. Here, we extend this approach using the vector form of Green’s theorem to relate the extracellular magnetic field to the current density on the membrane’s inner surface (see Supplementary Information, S1 Appendix for derivation).

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Fig 1. Application of Green’s theorem and neural currents diagram.

a, Application of Green’s theorem to the intracellular space, , of a neuron—characterized by the potential , electric field E_i, and magnetic field B_i, bounded by the surface with local normal —and to the extracellular space, —characterized by the potential , electric field E_e, and magnetic field B_e, bounded by the surface with local normal . Membrane created in BioRender. Ali, Z. (2026) https://BioRender.com/fmjdzjs. b, Illustration of longitudinal and transmembrane currents generated by a neuron during an action potential, along with the direction of the extracellular magnetic field induced by longitudinal currents. Shading is normalized separately for the transmembrane and longitudinal currents to highlight their relative, rather than absolute, distributions. Cell created in BioRender. Ali, Z. (2026) https://BioRender.com/rbcfhct.

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For a source-free, extended, and non-magnetic extracellular space, the magnetic flux density generated by a neuron with intracellular electrical potential can be approximated as

(1)

where is the normal to the membrane surface and is the intracellular conductivity. Equation 1 is similar to the equation for the extracellular magnetic field found in [32] but is obtained without making any assumptions about the neuron’s geometry; the full expression in Supplementary Information Equation 9 (S1 Appendix) therefore includes additional terms absent from [32]. In comparison, the extracellular electrical potential is given by

(2)

where is the extracellular conductivity. Therefore, the magnetic field depends on the tangential (longitudinal) component of the intracellular surface current density on , while the electric potential depends on the normal (transmembrane) component, on . Thus, as suggested in [33,34], magnetic and electric measurements capture complementary aspects of neural activity—currents flowing along and across the membrane, respectively (Fig 1b). We therefore hypothesize that simultaneous recordings of both modalities can provide a more complete picture of neuronal physical structure and a deeper understanding of neuronal dynamics across large populations.

2.2 Neural signal scaling

While the scaling of extracellular potentials generated by spiking neurons has been well studied [35–38], the distance-dependent behavior of extracellular magnetic fields remains relatively underexplored. Having established that extracellular magnetic fields and potentials arise from orthogonal components of the intracellular surface current density— (longitudinal) and (transmembrane), respectively—we now examine how these components reverse sign along the neuron. Since, in the quasi-magnetostatic regime, both current sources generate fields with equivalent scaling, the spatial frequency of the sign changes of these currents determines the multipole order of the resulting fields, which in turn governs how the signal amplitude decays with distance (see Supplementary Information, S1 Appendix for detailed explanation).

To analyze the sign reversal in current components, we begin by computing the intracellular potential, from which both the longitudinal () and transmembrane () surface current densities are derived. We model the neuron as an infinitely long cylindrical axon of radius a, embedded in an unbounded volume conductor. Under these assumptions, the intracellular and extracellular potentials, and , respectively, satisfy Laplace’s equation in cylindrical coordinates. The general solutions involve modified Bessel functions and are expressed as integrals over spatial frequency. The weighting terms for these Bessel components are determined by applying boundary conditions at the membrane.

To model the neuron’s activity during firing, we follow the approach in [39–41] assuming the transmembrane potential , which represents the action potential, is given. At the membrane (), we impose the boundary condition , along with continuity of the normal component of current across the boundary. These constraints determine the weighting terms in the Bessel-function solutions. Since the axon is infinitely long, we perform the analysis in the Fourier domain, where axial variations simplify algebraically. Given that the length of the axon is much greater than its radius, we apply small-argument approximations to the Bessel functions to simplify the solution. This yields a tractable expression for the intracellular potential , from which the surface current components follow. Taking the inverse Fourier transforms, we obtain:

(3a)

(3b)

where * denotes convolution and is the Dirac delta function (see Supplementary Information, S1 Appendix for derivation).

When is a unimodal function, such as a Gaussian, it does not change sign. Thus, the number of sign reversals in the current components is entirely determined by the order of differentiation. The longitudinal current , involving the first derivative of , reverses sign once, while the transmembrane current , involving the second derivative, reverses sign twice. In general, higher-order derivatives produce more sign changes, corresponding to higher-order multipoles, which decay more rapidly with distance. Specifically, the extracellular magnetic field—generated by the dipole-like structure of —scales with distance R as 1/R³. In contrast, the extracellular electric field and potential, shaped by the quadrupolar structure of , scale as 1/R⁴ and 1/R³, respectively, when referenced to a distant point. In practice, extracellular potentials are measured relative to a local reference electrode; if this reference is near the measurement electrode, the recorded potential approaches the electric field’s scaling behavior, 1/R⁴. As a result, the observed scaling typically falls between these two limits and can be approximated as , where , with reflecting the proximity of the reference electrode.

To validate and illustrate these theoretical predictions, we now proceed with three numerical evaluations. First, using a Gaussian function for to represent an action potential, Fig 2a shows the distribution of and along a cylindrical axon, as derived from (3). These currents form two current loops, n, (black dashed arrows in Fig 2a), corresponding to traveling waves of depolarization, which indicate the number of sign flips for the longitudinal (n) and transmembrane (n + 1) current components. These results confirm the dipole behavior of an extended neural process’s magnetic field and the quadrupole behavior of its electrical potential during an action potential.

