  The different active roles of neurons and astrocytes during neuronal
activation are associated with the metabolic processes necessary to supply the
energy needed for their respective tasks at rest and during neuronal
activation. Metabolism, in turn, relies on the delivery of metabolites and
removal of toxic byproducts through diffusion processes and the cerebral blood
flow. A comprehensive mathematical model of brain metabolism should account not
only for the biochemical processes and the interaction of neurons and
astrocytes, but also the diffusion of metabolites. In the present article, we
present a computational methodology based on a multidomain model of the brain
tissue and a homogenization argument for the diffusion processes. In our
spatially distributed compartment model, communication between compartments
occur both through local transport fluxes, as is the case within local
astrocyte-neuron complexes, and through diffusion of some substances in some of
the compartments. The model assumes that diffusion takes place in the
extracellular space (ECS) and in the astrocyte compartment. In the astrocyte
compartment, the diffusion across the syncytium network is implemented as a
function of gap junction strength. The diffusion process is implemented
numerically by means of a finite element method (FEM) based spatial
discretization, and robust stiff solvers are used to time integrate the
resulting large system. Computed experiments show the effects of ECS
tortuosity, gap junction strength and spatial anisotropy in the astrocyte
network on the brain energy metabolism.
