Computing Extracellular Electric Potentials from Neuronal Simulations.

Measurements of electric potentials from neural activity have played a key role in neuroscience for almost a century, and simulations of neural activity is an important tool for understanding such measurements. Volume conductor (VC) theory is used to compute extracellular electric potentials stemming from neural activity, such as extracellular spikes, multi-unit activity (MUA), local field potentials (LFP), electrocorticography (ECoG), and electroencephalography (EEG). Further, VC theory is also used inversely to reconstruct neuronal current source distributions from recorded potentials through current source density methods. In this book chapter, we show how VC theory can be derived from a detailed electrodiffusive theory for ion concentration dynamics in the extracellular medium, and we show what assumptions must be introduced to get the VC theory on the simplified form that is commonly used by neuroscientists. Furthermore, we provide examples of how the theory is applied to compute spikes, LFP signals, and EEG signals generated by neurons and neuronal populations.

Interstitial solute transport in 3D reconstructed neuropil occurs by diffusion rather than bulk flow.

The brain lacks lymph vessels and must rely on other mechanisms for clearance of waste products, including amyloid [Formula: see text] that may form pathological aggregates if not effectively cleared. It has been proposed that flow of interstitial fluid through the brain's interstitial space provides a mechanism for waste clearance. Here we compute the permeability and simulate pressure-mediated bulk flow through 3D electron microscope (EM) reconstructions of interstitial space. The space was divided into sheets (i.e., space between two parallel membranes) and tunnels (where three or more membranes meet). Simulation results indicate that even for larger extracellular volume fractions than what is reported for sleep and for geometries with a high tunnel volume fraction, the permeability was too low to allow for any substantial bulk flow at physiological hydrostatic pressure gradients. For two different geometries with the same extracellular volume fraction the geometry with the most tunnel volume had [Formula: see text] higher permeability, but the bulk flow was still insignificant. These simulation results suggest that even large molecule solutes would be more easily cleared from the brain interstitium by diffusion than by bulk flow. Thus, diffusion within the interstitial space combined with advection along vessels is likely to substitute for the lymphatic drainage system in other organs.

Astroglial endfeet exhibit distinct Ca2+ signals during hypoosmotic conditions.

Astrocytic endfeet cover the brain surface and form a sheath around the cerebral vasculature. An emerging concept is that endfeet control blood-brain water transport and drainage of interstitial fluid and waste along paravascular pathways. Little is known about the signaling mechanisms that regulate endfoot volume and hence the width of these drainage pathways. Here, we used the genetically encoded fluorescent Ca2+ indicator GCaMP6f to study Ca2+ signaling within astrocytic somata, processes, and endfeet in response to an osmotic challenge known to induce cell swelling. Acute cortical slices were subjected to artificial cerebrospinal fluid with 20% reduction in osmolarity while GCaMP6f fluorescence was imaged with two-photon microscopy. Ca2+ signals induced by hypoosmotic conditions were observed in all astrocytic compartments except the soma. The Ca2+ response was most prominent in subpial and perivascular endfeet and included spikes with single peaks, plateau-type elevations, and rapid oscillations, the latter restricted to subpial endfeet. Genetic removal of the type 2 inositol 1,4,5-triphosphate receptor (IP3R2) severely suppressed the Ca2+ responses in endfeet but failed to affect brain water accumulation in vivo after water intoxication. Furthermore, the increase in endfoot Ca2+ spike rate during hypoosmotic conditions was attenuated in mutant mice lacking the aquaporin-4 anchoring molecule dystrophin and after blockage of transient receptor potential vanilloid 4 channels. We conclude that the characteristics and underpinning of Ca2+ responses to hypoosmotic stress differ within the astrocytic territory and that IP3R2 is essential for the Ca2+ signals only in subpial and perivascular endfeet.

Ca2+ Signals in Astrocytes Facilitate Spread of Epileptiform Activity.

