Can ionic diffusion have an effect on extracellular potentials?

In computational neuroscience, it is common to use the simplifying assumption that diffusive currents are negligible compared to Ohmic currents. However, endured periods of intense neural signaling may cause local ion concentration changes in the millimolar range. Theoretical studies have identified scenarios where steep concentration gradients give rise to diffusive currents that are of comparable magnitude with Ohmic currents, and where the simplifying assumption that diffusion can be neglected does not hold. We here propose a novel formalism for computing (1) the ion concentration dynamics and (2) the electrical potential in the extracellular space surrounding multi-compartmental neuron models or networks of such (e.g., the Blue-Brain simulator). We use this formalism to explore the effects that diffusive currents can have on the extracellular (ECS) potential surrounding a small population of active cortical neurons. Our key findings are: (i) Sustained periods of neuronal output ...

PLoS computational biology, 2016

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 ...

Advances in Experimental Medicine and Biology

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.

PLOS Computational Biology

Many pathological conditions, such as seizures, stroke, and spreading depression, are associated with substantial changes in ion concentrations in the extracellular space (ECS) of the brain. An understanding of the mechanisms that govern ECS concentration dynamics may be a prerequisite for understanding such pathologies. To estimate the transport of ions due to electrodiffusive effects, one must keep track of both the ion concentrations and the electric potential simultaneously in the relevant regions of the brain. Although this is currently unfeasible experimentally, it is in principle achievable with computational models based on biophysical principles and constraints. Previous computational models of extracellular ion-concentration dynamics have required extensive computing power, and therefore have been limited to either phenomena on very small spatiotemporal scales (micrometers and milliseconds), or simplified and idealized 1-dimensional (1-D) transport processes on a larger scale. Here, we present the 3-D Kirchhoff-Nernst-Planck (KNP) framework, tailored to explore electrodiffusive effects on large spatiotemporal scales. By assuming electroneutrality, the KNP-framework circumvents charge-relaxation processes on the spatiotemporal scales of nanometers and nanoseconds, and makes it feasible to run simulations on the spatiotemporal scales of millimeters and seconds on a standard desktop computer. In the present work, we use the 3-D KNP framework to simulate the dynamics of ion concentrations and the electrical potential surrounding a morphologically detailed pyramidal cell. In addition to elucidating the single neuron contribution to electrodiffusive effects in the ECS, the simulation demonstrates the efficiency of the 3-D KNP framework. We envision that future applications of the framework to more complex and biologically realistic systems will be useful in exploring pathological conditions associated with large concentration variations in the ECS.

2018

Ionic homeostasis in the brain involves redistribution of ionic fluxes in several cell types and compartments, including neurons, astrocytes and the extracellular space. How the major ionic activity-dependent fluxes of potassium and sodium are individually regulated remains difficult to dissociate and to track experimentally. We here review recent progress in modeling the ionic fluxes evoked by neuronal activity based on mass conservation. Excitability of neurons indeed relies on inward sodium and outward potassium fluxes during action potential firing. Recently, we have developed a tri-compartment model based on mass-action kinetics equations that account for potassium dynamics between neurons, astrocytes and the extracellular space. This review describes how such type of model can be used to spatially and temporally predict potassium fluxes during various regimes of neuronal activity. In particular, the model initially showed that it takes several seconds for astrocytes to buffer ...

PLoS Computational Biology, 2013

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 + .

Journal of Computational Neuroscience, 2010

Recordings of extracellular potentials have been, and still are, a main workhorse in the quest for understanding how the brain works. The high-frequency part of the signal, the multi-unit activity (MUA), mainly reflects firing of action potentials in the vicinity of the electrode. The lowfrequency part, the local-field potential (LFP), appears largely to reflect subthreshold activity from a larger group of surrounding neurons. Despite their long history in neuroscience, these extracellular potentials are still not well understood. What can the LFP tell us about the underlying activity in neurons and neuronal networks? Which neurons dominate the MUA? These and similar questions need to be answered for us to take full advantage of the host of new types of multielectrodes with new geometries and ever more electrode contacts. Further, a better understanding of these issues will likely also be important for the development of good brain-machine interfaces. In all these problems, computational models and model-based analysis techniques are essential.

Two mathematical models are part of the foundation of Computational neurophysiology; (a) the Cable equation is used to compute the membrane potential of neurons, and, (b) volume-conductor theory describes the extracellular potential around neurons. In the standard procedure for computing extracellular potentials, the transmembrane currents are computed by means of (a) and the extracellular potentials are computed using an explicit sum over analytical point-current source solutions as prescribed by volume conductor theory. Both models are extremely useful as they allow huge simplifications of the computational efforts involved in computing extracellular potentials. However, there are more accurate, though computationally very expensive, models available where the potentials inside and outside the neurons are computed simultaneously in a self-consistent scheme. In the present work we explore the accuracy of the classical models (a) and (b) by comparing them to these more accurate schemes. The main assumption of (a) is that the ephaptic current can be ignored in the derivation of the Cable equation. We find, however, for our examples with stylized neurons, that the ephaptic current is comparable in magnitude to other currents involved in the computations, suggesting that it may be significant—at least in parts of the simulation. The magnitude of the error introduced in the membrane potential is several millivolts, and this error also translates into errors in the predicted extracellular potentials. While the error becomes negligible if we assume the extracellular conductivity to be very large, this assumption is, unfortunately, not easy to justify a priori for all situations of interest.

Advances in Cognitive Neurodynamics, 2014

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 resistivities.

Journal of Open Source Software