Analysis of potassium ion diffusion from neurons to capillaries: Effects of astrocyte endfeet geometry

Abstract

Neurovascular coupling (NVC) refers to a local increase in cerebral blood flow in response to increased neuronal activity. Mechanisms of communication between neurons and blood vessels remain unclear. Astrocyte endfeet almost completely cover cerebral capillaries, suggesting that astrocytes play a role in NVC by releasing vasoactive substances near capillaries. An alternative hypothesis is that direct diffusion through the extracellular space of potassium ions (K⁺) released by neurons contributes to NVC. Here, the goal is to determine whether astrocyte endfeet present a barrier to K⁺ diffusion from neurons to capillaries. Two simplified 2D geometries of extracellular space, clefts between endfeet, and perivascular space are used: (i) a source 1 μm from a capillary; (ii) a neuron 15 μm from a capillary. K⁺ release is modeled as a step increase in [K⁺] at the outer boundary of the extracellular space. The time-dependent diffusion equation is solved numerically. In the first geometry, perivascular [K⁺] approaches its final value within 0.05 s. Decreasing endfeet cleft width or increasing perivascular space width slows the rise in [K⁺]. In the second geometry, the increase in perivascular [K⁺] occurs within 0.5 s and is insensitive to changes in cleft width or perivascular space width. Predicted levels of perivascular [K⁺] are sufficient to cause vasodilation, and the rise time is within the time for flow increase in NVC. These results suggest that direct diffusion of K⁺ through the extracellular space is a possible NVC signaling mechanism.

Graphical Abstract

Introduction

The term neurovascular coupling (NVC) refers to the local increase in blood flow and oxygen delivery to regions of the brain during neuronal activity (Chaigneau et al., 2003; Cox et al., 1993; Freygang & Sokoloff, 1959). The mechanisms underlying NVC are the subject of active research and debate (Hosford & Gourine, 2019). Neurons release many vasoactive substances, including neurotransmitters and metabolites, which may play a role in NVC (Berne et al., 1974; Drake & Iadecola, 2007; Filosa et al., 2006). Multiple mechanisms have been proposed, whose relative importance remains unclear (Iadecola, 2017).

Astrocytes are the most abundant type of glial cells in the brain (Jäkel & Dimou, 2017). With multiple projecting processes, they come into close contact with many other cells. Where these processes lie close to capillaries, they possess endfeet that wrap around the vessels, forming almost continuous sheathes. This structural arrangement of astrocytes may suggest that they play a key role in the control of blood vessel functions, including NVC (Haydon & Carmignoto, 2006; Iadecola & Nedergaard, 2007; Masamoto et al., 2015). Several authors have proposed mechanisms by which astrocytes mediate NVC, with astrocyte endfeet releasing vasoactive molecules onto capillaries in response to neuronal activity (Filosa et al., 2006; Hösli et al., 2022; Iadecola, 2017; Kim et al., 2015; Paulson & Newman, 1987; Stackhouse & Mishra, 2021).

The observation that astrocyte endfeet form an almost complete covering of capillaries may also suggest that endfeet form a barrier to molecular diffusion between capillaries and the surrounding tissue (Kutuzov et al., 2018; Nuriya et al., 2013). The formation of a tight barrier by endfeet would have significant functional implications. Firstly, it would suggest that the endfeet contribute to the blood-brain barrier, a distinctive feature of brain vasculature (Kutuzov et al., 2018). Secondly, it would imply an essential role for astrocytes in NVC if extracellular pathways for molecular communication between neurons and vascular cells were effectively blocked by the endfeet.

Several studies have examined the structure and transport properties of the layer of astrocyte endfeet around microvessels. Quantification of astrocyte endfeet coverage of capillaries has produced a range of estimates (Ambrosi et al., 1995; Bertossi et al., 1997; Virgintino et al., 1997). Based on an electron microscope 3D reconstruction of astrocyte endfeet around microvessels, it is estimated that 0.3% of a 5-μm capillary’s circumference is exposed to the extracellular space (ECS) (Mathiisen et al., 2010). Using data from those and other structural studies, a recent theoretical analysis (Koch et al., 2023) provides estimates of endfoot sheath filtration coefficients in the range 2 to 3 × 10⁻¹¹ mPa⁻¹ s⁻¹ and diffusion membrane coefficients for small solutes in the range 5 to 6 × 10² m⁻¹. These values are much higher than corresponding values for the vessel wall, suggesting that the endfeet form a relatively permeable barrier to transport, and leaving open the possibility that extracellular diffusion of molecular signals directly from neurons to vessels contributes to NVC.

