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Open AccessArticle

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

Unidad de Sistemas Complejos, Biofásica y Fisiología, Academia Nacional de Investigación y Desarrollo, Cuernavaca, Morelos 62040, México
Evelyn F. McKnight Brain Institute, University of Arizona, Tucson, AZ 85724, USA
Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, Ciudad de México, DF 04510, México
School of Mathematical and Statistical Sciences & Mathematical, Computational and Modeling Sciences Center, Arizona State University, Tempe, AZ 85287–1804, USA
Department of Mathematics, Howard University, 223 Academic Support Building B, Washington, DC 20059, USA
Author to whom correspondence should be addressed.
Mathematics 2014, 2(3), 119-135;
Received: 18 January 2014 / Revised: 18 June 2014 / Accepted: 20 June 2014 / Published: 9 July 2014
(This article belongs to the Special Issue Mathematics on Partial Differential Equations)
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. View Full-Text
Keywords: cable equation; hyperbolic functions; Bessel functions; Ince’s equation cable equation; hyperbolic functions; Bessel functions; Ince’s equation
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Herrera-Valdez, M.A.; Suslov, S.K.; Vega-Guzmán, J.M. A Graphical Approach to a Model of a Neuronal Tree with a Variable Diameter. Mathematics 2014, 2, 119-135.

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