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Long-term potentiation and mathematical analysis of electrical models of dendritic spines

University of British Columbia

Dendritic spines are small evaginations of the dendrites of neurons first discovered late in the 19th century. Since their discovery, many theories have been put forth to explain the physiological role of the spine. However, only recently with the advent of new laboratory technology has data been available to test the various theories put forward. The two most compelling theories today are that spines are important mediators of a form of cellular memory known as long-term potentiation (LTP), and that spines may be involved in the conduction of regenerative electrical impulses within dendrites similar to action potentials in axons. We will review some of the major mathematical models put forth which attempt to explain the role spines may play in the induction of LTP. We will address the importance of calcium signals in LTP induction and suggest how the unique morphology of the spine may allow for transient, spatially localized increases in calcium within the spine head, but not elsewhere in the dendrite. This could help account for the associativity, cooperativity, and input specificity requirements of LTP. We will also review some of the major mathematical models on dendritic action potentials. These models generally assume the existence of voltage-gated ion channels with Hodgkin-Huxley (HH) type dynamics exist in the spine head. We will employ a continuum approach in which spines are modeled as having a certain uniform density. We will further make use of the FitzHugh-Nagumo (FHN) equations without recovery to approximate the HH equations. We will examine the new set of equations in the traveling wave frame and seek to determine how the various parameters influence the speed, and the shape of the traveling wave front solutions. We show that there is a certain balance between local excitation of the spine heads, and freedom for the electrical current to pass from spine head to dendrite required for traveling front solutions to exist. Furthermore, strict parameter spaces in which traveling front solutions exist are determined, as is the profile of the wave in the special case of a standing wave.

Thesis/Dissertation

application/pdf

2009-05-26

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http://hdl.handle.net/2429/8280

Mathematics

University of British Columbia

UBCV

DSpace