The goal of the present study is to examine the characteristics of such a signaling mechanism, including effects of the astrocyte endfoot geometry, to determine whether astrocyte endfeet would present a barrier to diffusing molecules. Potassium ions (K⁺) are considered as the prototype diffusing species, although a similar analysis would apply to other low-molecular-weight diffusible species including neurotransmitters and metabolites. One reason to focus on K⁺ is that it is ubiquitously released by active neurons as the membrane potential re-polarizes (Hodgkin & Huxley, 1952; Paulson & Newman, 1987). While neuronal activity also causes release of neurotransmitters, the specific neurotransmitter released varies with the type of neuron, and this might therefore represent a less robust mechanism of NVC. A second reason to focus on K⁺ is that it can act as a vasodilator by initiating upstream conducted responses along vessel walls from capillaries to arterioles (Caesar et al., 1999; Chen et al., 2014; Collins et al., 1998; Emerson & Segal, 2000; Haddy et al., 2006), and is therefore a likely candidate to participate in NVC. A notable feature of NVC is the speed of the response relative to other types of acute flow regulation, with the response reaching its maximal level within seconds of the initiation of a stimulus (Masamoto 2008). Therefore, the kinetics of any proposed mechanism should be considered when analyzing its potential contribution to NVC. This analysis does not exclude the possibility that astrocyte-mediated mechanisms also contribute to NVC, acting in parallel with direct extracellular diffusion.

In the present study, we simulate time-dependent diffusion of K⁺ released from an extravascular location to the perivascular space of a brain capillary, using simplified two-dimensional geometries. We examine the effects of varying the distance of the source from the capillary, the width of the clefts between astrocyte endfeet, and the width of the perivascular space between the endfeet and the vascular endothelial cells. The results are discussed in the context of the possible role of K⁺ diffusion through the extracellular space in neurovascular coupling.

Methods

Two idealized two-dimensional reference geometries are used to represent important features of the pathway for K⁺ diffusion from neurons to capillaries (Figure 1), including the ECS, intercellular clefts between adjacent astrocyte endfeet, and the perivascular space. These idealized geometries represent key features of pathways for diffusion from neurons to endothelial cells via endfoot clefts, namely transit from the extracellular space into a very narrow cleft, passage through the cleft, and transit from the cleft into a bounded perivascular space. The local neuronal release geometry represents K⁺ release in the ECS adjacent to a capillary. In this case, the short distance between the site of release and the capillary is represented by an ECS of length 1 μm and width 2 μm, which provides a minimal resistance to K⁺ diffusion. The distant neuronal release geometry represents K⁺ release from a site at the typical distance of a neuron to the nearest capillary. In this case, the ECS has length 15 μm and width 1 μm. The length of 15 μm is chosen to reflect the average distance between the neuronal soma to the nearest microvessel (Mabuchi et al., 2005; Tsai et al., 2009). Since many voltage-gated K⁺ channels are found on the soma and axon hillock (Safronov, 1999), using the average distance between the soma and microvessel is a reasonable assumption for a physiologically relevant K⁺ source. The width of 1 μm is larger than the typical width of extracellular spaces in the brain (Kinney et al., 2013) and is chosen to represent the combined effect of multiple diffusion pathways approaching a capillary from a given direction, whose combined width can be estimated as approximately 1 μm based on a capillary diameter d = 5 μm and a typical cerebral extracellular volume fraction of 0.2.

Figure 1.

Schematic illustrations of extracellular pathways for diffusion of K⁺ and representation using model geometries. A. Diffusion of K⁺ from a local neuron (purple shape, arrows show release of K⁺). Yellow shapes represent other cells (neurons or neuroglia). The capillary is shown in cross-section in red with an external diameter of 5 μm. Astrocytes are shown in blue, with endfeet wrapped around the capillary. Gray represents the extracellular space. B. Idealized two-dimensional geometry representing local neuronal release of K⁺. Bold arrows show the boundary where K⁺ is released into the extracellular space, represented by concentration = 9 mM for t > 0. w_c: width of cleft between astrocytic endfeet. w_p: width of perivascular space. C. Diffusion of K⁺ from a neuron at a typical distance of 15 μm from vessel. Dotted curves show K⁺ diffusion pathways; otherwise as in A. D. Idealized two-dimensional geometry representing distant neuronal release of K⁺, otherwise as in B.

