Theoretical
Neurobiology & Neuroengineering

An Active Membrane Model of the Cerebellar Purkinje Cell : I. Simulation of Current Clamps in Slice

JOURNAL OF NEUROPHYSIOLOGY
Vol. 71, No. 1, 375-400 January 1994, Printed in U.S.A
Copyright © The American Physiological Society, 1994

E. De Schutter, J.M. Bower

Division of Biology 216-76, California Institute of Technology, Pasedena, California 91125

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Abstract

1. A detailed compartmental model of a cerebellar Purkinje cell with active dendritic membrane was constructed. The model was based on anatomical reconstructions of single Purkinje cells and included 10 different types of voltage dependent channels described by Hodgkin Huxley equations, derived from Purkinje cell specific voltage clamp data where available. These channels included a fast and persistent Na⁺ channel, 3 voltage dependent K⁺ channels, T-type and P-type Ca²⁺ channels and 2 types of Ca²⁺-activated K⁺ channels.

2. The ionic channels were distributed differentially over 3 zones of the model, with Na⁺ channels in the soma, fast K⁺ channels in the soma and main dendrite and Ca²⁺ channels and Ca²⁺-activated K⁺ channels in the entire dendrite. Channel densities in the model were varied till it could reproduce Purkinje cell responses to current injections in the soma or dendrite, as observed in slice recordings.

3. As in real Purkinje cells, the model generated two types of spiking behavior. In response to small current injections, the model fired exclusively fast somatic spikes. These somatic spikes were caused by Na⁺ channels and repolarized by the delayed rectifier. When higher amplitude current injections were given, sodium spiking increased in frequency until the model generated large dendritic Ca²⁺ spikes. Analysis of membrane currents underlying this behavior showed that these Ca²⁺ spikes were caused by the P-type Ca²⁺ channel and repolarized by the BK-type Ca²⁺-activated K⁺ channel. As in pharmacological blocking experiments, removal of Na⁺ channels abolished the fast spikes and removal of Ca²⁺ channels removed Ca²⁺ spiking.

4. In addition to spiking behavior, the model also produced slow plateau potentials in both the dendrite and soma. These longer duration potentials occurred in response to both short and prolonged current steps. Analysis of the model demonstrated that the plateau potentials in the soma were caused by the window current component of the fast Na⁺ current which was much larger than the current through the persistent Na⁺ channels. Plateau potentials in the dendrite were carried by the same P-type Ca²⁺ channel that was also responsible for Ca²⁺ spike generation. The P channel could participate in both model functions because of the low-threshold K2-type Ca²⁺-activated K⁺ channel, which dynamically changed the threshold for dendritic spike generation through a negative feedback loop with the activation kinetics of the P-type Ca²⁺ channel.

5. These model responses were robust to changes in the densities of all of the ionic channels. For most of the channels modifying their density by factors of 2 or more resulted only in left or right shifts of the f-I curve. However, changes of more than 20% to the amount of P-type Ca²⁺ channel or of one of the Ca²⁺-activated K⁺ channels in the model either suppressed dendritic spikes or caused the model to always fire Ca²⁺ spikes. Modeling results were also robust to variations in Purkinje cell morphology. We simulated models of 2 other anatomically reconstructed Purkinje cells with the same channel distributions and got similar responses to current injections.

6. The model was used to compare the electrotonic length of the Purkinje cell in the presence and absence of active dendritic conductances. The electrotonic distance from soma to the tip of the most distal dendrite increased from 0.57 lambda in a passive model to 0.95 lambda in a quiet model with active membrane. During a dendritic spike generated by current injection, the distance increased even more to 1.57 lambda.

7. Finally, the model was used to study the probable accuracy of experimental voltage clamp data. Whole-cell patch clamp conditions were simulated by blocking most of the K⁺ currents in the model. The increased electrotonic length due to the active dendritic membrane caused space clamp to fail, resulting in membrane potentials in proximal and distal dendrites which differed critically from the holding potential in the soma.

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Introduction

The cerebellar Purkinje cell is one of the largest and most complex neurons in the mammalian nervous system. Purkinje cells have very active dendrites, generating massive Ca²⁺ signals in response to synaptic input (Sugimori and Llinás 1990, Miyakawa et al. 1992). Further, the unusual isoplanar anatomical organization of this cell's dendrite has been conserved to a remarkable degree through evolution (Ito 1984), suggesting that this specific morphology is essential to the function of the Purkinje cell.

In addition to its anatomical and physiological complexity and uniqueness, Purkinje cell activity also constitutes the sole output of the cerebellar cortex. While the precise computational role of the cerebellum is not yet known, it is clear that understanding the physiology of the Purkinje cell will be an essential part in unraveling the function of the cerebellar cortex as a whole. Given the complexity of this cell, we believe that computer modeling techniques are necessary to explore and analyze completely the properties of Purkinje cells.

In this paper we describe a large, detailed compartmental model (Rall 1962, 1964; Perkel et al. 1981) of the Purkinje cell based on real Purkinje cell morphology, which includes 10 active voltage dependent ionic conductances that have so far been demonstrated to exist in these neurons. Model parameters were established using the results of several recent voltage clamp studies of conductances in Purkinje cells (Gähwiler and Llano 1989; Hirano and Hagiwara 1989; Kaneda et al. 1990; Regan 1991; Wang et al. 1991). We tested whether the model was robust to changes in the densities of individual channels. The model was then used to explore the ionic mechanisms underlying the complex response properties of Purkinje cells to current clamp conditions in vitro (Llinás and Sugimori 1990a, b), which confirmed several postulates made by Llinás and Sugimori (1992). In addition, modeling results focus attention on the importance of low threshold Ca²⁺-activated K⁺ channels in controlling dendritic excitability. Finally, the model was used to explore the likely accuracy of whole-cell voltage clamping experiments in this neuron.

While several Purkinje cell models have previously been described in the literature, most have not included voltage dependent conductances in the dendrites (Llinás and Nicholson 1976; Shelton 1985; Rapp et al. 1992, 1994). Models that did include ionic conductances in the dendrites either did not include all channels now known to exist (Pellionisz and Llinás 1977) or were applied to a very limited set of questions (Bush and Sejnowski 1991). In addition to exploring the mechanisms underlying this cells response to current injection, the current model also lays the groundwork for modeling and experimental studies of Purkinje cell responses to synaptic input (De Schutter and Bower 1994).

Methods

Simulations were performed with the neuronal simulation program GENESIS (Wilson et al. 1989) on 8 Sun Sparc2 workstations. We used the Hines-algorithm (Hines 1984) available in GENESIS versions 1.3 and 1.4. The GENESIS implementation of this algorithm is fast and accurate (Bhalla et al. 1992). The simulations were run with a time step of 20 us. Initial control simulations at 10 us showed that the 20 us step produced numerically accurate results. For the model presented here 550 ms of Purkinje cell activity could be simulated in one hour. Several 1000 simulations were run, mostly to explore parameter space.

Morphology

The morphology of the models was based on a detailed light-microscopic reconstruction of HRP-filled guinea-pig Purkinje cells by M. Rapp of the Hebrew University of Jerusalem, Israel (Rapp et al. 1992, 1994). All simulations were done with a model using the morphology of cell 1 of Rapp et al. (1994), unless otherwise noted. We applied the same shrinkage factor of 10% as Rapp.

The final model described here contained 1600 compartments. As is standard for compartmental modeling (Rall 1962, 1964) this number was determined by the morphology of the cell and by simulation requirements for numerical accuracy. In particular, when active channels are used, it has been previously shown that the electrotonic length of compartments should be smaller than 0.05 lambda (Cooley and Dodge 1966).

In our model, the passive electrotonic length of single compartments ranged from 0.009 to 0.05 lambda. Because these electrotonic lengths were short, we were able to use more computationally efficient asymmetric compartments (this GENESIS object corresponds to a 3-element segment, as described by Segev et al. 1985). Simulations done with and without asymmetric compartments confirmed that the difference in input resistance (RN) and system time constant (tau-0) for a passive membrane model was less than 0.1%.

A rat Purkinje cell is known to have about 150,000 dendritic spines (Harvey and Napper 1991). However, because the morphological data used to build the current model was obtained with light microscopic techniques, the location and shapes of dendritic spines for the reconstructed cell were not known. Accordingly spines were not simulated. Instead, membrane surface was added to the spiny dendritic compartments (defined as dendrites with a diameter less than 3.17 um) to compensate for missing spines (Holmes and Woody 1989; Rapp et al. 1992). Based on published EM-reconstructions of rat Purkinje cell spines (Harris and Stevens 1988), we assumed a density of 13 spines per 1 um dendritic length, with a membrane surface for a spine of 1.33 um² (Harris and Stevens 1988).

Figure 1
Morphology of the Purkinje cell model (cell 1 of Rapp et al. 1993). The 3 zones with different channel densities (Table 2) are marked as soma (black), main dendrite (dark gray) and the rest of the dendrites (black). The stippled lines point to the recordings sites displayed in Figs 3-7, the asterisk high in the dendrite is the recording site for Fig. 12, 13.

Passive membrane parameters

To establish the passive properties of the Purkinje cell model we initially used the same parameters as previously published by Rapp et al. (1992). Accordingly membrane capacitance was set at 1.64 uF/cm², membrane resistance (Rm) was 0.44 kOmega.cm² in the soma and 110 kOmega.cm² in the dendrites and axial resistance was given a value of 250 Omega.cm. With these parameters we obtained a tau-0 of 46 ms and a RN of 12.6 MOmega, which is almost identical to the values of 46 ms and 12.9 MOmega reported by Rapp et al. (1992). However, initial simulations with active membrane showed that it was not possible to reproduce the characteristic firing pattern of Purkinje cells with these membrane parameters. In particular, the low Rm value in the soma caused a huge current sink, so that the model could not fire somatic Na⁺ spikes. For this reason, the model presented here had a Rm of 10 kOmega.cm² in the soma and 30 kOmega.cm² in the rest of the cell, which are comparable to values for Rm in other neuron models (Holmes and Rall 1992). This model had a RN of 19.6 MOmega under conditions of simulated external Cs⁺ block (i.e. Kdr, KM and Kh channels blocked; Hille 1991), which is a realistic RN value for Purkinje cells in slice (Llinás and Sugimori 1980a).

Equations for the voltage dependent conductances

The next stage of constructing the model was to incorporate voltage dependent ionic conductances into the model compartments. 10 different ionic channels were selected for this purpose, based on published in vitro recordings (Llinás and Sugimori 1980a, b; Hounsgaard and Midtgaard 1988; Llinás and Sugimori 1992), voltage clamp studies (Crepel and Penit-Soria 1986; Kaneda et al. 1990; Regan 1991; Wang et al. 1991) and single channel studies (Gähwiler and Llano 1989; Gruol et al. 1991) of Purkinje cells. Specifically, the channels were a fast and a persistent Na⁺ current, a low threshold (T-type) and a high threshold (P-type) Ca²⁺ current, an anomalous rectifier, a delayed rectifier, an A current, a non-inactivating K⁺ current, and a low-threshold (K2-type), and a high-threshold (BK-type) Ca²⁺-activated K⁺ current.

