Publications
An Active Membrane Model of the Cerebellar Purkinje Cell : II. Simulation of Synaptic Responses
JOURNAL OF NEUROPHYSIOLOGY
Vol. 71, No. 1, 401-419 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
Abstract
1. Both excitatory and inhibitory postsynaptic channels were added to a previously described complex compartmental model of a cerebellar Purkinje cell (De Schutter and Bower 1994), in order to examine model responses to synaptic inputs. All model parameters remained as described previously, leaving maximum synaptic conductance as the only parameter that was tuned in the studies described in this paper. Under these conditions, the model was capable of reproducing physiological recorded responses to each of the major types of synaptic input.
2. When excitatory synapses were activated on the smooth dendrites of the model, the model generated a complex dendritic Ca2+ spike similar to that generated by climbing fiber inputs. Examination of the model showed that activation of P-type Ca2+ channels in both the smooth and spiny dendrites augmented the depolarization during the complex spike and Ca2+-activated K+ channels in the same dendritic regions determined the duration of the spike. When these synapses were activated under simulated current clamp conditions, the model also generated the characteristic dual reversal potential of the complex spike (Llinás and Nicholson 1976). The shape of the dendritic complex spike could be altered by changing the maximum conductance of the climbing fiber synapse and thus the amount of Ca2+ entering the cell.
3. To explore the background simple spike firing properties of Purkinje cells in vivo, excitatory "parallel fiber" synapses were added to the spiny dendritic branches of the model. Continuous asynchronous activation of these granule cell synapses resulted in the generation of spontaneous sodium spikes. However, very low asynchronous input frequencies produced a highly regular, very fast rhythm (80 to 120 Hz), while slightly higher input frequencies resulted in Purkinje cell bursting. Both types of activity are uncharacteristic of in vivo Purkinje cell recordings.
4. Inhibitory synapses of the sort presumably generated by stellate cells were also added to the dendritic tree. When asynchronous activation of these inhibitory synapses was combined with continuous asynchronous excitatory input, the model generated somatic action potentials in a much more stochastic pattern typical of real Purkinje cells. Under these conditions, simulated interspike interval distributions resembled those found in experimental recordings. Also,as with in vivo recordings, the model did not generate dendritic bursts. This was mainly due to inhibition which suppressed the generation of dendritic Ca2+ spikes.
5. In the presence of asynchronous inhibition, changes in the average frequency of excitatory inputs modulated background simple spike firing frequencies in the natural range of Purkinje cell firing frequencies (30 to 100 Hz). This modulation was very sensitive to small changes in the average frequency of excitatory inputs. In addition, changes in inhibitory frequency caused a parallel shift of the relationship between excitatory input and spiking frequency. Because of the specific cerebellar circuitry, inhibitory inputs may allow Purkinje cells to detect small fluctuations in excitatory input at any mean frequency of input.
6. When climbing fiber input was given in the presence of background asynchronous excitatory and inhibitory inputs, the shape of the complex spike in the soma was significantly affected. However, the shape of the spike in the dendrites was almost constant. This difference reflected the more variable excitability of the soma compared to the dendrites.
7. Synchronous activation of basket cell inhibitory synapses during asynchronous activation of granule and stellate cell synapses interrupted somatic spiking. However, the hyperpolarization caused by the basket cell synapse did not penetrate far into the dendrite but stayed localized to the soma and main dendrite.
8. This simulation work demonstrates that a model based on voltage clamp data and tuned entirely on the response of Purkinje cells to current injection, is capable of reproducing a wide range of synaptically activated responses. In the presence of continuous granule cell excitation, the model showed a stable "in vivo" state, different from the silent, resting "in vitro" state. Further, the model suggests that there may be important functional interactions between different types of synaptic inputs. In particular, it makes several specific predictions about the role of stellate cell inhibition.
Introduction
The dendrite of the cerebellar Purkinje cell may be one of the most complex electrical structures in the mammalian nervous system (Llinás and Sugimori 1992). In the previous paper, we have used a relatively complex model of the Purkinje cell to explore the intrinsic electrical properties of this structure. In this paper, we begin the process of investigating the synaptic organization of this unusual dendrite.
There are many features of the organization of synaptic inputs onto the cerebellar Purkinje cell that make it unique. For example, with 150,000 to 175,000 excitatory synaptic connections, this neuron is known to receive more synaptic inputs than any other neuron in the brain (Ito 1984; Shepherd 1990; Harvey and Napper 1991). The vast majority of these synapses arise from the granule cells which are packed in an extremely dense cell layer below the Purkinje cell. In contrast to this tremendous number of inputs, the other source of excitatory input to each Purkinje cells consists of a single climbing fiber, i.e. an axonal projection from the inferior olive, a brainstem nucleus (Ito 1984). The possible interactions between the huge convergence of the granule cell pathway and the unitary influence of the climbing fiber has formed the basis for many theories of cerebellar function (Marr 1969; Albus 1971; Ito 1984).