Epileptic seizures are associated with increased astrocytic Ca2+ signaling, but the fine spatiotemporal kinetics of the ictal astrocyte-neuron interplay remains elusive. By using 2-photon imaging of awake head-fixed mice with chronic hippocampal windows we demonstrate that astrocytic Ca2+ signals precede neuronal Ca2+ elevations during the initial bout of kainate-induced seizures. On average, astrocytic Ca2+ elevations preceded neuronal activity in CA1 by about 8 s. In subsequent bouts of epileptic seizures, astrocytes and neurons were activated simultaneously. The initial astrocytic Ca2+ elevation was abolished in mice lacking the type 2 inositol-1,4,5-trisphosphate-receptor (Itpr2-/-). Furthermore, we found that Itpr2-/- mice exhibited 60% less epileptiform activity compared with wild-type mice when assessed by telemetric EEG monitoring. In both genotypes we also demonstrate that spreading depression waves may play a part in seizure termination. Our findings imply a role for astrocytic Ca2+ signals in ictogenesis.

Focal Local Field Potential Signature of the Single-Axon Monosynaptic Thalamocortical Connection.

A resurgence has taken place in recent years in the use of the extracellularly recorded local field potential (LFP) to investigate neural network activity. To probe monosynaptic thalamic activation of cortical postsynaptic target cells, so called spike-trigger-averaged LFP (stLFP) signatures have been measured. In these experiments, the cortical LFP is measured by multielectrodes covering several cortical lamina and averaged on spontaneous spikes of thalamocortical (TC) cells. Using a well established forward-modeling scheme, we investigated the biophysical origin of this stLFP signature with simultaneous synaptic activation of cortical layer-4 neurons, mimicking the effect of a single afferent spike from a single TC neuron. Constrained by previously measured intracellular responses of the main postsynaptic target cell types and with biologically plausible assumptions regarding the spatial distribution of thalamic synaptic inputs into layer 4, the model predicted characteristic contributions to monosynaptic stLFP signatures both for the regular-spiking (RS) excitatory neurons and the fast-spiking (FS) inhibitory interneurons. In particular, the FS cells generated stLFP signatures of shorter temporal duration than the RS cells. Added together, a sum of the stLFP signatures of these two principal synaptic targets of TC cells were observed to resemble experimentally measured stLFP signatures. Outside the volume targeted by TC afferents, the resulting postsynaptic LFP signals were found to be sharply attenuated. This implies that such stLFP signatures provide a very local measure of TC synaptic activation, and that newly developed inverse current-source density (CSD)-estimation methods are needed for precise assessment of the underlying spatiotemporal CSD profiles.SIGNIFICANCE STATEMENT Despite its long history and prevalent use, the proper interpretation of the extracellularly recorded local field potential (LFP) is still not fully established. Here we investigate by biophysical modeling the origin of the focal LFP signature of the single-axon monosynaptic thalamocortical connection as measured by spike-trigger-averaging of cortical LFPs on spontaneous spikes of thalamocortical neurons. We find that this LFP signature is well accounted for by a model assuming thalamic projections to two cortical layer-4 cell populations: one excitatory (putatively regular-spiking cells) and one inhibitory (putatively fast-spiking cells). The LFP signature is observed to decay sharply outside the cortical region receiving the thalamocortical projection, implying that it indeed provides a very local measure of thalamocortical synaptic activation.

Deletion of Aquaporin-4 Curtails Extracellular Glutamate Elevation in Cortical Spreading Depression in Awake Mice.

