Diffusing Substances during Spreading Depolarization: Analytical Expressions for Propagation Speed, Triggering, and Concentration Time Courses

Abstract

Spreading depolarization (SD) is an important phenomenon in stroke and migraine. However, the processes underlying the propagation of SD are still poorly understood, and an elementary model that is both physiological and quantitative is lacking. We show that, during the onset and propagation of SD, the concentration time courses of excitatory substances such as potassium and glutamate can be described with a reaction-diffusion equation. This equation contains four physiological parameters: (1) a concentration threshold for excitation; (2) a release rate; (3) a removal rate; and (4) an effective diffusion constant. Solving this equation yields expressions for the propagation velocity, concentration time courses, and the minimum stimulus that can trigger SD. This framework allows for analyzing experimental results in terms of these four parameters. The derived time courses are validated with measurements of potassium in rat brain tissue.

A Biophysical Model for Cytotoxic Cell Swelling

“…It reliably reproduced experimental data of ouabaininduced anoxic depolarization (Zandt et al, 2013b; Figs. 5A, 7B) and veratridine-induced neuronal swelling (Rungta et al, 2015; Fig.…”

“…This assumption holds in physiological conditions where the ATP supply is sufficient to maintain ion homeostasis and firing rates are modest. However, it loses its validity if the ATP supply does not meet its need (Zandt et al, 2011(Zandt et al, , 2013b, as observed in ischemia (Stokum et al, 2016), or pathological brain states that are intrinsically characterized by a massive redistribution of ions, such as seizures (Fröhlich et al, 2008;Raimondo et al, 2015) and spreading depolarization (Somjen, 2001;Zandt et al, 2013a). Recently, a similar approach was used in a threecompartment model to study cell swelling in astrocytes and neurons during spreading depolarization (Hübel and Ullah, 2016).…”

Imaging Reveals the Focal Area of Spreading Depolarizations and a Variety of Hemodynamic Responses in a Rat Microembolic Stroke Model

“…The duration of SD is indicative of the energy crisis within the tissue, 25 and the rate of SD propagation is predicted to increase when (1) extracellular K þ removal is hampered, (2) the extracellular space is smaller (cell swelling), and (3) extracellular K þ and/or glutamate concentrations are increased. 26 In line with the predictions, hypoxic SDs tended to propagate B1 mm/min more rapidly than normoxic SDs as observed in brain slices. 27 These data in combination with our findings suggest that the tissue at the SD focus may be at the highest risk for permanent injury.…”

Live, optical experimental neuroimaging reveals variations of the cerebral blood flow response to ischemic spreading depolarization

“…The rate of SD propagation is predicted to increase when (1) extracellular K + removal is hampered, (2) the extracellular space is smaller (cell swelling), and (3) extracellular K + and/or glutamate concentrations are increased [73]. In line with the predictions, hypoxic SDs tended to propagate approximately 1 mm/min more rapidly than normoxic SDs as observed in brain slices [74,75].…”