In both reference geometries, the assumed width of the cleft between astrocyte endfeet is w_c = 20 nm (Mathiisen et al. 2010). This value is consistent with observations that the average number of endfeet surrounding the microvessel is 2.5, and that the sum of the cleft widths represents 0.3% of the microvessel circumference (Mathiisen et al. 2010), according to the calculation that (2.5w_c)/(πd) ≅ 0.003. The assumed reference width of the perivascular space is w_p = 100 nm, in the intermediate range of experimentally observed values (Frank et al., 1990; Hawkes et al., 2013; Thomsen et al., 2017; Timpl, 1989). Effects of variations in w_c and w_p on the rise of perivascular [K⁺] are considered in the simulations.

The potassium ion concentration [K⁺] = c(x,y,t) in the ECS, intercellular cleft, and perivascular space is described by the time-dependent diffusion equation in two spatial dimensions (x,y) and time t:

$$\frac{\partial c}{\partial t}=D_{eff}{\nabla }^{2}c$$

where the effective diffusivity of K⁺ in the brain parenchyma is given by $D_{eff}=D_s/{\lambda }^2$, D_s = 2400 μm²/s is the diffusivity of K⁺ in free solution at 37 C (Lo et al., 2004), and λ is the tortuosity of the cerebral ECS. The tortuosity varies depending on brain region, and a typical value λ = 1.6 is assumed here, implying that $D_{eff}=937.5\mu {\mathrm{m}}^2/\text{s}$ (Syková & Nicholson, 2008). The intercellular cleft and the basement membrane in the perivascular space are assumed to be permeable to K⁺, with the same diffusivity as the ECS. A length of 200 nm is assigned to the intercellular cleft, which is assumed to be perpendicular to the wall of the arteriole. In reality, the astrocyte endfeet overlap, resulting in oblique cleft orientations and cleft lengths of 360 to 500 nm (Mathiisen et al., 2010). Because the diffusion constant in the cleft used in the simulation includes the effect of tortuosity, the assumed length of 200 nm is equivalent to an unobstructed cleft length of 200 λ² = 512 nm, in terms of the diffusive flux for a given concentration difference across its length.

The diffusion equation is solved using a finite-element method (MATLAB PDE Toolbox, R2022a, MathWorks, Natick MA). Controls on the mesh generator are used to ensure a very fine mesh in the narrow cleft regions, so that gradients of [K⁺] in the clefts are adequately resolved. Within the cleft, diffusion is effectively one-dimensional, with negligible concentration gradients across the cleft. Consequently, a mesh in which each element spans the cleft provides adequate resolution. Examples of the mesh used for each geometry are shown in Figure 2. Because the idealized computational domains have mirror symmetry about the vertical axis, the computation is performed in half of the domain, for computational efficiency (Figure 1). The initial condition is a baseline concentration c(x,y,0) = 3 mM. Zero-flux conditions are imposed on all boundaries of the computational domain except the outer boundaries of the ECS, where a step change in [K⁺] from 3 to 9 mM at time t = 0 is prescribed, to represent the release of K⁺ upon sustained neuronal activity. These are typical extracellular [K⁺] levels under quiescent conditions (3 mM) and when neurons fire continuously (9 mM) (Walz & Hertz, 1983). Effects of removal of K⁺ from the ECS by Na⁺/K⁺-ATPase and other transporters are neglected in the simulations. The observation that [K⁺] reaches an equilibrium between 8–12 mM when neurons fire continuously (Walz & Hertz, 1983) implies that K⁺ uptake is a relatively small effect during the short time intervals considered in the simulations. Significant vasodilation is observed when capillaries are exposed to [K⁺] levels of 8 mM or more (Dabertrand et al., 2015; Filosa et al., 2006; Knot et al., 1996). Therefore, the time to reach 8 mM in the perivascular space is considered indicative of the time delay in signaling associated with the diffusion of K⁺ from a source to the capillary wall.