Cortical spreading depression (CSD) is a phenomenon that challenges the homeostatic mechanisms on which normal brain function so critically depends. Analyzing the sequence of events in CSD holds the potential of providing new insight in the physiological processes underlying normal brain function as well as the pathophysiology of neurological conditions characterized by ionic dyshomeostasis. Here, we have studied the sequential progression of CSD in awake wild-type mice and in mice lacking aquaporin-4 (AQP4) or inositol 1,4,5-triphosphate type 2 receptor (IP3R2). By the use of a novel combination of genetically encoded sensors that a novel combination - an unprecedented temporal and spatial resolution, we show that CSD leads to brisk Ca2+ signals in astrocytes and that the duration of these Ca2+ signals is shortened in the absence of AQP4 but not in the absence of IP3R2. The decrease of the astrocytic, AQP4-dependent Ca2+ signals, coincides in time and space with a decrease in the duration of extracellular glutamate overflow but not with the initial peak of the glutamate release suggesting that in CSD, extracellular glutamate accumulation is extended through AQP4-dependent glutamate release from astrocytes. The present data point to a salient glial contribution to CSD and identify AQP4 as a new target for therapy.

Ion diffusion may introduce spurious current sources in current-source density (CSD) analysis.

Current-source density (CSD) analysis is a well-established method for analyzing recorded local field potentials (LFPs), that is, the low-frequency part of extracellular potentials. Standard CSD theory is based on the assumption that all extracellular currents are purely ohmic, and thus neglects the possible impact from ionic diffusion on recorded potentials. However, it has previously been shown that in physiological conditions with large ion-concentration gradients, diffusive currents can evoke slow shifts in extracellular potentials. Using computer simulations, we here show that diffusion-evoked potential shifts can introduce errors in standard CSD analysis, and can lead to prediction of spurious current sources. Further, we here show that the diffusion-evoked prediction errors can be removed by using an improved CSD estimator which accounts for concentration-dependent effects.NEW & NOTEWORTHY Standard CSD analysis does not account for ionic diffusion. Using biophysically realistic computer simulations, we show that unaccounted-for diffusive currents can lead to the prediction of spurious current sources. This finding may be of strong interest for in vivo electrophysiologists doing extracellular recordings in general, and CSD analysis in particular.

Effect of Ionic Diffusion on Extracellular Potentials in Neural Tissue.

Recorded potentials in the extracellular space (ECS) of the brain is a standard measure of population activity in neural tissue. Computational models that simulate the relationship between the ECS potential and its underlying neurophysiological processes are commonly used in the interpretation of such measurements. Standard methods, such as volume-conductor theory and current-source density theory, assume that diffusion has a negligible effect on the ECS potential, at least in the range of frequencies picked up by most recording systems. This assumption remains to be verified. We here present a hybrid simulation framework that accounts for diffusive effects on the ECS potential. The framework uses (1) the NEURON simulator to compute the activity and ionic output currents from multicompartmental neuron models, and (2) the electrodiffusive Kirchhoff-Nernst-Planck framework to simulate the resulting dynamics of the potential and ion concentrations in the ECS, accounting for the effect of electrical migration as well as diffusion. Using this framework, we explore the effect that ECS diffusion has on the electrical potential surrounding a small population of 10 pyramidal neurons. The neural model was tuned so that simulations over ∼100 seconds of biological time led to shifts in ECS concentrations by a few millimolars, similar to what has been seen in experiments. By comparing simulations where ECS diffusion was absent with simulations where ECS diffusion was included, we made the following key findings: (i) ECS diffusion shifted the local potential by up to ∼0.2 mV. (ii) The power spectral density (PSD) of the diffusion-evoked potential shifts followed a 1/f2 power law. (iii) Diffusion effects dominated the PSD of the ECS potential for frequencies up to several hertz. In scenarios with large, but physiologically realistic ECS concentration gradients, diffusion was thus found to affect the ECS potential well within the frequency range picked up in experimental recordings.

Power laws from linear neuronal cable theory: power spectral densities of the soma potential, soma membrane current and single-neuron contribution to the EEG.

