Home / Extensions of the Cable Equation Incorporating Spatial Dependent Variations in Nerve Cell Diameter

Extensions of the Cable Equation Incorporating Spatial Dependent Variations in Nerve Cell Diameter

Neuronal function relies on electrical signaling which, in turn, depends greatly on the morphology of neurons. Local changes in the diameter of neurites caused by deformations such as tumors or enlarged cerebral aneurysms may significantly affect electrical conduction. Morphologically accurate models are typically constructed by representing neurons as branched cables, where a system of coupled cable equations represents small neuronal segments as cylinders. Two extensions of the linear cable equation that allow the radius to change continuously in space are constructed here by assuming that the radius of a neuronal segment changes to form a section of a cone or a hyperbolic volume. The cable equation on a cone has an analytical solution involving modified Bessel functions. A more realistic representation of radial deformation is realized by writing the radius as a hyperbolic function of distance along the neurite. The steady state version of the hyperbolic equation is an adapted form of Ince's equation. The cable equation with a hyperbolic radius has an analytical solution only in special conditions, and its numerical solutions can be qualitatively compared to the steady state solutions of the cable equation on a series of cylinders. Unlike the cases for the cylindrical and frustum shapes, the steady state solution of the equation on the hyperbolic cable describes a decay followed by an amplification of voltage along the cable. This counter-intuitive result suggests that non-uniformities in the diameter of neurites have local amplifying effects that may have a non-negligible impact on the propagation of signals in neurons. This research shows how different geometrical representations used to derive cable equations have varying impacts on the calculation of solutions.

  • Poster session award recipient at the 2011 Emerging Researchers National Conference in STEM in Washington DC

Article Number:


Avis Foster, George Mason University
Emily Hendryx, Angelo State University
Anarina Murillo, Arizona State University
Michelle Salas, University of Portland
Emmanuel J. Morales-Butler, Arizona State University
Sergei Suslov, Arizona State University
Marco Herrera-Valdez, Arizona State University