Figure 2.

Meshes used for finite-element computations of K⁺ diffusion. A. Mesh used for local neuronal release geometry. B. Mesh used for distant neuronal release geometry. C. Enlarged view of cleft between astrocytic endfeet, showing fine mesh to resolve the cleft region.

Results

Local neuronal release geometry.

Figure 3 shows predicted spatial distributions of [K⁺] at time points from t = 0 s to t = 0.05 s in this geometry. At t = 0, [K⁺] in the entire space is 3 mM. The entire space shows changes in [K⁺] within t = 0.01 s and approaches 9 mM by t = 0.05 s. Average perivascular [K⁺] is calculated by averaging the concentration along the midline of the perivascular space. In reference conditions, average perivascular [K⁺] reaches 8 mM by t = 0.023 s.

Figure 3.

Distribution of [K⁺] at four time points after K⁺ release for local neuronal release geometry, color-coded as indicated at right, with w_c = 20 nm and w_p = 100 nm. Initial concentration is 3 mM throughout the domain.

Effects on the rise in average perivascular [K⁺] of varying the geometry of the intercellular cleft and perivascular space are shown in Figure 4, for local release. When cleft width is decreased from the reference value of w_c = 20 nm, perivascular [K⁺] rises more slowly. For a cleft width 6.67 nm, average perivascular space [K⁺] reaches 8 mM by t = 0.048 s. With an increased cleft width of 100 nm, the same level is reached around t = 0.013 s. Increasing the width of the perivascular space from the reference value w_p = 100 nm has the effect of slowing the rise in perivascular [K⁺]. For a perivascular space width 500 nm, the time to reach 8 mM is more than 0.05 s. Conversely, when the perivascular space width is reduced to 20 nm, the same level is reached before t = 0.012 s.

Figure 4.

Variation in average perivascular [K⁺] over time following release of K⁺ in local neuronal release geometry. A. Fixed perivascular space width w_p = 100 nm, cleft width w_c varies from 6.67 to 100 nm. B. Fixed cleft width w_c = 20 nm, perivascular space width w_p varies from 20 to 500 nm.

Distant neuronal release geometry.

Figure 5 shows the predicted [K⁺] levels in this geometry at time points from 0.1 s to 0.5 s. Changes in K⁺ occur throughout the ECS and perivascular space within 0.1 s. By t = 0.5 s, the concentration throughout the domain approaches 9 mM, with average perivascular [K⁺] reaching 8 mM by t = 0.23 s.

Figure 5.

Color-coded distribution of [K⁺] at four time points after K⁺ release for distant neuronal release geometry, color-coded as indicated at right, with w_c = 20 nm and w_p = 100 nm. Initial concentration is 3 mM throughout the domain.

Variations in the intercellular cleft width and perivascular space width have less effect on the rise in average perivascular [K⁺] in the neuronal geometry than in the local release geometry, as shown in Figure 6. When astrocyte endfeet cleft width w_c is varied from 6.67 nm to 100 nm, the trend is for the rise in [K⁺] to occur faster with increasing cleft width, but the variation is slight and the concentration of 8 mM is reached between t = 0.22 s and t = 0.27 s in all cases. When the perivascular space width w_p is increased from 20 nm to 500 nm, the time to reach 8 mM increases moderately, from t = 0.23 to 0.37 s.

Figure 6.

Variation in average perivascular [K⁺] over time following release of K⁺ in distant neuronal release geometry. A. Fixed perivascular space width w_p = 100 nm, cleft width w_c varies from 6.67 to 100 nm. B. Fixed cleft width w_c = 20 nm, perivascular space width w_p varies from 20 to 500 nm.

Discussion

The observation that astrocyte endfeet form an almost complete covering over the surface of capillaries in the brain has naturally led to the question of whether the endfeet represent a barrier to mass transport between vessels and the surrounding tissue. Here we address this question using a theoretical model of K⁺ diffusion in the extracellular space, including the effects of endfoot geometry. This model is based on well-established principles of molecular diffusion driven by concentration gradients. As is generally the case for theoretical models of biological phenomena, a number of simplifying assumptions are introduced, and parameter values are not precisely known. However, the present theoretical approach has the advantage that spatial distributions of concentration can be predicted on scales that are not accessible by currently available experimental techniques, as discussed below. Also, this approach permits investigation of the effects of geometrical parameters, such as the widths of the endfoot clefts and the perivascular space, that are not easily controllable in experimental settings.

