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+.
The interaction between neurons and glial cells has been the topic of many recent studies within the field of neuroscience (see reviews in [1]–[3]). Astrocytes (a species of glial cells) play an important role in modulating excitatory and inhibitory synapses by removal, metabolism, and release of neurotransmitters [4], homeostatic maintenance of extracellular K+, H+, and glutamate [5], supply of energy substrates for neurons [6], and neuronal pathfinding during development and regeneration [7]. Astrocytic cells seem to have key roles in many central nervous system disorders, ranging from neuropathic pain and epilepsy to neurodegenerative diseases such as Alzheimers, schizophrenia and depression [8]. Computational models of neuron-glia interactions is a prerequisite for understanding the dysfunctional situations, and for assessing glial cells as a potential therapeutic target [9]. To give a few examples, such models have been used to simulate glial regulation of extracellular K+-concentration [10]–[13], and the relation between extracellular K+-dynamics and epileptic seizures [14]–[16] and spreading depression [17], [18]. Regulation of the extracellular K+-concentration is considered one of the key cellular functions of astrocytes [2]. During normal conditions, the extracellular K+-concentration () is typically maintained close to the baseline level (). However, when neurons fire action potentials, they expel K+ into the extracellular space. During periods of intense neural activity, the local extracellular K+-concentration may increase by several millimolars, and may interfere with neural activity [10], [19], [20]. Concentrations between 8 and 12 mM are often considered a limit to pathological conditions [3], [12], [21]. Orkand (1966) [22] discovered that astrocytes can funnel out excess K+ from high concentration regions by a process coined spatial buffering [12], [21], [22]. According to this concept, K+ is taken up by the glial cell from high-concentration sites, evoking a local depolarization of the glial membrane. K+ is then transported longitudinally inside the glial cell (and possibly through several glial cells connected by gap junctions into a glial syncytium [10], [23]), and eventually expelled into the ECS at more distal cites where is lower. However, it has also been argued that astrocytes may reduce by local uptake and temporal storage, not necessarily including transport over distances [19], [24]. Furthermore, diffusion through the ECS is also involved in transporting excess K+ out from high concentration regions. The relative importance of these different clearance mechanisms are under debate [25]. Electrical neural signalling is typically modeled using the cable equation, where dendrites and axons are represented as one-dimensional, possibly branching, electrical cables, and the transmembrane potential is the key dynamical variable [26], [27]. With the possible exception of the signalling molecule Ca2+ (see e. g. , [28], [29]), ion concentrations are typically assumed to be constant. The effect of ionic diffusion (due to concentration gradients) on the net electrical currents is neglected in standard cable theory, and resistivities (which in reality depend on ion concentrations) are assumed to be constant. These are often good approximations, as concentrations of the main charge carriers (K+, Na+ and Cl−) in the extracellular- (ECS) or intracellular space (ICS) typically vary little at the short time-scale relevant for electrical neural activity (). Glial function typically involves processes that take place at a longer time-scale (), at which significant variations in ionic concentrations may occur. For example, the process of spatial K+-buffering involves local uptake, a local depolarization of the astrocytic membrane, and longitudinal electrodiffusive transports through the intracellular- (ICS) and extracellular space (ECS) propelled both by voltage- and concentration gradients [30]. A mechanistic understanding of glial function thus requires a modelling scheme that in a consistent way can capture the intricate interplay between ion concentration dynamics and the dynamics of. Physically, is determined by the total electrical charge on the inside (or outside) of the membrane, which in turn is uniquely determined by the concentrations () of all ionic species that are present there [31]. In some heart cell models, ion concentrations have been reported to drift to unrealistic values in long-term simulations, while maintain realistic values [32]–[34]. Whether the relationship between and is consistent, is a general concern with models that explicitly depend on both. If applied to general problems, and in particular in long-term simulations, models that do not ensure an internally consistent relationship may give erroneous predictions. Gardner-Medwin (1983) [10] proposed a pioneering computational model of the spatial buffering process, later re-analyzed by Chen and Nicholson (2000) [12]. In this model, spatial buffering was considered as an