K_HH1952#
- class braincell.channel.K_HH1952(size, g_max=Quantity(10., 'mS / cm^2'), temp=Quantity(309.15, 'K'), q10=3.0, temp_ref=Quantity(309.15, 'K'), V_sh=Quantity(-45., 'mV'), name=None)#
Hodgkin-Huxley 1952 delayed-rectifier potassium current.
The squid giant axon potassium current \(I_K\) of (Hodgkin & Huxley, 1952) [1], with \(p^4\) HH gating and an ohmic driving force:
\[\begin{split}\begin{aligned} \alpha_p &= \frac{0.1}{\mathrm{exprel}(-(V' + 10) / 10)} \\ \beta_p &= 0.125 \exp(-(V' + 20) / 80) \end{aligned}\end{split}\]where \(V' = (V - V_{sh}) / \mathrm{mV}\), \(\mathrm{exprel}(x) = (e^{x} - 1)/x\), and both rates are in \(\mathrm{ms}^{-1}\). Away from \(V' = -10\) the activation rate is exactly \(0.01 (V' + 10) / (1 - \exp(-(V' + 10)/10))\);
exprelonly removes that expression’s removable singularity, where the code returns \(0.1\ \mathrm{ms}^{-1}\) rather than0/0. The gate integrates \(\dot{p} = \phi (\alpha_p (1 - p) - \beta_p p)\) with \(\phi\) fromgate_phi().With the default
V_sh = -45.0 mV– which places rest at -65 mV in the modern absolute-potential convention – these expand to the published rates \(\alpha_n = 0.01 (V + 55)/(1 - \exp(-(V + 55)/10))\) and \(\beta_n = 0.125 \exp(-(V + 65)/80)\), with \(n^4\) gating.- Parameters:
size (
int|Sequence[int] |integer|Sequence[integer]) – Channel state shape.g_max (
Array|ndarray|bool|number|bool|int|float|complex|Quantity|Callable) – Maximal conductance density. Defaults to10.0 mS/cm2, which is not Hodgkin & Huxley’s own value (see Notes).temp (
Array|ndarray|bool|number|bool|int|float|complex|Quantity) – Absolute temperature driving the Q10 factor, default 36 degrees Celsius.q10 (
Array|ndarray|bool|number|bool|int|float|complex|Quantity|Callable) – Q10 scaling factor for the activation gate, default3.0.temp_ref (
Array|ndarray|bool|number|bool|int|float|complex|Quantity) – Reference temperature forq10, default 36 degrees Celsius (see Notes).V_sh (
Array|ndarray|bool|number|bool|int|float|complex|Quantity|Callable) – Threshold shift applied to both rates, default-45.0 mV.
See also
braincell.channel.sodium.Na_HH1952Sodium counterpart of the same model, whose
g_maxdefault does match the paper.K_TM1991Traub & Miles reparameterisation of the same \(p^4\) delayed rectifier.
Notes
Every rate constant was expanded by hand and compared with the classical Hodgkin-Huxley rate equations; the implementation reproduces them exactly.
exprelchanges no value: it removes the removable singularity at the linoid’s midpoint only.The default conductance is not the paper’s. Hodgkin & Huxley give \(\bar{g}_K = 36\ \mathrm{mS/cm^2}\); this class ships
10.0 mS/cm2. Document it as a BrainCell default, not as the published value.The default temperature is not the paper’s either. The rates above were measured at 6.3 degrees Celsius, whereas
tempandtemp_refboth default to 36 degrees Celsius, which makes the Q10 correction a no-op as shipped.q10 = 3.0is Hodgkin & Huxley’s own factor-of-three-per-ten-degrees, but the shipped defaults do not reproduce the paper’s 6.3 degrees Celsius behaviour.References