Mathematical and numerical modeling of coupled oxygen dynamics and neuronal electrophysiology

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.

💡 Research Summary

The manuscript presents a comprehensive multiscale mathematical and computational framework for investigating how oxygen supply modulates neuronal excitability, with a particular focus on ischemic pathology. The authors first introduce a volumetric variation law that links cellular swelling to local oxygen concentration and the degree of blood‑flow reduction. This law provides a mechanistic bridge from hypoxia to tortuosity‑driven transport impairment, as swelling reduces extracellular space and alters diffusion pathways.
The core of the model couples three sub‑systems: (i) a reduced metabolic network based on four well‑mixed compartments (blood, extracellular space, neurons, astrocytes) that tracks oxygen, ATP, ADP, NADH, and NAD⁺ dynamics; (ii) an electrophysiological model derived from the Barreto‑Cressman conductance formulation, extended to include astrocytic potassium clearance, calcium dynamics, and explicit ATP‑dependent scaling of Na⁺/K⁺‑ATPase activity; and (iii) the monodomain equation governing the propagation of the transmembrane potential across tissue. Energy demand is supplied by the metabolic subsystem through ATP dephosphorylation fluxes, while metabolic production is driven by oxygen delivery, which itself depends on blood flow and arterial oxygen concentration. The coupling is bidirectional: intense firing increases ATP consumption, accelerating local oxygen depletion, which in turn impairs ion‑pump function and can trigger pathological depolarization.
From a numerical standpoint, the authors adopt a high‑order p‑adaptive discontinuous Galerkin (DG) discretization on polygonal and polyhedral meshes (PolyDG). This choice accommodates complex brain geometries, resolves steep voltage gradients, and enables h‑, p‑, and hp‑adaptivity. Temporal integration uses a second‑order Crank‑Nicolson scheme combined with time‑step adaptivity to handle the disparate time scales of electrophysiology (milliseconds) and metabolism (seconds to minutes). Spatial p‑adaptivity selectively raises the polynomial degree in regions where the action potential front is sharp, reducing the overall degrees of freedom while preserving accuracy.
The computational experiments are conducted on two domains: an idealized two‑dimensional square and a realistic brain slice reconstructed from MRI data, both meshed with polygonal elements. Blood‑flow reduction is varied from 10 % to 70 % to emulate subclinical to severe ischemia. Results demonstrate that modest flow reductions cause only minor alterations in wave speed and shape, whereas severe reductions lead to pronounced cellular swelling, extracellular space shrinkage, increased tortuosity, and a marked slowdown of potassium diffusion. Consequently, the Na⁺/K⁺‑ATPase activity drops sharply, extracellular potassium accumulates, and the tissue exhibits spontaneous, high‑frequency spiking or depolarization block. Metabolic variables (