Ideal current dipoles are appropriate source representations for simulating neurons for intracranial recordings

Abstract

Objective

To determine whether dipoles are an appropriate simplified representation of neural sources for stereo-EEG (sEEG).

Methods

We compared the distributions of voltages generated by a dipole, biophysically realistic cortical neuron models, and extended regions of cortex to determine how well a dipole represented neural sources at different spatial scales and at electrode to neuron distances relevant for sEEG. We also quantified errors introduced by the dipole approximation of neural sources in sEEG source localization using standardized low-resolution electrotomography (sLORETA).

Results

For pyramidal neurons, the coefficient of correlation between voltages generated by a dipole and neuron model were >0.9 for distances >1 mm. For small regions of cortex (~0.1 cm²), the error in voltages between a dipole and region was <100 μV for all distances. However, larger regions of active cortex (>5 cm²) yielded >50 μV errors within 1.5 cm of an electrode when compared to single dipoles. Finally, source localization errors were <5 mm when using dipoles to represent realistic neural sources.

Conclusions

Single dipoles are an appropriate source model to represent both single neurons and small regions of active cortex, while multiple dipoles are required to represent large regions of cortex.

Significance

Dipoles are computationally tractable and valid source models for sEEG.

1. Introduction

Neural source localization is the process of using electrophysiological recordings to delineate active brain regions giving rise to the recorded signals. Traditionally, neural source localization is conducted using electroencephalography (EEG) recordings (Grech et al., 2008). Numerous studies have shown that source localization of epileptiform activity has clinical significance (Gavaret et al., 2009, Koessler et al., 2010, Nemtsas et al., 2017), and the modeling assumptions used in EEG source localization have been rigorously validated (Akalin Acar and Makeig, 2013, Song et al., 2015, Wang and Ren, 2013, Whittingstall et al., 2003). Recently, source localization has been successfully conducted with stereotaxic-electroencephalography (sEEG) electrode recordings (<2 cm localization error) (Cam et al., 2017, Caune et al., 2014, Satzer et al., 2022). In sEEG, up to 25 transcranial depth electrodes are implanted into widespread regions of the brain to record and localize epileptiform activity. In turn, the localizations can inform invasive surgical therapies such as resection and ablation. Previous sEEG source localization studies used the same modeling assumptions that were validated for EEG source localization, but it is not clear these assumptions are also valid for sEEG source localization. For example, there is uncertainty whether signals arising from discrete areas of neural activity can be recorded by multiple sEEG electrodes (von Ellenrieder et al., 2012), what is the appropriate complexity of the electrical properties of head models, and what is an appropriate simplified source model. In this work, we analyzed the validity of, and errors associated with, sEEG source localization using a dipole representation of active neural sources.

Populations of simultaneously active neurons in extended regions of cortex generate the signals recorded in EEG (Buzsaki et al., 2012). The spatially distributed currents into and out of a single neuron constitute a closed loop of current, and at sufficiently distant recording locations, the complex organization of currents can be approximated by a dipole. By extension, any spatially distributed set of dipoles can also be represented by a single dipole at distant recording locations. At the length scales (cm) of EEG, the neural sources are sufficiently far from the recording sites to make a dipole an appropriate source representation (Nunez and Srinivasan, 2006). However, it is unclear whether dipoles are an appropriate representation of neural sources for sEEG where active neurons generating appreciable signals can be as close as 100 μm from the electrode contacts.

Simplified source models are necessary for applying source localization methods to electrophysiological data. Source localization relies on a lead field matrix that maps the voltages recorded at all electrode locations to the neural signal generators (Grech et al., 2008, Liu et al., 2019, Pascual-Marqui, 2002). Without a simplified source representation, this lead field matrix would include a transfer function between the electrode contact voltage and the most basic unit of signal generation, i.e. the currents generated throughout each neuron’s spatial extents, thus rendering it computationally intractable. However, by establishing an appropriate simplified source model, extended regions of cortex (>6 cm² for epileptiform signals (Tao et al., 2005)) can be combined into a single source representation and thereby create a computationally tractable problem. Source localization methods are an important tool to identify the sources of epilepsy and aid in understanding human cognition (Abdallah et al., 2017, Alhilani et al., 2020, Sabeti et al., 2015). Therefore, an appropriate simplified source model is imperative to use inverse methods on intracranial recordings.

Previous work compared the extracellular voltages generated by single pyramidal neurons to a dipole. Milstein and Koch found that the dipole accounted for approximately 75% of the voltage generated 1 mm away from a two dimensional layer 5 pyramidal cell model during an action potential and accounted for nearly all the voltage at distances >1 cm (Milstein and Koch, 2008). Næss et al. found that a dipole adequately represented the voltages generated by a passive three dimensional layer 2/3 pyramidal cell model during a single synaptic event at EEG-relevant distances (> 1 cm) but not for ECoG-relevant distances (~200 μm) (Naess et al., 2021). Both studies concluded that a dipole is sufficient to model the voltages generated at EEG distances, but it remains unclear whether dipoles can also be used to represent the neural sources generating voltages recorded at intracranial recording distances. Further, in both studies, the authors compared the absolute voltages generated by dipoles and model neurons at different distances along their somatodendritic axes. In source localization, recording electrodes are rarely oriented directly above the neurons and the absolute voltages are not used to localize activity. Rather, the relative differences in recorded voltages between electrode contacts, i.e., the voltage distribution generated by a neural source, is used to localize the source of activity. Therefore, to understand the appropriateness of a dipole source model for sEEG, it is necessary to quantify how well a dipole represents the voltage distributions generated by neural sources as well as source localization errors at relevant sEEG distances.