The equations describing the kinetics of these channels are summarized in eq. 1 - 5 and in table 1. The channel conductance was determined by the product of voltage dependent activation (m) and inactivation (h) gates and for the Ca²⁺-activated channels a Ca²⁺-dependent activation gate (z):

(units: mV, uM, ms) (1)

The equations describing the voltage dependent gates were derived from the classic Hodgkin and Huxley (1952) scheme:

(2)

idem for h Eq. 2

        (3)

idem for alpha-h and beta-h Eq. 3

The activation rates for the Ca²⁺-dependent gates were determined by a dissociation constant A and a time constant B:

(4)

(5)

For the Ca²⁺ channels the Nernst potential (Hille 1991) was computed continuously. We did not model the rectification of Ca²⁺ channels using the Goldman-Hodgkin-Katz equation (GHK, Hille 1991), because dendritic membrane potentials in this study stayed within a range where Ca²⁺ channels can be considered ohmic (i.e. below -20 mV; fig. 4.15 in Hille 1991). Using the simulation results from the final model, we estimate that using the GHK equation with an appropriately scaled to compensate for differences in driving force, would cause only small changes in the amplitude of dendritic Ca²⁺ currents (mean difference 0.7%, maximum 4.5%).

Name	Abr	Er	factor	P	A	B	C	D	E	F	G	H
Fast sodium current	NaF	45	m	3	35.0	0	5	-10.0	7.0	0	65	20.0
			h	1	0.225	1	80	10.0	7.5	0	-3	-18.0
Persistent sodium current	NaP	45	m	3	200.0	1	-18	-16.0	25.0	1	58	8.0
P calcium current	CaP	135	m	1	8.5	1	-8	-12.5	35.0	1	74	14.5
			h	1	0.0015	1	29	8.0	0.0055	1	23	-8.0
T calcium current	CaT	135	m	1	2.60	1	21	-8.0	0.180	1	40	4.0
			h	1	0.0025	1	40	8.0	0.190	1	50	-10.0
Anomalous rectifier	Kh	-30
Delayed rectifier	Kdr	-85
Persistent potassium curr	KM	-85
A current	KA	-85	m	4	1.40	1	27	-12.0	0.490	1	30	4.0
			h	1	0.0175	1	50	8.0	1.30	1	13	-10.0
BK calcium-activated potassium current	KC	-85	m	1	7.5, αm is constant				0.110	0	-35	14.9
			z	2	4.00	10
K2 calcium-activated potassium current	K2	-85	m	1	25.0, αm is constant				0.075	0	5	10.0
			z	2	0.20	10

Table 1: E_r and the parameters to Eq. l-5 for the conductance of voltage- and Ca²⁺-dependent channels in the model, and abbreviations for the dzfirent ionic channel
Kh equations by Spain et al. (1987). Kdr equations by Yamada et al. (1989), rates multiplied by 5. Km equations by Yamada et al. (1989), rates multiplied by 5. E_r, reversal potential; m, voltage-dependent activation gate; h, voltage-dependent inactivation gate; z, Ca²⁺-dependent activation gate; α_m, voltage-independent activation.

Parameters for the channel kinetics

Fig. 2 summarizes the kinetics of the channels by showing curves for steady state activation and steady state inactivation and their time constants versus voltage. This figure also shows examples of simulated voltage clamp traces in a spherical cell for each channel, assuming a complete block of all other channels.

The Hodgkin Huxley equations just described require detailed information on channel kinetics and peak conductances. In principle, all of this data should be obtained from the cell being simulated. However, all the necessary data is rarely available for any particular neuron. For this reason it is usually necessary to obtain kinetic information from a variety of sources. The following sections describe the sources of experimental data used for each of the conductances included in this model.

Figure 2
Activation and inactivation properties of the ionic conductances in the model. For each conductance steady state activation (m-infinity) and inactivation (h-infinity) versus voltage are plotted at left, the time constants of activation (tau-m) and inactivation (tau-h) versus voltage in the middle (note the semi-logarithmic scale) and a simulation of representative voltage clamp currents at right.
A: Fast Na+ current (full lines) and persistent Na+ current (stippled lines, no inactivation). The voltage clamp simulation shows NaF only.
B: T-type Ca2+ current.
C: P-type Ca2+ current.
D: Anomalous rectifier; note that this current does not inactivate and that activation is determined by 2 time constants (tau-1 and tau-2; Spain et al. 1987).
E: Delayed rectifier (full lines) and persistent K+ current (stippled lines, no inactivation). The voltage clamp simulation shows Kdr only.
F: A current.
G: BK-type Ca2+-dependent K+ current.
H: K2-type Ca2+-dependent K+ current. Note that for both Ca2+-dependent K+ currents activation is controlled by the product of a voltage and a [Ca2+]-dependent factor, each with their own time constants (tau-V and tau-[Ca]). The voltage clamps simulate steps from a holding potential of -110 mV to -70 mV and up to 0 mV, in 10 mV increments except for D which steps from a holding potential of 0 mV to -80 mV and up to -10 mV. The voltage clamp current amplitude has been scaled arbitrarily, as we mainly wanted to demonstrate the current kinetics.

The Fast Sodium Current. The NaF current is responsible for the depolarization phase of the somatic action potential, there are no somatic spikes when this current is blocked by TTX (Llinás and Sugimori 1980a). The NaF current is difficult to voltage clamp because of its fast activation kinetics and its large conductance. Up until now nobody has attempted a complete voltage clamp study of this current in Purkinje cells. In modeling papers one can trace an evolution of modifications to the original Hodgkin and Huxley (1952) equations (Traub 1982; Wilson and Bower 1989; Lytton and Sejnowski 1991). Most of these modifications were to our knowledge not based on firm experimental data.

We have derived our equations for NaF from the original Hodgkin and Huxley (1952) equations. These were modified to accommodate data about steady state activation of NaF obtained from outside-out patch clamp recordings of cultured Purkinje cells (Gähwiler and Llano 1989, their fig. 4B). Another report of whole-cell voltage clamp recordings from cultured Purkinje cells (Hirano and Hagiwara 1989, their fig. 7B) provided an I-V relation for the peak NaF current with a comparable activation threshold, but corresponding to a somewhat shallower slope of the activation curve. Both reports also contained some data on the time to peak of the NaF current. The modifications based on these 2 reports resulted in a steeper steady state activation curve (Fig. 2A) with a higher threshold and slower kinetics (at room temperature) than the original Hodgkin and Huxley (1952) equations. The Gähwiler and Llano (1989, their fig. 4B) report also provided steady state inactivation data, which resulted in a shallower inactivation curve with slower kinetics.

The Persistent Sodium Current. The NaP current is assumed to cause the plateau potentials in the soma described by Llinás and Sugimori (1980a). The basis for the equations describing NaP were single electrode voltage clamp recordings of NaP in guinea pig hippocampal neurons (French et al., 1990). These recordings provided steady state activation data and we assumed the same number of gates as for the NaF channel (Hodgkin-Huxley, 1952). Time constants for activation and deactivation and the threshold of activation for NaP were obtained from Kay et al. (1990). Note that NaP-activation follows a similar slope as NaF-activation, but with a lower threshold of activation (Fig. 2A).

The T Calcium Current. Several groups have reported the presence of a low-threshold, inactivating Ca²⁺ channel in the Purkinje cell (Hirano and Hagiwara 1989; Gruol and Deal 1990; Kaneda et al. 1990) comparable to the T channel in other neurons (Fox et al. 1987). The whole-cell voltage clamp study of freshly isolated rat Purkinje cells by Kaneda et al. (1990, their figs. 1-3) provided all the data necessary to model activation of CaT (Fig. 2B). Hirano and Hagiwara (1989, their fig. 4C) have shown a higher threshold of activation for CaT in Purkinje cells, but Gruol and Deal (1990) reported the same threshold as Kaneda et al. (1990). The equations for CaT inactivation were based upon a combination of steady state inactivation data from Hirano and Hagiwara (1989, their fig. 5A) and time constant data from Kaneda et al (1990, their fig. 2B), which were almost identical to the data from Hirano and Hagiwara (1989, their fig. 5C).

The P Calcium Current. The P-type Ca²⁺ channel is a high-threshold, very slowly inactivating channel, first described in the Purkinje cell (Llinás et al. 1989a). A complete whole-cell patch clamp study of this channel in freshly dissociated rat Purkinje cells was done by Regan (1991), which provided us with all the data necessary to model CaP (Fig. 2C). Initial versions of the model were run with equations based on averaged data (Regan 1991, p. 2262), but confirming our experience in other systems (De Schutter et al. 1993), we found that equations based on data from a single preparation (Regan 1991, her figs. 5C and 6C) made Ca²⁺ spiking in the model more robust. These data do not support multiple activation states for the CaP channel as there does not seem to be any delay in activation (Hodgkin and Huxley 1952) and the steady state activation curve could be fitted by a Boltzmann style curve with power 1. Therefore, activation of CaP has been modeled with a single gate. This is in contrast to equations for other mammalian Ca²⁺ currents, which usually show some delay in activation (Kay and Wong 1987; Chen and Hess 1990).

Recently Usowicz et al. (1992a, b) also reported CaP channel data, based on cell-attached patch clamps of guinea pig Purkinje cells. These results seem to be in accord with the data reported by Regan for activation, especially the threshold of activation is very similar (-41 mV in 2 mM Ca²⁺ reported by Usowicz et al. (1992a) versus -45 to -40 mV in 5 mM Ba²⁺ (Regan 1991)). However, Usowicz et al. (1992b) show little or no inactivation of CaP, while Regan declares that there is a slow inactivation. Other authors claim that there might be several time constants of inactivation for CaP (Hockberger and Nam 1991) and to our knowledge a possible Ca²⁺-dependent inactivation, as found in other high threshold Ca²⁺ channels (Fox et al. 1987), has not been completely excluded. The model used the slow inactivation suggested by Regan (Fig. 2C).

The Anomalous Rectifier. Inward rectification has been shown to be present in the Purkinje cell (Crepel and Penit-Soria 1986; Hounsgaard and Midtgaard 1988) and a channel permeable to K⁺ and activated at hyperpolarized potentials has been identified in single channel recordings (Gruol et al. 1991, `K8'). However, kinetic information on this channel for the Purkinje cell is incomplete. Accordingly, we have used the equations for the anomalous rectifier in cortical neurons published by Spain et al. (1987). These equations have the same activation curve (Fig. 2D) as the voltage clamp data from a single Purkinje cell shown by Crepel and Penit-Soria (1986, their fig. 2).

The Delayed Rectifier. the Purkinje cell delayed rectifier is responsible for the repolarization of somatic action potentials. Whole-cell voltage clamp recordings (Hirano and Hagiwara 1989, their fig. 9) and patch-clamp recordings (Gähwiler and Llano 1989, their fig. 9) of the Kdr current are available, but incomplete. The I/V-relations of the currents in both reports are very similar. In the current model we have used the published equations for Kdr in bullfrog sympathetic ganglion cells (Yamada et al. 1989), as both the I/V-curve and the steady state inactivation (Fig. 2E) of the bullfrog data are comparable to the date from Hirano and Hagiwara (1989).

