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,

for

,

,

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 [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 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 [1] (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.

[1] B. Hille,

*Ion Channels of Excitable Membranes*, 3rd ed., Sunderland, MA: Sinauer, 2001.

[2] C. Koch,

*Biophysics of Computation: Information Processing in Single Neurons,* New York: Oxford University Press, 1999.