2.1.1. Generalized reciprocity approach.

BlueRecording permits the calculation of extracellular signals using the reciprocity theorem [6]. The reciprocity theorem establishes a simple relationship between the electric field induced at different locations in the brain by applying a current through two surface electrodes, and the voltage difference measured between these electrodes when placing a current source at the same brain locations. Intuitively, if a source at location A produces a voltage at location B, a source at location B will produce an identical voltage at location A – hence the term “reciprocity”. As a result, measurable signals from distributed and dynamic brain activity can be computed using a single electromagnetic simulation, rather than needing a distinct simulation per independent source activity component.

More formally, following Plonsey [6],let us suppose that we have a a volume current source I_U(t) in an arbitrary dielectric environment U (in the case of a neural current source, I_U(t) is the set of neural segment currents , where i is the index of each neuron and j is the index of each segment within a neuron). Further, let be the potential field resulting from that current source I_U(t). Now, suppose that instead of a volume current source, we have a surface current distribution K on the surface of the environment U. Let be the potential field resulting from K.

It can be shown (for a full derivation see [6]) that .

When K consists of a pair (a,b) of quasi-perfect electric conductors (e.g., metallic EEG electrodes) providing a (virtual) current J, the situation simplifies to , where corresponds to the measured signal.

For a neurite with electrodes sufficiently far away, we can treat each nodal current as point current sources at points , respectively. Thus, the left side of the equation reduces to and the potential between the two electrodes in this situation becomes

(2)

The coefficients C_j used to calculate the extracellular signal as described in Eq 1 are therefore given by

(3)

In order to calculate , a finite element method (FEM) simulation of a head model with a recording electrode array is performed, where a (virtual) current is applied between a pair of recording electrodes, and the potential field is interpolated at the locations of the neural segments . Information about the FEM rat head model setups created for this study is provided in S1 Methods.

2.1.2. Dipole-based reciprocity approach.

If, for each neuron i, we take the Taylor series expansion of around the vector – the geometric center of the neuron – we obtain, to first order

(4)

Substitution Eq 4 into Eq 2 and rearranging, we obtain

(5)

For a neural current source, by Kirchhoff’s Law. Thus,

(6)

where is the current dipole calculated for neuron i at each time step as

(7)

The normalized is the ‘lead-field’. can be computed using a FEM model, as described in S1 Methods, and is interpolated at the geometric center of each neuron, . Substituting into Eq 1 and rearranging, we find that

(8)

While the choice of is not relevant from an analytical perspective, it can be numerically advantageous to place it near the neuron center.

2.1.3. Calculation of coefficients with line-source approximation.

Unlike EEG or ECoG electrodes, LFP electrodes are small and located within the cortical tissue. If the LFP electrode can be treated as point-like, the tissue within the relevant environment is homogeneous and isotropic, and the ground reference sufficiently distant to be treated as being at infinity and omnidirectional, the computation of an ‘exposing’ potential required by the reciprocity theorem can be avoided and instead the line-source approximation can be used to compute the coefficients for LFP calculations without need for FEM simulation.

According to the line-source approximation [18], the contribution to the electric potential at a point electrode location of a neural segment situated in a homogeneous, isotropic environment is given by , where is the length of the segment, h is the distance from the near end of the segment to the recording electrode in the axial direction, and r is the absolute value of the distance from the near end of the segment to the electrode in the perpendicular direction. Evaluating the integral analytically and substituting into Eq 1, we find that:

(9)

With and as the endpoints of the segment and (x,y,z) as the location of the electrode,

(10)

(11)

(12)

2.1.4. Calculation of coefficients with point-source approximation.

A further simplification can be made (in addition to the assumptions of an infinite homogeneous medium, a point-like electrode, and reference at infinity) by treating each neural compartment as a point rather than a line. In this case, the potential at the recording electrode is given by

(13)

where is the position of the electrode and is the position of the compartment. Substituting into Eq 1 we obtain:

(14)

2.1.5. Choice of voltage and spatial reference.

The extracellular signals produced using the reciprocity approaches are gauge-independent with respect to the potential field (i.e., the location to which a potential of 0V is assigned is arbitrary). This is because the sum of membrane currents of the compartments over a neuron must be zero. Thus, any constant added to the compartment weights does not affect the signal, and only the difference in compartment weights matters. For better visualization, we can therefore add an offset to the compartment weights in each neuron such that the minimum weight per neuron is 0.

