A Graphical Approach to a Model of a Neuronal Tree with a Variable Diameter

Marco A. Herrera-Valdez, Sergei K. Suslov and José M. Vega-Guzmán
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Marco A. Herrera-Valdez: Unidad de Sistemas Complejos, Biofásica y Fisiología, Academia Nacional de Investigación y Desarrollo, Cuernavaca, Morelos 62040, México
Sergei K. Suslov: School of Mathematical and Statistical Sciences & Mathematical, Computational and Modeling Sciences Center, Arizona State University, Tempe, AZ 85287–1804, USA
José M. Vega-Guzmán: Department of Mathematics, Howard University, 223 Academic Support Building B, Washington, DC 20059, USA

Mathematics, 2014, vol. 2, issue 3, 1-17

Abstract: Tree-like structures are ubiquitous in nature. In particular, neuronal axons and dendrites have tree-like geometries that mediate electrical signaling within and between cells. Electrical activity in neuronal trees is typically modeled using coupled cable equations on multi-compartment representations, where each compartment represents a small segment of the neuronal membrane. The geometry of each compartment is usually defined as a cylinder or, at best, a surface of revolution based on a linear approximation of the radial change in the neurite. The resulting geometry of the model neuron is coarse, with non-smooth or even discontinuous jumps at the boundaries between compartments. We propose a hyperbolic approximation to model the geometry of neurite compartments, a branched, multi-compartment extension, and a simple graphical approach to calculate steady-state solutions of an associated system of coupled cable equations. A simple case of transient solutions is also briefly discussed.

Keywords: cable equation; hyperbolic functions; Bessel functions; Ince’s equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2014
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