Persistent potassium current.on-inactivating K⁺ channels in Purkinje cells have been reported by Bossu et al. (1988) and are also apparent in recordings from Hirano and Hagiwara (1989). No kinetic data were available from Purkinje cells, so we have again used the equations of Yamada et al. (1989) for the non-inactivating muscarinic K⁺ channel (Fig. 2E).

The a Current. The presence of a conductance like an A current in the Purkinje cell was shown by Hounsgaard and Midtgaard (1988). We derived our equations for KA (Fig. 2F) from the original reports on A currents (Connor and Stevens 1971; De Schutter 1986), modified to fit the data in the two reports on Purkinje cell A currents that were available at that time. The whole-cell voltage clamp study of cultured Purkinje cells by Hirano and Hagiwara (1989, their fig. 9) provided data about the activation and inactivation time constants and the steady state inactivation, and a single electrode voltage clamp study in slice by Li et al. (1990) supplied steady state activation data.

Recently Wang et al. (1991) have published a more complete report on a single electrode voltage clamp study of the Purkinje cell. The average threshold for activation they report is lower than the value used in our model (Fig. 2F), but there seems to be a large natural variability in the activation curves (compare their fig. 1 with their table 1). The steady state inactivation curve of Wang et al. (1991) is similar to ours and to the kinetics of A currents in other systems (Connor and Stevens 1971; Rogawski 1985), but they report a much slower inactivation time constant than Hirano and Hagiwara (1989).

The BK Calcium-Activated Potassium Current. Ca²⁺-activated K⁺ channels are assumed to be responsible for the repolarization of dendritic Ca²⁺-spikes (Llinás and Sugimori 1980b). Several Ca²⁺-activated K⁺ channels have been identified in single channel studies of Purkinje cells (Gruol et al. 1989; Gähwiler and Llano 1989; Gruol et al. 1991), among them a large conductance channel corresponding to the BK or maxi-K channel (Latorre et al. 1989). The macroscopic current carried by this channel is called the C current and is characterized by a voltage dependence and TEA-sensitivity (Adams et al. 1982). 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 (Latorre et al. 1989).

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 (Moczydklowski and Latorre, 1983; Smart, 1987; McManus and Magleby 1989). 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 (Hudspeth and Lewis 1988; Ikemoto et al. 1989; Landò and Zucker 1989; Gola et al. 1990). Most reports agree that a minimal model of the BK channel requires at least 3 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 (Moczydklowski and Latorre 1983; Gola et al. 1990; Smart 1987). However, there is no agreement on the details of these models, as for example reported Hill coefficients for Ca²⁺ dependent opening vary between 1 to 2 (Moczydklowski and Latorre 1983; Franciolini 1988), exactly 2 (Hudspeth and Lewis 1988; Reinhart et al. 1989) and 3 (Ikemoto et al. 1989), and some authors assume more than 1 open state (Smart 1987; McManus and Magleby 1989). Most BK channels studied in adult neurons require concentrations of internal Ca²⁺ in the uM range to fully activate (Smart 1987; Franciolini 1988; Reinhart et al. 1989; Lancaster et al. 1991), and the dependence on Ca²⁺ concentration seems to be non-linear (Moczydklowski and Latorre 1983; Franciolini 1988; also see however Landò and Zucker 1989).

The conflicting experimental data on the BK channel is 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 (Traub 1982; Moczydklowski and Latorre 1983; Hines 1989; Yamada 1989). Following the example of Traub et al. (1991) we have described this channel with 2 independent state variables (named 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 Gola et al. (1990) with a voltage-independent activation (alpha-m) and a voltage-dependent inactivation (beta-m), with a typical 15 mV per e-fold change in conductance (Franciolini 1988; Latorre et al. 1989). 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 Gruol et al. (1991, `K1'). The Ca²⁺-binding step was modeled along Franciolini (1988) as an adsorption isotherm distribution with a half-activation at 4 uM and a Hill coefficient of 2 (Fig. 2G; eq. 5). The delay in activation was modeled explicitly by a time constant of activation of 10 ms (Ikemoto et al., 1989; Landò and Zucker, 1989; Gola et al., 1990).

The K2 Calcium-Activated Potassium Current. In their single channel study of K⁺ channels in Purkinje cells, Gruol et al. (1991) describe 2 Ca²⁺-activated K⁺ channels. One is a 134 pS (symmetrical K⁺ conditions) channel, which they identify as the K2 channel. These authors report that this channel comprises about 13 % of the total Ca²⁺-activated K⁺ conductance, activates at lower potentials than the BK channel and is blocked by TEA. However, the Gruol et al. (1991) study does not contain enough data to model this channel. The K2 channel is not an SK channel, as SK channels have a lower slope conductance (10 to 20 pS), are not voltage sensitive (Lang and Ritchie, 1987; Lancaster et al., 1991) and are blocked by apamin (Latorre et al. 1989). In rat brain synaptosomal membranes several types of Ca²⁺-activated K⁺ channels have been characterized, some of which have a medium size conductance, are TEA-sensitive and are not blocked by apamin. A 110-125 pS channel described by Farley and Rudy (1988) is sensitive to low concentrations of Ca²⁺, so that most of the channels are open at 0.1 uM, is activated by 30 mV depolarizations and is blocked by charybdotoxin and by high concentrations of TEA. A 135 pS channel described by Reinhart et al. (1989) is sensitive to submicromolar concentrations of Ca²⁺, has a Hill coefficient of about 2, several closed states, faster kinetics than the BK channel and is blocked by charybdotoxin.

As we did not have enough data available to describe the K2 channel completely, we used the same kinetic scheme as for the BK channel but with a lower voltage threshold, faster kinetics and a lower half-activation Ca²⁺ concentration of 0.2 uM (Fig. 2H, table 1). Thus despite the similar form of the equations, the K2 channel is quite different from the KC channel in voltage- and Ca²⁺-dependence.
LEAK CURRENT. In addition to the voltage dependent currents just described, it was also necessary to include a leak current. The leak conductance was determined by Rm and the reversal potential was -80 mV. This resulted in a small hyperpolarizing current at rest, which compensated for subthreshold inward currents (mainly NaP and CaT), so that a stable resting potential of -68 mV was achieved.

Estimates of other parameters for ionic channels

Eestimating Channel Conductances. Constructing an accurate representation of an ionic channel also requires an estimate of the maximum channel conductance. However, channel conductance values measured in voltage clamp experiments are often quite variable (McCormick and Huguenard 1992) and are only indicative. For this reason, maximum channel conductances in the model were not based on experimental measurements (De Schutter et al. 1993; Bhalla and Bower 1993).

Temperature Dependence of Channel Kinetics. Temperature is a well known critical parameter for channel kinetics (Hille 1991). In the current simulations, the in vivo and in vitro data used to tune the model were collected at 37 C while all of the published voltage clamp data were obtained at room temperature. Assuming a Q10 factor of 3 (Hodgkin and Huxley 1952), we therefore multiplied all the rate constants by a factor of 5.

Modeling Calcium Concentrations. In the present model we did not try to simulate Ca²⁺ concentrations realistically. We needed, however, to compute Ca²⁺ concentrations to activate the KC- and K2 channels. Because these channels are assumed to be sensitive to the fast changing, high Ca²⁺ concentrations just below the membrane surface (Landò and Zucker 1989), we only computed Ca²⁺ concentrations in a thin submembrane shell (Traub 1982). These shells integrated the full Ca²⁺-inflow through the CaP and CaT channels. Their volume and decay time constants were initially free parameters in the model. The model tuning resulted in shells 0.2 um deep with a decay time of 0.1 ms.
The basal internal Ca²⁺ concentration was 0.040 uM, the outside concentration was constant at 2.4 mM. These concentrations were also used to compute Nernst potentials (Hille 1991) for the CaP and CaT currents.

Estimating channel densities

Having established and parameterized the equations governing the voltage dependent behavior of the ionic conductances in this cell, it was necessary to determine the channel densities in the model. Because precise information on channel densities is not technically obtainable, channel densities, usually expressed as maximum conductance (), are largely a free parameter in detailed single cell models (Lytton and Sejnowski 1991; Traub et al. 1991; Calabrese and De Schutter 1992; Bhalla and Bower 1993).

In the present model, simulations were started using initial guesses for the of the different channels in 4 zones of the cell, i.e. the soma, the main dendrite, the smooth dendrites and the spiny dendrites (see Fig. 1). Maximum conductances were then altered until the model could reproduce the characteristic firing behavior of Purkinje cells during current injection in vitro (Llinás and Sugimori 1980a, b), as described in the Results section (Figs 3-7). During this optimization process, we found that it was not necessary to distinguish between the smooth and spiny dendrites. In addition, optimal performance required that the delayed rectifier and A currents be present in the main dendrite as well as in the soma.

PM9 model				PM10 model
Name	soma	main dendrite	rest of dendr	soma	main dendrite	rest of dendrite
NaF	7500	0.0	0.0	7500	0.0	0.0
NaP	1.0	0.0	0.0	1.0	0.0	0.0
CaP	0.0	4.5	4.5	0.0	4.0	4.5
CaT	0.5	0.5	0.5	0.5	0.5	0.5
Kh	0.3	0.0	0.0	0.3	0.0	0.0
Kdr	600.0	60.0	0.0	900.0	90.0	0.0
KM	0.040	0.010	0.013	0.140	0.040	0.013
KA	15.0	2.0	0.0	15.0	2.0	0.0
KC	0.0	80.0	80.0	0.0	80.0	80.0
K2	0.0	0.39	0.39	0.0	0.39	0.39

Table 2: for the voltage and Ca⁺-dependent channels in 2 different versions of the Purkinje cell mode
Extent of the main dendrite is shown in Fig. 1. , maximum conductance, in mS/cm². For other abbreviations, see Table 1.

Also, we found that several combinations of could account for the responses of Purkinje cells to current injection. For contrast, this paper presents versions PM9 and PM10, where PM10 represents a Purkinje cell with higher K⁺ channel densities, i.e. a leakier cell. The final values for the 2 versions of the model are shown in table 2.

Electrotonic length of active membrane compartments

In an active membrane compartment the total membrane resistance is variable, resulting in a continuously changing space constant and electrotonic length (Bernander et al. 1991; Rapp et al. 1992).

The total membrane resistance RTMn of a compartment n at a specific time was computed as the sum of all conductances (G(V,[Ca²⁺],t) from Eq. 1) and the passive component RMn determined by Rm:

(6)

The compartment's space constant lambda-n could be derived from RTMn, its axial resistance RIn and its length Ln:

(7)

The compartment's electrotonic length was the ratio of its length Ln over the space constant lambda-n. The electronic distance between 2 compartments was computed as the sum of the electronic lengths of all intervening compartments.

Results

Tuning the model to replicate in vitro current injections.

The final stage of constructing a complex realistic model involves tuning model parameters to replicate well described properties of the neuron in question (Bhalla and Bower 1993). Once the model is tuned, additional questions can be explored by holding model parameters fixed and applying different types of inputs (see De Schutter and Bower 1994).