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Fig 2. Scaling analysis of intracellular currents and extracellular fields in neuron models.

a, Analytically-derived distribution of longitudinal and transmembrane components of the intracellular surface current along an infinitely extended cylindrical axon with a Gaussian-distributed membrane potential , characterized by a standard deviation of 167 m. The black dashed lines indicate current loops composed of the transmembrane and longitudinal currents which propagate down the length of the axon during the action potential. b, Cellular current distributions for a ball-stick cell model when the action potential peak travels half of the axon length. Parameters: axon length = 1 mm; axon diameter = 2 m; soma diameter = 20 m; segment length = 0.5 m. c, Scaling exponent n characterizing how the extracellular potential and magnetic field B_e, including the planar (B_e,y) and normal (B_e,z) components relative to the sensor array plane, vary as a function of distance from the cell (Rⁿ). The upper part of the figure shows the cell model (Cell 2 from S1 Fig) and measurement locations used to evaluate the scaling, marked by the black dashed line. Scale bar, 100 m. d, Total area of supra-threshold signal for electrical potential and magnetic field components B_e,y, B_e,z, averaged for cells 0–5 (S1 Fig). Threshold is defined relative to the absolute value of the peak signal for each cell across the total time duration of a spike. Measurement point pitch set at 5 m, so each location with a supra-threshold signal corresponds to an area of 25 . p = 0.042, 0.013, and 0.002 for thresholds 70%, 50%, and 30% respectively, computed using paired-samples t-test.

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Next, we implement a ball-stick model to compute the corresponding transmembrane and longitudinal currents. Fig 2b shows the current distribution at the time when the action potential peaks at one-half of the length of the axon. Unlike a cylindrical axon, the presence of the soma significantly amplifies the transmembrane current at the proximal end relative to the rest of the axon. As a result, the quadrupole effect on the electric potential is reduced, suggesting that the scaling behavior for the extracellular electric potential may be better than . Similarly, the forward longitudinal current in the axon is stronger than the reverse, indicating that the dipole effect on the magnetic field is diminished, and its scaling behavior may be better than 1/R³.

To evaluate how the non-uniform distribution of transmembrane and longitudinal currents in real cells skews electrical potential and magnetic field scaling away from the 1/R³ first-order approximation, we simulate six morphologically realistic layer V rat somatosensory cortical neurons (S1 Fig), all sourced from the Blue Brain Project [42]. These models were selected for their highly detailed morphology and biophysical realism, as well as their resemblance to the information-rich neurons commonly targeted in human studies. The set includes at least one representative of each pyramidal neuron type in the Blue Brain Project’s layer V repository (TTPC1, TTPC2, UTPC, STPC) and two interneuron types (DBC, MC). While all of the chosen cells are cortical neurons, the scaling trends in Equation 3 apply to all extended neural processes irrespective of cell type, including giant axons and peripheral nerves.

Fig 2c presents the scaling exponent n of the extracellular electrical potential and magnetic field B_e calculated along a radial line of measurement points extending outward from near the soma and adjacent to the apical dendrite of Cell 2 from S1 Fig during a spike (Fig 2c, inset). We also report the exponents for the planar, B_e,y, and normal, B_e,z, components of B_e, as most magnetic sensors are sensitive to only one directional axis. The simulations reveal that the asymmetry in transmembrane and longitudinal current distributions leads to deviations from the expected 1/R³ scaling. This deviation is more pronounced for the magnetic field, which decays approximately as 1/R² at large distance, compared to the electrical potential, which decays around 1/R^2.5. Notably, the two magnetic field components exhibit distinct scaling behaviors: the planar component B_e,y decreases roughly an order of magnitude faster than the normal component B_e,z. This discrepancy arises from the geometrical arrangement of the neuron relative to the measurement points. In conventional in vitro setups, neurons lie flat on top of probes composed of an array of sensors (e.g., Hall effect, magnetoresistive) that measure either B_e,x, B_e,y, or B_e,z fields. If the array is positioned a small distance beneath the neuron, then for , the sensors are effectively coplanar with the neuron. In this geometry, the magnetic field for points primarily in the normal direction, as shown in S2 Fig, rendering the planar field components negligible and favoring arrays composed of B_e,z sensors. These asymptotic scaling trends are validated for the other morphological cell models in S3 Fig.

To understand the practical implications of these scaling trends for sensing, we evaluate, for each cell, the total area around the cell in the measurement plane where the signal magnitude exceeds some fraction of the overall maximum signal for , B_e,y, and B_e,z. Fig 2d presents the results of this analysis for thresholds of 30%, 50%, and 70% of the peak signal. We observe that across thresholds, the total supra-threshold area for the B_e,z signal is approximately 2× that of the signal. This is due to the asymptotic scaling behavior illustrated in Fig 2c, but also to the distribution of longitudinal currents in the cell. Whereas the transmembrane current distribution is concentrated heavily in the soma, with weak return currents distributed across the entire rest of the cell (that do not generate strong signals), the propagation of the longitudinal currents down the major cellular processes, as shown in Fig 2a and 2b, generate a correspondingly widely distributed magnetic field signal. This can be seen in S1 Video as well as S4 Fig; while the electrical potential is largely radially symmetric around the soma, the B_e,y and B_e,z fields are more reflective of cellular morphology. This also explains why the B_e,y field, despite having worse radial scaling behavior than , has approximately similar supra-threshold signal areas in Fig 2d, because the longitudinal current distribution offsets the poor radial scaling property. These results suggest that for in vitro measurements where sensor resolution is often limited, it may be more advantageous to measure normal magnetic fields than electrical potentials. This advantage has important implications for sensor design: targeting the normal field component can help optimize sensitivity and improve signal detectability, particularly in systems constrained by spatial sampling density or noise thresholds.