We used morphologically realistic compartmental models of cortical neurons (Aberra et al., 2018) to quantify how well a dipole represents single neurons over a large range of source to electrode distances and over the time course of neural activity. Subsequently, we quantified the error associated with using a single dipole to represents a spatially extended region of cortex composed of many dipole sources. Finally, we quantified the error associated with using dipoles for source localization with sEEG electrodes to determine whether dipoles are an appropriate source model for sEEG source localization.

2. Methods

Models used in the study are available on ModelDB (https://senselab.med.yale.edu/modeldb/ Accession No. 241165 and 244262). All code and models developed in this study are available on gitlab (https://gitlab.oit.duke.edu/bjt20/dipolesareappropriate2021.git).

2.1. NEURON models

To simulate realistic single neuron activity, we implemented versions of published biophysically realistic cortical neuron models (Aberra et al., 2018) in NEURON v7.4 (Figure 1A). These models were adapted from the models originally developed by the Blue Brain Project to capture better the properties of human cortical neurons, including the addition of morphologically realistic axonal arbors. We simulated one cell type from each cortical layer: layer 1 neurogliaform cells (L1 NGC), layer 2/3 pyramidal cells (L2/3 PC), layer 4 large basket cells (L4 LBC), layer 5 pyramidal cells (L5 PC), and layer 6 pyramidal cells (L6 PC). We simulated five virtual “clones” of each cell type, and each clone had the same ion channel conductances but randomly varied axonal and dendritic branch lengths and rotations to capture morphological variability (Markram et al., 2015). Each model neuron had a full myelinated axonal arbor according to the diameter and branch length cutoffs specified previously (Aberra et al., 2018). The morphologies were discretized into compartments with a maximum length of 20 μm, and we used a numerical integration time step of 25 μs.

Figure 1.

Morphology of cortical neuron models and example currents for action potential and PSC simulations. A) Morphologies of the 5 clones of each neuron type used in this study reproduced from (Aberra et al., 2018). We included Layer 1 neurogliaform cells (L1 NGC), Layer 2/3 pyramidal cells (L2/3 PC), Layer 4 large basket cells (L5 LBC), Layer 5 pyramidal cells (L5 PC), and Layer 6 pyramidal cells (L6 PC) in our analysis. B) Example simulated extracellular voltages generated by either a single postsynaptic current (blue) or an action potential (yellow) of a L5 PC model neuron. C) Example simulated membrane currents summed in all dendritic compartments (top), the soma (middle), or all axonal compartments (bottom) during either a postsynaptic current (blue) or action potential (yellow). The membrane currents are used to calculate the extracellular spatial distribution of voltages.

We stimulated both action potentials and postsynaptic currents (Figure 1B, C) to determine how well dipoles represent different types of neuronal activity. We activated action potentials in each model neuron using intracellular current injection by inserting a current clamp into the axon initial segment and delivering a 2 ms pulse at threshold intensity. To simulate postsynaptic currents, we used the network connectivity and synaptic strengths defined for each neuron in the cortical microcircuit model developed by the Blue Brain Project. Each presynaptic neuron can make multiple synaptic connections onto another neuron that are all of the same type (excitatory or inhibitory). Therefore, we simulated postsynaptic currents in each model neuron by iteratively activating each neuron that was presynaptic to the model neuron in the cortical column network. The excitatory synaptic conductances were modeled using AMPA/NMDA receptor kinetics, and the inhibitory synapses used GABA_A/GABA_B receptor kinetics (Markram et al., 2015). While neurons receive inputs from many presynaptic neuron simultaneously, the voltage generated by postsynaptic currents onto the same neuron from multiple input neurons sum near linearly (Supplemental Figure 1). Therefore, how well a network is represented by a dipole will depend on how well the single synapses are represented by a dipole. For both action potentials and postsynaptic currents, we recorded transmembrane currents from each compartment of the model neurons for subsequent analyses in which we quantified the extracellular voltages generated by each neuron. For simulations with intracellular current injection, we subtracted the injected current from the transmembrane current in the injected segment to minimize the stimulation artifact.

2.2. Correlation with analytical solution

We determined how well a dipole represented each of the neuron models by calculating the correlation between voltages obtained at 5000 randomly seeded points per radii on concentric spheres with radii between 50 μm and 1 cm. We placed each neuron model in a homogenous, isotropic extracellular medium. When comparing the voltage generated by the neuron models to an ideal current dipole the medium conductivity (σ) cancels out, so we set σ = 1 S/m for our analyses. We compared the voltages computed with the analytical solution for an ideal current dipole (eq. 1) and the voltages generated by the sum of the transmembrane currents in each segment of the spatially extended neuron models (eq. 2), defined as

$$V=\frac{I\cos (\theta )}{4\pi \sigma R^2}$$ (Eq. 1)

$$V={\sum }_{seg}\frac{I_{seg}}{4\pi \sigma R_{seg}}$$ (Eq. 2)

where I is current amplitude, θ is angle of dipole displacement vector, R is source to recording location distance, and the seg subscripts indicate segment specific terms. We treated each compartment of the neural models as a point current source (Holt and Koch, 1999) to compute the generated voltages. For action potential simulations in the realistic neuron models, we positioned the dipole at the location of the somatic segment. For simulations of postsynaptic currents, we positioned the dipole at the center of electrical activity during the peak of an extracellularly recorded voltage oscillation. We determined the center of electrical activity as the point halfway between the center of the positive and negative current flows at the peak of the recorded postsynaptic current. This location varied for the postsynaptic current simulations given that the input could arrive anywhere along the dendritic tree. We used fixed dipole orientations along the somatodendritic axis of the neuron models when assessing how well a canonical dipole represented a model. We used the Pearson correlation coefficient between the voltages generated by the ideal dipole and the neural models as the metric for how well the dipole represented the models at a given distance. We also found the dipole moment generated by the neural models by determining the slope of a linear fit between the voltages generated by the unit ideal dipole and the voltages generated by the neural models.