Power laws, that is, power spectral densities (PSDs) exhibiting 1/f(α) behavior for large frequencies f, have been observed both in microscopic (neural membrane potentials and currents) and macroscopic (electroencephalography; EEG) recordings. While complex network behavior has been suggested to be at the root of this phenomenon, we here demonstrate a possible origin of such power laws in the biophysical properties of single neurons described by the standard cable equation. Taking advantage of the analytical tractability of the so called ball and stick neuron model, we derive general expressions for the PSD transfer functions for a set of measures of neuronal activity: the soma membrane current, the current-dipole moment (corresponding to the single-neuron EEG contribution), and the soma membrane potential. These PSD transfer functions relate the PSDs of the respective measurements to the PSDs of the noisy input currents. With homogeneously distributed input currents across the neuronal membrane we find that all PSD transfer functions express asymptotic high-frequency 1/f(α) power laws with power-law exponents analytically identified as α∞(I) = 1/2 for the soma membrane current, α∞(p) = 3/2 for the current-dipole moment, and α∞(V) = 2 for the soma membrane potential. Comparison with available data suggests that the apparent power laws observed in the high-frequency end of the PSD spectra may stem from uncorrelated current sources which are homogeneously distributed across the neural membranes and themselves exhibit pink (1/f) noise distributions. While the PSD noise spectra at low frequencies may be dominated by synaptic noise, our findings suggest that the high-frequency power laws may originate in noise from intrinsic ion channels. The significance of this finding goes beyond neuroscience as it demonstrates how 1/f(α) power laws with a wide range of values for the power-law exponent α may arise from a simple, linear partial differential equation.

Arterial stiffening provides sufficient explanation for primary hypertension.

Hypertension is one of the most common age-related chronic disorders, and by predisposing individuals for heart failure, stroke, and kidney disease, it is a major source of morbidity and mortality. Its etiology remains enigmatic despite intense research efforts over many decades. By use of empirically well-constrained computer models describing the coupled function of the baroreceptor reflex and mechanics of the circulatory system, we demonstrate quantitatively that arterial stiffening seems sufficient to explain age-related emergence of hypertension. Specifically, the empirically observed chronic changes in pulse pressure with age and the impaired capacity of hypertensive individuals to regulate short-term changes in blood pressure arise as emergent properties of the integrated system. The results are consistent with available experimental data from chemical and surgical manipulation of the cardio-vascular system. In contrast to widely held opinions, the results suggest that primary hypertension can be attributed to a mechanogenic etiology without challenging current conceptions of renal and sympathetic nervous system function.

Frequency dependence of signal power and spatial reach of the local field potential.

Despite its century-old use, the interpretation of local field potentials (LFPs), the low-frequency part of electrical signals recorded in the brain, is still debated. In cortex the LFP appears to mainly stem from transmembrane neuronal currents following synaptic input, and obvious questions regarding the 'locality' of the LFP are: What is the size of the signal-generating region, i.e., the spatial reach, around a recording contact? How far does the LFP signal extend outside a synaptically activated neuronal population? And how do the answers depend on the temporal frequency of the LFP signal? Experimental inquiries have given conflicting results, and we here pursue a modeling approach based on a well-established biophysical forward-modeling scheme incorporating detailed reconstructed neuronal morphologies in precise calculations of population LFPs including thousands of neurons. The two key factors determining the frequency dependence of LFP are the spatial decay of the single-neuron LFP contribution and the conversion of synaptic input correlations into correlations between single-neuron LFP contributions. Both factors are seen to give low-pass filtering of the LFP signal power. For uncorrelated input only the first factor is relevant, and here a modest reduction (<50%) in the spatial reach is observed for higher frequencies (>100 Hz) compared to the near-DC ([Formula: see text]) value of about [Formula: see text]. Much larger frequency-dependent effects are seen when populations of pyramidal neurons receive correlated and spatially asymmetric inputs: the low-frequency ([Formula: see text]) LFP power can here be an order of magnitude or more larger than at 60 Hz. Moreover, the low-frequency LFP components have larger spatial reach and extend further outside the active population than high-frequency components. Further, the spatial LFP profiles for such populations typically span the full vertical extent of the dendrites of neurons in the population. Our numerical findings are backed up by an intuitive simplified model for the generation of population LFP.

Electrodiffusive model for astrocytic and neuronal ion concentration dynamics.