essentially one-dimensional transport process. The complex composition of the tissue (Fig. 1A) could then be simplified to a two-domain model as that illustrated in Fig. 1B [10], [12]. There, the ICS of all cells participating in the transport process (i. e. the astrocytes) have been represented as an equivalent cable (I-domain) which is coated by ECS (E-domain). The I-E system could be pictured phenomenologically as an representative single astrocyte, coated with the average proportion of available ECS per astrocyte. This geometrical simplification was motivated for one-dimensional transport phenomena through the glial syncytium [10], [12], but could in principle apply to any transport phenomena that justifies a geometrical simplification as that in Fig. 1. A limitation with these modelling studies [10], [12], and related modelling studies by Newman and coworkers [11], [21], is that was derived from standard cable theory, which neglects effect from diffusive currents on. The concern regarding a consistent relationship between and the ionic concentrations thus also applies to these models. Qian and Sejnowski (1989) have previously developed a consistent, electrodiffusive scheme for modelling the dynamics on and ion concentrations [31]. Like the standard cable model, the electrodiffusive model assumes that transport phenomena are essentially one-dimensional. Unlike the standard cable model, the electrodiffusive model derived from the ion concentration dynamics, accounting for all ionic movements (membrane fluxes, longitudinal diffusion, and longitudinal electrical migration), as well as for the concentration-dependent variation of the intracellular resistivities. An important limitation with this previous electrodiffusive model is that it only includes intracellular dynamics, whereas the ECS was assumed to be isopotential and with constant ion concentrations [31]. This was a useful simplification for simulating a small intracellular compartment, such as a dendritic spine [31], but is not generally applicable to macroscopic transport mechanisms. In particular, it can not be applied for modelling the spatial buffering process, where ion concentration dynamics in the ECS plays a paramount role. In reality, the ECS comprises about 20% of the total neural tissue volume, while the remaining 80% is the ICS of various cells [12]. When a large number of cells participate in simultaneous ion exchange with the ECS, the impact on the ion concentrations in the ICS and ECS may be of the same order of magnitude. The aim of this work is twofold: First, we generalize the electrodiffusive formalim [31] to a explicitly include the ECS. The result is a general mathematical framework for consistently modelling the dynamics of the membrane potential (), the intra- () and extracellular () ion concentrations for a set () of ionic species. We believe that this framework will be of general value for the field of neuroscience, as it can be applied to any system that justifies a geometrical description as that in Fig. 1B. Next, we apply the electrodiffusive formalism in a spatially explicit model of astrocytes exchanging ions with the ECS. We run simulations to investigate the efficiency of the spatial K+-buffering process, relative to that of local uptake/storage by astrocytes, and that of diffusion in the ECS alone. Unlike the previous models [10]–[12], [21], our astrocyte model is based on the prevailing view that Na+/K+/ATPase-pump is the main uptake mechanism for K+ [3]. Furthermore, as our model was based on a physically consistent electrodiffusive formalism, we arrive at a full mechanistic description of the buffering process, which quantitatively describes the intricate interplay between and the dynamics of ion concentrations. This article is organized in the following way: The Model section contains two main parts. In the first part, we present the electrodiffusive formalism for computing the ion concentration dynamics in a system described by the geometry depicted in Fig. 1B. We consider this theoretical framework a key contribution of this work. However, the key concepts introduced in this part are summarized in Table 1, and with this in hand, the reader who is mainly interested the biological process of spatial K+-buffering by astrocytes may therefore skip to second part of the Model-section. There, the model for astrocytes exchanging ions with the ECS is presented. The Results section is devoted to simulations on the astrocyte model, and provides an improved biophysical insight in the electrodiffusive mechanisms utilized by astrocytes to spatially buffer K+. By comparing different versions of the model, we also assessed the importance of spatial buffering, relative to that of other clearance mechanisms such as local uptake/storage by astrocytes and diffusion through the ECS alone. Finally, in the Discussion section we address how our mathematical framework relates to previous electrodiffusive modeling frameworks. We also summarize the new insights that our simulations have given in the process of spatial K+-buffering by astrocytes.
