In neurophysiology, extracellular signals-as measured by local field potentials (LFP) or electroencephalography-are of great significance. Their exact biophysical basis is, however, still not fully understood. We present a three-dimensional model exploiting the cylinder symmetry of a single axon in extracellular fluid based on the Poisson-Nernst-Planck equations of electrodiffusion. The propagation of an action potential along the axonal membrane is investigated by means of numerical simulations. Special attention is paid to the Debye layer, the region with strong concentration gradients close to the membrane, which is explicitly resolved by the computational mesh. We focus on the evolution of the extracellular electric potential. A characteristic up-down-up LFP waveform in the far-field is found. Close to the membrane, the potential shows a more intricate shape. A comparison with the widely used line source approximation reveals similarities and demonstrates the strong influence of membrane currents. However, the electrodiffusion model shows another signal component stemming directly from the intracellular electric field, called the action potential echo. Depending on the neuronal configuration, this might have a significant effect on the LFP. In these situations, electrodiffusion models should be used for quantitative comparisons with experimental data.

The extracellular space has an ambiguous role in neuroscience. It is present in every physiologically relevant system and often used as a measurement site in experimental recordings, but it has received subordinate attention compared to the intracellular domain. In computational modeling, it is often regarded as a passive, homogeneous resistive medium with a constant conductivity, which greatly simplifies the computation of extracellular potentials. However, novel studies have shown that local ionic diffusion and capacitive effects of electrically active membranes can have a substantial impact on the extracellular potential. These effects can not be described by traditional models, and they have been subject to recent theoretical and experimental analyses. We strive to give an overview over current progress in modeling the extracellular space with special regard towards the concentration and potential dynamics on different temporal and spatial scales. Three models with distinct assumptions and levels of detail are compared both theoretically and by means of numerical simulations: the classical volume conductor (VC) model, which is most frequently used in form of the line source approximation (LSA); the biophysically detailed, but computationally intensive Poisson-Nernst-Planck model of electrodiffusion (PNP); and an intermediate model called the electroneutral model (EN). The results clearly show that there is no one model for all applications, as they show significantly different responses - especially close to neuronal membranes. Finally, we list some common use cases for model simulations and give recommendations on which model to use in each situation.

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We discovered that, when comparing our model to the line source approximation (LSA), we missed including the capacitive current I C as part of the total transmembrane current I in the LSA equation ( 13) in the original article. While the results of our simulations are not affected by this -the electrodiffusion model includes the influence of capacitive currents implicitly -

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