This section describes the results of optimizing the Purkinje cell model to replicate the characteristic firing patterns of Purkinje cells in slices during current injection in the soma or dendrites. This optimization, or tuning, primarily involved changing the densities and distributions of the different voltage sensitive channels and the decay time constant of the Ca²⁺ concentrations until the model generated biologically realistic output. Due primarily to the work of Llinás and Sugimori (1980a, b), considerable information is available concerning the intracellular response properties of cerebellar Purkinje cells to current injection in vitro in the slice preparation in the presence and absence of different channel blockers. Accordingly, all the following statements about normal Purkinje cell behavior refer to their work unless a different study is cited.

Figure 3
Simulation of current injection in the soma, model PM9. For each current injection amplitude (A: 0.5 nA, B: 1.0 nA, C: 2.0 nA, D: 3.0 nA) 3 recording sites are shown: the soma (upper trace), the main dendrite (middle trace) and a spiny dendrite (bottom trace). Duration of current injection is shown by the bar below the traces. ERROR IN FIG. 3: the time bar should read 500 ms instead of 200 ms.

Fast Sodium Spikes. In intracellular recordings in brain slice preparations, some Purkinje cells are silent, others fire spontaneously. Our model is quiet without stimulation, with a stable resting potential of -68 mV. For intrasomatic current injections below the spiking threshold, no somatic spikes could be generated regardless of the length of the current pulses (simulation results not shown). Low amplitude current injections in Purkinje cell somata above this threshold caused firing of fast somatic spikes, with the relatively high minimum spike frequency (30-40 Hz) characteristic of these cells (Figs. 3A, 4A-B and 6A). Somatic firing frequency increased relatively rapidly with current amplitude (range 22 to 250 Hz).
In in vitro intracellular recordings, there is often a delay between the onset of the current injection and the occurrence of the first somatic spike, which decreases with increasing amplitude of current. This variable delay was also present in the simulations (Fig. 3-5, 6B). Note also that in Fig 4D we reproduced another phenomenon observed in slice. When the injected current amplitude became too high, somatic spiking saturated at a depolarized level of about -30 mV. This saturation could be reversed either spontaneously (at this current amplitude) or by shutting off the current.

Figure 4
Simulation of current injection in the soma, model PM10. For each current injection amplitude (A: 1.0 nA, B: 2.0 nA, C: 3.0 nA, D: 4.0 nA) 3 recording sites are shown: the soma (upper trace), the main dendrite (middle trace) and a spiny dendrite (bottom trace). Duration of current injection is shown by the bar below the traces. ERROR IN FIG. 4: the time bar should read 500 ms instead of 200 ms.

Calcium Spikes. As the current amplitude injected in a Purkinje cell increases, dendritic Ca²⁺ spikes become apparent as depolarizing spike bursts in the soma. Again, the model reproduced this behavior reasonably well (Fig. 3B-C, Fig. 4C-D). In experimental somatic recordings, the transition between simple spikes and depolarizing spike bursts is abrupt, while the model often showed one or two small bursts before a full dendritic spike became apparent.
The transitions between the firing of simple somatic spikes and the firing of dendritic spikes causes a break in the linearity of the frequency current curve (f-I), with a second, shallower slope above this transition. Our f-I curves (Fig. 6A) were remarkably similar to those published by Llinás and Sugimori (1980a, their fig. 5). In the model, dendritic spike firing frequency remained relatively constant (16 to 19 Hz in both models) once the threshold was crossed and was rather insensitive to the current amplitude.

Figure 5
Simulation of short current steps just below and above firing threshold in the soma (model PM9). Amplitude of current injection for each trace is indicated and duration is shown by the bar below the traces.

Plateau Potentials. Current injection steps are often followed by a prolonged plateau potential. In our simulations this phenomenon was always observed after short current steps around firing threshold (Fig. 5) and sometimes after longer current steps also (Fig. 4A). In the model these plateau potentials resulted in depolarizations of between -49 and -55 mV. Similar prolonged plateau potentials after the end of a current injection were found by Llinás and Sugimori (1980a, compare their fig. 5D with our Fig. 3A). However, in the experimental data these plateaus decay over a time course of about 100 ms, while in the model they often did not decay at all (Fig. 5).

Comparison of Somatic and Dendritic Recordings. Paired intracellular somatic and dendritic recordings in slice have demonstrate that somatic action potentials do not propagate far into the dendrite. Thus recordings of the main dendrite of Purkinje cells often reveal somatic action potentials as only small wavelets. When the membrane potential in the soma and large proximal dendrites were compared in the model, a similar result was observed (Figs. 3, 4, 7). Further, using the model it was possible to demonstrate that the action potential did not propagate into spiny dendrites (a location that electrophysiologists cannot record from), even those relatively close to the soma (Figs. 3, 4, 10A).

Paired intracellular somatic and dendritic recordings have also suggested that the Ca²⁺ spiking, responsible for the depolarizing spike bursts observed in the soma during higher amplitude current injections, have a dendritic origin. As shown in Fig. 3B-C and Fig. 4C-D, this was also the case in the model.

Figure 6
Relation of somatic firing response to the amplitude of injected current in models PM9 (m) and PM10 (l). A: Firing frequency versus amplitude of injected current (f-I curve). Frequencies were measured in an interval from 300 to 700 ms after onset of the current injection, before any clear depolarizing spike bursts in the soma. B: Delay between onset of current injection and first somatic spike versus amplitude of injected current.

Response to Dendritic Current Injections. The model was also able to replicate previously reported responses to direct current injection into Purkinje cell dendrites (Fig. 7). As in the physiological recordings, low amplitude current injection resulted only in attenuated somatic spikes, while these spikes were interrupted by dendritic Ca²⁺ spikes with larger current injections. Under these circumstances, the onset of the bursting behavior was almost immediate as compared to somatic recordings. Note also the burst occurring after the end of the 2.0 nA current injection.

Figure 7
Recordings of membrane potential in the main dendrite in response to 3 different levels of current injection at the same location, model PM9. Amplitude of current injection is indicated and duration is shown by the bar below the traces.

Variability in Purkinje Cell Response Properties. In slice preparations, different Purkinje cells often have quite different firing patterns, especially as concerns the pattern of interacting somatic and dendritic spikes (Llinás and Sugimori 1980a, their fig. 11) and the slope of the f-I curve. However, a compartmental model is deterministic, i.e. always producing the same output for a given input. Variability in response properties comes about only as a result of varying parameters in the model.

In the case of the current model, we found that all of the reported variations in firing patterns could be obtained by simply changing the density of K⁺ conductances in the soma and the main dendrite of the model. Figures 3 and 4 compare the firing of a model with low K⁺ conductance (PM9) with a model with high K⁺ conductance (PM10). While these models were similar in all responses described so far, they showed richly different patterns of somatic firing, which encompassed most of the observed ones. Compare, for example, the somatic responses towards the end of the current injections in Figs. 3B, C and 4D and after the current step in Fig. 3D. The higher K⁺ channel densities in model PM10 made the soma also less excitable, resulting in a shallower f-I curve (Fig. 6A).

Robustness of modeling results

Having tuned the model to replicate in vitro responses to current injection, several simulations were run to test the robustness of the basic model. These tests helped to build confidence that the model has biological validity.

Variations in Purkinje Cell Morphology. Initial modeling experiments (and the results presented so far) were based on the anatomical reconstruction of one particular Purkinje cell. Accordingly, it was important to determine the sensitivity of the results to this particular morphology. To do this identical channel equations and densities (PM9) were placed into two additional Purkinje cells, reconstructed by Rapp et al. (1994), the results are shown in Fig. 8.

Figure 8
Simulation of response to current injection in models of other Purkinje cells. A: morphology of cell 2 of Rapp et al. (1994). B: membrane potential in the soma of cell 2 during current injections of indicated amplitude. C: morphology of cell 3 of Rapp et al. (1994). D: membrane potential in the soma of cell 3 during current injections of indicated amplitude.

In both cells the response properties of the model fell well within the normal variation seen in Purkinje cell recordings. Simulations of current injections in the soma produced the same, typical pattern as in our model of Purkinje cell 1 (Fig. 3), i.e. at low intensities steady firing of fast somatic spikes superimposed on an increasing plateau potential and at higher intensities the presence of dendritic Ca²⁺ spikes. The details of these firing patterns were, however, quite different for the 3 cells. Cell 2 is smaller than cell 1 (Fig. 1) and thus had a larger input resistance (Rapp et al. 1994). As a consequence, the Na⁺ currents caused a more pronounced plateau potential and the somatic spikes were less well repolarized. This resulted in a progressive attenuation of spike amplitude due to incomplete removal of inactivation of NaF. Note however, that despite this build up of inactivation, spiking did not saturate at a depolarized level during the 1.5 nA current injection, while it did in cell 3 (not shown). The firing pattern of cell 3 is more similar to cell 1, but with a shift in the f-I curve. This can be explained by the small soma and short, thin main dendrite, which caused a smaller total Kdr and KM conductance in the model of this cell (table 2).

Responses to Channel Blockers. A second measure of model robustness is its ability to replicate physiological data in response to standard channel blockers. In this case model results were compared to those of Llinás and Sugimori (1980a) in response to both Na⁺ and Ca²⁺ channel blockers while holding all other model parameters fixed.

Figure 9
Simulation of channel blocking experiments. A. Simulation of the effect of blocking Na+ currents by TTX application. B. Simulation of the effect of blocking Ca2+ currents by Co2+ or Cd2+ application. Membrane potential in the soma during current injections of indicated amplitudes in the soma.

Figure 9A shows a simulation of TTX block (the conductance of the NaF and NaP channels was set to zero) during a 1.0 nA current injection. Note that the somatic recording showed only attenuated dendritic spikes in the soma, there were no fast somatic spikes, and that there was a sustained plateau potential after the end of the current injection. Subthreshold current injection (0.1 nA) only evoked the plateau current. Similar to the results of Llinás and Sugimori (1980a, their fig. 6), Na⁺ channel blockers selectively interfered with somatic spikes while having little effect on dendritic spiking and plateau potentials. However, the dendritic spikes in the model were smaller and less sharp than those in experimental recordings. Also, contrary to experimental results, the plateau potential did not decay.

From these simulations, one can conclude that dendritic spiking in the model was caused by the slow progressive depolarization induced by the current injection, not by the larger depolarizations during fast somatic spikes.
A simulation of Co²⁺ block (the conductance of CaT and CaP was set to zero) is shown in Fig. 9B. Current injection caused a slow depolarizing response which generated fast, somatic action potentials. With increasing amplitude of currents, there was less delay before the first action potential, but the spikes also decreased in amplitude and saturated at a sustained plateau depolarization of about -30 mV. These results are comparable to experimental recordings, except that higher amplitude current injections were necessary in the model to obtain saturation of spiking.