2.3. Forward modeling in volume conductor models

We built a patient-specific finite element head model (FEM) in SCIRUN v5.0 (SCI Institute, University of Utah, Salt Lake City, UT) to compare source localization error for dipoles and morphologically realistic neuron models, and the head model was generated fully from the patient’s neuroimaging data (Cartmell et al., 2019).

2.3.1. Patient-specific head model construction

The Duke University Health System IRB approved the use of clinical neuroimaging in this study, and the participant whose neuroimaging was used provided written informed consent. T1-weighted MRI scans were obtained using a 3T Siemens Skyra scanner with in-plane resolution and slice thickness of 1 mm. Post-op CT images were obtained with a GE Discovery CT 750 HD scanner using a tube current of 335 mA and x-ray tube voltage of 120 kV. Diffusion weighted (DW)-MRI scans were acquired using a 3T Siemens Skyra scanner with in-plane resolution of 1.72 mm and slice thickness of 2.4 mm. DW-MRIs were collected using 30 non-collinear directions (bval = 1000 s/mm²) and a single T2 weighted low-b image (bval=0 s/mm²).

To create the patient-specific FEM, we used a patient’s T1-weighted MRI, post-op CT, and DW-MRI. We automatically segmented the T1-weighted MRI in FMRIB Software Library (FSL, https://fsl.fmrib.ox.ac.uk) using the “BET” command to obtain the outer skin, skull, and brain masks and “fsl_anat” to obtain masks of white matter, gray matter, and cerebrospinal fluid. The outer skin layer was used to define the base geometry of the model while the skull, white matter, gray matter, and cerebrospinal fluid layers were used to define the boundaries between different tissue types when defining conductivity tensors. We registered the post-op CT to T1 space using a rigid transformation with a mutual information cost function. We used the load preservation technique (Howell and McIntyre, 2016) to define the anisotropic conductivity tensors in the patient-specific model. The patient was implanted with a set of Adtech RD08R, RD10R, and RD12R electrodes which have a contact diameter of 0.86 mm, contact spacing of 5 mm, and contact length of 2.29 mm. We modeled the leads as a cylinder with axis defined by a line between the coordinates of the first two contacts, and the locations of the contacts were determined manually from the CT images.

2.3.2. Boundary Conditions

To define sources, we chose to add mesh nodes to our head model associated with the locations in the NEURON model to ensure exact solutions. We set the mesh nodes defined from the segment locations in the NEURON model to have a current value corresponding to the net transmembrane current in that segment at a given time. This led to a current balanced distribution of point current sources in space. We used the mesh element locations generated from the Freesurfer (https://surfer.nmr.mgh.harvard.edu/) segmentation of the white-grey matter boundary to position neural sources. For source localization, we randomly sampled 10 points on the white-grey matter boundary as the locations of the forward solution generators. We used the implanted electrode locations recovered from the CT images as the recording locations and grounded one electrode contact to act as a common reference for the recordings on the other electrode contacts. Since we moved the source and fixed the recording positions, we had to remesh and solve the FEM for each forward simulation.

2.3.3. Mesh Convergence

We compared two meshes with 28,158,497 elements and 47,499,955 elements and obtained <1% voltage errors. We used a mesh with 34,664,575 elements for forward simulations while the mesh with 28,158,497 elements was used to generate the lead field. This was done to avoid committing the “inverse crime” of seeding data and solving the inverse problem with the same mesh.

2.3.4. Lead-field

An inverse solution calculates an estimate of the current sources that generated a given set of recorded voltages. We generated the lead field matrix for each FEM model using the reciprocity theorem (Weinstein et al., 2000). Because we used purely ohmic conductivities, we could calculate the voltage recorded at each electrode contact by a current source anywhere in space by simply calculating the voltage distribution generated by a current source at the recording electrode contact. We then took the spatial gradient of each voltage distribution. Each set of gradients represented the potential generated by three orthogonal unit dipoles at an electrode contact (Weinstein et al., 2000). Therefore, determining the voltages recorded at each electrode contact for dipoles in all orientations anywhere in space required calculation of one forward solution for each contact. We sampled the head model at all the points on the white-gray matter boundary mesh elements. We repeated the simulation for each electrode contact and appended the results into a lead field matrix for each FEM.

2.4. Extended source modeling

To understand how well a single dipole represents extended regions of active cortex for intracranial recordings, we calculated the voltages generated by patches of active cortex in the patient-specific head model. We created patches of active cortex by seeding 40 points on the cortex and iteratively recruiting adjacent mesh elements from the cortical mesh to our patch. We continued the process until the patch reached our desired area (0.1 - 9.5 cm²), based on the area of cortex believed to be active during an interictal spike. For each location, we created 20 patches of various sizes. We calculated the voltages generated by these patches by assuming that a patch of active cortex was represented by a set of dipoles, each oriented orthogonally to the cortical surface. For each element in the patch, we found the corresponding column of the lead field matrix and scaled the voltage values by the area of the mesh element and an established estimate of the dipole moment density (1 nA-m/mm²) (Murakami and Okada, 2015). Finally, we summed all the scaled voltage values to determine the voltages at all electrode contacts generated by each patch.