2.3 Cellular spatial resolution limit

Building on the complementary information and scaling analysis, we now turn to a core functional question: how distinguishable are neurons based on their extracellular magnetic fields compared to their electrical potentials? Distinguishability is fundamental to decoding neural activity from extracellular recordings; however, these recordings compress the rich, 3D spatial information generated by the unique morphological distribution of longitudinal and transmembrane currents for each neuron into a sparse, 2D spike template captured by a planar sensor array. This dimensionality reduction complicates the differentiation of neurons that are spatially proximate and/or morphologically similar.

We assess neuronal distinguishability across three levels of increasing complexity—pairwise cell discrimination, network-level separability, and spike sorting performance under realistic conditions. We hypothesize that magnetic field signatures may offer superior distinguishability due to three key differences: (1) magnetic fields decay more slowly with distance, potentially extending the spatial footprint of each neuron’s signal; (2) they are vectorial and sensitive to both position and orientation, unlike scalar electric potentials; and (3) they reflect longitudinal currents along axonal and dendritic branches, rather than being dominated by somatic transmembrane currents alone. To test this hypothesis, we begin by quantifying the fundamental spatial resolution limit: the minimum intercellular distance required to reliably distinguish two neurons based on their extracellular electric or magnetic signals.

We analyze the six previously described layer V neuron models by generating simulated spike templates for both the extracellular electrical potential, , and the three Cartesian components of the magnetic field, B_e,x, B_e,y, and B_e,z. For brevity, we refer to the extracellular potential as and the magnetic field components as B_x, B_y, and B_z henceforth. We define the base templates—which are spike templates in which the cell is located at the origin—as for neuron n and signal . These base templates are N_R × N_T matrices, where N_R denotes the number of spatial measurement points (sensors) and N_T is the number of time steps over the course of a single spike event. In our simulations, each neuron is positioned at the center of a dense grid of measurement points, with its major process oriented along the positive x-axis. The grid consists of 1,500 × 1,500 measurement points with a 2-m pitch, effectively approximating a continuous measurement space with near infinitesimal spatial resolution. This setup allows us to isolate the effects of neuronal morphology and field patterns, independent of constraints imposed by real-world sensor spacing.

We investigate how overlapping spike signatures from two neurons, separated by a displacement , may appear similar due to spatial proximity and orientation. To simulate this effect, we generate new spike templates from the base template by applying spatial transformations: a rotation by angle and a translation by (S5 Fig). These transformed templates are denoted as , where and represent the respective rotation and translation operators. To quantify the minimum resolvable intercellular distance, we introduce a cosine similarity metric that measures the similarity between the spike signature of neuron n and that of a spatially transformed neuron m for signal s:

(4)

By averaging over all rotation angles, we account for variations in neuronal orientation. A high value of , approaching 1, indicates that the spike signatures of neurons n and m are highly similar for signal type s at the displacement , suggesting that the two neurons produce nearly indistinguishable extracellular fields and are therefore difficult to resolve. Conversely, a low or negative signifies a high degree of dissimilarity between the spike signatures, indicating that the neurons can be more easily distinguished.

Under the assumption of infinitesimal pitch, the translation operator can be factored outside the integral, and the rotation integral interpreted as an orientation-averaging operator T_sweep, yielding

(5)

where

(6)

We refer to as the spread template, as it characterizes how neuronal signals disperse through space irrespective of orientation, providing a basis for evaluating neuronal distinguishability based solely on spatial separation. Fig 3a displays the spread templates for Cell 2 at the time corresponding to the maximum overall template signal power. For the magnetic field, the rotation transformation is applied directly to the full vector field B_e. Consequently, the vector-valued spread template is invariant to orientation. However, the individual components—specifically B_x and B_y—are not, as they depend on how the rotated field projects onto their respective sensing axes. For visualization, the electrical potential template is normalized to its maximum value, while the magnetic field components are scaled by the maximum value across all three components to ensure consistent relative amplitudes. This normalization results in a lower relative magnitude for the B_z component compared to B_x and B_y.

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Fig 3. Spatial resolution limits of simulated neural extracellular electrical potentials and magnetic fields.

a, Spread templates for cell n = 2 and signals . b, Spatial auto-similarity for cell n = 2 and signals . c, Mean spatial auto-similarity for Cells 0–5 from S1 Fig, averaged over displacements along both x- and y-axes within ±400 m. d, Effective radius for high () and moderate () similarity thresholds, averaged over all cell pairs (n,m) from S1 Fig.

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Fig 3b shows the auto-similarity, for Cell 2, evaluated for each signal type . This auto-similarity is essentially the convolution of with across the 2D space. The electrical potential exhibits high similarity with a radially symmetric distribution centered at the soma—consistent with the complementary information section, where we identified the soma as the primary source of transmembrane currents generating uniform-polarity extracellular signals. This symmetry is also evident in the spread template for (Fig 3a), indicating that neurons positioned around and near the soma produce highly similar extracellular potentials. In contrast, magnetic fields show lower spatial similarities. The planar magnetic fields B_x and B_y exhibit antisymmetric spread templates with respect to their sensing axes (Fig 3a), due to their sensitivity to neuronal orientation. For instance, rotating a neuron by 180° reverses the B_y signal, flipping the spread template across the y-axis. This leads to lower similarity and a triphasic spread template pattern slightly elongated along the sensing axis. The normal magnetic field (B_z) is even more sensitive to rotation and translation. Sensors on opposite sides of the neuron’s extended structure detect signals of opposite polarity, as shown in Fig 1b, and even small rotations around the soma or translations perpendicular to the neuron’s major process can render the B_z template nearly anti-similar with the original. Consequently, the normal magnetic field results in minimal spatial similarity, substantially lowering the separation limit for distinguishing neurons—outperforming planar magnetic fields and, even more so, electrical potentials.