There are several other interesting points of contrast between the parallel fiber and climbing fiber systems. For example, each contacts a different and non-overlapping region of the Purkinje cell dendrite. Climbing fiber synapses are made on the thick, smooth dendrites of the cell while the granule cell synapses are restricted to the small spine covered tertiary dendrites referred to as the spiny branchlets (Palay and Chan-Palay 1974). It is also known that each of these systems generates different postsynaptic effects in the Purkinje cell. Climbing fiber activation evokes a large all or none Ca2+-dependent dendritic action potential (Llinás and Sugimori 1980b; Knöpfel et al. 1991; Miyakawa et al. 1992), while granule cell inputs are believed to generate more classical dendritic EPSPs and to produce fast sodium spikes as somatic output.
When extracellular recordings are made from Purkinje cells in vivo, both types of spiking behavior are observed. Complex dendritic action potentials resulting from climbing fiber activation are seen occurring at relatively low frequencies of 1.0-2.5 Hz (Armstrong and Rawson 1979), while simple spike responses influenced by granule cell synaptic input occur over the range of 30 to 100 Hz (Murphy and Sabah 1970; Ferin et al. 1971; Armstrong and Rawson 1979). Both types or responses are quite variable in their frequency over time. Both also occur "spontaneously" i.e. in the absence of obvious peripheral stimulation, although they can also be driven by peripheral stimulation (Bower and Woolston 1983). Taken together, these two types of Purkinje cell responses constitute the output of each cell, and therefore the sole output of cerebellar cortex.
In this paper, we have sought to replicate Purkinje cell activity in response to each of these classical excitatory synaptic inputs by adding climbing fiber and parallel fiber synaptic inputs to the previously described model (De Schutter and Bower 1994). To succeed, however, it was also necessary to add inhibitory synaptic inputs originating from cerebellar cortical stellate and basket cells (Ito 1984). The results reveal that the basic response properties of the Purkinje cell can be obtained from the original model tuned only on current and voltage clamp data, by simply adding these synaptic conductances. In this form, the model was able to replicate the complex dendritic response to climbing fiber activation, predicting the combination of dendritic membrane currents likely to be involved. The model was also able to replicate the basic spontaneous simple spike firing properties of Purkinje cells in vivo. predicting that this behavior results from a complex interaction of both excitatory and inhibitory synaptic influences. As such, this paper further establishes the capacity of this model as well as laying the groundwork for exploring the much more complex stimulus activated responses of the Purkinje cell (Bower and Woolston 1983).
Methods
All simulations described here were done with the model described in the accompanying paper (De Schutter and Bower 1994). Most simulation runs were performed using the parameters of model PM9, while some were performed using PM10. However, except for these two variations, the kinetics of voltage dependent channels, and the distribution and densities of these channels were unchanged for all the present simulation runs. For this reason, the reader is referred to the accompanying paper for a detailed description of these model features.
The following sections briefly recapitulates the basic properties of each type of synaptic input and then the features added to the model to explore Purkinje cell responses following its activation. In particular, we have added conductances intended to represent excitatory input from granule cells and climbing fibers and inhibitory inputs from intrinsic inhibitory neurons. Both excitatory and inhibitory synaptic conductances were described as dual exponential functions (Wilson and Bower 1989) and the amplitude of these conductances were the only parameters modified between simulation runs. All simulations were build in GENESIS and were run on 8 Sun Sparc2 workstations.
Climbing fiber synapses
Synaptic locations.
It is well known that in adults the climbing fiber input to a single Purkinje cell is provided by a single neuron in the inferior olive (Ito 1984) and that this contact is made on the smooth branches of the Purkinje cell dendritic tree, from close to the soma up to almost the top of the tree (Palay and Chan-Palay 1974). We modeled the climbing fiber input by placing synapses from a single axon on all compartments of the main and smooth dendrites of the cell (see De Schutter and Bower 1994, Fig. 1).
Synaptic pharmacology and kinetics.
Activation of the climbing fiber synapse evokes an all or none massive response, called the complex spike (Eccles et al. 1966; Llinás and Nicholson 1976) which involves activation of dendritic Ca2+ channels (Llinás and Sugimori 1980b; Knöpfel et al. 1991; Konnerth et al. 1992; Miyakawa et al. 1992). Climbing fiber inputs are known to be mediated by AMPA-receptors (Knöpfel et al. 1990; Llano et al. 1991b). In the model, activation kinetics were based upon data from spinal cord and hippocampal neurons (Nelson et al. 1986; Forsythe and Westbrook 1988; Holmes and Levy 1990), with an opening time constant of 0.5 ms and a closing time constant of 1.2 ms (peak at 0.8 ms). Ca2+ inflow through this receptor was not modeled though it might be present (Brorson et al. 1992). This synapse had a reversal potential of 0 mV (Mayer and Westbrook 1987; Cull-Candy and Usowicz 1989).
