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

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.

A Multi-Domain Continuum Model of Electrical Stimulation of Healthy and Degenerate Retina

“…A Cable Equation for Neuronal Modelling Non-uniformity in neuron geometry has been investigated in many mathematical studies. Several axonal or dendritic geometries were explored and analytical solutions were proposed for steady or transient state, and with passive or active properties (Goldstein and Rall, 1974;Herrera-Valdez et al, 2014;Joyner et al, 1978;Khodorov and Timin, 1976;Manor et al, 1991;Ohme and Schierwagen, 1998;Rattay et al, 2001;Schierwagen, 1989;Zhou and Bell, 1994). Most of these studies aimed to reduce the dendrites with their branches into one dendritic cylinder, with Rall's "equivalent cylinder" models being utilised as a basis for many of these studies.…”

Electrophysiological effects of low frequency electrical radiation on the neural compartment: a theoretical investigation

“…To describe the excitability of a neuron and the active membrane dynamics, the modified cable equation along with the Connor-Stevens model is employed [37][38][39]. In its simplest case, the cable equation is written in form of…”

“…Details of the rate functions used in the gating variables are included in the supplementary information that is available online at stacks.iop.org/BPEX/4/025040/mmedia. In order to consider the effect of deviation of geometry from a uniform cylinder, we have used the modified cable equation [37,39]. Considering the variation of radius of the cylinder, the modified cable equation will be…”

Effect of geometry on propagation of action potentials in neurons; an in silico analysis

“…In order to consider the effect of change in geometry of neuron and its effect on the propagation of action potential, we have exploited the model in which the spatial variations have been incorporated and an extended cable equation has been obtained [9]. As depicted in Fig.…”