With respect to the transport of K⁺, a small highly diffusible ion, the main prediction of this study, based on available data on endfeet geometry, is that the endfeet represent a relatively insignificant barrier to diffusive transport between neurons and capillaries. For typical levels of extracellular potassium released by neurons, the predicted time taken to reach a level in the perivascular space sufficient to cause vasodilation is 0.023 s for a source close to the capillary and 0.23 s for a source 15 μm from the capillary, a typical average distance from a neuronal soma to the nearest capillary. These times are shorter than the typical experimentally observed time taken for increase in blood flow by NVC, which is 1 to 2 s (Masamoto et al., 2008).

The model was used to examine the effects of changes in astrocyte endfeet geometry on K⁺ diffusion into the perivascular space. As the cleft width between astrocyte endfeet is decreased, the predicted rise in [K⁺] slows. However, even at the smallest tested width (6.67 nm), [K⁺] reaches levels sufficient to cause vasodilation within 0.05 seconds in the local neuronal release geometry and 0.5 s in the distant neuronal release geometry. Increasing the width of the perivascular space slows the rise in [K⁺], but even with the greatest width tested (500 nm), the concentration reaches 8 mM within 0.5 s in both cases.

The predicted rise time for perivascular [K⁺] is approximately 10 times longer for K⁺ at a typical distance from a neuron to the nearest capillary (15 μm) than it is for a source adjacent to the capillary. This result implies that the dynamics of potassium diffusion from neurons to capillaries are generally determined by the distance between the neuron and the capillary, and not by the presence of astrocyte endfeet. Support for this conclusion also comes from the result that varying the cleft and perivascular space width has little effect on [K⁺] in the perivascular space when neuronal release at a distance of 15 μm from the capillary is assumed.

In reality, the distances of neuronal somas from the nearest capillary vary considerably, and axons also release K⁺ into the ECS, so that each neuron represents a distributed source. Including these factors would not, however, alter the conclusion that the arrival of K⁺ in the perivascular space following onset of neuronal activity is relatively rapid compared to the observed time scale of NVC.

Estimates of the changes in extracellular [K⁺] during normal neuronal activity vary widely. Several studies report increases of 0.2–2 mM (Rasmussen et al., 2020). Most of these experiments were done with K⁺-sensitive microelectrodes, which may underestimate localized changes as a result of their limited spatial resolution (Walz & Hertz, 1983). The assumed increase by 6 mM results in a concentration of 9 mM, which is within the range 8–12 mM reported with sustained stimulation (Walz & Hertz, 1983). Further evidence for values in this range comes from observations that [K⁺] levels of 8–15 mM are needed to evoke vasodilation (Dabertrand et al., 2015; Filosa et al., 2006; Knot et al., 1996).

The present model for K⁺ diffusion neglects interactions with other ionic species and with the electric field. The electrodiffusion equation can be used to analyze such interactions (Halnes et al., 2013). In an analysis of ion transport in an extracellular space of length 300 μm, the K⁺ flux due to the electric field was found to be 17–28% of the flux due to the concentration gradient (Halnes et al., 2013). We applied the same analysis to a channel of length 15 μm and found that the electrically driven flux was 1–4% of the flux due to the concentration gradient. This implies that neglect of electrodiffusive effects is justified.

In the present model, changes in the geometry of the extracellular space due to osmotic fluid shifts are neglected. Observed decreases in the volume fraction of interstitial space with increasing neuronal activity are believed to result from swelling of astrocytes as they take up excess K⁺ (Rasmussen et al., 2020). However, as already discussed, K⁺ uptake is relatively small during the short time intervals considered in the present simulations, and so the resulting volume shifts are also likely to be small, becoming significant only on time scales of 10 to 20 s (Dietzel et al., 1982; Syková et al., 2003).