Synaptic conductance.
The conductance of each climbing fiber synaptic connection is unknown and was therefore treated
as a free parameter in the model. Model tuning resulted in the selection of two different values
of maximum synaptic conductance
(
): 7.5 mS/cm2 for the main
dendrite and 15 mS/cm2 for the smooth dendrites. Assuming a total of 300 synaptic contacts (Ito
1984), each with the same synaptic conductance, our
values correspond to a
conductance of 6.2 nS at each synaptic contact.
Synaptic specializations.
Climbing fiber synapses are known to occur on a specialized "stubby protuberance" with a short, smooth stem (Palay and Chan-Palay 1974). We did not, however, explicitly model these structures as we do not expect their absence in the model to affect the voltage transients caused by climbing fiber activation. Such smooth spines are unlikely to result in much signal attenuation (Rall and Segev 1990) and all spines on any particular compartment receive synchronously the same synaptic input from a single climbing fiber (Ito 1984).
Synaptic activation.
The climbing fiber synapses were fired as a volley, with the main dendrite being activated before the more distal smooth dendrites. The delay between first and last activated synaptic conductance was 0.9 ms (Llinás and Nicholson 1976).
Granule cell synapses.
Synaptic locations.
In the model we have placed granule cell synapses on all dendrites with diameters less than 3.17 um (Palay and Chan-Palay 1974). These synapses are made on dendritic spines (Harris and Stevens 1988) which have been explicitly modeled for each synaptic contact (see below).
Synaptic pharmacology and kinetics.
These excitatory inputs to the Purkinje cell are also mediated by AMPA-receptors (Garthwaite and Beaumont 1989; Farrant and Cull-Candy 1991; Lambolez et al. 1992), as well as metabotropic receptors (Blackstone et al. 1989; Vranesic et al. 1991; Glaume et al. 1992). Accordingly, we modeled this excitatory input with the same kinetics as for the climbing fiber, again ignoring possible Ca2+ associated effects.
Synaptic conductance.
The maximum conductance of a single parallel fiber synapse is also unknown, but seems to be
variable (Ito 1989; Hirano and Ohmori 1989). We used a value of 0.7 nS for
, which falls in the range of
values (0.2 nS to 50 nS) which have been used in other models using glutamate synaptic
transmission (Zador et al. 1989; Miller et al. 1985; Rapp et al. 1992; Holmes and Woody 1989;
Lytton and Sejnowski 1991; Wehmeier et al 1989; Wilson and Bower 1989). This synapse also had a
reversal potential of 0 mV (Mayer and Westbrook 1987; Cull-Candy and Usowicz 1989).
Numbers of synapses.
It is known that Each Purkinje cell receives about 150,000 granule cell synapses (Harvey and Napper 1991). Given computing resources, it was not possible in the current simulation to model all these inputs. Accordingly, in most of the simulations presented in this paper, granule cell inputs were delivered on 1474 spines (1% of real), i.e. one on each modeled spiny dendritic compartment. This simplification has two consequences. First, there are a large number of dendritic spines missing from the model. This was compensated for by adding dendritic membrane (see Holmes and Woody 1989; Rapp et al. 1992). Second, many synaptic inputs are missing. Under the conditions of random, asynchronous inputs simulated here, we have compensated for this missing input by increasing the firing rate of each synapse. A similar approach has been taken by other Purkinje cell modelers (Rapp et al. 1992). Assuming a linear scaling and simulating 1% of the inputs, an asynchronous firing rate of 1 Hz in the model would thus correspond to an average firing rate of about 0.01 Hz for real parallel fibers. We will show in the Results section that this scaling was appropriate for the current model. All synaptic input firing rates mentioned are unscaled unless explicitly stated.
Synaptic specializations.
Granule cell synaptic inputs are known to terminate on dendritic spines (Palay and Chan-Palay 1974; Harvey and Napper 1991). Based on EM-reconstructions of rat Purkinje cell spines (Harris and Stevens 1988) we assumed a density of 13 spines per 1 um dendritic length, resulting in a total of 144,456 spines. Most of these spines have narrow diameter spine necks and it is likely that they have a significant effect on the electrical effects of granule cell synaptic inputs (Rall and Segev 1990). For this reason, when granule cell synaptic effects were studied in the model, spines were explicitly simulated using 2 compartments each. One compartment represented a spherical spine head with a diameter of 0.54 um, while the second compartment represented the cylindrical spine neck with a diameter of 0.20 um and a length of 0.66 um (Harris and Stevens 1988). These dimensions resulted in a membrane surface per spine of 1.33 um2.