As in the case of Llinás and Sugimori (1980a, their fig. 8), Ca²⁺ channel blockers selectively interfered with dendritic spikes while leaving somatic plateau potentials and somatic spiking intact. However, because the depolarization could not evoke dendritic Ca²⁺ spikes, there was also no Ca²⁺-activated hyperpolarization to repolarize the cell and the model settled in sustained plateaus with high intensity current injection. The model thus confirmed the importance of dendritic Ca²⁺-activated K⁺ conductances in repolarizing the somatic depolarizing spike bursts.

Robustness to Changes in Channel Densities. As described above, the principle parameters used to tune this model to the current injection data were the distribution and density of specific ion channels. While table 2 described the final distribution for this model, it was also important to determine how sensitive the modeling results were to these particular density values.
A full search of the 10,000 dimensional parameter space for this model would have been computationally prohibitive (see Bhalla and Bower 1993). Accordingly, the approach we have taken involved examining the effect of changing the density of a single channel type on the physiological responses to current injection in the soma. The results showed, perhaps not surprisingly, that small amplitude currents like NaP, CaT, KA or KM could be completely removed with little effect, except for small changes in firing frequency. The same was true for increasing KA or KM by a factor of 2 or 3. Similar increases in NaP or CaT, however, caused fast spiking in the soma to saturate (as in Fig. 3D) at lower current amplitudes and could turn the cell into a spontaneously firing or bursting neuron.

The model was more sensitive to changes of the conductances involved in spike generation and repolarization. Small changes resulted in a shift of the f-I curve and of the current amplitude at which dendritic spiking started. Changes in the dendritic currents involved in spiking could suppress all dendritic spiking (reducing CaP by 20% or more, increasing Kdr by at least 40%, KC by 25% or K2 by 10%) or make dendritic bursting the unique mode of firing of the cell (reducing NaF by 50%, increasing CaP by 50% or decreasing Kdr by 20% or KC or K2 by 10%). NaF could be reduced by 70% before fast sodium spikes disappeared.

The model was thus sensitive to small changes in density of CaP, KC and K2; in other words the region of parameter space which generated correct model responses was not very large for these currents. Note, however that larger changes could be applied to these channel densities if one of the other current densities was changed in the opposite direction. The model was less sensitive to changes in the other currents, like for example the densities of voltage dependent K⁺ currents (c.f. table 2).

Analysis of the ionic currents responsible for somatic and dendritic spikes

Results from parameter variations, of the sort just mentioned, lead to the question of which ionic currents were most responsible for each of the response properties of the Purkinje cell. In fact, the parameter sensitivity of model output to ion channel densities followed directly from an analysis of the role of the different currents in the Purkinje cell firing behavior. In some cases the relationship between ion channels and response properties was straightforward, in other cases the response of the cell was a result of the interaction between different currents. Sometimes this was further complicated by the fact that the types of channels and their densities varied between the 3 different regions of the model. The following sections first describe the spatial distribution of the patterns of activity and then consider in detail the currents that cause each component of Purkinje cell responses to current injection in the model.

Fig. 10 shows images representing the membrane potential distribution and calcium concentration in all parts of the model during a somatic action potential and a dendritic spike. Fig. 11 shows the change in membrane potential, submembrane Ca²⁺ concentration and amplitude of all the ionic currents at 4 representative locations in the model during a current injection in the soma. The respective contribution of different channels to the somatic spikes, dendritic spikes and the corresponding depolarizing spike bursts in the soma can be determined from these figures. In each case, the data shown was obtained long after the initiation of the current injection so that the repetitive firing properties of the model had settled into a steady state.

Figure 10
False color representation of membrane potential and Ca2+ concentration in the complete model during a 2.0 nA current injection in the soma. A: membrane potential distribution during a somatic action potential. B: membrane potential distribution at the begin of a dendritic spike. C: membrane potential distribution 1.6 ms later, when the dendritic spike had peaked. D: somatic (red) and dendritic (green) recordings with the times when images A-C were taken indicated. E: submembrane Ca2+ concentration at same time as B. F: submembrane Ca2+ concentration at same time as C. Note the non-linear voltage scale, which is expanded between -60 and -20 mV.

Spatial Distribution Of Membrane Potential. Several spatial features of the response are clear. First, Fig. 10A shows dramatically how little somatic action potentials penetrated into the dendritic tree, as has long been known from dendritic recordings (Llinás and Sugimori 1980b). Essentially, only the largest proximal dendrites are at all effected by these fast somatic events.

In contrast, the generation of dendritic spikes affected large parts of the dendrite. Fig. 10B and C show, however, that these dendritic spikes did not fire synchronously in all regions of the dendritic tree. In this cell they tended to originate in the upper right branch, which is electrotonically the most distant from the soma, and spread as a wave toward the left part of the dendrite.

Calcium Concentration. The distribution of Ca²⁺ during Purkinje cell activity gives clues to the underlying ionic mechanisms. This is most true for the dendrites where large Ca²⁺ fluctuations were seen. The Ca²⁺ concentration in the soma remained very low (Figs. 10E, F, 11A), because only CaT channels were present in the soma. We do not claim that our simulation of the Ca²⁺ concentration in the soma was realistic. However, because there were no Ca²⁺-activated K⁺ channels in the soma, it did not affect the behavior of the model and we have not tried to optimize it further.

During dendritic spikes, the submembrane Ca²⁺ concentration attained a sharper peak than the membrane voltage (Fig. 11C, D). It started to increase with a delay compared to the depolarization and the CaP channel activation, because Ca²⁺ concentration under these conditions was mainly determined by the integral of Ca²⁺ inflow. It also decayed rapidly, so that it returned to baseline together with the membrane potential. The Ca²⁺ images in Fig. 10 (E, F) show the same pattern, i.e. the Ca²⁺ rise was delayed compared to the depolarization and showed steeper distributions over the dendrite. This specific temporal and spatial pattern of Ca²⁺ elevation restricted the activation of Ca²⁺-activated K⁺ currents (at these elevated concentrations mainly KC).

Figure 11
Dynamic change in model variables during somatic and dendritic spikes. The same 100 ms sequence is shown at 4 representative locations in the model during a 2.0 nA current injection in the soma, 900 ms after the start of the current injection (model PM9). A: soma. B: main dendrite. C: smooth dendrite. D: spiny dendrite. Each part of the FIG. shows the membrane potential (V, upper trace), the Ca2+ concentration ([Ca2+], middle trace) and the amplitude of all ionic currents (superimposed lower traces) in the compartment. In A and C part of the current traces is shown enlarged in a small box. The scale bars for voltage and concentration are identical in all sections, for the ionic currents each section is scaled differently (outward currents upward). The small horizontal bar at the left of the V traces indicates -50 mV membrane potential, at [Ca2+] traces it indicates 0 uM concentration.

Somatic Currents and the Fast Sodium Spike. In the soma (Fig. 11A), the main active currents were the fast and persistent Na⁺ channels and the delayed rectifier. NaF and Kdr played their classic role in generating the fast action potentials (Hodgkin and Huxley 1952). Note also that the amplitude of these currents was quite large compared to currents in the dendritic compartments; this was caused by the large size of the soma and the large in the model. The other somatic currents did not play a large role in the firing patterns shown here. KA and CaT were transiently activated when the cell was depolarized from resting membrane or hyperpolarized potentials, but inactivated to a large degree during long current steps (data not shown).

Dritic Currents and the Calcium Spike.he channels contributing to the dendritic spike operated in the same manner in the smooth (Fig. 11C) and spiny (Fig. 11D) dendrites, but the spike itself was much bigger in the spiny dendrites because of the higher input impedance of the smaller dendritic branches. The dendritic spikes were generated by the CaP channel, the CaT channel did not play a role in spike generation (except during a rebound spike from hyperpolarization). The dendritic spikes were repolarized by the KC channel, which activated toward the end of the spike and deactivated rapidly afterwards. While the K2 current amplitude increased a bit during the spike, it was already quite active before the spike, when it was a stronger outward current than the KC channel (Fig. 11C, inset).

Currents in the Main Dendrite. The currents in the main dendrite were a mixture of somatic and dendritic currents, and like the membrane potential, they combined the characteristics of both parts of the cell in their dynamical changes (Fig. 11B). Two aspects are noteworthy. The CaP channel in the main dendrite was activated during the somatic action potential and thus contributed to the broadened base of these Na⁺ spikes. Correspondingly, there were small activations of the Ca²⁺-activated K⁺ currents during each somatic spike. However, the delayed rectifier was a more important current in the main dendrite during the depolarizing spike burst than the K2 current.

The Generation of Plateau Potentials. In addition to the somatic and dendritic spiking events just described, the model also clearly generated plateau-like events in both the soma and dendrite. Analysis of the mechanisms underlying the generation of these plateaus is interesting in that it involves an interaction of ionic channels differentially distributed in soma and dendrite.

From our analysis of the model under conditions of somatic current injection with TTX block, it was clear that the previously described dendritic plateau potential could be induced in the absence of somatic spikes. During depolarizing current injection under normal conditions, the dendrite did not repolarize completely as the soma did in between somatic spikes, presumably because no Kdr channels were present beyond the main dendrite. This slight, continuous dendritic depolarization caused a low number of both CaP and CaT channels to open, thus generating the dendritic plateau. With longer duration current injections, this plateau slowly built up (Llinás and Sugimori 1980a) causing a continuous inflow of Ca²⁺ into the dendrite, which activated, as already mentioned, mainly the K2 channel.

In the soma, depolarizing spike bursts were caused by an interesting interaction between inward and outward currents. While these bursts were generated by depolarizing current flowing into the soma from the dendrites during dendritic Ca²⁺ spikes (Fig. 10B), their final amplitude in the soma was determined by the interaction between NaF, NaP and Kdr (Fig. 11A), which were all continuously activated during the depolarization. Note that NaF was much bigger than NaP. Such a steady NaF current is usually called the window current (French et al. 1990) and is caused by an overlap of the steady state activation and inactivation curves (Fig. 2A), so that current can flow at a depolarized potential where the channel does not inactivate completely. None of these 3 currents deactivated completely in between spikes, so that the somatic membrane did not repolarize beyond -55 mV after a burst. In comparison, in the distal dendrites the membrane potential reached -62 mV after a dendritic spike.

Control of Dendritic Spike Generation. Control of the alternation of somatic and dendritic spiking that determined the overall behavior of the cell in the simulation was a complex combination of all the events discussed so far. However, the important point is that the model clearly demonstrated that the Ca²⁺-activated K⁺ channels interacted with the steady, plateau-generating CaP activation as a negative feedback loop. The K2 channel dominated this effect, because it had a much lower threshold of Ca²⁺-activation than the KC channel, resulting in K2 currents larger than the KC currents in between dendritic spikes (Fig. 10C, inset).

One of the clear characteristics of Purkinje cell responses to large somatic current injections is the delay seen in activation of full blown calcium spikes (Fig. 3B, C) and then the alternation of dendritic calcium spikes and periods of somatic action potential firing. In the model, this phenomenon was directly related to the interaction between the CaP and the K2 channels. Initially, depolarization resulted in a slow build up of CaP current which resulted in an increase of the internal dendritic Ca²⁺ concentration (e.g. 0.17 uM at 200 ms). This activated a large number of K2 channels effectively countering the depolarization caused by the CaP channel. As a consequence following the onset of current injection there was a fluctuation of the baseline membrane potential in the dendrites (Fig. 3B), but no true dendritic spikes.