In Fig. 1B, particles in I or E may move along the x-axis or across the membrane. In a segment of I, centered at x, and with volume, the particle concentration dynamics of an ion species is determined by: (1) where the transmembrane- (), the intracellular- () and the extracellular () flux densities of particle species, have units mol/ (m2s). The first term on the left represents the ionic flux that enter this segment through the piece of the membrane with area. The negative sign follows from (by convention) being defined as positive in the direction from I to E. The second and third terms represent the ionic fluxes that enter (+) /leave (−) the section through the left/right boundaries, with cross section areas. If the net flux into the segment is nonzero, the ion concentration will build up over time, according to the right hand side of Eq. 1. We divide Eq. 1 by, and take the limit, to obtain the continuity equation on differential form: (2) (3) We have also written up the continuity equation for the extracellular domain. The axial flux densities are described by the generalized Nernst-Planck equation: (4) where is the valence of ion species, and the index n represents I or E. The first term on the right in Eq. 4 is the diffusive flux density (), driven by the concentration gradients, and the last term is the field flux density (), i. e. , the flux density due to ionic migration in the electrical field. The effective diffusion constant is composed of the diffusion constant in dilute solutions and the tortuosity factor, which summarizes the hindrance imposed by the cellular structures [12], [35]. We use, where is the gas constant, the absolute temperature, and is Faraday' s constant. The formalism is general to the form of, which may include contribution from multiple membrane mechanisms, such as ion pumps, co-transporters and ion channels. It is sufficient to require that is known at any point in time given the voltage across the membrane, the ionic concentrations on either side of the membrane, and possibly some additional local information () reflecting the local state of the membrane: (5) As boundary conditions, we shall apply the sealed-end condition, i. e. , we assume that no fluxes enter or leave through the ends (and) of I or E: (6) Equations 2–3, together with with Eqs. 4,5 and 6, specify the system we want to solve. Before we derive the electrodiffusive formalism for this problem, we recall how the standard cable equation can be derived from the principles of particle conservation. We here present a model of astrocytes exchanging ions with the ECS, as sketched in Fig. 3, and defined in further detail below. The astrocyte model was developed for macroscopic transport processes, involving a collection of astrocytes (possibly connected via gap junction into a syncytium) in a piece of tissue. For this problem, we used the geometrical simplification motivated in Fig. 1, i. e. , we applied the geometry in Fig. 1B. We took the intracellular domain to represents a phenomenological “average” astrocyte (the cable,), surrounded by a sheet of ECS (the coating,). We used the empirical estimates that a fraction of neural tissue volume is ECS, while astrocytes take up a fraction of about of the total tissue volume [12]. The intracellular domain was therefore twice as voluminous as the intracellular. Table 1 contains a list of definitions that are necessary for the reader to follow the remainder of the paper. The dynamics in the system was due to fluxes of ions crossing the membrane, or axial fluxes in the ECS or ICS due to diffusion () or migration in the electrical field (). We assumed that only the three main charge carriers (K+, Na+ and Cl−) contributed to electrodiffusive transport. For the diffusion constants (), we used values valid for electrodiffusion in diluted media [36], modified with the tortuosities () estimated in [12]. The same values have also been used in earlier, related studies [31], [37]. All relevant model parameters are listed in Table 2. The system input, and the astrocytic membrane mechanisms are defined in further details below.