Reducing the number of granule cell inputs in the model also had the effect of reducing the total number of spines. Given the enormous number of these spines on a normal Purkinje cell, simply leaving their membrane area out of the model would significantly affect simulation results. In our previous efforts to model the electrical (not synaptic) properties of the Purkinje cell (De Schutter and Bower 1994), the influence of these dendritic spines was approximated by increasing the membrane surface of compartments of the region of the dendrite on which spines occur (Holmes and Woody 1989; Jaslove 1992; Rapp et al. 1992). The same approach was used here when granule cell synaptic effects were not being simulated. However, for those simulations that did relate to granule cell input, the total membrane surface was kept constant by subtracting the membrane surface of modeled spines (i.e. 1.33 um2 for each spine) from the membrane surface of the dendritic compartment they were connected to (eq. 1). For a spiny dendritic compartment with length L and diameter D , the membrane surface (S in um2), used for the computation of compartment membrane capacitance and resistance (Holmes and Woody 1989; Rapp et al. 1992), depended thus also on the number of simulated spines (Ns) connected to that compartment.
Eq. 1
As already mentioned, when no granule cell inputs were modeled, the simulation was run without spines (i.e. Ns = 0), as was done in the preceding paper (De Schutter and Bower 1994).
Synaptic activation.
Parallel fiber synaptic input was modeled as an asynchronous process (Bernard and Axelrad 1991). Activation of each modeled synapse was triggered independently by a random number generator, which generated a Poisson distribution around a mean frequency of input.
Stellate cell synapses
Synaptic locations.
Morphological data about stellate cell synapse distributions on Purkinje cell dendrites are incomplete. Based on EM-data they seem to synapse on both small spiny and thicker dendritic branches (Palay and Chan-Palay 1974) and in general they do not contact the innervation sites of basket cell synapses (ibid.), but the density of these synaptic contacts or their exact sites of termination are not known. Because there are more stellate cells than Purkinje cells (Ito 1984), we assumed that stellate cell synapses were present on all the smooth and spiny dendritic compartments. We placed 2 stellate cell synaptic contacts on the shaft of every smooth dendritic compartment and 1 contact on every spiny dendritic compartment, resulting in a total of 1695 stellate cell synapses in the model.
Synaptic pharmacology and kinetics.
Stellate cell inhibition is mediated by GABAA-receptors (Ito 1984; Gabbot et al. 1986; Llano et al. 1991a). The kinetics for this inhibitory synaptic conductance were based on recordings of miniature inhibitory synaptic currents in pyramidal neurons of the hippocampus (Ropert et al. 1990), with an opening time constant of 0.9 ms, a closing time constant of 26.5 ms (peak at 3.2 ms) and a reversal potential of -80 mV.
Synaptic conductances.
Postsynaptic conductances on smooth dendritic compartments had a
of 1.4 mS/cm2, conductances on
a spiny dendritic compartment had a
of 7 mS/cm2.
Synaptic activation.
Stellate cell synapses were fired asynchronously, following a Poisson distribution around a mean frequency of input. There was no relation between the timing of excitatory and inhibitory inputs.
Basket cell synapses
Synaptic locations.
About 30 to 50 basket cells terminate on each Purkinje cell on the soma and dendritic trunk (Palkovits et al. 1971; Palay and Chan-Palay 1974). Basket cell synaptic receptor channels were placed on the soma and on all compartments of the main dendrite (see De Schutter and Bower 1994, Fig. 1).
Synaptic pharmacology and kinetics.
Basket cell inhibition is also mediated by GABAA-receptors (Ito 1984; Gabbot et al. 1986). We used the same kinetics as for the stellate cell synapses.
Synaptic conductances.
As with the excitatory synaptic inputs, the amplitude of the conductance of inhibitory synapses
are not known but the strength of these inputs is assumed to vary (Bishop 1992). Basket cell
synapses on the soma had a of
100 uS/cm2 (total conductance 139 uS) and on the dendrite a
of 50 uS/cm2 (total
conductance for main dendrite 47 uS) for a single input.
Synaptic activation.
In simulations of basket cell activity 20 such basket cell synapses were fired synchronously to provide a strong inhibitory input.
Results
The basic anatomical and physiological features of the model described here were tuned to simulate the response of Purkinje cells to current injection in slice. This process and its results are described in detail in the accompanying paper (De Schutter and Bower 1994) and will not be repeated here.