Fig 3c presents the spatial auto-similarity, averaged over displacements along both the x and y axes within a range of ±400 m, for all six cells in S1 Fig. On average, electrical potential templates exhibit similarity values several orders of magnitude higher than those of magnetic field templates—even for cells n = 2, 3, 4, where extensive dendritic branching reduces radial symmetry. Planar magnetic field templates occasionally show non-negligible similarity or even anti-similarity, due to their antisymmetric spread templates and sensitivity to neuronal orientation. In contrast, normal magnetic field templates consistently exhibit near-zero similarity, making them particularly well-suited for recording and analyzing densely packed neuronal populations.

Finally, we quantify the spatial resolution limit—the minimum intercellular distance required for reliable discrimination—by defining an effective radius as the radius of a circle with an area equal to the region where the similarity exceeds a threshold :

(7)

Here, denotes the cardinality of the set. Fig 3d shows the effective radii averaged across all cell pairs (n, m) for two thresholds: 0.25 (high similarity) and 0.1 (moderate similarity). As expected, magnetic fields achieve higher spatial resolution than electrical potentials. Planar magnetic field templates yield effective radii approximately three times smaller than those of electrical potentials, while normal magnetic fields provide even greater resolution—up to five- to nine-fold improvements. These results indicate that magnetic sensors, especially those measuring the normal field, are fundamentally better suited for distinguishing neurons in densely populated regions than traditional electrical recordings.

2.4 Multi-source discrimination in practical sensor arrays

Having shown that magnetic fields offer superior pairwise discrimination under ideal spatial sampling, we now examine how this advantage scales to more realistic network-level settings. In particular, we investigate how sensor sparsity—an unavoidable constraint in practical arrays [43]—affects the ability to distinguish multiple simultaneously active neurons across space. Achieving broader coverage to monitor more neurons requires reducing sensor density, which in turn lowers spatial resolution and degrades the measurement quality for individual neurons. To evaluate how signal modality affects this trade-off, we adopt a metric from communication theory—the condition number—to quantify network-wide neuronal distinguishability. In addition to analyzing individual field components, we also examine combinations, such as , to assess whether integrating complementary electrical and magnetic recordings improves performance. In these multimodal configurations, each measurement location includes multiple sensors capable of recording different fields.

We use the condition number as a measure of signal separability, drawing from its established role in communication systems like massive MIMO and spread-spectrum CDMA [44,45]. In these systems, multiple overlapping signals are distinguished using non-orthogonal codes. If these codes are combined into a channel matrix, the condition number—defined as the ratio of the largest to smallest singular values of the channel matrix’s singular value decomposition—quantifies how well these signals can be separated. A value close to 1 indicates good separability, while a large value signals poor separation, especially in noisy environments. Similarly, if we create channel matrices consisting of the flattened spike templates of N_C randomly oriented and positioned neurons, we can compute the resultant condition number to assess how well different extracellular signal types support neuronal discrimination in crowded networks.

We use the condition number to evaluate neuronal distinguishability in two experimental scenarios. First, we fix the number of sensors and vary the sensor pitch—50, 75, and 100 m—in a 10 × 10 array to examine how increasing the coverage area affects distinguishability, while maintaining a constant cell density of 340 cells/mm² [46]. Second, we fix the sensor pitch at 75 m—thereby holding the coverage area constant—and vary the number of neurons to assess how many cells an array can distinguish within a given condition number threshold.

Fig 4a presents the trade-off between neuronal distinguishability—measured as the inverse of the condition number—and array coverage using channel matrices constructed from Cells 2 and 3 in S1 Fig. These neurons were chosen due to their strong, linear processes which can most clearly demonstrate the distinction between electrical and magnetic field information; an analysis of all cell types can be found in S6a Fig. Combining electrical potential and normal magnetic field measurements achieves the optimal trade-off, supporting our hypothesis that these signals provide complementary information. The B_z array significantly outperforms the array, aligning with our pair-wise analysis, whereas the B_x and B_y arrays under-perform the array, contradicting that analysis. This discrepancy arises as an artifact of the in vitro measurement setup because neuronal processes lying flat above a plane of sensors and aligned parallel with the sensing axis generate magnetic fields orthogonal to it (via the right-hand rule), making such neurons nearly undetectable. Measuring both planar components (B_x, B_y) mitigates this orientation sensitivity and yields performance comparable to the B_z-only array, which is inherently less susceptible to orientation-dependent signal loss on a planar array. Simultaneous measurement of all three magnetic components (B_x, B_y, B_z) captures additional information and further improves performance, closely approaching that of the hybrid array.

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Fig 4. Multi-source discrimination analysis of neural electrical potential and magnetic field templates.

a, Negative logarithm of the average condition number for channel matrices constructed from constant-density populations of N_C = 50, 113, and 200 cells (Cells 2–3 from S1 Fig) corresponding to sensor array pitch of 50, 75, and 100 m, respectively, for signal types , B_y, B_z, (B_x, B_y), (B_x, B_y, B_z), (, B_z)} (N = 30). The dashed arrow highlights the direction of a more favorable trade-off between neuronal distinguishability and coverage area. b, Maximum number of distinguishable neurons at various condition number thresholds, estimated via bootstrap resampling of the data shown in S6b Fig.

