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

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Unidad de Sistemas Complejos, Biofásica y Fisiología, Academia Nacional de Investigación y Desarrollo, Cuernavaca, Morelos 62040, México

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Evelyn F. McKnight Brain Institute, University of Arizona, Tucson, AZ 85724, USA

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Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, Ciudad de México, DF 04510, México

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School of Mathematical and Statistical Sciences & Mathematical, Computational and Modeling Sciences Center, Arizona State University, Tempe, AZ 85287–1804, USA

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Department of Mathematics, Howard University, 223 Academic Support Building B, Washington, DC 20059, USA

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Author to whom correspondence should be addressed.

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.