Response to climbing fiber stimulation
After constructing a complex model, it is essential to first test if the model can reproduce a
well defined physiological response, different from the neuronal properties used to initially tune
the model. Ideally, this response should also involve the manipulation of relatively few model
parameters. We have used the fairly well characterized Purkinje cell response to a climbing fiber
input (Llinás and Nicholson 1976; Ito 1984) to test the basic response properties of the
model. Because both the distribution and the kinetics of the climbing fiber synaptic receptors are
well known, the only free parameter in implementing a climbing fiber synapse in the model was the
maximum conductance () of the
synaptic current. Further, there is Ca2+ imaging data to compare to model results
(Konnerth et al. 1992; Miyawaka et al. 1992).
Voltage responses.
Fig. 1A shows the response of the model to a climbing fiber input. The model reproduced the typical voltage records of the complex spike well. In the soma there was a depolarization of about 20 ms, with 3 somatic action potentials riding on top during the first 10 ms. In the distal smooth and spiny dendrites a large Ca2+ spike could be observed. The main dendrite showed a combination of the 2 patterns. Such combinations have been reported experimentally using intradendritic recordings in vitro (Llinás and Sugimori 1980b; compare with their fig. 1A) and in vivo (Ekerot and Oscarsson 1981).
Figure 1
Simulation of complex spike in slice.
A: Membrane potential before (-68 mV) and during activation of the climbing fiber synapse
is shown as it would be recorded in (from top to bottom) the soma, the main dendrite, a smooth
dendrite and a spiny dendrite. See De Schutter and Bower (1993), Fig. 1 for the specific locations
of these recording sites in the model.
B: Effect of changing
of the climbing fiber synapse. Complex spike in the soma (top) and smooth dendrite with
at half of normal. C:
Same as B with double
of normal.
Complex spikes could be evoked over a wide range of
values for the synaptic
conductance, but for the complex spike to result in the generation of a few somatic action
potentials, we had to constrain
to a value within 50 % of the final value used and
in the main dendrite had to be
smaller than in the more distal dendrite. Fig. 1 also shows how the shape of the complex spike
changed in both the soma and the dendrite when
of the climbing fiber synaptic
conductance was halved (Fig. 1B) or doubled (Fig. 1C). Such variability in the shape
of the complex spike has been observed in vivo (Campbell and Hesslow 1986).
Dual reversal of the complex spike.
Because the complex spike results from activation of a group of distributed synapses, it does not reverse at a unique membrane potential (Llinás and Nicholson 1976; Llinás and Sugimori 1980a; Kimura et al. 1985). The long electrotonic length of the Purkinje cell dendrite, caused to a large degree by the presence of active channels (De Schutter and Bower 1994), attenuates steady state depolarizations so that proximal parts of the dendrite will attain the reversal potential of the synapse at lower somatic depolarization levels than distal parts. This gives rise to a biphasic reversal potential (Llinás and Nicholson 1976).
Figure 2
Simulation of the dual reversal of the climbing fiber synapse. Membrane potential in the soma
before and during a complex spike is shown under continuous current injections in the soma
(current level indicated for each trace). See text for an explanation of the dual reversal.
Fig. 2 shows simulations of the dual reversal potential with our model. The model was allowed to stabilize after the start of the depolarizing current injection (more than 5 sec simulated time) and then the climbing fiber synapses were activated. The model behaved exactly like Purkinje cells in slice, even the current amplitudes necessary to reverse the complex spike were similar (Llinás and Nicholson 1976). At low levels of current injection, the peak of the EPSP was at first delayed. When a 14.5 nA current was simulated, the rising part of the EPSP started to reverse while it required about 20 nA to complete the reversal of the EPSP. Note that in the model the reversal potential of the climbing fiber synapse is 0 mV, but the EPSP nevertheless only started to reverse at a depolarization of the soma to 30 mV. This was similar to the drop in holding potential seen during voltage clamps of the soma, see Fig. 13C of De Schutter and Bower (1994). Note also that the large depolarizations used in these reversal simulations resulted in an inactivation of the Na+ currents and therefore no somatic spikes were generated by the climbing fiber input. This is also similar to experimental data (Llinás and Nicholson 1976).
Figure 3
Direct comparison of the response to climbing fiber activation in an active membrane and passive
membrane model. The responses of the 2 models were superimposed. The active membrane responses are
identical to those in Fig. 1 and the same recording sites were used.
Contribution of active membrane.
It is known that dendritic Ca2+ channels are activated during a complex spike (Knöpfel et al. 1991; Konnerth et al. 1992; Miyawaka et al. 1992), but it has not been possible to show experimentally how much they contribute to the actual voltage transient. Fig. 3 compares the response evoked by a climbing fiber input of the model with active membrane to the same model with passive membrane (i.e. the conductance of all ionic channels was set to zero). In the passive membrane model, the climbing fiber input caused a large EPSP, which had a lower amplitude and a longer duration than the complex spike. The dendritic Ca2+ conductance in the active membrane model therefore resulted in a larger depolarization and, by activating Ca2+-activated K+ channels, a faster repolarization.