2.4.1. Comparison of extended source and single dipole models

We compared the voltages produced by each patch of active cortex to those produced by a single dipole oriented in the vector summed directions of the dipoles in each patch. We fit a line to the relationship between the voltages generated by the patch and the voltages generated by a single dipole by solving a system of linear equations in MATLAB. We scaled the voltage generated by the single dipole by the slope of the line and quantified the root mean squared and maximum errors between the voltages generated by the patch and those generated by a single dipole. For patches at all 40 locations in the head, we also quantified the minimum distance between any element in the patch and any electrode contact.

2.5. Source localization

We quantified the performance of a dipole representation of active neural populations by comparing the dipole localization error of standardized low resolution electrotomography (sLORETA) (Pascual-Marqui, 2002) when the simulated recordings were generated by an ideal dipole and a morphologically realistic neuron. Because sLORETA is a static source localization algorithm, we conducted source localization with the voltages recorded at a single time point. To find the time point to use in our localization, we simulated a recording 1 cm away from the active source using the analytic solution for the recorded voltages and used the set of current source amplitudes associated with the peak in the recorded signal for source localization. A brief summary of sLORETA can be found below, and technical details of sLORETA can be found in (Pascual-Marqui, 2002).

2.5.1. sLORETA

Source localization is inherently ill posed because there are orders of magnitude more dipole sources than recording sites, and thus the set of dipoles that can recreate the recorded potentials is not unique. However, regularization based source localization algorithms constrain the problem to yield adequate localizations. We conducted source localization using sLORETA. There are 3 step to conducting sLORETA:

1). Centering

The lead field matrix and the recorded potentials were multiplied by the centering matrix.

$$H=I-\left(\frac{1}{m}\right)1^T$$ (Eq. 3)

I (#Recording sites x # Recording sites) is the identity matrix, 1 (Recording Sites x 1) is a vector of ones, and m is the number of recording sites. This process simplifies the functional of interest to:

$$F=\Vert \Phi -KJ{\Vert }^2+\alpha \Vert J{\Vert }^2$$ (Eq. 4)

Φ is the centered recorded potentials, J is the current density, K is the centered lead field and α is a regularization constant.

2). Current density estimate

We found a current density estimate using the following equation:

$$\widehat{j}=K^T(K{K}^T+\alpha H{)}^{+}\Phi$$ (Eq. 5)

where ⁺ denotes the Moore-Penrose pseudoinverse. Additionally, the variance of the estimated current density is defined by:

$$S_{\widehat{j}}=K^T(K{K}^T+\alpha H{)}^{+}\mathrm{K}$$ (Eq. 6)

3). Defining the Localization inference map (LIM)

A localization inference map was generated through the process of standardization.

$$LIM={\widehat{j}}^T{S}_{\widehat{j}}^{-1}\widehat{j}$$ (Eq. 7)

where $\widehat{j}$ is the current density estimate of a certain dipole location and $\mathrm{S}\widehat{j}$ is the corresponding covariance matrix. We used the location with the maximum LIM value as the sLORETA determined source location.

3. Results

We provide an overview of the models and simulations conducted in this study in Table 1.

Table 1.

Overview of simulations completed in each results section.

Section	Analysis	Source Type	Number of Simulations
3.1/3.2	Single neuron	AP	5 Cell types x 5 Clones
		PSC	5 Cell types x 5 Clones x NumSynapses
3.3	Multiple dipole	Peak of Activity (Dipole Moment Density)	40 Source locations x 20 Surface areas
3.4	Single neuron source localization	AP	10 Source locations x 5 Types x 5 Clones
		PSC	10 Source locations x 5 Cell types x 5 Clones x 10 PSCs

AP – Action potential, PSC – Postsynaptic current, Type – Neuron model type, Clone – Different neuron models of the same type, NumSynapses – Number of synapses (differ based on neuron type and clone).

3.1. Ideal current dipoles represent well individual neurons at mm scale distances

To determine how well an ideal current dipole represents neural sources, we first quantified how well an ideal current dipole oriented along the somatodendritic axis represented the voltage distribution generated by a morphologically realistic neuron model at the time of peak extracellular voltage for both action potentials and postsynaptic currents (Morphology of cortical neuron models and example currents for action potential and PSC simulations. A) Morphologies of the 5 clones of each neuron type used in this study reproduced from (Aberra et al., 2018). We included Layer 1 neurogliaform cells (L1 NGC), Layer 2/3 pyramidal cells (L2/3 PC), Layer 4 large basket cells (L5 LBC), Layer 5 pyramidal cells (L5 PC), and Layer 6 pyramidal cells (L6 PC) in our analysis. B) Example simulated extracellular voltages generated by either a single postsynaptic current (blue) or an action potential (yellow) of a L5 PC model neuron. C) Example simulated membrane currents summed in all dendritic compartments (top), the soma (middle), or all axonal compartments (bottom) during either a postsynaptic current (blue) or action potential (yellow). The membrane currents are used to calculate the extracellular spatial distribution of voltages.

Figure 2). If the dipole represented the neural source well, then the correlation between the voltages generated by the dipole and those generated by neuron model would be high. For action potentials, the correlation coefficient between the voltages generated by ideal current dipoles and those generated by realistic model neurons increased with distance from the soma to an asymptotic value. For the larger pyramidal cell models (L5 and L6 PC), the correlation coefficients reached 90% of their respective asymptotic values within a 100 μm recording radius (Morphology of cortical neuron models and example currents for action potential and PSC simulations. A) Morphologies of the 5 clones of each neuron type used in this study reproduced from (Aberra et al., 2018). We included Layer 1 neurogliaform cells (L1 NGC), Layer 2/3 pyramidal cells (L2/3 PC), Layer 4 large basket cells (L5 LBC), Layer 5 pyramidal cells (L5 PC), and Layer 6 pyramidal cells (L6 PC) in our analysis. B) Example simulated extracellular voltages generated by either a single postsynaptic current (blue) or an action potential (yellow) of a L5 PC model neuron. C) Example simulated membrane currents summed in all dendritic compartments (top), the soma (middle), or all axonal compartments (bottom) during either a postsynaptic current (blue) or action potential (yellow). The membrane currents are used to calculate the extracellular spatial distribution of voltages.

