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

Mathematics

Marco A. Herrera-Valdez¹  Sergei K. Suslov² 

[1] Unidad de Sistemas Complejos, Biofísica y Fisiología, Academia Nacional de Investigación y Desarrollo, Cuernavaca, Morelos 62040, México; E-Mail:;School of Mathematical and Statistical Sciences & Mathematical, Computational and Modeling Sciences Center, Arizona State University, Tempe, AZ 85287–1804, USA; E-Mail:

关键词: cable equation;    hyperbolic functions;    Bessel functions;    Ince’s equation;   

DOI  :  10.3390/math2030119

来源: mdpi

【 摘 要 】

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.

【 授权许可】

CC BY   
© 2014 by the authors; licensee MDPI, Basel, Switzerland.