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 equilbrium. 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

) at position

and time

as

,

where

(

) is the diffusion constant,

is the ionic concentration,

is the ion's charge (unitless),

is the Faraday constant,

is the universal gas constant,

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

.

In this demonstration, this equation is considered in a one-dimensional form 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

identify the interior, exterior of the neuron. Divide this distance into

compartments of width

. Furthermore, discretize time into time 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

As parameter values, we take the membrane width

, the spacing

, the temperature

. It is assumed that the electric field is constant across the membrane, i.e. 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 [3]

where

is the magnitude of the electric field.

Four ion species may be examined, with diffusion constants taken from [1] (Table 10.1) at

in units of

as

.

Note that these values are approximate, and the diffusion constant may in general vary with temperature and across the membrane. The default values [1] (Table 1.3) of the initial interior concentrations in

are

,

and exterior concentrations are

.

These values may be adjusted using the sliders which add/subtract a percentage of the default concentrations.

Three initial concentration shapes are possible:

(i) 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 equilbrium, indicated by a solid brown line.

(ii) a linear shape between the interior and exterior concetrations.

(iii) 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.

[1] B. Hille,

*Ion channels of excitable membranes, *Sunderland, Mass: Sinauer, 2001. Print.

[2] Koch, Christof,

*Biophysics of computation information processing in single neurons,* New York: Oxford University Press, 1999. Print.