Biology subjects

Einevoll, G. T.

Publications and source records attributed to Einevoll, G. T..

33 records · Page 2Linked to original sources

bioRxiv2020

Evidence for reduced long-term potentiation-like visual cortical plasticity in schizophrenia and bipolar disorder

BackgroundSeveral lines of research suggest that impairments in long-term potentiation (LTP)-like synaptic plasticity might be a key pathophysiological mechanism in schizophrenia (SZ) and bipolar disorder type I (BDI) and II (BDII). Using modulations of visually evoked potentials (VEP) of the electroencephalogram, impaired LTP-like visual cortical plasticity has been implicated in patients with BDII, while there has been conflicting evidence in SZ, a lack of research in BDI, and mixed results regarding associations with symptom severity, mood states, and medication. MethodsWe measured the VEP of patients with SZ spectrum disorders (n=31), BDI (n=34), BDII (n=33), and other BD spectrum disorders (n=2), and age-matched healthy control participants (n=200) before and after prolonged visual stimulation. ResultsCompared to healthy controls, modulation of VEP component N1b, but not C1 or P1, was impaired both in patients within the SZ spectrum ({chi}2=35.1, p=3.1x10-9) and BD spectrum ({chi}2=7.0, p=8.2x10-3), including BDI ({chi}2=6.4, p=0.012), but not BDII ({chi}2=2.2, p=0.14). N1b modulation was also more severely impaired in SZ spectrum than BD spectrum patients ({chi}2=14.2, p=1.7x10-4). The reduction in N1b modulation was related to PANSS total scores ({chi}2=10.8, p=1.0x10-3), and nominally to number of psychotic episodes ({chi}2=4.9, p=0.027). Conclusions. These results suggest that LTP-like plasticity is impaired in SZ and BDI, but not BDII, and related to psychotic symptom severity. Adding to previous genetic, pharmacological, and anatomical evidence, these results implicate aberrant synaptic plasticity as a mechanism underlying SZ and BD.

neuroscience↗

bioRxiv2020

Corticothalamic feedback sculpts visual spatial integration in mouse thalamus

En route from retina to cortex, visual information passes through the dorsolateral geniculate nucleus of the thalamus (dLGN), where extensive corticothalamic (CT) feedback has been suggested to modulate spatial processing. How this modulation arises from direct excitatory and indirect inhibitory CT feedback pathways remains enigmatic. Here we show that in awake mice, retinotopically organized cortical feedback sharpens receptive fields (RFs) and increases surround suppression in the dLGN. Guided by a network model indicating that widespread inhibitory CT feedback is necessary to reproduce these effects, we targeted the visual sector of the thalamic reticular nucleus (visTRN) for recordings. We found that visTRN neurons have large receptive fields, show little surround suppression, and exhibit strong feedback-dependent responses to large stimuli. These features make them an ideal candidate for mediating feedback-enhanced surround suppression in the dLGN. We conclude that cortical feedback sculpts spatial integration in dLGN, likely via recruitment of neurons in visTRN.

neuroscience↗

bioRxiv2020

RippleNet: A Recurrent Neural Network for Sharp Wave Ripple (SPW-R) Detection

Hippocampal sharp wave ripples (SPW-R) have been identified as key bio-markers of important brain functions such as memory consolidation and decision making. SPW-R detection typically relies on hand-crafted feature extraction, and laborious manual curation is often required. In this multidisciplinary study, we propose a novel, self-improving artificial intelligence (AI) method in the form of deep Recurrent Neural Networks (RNN) with Long Short-Term memory (LSTM) layers that can learn features of SPW-R events from raw, labeled input data. The algorithm is trained using supervised learning on hand-curated data sets with SPW-R events. The input to the algorithm is the local field potential (LFP), the low-frequency part of extracellularly recorded electric potentials from the CA1 region of the hippocampus. The output prediction can be interpreted as the time-varying probability of SPW-R events for the duration of the input. A simple thresholding applied to the output probabilities is found to identify times of events with high precision. The reference implementation of the algorithm, named RippleNet, is open source, freely available, and implemented using a common open-source framework for neural networks (tensorflow.keras) and can be easily incorporated into existing data analysis workflows for processing experimental data.

neuroscience↗

bioRxiv2020

Brain Modeling ToolKit: an Open Source Software Suite for Multiscale Modeling of Brain Circuits