2.1.6. Numerical considerations.

Calculation of the extracellular signal involves summation over a large number of terms that typically compensate, potentially leading to important numerical errors as a large number of digits can be significant. Because of gauge freedom and current conservation, this can be mitigated by selecting good reference values for and , and if that proves insufficient, by using a compensation algorithm such as Kahan summation. Our implementation does not use Kahan summation, as we have not found it to be necessary for the current study. However, awareness that compensated summation might be required in other situations, is necessary.

2.3.1. Extension to SONATA-format.

We defined an extension to the SONATA format [9] to store the coefficients C_i,j. Briefly, the SONATA format treats each neuron in the network as a node, and nodes are organized into populations. The coefficients are stored in an HDF5 file (Table 1), which also includes metadata on the type and location (both in Cartesian space and with respect to the anatomy) for each electrode. For each population, the coefficients for all nodes and all electrodes are listed in a single array; for each node, the coefficients (one per compartment) are listed consecutively. For each population, there is also a dataset listing the IDs of each node therein, as well as the index in the coefficient array at which the coefficients for that node begin.

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Table 1. The format of the weights file

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2.3.2. Electrode array file format.

The electrode array for which the coefficients are calculated is defined in a csv file. The csv file begins with the header name,x,y,z, layer,region,type. Each column of the file is described in Table 2. Each row corresponds to an additional electrode. The electrode csv file is read when the weights file is created, in order to populate the metadata.

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Table 2. The format of the electrode csv file

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2.3.3. Generation of weights files.

Discretization of the neurons into electrical compartments is performed at runtime. Therefore, to extract the number of compartments per neuron, a single-timestep simulation is run, from which a compartment report is produced. For each compartment, which are uniformly spaced, the position in Cartesian coordinates is interpolated from the 3d points in the morphology file, which do not necessarily correspond to the discretization.

After interpolation of segment positions, the H5 file defined in Sect 2.3.1 is created, with all coefficients initially set to 1. In a subsequent step, coefficients are computed, e.g., based on interpolated potential field calculated using Sim4Life (ZMT Zurich MedTech AG, Zurich, Switzerland; c.f. S1 Methods), or using the line-source approximation (c.f. Sect 2.1.3) or the point-source approximation (c.f. Sect2.1.4).

While the weights files are intended for runtime calculation of extracellular signals in the Neurodamus simulation control application (see Sect 2.3.4), it is also conceivable that other simulators, such as LFPy [19] or HNN [1], could be extended to accept BlueRecording weights files. This would allow these simulators to use the full set of extracellular signal calculation methods that BlueRecording provides, particularly the generalized reciprocity approach, which is not available in other simulators. Alternatively, tools such as LFPy could be extended to produce weights files in the format described in Table 1, which could then be used with Neurodamus (see below). As LFPy provides signal calculation methods that are not available in BlueRecording (c.f. Sect 3.1.2), this would allow those methods to be used in conjunction with the efficient neural simulation toolchain used by BlueRecording.

2.3.4. Online calculation of EEG/LFP signals.

The calculation of extracellular signals is a multi-step process that begins with the launch of a Neurodamus [7] simulation. If a weights file, as defined in Sect 2.3.1, is present in the SONATA simulation config file, the weights and configuration are loaded, providing the necessary information for the subsequent extracellular signal calculation.

