A Mathematical Model of Cerebral Blood Flow Control: Role of Kir Channels
Authors: Moshkforoush Arash , Longden Thomas , Dabertrand Fabrice , Nelson Mark , Tsoukias Nikolaos 
Published: November 2023

Introduction

In‐time supply of oxygen and nutrients in response to elevation in neuronal activity is essential for proper functioning of the brain. This process, referred to as neurovascular coupling (NVC), is vital for the function and survival of neurons and is impaired in various clinical conditions including Alzheimer's disease, stroke and aging (Phillips, et al., JCBFM, 2016). K+ release at astrocytic endfeet is considered a key mediator of NVC. The elevated extracellular K+ concentration ([K+]o) activates vascular inward rectifying K+ channels (Kir) generating hyperpolarizing signals that propagate upstream and dilate feeding arteries and arterioles (Dunn and Nelson, Circulation J. Japanese, 2010; Longden and Nelson, Microcirculation, 2015). Kir channels are expressed in endothelial (EC) and smooth muscle (SMC) cells of parenchymal arterioles (PAs) as well as in capillary endothelial cells (CaP‐ECs), thus, both sites of the vasculature may act as sensors of neuronal activity and contribute to NVC (Longden, et al. J. Gen. Phys. 2015). In this study, we outline a mathematical model of the brain vasculature containing a network of capillaries and PAs. We use the model to investigate the ability of capillaries and PAs to sense changes in [K+]o and module functional hyperemia by conducting dilatory signals upstream the vascular network.

Methods

Mathematical models of a CaP‐EC and of EC and SMC of PAs are constructed. The cell models contain expressions of ionic channels and intracellular components, and predict membrane potential (Vm) and Ca2+ dynamics. 20 CaP‐ECs, connected via gap junctions, form a model of a branched capillary network connected to a PA. Local and conducted hyperpolarization are predicted following stimulation with elevated [K+]o (from 3 to 10 mM) at different network sites. Models are coded in MATLAB and solved using backward Gear's method.

Results and Discussion

In response to [K+]o stimulation, an isolated CaP‐EC exhibits significant hyperpolarization that depends on the Kir conductance and resting membrane resistivity (DVm = −32 mV for GKir,max = 0.05 [K+]o0.5 nS and Rm = 15GΩ). In a vascular network, this hyperpolarizing current spreads to neighboring cells, reducing local hyperpolarization (DVm = −12 mV for gap junctional resistance, Rgj = 200MΩ and CaP‐ECs stimulated, n=2 cells (Fig. 1B)). Under these conditions, minimal hyperpolarization and dilation is predicted in upstream PAs (DVm < −1mV). However, if Kir current is more pronounced (GKir,max = 0.15 [K+]o0.5 nS), a higher local hyperpolarization is predicted, with Vm approaching K+ Nernst potential (EK) (Fig. 1B). The electrical signal is then conducted along the capillary network in a self‐amplifying, regenerative fashion. Kir channels in PA‐EC and PA‐SMCs can amplify it further yielding significant hyperpolarization (DVm ≈ −5 mV) and dilation of the PA (Fig. 1C).

Conclusion

A multiscale mathematical approach was utilized to investigate the role of Kir in NVC. Simulations suggest that Kir channels allow vascular cells to sense neuronal activity and enable them to efficiently transmit electrical signals and dilate upstream PAs.

Support or Funding Information

This work was supported by NIH award R01‐HL131181.