While the focus here is on K⁺ due to its high diffusivity and ubiquitous release by neurons, other small molecules could also diffuse through the astrocytic layer to reach capillaries within the 1–2 second time frame of NVC responses. Neurotransmitters spilling over from synaptic clefts are one potential example. Glycine, one of the major inhibitory neurotransmitters of the brain and spinal cord, has D_s = 1409 μm²/s (all diffusivities at 37°C) and glutamate, the major excitatory neurotransmitter in the nervous system, has D_s = 1015 μm²/s (Longsworth, 1953). Dopamine has D_s = 750 μm²/s (Rice et al., 1985). Another metabolite from active neurons, ATP, has D_s = 434 μm²/s (Hubley et al., 1996). The present results indicate that such molecules could readily gain access to the perivascular space despite the presence of astrocyte endfeet. The time course of their arrival could be predicted from the present results, to a first approximation, by scaling up the arrival times computed for K⁺ by the factor D_sK/D_sX, where D_sK and D_sX are the diffusivities in free solution of K⁺ and the solute under consideration. Their dynamics would be slower than those of K⁺, but they could still potentially contribute to NVC.

According to the present results, blockage of K⁺ diffusion to capillaries by the layer of astrocyte endfeet would require much narrower clefts than are observed experimentally under normal conditions. This could be the case if tight junctions, such as are present between endothelial cells, formed between astrocyte endfeet. Presence of tight junctions between astrocyte endfeet has been reported in disease states (Horng et al., 2017), and may provide a possible explanation for reduced NVC seen in neurodegenerative diseases. Thickening of the cerebral basement membrane, resulting in increased width and tortuosity of perivascular space, has been observed in some neurodegenerative disorders (Castejón, 1988; Farkas et al., 2000; Johnson et al., 1982; Junker et al., 1985; Liwnicz et al., 1990; Mancardi et al., 1980). Such an increase in perivascular space width could affect NVC in neurodegenerative diseases, by slowing the rise of K⁺ or other signaling molecules.

A number of investigators have developed theories focused on astrocytes as key mediators of NVC, due to their almost complete coverage of cerebral microvessels (Haydon & Carmignoto, 2006; Iadecola & Nedergaard, 2007; Masamoto et al., 2015). According to these theories, K⁺ released by astrocyte endfeet acts as a signal causing vasodilation. Paulson and Newman (1987) analyzed the rise of extracellular [K⁺] at arterioles following K⁺ release associated with neuronal activity, and compared extracellular diffusion, in a geometry similar to that assumed here, with “siphoning” via astrocytes and K⁺ release at endfeet. For K⁺ release distributed over a region 2.5 μm or more from the arteriole, they estimated that the rise of [K⁺] at arterioles takes about 0.3 s. For K⁺ release at distances of 50 μm or more from the arteriole, the estimated rise time was about 2 s. In the present study, diffusion from neurons to capillaries is considered, because an increase in [K⁺] at capillaries can initiate conducted responses that travel to the arterioles, resulting in vasodilation. Since capillaries are more closely spaced than arterioles, the typical diffusion distance is less (about 15 μm), resulting in a predicted [K⁺] rise time of about 0.2 s in the present study. In the analysis of Paulson and Newman, the release of K⁺ from astrocytes at the endfeet by siphoning was predicted to occur in less than 0.05 s. The mechanisms by which astrocytes might respond so quickly to a remote stimulus are unclear. Intracellular Ca²⁺ signaling has been considered to play a role, but conflicting information about the timing and amplitude of changes in [Ca²⁺] has been presented (Hillman, 2014; Stackhouse & Mishra, 2021; Tran et al., 2018). In summary, the relative importance of direct diffusion through the extracellular space and astrocyte-mediated signaling mechanisms remains unknown. The present results do not preclude a role for astrocyte-mediated mechanisms. Further experimental and theoretical studies are needed to obtain improved understanding of the mechanisms of neurovascular coupling.

Main points:

• Despite coverage of capillaries by astrocytic endfeet, sufficient K⁺ can diffuse rapidly from neurons to capillaries to cause vasodilation.

• Direct diffusion of K⁺ through the extracellular space is a possible mechanism of neurovascular coupling.

Abbreviations:

ECS

extracellular space

K⁺

potassium ions

NVC

neurovascular coupling

PVS

perivascular space

Data sharing

Computer codes and data files are available at https://github.com/secomb/Endfeet2023