  The degrees of freedom of multi-compartment mathematical models for energy
metabolism of a neuron-astrocyte complex may offer a key to understand the
different ways in which the energetic needs of the brain are met. In this paper
we address the problem within a steady state framework and we use the
techniques of linear algebra to identify the degrees of freedom first in a
lumped model, then in its extension to a spatially distributed case. The
interpretation of the degrees of freedom in metabolic terms, more specifically
in terms of glucose and oxygen partitioning, is then leveraged to derive
constraints on the free parameters needed to guarantee that the model is
energetically feasible. We also demonstrate how the model can be used to
estimate the stoichiometric energy needs of the cells as well as the household
energy based on observed oxidative cerebral metabolic rate (CMR) of glucose,
and the glutamate cycling. Moreover, our analysis shows that in the lumped
model the direction of lactate dehydrogenase (LDH) in the cells can be deduced
from the glucose partitioning between the compartments. The extension of the
lumped model into a spatially distributed multi-compartment setting that
includes diffusion fluxes from capillary to tissue increases the number of
degrees of freedom, requiring the use of statistical sampling techniques. The
analysis of distributed model reveals that some of the conclusions, e.g.,
concerning the LDH activity and glucose partitioning, based on a spatially
lumped model may no longer hold.