Experimental studies in neuroscience are producing data at a rapidly increasing rate, providing exciting opportunities and formidable challenges to existing theoretical and modeling approaches. To turn massive datasets into predictive quantitative frameworks, the field needs software solutions for systematic integration of data into realistic, multiscale models. Here we describe the Brain Modeling ToolKit (BMTK), a software suite for building models and performing simulations at multiple levels of resolution, from biophysically detailed multi-compartmental, to point-neuron, to population-statistical approaches. Leveraging the SONATA file format and existing software such as NEURON, NEST, and others, BMTK offers consistent user experience across multiple levels of resolution. It permits highly sophisticated simulations to be set up with little coding required, thus lowering entry barriers to new users. We illustrate successful applications of BMTK to large-scale simulations of a cortical area. BMTK is an open-source package provided as a resource supporting modeling-based discovery in the community.

neuroscience↗

bioRxiv2020

Experience-dependent modulation of the visual evoked potential: testing effect sizes, retention over time, and associations with age in 415 healthy individuals

Experience-dependent modulation of the visual evoked potential (VEP) is a promising proxy measure of synaptic plasticity in the cerebral cortex. However, existing studies are limited by small to moderate sample sizes as well as by considerable variability in how VEP modulation is quantified. In the present study, we used a large sample (n = 415) of healthy volunteers to compare different quantifications of VEP modulation with regards to effect sizes and retention of the modulation effect over time. We observed significant modulation for VEP components C1 (Cohens d = 0.53), P1 (d = 0.66), N1 (d = -0.27), N1b (d = -0.66), but not P2 (p = 0.1), and in one time-frequency cluster (~30 Hz and ~70 ms post-stimulus; d = -0.48), 2-4 minutes after 2 Hz prolonged visual stimulation. For components N1 (d = -0.21) and N1b (d = -0.38), as well for the time-frequency cluster (d = -0.33), this effect was retained after 54-56 minutes. Moderate to high correlations ({rho} = [0.39, 0.69]) between modulation at different postintervention blocks revealed a relatively high temporal stability in the modulation effect for each VEP component. However, different VEP components also showed markedly different temporal retention patterns. Finally, P1 modulation correlated positively with age (t = 5.26), and was larger for female participants (t = 3.91), with no effects of either age or sex on N1 and N1b potentiation. These results provide strong support for VEP modulation, and especially N1b modulation, as a robust measure of synaptic plasticity, but underscore the need to differentiate between components, and to control for demographic confounders.

neuroscience↗

bioRxiv2020

An electrodiffusive, ion conserving Pinsky-Rinzel model with homeostatic mechanisms

Most neuronal models are based on the assumption that ion concentrations remain constant during the simulated period, and do not account for possible effects of concentration variations on ionic reversal potentials, or of ionic diffusion on electrical potentials. Here, we present what is, to our knowledge, the first multicompartmental neuron model that accounts for electrodiffusive ion concentration dynamics in a way that ensures a biophysically consistent relationship between ion concentrations, electrical charge, and electrical potentials in both the intra- and extracellular space. The model, which we refer to as the electrodiffusive Pinsky-Rinzel (edPR) model, is an expanded version of the two-compartment Pinsky-Rinzel (PR) model of a hippocampal CA3 neuron, where we have included homeostatic mechanisms and ion-specific leakage currents. Whereas the main dynamical variable in the original PR model is the transmembrane potential, the edPR model in addition keeps track of all ion concentrations (Na+, K+, Ca2+, and Cl-), electrical potentials, and the electrical conductivities in the intra- as well as extracellular space. The edPR model reproduces the membrane potential dynamics of the PR model for moderate firing activity, when the homeostatic mechanisms succeed in maintaining ion concentrations close to baseline. For higher activity levels, homeostasis becomes incomplete, and the edPR model diverges from the PR model, as it accounts for changes in neuronal firing properties due to deviations from baseline ion concentrations. Whereas the focus of this work is to present and analyze the edPR model, we envision that it will become useful for the field in two main ways. Firstly, as it relaxes a set of commonly made modeling assumptions, the edPR model can be used to test the validity of these assumptions under various firing conditions, as we show here for a few selected cases. Secondly, the edPR model is a supplement to the PR model and should replace it in simulations of scenarios in which ion concentrations vary over time. As it is applicable to conditions with failed homeostasis, the edPR model opens up for simulating a range of pathological conditions, such as spreading depression or epilepsy. Author summaryNeurons generate their electrical signals by letting ions pass through their membranes. Despite this fact, most models of neurons apply the simplifying assumption that ion concentrations remain effectively constant during neural activity. This assumption is often quite good, as neurons contain a set of homeostatic mechanisms that make sure that ion concentrations vary quite little under normal circumstances. However, under some conditions, these mechanisms can fail, and ion concentrations can vary quite dramatically. Standard models are thus not able to simulate such conditions. Here, we present what to our knowledge is the first multicompartmental neuron model that in a biophysically consistent way does account for the effects of ion concentration variations. We here use the model to explore under which activity conditions the ion concentration variations become important for predicting the neurodynamics. We expect the model to be of great use for simulating a range of pathological conditions, such as spreading depression or epilepsy, which are associated with large changes in extracellular ion concentrations.

