### 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.

Original language | English (US) |
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Pages (from-to) | 119-135 |

Number of pages | 17 |

Journal | Mathematics |

Volume | 2 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 2014 |

### Keywords

- Bessel functions
- Cable equation
- Hyperbolic functions
- Ince's equation

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Mathematics*,

*2*(3), 119-135. https://doi.org/10.3390/math2030119