Measurements of electric potentials from neural activity have played a key role in neuroscience f... more 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 such as extracellular spikes, MUA, LFP, ECoG and EEG surrounding neurons, and also 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 show what assumptions that 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.
Springer Series in Computational Neuroscience, 2019
We review modeling of astrocyte ion dynamics with a specific focus on the implications of socalle... more We review modeling of astrocyte ion dynamics with a specific focus on the implications of socalled spatial potassium buffering where excess potassium in the extracellular space (ECS) is transported away to prevent pathological neural spiking. The recently introduced Kirchoff-Nernst-Planck (KNP) scheme for modeling ion dynamics in astrocytes (and brain tissue in general) is outlined and used to study such spatial buffering. We next describe how the ion dynamics of astrocytes may regulate microscopic liquid flow by osmotic effects and how such microscopic flow can be linked to whole-brain macroscopic flow. The chapter thus describes key elements in a putative multiscale theory with astrocytes linking neural activity on a microscopic scale to macroscopic fluid flow.
Hippocampal sharp wave ripples (SPW-R) have been identified as key bio-markers of important brain... more 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 fo...
The Journal of neuroscience : the official journal of the Society for Neuroscience, Jan 17, 2017
A resurgence has taken place in recent years in the use of the extracellularly recorded local fie... more 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 cont...
Current-source density (CSD) analysis is a well-established method for analyzing recorded local f... more 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 realis...
Recorded potentials in the extracellular space (ECS) of the brain is a standard measure of popula... more 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 ...
Conference proceedings : ... Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE Engineering in Medicine and Biology Society. Annual Conference, 2010
Extracellular recordings are a key tool to record the activity of neurons in vivo. Especially in ... more Extracellular recordings are a key tool to record the activity of neurons in vivo. Especially in the case of experiments with behaving animals, however, the tedious procedure of electrode placement can take a considerable amount of expensive and restricted experimental time. Furthermore, due to tissue drifts and other sources of variability in the recording setup, the position of the electrodes with respect to the recorded neurons can change causing low recording quality. The contributions of this work are threefold. We introduce a quality measure for the recording position of the electrode which should be maximized during recordings and is especially suitable for the use of multi-electrodes. An automated positioning system based on this quality measure is proposed. The system is able to find favorable recording positions and adapts the electrode position smoothly to changes of the neuron positions. Finally, we evaluate the system using a new simulator for extracellular recordings b...
Introduction Extracellular recordings have been, and still are, the main workhorse when measuring... more Introduction Extracellular recordings have been, and still are, the main workhorse when measuring neural activity in vivo. In single-unit recordings sharp electrodes are positioned close to a neuronal soma, and the firing rate of this particular neuron is measured by counting spikes , that is, the standardized extracellular signatures of action potentials (Gold et al., 2006). For such recordings the interpretation of the measurements is straightforward, but complications arise when more than one neuron contributes to the recorded extracellular potential. For example, if two firing neurons of the same type are at about the same distance from their somas to the tip of the recording electrode, it may be very difficult to sort the spikes according to from which neuron they originate. The use of two ( stereotrode (McNaughton et al., 1983)), four (tetrode (Recce and O'Keefe, 1989;Wilson andMcNaughton, 1993; Gray et al., 1995; Jog et al., 2002)) or more (Buzsaki, 2004) close-neighbored recording sites allows for improved spike sorting, since the different distances from the electrode tips or contacts allow for triangulation. With present recording techniques and clustering methods one can sort out spike trains from tens of neurons from single tetrodes and from hundreds of neurons with multi-shank electrodes (Buzsaki, 2004). Information about spiking is typically extracted from the high-frequency band (≳500 Hz) of extracellular potentials. Since these high-frequency signals generally stem from an unknown number of spiking neurons in the immediate vicinity of the electrode contact, this is called multi-unit activity (MUA) .
