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

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.