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Fig 4b presents results for the second experimental scenario, where the sensor pitch is fixed at 75 m and 20–200 neurons are sampled from Cells 2 and 3 from S1 Fig to assess the effect of increasing cell density. For each array type, we use bootstrapping of the data in S6b Fig to evaluate the maximum number of cells each array can distinguish at the same distinguishability, or condition number, threshold. At nearly all tested thresholds, the B_z array supports approximately twice as many neurons as the array, while the and (B_x, B_y, B_z) arrays support up to three to four times more neurons. However, these multimodal arrays require two to three times as many sensors as the conventional -only configuration. In contrast, the B_z array outperforms the array even when constrained to the same number of sensors. These findings suggest that B_z arrays provide an efficient and scalable solution for large-scale neural recordings, particularly under practical constraints such as limited data bandwidth and sensor fabrication complexity.

2.5 Spike sorting

Building on our previous findings—namely, that dense populations of cortical neurons are more readily distinguishable via their extracellular magnetic fields than their electrical potentials, and that combining complementary electrical and magnetic recordings further enhances separability—we now investigate how these metric-based differences translate to performance differences in realistic scenarios using in silico spike sorting. We simulate spike sorting using both in vitro and in vivo array configurations. The in vitro arrays are square, planar sensor arrays with varying pitches (S7a Fig) whereas the in vivo array replicates the configuration of a Neuropixel 1.0 probe [6] (S7b Fig). This setup allows us to compare sorting performance across different spatial layouts and experimental conditions. Simulated recordings are generated by randomly distributing a population of neurons across the arrays, summing and concatenating their Poisson-generated spikes, and subsequently adding noise. Spike sorting is performed using the Kilosort2 algorithm [47]. The full sorting pipeline is shown in S8 Fig. Performance is evaluated using two metrics: overall sorting accuracy, defined as the match rate between predicted and ground truth spike trains, and the number of well-detected cells, defined as units with match rates exceeding 80% [48]. These well-detected cells, which typically correspond to one-to-one matches with ground truth neurons, reduce the need for manual curation, such as merging or splitting spike templates [49].

Fig 5a presents the trade-off between spike sorting accuracy and coverage area for in vitro arrays with a fixed number of sensors. Across all coverage areas, the normal magnetic field B_z consistently yields higher sorting accuracy than the electrical potential , while the planar magnetic components B_x and B_y perform worse than except at the smallest coverage area. To further examine these trends, we evaluate how recording duration and noise level affect sorting performance. As shown in Fig 5b, the relative ranking of spike sorting accuracy for all signal types remains consistent as a function of these variations. As expected, accuracy improves with longer recording durations and degrades with increased noise across all signal types. While the performance of and B_z declines gradually with increasing noise, the degradation in B_x and B_y is significantly steeper, reflecting the same pronounced sensitivity to coverage area (Fig 5a).

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Fig 5. Simulated spike sorting for in vitro neural probes.

a, Spike sorting accuracy for as a function of sensor pitch (50, 75, and 100 m; corresponding to 50, 113, 200 cells at constant density; 60 s duration, 15 noise level). The noise level for magnetic templates is in units of tesla and is normalized such that their average SNR matches that of the template. b, Sorting accuracy as a function of recording duration (20, 60, 120 s) and noise level (5, 15, 30 ) for each signal type (200 cells, 75 m pitch). c, Percent difference in spike sorting accuracy relative to for signal types as a function of cell count (75 m pitch, 60 s duration, 15 noise level). d, Percent difference in the number of well-detected cells (match rate >80%) under the same conditions as in c. N = 10 for all panels.

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These findings align with our earlier multi-source discrimination analysis, in which B_x and B_y exhibited significantly larger condition numbers—indicating poor signal separability—while the normal magnetic field B_z is consistently better conditioned. Although noise was not explicitly considered in that analysis, the sharper decline in sorting accuracy for B_x and B_y under noise reinforces their ill-conditioned nature. This demonstrates that although poorly conditioned channels are more susceptible to noise, their performance can be improved with sufficiently small sensor pitch. Overall, these findings support the condition number as a strong predictor of spike sorting algorithms’ ability to reliability separate and identify neural signals in practical settings.

Fig 5c evaluates how increasing cell density within a fixed coverage area affects sorting accuracy for in vitro arrays. To facilitate comparison, we plot the percentage difference in spike sorting accuracy between each signal type and the array. At low cell densities, the B_z and (B_x, B_y, B_z) arrays perform comparably to the array; however, at the highest evaluated density, they yield nearly a 20% increase in sorting accuracy. In contrast, the B_x and B_y arrays consistently underperform the array, while the and arrays remain comparable to it. A similar trend appears in Fig 5d, where the B_z and (B_x, B_y, B_z) arrays yield 20–30% more well-detected cells than the array, while other multimodal arrays show marginal or no improvement.

These results reaffirm the effectiveness of measuring the normal magnetic field component B_z, alone or in combination with other fields, which is consistent with our distinguishability analysis identifying B_z as best suited for recording from dense neural networks. However, multimodal arrays—particularly —underperform relative to predictions from the distinguishability analysis. This discrepancy likely reflects limitations of the spike sorting algorithm rather than the signal content itself. Since Kilosort incorporates sensor position into its model, it may struggle with co-located heterogeneous signals. Additionally, the two- to three-fold increase in sensor count for multimodal arrays may further challenge algorithmic performance.

