Ca2+-activated K+ channels are assumed to be responsible for the repolarization of dendritic Ca2+ spikes . Several Ca2+-activated K+ channels have been identified in single channel studies of Purkinje cells [3, 5, 6] among them a large conductance channel corresponding to the BK or maxi-K channel . The macroscopic current carried by this channel is called the C current (KC) and is characterized by a voltage dependence and tetraethylammonium (TEA) sensitivity . This channel is widely distributed in different tissues in both vertebrate and invertebrate preparations, with apparently similar voltage dependence but a variable Ca2+ dependence in all the cells studied .
No experimental studies on the kinetics of KC in Purkinje cells were available. Technically it is difficult to characterize the kinetics of KC because the Ca2+ activation cannot be controlled by a “Ca2+ clamp” comparable to voltage clamps. So most experimental investigations have sacrificed temporal resolution by investigating channel activation at steady, well-controlled Ca2+ concentrations [14, 15, 17]. Several groups that have tried to study the temporal dynamics of Ca2+ activation, i.e., how fast the channel reacts to a sudden jump in Ca2+ concentration, have concluded that there was a significant lag in response [4, 8, 9, 11]. Most reports agree that a minimal model of the BK channel requires at least three closed states and one open state, that the open-closed transitions include at least two Ca2+ binding steps and a voltage-independent step, and that the channel does not inactivate [4, 15, 17]. However, there is no agreement on the details of these models because, for example, reported Hill coefficients for Ca2+-dependent opening vary between 1–2 [2, 15], exactly 2 [8, 16] and 3  and some authors assume more than one open state [14, 17]. Most BK channels studied in adult neurons require concentrations of internal Ca2+ in the micromolar range to fully activate [2, 10, 16, 17] and the dependence on Ca2+ concentration seems to be nonlinear [2, 15] (also see however ).
The conflicting experimental data on the BK channel are reflected by the multiple approaches used by different modelers to describe this channel. Most models lump all the open-closed transitions together into one differential equation [7, 15, 18, 20]. Following the example of  we have described this channel with two independent state variables (m and z in Eq. 1), but we have used a different model for the Ca2+-dependent step. The Ca2+-independent gate was modeled along data from  with a voltage-independent activation (αm) and a voltage-dependent inactivation (βm), with a typical 15 mV per e-fold change in conductance [2, 12]. We shifted the deactivation to more positive potentials to fit the strong depolarizations (>50 mV) required to activate KC in Purkinje cells, as reported by . The Ca2+-binding step was modeled along  as an adsorption isotherm distribution with a half-activation at 4 μM and a Hill coefficient of 2 (Eq. 5). The delay in activation was modeled explicitly by a time constant of activation of 10 ms [4, 9, 11].
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