Figure 2.

Correlation of extracellular voltages generated by dipoles and realistic neuron models. A) Layer 5 pyramidal cell model with surrounding ring depicting the sampling of normalized voltages on concentric spheres surrounding the model neuron. Colors on the neuron visualization delineate the apical dendrites (blue), basal dendrites (green), and axon (red) B) Extracellular voltage generated by model layer 5 pyramidal cell recorded 1 cm away from the neuron. The correlation between the extracellular voltages generated by a dipole and the model neurons were quantified at the time of the peak of the extracellular action potential (arrow) for all recording distances. C) Correlation between extracellular voltages generated by a dipole and realistic model neurons for action potentials in 5 neuron types (Layer 1 neuralgiform cells (L1 NGC), Layer 2/3 pyramidal cells (L2/3 PC), Layer 4 large basket cells (L5 LBC), Layer 5 pyramidal cells (L5 PC), and Layer 6 pyramid/al cells (L6 PC)). Error bars are the standard error of the correlation (n=5 clones of each neuron type). D) Angle difference between a dipole oriented along the somatodendritic axis and the orientation of a dipole oriented in the best fit direction compared to the correlation coefficient between the voltage distributions on a recording sphere at radius 1 cm. Each dot represents one clone of each model neuron type, denoted by the same color scheme as in C. E) Correlation between extracellular voltages generated by a dipole oriented along the somatodendritic axis and realistic model neuron for postsynaptic currents in 5 neuron types. F) Correlation between extracellular voltages generated by a realistic model neuron and a dipole oriented in the direction that maximizes correlation for postsynaptic currents in 5 neuron types. G) Percent differences in absolute voltages generated by an ideally oriented dipole and model neuron at increasing distances from the apical end of the model neurons in the z direction. H) Distribution of percent differences in absolute voltages generated by an ideally oriented dipole and model neuron for postsynaptic currents in 5 neuron types. Histograms for each neuron type are stacked on each other.

Figure 2C). Correlations for all neurons reached 90% of their asymptotic values within a 1 mm recording radius, and correlation coefficients ranged from 0.52 to 0.99 across the neuron types at a 1 cm recording radius. For both L5 and L6 PCs, the far field (1 cm) correlation coefficient was >0.98 while L1 NGC (0.94±0.03), L4 LBCs (0.80±0.08), and L2/3 PC (0.91±0.04) were not as well represented by ideal current dipoles oriented along the somatodendritic axis due to the diffuse branching of interneuron (L1 NGC and L4 LBC) dendrites and the short apical dendrite of small PCs (L2/3 PC).

Although some model neurons were not well represented by a dipole oriented along the somatodendritic axis, allowing the dipole to point in the orientation that maximized the correlation coefficient increased the correlation with an ideal dipole to ~0.99 for all model neurons for source to recording distances >1 mm (Supplemental Figure 2). Additionally, the angle difference between a dipole oriented along the somatodendritic axis and the dipole orientation that best fit the voltage distribution generated by the model neurons decreased monotonically to near 0 as the correlation coefficient trended towards 1 (Morphology of cortical neuron models and example currents for action potential and PSC simulations. A) Morphologies of the 5 clones of each neuron type used in this study reproduced from (Aberra et al., 2018). We included Layer 1 neurogliaform cells (L1 NGC), Layer 2/3 pyramidal cells (L2/3 PC), Layer 4 large basket cells (L5 LBC), Layer 5 pyramidal cells (L5 PC), and Layer 6 pyramidal cells (L6 PC) in our analysis. B) Example simulated extracellular voltages generated by either a single postsynaptic current (blue) or an action potential (yellow) of a L5 PC model neuron. C) Example simulated membrane currents summed in all dendritic compartments (top), the soma (middle), or all axonal compartments (bottom) during either a postsynaptic current (blue) or action potential (yellow). The membrane currents are used to calculate the extracellular spatial distribution of voltages.

Figure 2D). Therefore, the correlation coefficient is inversely related to the difference in direction between the best fitting dipole and somatodendritic axis.

For postsynaptic currents, the correlation of voltages generated by a model neuron and a dipole oriented along the somatodendritic axis also rapidly reached 90% of the asymptotic value within a 1 mm recording radius (Morphology of cortical neuron models and example currents for action potential and PSC simulations. A) Morphologies of the 5 clones of each neuron type used in this study reproduced from (Aberra et al., 2018). We included Layer 1 neurogliaform cells (L1 NGC), Layer 2/3 pyramidal cells (L2/3 PC), Layer 4 large basket cells (L5 LBC), Layer 5 pyramidal cells (L5 PC), and Layer 6 pyramidal cells (L6 PC) in our analysis. B) Example simulated extracellular voltages generated by either a single postsynaptic current (blue) or an action potential (yellow) of a L5 PC model neuron. C) Example simulated membrane currents summed in all dendritic compartments (top), the soma (middle), or all axonal compartments (bottom) during either a postsynaptic current (blue) or action potential (yellow). The membrane currents are used to calculate the extracellular spatial distribution of voltages.

