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De Schutter: Purkinje Cell Model


Figure 1: Activation and inactivation properties of the high-threshold Ca2+-activated K+ current (KC, —) ionic conductances in the model. Seady-state activation and inactivation vs. voltage are plotted at the left, the time constants of activation (τm) and inactivation (τh) vs. voltage in the middle (Note: Semilogarithmic scale), and a simulation of representative voltage-clamp currents at the right, obtained from a spherical cell and assuming a complete block of all other channels. Note: Activation is controlled by the product of a voltage and a [Ca2+]-dependent factor, each with their own time constants (τv and τ[Ca]). The voltage clamps simulate steps from a holding potential of -110 to -70 mV up to 0 mV in 10 mV increments. The voltage-clamp current amplitude has been scaled arbitrarily because we mainly wanted to demonstrate the current kinetics.

High-threshold Ca2+-activated K+ (KC) Current

Ca2+-activated K+ channels are assumed to be responsible for the repolarization of dendritic Ca2+ spikes [13]. Several Ca2+-activated K+ channels have been identified in single channel studies of Purkinje cells [356] among them a large conductance channel corresponding to the BK or maxi-K channel [12]. The macroscopic current carried by this channel is called the C current (KC) and is characterized by a voltage dependence and tetraethylammonium (TEA) sensitivity [1]. 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 [12].

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 [141517]. 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 [48911]. 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 [41517]. 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 [215], exactly 2 [816] and 3 [9] and some authors assume more than one open state [1417]. Most BK channels studied in adult neurons require concentrations of internal Ca2+ in the micromolar range to fully activate [2101617] and the dependence on Ca2+ concentration seems to be nonlinear [215] (also see however [11]).

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 [7151820]. Following the example of [19] 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 [4] with a voltage-independent activation (αm) and a voltage-dependent inactivation (βm), with a typical 15 mV per e-fold change in conductance [212]. 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 [6]. The Ca2+-binding step was modeled along [2] 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 [4911].


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