  Cortical spreading depression (CSD) is a slow-moving ionic and metabolic
disturbance that propagates in cortical brain tissue. In addition to massive
cellular depolarization, CSD also involves significant changes in perfusion and
metabolism -- aspects of CSD that had not been modeled and are important to
traumatic brain injury, subarachnoid hemorrhage, stroke, and migraine.
  In this study, we develop a mathematical model for CSD where we focus on
modeling the features essential to understanding the implications of
neurovascular coupling during CSD. In our model, the sodium-potassium--ATPase,
mainly responsible for ionic homeostasis and active during CSD, operates at a
rate that is dependent on the supply of oxygen. The supply of oxygen is
determined by modeling blood flow through a lumped vascular tree with an
effective local vessel radius that is controlled by the extracellular potassium
concentration. We show that during CSD, the metabolic demands of the cortex
exceed the physiological limits placed on oxygen delivery, regardless of
vascular constriction or dilation. However, vasoconstriction and vasodilation
play important roles in the propagation of CSD and its recovery. Our model
replicates the qualitative and quantitative behavior of CSD --
vasoconstriction, oxygen depletion, extracellular potassium elevation,
prolonged depolarization -- found in experimental studies.
  We predict faster, longer duration CSD in vivo than in vitro due to the
contribution of the vasculature. Our results also help explain some of the
variability of CSD between species and even within the same animal. These
results have clinical and translational implications, as they allow for more
precise in vitro, in vivo, and in silico exploration of a phenomenon broadly
relevant to neurological disease.