neuroscience↗

bioRxiv2019

Spatially Resolved Estimation of Metabolic Oxygen Consumption From Optical Measurements in Cortex

The cerebral metabolic rate of oxygen (CMRO2) is an important indicator of brain function and pathology. Knowledge about its magnitude is also required for proper interpretation of the blood oxygenation level dependent (BOLD) signal measured with functional MRI (fMRI). The ability to measure CMRO2 with high spatial and temporal accuracy is thus highly desired. Traditionally the estimation of CMRO2 has been pursued with somewhat indirect approaches combining several different types of measurements with mathematical modeling of the underlying physiological processes. Given the numerous assumptions involved, questions have thus been raised about the accuracy of the resulting CMRO2 estimates. The recent ability to measure the level of oxygen (pO2) in cortex with high spatial resolution in in vivo conditions has provided a more direct way for estimating CMRO2. CMRO2 and pO2 are related via the Poisson partial differential equation. Assuming a constant CMRO2 and cylindrical symmetry around the blood vessel providing the oxygen, the so-called Krogh-Erlang formula relating the spatial pO2 profile to a constant CMRO2 value can be derived. This Krogh-Erlang formula has previously been used to estimate the average CMRO2 close to cortical blood vessels based on pO2 measurements in rats.\n\nHere we introduce a new method, the Laplace method, to provide spatial maps of CMRO2 based on the same measured pO2 profiles. The method has two key steps: First the measured pO2 profiles are spatially smoothed to reduce effects of spatial noise in the measurements. Next, the Laplace operator (a double spatial derivative) in two spatial dimensions is applied on the smoothed pO2 profiles to obtain spatially resolved CMRO2 estimates. The smoothing introduces a bias, and a balance must be found where the effects of the noise are sufficiently reduced without introducing too much bias. In this model-based study we explore this balance in situations where the ground truth, that is, spatial profile of CMRO2 is preset and thus known, and the corresponding pO2 profiles are found by solving the Poisson equation, either numerically or by taking advantage of the Krogh-Erlang formula. MATLAB code for using the Laplace method is provided.

neuroscience↗

bioRxiv2019

MEArec: a fast and customizable testbench simulator for ground-truth extracellular spiking activity

When recording neural activity from extracellular electrodes, both in vivo and in vitro, spike sorting is a required and very important processing step that allows for identification of single neurons activity. Spike sorting is a complex algorithmic procedure, and in recent years many groups have attempted to tackle this problem, resulting in numerous methods and software packages. However, validation of spike sorting techniques is complicated. It is an inherently unsupervised problem and it is hard to find universal metrics to evaluate performance. Simultaneous recordings that combine extracellular and patch-clamp or juxtacellular techniques can provide ground-truth data to evaluate spike sorting methods. However, their utility is limited by the fact that only a few cells can be measured at the same time. Simulated ground-truth recordings can provide a powerful alternative mean to rank the performance of spike sorters. We present here MEArec, a Python-based software which permits flexible and fast simulation of extracellular recordings. MEArec allows users to generate extracellular signals on various customizable electrode designs and can replicate various problematic aspects for spike sorting, such as bursting, spatio-temporal overlapping events, and drifts. We expect MEArec will provide a common testbench for spike sorting development and evaluation, in which spike sorting developers can rapidly generate and evaluate the performance of their algorithms.

neuroscience↗

bioRxiv2019

LFPy - multimodal modeling of extracellular neuronal recordings in Python

1.LFPy is an open-source tool for calculating brain signals such as extracellular potentials and magnetic signals from simulated activity in multicompartment neuron models, ranging from single cells to large neuronal networks. The tool is provided as a Python package, and relies on the NEURON simulation environment.

