Mathematical and numerical modeling of coupled oxygen dynamics and neuronal electrophysiology

Abstract

Modeling and simulating how oxygen supply shapes neuronal excitability is crucial for advancing the understanding of brain function in pathological scenarios, such as ischemia. This condition is caused by a reduced blood supply, leading to the deprivation of oxygen and other metabolites; this energy deficit impairs ionic pumps and causes cellular swelling. In this work, this phenomenon is modeled through a volumetric variation law that links cell swelling to local oxygen concentration and the percentage of blood flow reduction. The swelling law links volume changes to local oxygen and the degree of blood-flow depression, providing a simple mechanistic pathway from hypoxia to tortuosity-driven transport impairment. The interplay between oxygen supply and excitability in brain tissue is described by coupling the monodomain model with specific neuronal ionic and metabolic models that characterize ion and metabolite concentration dynamics. The numerical approximation of this coupled multiscale problem is particularly challenging, owing to the presence of sharp and fast-propagating wavefronts and complex geometrical domains. To address these challenges, suitable space- and time-adaptive schemes are employed to capture the action potential dynamics accurately. This multiscale model is discretized in space with the high-order p-adaptive discontinuous Galerkin method on polygonal and polyhedral grids (PolyDG) and integrated in time with a Crank-Nicolson scheme. We numerically investigate different pathological scenarios on a two-dimensional idealized square domain and on a realistic geometry, both discretized with a polygonal grid, analyzing how subclinical and severe ischemia can affect brain electrophysiology and metabolic concentrations.

Figures (15)

• Figure 1: Schematic representation of electro-metabolic model. The equilibrium between metabolic and electrophysiological dynamics is preserved by the demand/supply mechanism of ATP production. $p_n$ and $p_a$ denote phosphorylation states of neuron and astrocyte, respectively, while outputs of ATP demand are the ATP dephosphorylation fluxes, which are additional inputs of the metabolic processes. Figure adapted from Servier Medical Art (https://smart.servier.com), licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
• Figure 2: Sensitivity to ischemic severity $\delta$ for temporal zoom $t\in[41.0,\,41.35]$ s (first row) and extended window $t\in[40,\,47]$s (second row). Left: transmembrane potential $\mathrm{u}$. Middle: extracellular potassium $\text{[K$^+$]}_o$. Right: intracellular sodium $\text{[Na$^+$]}_i$.
• Figure 3: Coupled electro–metabolic response at increasing ischemic severity $\delta$. Columns correspond to $\delta=0$ (first column), $\delta=0.3$ (second column, subclinical), $\delta=0.6$ (third column, subclinical), and $\delta=0.7$ (right, severe). First row: transmembrane potential $\mathrm{u}$ (blue, left axis) and instantaneous firing frequency $f$ (right axis). Second row: ATP concentration. Third row: blood oxygen concentration $[{\rm O}_2]_b$. Fourth row: oxygen concentration in extracellular space (ECS), neuron, and astrocyte.
• Figure 4: Volume evolution of neurons, astrocytes and extracellular space with respect to different values of $\delta$.
• Figure 5: Sensitivity analysis with respect to potassium clearance rate ($\varepsilon$) in severe ischemic condition. (a) $\varepsilon=2.7$. (b) $\varepsilon=5$. (c) $\varepsilon=9.33$. (d) $\varepsilon=13$.