For each neuron distributed to a given processor, Neurodamus iterates through their corresponding sections, loading the factors associated with each section from the weights file. Neurodamus calls Neuron’s nbbcore_register_mapping function, which writes the factors to an NrnThreadMappingInfo object – a Neuron datatype which stores (among other data) the segment ids and corresponding LFP factors for each neuron on a given thread. On each thread at runtime, for each timestep CoreNEURON (the compute engine used by the NEURON simulation environment [8]) iterates through the neurons in the NrnThreadMappingInfo object, and calculates the contribution to the LFP for each neuron by summing the products of the segment transmembrane currents and their corresponding segment weights. The calculation of the contributions of the neurons to the extracellular signal is performed independently on each thread; these individual contributions are then written to a SONATA report file as described in [9]. We estimate that the online calculation of the extracellular signals adds a computational cost of 14% for 35 electrodes.

BlueRecording was developed for the CoreNEURON engine, and is not compatible with the legacy NEURON engine. Therefore, BlueRecording cannot be used in simulations of extracellular electrical stimulation or ephaptic coupling, which rely on NEURON’s extracellular mechanism, which is not supported by CoreNEURON. However, rather than modeling ephaptic coupling explicitly, it is also possible to model the effects of ephaptic coupling by injecting a current into the neuron, corresponding to the impact of the activating function. In this way, it is theoretically conceivable that BlueRecording could be used to model ephaptic coupling. The extracellular potential induced by a current from each of the n neural compartments would need to be calculated at the location of every other neural compartment, thus requiring weights to be computed. In a heterogeneous dielectric environment, this would require as many finite element simulations as there are neural compartments, which would be computationally infeasible. However, if we assume a homogeneous environment, then an analytic solution can be used to calculate the weights. It may be possible to reduce these computations to a reasonable degree of complexity using, for example, the fast multipole method. Injecting the appropriate currents into each compartment would also require the creation of a new current clamp mechanism, which would need to have access to the recorded extracellular potential. Thus, the implementation of ephaptic coupling using BlueRecording is conceivable but technically very difficult.

2.3.5. Reports.

For storing simulation reports, the SONATA format [9] is utilized, which organizes the data in an HDF5 file (Table 3). This format is specifically designed to handle various types of data related to neural simulations, providing a structured and efficient way for storing and retrieving the data.

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Table 3. The format of the simulation reports

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For extracellular reports, a unique aspect of the SONATA format is the use of the field “element ids” to represent electrode IDs, as opposed to representing compartments as in NEURON. This means that reports capture the electrical activity at specific electrode locations, rather than at individual compartments within the neural model.

An extracellular report is requested by adding an lfp block to the report section of the simulation configuration file (see [9] for a full description of this file).

"reports":{

 "report_name":{

  "type":"lfp",

  "cells":"Mosaic",

  "variable_name":"v",

  "dt":0.1,

  "start_time": 0.0,

  "end_time": 5000.0,

  "sections":"all"

 }

}

The name-value pairs "type":"lfp" and "variable_name":"v" must be present verbatim; the others are user-configurable.

3.1.1. Unit tests.

Unit tests, with 70% code coverage, are provided in our Github repository, in the folder tests. These ensure, for example, that the weights files are outputted with the correct format, that segment positions are interpolated correctly for a morphology with 10 sections, and that weights are calculated correctly for a simple two-compartment model. They can be run with pytest.

3.1.2. Comparison of calculation methods.

We provide a small verification example in our Github repository (https://github.com/BlueBrain/BlueRecording), comparing the generalized reciprocity approach, dipole-based reciprocity approach, line-source approximation, and point-source approximation, in a large homogeneous medium. We simulate the signal from a single Layer 5 pyramidal cell from the BBP somatosensory cortex (SSCx) model. The cell is driven with Ornstein-Uhlenbeck noise with an amplitude of 100% of the spiking threshold and a standard deviation of 1% (c.f. S2 Methods). We chose such a low standard deviation in the noise process to facilitate visualization of the differences between the signal calculation methods. In principle, a deterministic input could also have been used, but as the goal of this experiment is to verify that our calculation methods perform as expected (insofar as the differences between them match theoretical considerations), and given that Ornstein-Uhlenbeck noise input is used in larger-scale simulations described in Sects 3.2–3.4, we elect to use Ornstein-Uhlenbeck noise here as well.

