Finite Element Simulation of Ionic Electrodiffusion in Cellular Geometries.
TL;DR: This paper introduces and numerically evaluates a new, finite element-based numerical scheme for the KNP-EMI model, capable of efficiently and flexibly handling geometries of arbitrary dimension and arbitrary polynomial degree and studies ephaptic coupling induced in an unmyelinated axon bundle.
Abstract: Mathematical models for excitable cells are commonly based on cable theory, which considers a homogenized domain and spatially constant ionic concentrations. Although such models provide valuable insight, the effect of altered ion concentrations or detailed cell morphology on the electrical potentials cannot be captured. In this paper, we discuss an alternative approach to detailed modeling of electrodiffusion in neural tissue. The mathematical model describes the distribution and evolution of ion concentrations in a geometrically-explicit representation of the intra- and extracellular domains. As a combination of the electroneutral Kirchhoff-Nernst-Planck (KNP) model and the Extracellular-Membrane-Intracellular (EMI) framework, we refer to this model as the KNP-EMI model. Here, we introduce and numerically evaluate a new, finite element-based numerical scheme for the KNP-EMI model, capable of efficiently and flexibly handling geometries of arbitrary dimension and arbitrary polynomial degree. Moreover, we compare the electrical potentials predicted by the KNP-EMI and EMI models. Finally, we study ephaptic coupling induced in an unmyelinated axon bundle and demonstrate how the KNP-EMI framework can give new insights in this setting.
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TL;DR: The electrodiffusive Pinsky-Rinzel (edPR) model is believed to be 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.
Abstract: In most neuronal models, ion concentrations are assumed to be constant, and effects of concentration variations on ionic reversal potentials, or of ionic diffusion on electrical potentials are not accounted for. Here, we present the electrodiffusive Pinsky-Rinzel (edPR) model, which we believe is 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 edPR model is an expanded version of the two-compartment Pinsky-Rinzel (PR) model of a hippocampal CA3 neuron. Unlike the PR model, the edPR model includes homeostatic mechanisms and ion-specific leakage currents, and keeps track of all ion concentrations (Na+, K+, Ca2+, and Cl-), electrical potentials, and electrical conductivities in the intra- and extracellular space. The edPR model reproduces the membrane potential dynamics of the PR model for moderate firing activity. For higher activity levels, or when homeostatic mechanisms are impaired, the homeostatic mechanisms fail in maintaining ion concentrations close to baseline, and the edPR model diverges from the PR model as it accounts for effects of concentration changes on neuronal firing. We envision that the edPR model will be useful for the field in three main ways. Firstly, as it relaxes 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 should supplement the PR model when simulating scenarios where ion concentrations are expected to vary over time. Thirdly, being applicable to conditions with failed homeostasis, the edPR model opens up for simulating a range of pathological conditions, such as spreading depression or epilepsy.
19 citations
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..., the previously developed Kirchhoff-Nernst-Planck framework [31, 32, 60, 62]....
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Posted Content•
TL;DR: In this paper, new electrical laws in a cell, which follow from a nonlinear electro-diffusion model, are derived from the geometrical cell-membrane properties, such as membrane curvature, volume, and surface area.
Abstract: Voltage and charge distributions in cellular microdomains regulate communications, excitability, and signal transduction. We report here new electrical laws in a cell, which follow from a nonlinear electro-diffusion model. These newly discovered relations derive from the geometrical cell-membrane properties, such as membrane curvature, volume, and surface area. These electro-diffusion laws can now be used to predict and interpret voltage distribution in cellular microdomains.
14 citations
TL;DR: The edNEG model as mentioned in this paper combines compartmental neuron modeling with an electrodiffusive framework for intra-and extracellular ion concentration dynamics in a local piece of neuro-glial brain tissue.