neuroscience↗

bioRxiv2019

Estimation of neural network model parameters from local field potentials (LFPs)

Most modeling in systems neuroscience has been descriptive where neural representations, that is, receptive fields, have been found by statistically correlating neural activity to sensory input. In the traditional physics approach to modelling, hypotheses are represented by mechanistic models based on the underlying building blocks of the system, and candidate models are validated by comparing with experiments. Until now validation of mechanistic cortical network models has been based on comparison with neuronal spikes, found from the high-frequency part of extracellular electrical potentials. In this computational study we investigated to what extent the low-frequency part of the signal, the local field potential (LFP), can be used to infer properties of the neuronal network. In particular, we asked the question whether the LFP can be used to accurately estimate synaptic connection weights in the underlying network. We considered the thoroughly analysed Brunel network comprising an excitatory and an inhibitory population of recurrently connected integrate-and-fire (LIF) neurons. This model exhibits a high diversity of spiking network dynamics depending on the values of only three synaptic weight parameters. The LFP generated by the network was computed using a hybrid scheme where spikes computed from the point-neuron network were replayed on biophysically detailed multicompartmental neurons. We assessed how accurately the three model parameters could be estimated from power spectra of stationary background LFP signals by application of convolutional neural nets (CNNs). All network parameters could be very accurately estimated, suggesting that LFPs indeed can be used for network model validation. Significance statementMost of what we have learned about brain networks in vivo have come from the measurement of spikes (action potentials) recorded by extracellular electrodes. The low-frequency part of these signals, the local field potential (LFP), contains unique information about how dendrites in neuronal populations integrate synaptic inputs, but has so far played a lesser role. To investigate whether the LFP can be used to validate network models, we computed LFP signals for a recurrent network model (the Brunel network) for which the ground-truth parameters are known. By application of convolutional neural nets (CNNs) we found that the synaptic weights indeed could be accurately estimated from background LFP signals, suggesting a future key role for LFP in development of network models.

neuroscience↗

bioRxiv2018

BK channels have opposite effects on sodium versus calcium-mediated action potentials in endocrine pituitary cells.

Pituitary endocrine cells fire action potentials (APs) to regulate their cytosolic Ca2+ concentration and hormone secretion rate. Depending on animal species, cell type, and biological conditions, pituitary APs are generated either by TTX-sensitive Na+ currents (INa), high-voltage activated Ca2+ currents (ICa), or by a combination of the two. Previous computational models of pituitary cells have mainly been based on data from rats, where INa is largely inactivated at the resting potential, and spontaneous APs are exclusively mediated by ICa. As a part of the previous modeling studies, a paradoxical role was identified for the big conductance K+ current (IBK), which was found to prolong the duration of ICa-mediated APs, and sometimes give rise to pseudo-plateau bursts, contrary to what one would expect from a hyperpolarizing current. Unlike in rats, spontaneous INa-mediated APs are consistently seen in pituitary cells of several other animal species, including several species of fish. In the current work we develop the, to our knowledge, first computational model of a pituitary cell that fires INa-mediated APs. Although we constrain the model to experimental data from gonadotrope cells in the teleost fish medaka (Oryzias latipes), it may likely provide insights also into other pituitary cell types that fire INa-mediated APs. In the current work, we use the model to explore how the effect of IBK depends on the AP generating mechanisms of pituitary cells. We do this by comparing simulations on the medaka gonadotrope model (two versions thereof) with simulations on a previously developed model of a rat pituitary cell. Interestingly, we find that IBK has the opposite effect on APs in the two models, i.e. it reduces the duration of already fast INa-mediated APs in the medaka model, and prolongs the duration of already slow ICa-mediated APs in the rat model.\n\nAuthor summaryExcitable cells elicit electrical pulses called action potentials (APs), which are generated and shaped by a combination of ion channels in the cell membrane. While neurons use APs for interneuronal communication and heart cells use them to generate heart-beats, pituitary cells use APs to regulate their cytosolic Ca2+ concentration, which in turn controls their hormone secretion rate. The amount of Ca2+ that enters the pituitary cell during an AP depends strongly on how long it lasts, and it is therefore important to understand the mechanisms that control this. Depending on animal species and biological conditions, pituitary APs may be initiated either by Ca2+ channels or Na+ channels. Here, we explore the differences between the two scenarios by comparing simulations on two different computer models: (i) a previously developed model which fires Na+-based APs, adapted to data from pituitary cells in rats, and (ii) a novel model that fires Ca2+-based APs, adapted to data from pituitary cells in the fish medaka. Interestingly, we find that the role of big conductance K+ (BK) channels, which are known to affect the duration of the AP, are opposite in the two models, i.e., they act to prolong Ca2+-based APs while they act to shorten Na+-based APs.