Power laws, that is, power spectral densities (PSDs) exhibiting 1/f α behavior for large frequenc... more Power laws, that is, power spectral densities (PSDs) exhibiting 1/f α behavior for large frequencies f , have commonly been observed in neural recordings. Power laws in noise spectra have not only been observed in microscopic recordings of neural membrane potentials and membrane currents, but also in macroscopic EEG (electroencephalographic) 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. The corresponding powerlaw exponents are 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. These power-law exponents are found for arbitrary combinations of uncorrelated and correlated noisy input current (as long as both the dendrites and the soma receive some uncorrelated input currents). Comparison with available data suggests that the apparent power laws observed in experiments may stem from uncorrelated current sources, presumably intrinsic ion channels, which are homogeneously distributed across the neural membranes and themselves exhibit pink (1/f) noise distributions. 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. We find here that the well-known cable equation describing the electrical properties of membranes transfers white-noise current input into 'colored' 1/f α-noise where α may have any half-numbered value within the interval from 1/2 to 3 for the different measurement modalities. Intuitively, the physical origin of these novel power laws can be understood in terms of the superposition of numerous low-pass filtered contributions with different cutoff frequencies (i.e., different time constants) due to the different spatial positions of the various current inputs along the neuron. As our model system is linear, the results directly generalize to any colored input noise, i.e., transferring 1/f β spectra of input currents to 1/f β+α output spectra.
Electrical extracellular recordings, i.e., recordings of the electrical potentials in the extrace... more Electrical extracellular recordings, i.e., recordings of the electrical potentials in the extracellular medium between cells, have been a main work-horse in electrophysiology for almost a century. The high-frequency part of the signal (500 Hz), i.e., the multi-unit activity (MUA), contains information about the firing of action potentials in surrounding neurons, while the low-frequency part, the local field potential (LFP), contains information about how these neurons integrate synaptic inputs. As the recorded extracellular signals arise from multiple neural processes, their interpretation is typically ambiguous and difficult. Fortunately, a precise biophysical modeling scheme linking activity at the cellular level and the recorded signal has been established: the extracellular potential can be calculated as a weighted sum of all transmembrane currents in all cells located in the vicinity of the electrode. This computational scheme can considerably aid the modeling and analysis of MUA and LFP signals. Here, we describe LFPy, an open source Python package for numerical simulations of extracellular potentials. LFPy consists of a set of easy-to-use classes for defining cells, synapses and recording electrodes as Python objects, implementing this biophysical modeling scheme. It runs on top of the widely used NEURON simulation environment, which allows for flexible usage of both new and existing cell models. Further, calculation of extracellular potentials using the line-source-method is efficiently implemented. We describe the theoretical framework underlying the extracellular potential calculations and illustrate by examples how LFPy can be used both for simulating LFPs, i.e., synaptic contributions from single cells as well a populations of cells, and MUAs, i.e., extracellular signatures of action potentials.
ABSTRACT In processes where ionic concentrations vary significantly, the standard cable equation ... more ABSTRACT In processes where ionic concentrations vary significantly, the standard cable equation fails to accurately predict the transmembrane potential. Such processes call for 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. We here derive a general electrodiffusive formalism for consistently modeling the dynamics of ion concentration and the transmembrane potential in a one-dimensional geometry, including both the intra-and extracellular domains. Unlike standard cable theory, the electrodiffusive formalism accounts for diffusive currents and concentration-dependent variation of the longitudinal resis-tivities.
Power laws, that is, power spectral densities (PSDs) exhibiting 1/f(α) behavior for large frequen... more 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 neur...
Electrical neural signalling typically takes place at the timescale of milliseconds, and is typic... more Electrical neural signalling typically takes place at the timescale of milliseconds, and is typically modeled using the cable equation. This is a good approximation when ionic concentrations are expected to vary little during the time course of a simulation. During periods of intense neural signalling, however, the local extracellular K +-concentration may increase by several millimolars. Clearance of excess K + likely depends partly on diffusion in the extracellular space, partly on local uptake by-and intracellular transport within astrocytes. The processes that maintain the extracellular environment typically takes place at the time scale of seconds, and cannot be modeled accurately without accounting for the spatiotemporal variations in ion concentrations. This work presented here consists of two main parts: First, we developed a general electrodiffusive formalism for modeling ion concentration dynamics in a one-dimensional geometry, including both an intra-and extracellular domain. The formalism was based on the Nernst-Planck equations. It ensures (i) that the membrane potential and ion concentrations are in consistency, (ii) global particle/charge conservation, and (iii) accounts for diffusion and concentration dependent variations in resistivities. Second, we applied the formalism to a model of astrocytes exchanging ions with the ECS. Through simulations, we identified the key astrocytic mechanisms involved in K + removal from high concentration regions. We found that a local increase in extracellular K + evoked a local depolarization of the astrocyte membrane, which at the same time (i) increased the local astrocytic uptake of K + (by locally inactivating the outward Kir-current), (ii) suppressed extracellular transport of K + , (iii) increased transport of K + within astrocytes, and (iv) facilitated astrocytic relase of K + in regions where the extracellular concentration was low. In summary, these mechanisms seem optimal for shielding the extracellular space from excess K + .
