Electrodiffusive Model for Astrocytic and Neuronal Ion Concentration Dynamics

Abstract

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+.

Effect of Ionic Diffusion on Extracellular Potentials in Neural Tissue

“…Whereas K + buffering currents through the glia-cell membranes are believed to be the main source of these slow potential shifts [3, 7], it has been estimated that also diffusive currents along extracellular concentration gradients could contribute by shifting ECS potentials by up to 0.4 mV [3]. As ion concentrations in the ECS typically vary on the time scale of seconds [3, 4, 28], it is nevertheless a priori unclear whether diffusion-evoked potential shifts would be picked up by the electrode measurement systems applied in most experiments, which typically have cut-off frequencies of about 0.1–0.2 Hz or higher (see e.g., [29, 30]).…”

“…This requires an extremely high spatiotemporal resolution, which makes PNP models computationally expensive and unsuited for predictions at the tissue/population level [52]. However, a series of modelling schemes have been developed that circumvent the charge relaxation processes, essentially by replacing Poisson’s equation by the constraint that the bulk solution is electroneutral [28, 52–58]. The electroneutrality condition is a physical constraint valid at a larger spatiotemporal scale, and thus allows for a dramatic increase in the spatial and temporal grid sizes in the numerical simulations.…”

“…The electroneutrality condition is a physical constraint valid at a larger spatiotemporal scale, and thus allows for a dramatic increase in the spatial and temporal grid sizes in the numerical simulations. One of these simpler models were previously developed by our group [28, 57], and is here referred to as the Kirchhoff-Nernst-Planck (KNP) scheme. The KNP scheme is a means of deriving the local potential in the intra- and extracellular bulk solution from the constraint that Kirchhoff’s current law should be fulfilled for all finite volumes (the sum of currents into a finite subvolume of bulk solution should be zero).…”

“…First, it utilizes the NEURON simulator [59, 60], which is a standard tool for simulating morphologically complex neurons, to simulate the activity of a neural population and its exchange of ions with the ECS (Fig 1A). Second, it utilizes the KNP formalism [28, 57] to compute the dynamics of ion concentrations and the electrical potential in the ECS surrounding the neurons (Fig 1B). The KNP scheme accounts for all electrical currents entering an ECS subvolume in the system (i.e., transmembrane ionic currents, transmembrane capacitive currents, diffusive currents through the ECS, and field currents through the ECS), as well as for concentration-dependent variations in the ECS conductivity (see Methods).…”

Electrodiffusion phenomena in neuroscience: a neglected companion

“…Subsequently, there had been several attempts to improve the solution accuracy for small spaces by introducing fractional Nernst-Planck equations combined with the corresponding fractional cable equations, to model ion electrodiffusion in nerve cells 77,78 . Whilst the cable equation still provides a well-trodden path to study the cell membrane electrodynamics efforts are being made at adopting the Nernst-Planck equations more widely, to model electrogenic processes in neurons and glia more accurately 79,80 .[H3]Cell-impermeable anions (CIAs) and perturbation of electroneutrality. The cell cytoplasm hosts a variety of CIAs -proteins, hydrogen phosphate groups, sulphates, other organic macromolecules -that remain negatively charged at intracellular pH 81 .…”