Traveling waves in a model for cortical spreading depolarization with slow-fast dynamics

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 with 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. 111(2), 452–462, 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 a 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.Work produced with support of the grant PID-2021-122954NB-100 funded by MCIN/AEI/ 10.13039/501100011033 and “ERDF: A way of making Europe.” T.M.S and G.H acknowledge the Maria de Maeztu Award for Centers and Units of Excellence in R&D (No. CEX2020-001084-M). T.M.S. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2019. We also acknowledge the use of the UPC Dynamical Systems group’s cluster for research computing (https://dynamicalsystems.upc.edu/en/computing/)Peer ReviewedPostprint (published version