Despite its century-old use, the interpretation of local field potentials (LFPs), the low-frequen... more 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 activation, 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 frequency dependence of LFP are (1) the spatial decay of the singleneuron LFP contribution and (2) the translation 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 in the spatial reach is observed for higher frequencies compared to the near-DC value (∼ 0 Hz) of about 200 µm. Much larger frequency-dependent effects are seen when populations of pyramidal neurons receive correlated and spatially asymmetric inputs (basally or apically): the low-frequency (∼ 0 Hz) LFP power can here be an order of magnitude or more larger than the LFP power at, say, 60 Hz. Moreover, the low-frequency LFP components are found to have larger spatial reach and extend further outside the active population than high-frequency components. Our numerical findings are backed up by an intuitive simplified model for the generation of population LFP.
The local field potential (LFP) reflects activity of many neurons in the vicinity of the recordin... more The local field potential (LFP) reflects activity of many neurons in the vicinity of the recording electrode and is therefore useful for studying local network dynamics. Much of the nature of the LFP is, however, still unknown. There are, for instance, contradicting reports on the spatial extent of the region generating the LFP. Here, we use a detailed biophysical modeling approach to investigate the size of the contributing region by simulating the LFP from a large number of neurons around the electrode. We find that the size of the generating region depends on the neuron morphology, the synapse distribution, and the correlation in synaptic activity. For uncorrelated activity, the LFP represents cells in a small region (within a radius of a few hundred micrometers). If the LFP contributions from different cells are correlated, the size of the generating region is determined by the spatial extent of the correlated activity.
A new method for estimation of current-source density (CSD) from local field potentials is presen... more A new method for estimation of current-source density (CSD) from local field potentials is presented. This inverse CSD (iCSD) method is based on explicit inversion of the electrostatic forward solution and can be applied to data from multielectrode arrays with various geometries. Here, the method is applied to linear-array (laminar) electrode data. Three iCSD methods are considered: the CSD is assumed to have cylindrical symmetry and be (i) localized in infinitely thin discs, (ii) step-wise constant or (iii) continuous and smoothly varying (using cubic splines) in the vertical direction. For spatially confined CSD distributions the standard CSD method, involving a discrete double derivative, is seen in model calculations to give significant estimation errors when the lateral source dimension is comparable to the size of a cortical column (less than ∼1 mm). Further, discontinuities in the extracellular conductivity are seen to potentially give sizable errors for even wider source distributions. The iCSD methods are seen to give excellent estimates when the correct lateral source dimension and spatial distribution of conductivity are incorporated. To illustrate the application to real data, iCSD estimates of stimulus-evoked responses measured with laminar electrodes in the rat somatosensory (barrel) cortex are compared to estimates from the standard CSD method.
We present a new method, laminar population analysis (LPA), for analysis of laminar-electrode (li... more We present a new method, laminar population analysis (LPA), for analysis of laminar-electrode (linear multielectrode) data, where physiological constraints are explicitly incorporated in the mathematical model: the high-frequency band [multiunit activity (MUA)] is modeled as a sum over contributions from firing activity of multiple cortical populations, whereas the low-frequency band [local field potential (LFP)] is assumed to reflect the dendritic currents caused by synaptic inputs evoked by this firing. The method is applied to stimulus-averaged laminar-electrode data from barrel cortex of anesthetized rat after single whisker flicks. Two sample data sets, distinguished by stimulus paradigm, type of applied anesthesia, and electrical boundary conditions, are studied in detail. These data sets are well accounted for by a model with four cortical populations: one supragranular, one granular, and two infragranular populations. Population current source densities (CSDs; the CSD signat...