Figure 2E). However, the asymptotic correlation values were more variable and smaller than their action potential counterparts, meaning that the optimal direction of a postsynaptic current dipole was not in the direction of the somatodendritic axis. When allowing the dipole to point in the best direction, the correlation of voltages generated by a dipole and the neuron models reached a correlation coefficient of >0.9 within a 1 mm recording radius (Morphology of cortical neuron models and example currents for action potential and PSC simulations. A) Morphologies of the 5 clones of each neuron type used in this study reproduced from (Aberra et al., 2018). We included Layer 1 neurogliaform cells (L1 NGC), Layer 2/3 pyramidal cells (L2/3 PC), Layer 4 large basket cells (L5 LBC), Layer 5 pyramidal cells (L5 PC), and Layer 6 pyramidal cells (L6 PC) in our analysis. B) Example simulated extracellular voltages generated by either a single postsynaptic current (blue) or an action potential (yellow) of a L5 PC model neuron. C) Example simulated membrane currents summed in all dendritic compartments (top), the soma (middle), or all axonal compartments (bottom) during either a postsynaptic current (blue) or action potential (yellow). The membrane currents are used to calculate the extracellular spatial distribution of voltages.

Figure 2F). Previous work quantifying the dipole assumption of a neuron compared the voltages generated by a postsynaptic current and dipole at various distance above a model neuron (Næss et al., 2021). They found that the errors in voltage between a single dipole and more sophisticated representations were between 20% and 60% at 1 mm. We simulated the voltages generated by a dipole and neuron models at increasing distances from the top (apical end) of the neuron models, and the mean and median absolute percent differences were <20% at 2 mm from the neuron models (Morphology of cortical neuron models and example currents for action potential and PSC simulations. A) Morphologies of the 5 clones of each neuron type used in this study reproduced from (Aberra et al., 2018). We included Layer 1 neurogliaform cells (L1 NGC), Layer 2/3 pyramidal cells (L2/3 PC), Layer 4 large basket cells (L5 LBC), Layer 5 pyramidal cells (L5 PC), and Layer 6 pyramidal cells (L6 PC) in our analysis. B) Example simulated extracellular voltages generated by either a single postsynaptic current (blue) or an action potential (yellow) of a L5 PC model neuron. C) Example simulated membrane currents summed in all dendritic compartments (top), the soma (middle), or all axonal compartments (bottom) during either a postsynaptic current (blue) or action potential (yellow). The membrane currents are used to calculate the extracellular spatial distribution of voltages.

Figure 2G). In the worst case scenario, using dipoles to represent the potentials generated by postsynaptic currents had errors up to 60% at 1 mm (Morphology of cortical neuron models and example currents for action potential and PSC simulations. A) Morphologies of the 5 clones of each neuron type used in this study reproduced from (Aberra et al., 2018). We included Layer 1 neurogliaform cells (L1 NGC), Layer 2/3 pyramidal cells (L2/3 PC), Layer 4 large basket cells (L5 LBC), Layer 5 pyramidal cells (L5 PC), and Layer 6 pyramidal cells (L6 PC) in our analysis. B) Example simulated extracellular voltages generated by either a single postsynaptic current (blue) or an action potential (yellow) of a L5 PC model neuron. C) Example simulated membrane currents summed in all dendritic compartments (top), the soma (middle), or all axonal compartments (bottom) during either a postsynaptic current (blue) or action potential (yellow). The membrane currents are used to calculate the extracellular spatial distribution of voltages.

Figure 2H), but most errors were <10% even at 1 mm.

3.2. Ideal current dipoles represent well action potentials over time

To quantify how well a dipole represented neural sources over the full time course of an action potential, we compared the voltages generated by a dipole source oriented along the somatodendritic axis and realistic model neurons. We did not compare the voltages generated by postsynaptic currents because the postsynaptic current dipoles at the peak of activity did not exhibit consistent directions, while the voltages generated by action potentials did. If the dipole represented well the neural source, then we would observe strong correlations (>0.9) throughout the time course of the action potential. The voltages from the L5/6 PCs and the dipole exhibited strong (>0.95) negative, then positive, then negative correlations with the voltages from the ideal dipole. Thus, L5/6 PCs were well represented by a dipole oriented along the somatodendritic axis opposite the conventional dipole direction (apical dendrite to soma) before and after the action potential, and, at the peak of the extracellular spike, by a dipole in the conventional direction (soma to apical dendrite) (Figure 3B). We observed a similar oscillation between positive and negative correlations for small PCs and interneurons over the time course of an action potential. The large PCs always oscillated from a negative correlation before the spike, to a positive correlation during the spike, and back to a negative correlation after the spike, while other neuron types exhibited more variable changes in dipole direction over time (Figure 3B and Supplemental Figure 3). Additionally, the dipole moments of the L5 PCs were approximately 4 times larger than those of any other neuron type, both at the peak of the recorded spike (Figure 3C and Supplemental Figure 4) and at the trough of the afterpotential (Figure 3D and Supplemental Figure 4). Therefore, L5 PCs are likely a dominant contributor to the orientation of the net neural sources normal to the cortical surface.

Figure 3.

Correlation between the extracellular voltages generated by action potentials in realistic model neurons and a dipole over time. Dipole moments (A) and correlation between extracellular voltages from a dipole (B) for layer 5 pyramidal cell (Left) or layer 4 large basket cell (Right) over time. Maximum dipole moment (C) and minimum dipole moment (D) for each neuron type (Layer 1 neuralgiform cells (L1 NGC), Layer 2/3 pyramidal cells (L2/3 PC), Layer 4 large basket cells (L5 LBC), Layer 5 pyramidal cells (L5 PC), and Layer 6 pyramid/al cells (L6 PC)).