During the current injection, the persistent CaP current was progressively reduced by about 50%, due to the slow inactivation of the CaP channel, until a steady state was reached. This caused a corresponding reduction in the Ca²⁺ concentration (e.g. 0.11 uM between dendritic spikes at 900 ms) and less activation of the K2 channel, so that the CaP channel conductance could overcome the counteracting influence of the K2 channels, resulting in a full dendritic spike. Once initiated, the large influx of Ca²⁺ then caused the sequence described above for the Ca²⁺ spike, with the dendrite being repolarized by the KC channel.

Thus it appears that the CaP channel effectively controlled dynamically the threshold of activation of dendritic spike firing, through a negative feedback from the K2 channel. Because the K2 channel was sensitive to small changes in Ca²⁺ concentration caused by the beginning activation of CaP channels and because the opening of K2 channels counteracted CaP activation, K2 channel activation could effectively increase the threshold for Ca²⁺ spike generation. In other words, when the K2 current was high the CaP channel could only cause plateaus. To generate dendritic spikes the K2 current had to be low and the CaP channel had to activate fast. Vice versa, high CaP plateau currents suppressed dendritic spiking because they activated most of the K2 channels. This also explained the lack of full blown dendritic spikes during the prolonged episodes of somatic depolarization with high amplitude current injections (Fig. 3D).

Effect of active membrane on the electrical properties of Purkinje cells

Channel Activation and Electrotonic Length. We have used the model to contrast the electrical properties of a passive Purkinje cell dendrite with those of a dendrite with the active properties just described.

Figure 12
Effect of voltage dependent conductances on the electrotonic length of dendritic segments.
A: full Sholl diagram in units of electrotonic length of the cell with active conductances in the dendrites, during a pause in between action potentials.
B-D: enlargement of the Sholl diagram showing the same subset of branchlets (original location indicated by the circle in A) under different conditions of the ionic channels.
B: all passive membrane.
C: active membrane, the cell is in a quiet state in between action potentials (same as A).
D: active membrane, at the peak of a dendritic spike.
E: recording of activity in a spiny dendrite (upper trace, location marked on the Sholl diagrams) and soma (bottom trace). Dendrogram D was taken at the time indicated by the asterisk, dendrogram C was taken at the time indicated by the circle. Simulation of 2.0 nA current injection in the soma, model PM9.

Fig. 12 shows the effect of including voltage dependent conductance on the electrotonic structure of the model. In this figure, Sholl diagrams (Sholl 1953) are presented in units of electrotonic length of the Purkinje cell under different conditions. At the right side, enlargements show the electrotonic lengths of a few distal branchlets in a passive membrane model (no active conductances in the model; Fig. 12B) and in an active membrane model during and after a dendritic spike. One can see that adding active membrane to the cell roughly doubled its electrotonic size during the resting state shown in Fig. 12C (the dendritic membrane had a potential of about -60 mV) and that during a dendritic spike the electrotonic size was roughly doubled again (Fig. 12D). The electrotonic distance from the soma to the marked dendritic tip (morphological distance 392 um) was 0.57 lambda in the passive model; it was 0.95 lambda in the active model in the resting state and 1.57 lambda in the active model during the dendritic spike.

Effect on Voltage Clamp Experiments. Ultimately, the significance of active membrane properties on the electrotonic length of a Purkinje cell is related to its integration of synaptic input. That is the subject of the accompanying paper. However, the electrotonic structure of the Purkinje cell is also relevant to the feasibility and interpretation of voltage clamp experiments. Recently, several groups have reported results of whole-cell patch clamps of Purkinje cells (Hirano and Hagiwara 1989; Kaneda et al. 1990; Llano et al. 1991b; Regan 1991; Regehr et al. 1992). An important question in interpreting the data from such studies is whether an adequate space clamp was achieved (Rall and Segev 1985).

To test the general difficulty of having a good space clamp of a cell the size and complexity of the Purkinje cell, we simulated a voltage clamp in our basic model. The results shown in Fig. 13 clearly indicate that it is very difficult and likely impossible to establish a good space clamp of an adult Purkinje cell.

Figure 13
Simulation of voltage clamp steps.
A: the model was clamped at resting potential (-68 mV) and then stepped to -100 mV or +40 mV. Superimposed upper traces are the membrane potential in the soma (s), main dendrite (m) and a distal spiny dendrite (d, marked with the star in FIG. 1), lower trace is the clamp current.
B: as in A, but stepped to -40 mV.
C: membrane potential under steady state voltage clamp versus distance from the soma. Data from the same voltage clamp steps as shown in A, B. Data points are exclusively from compartments linking the soma with the distal spiny dendrite. Because the potential oscillated during steps to -40 mV, two traces are shown for that voltage step (corresponding to the minimum and maximum voltage in the spiny dendrite). Simulation time step was 5 us.

To model voltage clamps under conditions similar to the ones used by experimentalists (Llano et al. 1991b), we simulated external Cs²⁺ block of Kdr, KM and Kh channels (Hille 1991) and inactivation of all Ca²⁺-activated K⁺ channels because of the presence of EGTA in the internal solution. The model was clamped at 4 voltages: -100 mV, -68 mV (resting potential), -40 mV and +40 mV and allowed to stabilize to a steady state, which took about one simulated second. The space clamp was always bad, as the cell was never isopotential (even at -68 mV the distal spiny dendrite had a potential of -67.47 mV). The clamp was, however, especially bad in the depolarized direction. Any step above -50 mV resulted in activation of the Ca²⁺ channels in the model, causing a clear spike in the dendrites (Fig. 13A, B). In the steady state, the distal dendrite had potentials that diverged up to +/-40 mV from the holding potential in the soma, because the CaP channel did not inactivate completely. The model did not achieve a steady state for voltage steps between -50 mV and -25 mV, because the membrane potential oscillated in the whole dendritic tree.
Fig. 13C shows the membrane potential at increasing distances from the soma for different control voltages. It demonstrates that even the proximal dendrite was not clamped well. During depolarizing voltage clamp steps, the membrane potential differed from the holding potential by 10 mV or more at a distance of only 25 um from the soma.

Discussion

This paper describes a detailed compartmental model of the cerebellar Purkinje cell. While this cell has been the subject of numerous previous modeling efforts (see below), the current model includes important features of this neuron that were not present in previous ones. For example, several authors have explored the passive properties of the Purkinje cell (Llinás and Nicholson 1976; Shelton, 1985; Rapp et al. 1992, 1994). However, these models did not include voltage dependent conductances in the dendrites which are known to be extensive in Purkinje cells (Llinás and Sugimori 1992) and which play an important role in the response to both current injections and synaptic inputs (most spectacular for the climbing fiber synapse). Two models that do include ionic conductances in the dendrites have been reported in the literature. The first one (Pellionisz and Llinás 1977) was quite innovative, in that it was one of the first models with active membrane in the dendrites ever published. Unfortunately, much less was known at that time about the dendritic conductances within this cell so that the model used fast Na⁺ channels and a delayed rectifier in the dendrites instead of the Ca²⁺ and Ca²⁺-activated K⁺ channels which are now known to be present (Llinás and Sugimori 1980a, b). The other active membrane model (Bush and Sejnowski 1991) used a more appropriate set of ionic channels in the soma and dendrite and some data was presented indicating that it replicated responses to current injections. However, neither the model nor its responses were described in any detail. Instead, the main emphasis of the report was on describing a new set of equations to simulate ionic channels in single cell models.

Replication of current injection results.

This paper principally concerns the simulation of Purkinje cell responses to current injection in the slice preparation (Llinás and Sugimori 1980a, b; Hounsgaard and Midtgaard 1988; Llinás and Sugimori 1992). We believe that overall, the model does a good job of simulating these experimental results.

Somsyiv snf Dendritic Firing Patterns. As demonstrated in Figs. 3-7 both the somatic and dendritic firing patterns were reproduced well. In particular, at low level current intensities the model fired only fast somatic spikes after a delay. At higher current amplitudes these fast Na⁺ spikes were interrupted by dendritic Ca²⁺ spikes which were caused by activation of the CaP channel. The model also replicated the experimental f-I curves well.

One aspect of Purkinje cells in slice preparations that we have not addressed explicitly is the tendency for Purkinje cells to become spontaneously active (Llinás and Sugimori 1980a). This could be achieved in the model by introducing a bit of bias current. However, there are numerous ways in which this could be done. For example, increasing the amount of NaP or CaT channels would introduce such a current, as would decreasing any of the K⁺ currents or the leak in the model. In other words, there might be many ways in which Purkinje cells could become spontaneously firing cells, as modulating the conductivity of any of 6 different channels would be sufficient.

Plateau Potentials. The model also generated longer duration plateau-type potentials that have been shown to exist in this neuron (Llinás and Sugimori 1980a; Llinás and Sugimori 1992; Jaeger and Bower 1991). It has been demonstrated physiologically that current injection generates a plateau potential in the soma of the cell which is dependent on Na⁺ conductances. The model generated Na⁺ dependent somatic plateaus (Fig. 9B, 11). However, it is debated in the literature if such persistent Na⁺ currents rely on a special channel (NaP) or depend on the NaF channel itself (French et al. 1990; Kay et al. 1990; Alzheimer et al. 1993). In our model the somatic plateau potential was mainly carried by NaF channels in the form of a so-called window current, although the model also included the experimentally demonstrated NaP channel (Kay et al. 1990). The plateau related current in the model flowed through NaF channels at a potential where the steady state activation and inactivation curves (Fig. 2A) overlap. While this mechanism was robust in the model, one argument against the existence of such a window current is that it is based on a wrong model of inactivation, as Na⁺ channel inactivation is probably not voltage-dependent (Aldrich and Stevens 1987). If this is in fact correct, then the term in our equations for steady state inactivation would be meaningless. While this possible inaccuracy in the Hodgkin and Huxley (1952) model of inactivation means that our results are not conclusive, the model at least suggests that there may not be a need for separate NaP channels to explain somatic plateaus. In our simulations the NaP actually primarily affected the f-I curve.

The model also generated dendritic plateau potentials (Fig. 9, 11) which have recently been described in more detail (Jaeger and Bower 1991) and which may be particularly important during synaptic activation of the Purkinje cell by peripheral stimuli (Thompson and Bower 1991). In the model, these dendritic plateau potentials were carried largely by the CaP channels, as the CaT channel inactivated too rapidly to play a major role. In this case again, the plateau response resulted from a window current-like mechanism very similar to that found with the NaF channels in the soma, because the dendritic plateau current through the CaP channel occurred at a potential where the CaP channel does not inactivate completely. As a consequence, these dendritic plateau potentials did not wane in the model as they do in slice preparations.

Interaction Between Ionic Currents. One of the principle benefits of modeling neurons at this level of detail is that the interactions between different currents can be explored. All too frequently, experimentalists assign very specific roles to particular channel types without taking into account the often complex interactions between different conductances that are actually responsible.

