braincell.channel.ghk_flux

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braincell.channel.ghk_flux#

braincell.channel.ghk_flux(V, ci, co, z, temp)[source]#

Compute the Goldman-Hodgkin-Katz constant-field flux, per permeability.

Evaluates the constant-field (GHK) electrodiffusion equation for a single permeant ion species, returning the flux divided through by the membrane permeability. Channels in the catalogue that use GHK-style currents (calcium channels, most notably) multiply this result by their own g_max, which occupies the permeability slot P_s that a textbook GHK flux equation would otherwise carry explicitly.

Parameters:

  • V (ArrayLike) – Membrane potential, as a voltage (for example u.mV).

  • ci (ArrayLike) – Intracellular concentration of the permeant ion, as a molar concentration (for example u.mM).

  • co (ArrayLike) – Extracellular concentration of the permeant ion, as a molar concentration (for example u.mM).

  • z (ArrayLike) – Valence of the permeant ion. Dimensionless.

  • temp (ArrayLike) – Absolute temperature, as a temperature (for example u.kelvin).

Returns:

The constant-field flux, divided through by the permeability P_s that a caller is expected to supply separately (typically as g_max). Not yet a current or a conductance on its own.

Return type:

ArrayLike

Notes

Let zeta = z * F * V / (R * T), with F the Faraday constant and R the gas constant. The function evaluates

\[\text{regular\_branch} = \frac{z \zeta F (c_i - c_o e^{-\zeta})}{1 - e^{-\zeta}}\]

which is undefined at zeta = 0 because 1 - exp(-zeta) vanishes there and the branch divides by it. To keep the function stable as zeta -> 0, a second, algebraically equivalent branch

\[\text{small\_branch} = z F (c_i - c_o e^{-\zeta}) \left(1 + \frac{\zeta}{2}\right)\]

is selected instead via u.math.where whenever abs(1 - exp(-zeta)) <= 1e-6, which both avoids the division and stays numerically well conditioned near that singularity.

Goldman (1943) [1] and Hodgkin & Katz (1949) [2] are together the origin of the name “GHK”, but they are the primary sources for two different equations. This function computes the constant-field flux/current equation, whose primary source is Goldman (1943) – hence .. [1] first. It does not compute the constant-field voltage equation, for which Hodgkin & Katz (1949) is the primary source.

References

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