A three-dimensional mathematical model of active signal processing in axons

Konstantinos Xylouris*, Gillian Queisser, Gabriel Wittum

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


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 languageEnglish (US)
Pages (from-to)409-418
Number of pages10
JournalComputing and Visualization in Science
Issue number8
StatePublished - Dec 2010
Externally publishedYes


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

ASJC Scopus subject areas

  • Software
  • General Engineering
  • Theoretical Computer Science
  • Computer Vision and Pattern Recognition
  • Computational Theory and Mathematics
  • Modeling and Simulation


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