The noise input triggers an action potential in the neuron, which is observed as a large transient in the recorded signal (Fig 2A–2B). Based on theoretical considerations outlined in the introduction, we expect a weaker signal and disappearing differences between the approaches at larger distances. We confirm that when the recording and reference electrode are located far from the neuron (30 mm and 40 mm, respectively), the signals recorded using each of the methods are very similar (Fig 2A). In contrast, when the recording electrode is placed 218.5 m from the neuron, the signals recorded using the different methods begin to diverge (Fig 2B), with the signal calculated using the dipole-reciprocity method diverging the most. This is the results of differences in the weights calculated for individual compartments (Fig 2C and 2D). For the electrode closer to the neuron, the full reciprocity approach yields more positive weights (relative to the soma) for the apical tuft and more negative weights for the basal dendrites than the simplified dipole approach. The higher polarity between tuft and basals explains the larger amplitude of the signal observed.

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Fig 2. Extracellular signals recorded in a large homogeneous medium.

A: Signals recorded with recording electrode (not visible in panel C) distant from the neuron. B: Signals recorded with recording electrode (green dot in panel D) near the neuron. C: Difference in per-compartment weight between generalized reciprocity and dipole-based signal calculations, for electrode far from the neuron (adjusted for constant offset, and normalized to the range of compartment weights in general reciprocity approach). D: The same, for electrode close to the neuron.

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We further observe, as expected, that the signal recorded with distant electrodes (Fig 2A) has substantially lower amplitude than the signal recorded with nearby electrodes (Fig 2B). We do not expect that the former signal would be detectable in the presence of activity from other cells, but as the intent of this experiment is merely to compare the outputs of the different signal calculation methods, we do not consider this a limitation.

3.2.1. Utility of the generalized reciprocity approach.

We compare the EEG, ECoG, and LFP signals obtained with the generalized reciprocity approach to those obtained with the dipole-based reciprocity approach and the line- and point-source approximations (Fig 5A–5C). The dipole-based approach overestimates the amplitude of the EEG and ECoG signal by a factor of 1.5 relative to the generalized reciprocity approach (Fig 5A). This is attributable to differences in the contribution from the upper lip and hindlimb regions (Fig 5D). These differences can in turn be explained by differences between the generalized and dipole approach in the compartment weights in the upper lip and hindlimb regions (Fig 5G vs. 5J). The differences in compartment weights between the two methods arise due to the spatial heterogeneity of the E-field on the scale of the transmembrane current sources represented by a single dipole, which is accounted for by the generalized reciprocity approach (based on the finite element model of a virtual current applied between the recording electrodes), while the dipole approach assumes negligible field homogeneity , which can be inaccurate (S2 Fig). The field heterogeneity depends on the geometry of the dielectric environment.

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Fig 5. Simplified calculation methods produce inaccurate signals.

EEG (A), ECoG (B), and LFP (C) signals, recorded over, or within, the somatosensory cortex, calculated with the point-source and line-source approximation, with the generalized reciprocity theorem (ground truth), and with the simplified dipole-based approaches. D-F: Contribution of each region to EEG, ECoG, and LFP signals, respectively. Solid lines indicate general reciprocity approach, dashed lines indicate dipole approach. G-I: Weights for EEG, ECoG, and LFP recordings, respectively, calculated using the reciprocity approach, for a sample of L5 pyramidal cells in the forelimb (compartments represented as circles), hindlimb (compartments represented as triangles, and enclosed in pink box) and upper lip (compartments represented as squares, and enclosed in red box) regions. Electrodes are represented as red circles. Note the varying color scale ranges. J-L: As in G-I, but calculated using the dipole approach

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Surprisingly, the dipole approximation provides a better fit to the generalized reciprocity signal for ECoG (Fig 5B) than for EEG. This is because, in ECoG, the dipole approximation induces a larger error in the signal contribution from the forelimb region, in the opposite direction as the error in the contribution from the other regions (Fig 5E). Thus, while the general reciprocity and dipole approaches produce similar signals, their interpretation is very different.