Figure 4
Dynamic change in model variables during a complex spike at 4 representative locations in the
model. A: soma.
B: main dendrite.
C: smooth dendrite.
D: spiny dendrite.
Each part of the figure shows the membrane potential (upper trace), the Ca2+
concentration (middle trace) and the amplitude of all ionic currents (superimposed lower traces)
in the compartment. In B and C the synaptic current is also shown (Syn). The scale
bars for voltage and concentration are identical in all sections, for the currents each section is
scaled differently (outward currents upward). The model was at a stable resting membrane potential
of -68 mV before the complex spike.
The specific interaction of modeled synaptic and ionic currents and the resulting change in
Ca2+ concentration over time during a climbing fiber potential are shown in more detail
in Fig. 4. The synaptic currents (Syn in Fig. 4B, C) causing the climbing fiber were
large, as could be expected from the synchronous activation of 300 synapses (Ito 1984). This
synaptic activation resulted directly in the activation of CaP channels over large regions of the
dendrite. The CaP channels were responsible for almost all of the resulting additional
depolarization, as the small number of CaT channels remained inactive. It should also be noted
that the CaP currents activated with a delay compared to the synaptic current which had a short
time course and was shut off before the second somatic spike. The complex spike ended by
activation of the KC channel in the whole dendritic tree (Fig. 4B, C, D); the
other Ca2+-activated K+ channel (K2) did not play an important role during
the complex spike, due to its comparatively small
(De Schutter and Bower 1994,
table 2).
The contribution of the soma during the generation of a complex spike was mainly the firing of action potentials, as can be seen in Fig. 4A. The synaptic currents generated by the distributed climbing fiber input depolarized the Purkinje cell soma fast enough to cause an initial somatic spike with almost no delay. This can be seen in the nick in the synaptic current profile in Fig. 4B which was caused by a sudden drop in the driving potential for the synapses during the first somatic spike. The somatic currents activated during the somatic spike were the NaF and Kdr channels, the other somatic channels were mostly inactive during the complex spike.
Distribution of Ca2+ channel activation.
Figures 1, 3, and 4 demonstrate that while climbing fiber synapses are located on the largest dendritic branches, activation of the CaP channel occurs not only in these regions but also in the smaller spiny dendrites. Fig. 5 demonstrates the spatial distribution of the membrane depolarization and Ca2+ entry during the complex spike in more detail. This figure also shows that the Ca2+ spike in the spiny branchlets peaked at a time different from the spike in the smooth dendrite.
Figure 5
False color representation of membrane potential and Ca2+ concentration in the complete
model during complex spike.
A: membrane potential 1.4 ms after begin of complex spike.
B: membrane potential 4.0 ms after begin of complex spike.
C: membrane potential 10.0 ms after begin of complex spike (after the last somatic action
potential).
D and E: submembrane Ca2+ concentration at same times as A and
B respectively.
F: the complex spike in the soma (red) and distal dendrite (green) with the time at which
images A-C were taken indicated by white bars. Note the non-linear [Ca2+]
scales.
Fig. 5A shows the membrane potential 1.4 ms after the firing of the synapse, the yellow and green parts of the dendrite correspond approximately to the distribution of the synaptic contacts over the model. Fig. 5B shows that 2.6 ms later the whole dendrite had depolarized, although not evenly as, especially in the more distal dendrites, there was a mosaic of different levels of depolarization. This was partially a snapshot effect, i.e. these locations in the dendrite would peak at different times (cf. Fig. 1). However, Ca2+ spikes tended to be relatively localized, with different time profiles depending on the location. Similarly, some dendritic branchlets repolarized before others (Fig. 5C).
The submembrane Ca2+ concentrations revealed an uneven pattern of elevation, similar to the mosaic pattern of dendritic depolarization, 4 ms after the climbing fiber input (Fig. 5E). Note, however, that in the dendrites there was a noticeable difference in the details of the voltage profile compared to the Ca2+ profile. For example, the bump in the membrane potential seen in Fig. 4C was absent in the corresponding Ca2+ concentration record. Conversely the first peak in Ca2+ concentration in Fig. 4D was not present in the voltage trace.
Simulating of Purkinje cell background activity in vivo
All the simulations presented so far showed the response of our model when it was stimulated from a stable resting membrane potential. We propose that this state is comparable to slice conditions, where it is reasonable to assume that there is little or no background synaptic granule cell input and most cells do not fire unless stimulated by current injection or synaptic input (Llinás and Sugimori 1980a; Hounsgaard and Midtgaard 1988). Extracellular recordings of Purkinje cells in vivo,however, are quite different (Llinás 1981; Bower and Woolston, 1983). In vivo, Purkinje cells fire simple spikes continuously with an irregular rhythm of between about 30 to 100 Hz (Murphy and Sabah 1970; Ferin et al. 1971; Armstrong and Rawson 1979; Llinás 1981). Also, while Purkinje cells often burst in slice recordings, spontaneously or during current injection (Llinás and Sugimori 1980a), they never burst in vivo under normal conditions except during a complex spike (Murphy and Sabah 1970; Crepel 1972; Armstrong and Rawson 1979; Montaloro et al. 1982; Ito 1984). Having established that the model is capable of generating complex dendritic spikes in response to climbing fiber input, the following sections demonstrate that the model was also capable of generating these in vivo activity patterns in the presence of continuous asynchronous background synaptic input from granule cells and stellate inhibitory interneurons.
