Electrodiffusion of Ions across a Neural Cell Membrane
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The Nernst–Planck equation describes the diffusion of ions under the influence of an electric field. Here, it is applied to describe the movement of ions across a neural cell membrane. The top half of the Demonstration sets up the simulation, while the bottom displays the results.[more]
You can select the ions , which play key roles in neural dynamics, and vary their interior and exterior concentrations using the sliders (shown in dotted and dashed green). The electric field across the membrane is assumed constant so that the potential is linear, as shown in the left plot. You can choose from three different initial concentration distributions, shown by a dashed blue line in the right plot. The total simulation time can be set to either a relatively short or a longer time (slight delay to update).
Once the parameters are set, click the "run simulation" button to generate the time development of the ion concentration, shown in solid blue. Move the simulation time slider to view the evolution of the distribution in time. If the parameters are changed, the curve turns to a dashed gray to indicate that the simulation should be run again. Equilibrium for the concentration of a single ion occurs when the potential is set to the Nernst reversal potential and the initial distribution is set such that there is zero ionic current, each shown in brown in the left and right plots.[less]
Snapshot 1: the time evolution of sodium ions from an initially linear concentration distribution
Snapshot 2: at the Nernst reversal potential, for an appropriate initial concentration distribution, the system is at equilibrium
Snapshot 3: the initial delta spike of potassium ions decays as ions drift due to diffusion and the electric field
Ions undergoing diffusion in the presence of an electric field give rise to an ionic current flux (here in units of ) at position and time as
where (in ) is the diffusion constant, (in mM) is the ionic concentration, is the ion's charge (unitless), is the Faraday constant, is the universal gas constant, (in K) is the temperature, and is the electric potential [1, 2]. Combined with the continuity equation
the Nernst–Planck equation describing the evolution of the ionic concentration in time is obtained as
This Demonstration assumes one-dimensional motion to describe the diffusion of ions across a neural membrane. Let denote the direction through the membrane, perpendicular to the surface, where is the width of the membrane, such that and identify the interior and exterior of the neuron. Divide this distance into compartments of width . Furthermore, discretize time into steps such that for and . Let and denote the constant ion concentration and potential in compartment at time step . A finite-difference approximation of the Nernst–Planck equation with zero-flux boundary conditions at the membrane boundaries is,
and for and ,
As parameter values, take the membrane width , the spacing , and the temperature . Assume that the electric field is constant across the membrane; that is, the potential is a linear function of the distance, with a potential set at . The time step is chosen to satisfy the stability criteria obtained by a von Neumann stability analysis 
where is the magnitude of the electric field.
Four ion species may be examined, with diffusion constants taken from  (Table 10.1) at 25 °C in units of as
Note that these values are approximate, so that the diffusion constant can, in general, vary with temperature and across the membrane. The default values  (Table 1.3) of the initial interior concentrations in mM are
and exterior concentrations are
You can vary these values using the sliders that add or subtract a percentage of the default concentrations.
Three initial concentration shapes are possible:
1. The equilibrium shape at the Nernst reversal potential. If the membrane potential is also set to the Nernst reversal potential, the ionic flux is zero and the initial distribution is the equilibrium, indicated by a solid brown line.
2. A linear shape between the interior and exterior concentrations.
3. A delta function shape, with zero initial concentration everywhere across the membrane except the endpoints.
Note finally the connection between the Nernst–Planck equation and the Goldman–Hodgkin–Katz equation, which may be derived as the solution to the first differential equation above for constant ionic current flux.
 B. Hille, Ion Channels of Excitable Membranes, 3rd ed., Sunderland, MA: Sinauer, 2001.
 C. Koch, Biophysics of Computation: Information Processing in Single Neurons, New York: Oxford University Press, 1999.
 Wikipedia. "Von Neumann Stability Analysis." (Sep 23, 2015) en.wikipedia.org/w/index.php?title=Von_Neumann _stability _analysis&oldid=674227751.