We investigated the ion concentration dynamics in the astrocyte model (Fig. 3) in full detail. Fig. 4A–D shows the dynamics of selected variables in the input zone (at). Fig. 4E–H shows how the same variables depend on at a time when the system was in SS. We explain this further below. The input was applied from to in the input zone (). This is illustrated in Fig. 4A (solid line), which shows the flux density of K+ () entering the system in the input zone. We recall that the input was a cation exchange, so that there was an equal flux density of Na+ leaving the system (). For simplicity, was not included in the figure, but we keep in mind that whenever K+ entered/left the system, an equal amount of Na+ left/entered. The cation-exchange input thus caused an increase in and a decrease in in the input zone. This can be seen in Fig. 4B. The notation represents the deviations from baseline concentration (cf. Table 2). As increased, the output from the system (being proportional to) increased. Also this is illustrated in Fig. 4A (dashed line), which shows the flux density of K+ () leaving the system from a point in the input zone. We recall that also the output was a cation exchange, so that the efflux of K+ implied a corresponding influx Na+. The input was given in the input zone, while the output occurred over the full axis, depending on the local value of. During a transient period, the constant input changed the ion concentrations in the system. The system reached steady state (SS) when became sufficiently high. Then, the total amount of K+ entering the system per second, and the total amount of K+ leaving the system per second, coincided (with the same being true for Na+). This is illustrated in Fig. 4E, which shows how the and are distributed over the -axis at a time, when the system was in SS. The areas under the curves for and were then equal. In the input zone, however, the output rate was about 1/3 of the input rate (Fig. 4A). This means that about 2/3 of the K+ that entered the system was transported in the positive -direction, and left the system from the decay zone. (We recall from Fig. 3 that the decay zone is defined as any part of the -axis outside the input-zone). Fig. 4B–D shows how the local (at) intracellular ion concentrations, the extracellular ion concentrations and changed from the input had been turned on until the system reached SS. For the present example it took 49 s from the constant input had been turned on until the slowest variable () reached 99% of its SS value. The other variables approached SS faster than this (e. g. , 12 s for and 19 s for). During SS, was about 7. 7 mM, corresponding to a concentration (as the baseline concentration was). Although the input was applied to the ECS of the input zone, the local intracellular K+-concentration had increased even more (). This reflects the astrocyte' s propensity for local K+-uptake. The changes in ionic concentrations in the ECS and ICS coincided with a local depolarization of the astrocytic membrane, from the resting potential () to about, reflecting concentration dependent changes in the reversal potentials of the involved ionic species. From here on, we focus on the SS-situation, i. e. , on the activity of astrocytes during periods of on-going intense neural activity. For all system variables, the devition from the baseline (resting) conditions were generally biggest at the point, i. e. , in the part of the input zone which is furthest away from the decay zone (Fig. 4E–H). The average value of, taken over the input zone () was approximately 10 mM (about 6. 9 mM above the resting concentration). During the model calibration, the constant input rate () was tuned to obtain this value, which is on the threshold between functional and pathological conditions [3], [12], [21]. During SS, the gradients in ionic concentrations (Fig. 4F–G) and (Fig. 4H) were quite pronounced. We thus expect that both diffusive and electrical forces contribute to transporting ions through the system (from entering to leaving). This is explored further in the following section. In addition to spatial buffering, K+ may also be buffered by diffusion through the ECS alone, or by local (space independent) storage by the glial cell, to be later released in the same region of the ECS [19], [24]. To investigate the relative importance these clearance mechanisms, we compared the 6 six model versions depicted in Fig. 8A, including one group of three spatially extended models (solid lines), and one group of three point models (dashed lines). Both groups included one model version with an active astrocyte, one model version where the astrocyte had been replaced by a corresponding increase in the ECS volume (the total ECS volume fraction increased to), and one version where the original ECS volume fraction () was kept when the astrocyte was removed. The spatially extended model including the astrocyte, is the one we studied in the previous sections. The other models were reduced versions of this. All model versions were exposed to the input signal described by Eq. 30, causing an increase in. The input was applied in the