Figure 6
Response of the Purkinje cell model to asynchronous excitation by 1474 granule cell synapses,
firing at the average frequencies indicated. Membrane potential in the soma is shown, bars at the
left of each trace indicate -50 mV membrane potential.
Response to asynchronous excitation alone.
Fig. 6 shows the response of the model when only random excitatory granule cell inputs were applied at the average frequencies shown. The results demonstrate that the model generated spontaneous simple spikes under these conditions. However, while the frequency of simple spike output was sensitive to the rate of spontaneous parallel fiber input, several other features of this response were not realistic. First, the minimum firing frequency was much too high, i.e. inputs at only 0.1 Hz that were large enough to activate somatic spiking caused the model to fire at about 80 Hz, while increasing the frequency of the input to 1.0 Hz forced the Purkinje cell firing frequency up to 120 Hz only. Considering the scaling by 100 of the synaptic input frequency (see Methods), this would mean that if parallel fibers had an average real firing frequency of 0.01 Hz or more, Purkinje cells would be maximally activated all the time under these conditions. Fig. 6 also demonstrates that when even faster input frequencies were applied (3 Hz or more), the Purkinje cell started bursting. This type of behavior is never seen in healthy Purkinje cells recorded in vivo, although, as already discussed it is seen in slice preparations (Midtgaard 1992).
A second important difference between the model and real Purkinje cell activity under these conditions was that the model did not fire stochastically. Fig. 6 shows that there was very little variability in the interspike intervals when only granule cell inputs were present. Spontaneous activity recorded from Purkinje cells in vivo actually reveal a Poisson distribution in interspike intervals (Figure 8B and Bower and Woolston, unpublished observations).
Figure 7
Response of the Purkinje cell model to combined asynchronous excitation and inhibition. All the
stellate cell synapses fired at an average rate of 1 Hz, the 1474 granule cell synapses fired at
the average frequencies indicated. Membrane potential in the soma is shown, bars at the left of
each trace indicate -50 mV membrane potential.
Response to asynchronous excitation and inhibition.
Fig. 7 presents the model's response when both asynchronous excitatory and inhibitory inputs were provided. Under these conditions, it can be seen that the model behaved quite differently and much more realistically. For example, the model showed a complete range of firing frequencies, from less than 1 Hz to 190 Hz (not shown, 250 Hz excitatory input). For firing rates between 0 and 100 Hz the firing pattern was stochastic, with quite variable interspike intervals while intervals became more regular at higher rates of spiking. While it is true that even relatively low levels of inhibitory input frequencies (1 Hz), required much higher excitatory input frequencies to cause somatic spikes (21 Hz in Fig. 7), these frequencies are reasonable considering the scaling factor (see Methods).
Figure 8
Comparison of simulated and experimental interspike interval distributions.
A: ISI of simulated response to asynchronous excitation and inhibition.
B: ISI constructed from in vivo extracellular Purkinje cell recording in crus IIa of
the rat. Both ISIs have 1 ms bin widths.
Fig. 8 compares an ISI constructed from spontaneous Purkinje cell activity to the results from a simulation. It can be seen that the distribution of the experimental ISI was very similar to that generated by the model. The fact that the model could generate the normal stochastic firing pattern of Purkinje cells and reproduce experimental ISIs, made us confident that our simulation of background parallel fiber and inhibitory inputs was realistic.
Finally, in the absence of climbing fiber input, Purkinje cells are not generally believed to fire dendritic Ca2+ spikes in vivo. When both excitatory and inhibitory asynchronous input was provided, the model only generated dendritic Ca2+ events in response to very high and almost certainly unrealistic input rates (300 Hz or more continuously). Further, these bore little resemble to classical Purkinje cell responses to climbing fiber input. Accordingly, the presence of both excitatory and inhibitory synaptic input, alleviated the abnormal dendritic spiking generated in the model by excitatory granule cell input alone (Fig. 6).
Figure 9
Frequency response of the Purkinje cell model to different levels of excitation and inhibition.
