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Stochastic Modeling of Spreading Cortical Depression

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Stochastic Biomathematical Models

Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 2058))

Abstract

The nonlinear wave phenomenon of cortical spreading depression (SD), which occurs in many brain structures, has mathematical similarities to neuronal spiking but on very different space and time scales. Its properties and previous modeling are briefly reviewed. A model consisting of a 6-component reaction–diffusion system in two space dimensions is described. With 3-parameter Poisson process sources of potassium ions representing extrusions due to the random firings of neurons, the model takes the form of a multi-component set of nonlinear stochastic partial differential equations. Assuming that in a restricted small area the sources have greater strength than background, the probability of an SD wave is found as a function of the patch size. Also investigated is the probability of elicitation of SD through the occurrence of a patch with compromised metabolic activity, as may occur by virtue of an infarct after stroke. The analysis proceeds in terms of the effect of relative decreases in the strength of ATP-dependent sodium–potassium exchange pump.

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Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, Leipzig, 04103, Germany

    Henry C. Tuckwell

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  1. Henry C. Tuckwell
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Correspondence to Henry C. Tuckwell .

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Editors and Affiliations

  1. College of Sciences, Department of Mathematics, King Saud University, King Khalid Road, Diriyah Campus, Riyadh, 11451, Saudi Arabia

    Mostafa Bachar

  2. Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, Graz, 8010, Austria

    Jerry Batzel

  3. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, Copenhagen, 2100, Denmark

    Susanne Ditlevsen

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Tuckwell, H.C. (2013). Stochastic Modeling of Spreading Cortical Depression. In: Bachar, M., Batzel, J., Ditlevsen, S. (eds) Stochastic Biomathematical Models. Lecture Notes in Mathematics(), vol 2058. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32157-3_8

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