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Limitations of the model

The current model is an advancement in modeling the Purkinje cell, but as any computer model, it must be seen as a limited representation of reality [3]. Thus, although the model replicated the basic features of the response of this cell to current injection quite well, it did not simulate all details of the several generated response properties perfectly. For example, the simulated dendrite tended to be a bit too excitable, so that a small spike often occurred at the beginning of a low-amplitude current injection, followed by a plateau potential (Figs. 3A and 4A,B). In experimental data, low-amplitude current injections do not result in this firing pattern [17, 16]. Similarly, at higher current amplitudes fullblown dendritic spikes were often preceded by smaller Ca²⁺ spikes (Fig. 3B,C). Also, the prolonged plateau potentials after current injections did not decay properly (Fig. 5).

We believe that an important reason for limitations in model performance is the lack of critical experimental data. Because this is the first description of this model, the following sections discuss the limitations posed by the lack of data in some detail.

MORPHOLOGICAL DATA

At the most basic level, the model was based on a light microscopic reconstruction of the Purkinje cell, effectively limiting the resolution of the model to branches ~ 1 μm in diameter. However, it is known that Purkinje cells have many branches smaller than this diameter [8, 22]. On the basis of this information we estimate that our model may be lacking ~ 10 % of the total dendrite. However, we also believe that for the purposes of the current paper this inaccuracy is relatively unimportant. First, this paper primarily concerns Purkinje cell responses to current injections into the soma or the smooth dendrites. Second, an analysis of previously published EM data that examined the dendritic spines in detail [9, 10] suggests that the inaccuracies in the total spine surface area and in the shrinkage factor used in the original reconstruction [24] were of the same order as the missing dendritic membrane.

DATA FROM DIFFERENT ANIMALS

A second limitation of the current model is that model parameters were based on a mixing of data obtained from the two different species used most commonly in cerebellar physiology and anatomy, rats and guinea pigs. For example, the morphological reconstruction on which the basic structure of the model is based represents a guinea pig Purkinje cell [24], whereas the EM data on which we based the size and distribution of spines were obtained in the rat [9]. Likewise, most of the voltage-clamp data came from rat Purkinje cells [5, 11, 13, 25], although we compared model outputs to current injection data obtained in the guinea pig slice [17, 16]. Although in principle one would prefer to base the entire model on data from a single species, the lack of available data makes species mixing quite common in modern modeling efforts (cf. [2, 18, 19, 29]. Furthermore, in the current case there is no evidence that these species differ substantially in the properties of their Purkinje cells.

NaF CHANNELS

As was pointed out in METHODS, voltage- clamp data on Purkinje cell fast Na⁺ channels are incomplete. This resulted in a number of anomalies in our equations for the NaF current that could not be resolved without better experimental data. Because of the potential importance of the window current in the generation of Na⁺ plateau potentials (see above) a better description of the NaF current would be useful.

First of all, the equations were accurate for voltages up to -10 mV, but beyond this level the activation time constant became too fast (Fig. 2A). However, because of the all-or none nature of action potentials this discrepancy did not affect modeling results.

Second, at rest (-68 mV) 73 % of the NaF channels were inactivated and during current injections 294 % of the channels were always inactivated. This necessitated an unphysiologically high density of NaF channels in the soma (total conductance 7,500 mS/cm²). However, because of the continuous inactivation, the real was ~ 2,000 mS/cm², comparable with, for example, an Na⁺ conductance of ~ 1,000 mS/cm² (20^∘ C) for a rat node of Ranvier [21], but much larger than the 15 mS/cm² (22^∘ C) reported for the somata of freshly dissociated hippocampal neurons [27]. This difference can be explained by the absence of an axon initial segment in the model. This was not included in the model because the reconstruction of the Purkinje cell did not contain the axon. During the tuning phase of the model some simulations were performed with a model including an axon initial segment [28] but these showed no qualitative differences compared with the standard model.

Ca²⁺-ACTIVATED K⁺ CHANNELS

It is of more concern that the Ca²⁺-activated K⁺ currents that are known to exist in the Purkinje cell and that have a substantial influence on model behavior have not been described adequately in this cell with voltage-clamp techniques.

From experimental data it is quite likely that the Purkinje cell membrane contains several types of Ca²⁺-activated K⁺ channels [7]. This is not surprising considering that the BK channel alone seems to have > 100 expression variants [1]. Without more detailed Purkinje cell data, in the current model we have attempted to cover the effects of these diverse channels by including only two Ca²⁺-activated K⁺ channels, a KC and a K2 channel. As described in KC and K2, these channels have quite different activation characteristics, with the KC channel requiring high Ca²⁺ concentrations and large depolarizations whereas the K2 channel activates at low Ca²⁺ concentrations and small depolarizations. By using two relative extremes of what could very well be a continuous distribution of slightly different Ca²⁺-activated K⁺ channels [15] we hoped to approximate the behavior of the whole population.

In the absence of good Purkinje cell data on these channels we initially borrowed the kinetic description for the BK conductance (KC) from simulations of the bullfrog sympathetic ganglion [30]. However, in Purkinje cell simulations these channels activated too quickly, so that dendritic Ca²⁺ channels could not generate full Ca²⁺ spikes. Accordingly, we modified the channel description equations to include an explicit time constant for Ca²⁺ activation that mimicked the experimentally observed delay in onset of this conductance [12]. The resulting KC equation reproduced several characteristics of the BK channel well, among them the two separate Ca²⁺ activation steps with a delay, the additional voltage-independent open-closed transition with typical time constants, and voltage threshold. However, some other reported characteristics are not captured by our equations. For example, a simple horizontal shift of the open probability versus voltage (P_O∕V ) curve on changes in Ca²⁺ concentration has been reported [6, 20], whereas our equations only scale the amplitude of the P_O∕V curve (Fig. 2G). In addition, the BK channel also seems to have a fast Ca²⁺-related deactivation [12], whereas in our model Ca²⁺ activation and Ca²⁺ deactivation had the same time constants. The most substantial difference between the data and our equations is that the experimental data predict that the does not depend on Ca²⁺ concentration, so that at very low Ca²⁺ concentrations extremely high depolarizations can still fully open the channel and vice versa. In our equations g was limited by the Ca²⁺ concentration. However, because the Purkinje cell dendrite never depolarized enough to open BK channels at low Ca²⁺ concentrations and because Ca²⁺ never reached the saturating concentration for opening all BK channels, the model operated in a region of parameter space where the effective difference between equations and experimental data is minimal.

The presence of the K2 channel was based on singlechannel recordings [7], but few kinetic data were available. We have therefore combined data from similar channels in synaptosomal membranes [4, 26]. However, the experimental data remain very incomplete. On the basis of our results, we believe that a better characterization of this channel is essential for a further refinement of the model. There is also evidence in Purkinje cells for a K⁺ channel that causes slow afterhyperpolarizations (referred to as K7 by [7]), but we did not model this conductance. This channel resembles the SK channel or AHP conductance that causes slow afterhyperpolarizations in other neurons [14, 15]. It is believed to be sensitive to low Ca²⁺ concentrations with slow activation kinetics [23]. We did not include this channel in the model because the model computed only quickly changing Ca²⁺ concentrations, but the SK channel would be expected to sense mainly slower transients. The fact that the model could reproduce most of the physiological characteristics of Purkinje cells leads us to suggest that, if present, SK channels play a minor role in the short-term response to current injections.

References