This plots the sKCa current function, which is describes the small-conductance calcium-activated potassium channel, using the equations described in Fujita et al (2012) based on Gunay et al (2008), (also Muddapu & Chakravarthy, 2021). There is a gating factor M that depends on the Ca concentration, modeled using an X / (X + C50) form Hill equation:
M_hill(ca) = ca^h / (ca^h + c50^h)This function is .5 when ca == c50, and the h default power (Hill param) of 4 makes it a sharply nonlinear function.
SKCa can be activated by intracellular stores in a way that drives pauses in firing, and can require inactivity to recharge the Ca available for release. These intracellular stores can release quickly, have a slow decay once released, and the stores can take a while to rebuild, leading to rapidly triggered, long-lasting pauses that don't recur until stores have rebuilt, which is the observed pattern of firing of STNp pausing neurons, that open up a window for BG gating.
CaIn = intracellular stores available for release; CaR = released amount from stores; CaM = K channel conductance gating factor driven by CaR binding.
CaR -= CaR * CaRDecayDt
if spike {
CaR += CaIn * KCaR
}
if CaD < CaInThr {
CaIn += CaInDt * (1 - CaIn)
}
Figure 1: M gating as a function of Ca, using Hill function with exponent 4, C50 = .5.
Figure 2: Time plot showing pausing and lack of recovery. The spiking input to the neuron is toggled every 200 msec (theta cycle), with 3 cycles shown. The CaIn level does not recover during the off phase -- 2 or more such phases are required.