The cable equation is a proper framework for modeling electrical neural signalling that takes place at a timescale at which the ionic concentrations vary little. However, in neural tissue there are also key dynamic processes that occur at longer timescales. For example, endured periods of intense neural signaling may cause the local extracellular K(+)-concentration to increase by several millimolars. The clearance of this excess K(+) depends partly on diffusion in the extracellular space, partly on local uptake by astrocytes, and partly on intracellular transport (spatial buffering) within astrocytes. These processes, that take place at the time scale of seconds, demand a mathematical description able to account for the spatiotemporal variations in ion concentrations as well as the subsequent effects of these variations on the membrane potential. Here, we present a general electrodiffusive formalism for modeling of ion concentration dynamics in a one-dimensional geometry, including both the intra- and extracellular domains. Based on the Nernst-Planck equations, this formalism ensures that the membrane potential and ion concentrations are in consistency, it ensures global particle/charge conservation and it accounts for diffusion and concentration dependent variations in resistivity. We apply the formalism to a model of astrocytes exchanging ions with the extracellular space. The simulations show that K(+)-removal from high-concentration regions is driven by a local depolarization of the astrocyte membrane, which concertedly (i) increases the local astrocytic uptake of K(+), (ii) suppresses extracellular transport of K(+), (iii) increases axial transport of K(+) within astrocytes, and (iv) facilitates astrocytic relase of K(+) in regions where the extracellular concentration is low. Together, these mechanisms seem to provide a robust regulatory scheme for shielding the extracellular space from excess K(+).

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. Understanding their underlying mechanisms in healthy and pathological brain function and behaviour rely on accurate SPW-R detection. In this multidisciplinary study, we propose a novel, self-improving artificial intelligence (AI) detection 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 approach contrasts conventional routines that typically relies on hand-crafted, heuristic feature extraction and often laborious manual curation. The algorithm is trained using supervised learning on hand-curated data sets with SPW-R events obtained under controlled conditions. 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. Its output predictions can be interpreted as time-varying probabilities of SPW-R events for the duration of the inputs. A simple thresholding applied to the output probabilities is found to identify times of SPW-R events with high precision. The non-causal, or bidirectional variant of the proposed algorithm demonstrates consistently better accuracy compared to the causal, or unidirectional counterpart. Reference implementations of the algorithm, named 'RippleNet', are 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.

Begonia-A Two-Photon Imaging Analysis Pipeline for Astrocytic Ca2+ Signals.

Imaging the intact brain of awake behaving mice without the dampening effects of anesthesia, has revealed an exceedingly rich repertoire of astrocytic Ca2+ signals. Analyzing and interpreting such complex signals pose many challenges. Traditional analyses of fluorescent changes typically rely on manually outlined static region-of-interests, but such analyses fail to capture the intricate spatiotemporal patterns of astrocytic Ca2+ dynamics. Moreover, all astrocytic Ca2+ imaging data obtained from awake behaving mice need to be interpreted in light of the complex behavioral patterns of the animal. Hence processing multimodal data, including animal behavior metrics, stimulation timings, and electrophysiological signals is needed to interpret astrocytic Ca2+ signals. Managing and incorporating these data types into a coherent analysis pipeline is challenging and time-consuming, especially if research protocols change or new data types are added. Here, we introduce Begonia, a MATLAB-based data management and analysis toolbox tailored for the analyses of astrocytic Ca2+ signals in conjunction with behavioral data. The analysis suite includes an automatic, event-based algorithm with few input parameters that can capture a high level of spatiotemporal complexity of astrocytic Ca2+ signals. The toolbox enables the experimentalist to quantify astrocytic Ca2+ signals in a precise and unbiased way and combine them with other types of time series data.

Intrinsic dendritic filtering gives low-pass power spectra of local field potentials.