Measurements of electric potentials from neural activity have played a key role in neuroscience f... more 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 such as extracellular spikes, MUA, LFP, ECoG and EEG surrounding neurons, and also 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 show what assumptions that 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.
Springer Series in Computational Neuroscience, 2019
We review modeling of astrocyte ion dynamics with a specific focus on the implications of socalle... more We review modeling of astrocyte ion dynamics with a specific focus on the implications of socalled spatial potassium buffering where excess potassium in the extracellular space (ECS) is transported away to prevent pathological neural spiking. The recently introduced Kirchoff-Nernst-Planck (KNP) scheme for modeling ion dynamics in astrocytes (and brain tissue in general) is outlined and used to study such spatial buffering. We next describe how the ion dynamics of astrocytes may regulate microscopic liquid flow by osmotic effects and how such microscopic flow can be linked to whole-brain macroscopic flow. The chapter thus describes key elements in a putative multiscale theory with astrocytes linking neural activity on a microscopic scale to macroscopic fluid flow.
Hippocampal sharp wave ripples (SPW-R) have been identified as key bio-markers of important brain... more 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 fo...
The Journal of neuroscience : the official journal of the Society for Neuroscience, Jan 17, 2017
A resurgence has taken place in recent years in the use of the extracellularly recorded local fie... more 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 cont...
Current-source density (CSD) analysis is a well-established method for analyzing recorded local f... more 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 realis...
Recorded potentials in the extracellular space (ECS) of the brain is a standard measure of popula... more 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 ...
Conference proceedings : ... Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE Engineering in Medicine and Biology Society. Annual Conference, 2010
Extracellular recordings are a key tool to record the activity of neurons in vivo. Especially in ... more Extracellular recordings are a key tool to record the activity of neurons in vivo. Especially in the case of experiments with behaving animals, however, the tedious procedure of electrode placement can take a considerable amount of expensive and restricted experimental time. Furthermore, due to tissue drifts and other sources of variability in the recording setup, the position of the electrodes with respect to the recorded neurons can change causing low recording quality. The contributions of this work are threefold. We introduce a quality measure for the recording position of the electrode which should be maximized during recordings and is especially suitable for the use of multi-electrodes. An automated positioning system based on this quality measure is proposed. The system is able to find favorable recording positions and adapts the electrode position smoothly to changes of the neuron positions. Finally, we evaluate the system using a new simulator for extracellular recordings b...
Introduction Extracellular recordings have been, and still are, the main workhorse when measuring... more Introduction Extracellular recordings have been, and still are, the main workhorse when measuring neural activity in vivo. In single-unit recordings sharp electrodes are positioned close to a neuronal soma, and the firing rate of this particular neuron is measured by counting spikes , that is, the standardized extracellular signatures of action potentials (Gold et al., 2006). For such recordings the interpretation of the measurements is straightforward, but complications arise when more than one neuron contributes to the recorded extracellular potential. For example, if two firing neurons of the same type are at about the same distance from their somas to the tip of the recording electrode, it may be very difficult to sort the spikes according to from which neuron they originate. The use of two ( stereotrode (McNaughton et al., 1983)), four (tetrode (Recce and O'Keefe, 1989;Wilson andMcNaughton, 1993; Gray et al., 1995; Jog et al., 2002)) or more (Buzsaki, 2004) close-neighbored recording sites allows for improved spike sorting, since the different distances from the electrode tips or contacts allow for triangulation. With present recording techniques and clustering methods one can sort out spike trains from tens of neurons from single tetrodes and from hundreds of neurons with multi-shank electrodes (Buzsaki, 2004). Information about spiking is typically extracted from the high-frequency band (≳500 Hz) of extracellular potentials. Since these high-frequency signals generally stem from an unknown number of spiking neurons in the immediate vicinity of the electrode contact, this is called multi-unit activity (MUA) .