In the current model there are numerous examples of this type of interaction. One of the most striking, involves the interaction of the non-inactivating K⁺ channels and the Na⁺ and Ca²⁺ channels responsible for the plateaus. At the somatic level, these K⁺ conductances served to counteract the depolarizing effect of the Na⁺ channels with their balance determining the voltage of the plateau potential (Fig. 11A).

In the dendrite, the Ca²⁺-inactivated K⁺ channels played a critical role in the dual function of the CaP channel in generating Ca²⁺spikes as well as dendritic plateaus. Because the K2 channel is sensitive to small changes in Ca²⁺ concentration, it could effectively increase the threshold for Ca²⁺ spike generation by counteracting the activation of small numbers of CaP channels. The involvement of the K2 channel in these two very different forms of dendritic response, and the presence of other mechanisms in the Purkinje cell dendrite which through changes of the Ca²⁺ concentration (Llano et al. 1991a; Takei et al. 1992) could activate this channel, suggest interesting possibilities for regulation of more global dendritic response properties by the K2 channel and other Ca²⁺-inactivated K⁺ channels that were not included in the model (like the SK or AHP channel, Lancaster et al. 1991).

The model supports the suggestion that the channels responsible for generating plateau potentials are different in the soma and dendrite (Llinás and Sugimori 1980a). However, examination of the model also makes clear that these different plateaus are not physiologically isolated. A somatic plateau potential was always accompanied by a dendritic plateau and vice versa, because any depolarization will spread throughout the cell.

Other interactions between channels, like the activation of the anomalous rectifier during prolonged hyperpolarizations, followed by an activation of CaT channels during the rebound spike, were not explored in detail. The anomalous rectifier did not affect the repetitive firing properties of the model, as it was completely deactivated beyond spiking threshold.

Response Variability. While any particular set of parameters in the model generated an identical (deterministic) output, we have also found that slight variations in parameters could produce the kinds of subtle variations seen in Purkinje cell recordings. Thus, different levels of current injection, small changes in the densities of outward currents (PM9 versus PM10), or slight changes in morphology generated subtle changes in model output. Under these conditions, the Purkinje cell model responded generally in the same way but also showed subtle variations in the details of its responses (e.g. in the sequence of action potentials). We believe this variability in the model is important, as the specific objective of this effort was to represent the entire population of Purkinje cells rather than just one individual cell (Bower and Koch 1992).

One of the variable aspects of Purkinje cells that has been previously described involves the details of the f-I curve (Llinás and Sugimori 1980a). Further, this curve can change during long recordings of the same cell (D. Jaeger, personal communication). It is interesting that changes in the density of the delayed rectifier and non-inactivating K⁺ channels in the model alone could cause a lot of this variability, as can be seen by comparing Fig. 3 with Fig. 4 and in the f-I curves of Fig. 6A. The results from patch clamp recordings demonstrating that Purkinje cell Kdr channels are under metabolic control by protein kinase C, which attenuates Kdr current (Linden et al. 1992), could provide one mechanism for short term changes in f-I relationships. Further, our finding that somatic action potentials only repolarized completely when Kdr channels were added to the main dendrite, showed that the Kdr channel also controls the coupling between the soma and more distal dendrite which may influence these f-I relationships. This distribution of Kdr channels matches data obtained with voltage-sensitive dye imaging, where a decoupling between the soma and the more distal dendrite could be removed by blocking K⁺ channels (Knöpfel et al. 1990).

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 (Bower and Koch 1992). Thus, while 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 generated several 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 (Fig 3A, 4A, B). In experimental data, low amplitude current injections do not result in this firing pattern (Llinás and Sugimori 1980a, b). Similarly, at higher current amplitudes fullblown dendritic spikes were often preceded by smaller Ca²⁺ spikes (Fig 3B, C). Also, the prolonged plateau potentials following 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. As 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.t the most basic level, the model was based on a light microscopy reconstruction of the Purkinje cell, effectively limiting the resolution of the model to about 1 um diameter branches. However, it is known that Purkinje cells have many branches smaller than this diameter (Palay and Chan-Palay 1974; Gundappa-Sulur and Bower 1990). Based on this information, we estimate that our model may be lacking about 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 (Harris and Stevens 1988; Harvey and Napper 1991) suggests that the inaccuracies in the total spine surface area and in the shrinkage factor used in the original reconstruction (Rapp et al. 1994) 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 (Rapp et al. 1994), while the EM data on which we based the size and distribution of spines was obtained in the rat (Harris and Stevens 1988). Likewise, most of the voltage clamp data came from rat Purkinje cells (Gähwiler and Llano 1989; Hirano and Hagiwara 1989; Kaneda et al. 1990; Regan 1991) although we compared model outputs to current injection data obtained in the guinea pig slice (Llinás and Sugimori 1980a, b). While 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. Lytton and Sejnowski 1991; Traub et al. 1991; McCormick and Huguenard 1992; Bhalla and Bower 1993). 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 the Methods section, 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, which 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 about 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 at least 94% of the channels were always inactivated. This necessitated an unphysiologically high density of NaF channels in the soma (total conductance 7500 mS/cm²). However, because of the continuous inactivation, the real maximum conductance was less than 2000 mS/cm², comparable to for example a Na⁺ conductance of about 1000 mS/cm² (20 C) for a rat node of Ranvier (Neumcke and Stämpfli 1982), but much larger than the 15 mS/cm² (22 C) reported for the somata of freshly dissociated hippocampal neurons (Sah et al. 1988). 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 (Somogyi and Hamori 1976), but these showed no qualitative differences compared to the standard model.

Ca²⁺-Activated K⁺ Channels. It is of more concern that the Ca²⁺-activated K⁺ currents known to exist in the Purkinje cell and which 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 (Gruol et al. 1991). This is not surprising considering that the BK channel alone seems to have more than 100 expression variants (Adelman et al. 1992). Without more detailed Purkinje cell data, in the current model we have attempted to cover the effects of these diverse channels by including only 2 Ca²⁺-activated K⁺ channels, a KC and a K2 channel. As described in the Methods section, these channels have quite different activation characteristics, with the KC channel requiring high Ca²⁺ concentrations and large depolarizations while 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 (Latorre et al. 1989), 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 (Yamada et al. 1989). 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 (Ikemoto et al. 1989). The resulting KC equation reproduced several characteristics of the BK channel well, among them the 2 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 P0-V curve upon changes in Ca²⁺ concentration has been reported (Moczydklowski and Latorre, 1983; Gola et al., 1990), while our equations only scale the amplitude of the P0-V curve (Fig. 2G). In addition, the BK channel also seems to have a fast Ca²⁺ related deactivation (Ikemoto et al., 1989), while 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 maximum conductance 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 maximum conductance 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 upon single channel recordings (Gruol et al. 1991), but little kinetic data were available. We have therefore combined data from similar channels in synaptosomal membranes (Farley and Rudy 1988; Reinhart et al. 1989). However, the experimental data remains very incomplete. Based on 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 Gruol et al. 1991), but we did not model this conductance. This channel resembles the SK channel or AHP conductance that causes slow afterhyperpolarizations in other neurons (Latorre et al. 1989; Lancaster et al. 1991). It is believed to be sensitive to low Ca²⁺ concentrations with slow activation kinetics (Pennefather et al. 1990). We did not include this channel in the model because the model computed only fast changing Ca²⁺ concentrations, while 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.

Modeling Calcium Concentrations. For a neuron with Ca²⁺ activity like the Purkinje cell, the detailed modeling of Ca²⁺ represents one of the biggest technical challenges. Modeling the diffusion, pumping, internal uptake and release, cytosolic buffering, etc. of Ca²⁺ is a formidable computational challenge (Yamada et al. 1989; Sala and Hernández-Cruz 1990). In the current version of the model we used a very simple one shell scheme to model Ca²⁺ concentration. Calcium concentration in this shell decreased with a fast exponential decay, despite the fact that Ca²⁺ concentration is regulated by a complex interaction of, among others, metabotropic receptors (Llano et al. 1991a; Staub et al. 1992), cytoplasmic stores with IP3 and ryanodine receptors (Takei et al. 1992), Ca²⁺ binding proteins like calbindin and parvalbumin (Kadowaki et al. 1993) and Ca²⁺ inflow through the Ca²⁺ channels (Hockberger et al. 1989; Ross et. al. 1989; Lev-Ram et al. 1992). Clearly there is tremendous room for improvement of the model in this regard.

Our principle reason for modeling internal Ca²⁺ concentration involved the regulation of conductance in the Ca²⁺-activated K⁺ channels. In this case the most relevant region of the cell is the area immediately adjacent to the internal membrane which is presumably where the mechanism for activating the K⁺ channels operates. As presented in the results section, the Ca²⁺ concentration we modeled provided both the amplitude and fast changes necessary to activate the BK channel properly. However, at 6 uM peak concentrations, Ca²⁺ was much higher, and the change in concentration much faster, than those reported for the Purkinje cell in the literature (Tank et al. 1988; Hockberger et al. 1989; Ross et al. 1990; Sugimori and Llinás 1990). One likely explanation for this discrepancy is the time constants involved in experimentally measuring Ca²⁺ concentration. Each of these measurements were made with the Ca indicator fura-2 whose slow kinetics filter out fast transients of internal calcium (Vranesic and Knöpfel 1991). However, from the Ca²⁺ sensitivity of the BK channel (Smart 1987; Franciolini 1988; Reinhart et al. 1989; Lancaster et al. 1991) we know that the Ca²⁺ concentrations must reach the uM range. In other neuronal systems Ca²⁺ concentrations in the uM range (Müller and Connor 1991) and even in the 200 to 300 uM range (Llinás et al. 1992) have been reported. For these reasons, we suspect that the rapid, high amplitude changes in Ca²⁺ concentration that are necessary in the model to activate the BK channel properly may very well more accurately resemble the real submembrane peaks during Ca²⁺ spikes. However, there is ample evidence for a slower build-up of Ca²⁺ concentration during repetitive Ca²⁺ spiking (Ross et al. 1990; Lev-Ram et al. 1992). These slower transients are not reproduced by our model (Fig. 11C, D), because of the fast time constant of Ca²⁺ removal in the shell. A more sophisticated model of Ca²⁺ removal (Yamada et al. 1989; Sala and Hernández-Cruz 1990) will be necessary to investigate the consequences of this slow increase in Ca²⁺.

Model Robustness. The primary parameters searched in this model involved the locations and densities of the different channels. While the complexity of the current model has not allowed us to undertake a full parameter search as has been done in some other recent modeling studies (Bhalla and Bower 1993), we have explored in detail variations on key parameters to examine the overall robustness of the results. As can be expected from a model that shows so much response variability, it was found to be quite robust to changes in channel densities.

Results were most sensitive to changes in the densities of the CaP, KC and K2 channels. However, these 3 channels are actually tied together, i.e. modifying the density of one of them will change the activation of the other 2 during the dendritic spiking cycle. They are also linked through our simple representation of Ca²⁺ concentration, which drives activation of the KC and K2 channels. For this reason, robustness of the model could be maintained over a wider range of channel densities for even these conductances, provided more than one channel density was changed at once. In fact, the initial process of model tuning primarily involved manually changing the densities of these 3 channels, in an effort to obtain correct levels of dendritic excitability.

