Goldman-style membrane potential tools
The Goldman-style membrane potential model explains how the whole-cell voltage depends on multiple ions at the same time, not just one ion in isolation. This is the natural quantitative extension beyond the single-ion Nernst equation because real membranes are influenced by K+, Na+, Cl−, and their relative permeabilities.
Core idea and main equation
The key concept is that membrane voltage is pulled toward the equilibrium potentials of the ions that have the greatest permeability. When K+ permeability is dominant, Vm usually stays closer to EK. When Na+ permeability rises, the membrane potential shifts in the positive direction. Chloride contributes through the same weighted balance, but because it is an anion, its concentration terms are reversed in the Goldman-Hodgkin-Katz form.
\[
\begin{aligned}
V_m &= \frac{R \cdot T}{F}\ln\!\left(
\frac{P_K[K^+]_{out} + P_{Na}[Na^+]_{out} + P_{Cl}[Cl^-]_{in}}
{P_K[K^+]_{in} + P_{Na}[Na^+]_{in} + P_{Cl}[Cl^-]_{out}}
\right)\cdot 1000
\end{aligned}
\]
For comparison, each single-ion equilibrium potential can still be computed with the Nernst equation:
\[
\begin{aligned}
E_{ion} &= \frac{R \cdot T}{z \cdot F}\ln\!\left(\frac{[ion]_{out}}{[ion]_{in}}\right)\cdot 1000
\end{aligned}
\]
Symbols: \(P\) represents relative permeability, \(R\) is the gas constant, \(T\) is absolute temperature in kelvin, and \(F\) is Faraday’s constant. Square brackets represent ion concentrations inside or outside the membrane.
How to interpret the result
A more negative Vm usually means the membrane is being pulled more strongly toward EK, which is common in resting neurons. A less negative or more positive Vm often means Na+ permeability has increased. If chloride permeability becomes more influential, Vm may move toward ECl depending on the concentration gradient and cell type.
The most useful interpretation is not only the final voltage, but also why it moved. A Goldman-style simulator is strongest when permeability is changed interactively and the voltage shift is observed immediately. That makes the weighted contribution of each ion much easier to understand than a static formula alone.
- Treating the Goldman model like a single-ion equation.
- Forgetting that Cl− uses reversed concentration placement in the GHK form.
- Changing permeability without noticing which equilibrium potential Vm is approaching.
- Mixing concentration values or temperature units incorrectly.
Micro example: if PK is much larger than PNa, Vm stays closer to EK. If PNa increases, the membrane potential shifts away from EK and toward ENa, which is exactly the kind of behavior expected during depolarization.
This tool is useful for interactive membrane-potential simulation and for comparing resting, depolarizing, and repolarizing states. It is not a full time-dependent action-potential model; the next step beyond this topic is conductance changes over time, voltage-gated channels, and dynamic membrane-current models.