Power laws, that is, power spectral densities (PSDs) exhibiting 1/f α behavior for large frequenc... more Power laws, that is, power spectral densities (PSDs) exhibiting 1/f α behavior for large frequencies f , have commonly been observed in neural recordings. Power laws in noise spectra have not only been observed in microscopic recordings of neural membrane potentials and membrane currents, but also in macroscopic EEG (electroencephalographic) 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. The corresponding powerlaw exponents are 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. These power-law exponents are found for arbitrary combinations of uncorrelated and correlated noisy input current (as long as both the dendrites and the soma receive some uncorrelated input currents). Comparison with available data suggests that the apparent power laws observed in experiments may stem from uncorrelated current sources, presumably intrinsic ion channels, which are homogeneously distributed across the neural membranes and themselves exhibit pink (1/f) noise distributions. 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. We find here that the well-known cable equation describing the electrical properties of membranes transfers white-noise current input into 'colored' 1/f α-noise where α may have any half-numbered value within the interval from 1/2 to 3 for the different measurement modalities. Intuitively, the physical origin of these novel power laws can be understood in terms of the superposition of numerous low-pass filtered contributions with different cutoff frequencies (i.e., different time constants) due to the different spatial positions of the various current inputs along the neuron. As our model system is linear, the results directly generalize to any colored input noise, i.e., transferring 1/f β spectra of input currents to 1/f β+α output spectra.
Electrical extracellular recordings, i.e., recordings of the electrical potentials in the extrace... more Electrical extracellular recordings, i.e., recordings of the electrical potentials in the extracellular medium between cells, have been a main work-horse in electrophysiology for almost a century. The high-frequency part of the signal (500 Hz), i.e., the multi-unit activity (MUA), contains information about the firing of action potentials in surrounding neurons, while the low-frequency part, the local field potential (LFP), contains information about how these neurons integrate synaptic inputs. As the recorded extracellular signals arise from multiple neural processes, their interpretation is typically ambiguous and difficult. Fortunately, a precise biophysical modeling scheme linking activity at the cellular level and the recorded signal has been established: the extracellular potential can be calculated as a weighted sum of all transmembrane currents in all cells located in the vicinity of the electrode. This computational scheme can considerably aid the modeling and analysis of MUA and LFP signals. Here, we describe LFPy, an open source Python package for numerical simulations of extracellular potentials. LFPy consists of a set of easy-to-use classes for defining cells, synapses and recording electrodes as Python objects, implementing this biophysical modeling scheme. It runs on top of the widely used NEURON simulation environment, which allows for flexible usage of both new and existing cell models. Further, calculation of extracellular potentials using the line-source-method is efficiently implemented. We describe the theoretical framework underlying the extracellular potential calculations and illustrate by examples how LFPy can be used both for simulating LFPs, i.e., synaptic contributions from single cells as well a populations of cells, and MUAs, i.e., extracellular signatures of action potentials.
ABSTRACT In processes where ionic concentrations vary significantly, the standard cable equation ... more ABSTRACT In processes where ionic concentrations vary significantly, the standard cable equation fails to accurately predict the transmembrane potential. Such processes call for 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. We here derive a general electrodiffusive formalism for consistently modeling the dynamics of ion concentration and the transmembrane potential in a one-dimensional geometry, including both the intra-and extracellular domains. Unlike standard cable theory, the electrodiffusive formalism accounts for diffusive currents and concentration-dependent variation of the longitudinal resis-tivities.
Power laws, that is, power spectral densities (PSDs) exhibiting 1/f(α) behavior for large frequen... more 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 neur...
Electrical neural signalling typically takes place at the timescale of milliseconds, and is typic... more Electrical neural signalling typically takes place at the timescale of milliseconds, and is typically modeled using the cable equation. This is a good approximation when ionic concentrations are expected to vary little during the time course of a simulation. During periods of intense neural signalling, however, the local extracellular K +-concentration may increase by several millimolars. Clearance of excess K + likely depends partly on diffusion in the extracellular space, partly on local uptake by-and intracellular transport within astrocytes. The processes that maintain the extracellular environment typically takes place at the time scale of seconds, and cannot be modeled accurately without accounting for the spatiotemporal variations in ion concentrations. This work presented here consists of two main parts: First, we developed a general electrodiffusive formalism for modeling ion concentration dynamics in a one-dimensional geometry, including both an intra-and extracellular domain. The formalism was based on the Nernst-Planck equations. It ensures (i) that the membrane potential and ion concentrations are in consistency, (ii) global particle/charge conservation, and (iii) accounts for diffusion and concentration dependent variations in resistivities. Second, we applied the formalism to a model of astrocytes exchanging ions with the ECS. Through simulations, we identified the key astrocytic mechanisms involved in K + removal from high concentration regions. We found that a local increase in extracellular K + evoked a local depolarization of the astrocyte membrane, which at the same time (i) increased the local astrocytic uptake of K + (by locally inactivating the outward Kir-current), (ii) suppressed extracellular transport of K + , (iii) increased transport of K + within astrocytes, and (iv) facilitated astrocytic relase of K + in regions where the extracellular concentration was low. In summary, these mechanisms seem optimal for shielding the extracellular space from excess K + .