Abstract: Within the computational neuroscience community, there has been a focus on simulating the electrical activity of neurons, while other components of brain tissue, such as glia cells and the extracellular space, are often neglected. Standard models of extracellular potentials are based on a combination of multicompartmental models describing neural electrodynamics and volume conductor theory. Such models cannot be used to simulate the slow components of extracellular potentials, which depend on ion concentration dynamics, and the effect that this has on extracellular diffusion potentials and glial buffering currents. We here present the electrodiffusive neuron-extracellular-glia (edNEG) model, which we believe is the first model to combine compartmental neuron modeling with an electrodiffusive framework for intra- and extracellular ion concentration dynamics in a local piece of neuro-glial brain tissue. The edNEG model (i) keeps track of all intraneuronal, intraglial, and extracellular ion concentrations and electrical potentials, (ii) accounts for action potentials and dendritic calcium spikes in neurons, (iii) contains a neuronal and glial homeostatic machinery that gives physiologically realistic ion concentration dynamics, (iv) accounts for electrodiffusive transmembrane, intracellular, and extracellular ionic movements, and (v) accounts for glial and neuronal swelling caused by osmotic transmembrane pressure gradients. The edNEG model accounts for the concentration-dependent effects on ECS potentials that the standard models neglect. Using the edNEG model, we analyze these effects by splitting the extracellular potential into three components: one due to neural sink/source configurations, one due to glial sink/source configurations, and one due to extracellular diffusive currents. Through a series of simulations, we analyze the roles played by the various components and how they interact in generating the total slow potential. We conclude that the three components are of comparable magnitude and that the stimulus conditions determine which of the components that dominate.
10 citations
TL;DR: In this article, the authors proposed the EMI model for cardiac electrophysiology, which allows for detailed analysis of the biophysical processes going on in functionally important spaces very close to individual myocytes, although at the cost of significantly increased CPU-requirements.
Abstract: Computational modeling has contributed significantly to present understanding of cardiac electrophysiology including cardiac conduction, excitation-contraction coupling, and the effects and side-effects of drugs. However, the accuracy of in silico analysis of electrochemical wave dynamics in cardiac tissue is limited by the homogenization procedure (spatial averaging) intrinsic to standard continuum models of conduction. Averaged models cannot resolve the intricate dynamics in the vicinity of individual cardiomyocytes simply because the myocytes are not present in these models. Here we demonstrate how recently developed mathematical models based on representing every myocyte can significantly increase the accuracy, and thus the utility of modeling electrophysiological function and dysfunction in collections of coupled cardiomyocytes. The present gold standard of numerical simulation for cardiac electrophysiology is based on the bidomain model. In the bidomain model, the extracellular (E) space, the cell membrane (M) and the intracellular (I) space are all assumed to be present everywhere in the tissue. Consequently, it is impossible to study biophysical processes taking place close to individual myocytes. The bidomain model represents the tissue by averaging over several hundred myocytes and this inherently limits the accuracy of the model. In our alternative approach both E, M, and I are represented in the model which is therefore referred to as the EMI model. The EMI model approach allows for detailed analysis of the biophysical processes going on in functionally important spaces very close to individual myocytes, although at the cost of significantly increased CPU-requirements.
10 citations
TL;DR: The edPR model is an expanded version of the two-compartment Pinsky-Rinzel (PR) model of a hippocampal CA3 neuron 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.
Abstract: 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 summary Neurons 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.
10 citations
References
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TL;DR: This article concludes a series of papers concerned with the flow of electric current through the surface membrane of a giant nerve fibre by putting them into mathematical form and showing that they will account for conduction and excitation in quantitative terms.
Abstract: This article concludes a series of papers concerned with the flow of electric current through the surface membrane of a giant nerve fibre (Hodgkinet al, 1952,J Physiol116, 424–448; Hodgkin and Huxley, 1952,J Physiol116, 449–566) Its general object is to discuss the results of the preceding papers (Section 1), to put them into mathematical form (Section 2) and to show that they will account for conduction and excitation in quantitative terms (Sections 3–6)
19,800 citations
Book•
16 Jul 2001
TL;DR: The superfamily of voltage-gated channels was studied in this paper, where a classical biophysics of the squid giant axon was discussed. But the superfamily was not considered in this paper.