Next, we turn to in vivo settings, where probes sample from a three-dimensional (3D) volume (S7b Fig), and only neurons near the probe are reliably detected, while distant cells contribute background noise [50]. Because sorting accuracy depends on the number of detectable neurons, we define an effective sensing radius—the distance beyond which neurons are excluded and treated as noise—and evaluate sorting accuracy across radial distances (S7c Fig). Fig 6a shows that the sorting accuracy for individual components (, B_x, B_y, B_z) decreases linearly with radial distance, dropping below 25% between 75 and 100 m. We therefore define 100 m as the effective sensing radius, using 25% as a conservative threshold below which spike assignments are considered unreliable. To determine whether neurons beyond this radius should be treated as noise, we vary their number within a probe distance between 100–200 m. Sorting accuracy remain largely unaffected across signal types, supporting the exclusion of biological noise from further simulations (S9 Fig).

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Fig 6. Simulated spike sorting for in vivo neural probes.

a, Spike sorting accuracy as a function of distance from the probe for (400 cells, 120 s duration, 15 noise level). b, Percent difference in sorting accuracy relative to for signal types , as a function of cell count (120 s duration, 15 noise level, 10–100 m cell radius). c, Percent difference in the number of well-detected cells under the same conditions as b. N = 10 for all panels.

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Unlike the in vitro case, where we examine trade-offs between sorting accuracy and both coverage and cell density, the in vivo analysis focuses solely on the trade-off with cell density, as coverage is fixed by the probe geometry. As shown in Fig 6b, individual magnetic field arrays achieve up to 25–35% higher sorting accuracy than the array, even at the highest evaluated cell densities. In contrast, multimodal arrays underperform relative to the individual components, likely due to the same algorithmic limitations observed in vitro. While the (B_x, B_y, B_z) array provides a modest 10% improvement over , the array performs significantly worse. Fig 6c shows that well-detected cell counts follow similar trends.

Additionally, while the normal magnetic field B_z continues to outperform as in the in vitro case, it is no longer a clear outlier relative to the planar components. This shift likely stems from the three-dimensional cellular arrangement in vivo, where a neuron’s magnetic field may be oriented either normal or tangential to the probe surface—unlike the in vitro setting, where neurons predominantly lie in a plane orthogonal to the B_z sensing axis.

Together, these results demonstrate that the theoretical advantages of magnetic field sensing for neuronal discrimination extend to practical spike sorting scenarios, even when using algorithms originally developed for electrical potential recordings. Across both in vitro and in vivo settings, the normal magnetic field component B_z consistently enhances sorting accuracy and increases the number of reliably identified neurons, particularly in dense neural populations. While performance gains are partially limited by algorithmic assumptions–especially in multimodal, co-located configurations—the robustness of B_z across spatial scales and noise conditions highlights its value for next-generation neural interfaces. Future improvements may be realized through dedicated algorithmic development tailored to the unique characteristics of vectorial and composite magnetic recordings.

2.6 Morphological reconstruction

Thus far, we have examined how fundamental differences between neural electrical potentials and magnetic fields affect the separability of cells within large populations. We now shift focus to the single-cell level, exploring how the distinctive properties of neural magnetic fields can also inform the structural characterization of individual neurons. In particular, we explore in vitro morphology reconstruction, the task of estimating the spatial configuration of a neuron and its major processes. Traditionally, this task requires dense electrode arrays with inter-electrode spacing below 20 m, along with complex algorithms or antidromic stimulation, due to the small radius and weak extracellular potential signal of neuronal processes relative to the soma [51–53]. Here, we investigate whether magnetic field measurements can facilitate morphology reconstruction using sparser arrays, which are more compatible with large-scale network recordings.

As shown in Fig 1b, a planar in vitro array measuring normal magnetic fields from a closely positioned cell shows opposite polarities on either side of a major process, allowing its localization by detecting the boundary between positive and negative versions of the extracellular spike waveform. To demonstrate the feasibility with which this boundary can be estimated, we repurpose support vector machines (SVMs) for single-shot estimation—deviating from their conventional use in classification—and refer to this approach as General Neural Boundary Estimation (GNBE). SVMs are well-suited due to their off-the-shelf availability, training-free use in this context, and ability to compute contiguous, non-linear hyperplanes that better match the geometry of realistic neural processes compared to alternatives such as logistic regression or random forests [54]. In GNBE, a noisy B_z spike template—representing a single neuron’s estimated signal after spiking sorting—is input into the SVM (actual spike-sorted templates are not used so that the added noise in the template can be precisely controlled). Using sensor locations and signal polarities, the SVM computes a boundary curve interpreted as the dominant process’s orientation. This estimated boundary is then compared to the ground-truth morphology, as illustrated in Fig 7a. To assess feasibility, we simulate neurons on planar sensor arrays across varying sensor pitches and signal-to-noise ratios (SNRs), reflecting different levels of template quality.