Modeling Predictions

Channel Distributions. As mentioned above, the density and distribution of channels in the model were the main uncontrolled variables in these simulations. Experimental techniques do not yet exist to give detailed distribution and density information for all ionic channels in a given cell. Accordingly, by searching parameter space, detailed single cell models can make predictions concerning this critical information (Bhalla and Bower 1993). In the current case, the initial channel distributions were primarily based on the speculations of Llinás and Sugimori (1980a, b). The results of the model largely confirm these predictions with a few modifications.

A CURRENT. One point of contention in the literature about the distribution of ionic channels involves the presence or absence of an A current in the distal dendrites (Hounsgaard and Midtgaard 1988; Chan et al. 1988). One argument for a more extensive distribution is that a depolarizing bias current, which would inactivate A currents also causes the dendrite to fire spikes more easily. However, a similar result would be expected if the depolarization also activated a plateau current as shown in Fig. 11. Moreover, in contrast to the expected 4-AP sensitivity of A currents (Rogawski 1985), the outward current described by Hounsgaard and Midtgaard (1988) was not blocked by 4-AP. In addition, voltage clamp data (Wang et al. 1991) do not support a distal dendritic location of the A current. Patch clamp studies of cultured Purkinje cells (Gruol et al 1991) have shown the A current to be present in both somatic and dendritic membrane, but it is likely that patches were obtained from smooth dendrites only. In the current model an A current was present in the soma and main dendrite. However, dendritic spiking was more influenced by the K2 Ca²⁺-activated K⁺ current, which allowed a finer control of dendritic excitability than the A current, which inactivated fast during long current injections.

Calcium Channel Types. The obvious prominence of Ca²⁺ conductances in the dendrites of Purkinje cells has lead to much discussion, and some disagreement, concerning their nature. In their original report Llinás and Sugimori (1980b) explained the electroresponsiveness of the Purkinje cell dendrite by suggesting that 2 Ca²⁺ conductances would be present, namely a plateau-generating one and a spike-generating one. When several years later different types of Ca²⁺ conductances were identified in other neural systems (Fox et al. 1987) and in the Purkinje cell (Bossu et al. 1989; Fortier et al. 1991) it was proposed that a low threshold Ca²⁺ channel (T type) would be responsible for generating the dendritic plateau. In the mean time, however, Llinás et al. (1989a, b) had discovered a new type of high threshold Ca²⁺ conductance in the Purkinje cell, which they named the P channel. Since then, there has been debate about whether other Ca²⁺ channels than the P channel are present in the Purkinje cell (Llinás et al. 1989a; Fortier et al. 1991; Usowicz et al. 1992b) and about the physiological role of these channels in generating the dendritic plateaus and spikes.

We believe that there is now ample experimental evidence for the presence of both a CaP and a CaT channel in the Purkinje cell. Voltage clamp studies have shown the CaT channel to be present in both young and adult animals, with similar I/V relations (Kaneda et al. 1991; Regan 1991). Also, channel blocking studies with funnel web spider toxin (FTX) have shown that the CaP channel constitutes only about 90% of the total Ca²⁺ conductance (Mintz et al. 1992). This corresponds well to the relative channel densities for CaP and CaT in the dendrites in our model (table 2), which were determined by trial and error before these data became available.

Perhaps of more importance than the presence of these channels however, is their relative contribution to Purkinje cell responses. While it is generally accepted that the CaP channel is responsible for the fast dendritic calcium spikes (Llinás et al. 1989b), it has been suggested that the generation of the longer duration plateau potentials requires the slower kinetics of a channel like the CaT channel (Fortier et al. 1991). Llinás and Sugimori (1992) on the other hand, have argued that the CaP channel is also capable of producing these prolonged potentials providing it with a dual role. As described in detail above, our modeling results clearly support the suggestion by Llinás and Sugimori. Both the dendritic plateaus and spikes in the model were carried by the CaP current. Recent Ca²⁺ imaging results also indicated a common mechanism for plateaus and spiking (Lev-Ram et al. 1992). Our model suggests that the CaP channel can generate plateaus because it has a relatively low threshold of activation (Regan 1991; Usowicz et al. 1992a) compared to other high threshold Ca²⁺ channels (Fox et al. 1987) and because of its incomplete inactivation. In our hands, conductance through the CaT channels contributed only to the rebound spike generation after hyperpolarizations.

Hot Spots. Another source of debate with respect to Ca²⁺ channels has been the uniformity of their distribution. It has been proposed that the Ca²⁺ channels in the Purkinje cell dendrites are clustered in so called hot spots (Llinás and Sugimori 1979; Tank et al. 1988; Bossu et al. 1989; Llinás and Sugimori 1992). The fact that different parts of the Purkinje cell dendritic tree seemed to be able to generate Ca²⁺ spikes with distinctly different shapes suggested this idea (Fujita 1968; Llinás and Nicholson 1971), but Hounsgaard and Midtgaard (1988) did not find any evidence in their dendritic recordings for hot spots at proximal branch points. The hot spot theory seemed subsequently to be supported by Ca²⁺ imaging data which showed some hot spots and phase shifts between the Ca²⁺ rise in spiny versus smooth dendrites (Tank et al. 1988). It was postulated that plateaus are generated in the spiny dendrites, leading to dendritic spikes that would originate at hot spots located at the bifurcations of the smooth dendrites (Sugimori and Llinás 1990; Llinás and Sugimori 1992). Thus the CaP channel would cause both Ca²⁺ plateaus and spikes, but plateaus would be generated in other parts of the dendrite than Ca²⁺ spikes.

While there has been much debate on this point, none of these studies provide direct evidence for clustering of Ca²⁺ channels as the experimental observations could also be explained by other mechanisms (for example dendritic electrotonic structure and local differences in Ca²⁺ release or removal mechanisms). More recent Ca²⁺ imaging data (Ross et al. 1990; Lev-Ram et al. 1992) confirmed that dendritic spikes sometimes occur in individual branches only, but these authors also claim to show no evidence for clustering of Ca²⁺ channels in hot spots. The Lev-Ram et al. (1992) study also showed clearly that the fine, spiny dendrites are always involved in generation of Ca²⁺ spikes. Further, recent immunohistochemical studies of FTX binding sites have shown the CaP channel to be present everywhere in the dendrite and even on spine heads (Hillman et al. 1991). Though there seemed to be higher immunolabeling at bifurcations of the smooth dendrites, there was no evidence in this study for well delineated hot spots.

Given the conflicting data on this issue, we have used uniform Ca²⁺ channel distributions in this model. Our parameters search showed that the model performance did not improve when CaP densities were higher in the smooth dendrites (which include the bifurcation regions) than in the spiny dendrites. With uniform channel densities, the model still generated spatially localized Ca²⁺ spikes (Fig. 10B, C) which subsequently propagated to the rest of the dendrite. Accordingly, our model demonstrated that the response properties of the Purkinje cell to current injection can be simulated without requiring the presence of hot spots.

Fidelity of voltage clamps of Purkinje cells

As described in detail above, constructing a realistic model of a neuron relies on the availability of high quality data, especially from voltage clamp experiments. However, once constructed, the model itself can be used to explore the likely value of the voltage clamp experiments themselves. For example, using voltage clamp techniques, Regehr et al. (1992) have recently proposed that Na⁺ channels are present in the (distal) dendrite as well as the soma of the Purkinje cell. They based this claim upon the fact that they could measure fast inward currents with whole-cell patch clamps of axotomized Purkinje cells, under conditions of good voltage control in the soma. However, it has been suggested that in these experiments the soma itself may not have been adequately clamped (Sugimori and Llinás 1992).

Using the model, we have found that the presence of active dendritic membrane had a dramatic effect on the electrotonic length of the Purkinje cell and thus the ability to achieve an adequate space clamp (Rall and Segev 1985). Passive membrane models predict that Purkinje cells are electrotonically compact, with a length of about 0.3 lambda (Shelton 1985). Rapp et al. (1992) have pointed out that the large synaptic input the Purkinje cell receives could increase this length by a factor of 2.4. Our model shows that during a dendritic spike, the electrotonic distance to the top of the dendrite increased by a factor of 3 compared to a passive membrane dendrite. In between spikes the length was also increased, but less. Thus the state of activation of the dendritic channels also made a significant difference in the electrotonic length of the dendrite. Similarly, activation of dendritic channels also caused a very bad space clamp, especially in the depolarizing direction. We have assumed that the application of Cs⁺ externally and EGTA internally blocked most K⁺ currents completely (Hille 1991). If this was not the case, K⁺ currents would contribute only a small conductance compared to the CaP conductance and similar results would be obtained.

Our results suggest that space clamp is extremely bad in adult Purkinje cells, with a significant divergence from the holding potential at only 50 um from the soma for depolarizing potentials. Therefore, it is possible that the Na⁺ currents recorded by Regehr et al. (1992) are carried by Na⁺ channels located in the very proximal parts of the main dendrite. In our present model, we have confined Na⁺ channels to the soma, because several other studies, including TTX binding (Llinás and Sugimori 1992), dendritic patch recordings (ibid.) and imaging with the fluorescent Na⁺ indicator SBFI (Lasser-Ross and Ross 1992), demonstrated Na⁺ channels only on the soma and axon of the Purkinje cell.

Using the model, we can predict that space clamps could be somewhat improved if Ca²⁺ channels were also blocked, but this was not the case in most reported whole-cell patch clamp studies (Kaneda et al. 1990; Llano et al. 1991b; Regan 1991; Regehr et al. 1992). Note that if the CaP channel does not inactivate, as has been reported by Usowicz et al. (1992b), one cannot remove the Ca²⁺ conductance by steady depolarization as has been claimed in some studies (Llano et al. 1991b). In dissociated Purkinje cells, where large parts of the dendritic tree may be amputated, space clamp can also assumed to be better. Thus the "whole-cell" patch clamp method may be appropriate to measure ionic currents in soma and proximal dendrite in dissociated, cultured Purkinje cells. Equations for several channels used in our model were based on this approach (Hirano and Hagiwara 1989; Kaneda et al. 1990; Regan 1991).

Finally, our ultimate objective in constructing this model was to explore synaptic influences on Purkinje cell firing patterns. The response of this model to different types of synaptic inputs will be presented in the following paper (De Schutter and Bower 1994).

Acknowledgements

We wish to thank especially M. Rapp, I. Segev and Y. Yarom for providing us with the morphological reconstruction of 3 Purkinje cells and for the prepublication release of the parameters for their passive Purkinje cell model, which were very valuable in the early stages of our modeling efforts. We thank also Ö. Bernander, D. Jaeger and J. H. Thompson for useful, often critical comments.

This work was supported by Fogarty fellowship F05 TW04368 to E. De Schutter and National Institutes of Neurological Disorders and Stroke Grant NS31378 and National Science Foundation Grant BIR-9017153 to J. M. Bower.

Address reprint requests to E. De Schutter.

Received 27 May 1993; accepted in final form 17 September 1993.

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