Abstract: PART I Classical biophysics of the squid giant axon The superfamily of voltage-gated channels Voltage-gated calcium channels Potassium channels and chloride channels Ligand-gated channels of fast chemical synapses Modulation, slow synaptic action, and second messengers Sensory transduction and excitable cells Calcium dynamics, epithelial transport, and intercellular coupling PART II Elementary properties of ions in solution Elementary properties of pores Counting channels Structure of channel proteins Selective permeability: Independence Selective permeability: Saturation and binding Classical mechanisms of block Structure-function studies of permeation and block Gating mechanisms: Kinetic thinking Gating: Voltage sensing and inactivation Modification of gating in voltage-sensitive channels Cell biology and channels Evolution and origins
3,678 citations
01 Jan 1996
TL;DR: The action potential is triggered when the membrane potential, which was at the resting level, depolarizes and reaches the threshold of excitation, which triggers the action potential.
Abstract: Excitability. Excitability of cell membranes is crucial for signaling in many types of cell. Excitation in the physiological sense means that the cell membrane potential undergoes characteristic changes which, in most cases, go in the depolarizing direction. Single depolarization from the resting potential to potentials near 0 mV has generally been called an action potential. A schematic representation of a neuronal action potential is given in Fig. 12.1 A. The action potential is triggered when the membrane potential, which was at the resting level, depolarizes and reaches the threshold of excitation. This depolarization, which triggers the action potential, is generated by depolarizing synaptic currents, or depolarizing current coming from a membrane region that is already excited (propagation of an action potential), or by pacemaker currents mediated by pacemaker channels, or by current injected externally by an electrode. The duration of different types of action potential varies from seconds to less than 1 ms.
3,016 citations
"Finite Element Simulation of Ionic ..." refers background in this paper
...…HH gating value (Na+ inactivation h0 0.688 Hodgkin and Huxley, 1952 Initial HH gating value (K+ activation) n0 0.276 Hodgkin and Huxley, 1952 Global time step 1t 1.0 · 10−5 s Local time step 1t∗ 1t/25 s The values are collected from Hodgkin and Huxley (1952), Hille (2001), and Pods et al. (2013)....
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...…· 104 C/mol Membrane capacitance CM 0.01 F/m Na+ diffusion coefficient DNar 1.33 · 10 −9 m2/s Hille, 2001 K+ diffusion coefficient DKr 1.96 · 10 −9 m2/s Hille, 2001 Cl− diffusion coefficient DClr 2.03 · 10 −9 m2/s Hille, 2001 Na+ leak conductivity gNaL 2.0 S/m 2 K+ leak conductivity gKL 8.0…...
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...…coefficient DNar 1.33 · 10 −9 m2/s Hille, 2001 K+ diffusion coefficient DKr 1.96 · 10 −9 m2/s Hille, 2001 Cl− diffusion coefficient DClr 2.03 · 10 −9 m2/s Hille, 2001 Na+ leak conductivity gNaL 2.0 S/m 2 K+ leak conductivity gKL 8.0 S/m 2 Cl- leak conductivity gClL 0 S/m 2 K+ HH max…...
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...…J/(K mol) Temperature T 300 K Faraday’s constant F 9.648 · 104 C/mol Membrane capacitance CM 0.01 F/m Na+ diffusion coefficient DNar 1.33 · 10 −9 m2/s Hille, 2001 K+ diffusion coefficient DKr 1.96 · 10 −9 m2/s Hille, 2001 Cl− diffusion coefficient DClr 2.03 · 10 −9 m2/s Hille, 2001 Na+ leak…...
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Book•
07 Aug 1998
1,924 citations
DOI•
07 Dec 2015
TL;DR: The FEniCS Project is a collaborative project for the development of innovative concepts and tools for automated scientific computing, with a particular focus on the solution of differential equations by finite element methods.
Abstract: The FEniCS Project is a collaborative project for the development of innovative concepts and tools for automated scientific computing, with a particular focus on the solution of differential equations by finite element methods. The FEniCS Projects software consists of a collection of interoperable software components, including DOLFIN, FFC, FIAT, Instant, UFC, UFL, and mshr. This note describes the new features and changes introduced in the release of FEniCS version 1.5.
1,628 citations
"Finite Element Simulation of Ionic ..." refers methods in this paper
...Implementation The numerical scheme was implemented using a mixed dimensional framework from the FEniCS finite element library (Alnæs et al., 2015)....
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