For LFP recordings, not only does the dipole-based approach overestimate the signal amplitude – it also changes the shape of the signal, leading to a loss of information and reduced interpretability. A smaller change in shape is also observed for the point- and line-source approximations (Fig 5C).

While the line-source and point source approaches provide a reasonable approximation of the EEG and LFP signals obtained using the generalized reciprocity approach (Figs 5A and 5C), they lead to even greater error than the dipole approximation for the ECoG signal. Therefore, the generalized reciprocity approach is the only method capable of producing realistic results in all cases.

We emphasize that the dipole reciprocity approach is derived from the generalized approach under the additional simplifying assumption that the E field is constant over the range of the neural source, which is not true near the electrode (S2 Fig). The point and line source approaches similarly involve approximations that are not required by the reciprocity-based approaches, namely, that the extracellular medium is infinite and homogeneous, and that the recording electrodes are infinitesimally small. The generalized reciprocity approach, which does not rely on such simplifications, is inherently more accurate. While the line-source approach has the benefit of accounting for the finite extent of neural segments, it is evident, e.g., from Fig 2B, that the associated error is negligible, even relatively close to neurons.

Several previous studies have investigated the adequacy of different methods for calculating extracellular signals in neural models. It has been established that, even for homogeneous tissues, within a millimeter of a neuron, the dipole approximation can produce substantial errors, but that it holds for electrode positions several millimeters from the neuron [5]. This is in accordance with our observations in Figs 2, 5C and 5F.

Naess et al. [5] found that in an analytically solvable four-sphere human head model, the dipole reciprocity approach performed well for EEG, but resulted in substantial errors compared to a multi-dipole approach for ECoG recordings.Halnes et al. [17] found that, unlike for the four-sphere human head model, the dipole reciprocity approach resulted in poor performance for EEG compared to a multi-dipole approach. In both human ECoG and mouse EEG, the dipole approximation fails when the electrode is too close to the neural current source. These findings are in accordance with our observation that in a realistic rat head model, the dipole approach results in substantial errors for both EEG and ECoG compared to the analytically correct generalized reciprocity method. Obviously, even in a realistically heterogeneous human head model, the dipole approach incurs less simplification error than in a rodent model, because of the much larger electrode-source distance. Furthermore, validation using the four-sphere human model, while advantageous because of the existence of an analytical solution, is very likely to underestimate the dipole method error: For example, scalp-to-cortex distances in real heads can vary by up to 10 mm depending on location [22] and bone thickness and composition also shows a strong variation, negatively affecting the agreement of EEG signals predicted using the dipole approximation and the ground truth [5].

Previous studies have also examined signal calculation methods that are not currently available in BlueRecording, but which may be implemented in the future. For example, Ness et al. [23] found that the Method of Images accurately approximates results using a forward finite element approach in a model of a microelectrode array. However, unlike reciprocity-based approaches, the Method of Images depends on highly restrictive assumptions on the structure of the dielectric medium: the neural source is assumed to be surrounded by semi-infinite homogeneous media. Beltrachini [24] showed that a finite element subtraction approach using a quadripolar neural source outperformed a similar approach using only dipoles. While this method was able to accurately approximate a distributed source for EEG recordings in a human head model, the error will necessarily be greater when the electrodes are closer to the source – which is the case for ECoG or rodent EEG – as the second-order Taylor series approximation that underlies the quadripolar approach, while superior to the lower-order dipolar approximation, will break down close to the neural source, where higher-order terms must be considered.