Potassium buffering in the neurovascular unit: models and sensitivity analysis

Abstract

Astrocytes are critical regulators of neural and neurovascular network communication. Potassium transport is a central mechanism behind their many functions. Astrocytes encircle synapses with their distal processes, which express two potassium pumps (Na-K and NKCC) and an inward rectifying potassium channel (Kir), whereas the vessel-adjacent endfeet express Kir and BK potassium channels. We provide a detailed model of potassium flow throughout the neurovascular unit (synaptic region, astrocytes, and arteriole) for the cortex of the young brain. Our model reproduces several phenomena observed experimentally: functional hyperemia, in which neural activity triggers astrocytic potassium release at the perivascular endfoot, inducing arteriole dilation; K(+) undershoot in the synaptic space after periods of neural activity; neurally induced astrocyte hyperpolarization during Kir blockade. Our results suggest that the dynamics of the vascular response during functional hyperemia are governed by astrocytic Kir for the fast onset and astrocytic BK for maintaining dilation. The model supports the hypothesis that K(+) undershoot is caused by excessive astrocytic uptake through Na-K and NKCC pumps, whereas the effect is balanced by Kir. We address parametric uncertainty using high-dimensional stochastic sensitivity analysis and identify possible model limitations.

Figure 1

Model overview. ${\text{Ω}}_{\text{S}}$ represents the synaptic space, where active synapses release glutamate and K⁺, and ${\text{Ω}}_{\text{Astr}}$ is the astrocyte intracellular space, where K⁺ enters the astrocyte through the Na-K pump, NKCC, and Kir_AS channels. Na⁺ enters via NKCC and exits via the Na-K pump. Glutamate binds to metabotropic receptors on the astrocyte endfoot, effecting IP₃ production inside the astrocyte wall, which leads to release of Ca²⁺ from internal stores, causing EET production. Ca²⁺ and EET open BK channels at the perivascular endfoot, releasing K⁺ into the perivascular space $\left({\text{Ω}}_{\text{P}}\right)$. Meanwhile, buildup of intracellular K⁺ in the astrocyte results in K⁺ efflux through the perivascular endfoot Kir_AV channel. ${\text{Ω}}_{\text{SMC}}$ is the arteriole smooth muscle cell intracellular space, where Kir_SMC channels are activated by the increase in extracellular K⁺. The resulting drop in membrane potential closes Ca²⁺ channels, which reduces Ca²⁺ influx, leading to SMC relaxation and arteriole dilation (strain, ϵ). The arteriole dilation (ϵ) stretches the membrane of the enclosing astrocyte perivascular endfoot, which activates Ca²⁺ influx through TRPV4 channels. The prohibition sign on the channel is meant to indicate the inhibition mechanism of the channel, as the TRPV4 channel is inhibited by intracellular and extracellular Ca²⁺. Note that the diagram here is not to scale. The perivascular endfoot is actually wrapped around the arteriole, but here we show them separated to make clear the ion flow at the endfoot-vessel interface. Dashed arrows indicate ion movement; solid arrows indicate causal relationships, and dotted arrows indicate inhibition. Thin dashed arrows in astrocyte Kir channels indicate the ion flux direction at baseline or, in the vessel Kir, the change in flux direction when extracellular K⁺ reaches over 15 mM. The potassium signaling pathway is highlighted by blue arrows. To see this figure in color, go online.

Figure 2

Astrocyte Kir effect on neurovascular coupling. Black curves represent the astrocyte model equations described in this article; gray curves represent the astrocyte model equations without Kir_AS or Kir_AV channels. (a) Extracellular K⁺ in the synaptic space. The thin red curve is the glutamate transient, represented by the ratio of bound to unbound glutamate receptors, ρ (see Eq. S2). (b) Solid lines indicate the intracellular astrocytic K⁺ concentration and dashed lines the astrocyte membrane potential. (c) Astrocyte intracellular IP₃ concentration. (d) Astrocyte intracellular Ca²⁺ concentration. (e) Astrocyte intracellular EET concentration. (f) Astrocyte perivascular endfoot BK (dashed lines) and Kir_AV (dash-dotted lines) currents. (g) Extracellular K⁺ concentration in the perivascular space. (h) Arteriole radius. To see this figure in color, go online.

Figure 3

K⁺ undershoot in the extracellular synaptic space after stimulus is more pronounced with increasing length of the activation period. The stimulus period is indicated by thick black bars. (a) Simulation results. (b) Experimental results interpolated from Fig. 3 in Chever et al. (5).

Figure 4

Astrocyte response to K⁺ channel blocker with short stimulus spike. Black curves indicate neural-induced astrocyte stimulation under control conditions; gray curves represent astrocyte stimulation in the presence of K⁺ channel blocker. (a) Intracellular astrocytic K⁺ concentration. (b) Astrocyte membrane potential. (c) Extracellular K⁺ concentration in the synaptic space. (d–f) Experimental results corresponding to simulation data in a–c, interpolated from Fig. 7 in Ballanyi et al. (30).

Figure 5

Sensitivity of K⁺ undershoot and effects of the Kir blockade. Diameters of small circles indicate single-parameter sensitivity; circle color indicates whether increasing parameter magnitude will increase (white) or decrease (black) the following values: the synaptic K⁺ undershoot (top left); the change in astrocytic K⁺ after Kir blockade at baseline, Δ[K⁺]_A,0 (upper right), and in the active state, Δ[K⁺]_A,max (lower left); the maximum astrocyte hyperpolarization, ΔV_A,max, due to activation during Kir blockade (lower right). Thickness of the gray lines indicates the sensitivity of two-parameter interaction pairs.