3.3. Extended source representations are necessary for source to recording distances <1.5 cm

We simulated the voltages generated by patches of active cortex composed of many dipoles in a patient-specific head model to understand how well a single dipole represents extended regions of active cortex and to determine whether it is necessary to model neural sources with extended representations (Figure 4). If a single dipole is sufficient to model extended regions of active cortex for sEEG, then the differences in voltages generated at each sEEG contact by a dipole and the extended source would be below the noise floor (50 μV) (Hill et al., 2018) at relevant distances and relevant areas of active cortex. We compared the voltages generated at the sEEG recording electrode contacts by patches between 0.1 to 9.5 cm² to the voltages generated by a single dipole at the centroid of the cortical patch oriented in the vector-summed direction of all the dipoles in the patch. When the patch was far from the nearest recording electrode (> 1.5 cm), the maximum error between the recorded voltages generated by a patch of any size and the single dipole was consistently below 50 μV. However, when the patch was closer to the recording electrodes (<1.5 cm), the maximum error was larger than 50 μV for larger patch sizes (>5 cm²) (Figure 4C). The maximum voltage error increased with the size of the patch, and smaller patches were better represented by a single dipole than large patches of active cortex (Figure 4D). When patches were small (~0.1 cm²), the maximum voltage error was <100 μV for all source to recording distances. Interictal spiking events are thought to be generated by neural sources on the order of 10 cm² (Tao et al., 2005), and thus within 1.5 cm of a recording site, there are likely substantial voltage errors caused by representing an extended region with a single dipole. However, for more local events, such as high frequency oscillations (Zelmann et al., 2014), where the neural signal generator is smaller, the voltage errors caused by using a single dipole to represent the region of cortex appear to be negligible.

Figure 4.

Single dipole representation of a patch of active cortex. A) 6 cm² patch of active cortex (pink region) and the location of the single dipole representation (green dot) of the source. B) All locations of simulated active patches of cortex. C) Maximum voltage difference between the voltages generated by patches of varying sizes (0.1 – 9.5 cm²) and locations and voltages generated by an ideal dipole at all of the patch locations for different electrode to patch distances. Each colored line is a different patch, points are individual electrode contacts, solid black line indicates 50 μV, and dashed black line indicates 1.5 cm. D) Maximum voltage difference between patch and dipole for various patch areas. Each colored line is a different patch and solid black line indicates 50 μV.

3.4. Ideal current dipoles are sufficient to localize spatially extended sources from sEEG recordings

To determine how well dipoles represent realistic sources for source localization, one of the primary applications of the dipole approximation, we quantified source localization errors using sLORETA to conduct dipole source localization with voltage inputs generated by either a dipole or a realistic model neuron. If a dipole is an appropriate source model for sEEG, then the difference in localization errors using a dipole or model neuron as the source model will be small (<1 cm). We simulated forward solutions by placing the dipole or realistic model neuron at 10 different locations in a patient-specific head model. The localization error with a dipole source ranged from 0.4 to 1.9 mm and increased with source to electrode distance (Figure 5A). A similar pattern was seen for action potentials and postsynaptic currents where the localization error increased with electrode to source distance (Figure 5A and B). Although the localization error was greater for the model neurons than localizations with single dipole forward solutions, the maximum error was ≤1.2 cm, even at distances where recording appreciable signals was unlikely. Considering realistic source to electrode distances (<3 cm) the average localization error was 0.92 mm, and the average difference between the localization error of a dipole and the model neuron was 0.36 mm. Thus, the source localization error introduced by using dipoles to approximate spatially extended neural sources was typically <1 mm.

Figure 5.

Localization error due to using dipoles to localize active model neurons. Localization errors for action potentials (A) and postsynaptic currents (B) when using different neural models to generate forward solutions in the patient-specific head model. Localization errors are shown as a function of distance to the nearest electrode contact. The number of neuron models that share the same localization error are shown by the size of the marker ranging from 1 to 24 for action potentials and 1 to 262 for post synaptic currents. 250 (10 locations, 5 neuron types, and 5 clones) action potentials were simulated while 2500 (10 locations, 5 neuron types, 5 clones, and 10 randomly chosen PSCs) postsynaptic currents were simulated.

4. Discussion

Simplified source models are necessary to represent active populations of neurons for source localization. For EEG, a dipole source model adequately represents voltages from active neurons because the recording sites are far from the sources (Nunez and Srinivasan, 2006). For sEEG, where the recording sites are potentially much closer to the active sources, an appropriate simplified source representation is unclear.

Using morphologically realistic model neurons to represent neural activity, we quantified how well dipoles represented a range of neural sources. For action potentials, dipoles oriented along the somatodendritic axis represented well L5/6 PCs (R>0.9) for source to recording distances >100 μm at the signal peak. However, the correlations between the voltages generated by the dipole and those generated by L2/3 PCs and interneurons were more variable. For postsynaptic currents, a dipole oriented along the somatodendritic axis did not consistently well represent the voltages generated by any model neuron. However, when allowing the dipole to point in any direction, the dipole was a good representation (R>0.9) of postsynaptic currents in all neuron models within 2 mm. Additionally, the error in voltages generated by a dipole relative to the model neurons was on average <20% at 2 mm recording distances for all model neurons. Over the time course of an action potential, the correlation between the voltages generated by a dipole and by model L5 PCs rapidly oscillated (<1 ms) between being well represented by a dipole in one direction and in the opposite direction. Other model neurons exhibited similar behavior, but the correlations were not as strong. Importantly, the dipole moments of the L5 PCs were four times larger than those of any other neuron type, suggesting that the signal recorded from a population of neurons will be dominated by L5 PCs.

