The discoverers of this are David E. Goldman of Columbia University, and the Medicine Nobel laureates Alan Lloyd Hodgkin and Bernard Katz.
is approximately 26.7 mV at human body temperature (37 °C); when factoring in the change-of-base formula between the natural logarithm, ln, and logarithm with base 10
The ionic charge determines the sign of the membrane potential contribution.
They are always very close to their respective concentrations when the membrane is at their resting potential.
Goldman's equation seeks to determine the voltage Em across a membrane.
Assuming that the system is symmetrical in the x and y directions (around and along the axon, respectively), only the z direction need be considered; thus, the voltage Em is the integral of the z component of the electric field across the membrane.
According to Goldman's model, only two factors influence the motion of ions across a permeable membrane: the average electric field and the difference in ionic concentration from one side of the membrane to the other.
For a given ion denoted A with valence nA, its flux jA—in other words, the number of ions crossing per time and per area of the membrane—is given by the formula The first term corresponds to Fick's law of diffusion, which gives the flux due to diffusion down the concentration gradient, i.e., from high to low concentration.
The second term reflects the flux due to the electric field, which increases linearly with the electric field; Formally, it is [A] multiplied by the drift velocity of the ions, with the drift velocity expressed using the Stokes–Einstein relation applied to electrophoretic mobility.
The constants here are the charge valence nA of the ion A (e.g., +1 for K+, +2 for Ca2+ and −1 for Cl−), the temperature T (in kelvins), the molar gas constant R, and the faraday F, which is the total charge of a mole of electrons.
This is a first-order ODE of the form y' = ay + b, with y = [A] and y' = d[A]/dz; integrating both sides from z=0 to z=L with the boundary conditions [A](0) = [A]in and [A](L) = [A]out, one gets the solution where μ is a dimensionless number and PA is the ionic permeability, defined here as The electric current density JA equals the charge qA of the ion multiplied by the flux jA Current density has units of (Amperes/m2).
Thus, to get current density from molar flux one needs to multiply by Faraday's constant F (Coulombs/mol).
By assumption, at the Goldman voltage Em, the total current density is zero (Although the current for each ion type considered here is nonzero, there are other pumps in the membrane, e.g. Na+/K+-ATPase, not considered here which serve to balance each individual ion's current, so that the ion concentrations on either side of the membrane do not change over time in equilibrium.)
If all the ions are monovalent—that is, if all the nA equal either +1 or -1—this equation can be written whose solution is the Goldman equation where If divalent ions such as calcium are considered, terms such as e2μ appear, which is the square of eμ; in this case, the formula for the Goldman equation can be solved using the quadratic formula.