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Fig 7. Morphology reconstruction for simulated B_z in vitro sensor arrays.

a, Example of a decision boundary computed using the GNBE method, overlaid on the ground-truth neuron morphology. Spike signals at each sensor are shown, with positive signals in indigo and negative in pink. Inset shows how magnetic field polarities differ on opposite sides of the soma (top versus bottom) due to opposing longitudinal current flows. b, Reconstruction error using the GNBE algorithm for B_z templates with 0, 10, and 20 dB average SNR, across 20 × 20 sensor arrays with 50, 75, and 100 m pitch. Dashed lines indicate corresponding values of sensor pitch. c, Average reconstruction error as a function of distance from the soma along the apical dendrite, comparing GNBE and CNBE methods for 20 dB-SNR, 50 m pitch sensor array. d, Reconstruction error versus SNR for a 20 × 20 B_z array with 50 m pitch, comparing GNBE and CNBE methods. e, Percent reduction in reconstruction error achieved by CNBE method under the same conditions as in a. N = 300 randomly oriented instances of Cell 2 from S1 Fig. Standard error is plotted for b, d, e, and standard deviation is shown for c.

https://doi.org/10.1371/journal.pcbi.1014283.g007

Fig 7b shows the reconstruction error, defined as the average distance between each neuronal segment and the predicted boundary curve, as a function of template SNR and sensor pitch. For moderate (10 dB) and high (20 dB) SNRs, the reconstruction error is smaller than the sensor pitch for all pitch values tested, demonstrating super-resolution accuracy. At low SNR (0 dB), performance degrades considerably but improves with decreasing senor pitch.

While off-the-shelf GNBE-based reconstructions can attain high accuracy, they also exhibit consistent inaccuracies near the soma due to branching dendrites connected to the soma and oriented away from the primary neural process generating magnetic fields that rotate in the opposite direction (Fig 7a inset). To address this, we introduce a second-pass algorithm, referred to as Customized Neural Boundary Estimation (CNBE), that leverages known properties of cortical magnetic fields. CNBE proceeds in four steps: apply GNBE to obtain an initial boundary estimate; use this estimate to approximate the soma location and the direction of the major process; discard sensors on the side opposite the major process—typically associated with dendritic fields (Fig 7a, insert)—and re-estimate the boundary using only the remaining sensors. Full algorithmic details are provided in the Methods section.

Fig 7c shows the reconstruction error along the major process as a function of distance from the soma, comparing GNBE and CNBE algorithms for a 50 m-pitch array. CNBE effectively eliminates the elevated error near the soma observed with the naive method. Fig 7d summarizes reconstruction error across all SNR levels for both algorithms. With CNBE, the error is reduced to approximately 20% of the pitch at moderate and high SNR, with modest improvement even at low SNR. Fig 7e quantifies the percentage reduction in reconstruction error achieved by the CNBE algorithm across all sensor pitches.

These results demonstrate that approximate morphology reconstruction at the cellular level, even using off-the-shelf tools as in GNBE, is enabled by the unique solenoidal properties of perpendicular neural magnetic fields. Furthermore, tailoring these tools to the known geometrical properties of neurons, as in the CNBE approach, yields significant improvement for dense arrays or high-SNR templates. While CNBE performs poorly on sparse arrays or low-SNR templates, as it then amplifies rather than corrects GNBE errors, it highlights the potential for more complex, custom algorithms to achieve even greater reconstruction accuracy. Neither GNBE nor CNBE are exhaustive — in particular, they are unlikely to work well for cells without major longitudinal processes, such as stellate cells or anaxonic neurons, as there is no single major boundary to estimate. However, their efficacy in this limited example provides an intuitive demonstration of how the unique informational content of perpendicular neural magnetic fields may facilitate new techniques.

4.1 Neuron simulation and extracellular calculations

To calculate neural action potentials we used the NEURON simulator library [58], and to calculate extracellular electrical potentials we used the line-source approximation [59] within the LFPy module [60]. Spikes were elicited by injecting current into the soma using NEURON’s IClamp point process. The IClamp stimulus was weighted to generate a burst of spikes, and the first spike was omitted from all further analyses to eliminate artifacts from the stimulus itself. Current dipoles were computed for all cells as a sanity check using the method in [61] and compared to the values from that work. We obtained dipoles between 0.1–0.9 pA m, which is in agreement with the literature.

We used the MEArec package [62] to generate simulated extracellular electrical potential recordings from large populations and modified it to calculate and incorporate extracellular magnetic field data. Reference electrodes were assumed to be located at infinity. Magnetic field calculations were made using the Biot-Savart Law approximation [63] updated for a neuron model as shown in Eqn. 8, which describes the magnetic field at an extracellular coordinate r in terms of the neuron compartmental axial current I_a, direction vector d, and relative position vector (pointing from the compartment to the calculation point) for each compartment n.

(8)

For both in vitro and in vivo arrays, cell morphologies were modified so as to neither intersect the plane of the array nor extend, unrealistically, upwards into space. For in vitro arrays, all cells were flattened to within a z-dimension of 5–25 m above the plane of the sensors. For the in vivo arrays, all cellular processes that would have intersected the shank were made to instead wrap around the nearest shank boundary. Cells were also flattened so as to not cross within 5 m of the front-face of the shank.

4.2 Similarity calculation and template creation

Simulated cells were suspended 15 m (soma as the origin) over a 1,500×1,500 grid of measurement points with 2 m pitch and flattened. This setup was intended to model a traditional in vitro electrophysiology experiment but with near-infinitesimal resolution. The signal traces of each sensor for the spike duration (224 timesteps, dt = 31.25 s, 7 ms total) were combined to form a spike template for each neuron n and signal type s (2,250,000×224, n = 0–5, ). To create , we defined the T_sweep transformation over from 0 to 360° in 15° increments. Translation coordinates for were defined over a coarse grid from (-400 m, -400 m) to (400 m, 400 m) with 50 m pitch and a fine grid from (-200 m, -200 m) to (200 m, 200 m) with 20 m pitch (in the x-y plane).

In generating templates for rotated cells, we had to contend with the fact that rotated sensor coordinates no longer necessarily aligned with the original sensor coordinates. Therefore, after obtaining the rotated coordinates, the K-nearest rotated sensors (K = 5) of each original sensor were determined. Then, the new template value at each original sensor location was calculated as the average of the data at the K-nearest neighbors inversely weighted by distance as described in (9) (for data p_k at the K-nearest neighbors with distance d_k and weights w_k).