time window, which was sufficient for to reach its SS-value in all models. Fig. 8B shows the dynamics of the K+-concentration in the ECS at the point where the concentration was the highest (). In the spatially extended models, this occurred at, i. e. in the part of the input zone furthest away from the decay zone. During SS, the net K+ efflux and influx from/to the system coincided. For the point models, having no spatial resolution, there was no distinction between the input zone and decay zone, as the input and output were injected to/subtracted from the same single compartment. The net output rate thus depended on in this single compartment. Therefore, all point models approached the same SS value (). For the spatially extended models was lower, as parts of the K+ could leave the system also outside the input zone. For these models, depended on how efficient they were in longitudinally transporting K+ out from input zone before (revisit Fig. 4 for more details). To gain insight in the importance of local K+-uptake by astrocytes, relative to diffusion in the ECS, we compared the performance of the point model including the astrocyte (black, dashed line in Fig. 8B) to that of the spatially extended model including only the ECS (blue, full line). During the first few seconds after the stimulus had been turned on, the point model with the astrocyte (representing local uptake) was most efficient in terms of limiting. However, local uptake was limited by the storing capacitance of the astrocyte. After seconds with constant K+-influx to the system, the spatially extended model (representing diffusion through the ECS) performed better, as it could redistribute K+ over a larger spatial region. The astrocyte' s ability to locally store excess K+ has been emphasized in previous investigations [19], [24]. Our simulations predicted that the local storage mechanism is mainly important in relatively short time spans after potassium release (a few seconds). A similar conclusion was also drawn from previous modelling studies [10], [12]. We here add an additional point to this discussion: The performance of the point model with extended ECS (dashed red lines) more or less coincided with that of the point model including the astrocyte (dashed black lines). In terms of local storage, the astrocyte (with its membrane being highly permeable to K+), essentially just acts to expand the local volume that the incoming flux of K+ enters into. It has been argued that because K+-transport is aided by transmembrane processes as well as internal processes in the glial cell, K+ can be cleared more effectively by glia than would be possible by a much enlarged extracellular space [40]. To investigate this claim, we compared the three spatially extended models (solid lines). We found that the model including the astrocyte (black, solid line) was more successful in limiting than any of the other model versions. It was significantly more successful than diffusion in the ECS alone, even in the (rather hypothetical) system where the extracellular volume had been increased by a factor 3. In conclusion: In terms of local storage, the astrocyte was not significantly more efficient than an increased enlarged extracellular space. In terms of spatial buffering, however, it was.
The astrocyte/ECS-model provided a mechanistic understanding of how astrocytes may remove K+ from high-concentration regions. In summary, the model astrocyte responded to a local extracellular increase in by a local depolarization of the membrane. At the same time, this depolarization (i) increased astrocytic K+ uptake in the input zone, (ii) increased astrocytic K+-release outside the input zone, (iii) decreased axial K+ transport in the ECS, and (iv) increased axial K+ transport inside the astrocyte. Furthermore, by comparing different versions of the model, we predicted that (v) local storage of K+ by astrocytes may play an important role at a short time scale, while (vi) at a longer time scale, the ability to distribute K+ spatially will be crucial for maintaining a low extracellular K+-concentration. In this regard, we found (viii) that the astrocyte was more efficient spatial buffering mechanism than diffusion in an enlarged extracellular space. The findings (i–ii) were due to well documented astrocytic membrane mechanisms that we implemented in the model. Uptake from high concentration regions was mediated by the K+/Na+-pump, while release into low-concentration regions of the ECS was mediated by the Kir-channel. This supports the dominant view of glial K+-buffering [2], [3], [25], [39], [41]. The findings (iii–iv) were due to an interplay between electrical and diffusive forces. When locally depolarized (in the region with high extracellular), longitudinal voltage gradients arose, and ions in the ICS and ECS were exposed to an electrical force. In the ICS, the electrical force and diffusive force both drove K+-ions in the same direction (out from the high concentration regions). In the ECS, the electrical force acted in the opposite direction from the diffusive force, and reduced the net longitudinal transport