## Abstract

Action potentials in neurons are generated on the plasma membrane through depolarization, i.e. exchange of charges through the membrane. Hodgkin and Huxley developed a mathematical model which describes the interaction of ions through an active plasma membrane. In Vossen et al. (Comput Visual Sci 10:107-121, 2007) we developed a passive three-dimensional (3D) model for signal propagation in dendrite. We now combine this model with a generalized Hodgkin-Huxley model to obtain a 3D-model that describes active signal processing on realistic cell morphologies. Time dependent changes of the neuron's intra- and extracellular potential is regulated by the Ohmic flux of charges. These fluxes are balanced in membrane-near areas by the capacitory and Hodgkin-Huxley flux. The active model we present consists of five non-linear, coupled integro-differential equations which are solved numerically with a finite volume approach, implicit time stepping and Newton's method for solving the underlying non-linear system of equations with multigrid solver methods. We present numerical results as well as axon behavior in a biological setting. This model can be considered as a three-dimensional expansion of existing state of the art one-dimensional models, with the significant advantage of being able to investigate the morphological influence of neuron cell types on their specific signaling properties.

Original language | English (US) |
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Pages (from-to) | 409-418 |

Number of pages | 10 |

Journal | Computing and Visualization in Science |

Volume | 13 |

Issue number | 8 |

DOIs | |

State | Published - Dec 1 2010 |

## Keywords

- 3D-model
- Action potential
- Dendrites
- Hodgkin-huxley
- Mathematical model
- Neuron
- Signal propagation

## ASJC Scopus subject areas

- Modeling and Simulation
- Theoretical Computer Science
- Software
- Computer Vision and Pattern Recognition
- Computational Theory and Mathematics
- Engineering(all)