(9a)

(9b)

4.3 Spike sorting recording generation

To generate the simulated recording data for spike sorting, we first created template libraries in which a sensor configuration was chosen (such as a 10 × 10 planar grid of sensors with 50, 75, or 100 m pitch, or a Neuropixel 1.0 probe), a cell type was specified (Cell 2 or 3 from S1 Fig), and then n different spike templates of signal s—for —for that cell were generated by randomly varying the location and rotation angle of the cell. For the Neuropixel, cells were placed only within some radius in front of the probe (volume defined by a half-cylinder) due to the inability of our simulation environment to adequately model transmembrane currents originating behind the probe (which would theoretically travel around it to reach microelectrodes rather than directly through it). All cells were flattened and/or wrapped around the sensor array.

Following the generation of a template library for each cell for a specified sensor configuration, N_C cell templates were randomly sampled across the combined templates of all selected cell types to represent the N_C neurons being simultaneously measured by the probe. A Poisson spike train was generated using MEArec for each template with a rate chosen from a normal distribution (mean = 5 Hz, standard deviation = 1 Hz) and the resultant extracellular signals were then combined to produce a recording. Spikes violating a set refractory period of 2 ms were removed.

Gaussian thermal noise was generated and added to voltage recordings according to a set noise power value. The range of noise powers used was chosen based on fundamental thermal noise properties of resistors. A typical impedance value (at 1 kHz) for an electrode with a size on the order of 10 m is generally between 100 kOhm to over 1 MOhm depending on the electrode material [6,64,65]. The power spectral density of thermal noise from a resistor is 4kTR, so for an electrode measuring a signal in a 32 kHz bandwidth (equivalent to our dt for spike generation; data was band-pass filtered during spike sorting), this impedance range approximately corresponds to a noise RMS voltage range of 7.35 to 32.9 . We chose to simulate from 5 to 30 . While realistic electrode impedances scale as a function of frequency, assuming a constant resistance provides us with a simple and conservative estimate. Notably, the Neuropixel 1.0 probe reports approximately 7.1 noise for an in vivo implant within a 300 Hz to 10 kHz bandwidth [66]—this corresponds to 12.9 when the bandwidth is changed to 32 kHz. We therefore chose 15 as our default noise level for the in vitro and in vivo simulations.

For magnetic sensor noise, to make a fair comparison considering the difference in units—volts vs. tesla—the average signal power across the sensors for the entire recording duration was calculated for each magnetic field component (B_x, B_y, B_z) and the data. The magnitude of the magnetic field signals was then scaled up so that the average signal power of the magnetic field data matched the average signal power of the voltage data (numerically, ignoring units). After the additive sensor noise was included, each field had an equivalent average SNR. This also ensured that the scale of the signals was approximately identical when fed into the sorting algorithm which was important to minimize unwanted variance. For multimodal measurements, the fields were scaled according to the signal type with the largest average power (so as to preserve the relative magnitudes of each field type).

4.4 Simulated spike sorting

Kilosort2 [47] was used within the SpikeInterface [48] framework for spike sorting. The only significant modification we made to the Kilosort algorithm was with regards to spike detection based on threshold crossing: whereas the original algorithm detected only negative spikes, we set it to detect either negative or positive spikes for magnetic fields. Implementing this change for voltage signals worsened the overall performance (the largest spikes, usually near the soma, typically have negative polarity), and so was not included, but it was essential for magnetic fields given that their largest spikes can have different polarities depending on the direction of longitudinal currents relative to the sensor positions.

We performed a grid search across Kilosort2’s detection threshold and minimum spiking frequency variables for each combination of recording parameters—recording duration, number of neurons, noise level, and sensor configuration—to attempt to find the close-to-optimal sorting performance for each parameter set. We randomly generated N = 10 recordings for each combination of parameters and then, for each recording, determined the average sorting accuracy, defined as true positive / (true positive + false positive + false negative) for predicted spikes, and the number of well-detected units, defined as having greater than 80% match between their ground-truth spike train and spike-sorted spike train [48]. The highest-accuracy sorts and highest well-detected-count sorts for each signal across the grid search were isolated and compared to get an estimate of the upper bound of sorting performance per field type, regardless of sorting condition. The full sorting pipeline is shown in S8 Fig.

4.5 Morphology reconstruction modified algorithm

The CNBE algorithm to enhance the performance of the GNBE approach for morphology reconstruction proceeds as follows. First, a naive SVM boundary is calculated. Next, the location of the axon hillock (the region of the cell typically responsible for the strongest magnetic field signal [67]) is estimated by finding the weighted center of the coordinates of the 4 sensors that measured the signals with the greatest average power. This procedure is repeated for the next 96 highest power sensors, producing 24 points that were estimated to lie close to, or in the general direction of, the neuron’s actual morphology. The 24 points along the naive SVM boundary lying closest to these points are then determined, and “outliers”—with an average distance to their 4 nearest neighbors greater than 50 m—are removed. A line of best fit that runs through the remainder of these 24 points and the axon hillock coordinate is then calculated (strongly weighted to pass close to the axon hillock). A line perpendicular to this best fit line and lying 10 m “behind” the axon hillock (the opposite direction of the vector pointing from the axon hillock to the 24 points) is then constructed—any sensor behind this line is then excluded from a second SVM calculation, which produces an updated morphology estimate.