A three-dimensional mathematical model for the signal propagation on a neuron's membrane

In order to be able to examine the extracellular potential's influence on network activity and to better understand dipole properties of the extracellular potential, we present and analyze a three-dimensional formulation of the cable equation which facilitates numeric simulations. When the neuron's intra- and extracellular space is assumed to be purely resistive (i.e., no free charges), the balance law of electric fluxes leads to the Laplace equation for the distribution of the intra- and extracellular potential. Moreover, the flux across the neuron's membrane is continuous. This observation already delivers the three dimensional cable equation. The coupling of the intra- and extracellular potential over the membrane is not trivial. Here, we present a continuous extension of the extracellular potential to the intracellular space and combine the resulting equation with the intracellular problem. This approach makes the system numerically accessible. On the basis of the assumed pure resistive intra- and extracellular spaces, we conclude that a cell's out-flux balances out completely. As a consequence neurons do not own any current monopoles. We present a rigorous analysis with spherical harmonics for the extracellular potential by approximating the neuron's geometry to a sphere. Furthermore, we show with first numeric simulations on idealized circumstances that the extracellular potential can have a decisive effect on network activity through ephaptic interactions.