Despite its century-old use, the interpretation of local field potentials (LFPs), the low-frequen... more 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 activation, 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 frequency dependence of LFP are (1) the spatial decay of the singleneuron LFP contribution and (2) the translation 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 in the spatial reach is observed for higher frequencies compared to the near-DC value (∼ 0 Hz) of about 200 µm. Much larger frequency-dependent effects are seen when populations of pyramidal neurons receive correlated and spatially asymmetric inputs (basally or apically): the low-frequency (∼ 0 Hz) LFP power can here be an order of magnitude or more larger than the LFP power at, say, 60 Hz. Moreover, the low-frequency LFP components are found to have larger spatial reach and extend further outside the active population than high-frequency components. Our numerical findings are backed up by an intuitive simplified model for the generation of population LFP.
The local field potential (LFP) reflects activity of many neurons in the vicinity of the recordin... more The local field potential (LFP) reflects activity of many neurons in the vicinity of the recording electrode and is therefore useful for studying local network dynamics. Much of the nature of the LFP is, however, still unknown. There are, for instance, contradicting reports on the spatial extent of the region generating the LFP. Here, we use a detailed biophysical modeling approach to investigate the size of the contributing region by simulating the LFP from a large number of neurons around the electrode. We find that the size of the generating region depends on the neuron morphology, the synapse distribution, and the correlation in synaptic activity. For uncorrelated activity, the LFP represents cells in a small region (within a radius of a few hundred micrometers). If the LFP contributions from different cells are correlated, the size of the generating region is determined by the spatial extent of the correlated activity.
A new method for estimation of current-source density (CSD) from local field potentials is presen... more A new method for estimation of current-source density (CSD) from local field potentials is presented. This inverse CSD (iCSD) method is based on explicit inversion of the electrostatic forward solution and can be applied to data from multielectrode arrays with various geometries. Here, the method is applied to linear-array (laminar) electrode data. Three iCSD methods are considered: the CSD is assumed to have cylindrical symmetry and be (i) localized in infinitely thin discs, (ii) step-wise constant or (iii) continuous and smoothly varying (using cubic splines) in the vertical direction. For spatially confined CSD distributions the standard CSD method, involving a discrete double derivative, is seen in model calculations to give significant estimation errors when the lateral source dimension is comparable to the size of a cortical column (less than ∼1 mm). Further, discontinuities in the extracellular conductivity are seen to potentially give sizable errors for even wider source distributions. The iCSD methods are seen to give excellent estimates when the correct lateral source dimension and spatial distribution of conductivity are incorporated. To illustrate the application to real data, iCSD estimates of stimulus-evoked responses measured with laminar electrodes in the rat somatosensory (barrel) cortex are compared to estimates from the standard CSD method.
We present a new method, laminar population analysis (LPA), for analysis of laminar-electrode (li... more We present a new method, laminar population analysis (LPA), for analysis of laminar-electrode (linear multielectrode) data, where physiological constraints are explicitly incorporated in the mathematical model: the high-frequency band [multiunit activity (MUA)] is modeled as a sum over contributions from firing activity of multiple cortical populations, whereas the low-frequency band [local field potential (LFP)] is assumed to reflect the dendritic currents caused by synaptic inputs evoked by this firing. The method is applied to stimulus-averaged laminar-electrode data from barrel cortex of anesthetized rat after single whisker flicks. Two sample data sets, distinguished by stimulus paradigm, type of applied anesthesia, and electrical boundary conditions, are studied in detail. These data sets are well accounted for by a model with four cortical populations: one supragranular, one granular, and two infragranular populations. Population current source densities (CSDs; the CSD signat...
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Papers by Klas Pettersen