Action Potentials

Summary Table: Ion Contributions to Resting Membrane Potential (~ -70 mV)

Ion

Typical Extracellular Concentration (mM)

Typical Intracellular Concentration (mM)

Approx. Nernst Equilibrium Potential (Eion​) (mV) at 37°C

Example Relative Permeability at Rest (Pion​)

Driving Force at Vm​=−70 mV (Vm​−Eion​) (mV)

Primary Role at Rest / Net Effect on Vrest​

K+ (Potassium)

5

140

-90

PK​=1.0 (Reference)

(−70 mV) - (−90 mV) = +20 mV

Primary determinant of Vrest​. High permeability and outward concentration gradient lead to K+ efflux, making the inside negative. The +20 mV driving force indicates a net outward movement of K+ at -70 mV, which is balanced by inward currents.

Na+ (Sodium)

145

15

+61

PNa​=0.05

(−70 mV) - (+61 mV) = −131 mV

Shifts Vrest​ to be less negative than EK​. Low permeability but very large inward driving force causes a small, steady Na+ influx, making Vrest​ more positive than EK​ alone would dictate.

Cl− (Chloride)

110

10

~$ -64

PCl​=0.45

(−70 mV) - (−64 mV) = −6 mV

Stabilizes Vrest​ near ECl​. With ECl​ close to Vrest​, the driving force is small. Its permeability helps buffer Vm​ against changes. If Vm​ becomes more positive than ECl​, Cl− influx will counter it (inhibitory). The -6 mV driving force suggests a small net inward Cl− movement at -70 mV

Note on Driving Force Sign: A negative driving force for a positive ion means influx; a positive driving force for a positive ion means efflux. For a negative ion like Cl−, a negative driving force means influx (making the inside more negative), and a positive driving force means efflux.

Goldman-Hodgkin-Katz (GHK) Equation Demonstration

The GHK equation allows us to calculate the membrane potential (Vm​) by considering the concentrations and relative permeabilities of multiple ions.

The equation is: Vm=\frac{RT}{F}\ln\left(\frac{PK\left\lbrack K^{+}\right\rbrack out+PNa\left\lbrack Na^{+}\right\rbrack out+PCl\left\lbrack Cl^{-}\right\rbrack dentro}{PK\left\lbrack K^{+}\right\rbrack dentro+PNa\left\lbrack Na+\right\rbrack dentro+PCl\left\lbrack Cl^{-}\right\rbrack out}\right)

Where:

  • Vm​ = Membrane potential

  • R = Ideal gas constant (8.314 J⋅mol−1⋅K−1)

  • T = Absolute temperature (in Kelvin)

  • F = Faraday constant (96485 C⋅mol−1)

  • Pion​ = Relative permeability of the membrane to that ion

  • [ion]out​ = Extracellular concentration of the ion

  • [ion]in​ = Intracellular concentration of the ion

Values for Calculation (at 37°C):

  • T=37 ∘C=310.15 K

  • FRT​≈0.0267 V=26.7 mV

Ion Concentrations (from table, in mM):

  • [K+]out​=5

  • [K+]in​=140

  • [Na+]out​=145

  • [Na+]in​=15

  • [Cl−]out​=110 (Note: for the GHK equation, the Cl− concentrations are inverted in the fraction shown above because of its negative charge. If you write the Cl− permeability term separately with z=−1, you'd use [Cl−]out​ in the numerator for the current driving Cl− in and [Cl−]in​ in the denominator. The form I've used above is standard and accounts for the charge by inverting the concentration terms for anions relative to cations).

  • [Cl−]in​=10

Relative Permeabilities (as discussed):

  • PK​=1.0

  • PNa​=0.05

  • PCl​=0.45

Calculation Steps:

  1. Calculate the numerator term (N): N=PK​[K+]out​+PNa​[Na+]out​+PCl​[Cl−]in​ N=(1.0×5)+(0.05×145)+(0.45×10) N=5+7.25+4.5 N=16.75

  2. Calculate the denominator term (D): D=PK​[K+]in​+PNa​[Na+]in​+PCl​[Cl−]out​ D=(1.0×140)+(0.05×15)+(0.45×110) D=140+0.75+49.5 D=190.25

  3. Calculate the ratio (N/D): Ratio = 16.75/190.25≈0.08804

  4. Calculate Vm​: Vm​=26.7 mV×ln(0.08804) ln(0.08804)≈−2.4299 Vm​=26.7 mV×(−2.4299) Vm​≈−64.88 mV