through the ECS. Hence, in addition to being an efficient transport route for K+ out from high-concentration regions, the astrocyte actively reduces the extracellular K+-transport. This represents a (to our knowledge) novel mechanism that astrocytes may utilize to shield the extracellular space from excess K+. All these effects (i–iv) taken together turned the astrocyte into an efficient sluice for removing K+ from the input zone. The findings (v–viii) shed light on the relative efficiency of spatial buffering and other K+-clearance mechanisms, such as local storage by astrocytes, or diffusion in the ECS alone. An interesting prediction was that, in terms of local storage, the astrocyte did not have a stronger effect on than an enlarged extracellular space would. In terms of longitudinal transports, however, the astrocyte performed much better (by spatial buffering) than diffusion in an enlarged extracellular space (Fig. 8). We do, however, wish to comment that these mechanisms are not mutually exclusive. In fact, an (initial) local accumulation of intracellular K+ is required for the astrocyte to initiate the spatial buffering process. It is this local accumulation that evokes the intracellular voltage- and concentration gradients that the astrocyte utilizes for intracellular K+-transport. It is likely that the mechanisms responsible for spatial buffering vary between brain regions and between different species of glial cells [30]. Previous literature has suggested several mechanisms for spatial buffering apart from the ones that were included in our model. K+-uptake by Na+K/K+/Cl−-cotransporters and K+/Cl−-cotransporters are two candidate mechanisms that likely could affect the simulated results [38]. Furthermore, regions in the endfoot processes of astrocytes have been shown to have an extremely high K+-conductance compared to the membrane in general [42]. Such high concentration regions could improve the buffering process by transferring (siphoning) excess K+ into the vitreos humor or vasculature [43]. The buffering process may also be affected by water influx and swelling experienced by the astrocyte during the uptake process [13], [30], [38]. Rather than increasing the biological complexity, by e. g. , including multiple candidate buffering mechanisms, we have in this study strived towards elucidating the fundamental physical processes involved in spatial K+-buffering. As our simulations demonstrated, K+-buffering is a highly complex process. It involves an intricate and sensitive interplay between and ionic concentrations, and between electrical and diffusive transports. We therefore highlight the importance of applying an electrodiffusive, physically consistent, modelling scheme which ensures a complete book-keeping of ion concentration dynamics and its effects on. In previous models of spatial buffering, was derived from standard cable theory [10]–[12], [21], where diffusive currents are assumed to have a negligible impact on, and where the resistivity is assumed to be constant (i. e. , not dependent on ion concentration variations). During our simulations, intra- and extracellular resistivities changed by as much as 10% and 20%, respectively, and the diffusive current was about 25–30% of the field current in the ECS. The assumptions underlying standard cable theory are therefore poorly justified if applied to the spatial K+-buffering process. The astrocyte/ECS-model was represented phenomenologically as a single astrocyte coated with the average proportion of available ECS per astrocyte (see Figs. 1 and 3). This geometrical representation is justified for macroscopic transport processes, when a large number of astrocytes perform the same function simultaneously [12]. For the current study, this was a reasonable assumption, as the input was a change in the ion-concentrations in the ECS, shared by all present astrocytes. If we instead wanted to study a cell specific signal, such as e. g. , the response of a single astrocyte to a transmembrane current injection, the geometrical representation in Fig. 1B would be less appropriate. Firstly, the notion of the ECS as a relatively thin coating following a single cell is only motivated at the macroscopic “average transport”-level. Secondly, if only a single cell was involved in a particular process, we would expect that. That is, a single active cell would have a significantly larger proportion of the ECS to its own disposal, compared to the macroscopic case, where all cells in a piece of tissue are active, and share the limited amount of available ECS. In single-cell models it is common to assume that conditions in E are constant, so that only I is modeled explicitly. In this limit, the electrodiffusive formalism reduces to the one-domain model presented previously by Qian and Sejnowski [31]. The framework presented here is essentially an expansion of the one-domain model by Qian and Sejnowski [31] to a two-domain model that includes both the ECS and ICS. Like the one-domain model, the framework ensures (i) a consistent relationship between and ionic concentrations. Unlike the one-domain model, the framework