The local field potential (LFP) is among the most important experimental measures when probing neural population activity, but a proper understanding of the link between the underlying neural activity and the LFP signal is still missing. Here we investigate this link by mathematical modeling of contributions to the LFP from a single layer-5 pyramidal neuron and a single layer-4 stellate neuron receiving synaptic input. An intrinsic dendritic low-pass filtering effect of the LFP signal, previously demonstrated for extracellular signatures of action potentials, is seen to strongly affect the LFP power spectra, even for frequencies as low as 10 Hz for the example pyramidal neuron. Further, the LFP signal is found to depend sensitively on both the recording position and the position of the synaptic input: the LFP power spectra recorded close to the active synapse are typically found to be less low-pass filtered than spectra recorded further away. Some recording positions display striking band-pass characteristics of the LFP. The frequency dependence of the properties of the current dipole moment set up by the synaptic input current is found to qualitatively account for several salient features of the observed LFP. Two approximate schemes for calculating the LFP, the dipole approximation and the two-monopole approximation, are tested and found to be potentially useful for translating results from large-scale neural network models into predictions for results from electroencephalographic (EEG) or electrocorticographic (ECoG) recordings.

Astrocytic Ca2+ signaling is reduced during sleep and is involved in the regulation of slow wave sleep.

Astrocytic Ca2+ signaling has been intensively studied in health and disease but has not been quantified during natural sleep. Here, we employ an activity-based algorithm to assess astrocytic Ca2+ signals in the neocortex of awake and naturally sleeping mice while monitoring neuronal Ca2+ activity, brain rhythms and behavior. We show that astrocytic Ca2+ signals exhibit distinct features across the sleep-wake cycle and are reduced during sleep compared to wakefulness. Moreover, an increase in astrocytic Ca2+ signaling precedes transitions from slow wave sleep to wakefulness, with a peak upon awakening exceeding the levels during whisking and locomotion. Finally, genetic ablation of an important astrocytic Ca2+ signaling pathway impairs slow wave sleep and results in an increased number of microarousals, abnormal brain rhythms, and an increased frequency of slow wave sleep state transitions and sleep spindles. Our findings demonstrate an essential role for astrocytic Ca2+ signaling in regulating slow wave sleep.

Amplitude variability and extracellular low-pass filtering of neuronal spikes.

The influence of neural morphology and passive electrical parameters on the width and amplitude of extracellular spikes is investigated by combined analytical and numerical investigations of idealized and anatomically reconstructed pyramidal and stellate neuron models. The main results are: 1), All models yield a low-pass filtering effect, that is, a spike-width increase with increasing distance from soma. 2), A neuron's extracellular spike amplitude is seen to be approximately proportional to the sum of the dendritic cross-sectional areas of all dendritic branches connected to the soma. Thus, neurons with many, thick dendrites connected to soma will produce large amplitude spikes, and therefore have the largest radius of visibility. 3), The spike shape and amplitude are found to be dependent on the membrane capacitance and axial resistivity, but not on the membrane resistivity. 4), The spike-amplitude decay with distance r is found to depend on dendritic morphology, and is decaying as 1/r(n) with 1 or= 2 far away.

Estimation of population firing rates and current source densities from laminar electrode recordings.

This model study investigates the validity of methods used to interpret linear (laminar) multielectrode recordings. In computer experiments extracellular potentials from a synaptically activated population of about 1,000 pyramidal neurons are calculated using biologically realistic compartmental neuron models combined with electrostatic forward modeling. The somas of the pyramidal neurons are located in a 0.4 mm high and wide columnar cylinder, mimicking a stimulus-evoked layer-5 population in a neocortical column. Current-source density (CSD) analysis of the low-frequency part (<500 Hz) of the calculated potentials (local field potentials, LFP) based on the 'inverse' CSD method is, in contrast to the 'standard' CSD method, seen to give excellent estimates of the true underlying CSD. The high-frequency part (>750 Hz) of the potentials (multi-unit activity, MUA) is found to scale approximately as the population firing rate to the power 3/4 and to give excellent estimates of the underlying population firing rate for trial-averaged data. The MUA signal is found to decay much more sharply outside the columnar populations than the LFP.