cell biology↗

bioRxiv2018

Multimodal modeling of neural network activity: computing LFP, ECoG, EEG and MEG signals with LFPy2.0

Recordings of extracellular electrical, and later also magnetic, brain signals have been the dominant technique for measuring brain activity for decades. The interpretation of such signals is however nontrivial, as the measured signals result from both local and distant neuronal activity. In volume-conductor theory the extracellular potentials can be calculated from a distance-weighted sum of contributions from transmembrane currents of neurons. Given the same transmembrane currents, the contributions to the magnetic field recorded both inside and outside the brain can also be computed. This allows for the development of computational tools implementing forward models grounded in the biophysics underlying electrical and magnetic measurement modalities.\n\nLFPy (LFPy.readthedocs.io) incorporated a well-established scheme for predicting extracellular potentials of individual neurons with arbitrary levels of biological detail. It relies on NEURON (neuron.yale.edu) to compute transmembrane currents of multicompartment neurons which is then used in combination with an electrostatic forward model. Its functionality is now extended to allow for modeling of networks of multicompartment neurons with concurrent calculations of extracellular potentials and current dipole moments. The current dipole moments are then, in combination with suitable volume-conductor head models, used to compute non-invasive measures of neuronal activity, like scalp potentials (electroencephalographic recordings; EEG) and magnetic fields outside the head (magnetoencephalographic recordings; MEG). One such built-in head model is the four-sphere head model incorporating the different electric conductivities of brain, cerebrospinal fluid, skull and scalp.\n\nWe demonstrate the new functionality of the software by constructing a network of biophysically detailed multicompartment neuron models from the Neocortical Microcircuit Collaboration (NMC) Portal (bbp.epfl.ch/nmc-portal) with corresponding statistics of connections and synapses, and compute in vivo-like extracellular potentials (local field potentials, LFP; electrocorticographical signals, ECoG) and corresponding current dipole moments. From the current dipole moments we estimate corresponding EEG and MEG signals using the four-sphere head model. We also show strong scaling performance of LFPy with different numbers of message-passing interface (MPI) processes, and for different network sizes with different density of connections.\n\nThe open-source software LFPy is equally suitable for execution on laptops and in parallel on high-performance computing (HPC) facilities and is publicly available on GitHub.com.

neuroscience↗

bioRxiv2018

Uncertainpy: A Python toolbox for uncertainty quantification and sensitivity analysis in computational neuroscience.

Computational models in neuroscience typically contain many parameters that are poorly constrained by experimental data. Uncertainty quantification and sensitivity analysis provide rigorous procedures to quantify how the model output depends on this parameter uncertainty. Unfortunately, the application of such methods is not yet standard within the field of neuroscience.\n\nHere we present Uncertainpy, an open-source Python toolbox, tailored to perform uncertainty quantification and sensitivity analysis of neuroscience models. Uncertainpy aims to make it easy and quick to get started with uncertainty analysis, without any need for detailed prior knowledge. The toolbox allows uncertainty quantification and sensitivity analysis to be performed on already existing models without needing to modify the model equations or model implementation. Uncertainpy bases its analysis on polynomial chaos expansions, which are more efficient than the more standard Monte-Carlo based approaches.\n\nUncertainpy is tailored for neuroscience applications by its built-in capability for calculating characteristic features in the model output. The toolbox does not merely perform a point-to- point comparison of the \"raw\" model output (e.g. membrane voltage traces), but can also calculate the uncertainty and sensitivity of salient model response features such as spike timing, action potential width, mean interspike interval, and other features relevant for various neural and neural network models. Uncertainpy comes with several common models and features built in, and including custom models and new features is easy.\n\nThe aim of the current paper is to present Uncertainpy for the neuroscience community in a user- oriented manner. To demonstrate its broad applicability, we perform an uncertainty quantification and sensitivity analysis on three case studies relevant for neuroscience: the original Hodgkin-Huxley point-neuron model for action potential generation, a multi-compartmental model of a thalamic interneuron implemented in the NEURON simulator, and a sparsely connected recurrent network model implemented in the NEST simulator.\n\nSIGNIFICANCE STATEMENTA major challenge in computational neuroscience is to specify the often large number of parameters that define the neuron and neural network models. Many of these parameters have an inherent variability, and some may even be actively regulated and change with time. It is important to know how the uncertainty in model parameters affects the model predictions. To address this need we here present Uncertainpy, an open-source Python toolbox tailored to perform uncertainty quantification and sensitivity analysis of neuroscience models.

