  Cortical spreading depression and spreading depolarization (CSD) are waves of
neuronal depolarization that spread across the cortex, leading to a temporary
saturation of brain activity. They are associated to various brain disorders
such as migraine and ischemia. We consider a reduced version of a biophysical
model of a neuron-astrocyte network for the initiation and propagation of CSD
waves (Huguet et al., Biophys. J. , 2016), consisting of reaction-diffusion
equations. The reduced model considers only the dynamics of the neuronal and
astrocytic membrane potentials and the extracellular potassium concentration,
capturing the instigation process implicated in such waves. We present a
computational and mathematical framework based on the parameterization method
and singular perturbation theory to provide semi-analytical results on the
existence of a wave solution and to compute it jointly with its velocity of
propagation. The traveling wave solution can be seen as an heteroclinic
connection of an associated system of ordinary differential equations with a
slow-fast dynamics. The presence of distinct time scales within the system
introduces numerical instabilities, which we successfully address through the
identification of significant invariant manifolds and the implementation of the
parameterization method. Our results provide a methodology that allows to
identify efficiently and accurately the mechanisms responsible for the
initiation of these waves and the wave propagation velocity.