ensures (ii) global particle/charge conservation, and (iii) that the charges on either side of a piece of membrane must be equal in magnitude and opposite in sign (). The latter constraint is implicit when the the membrane is assumed to be a parallel plate capacitor, an assumption made in most models of excitable cells (see e. g. , [26], [27], [31]). It is also related to the topic of electroneutrality. Electroneutrality in electrodiffusive models of biological tissue has been the topic of many discussions [44]–[46]. It is relevant for how the electrical potential (v), occurring in the Nernst-Planck equation, is derived. Generally (at sufficiently course spatial resolutions so that the charge density can be assumed to be continuous), v obeys Poisson' s equation: (43) where is the dielectric constant, and is the total charge density. In biological tissue, the charge relaxation time is very small in any region except in the thin Debye layer () surrounding a bio-membrane. Any nonzero net charge density in the bulk solution will decay very rapidly () to zero [36]. Several models have simulated electrodiffusion by solving the Nernst-Planck equations in one or more dimensions, with Poisson' s equation for (see e. g. , [45], [47]–[50]). The advantage with such a procedure is that the Poisson-Nernst-Planck (PNP) equations can be implemented in a general way in three-dimensional space. The challenge is then to specify the appropriate boundary conditions for solving Eq. 43 in the vicinity of membranes. Generally, PNP-solvers apply a fine spatial resolution near the membrane, and simulation time steps smaller than the charge-relaxation time [48]. For these reasons, they tend to be extremely computationally demanding [51]. The formalism presented in this work belongs to a class of of one-dimensional models, including the cable equation and several electrodiffusive models [10], [12], [13], [17], [31], [52], [53], which bypasses the computationally heavy PNP-scheme. The physical interpretation of these models is as follows: Any net charge in a volume is implicitly assumed to be located in the thin Debye-layer surrounding the capacitive membrane, and is identical to the charge that determines. The remainder of the space (i. e. , the bulk) will therefore be electroneural (). Note that any finite volume, enclosing a piece of membrane, will also be electroneutral. This follows from the charge symmetry condition (Eq. 18), constraining the charge on either side of the membrane to be equal in magnitude and opposite in sign. The charge symmetry condition and the electroneutrality condition are in this way closely related. In these electroneutral models, charge relaxation is implicit. This is a plausible assumption at time scales relevant for most biophysical processes. Accordingly, simulations may be run with time-steps ranging from 1 ms to 1 s, depending on the time course of the included membrane mechanisms. To our knowledge, the formalism summarized in Fig. 2 is the first biodiffusive model where the intra- and extracellular voltage gradients have been derived from the charge symmetry condition. Eqs. 25 and 26 can be interpreted as summarizing all local and global electrical forces driving the system towards electroneutrality. A natural future ambition would be to generalize the electrodiffusive formalism to 2 or 3 spatial dimensions, so it can address the same 3-dimensional transport problems as PNP-solvers. The challenge will be to formulate the system as a grid of coupled constraints (electroneutrality in the bulk and Eq. 12 for across the membrane) for which the Nernst-Planck equations can be solved with time steps much longer than those involved in the charge relaxation process.
When neurons generate electrical signals they release potassium ions (K+) into the extracellular space. During periods of intense neural activity, the local extracellular K+ may increase drastically. If it becomes too high, it can lead to neural dysfunction. Astrocytes (a kind of glial cells) are involved in preventing this from happening. Astrocytes can take up excess K+, transport it intracellularly, and release it in regions where the concentration is lower. This process is called spatial buffering, and a full mechanistic understanding of it is currently lacking. The aim of this work is twofold: First, we develop a formalism for modeling ion concentration dynamics in the intra- and extracellular space. The formalism is general, and could be used to simulate many cellular processes. It accounts for ion transports due to diffusion (along concentration gradients) as well as electrical migration (along voltage gradients). It extends previous, related formalisms, which have focused only on intracellular dynamics. Secondly, we apply the formalism to model how astrocytes exchange ions with the extracellular space. We conclude that the membrane mechanisms possessed by astrocytes seem optimal for shielding the extracellular space from excess K+, and provide a full mechanistic description of the spatial (K+) buffering process.
Electrodiffusive Model for Astrocytic and Neuronal Ion Concentration Dynamics