neuroscience↗

bioRxiv2018

A Kirchhoff-Nernst-Planck framework for modeling large scale extracellular electrodiffusion surrounding morphologically detailed neurons

1Many pathological conditions, such as seizures, stroke, and spreading depression, are associated with substantial changes in ion concentrations in the extracellular space (ECS) of the brain. An understanding of the mechanisms that govern ECS concentration dynamics may be a prerequisite for understanding such pathologies. To estimate the transport of ions due to electrodiffusive effects, one must keep track of both the ion concentrations and the electric potential simultaneously in the relevant regions of the brain. Although this is currently unfeasible experimentally, it is in principle achievable with computational models based on biophysical principles and constraints. Previous computational models of extracellular ion-concentration dynamics have required extensive computing power, and therefore have been limited to either phenomena on very small spatiotemporal scales (micrometers and milliseconds), or simplified and idealized 1-dimensional (1-D) transport processes on a larger scale. Here, we present the 3-D Kirchhoff-Nernst-Planck (KNP) framework, tailored to explore electrodiffusive effects on large spatiotemporal scales. By assuming electroneutrality, the KNP-framework circumvents charge-relaxation processes on the spatiotemporal scales of nanometers and nanoseconds, and makes it feasible to run simulations on the spatiotemporal scales of millimeters and seconds on a standard desktop computer. In the present work, we use the 3-D KNP framework to simulate the dynamics of ion concentrations and the electrical potential surrounding a morphologically detailed pyramidal cell. In addition to elucidating the single neuron contribution to electrodiffusive effects in the ECS, the simulation demonstrates the efficiency of the 3-D KNP framework. We envision that future applications of the framework to more complex and biologically realistic systems will be useful in exploring pathological conditions associated with large concentration variations in the ECS.\n\n2 Author summaryMany pathological conditions, such as epilepsy and cortical spreading depression, are linked to abnormal extracellular ion concentrations in the brain. Understanding the underlying principles of such conditions may prove important in developing treatments for these illnesses, which incur societal costs of tens of billions annually. In order to investigate the role of ion-concentration dynamics in the pathological conditions, one must measure the spatial distribution of all ion concentrations over time. This remains challenging experimentally, which makes computational modeling an attractive tool. We have previously introduced the Kirchhoff-Nernst-Planck framework, an efficient framework for modeling electrodiffusion. In this study, we introduce a 3-dimensional version of this framework and use it to model the electrodiffusion of ions surrounding a morphologically detailed neuron. The simulation covered a 1 mm3 cylinder of tissue for over a minute and was performed in less than a day on a standard desktop computer, demonstrating the frameworks efficiency. We believe this to be an important step on the way to understanding phenomena involving ion concentration shifts at the tissue level.

neuroscience↗

bioRxiv2017

Four-Sphere Head Model For EEG Signals Revisited

Electric potential recorded at the scalp (EEG) is dominated by contributions from current dipoles set by active neurons in the cortex. Estimation of these currents, called inverse modeling, requires a forward model, which gives the potential when the positions, sizes, and directions of the current dipoles are known. Di[ff]erent models of varying complexity and realism are used in the field. An important analytical example is the four-sphere model which assumes a four-layered spherical head where the layers represent brain tissue, cerebrospinal fluid (CSF), skull, and scalp, respectively. This model has been used extensively in the analysis of EEG recordings. Since it is analytical, it can also serve as a benchmark against which numerical schemes, such as the Finite Element Method (FEM), can be tested. While conceptually clear, the mathematical expression for the scalp potentials in the four-sphere model is quite cumbersome, and we observed the formulas presented in the literature to contain errors. We here derive and present the correct analytical formulas for future reference. They are compared with the results of FEM simulations of four-sphere model. We also provide scripts for computing EEG potentials in this model with the correct analytical formula and using FEM.

neuroscience↗