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Origination and Propagation of Reaction Diffusion Waves in Three Spatial Dimensions

Hahn, Philip James
http://rave.ohiolink.edu/etdc/view?acc_num=case1091809306

Abstract Details

2004, Doctor of Philosophy, Case Western Reserve University, Mathematics.
Propagation of functional or pathological ionic disturbances in biological systems plays an important role in normal regulatory mechanisms and in disease. Potassium diffusion in brain tissue is involved in spreading excitation. Models of this type of phenomenon often take the form of a reaction-diffusion system in one spatial dimension with continuous dynamic variables. Examined here is propagation in three spatial dimensions through a network of discrete dynamic elements coupled by diffusion. Conditions permissive of pulse origination and propagation can be determined analytically for systems in one spatial dimension. However, in three spatial dimensions or in dynamic systems containing discontinuities, explicit solutions may not exist. Instead, the local dynamics of the excitable system at a point in space are analyzed. The effective diffusive flux or current at a point is interpreted as a slowly varying parameter. The bifurcation structure of the dynamics with respect to this parameter and the effect of waveform on the time course of the parameter are examined. Propagation results when an excursion at a point produces a diffusion current sufficient to move its resting neighbor above some threshold value. The formation of a pulse back depends on the stability of equilibria of the local dynamics. Propagation in some cases may also depend on the geometry of the wavefront. Predictions are verified by numerical simulation using a software package developed by the author for this dissertation. A three dimensional lattice allows for description of the local dynamics at nodal elements and diffusion between elements and throughout the lattice. Three models are studied using the method developed. First, the Fitzhugh-Nagumo equation is used to illustrate the method. Second, the continuous Nelkin-Yaari model, describing spreading excitation in brain tissue, is examined. Third, a novel model of non-synaptic pulse propagation in hippocampal slices is developed and analyzed. Investigation of this new model shows that potassium wave behavior in the CA1 region can be explained using descriptions of only two phenomena: action potential spike dynamics in response to elevated potassium and simple sink functions that allow for the formation of a wave backside and refractory time.
James Alexander (Advisor)
Mathematics
Diffusion; excursion; Potassium

Recommended Citations

Citations

  • Hahn, P. J. (2004). Origination and Propagation of Reaction Diffusion Waves in Three Spatial Dimensions [Doctoral dissertation, Case Western Reserve University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=case1091809306

    APA Style (7th edition)

  • Hahn, Philip. Origination and Propagation of Reaction Diffusion Waves in Three Spatial Dimensions. 2004. Case Western Reserve University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=case1091809306.

    MLA Style (8th edition)

  • Hahn, Philip. "Origination and Propagation of Reaction Diffusion Waves in Three Spatial Dimensions." Doctoral dissertation, Case Western Reserve University, 2004. http://rave.ohiolink.edu/etdc/view?acc_num=case1091809306

    Chicago Manual of Style (17th edition)

case1091809306
973
© 2004, all rights reserved.
This open access ETD is published by Case Western Reserve University School of Graduate Studies and OhioLINK.