Extending the dipole source to represent a patch of active cortex, we found that a single dipole is only appropriate to represent a large patch of active cortex when the source to recording distance is >1.5 cm. However, a single dipole exhibited small voltage errors (<100 μV) when compared to small regions of active cortex at any source to recording distance. Finally, we quantified the effect of using the dipole source approximation on the accuracy of source localization. In a realistic FEM head model, there was, on average, < 1 mm error associated with using dipoles to represent extended sources at relevant electrode to source distances. Therefore, we conclude that dipoles are an adequate current source model to localize active populations of neurons for sEEG.

4.1. Utility of Dipole Representation

Postsynaptic currents are believed to generate local field potentials in the brain that can be recorded with intracranial electrodes (Nunez and Srinivasan, 2006). The neural sources contributing to the intracranial signals are predicted to be located within one cm of the recording site (Job et al., 2014). When modeling populations of active neurons for intracranial local field potential recordings, dipoles are an appropriate source model when the sources are millimeters from the recording electrode. However, when the source to electrode distance is <1 mm, the dipole approximation can break down. sEEG electrode contacts are spaced at least 1.5 mm apart, so a dipole is appropriate for most sEEG recordings that register an appreciable signal even if a single contact may be <1 mm from the active neural source.

When modeling extended regions of cortex, modeling the topology of the brain (i.e., patient-specific sulci and gyri undulations) is necessary for simulating invasive recordings. While not all single action potential or postsynaptic current events generate dipoles oriented along the somatodendritic axis, all PCs are oriented orthogonal to the cortical surface. Therefore, the transverse components of the dipoles generated during action potentials or postsynaptic currents will cancel for a population of neurons. That will leave only the dipole moments oriented along the somatodendritic axes to sum and generate a population dipole moment oriented orthogonal to the cortical surface (Nunez and Srinivasan, 2006, Pascual-Marqui, 2002). Therefore, to model an extended region of active cortex for sEEG, all the orthogonally oriented dipoles composing a patch of cortex need to be simulated.

For source localization, dipoles are a sufficient source representation given that electrodes are spaced at least 1.5 mm apart. In addition, source localization requires multiple contacts recording the same signal (Lantz et al., 2002). The error from simplifying the source representation of a neural population to a single dipole was small relative to other sources of error. For example, the algorithms used for source localization and the noise intrinsic to neural recordings typically result in errors on the order of 1 cm, compared to the mean additional error of 0.36 mm between localizations with a dipole and a realistic source (Liu et al., 2019).

4.2. Implications for source localization

Recent advances in source localization demonstrate that the spatial extent of a neural source generating a set of EEG recordings can be consistently determined without a priori knowledge of the signal generator’s source strength (Sohrabpour et al., 2016). However, at far distances, the relative voltages recorded at all EEG sites, generated by a set of neural sources centered at the same location in the brain, will be very similar and well approximated by a single dipole source. Therefore, an algorithm should not be able to determine accurately the size of the source without an estimate of the source’s strength. However, our results indicate that when the source to recording distance is ≤ 1.5 cm, the relative voltages recorded at all recording sites are not the same for varying source sizes and differ from a single ideal dipole with root mean squared differences >50 μV. Therefore, there is information that corresponds to the spatial extent of the neural source encoded in the relative voltages recorded at all recording sites, and source reconstruction algorithms can make use of this information to determine the spatial extent of a neural source. This is likely how previously developed algorithms replicated well the spatial extents of neural sources.

All source localization algorithms rely on a simplified representation of the neural sources generating electric fields in the brain. Previous work developed algorithms to compute a lead field matrix with current multipole sources under the assumption that a single dipole is insufficient for localizing spatially complex neural sources (Beltrachini, 2019). They found that for extended regions of cortex, source localization with multipole sources yielded superior localization compared to dipole sources. This result seems to challenge our findings; however, for our source localization results, we used dipoles to localize single neuron models, which are much smaller than an extended region of cortex. Additionally, we found the differences in recorded voltages generated by a dipole and large extended cortical regions were >50 μV for source to recording distances <1.5 cm, and this points to a consistent conclusion that dipoles are insufficient to model large extended regions of cortex. Thus, while dipoles are appropriate for modeling small populations of neurons (~0.1 cm²), extended regions of active cortex need to be modeled by either a population of dipoles, each representing a small population of simultaneously active neurons, or a current multipole. These results are consistent with previous experimental findings that a single or small populations of dipoles are insufficient for modeling interictal sources (Alarcon et al., 1994).

4.3. Modeling Limitations

There are several limitations with the models used in this study. First, when modeling the voltage distributions generated by the neural source models, we did not account for the effect of the electrode on the recorded voltages. Instead, we considered ideal point recording electrodes for ease of computation. For sEEG, the distribution of voltages from dipole sources is not distorted by an electrode contact when the distance between source and electrode is >0.5 mm (Kent and Grill, 2014, von Ellenrieder et al., 2012). Therefore, while inclusion of the electrode contacts would change the distribution of voltages within 0.5 mm of the recording site, the representation of a neural source with a dipole, which is only suitable ≥ 1 mm from the recording site, is likely not impacted. Additionally, when simulating source localization we used a single static source localization algorithm, sLORETA. SLORETA is a distributed source localization algorithm that is known to produce a diffuse estimate of the source location. While we used the maximum point of the source localization estimate to determine localization error, the certainty of sLORETA’s estimate changes based on distance from the source and the noise level. We did not account for the uncertainty in the sLORETA estimate, which has a direct utility in clinical diagnosis where the size of the sLORETA determined source is displayed on commercial analysis software.

Highlights.

• Spatiotemporal voltage distributions from dipoles match realistic model neurons.

• Multiple dipoles are required to represent extended regions (cm²) of active cortex.

• Minimal localization error using dipoles to represent neural sources for sEEG.