A: frequency response curve for the PM9 model. The average firing frequency of the model is
shown for different rates of asynchronous excitation and inhibition. The 5 curves correspond to
the different frequencies of stellate cell inhibition indicated. Firing frequencies were computed
over the interval 1-2 sec simulated time for firing frequencies above 50 Hz and over the interval
2-5 sec for lower frequencies. Note that these intervals were too short to completely average out
the noise.
B: frequency response curve for the PM10 model.
C: interspike distributions of model PM9, firing at the same average frequency of 67 Hz,
caused by 4 different combinations of asynchronous excitatory and inhibitory input frequencies
(10.4 Hz/0.5 Hz, 23.5 Hz/1 Hz, 37 Hz/1.5 Hz and 50 Hz/2 Hz). The 4 ISIs are superimposed to
facilitate comparison. Each ISI based on 1600 intervals, bin width was 0.5 ms.
Frequency response of the model.
We have used the model to explore in detail the possible relationship between the average rates of excitatory and inhibitory synaptic inputs and the frequency response of the Purkinje cell. This analysis has shown that the modeled neuron was extremely sensitive to these influences. Fig. 9A shows response curves of model PM9 to different rates of excitatory and inhibitory inputs. Each curve shows the firing rate for varying frequencies of excitation with a constant frequency of inhibition. The response curves were quite steep for Purkinje cell firing frequencies below 100 Hz, with a sharp onset. At about 100 Hz there was a rather pronounced inflection in the curves and the firing rate then slowly increased to a maximum of 190 Hz. Note that the steep part of the curves corresponds to the normal firing frequency of Purkinje cells in vivo (30 to 100 Hz; Llinás 1981). In this physiological firing range, the Purkinje cell model had a narrow window of maximal responsiveness, where small changes in excitatory inputs caused large shifts in firing frequency. Outside of this window the model was either silent, or fired at rates faster than 100 Hz and was much less sensitive to its inputs.
Changing the average frequency of the inhibitory inputs caused a parallel shift of the response curve for granule cell excitatory inputs, to the left for lower frequencies and to the right for higher ones (Fig. 9A, B). It also slightly changed the slope of the curve, making it less steep at higher inhibitory input frequencies. However, the differences in slope were minimal within the 30 to 100 Hz physiological range. If both excitatory and inhibitory input frequencies were changed concomitantly, so that the model fired at the same average frequency, then the detailed firing pattern of the model remained unchanged, resulting in almost identical ISIs (Fig. 9C).
Fig. 9B shows response curves for model PM10, which has more K+ channels in the soma and proximal dendrite (De Schutter and Bower 1994). It can be seen that even with this variation, the basic properties described for model PM9 were also present, i.e. there was an initial sharp rise to about 100 Hz and changing the rate of inhibition shifted the curve to left or right. The response curve was not as steep as for model PM9, because the membrane in PM10 is leakier and input resistance lower.
Scaling of parallel fiber excitation.
As mentioned in the Methods section, only a small number of the inputs present on a Purkinje cell were simulated in the model. We have assumed that one can compensate for missing inputs by increasing the average firing frequency (c.f. Rapp et al. 1992). Nevertheless we have explicitly used the model to examine whether the relation between average frequency and the number of excitatory inputs was linear. If the scaling was linear, we would expect that the relation between input and output frequency (Fig. 9) would be identical for all cases where the number of inputs multiplied by average input frequency was constant.
Figure 10
Effect of scaling of number of inputs and of
on simulated ISIs.
A: Interspike interval distributions for variable number of asynchronous modeled spines.
The ISIs have been superimposed to show similarity. Simulations were: 500 spines firing at 90 Hz,
1500 spines at 30 Hz, 3000 spines at 15 Hz, 4500 spines at 10 Hz, 9000 spines at 5 Hz and 14000
spines at 3.21 Hz. Each ISI is constructed of 800 events, bin size 1 ms.
B: Interspike distributions for 4 different maximum conductances of the granule cell
synapse. Simulations were: 1 nS at 21 Hz, 0.7 nS at 30 Hz, 0.35 nS at 60 Hz and 0.1 nS at 210 Hz.
Inhibition was always asynchronous at 1 Hz. Each ISI is constructed of 1600 events, bin size 1
ms.
To test if this was in fact the case, we simulated 6 versions of the model with from 500 to 14,000 of the spines and the corresponding synaptic inputs present. This represents between 0.3 to 9% of the 150,000 granule cell inputs on a Purkinje cell (Harvey and Napper 1991). For these measures, the cell was excited at an average firing frequency of 30 Hz when 1500 spines were used and inhibited at 1 Hz. We found that the 6 models fired at the same average firing frequency of 111 Hz. Fig. 10A shows the ISIs for these 6 models superimposed. This sensitive measure of neuronal response was nearly identical in each case. The only difference between the histograms was the exact location of the top of the peak. This difference, however, did not show any systematic variation with the number of spines simulated and could be explained by chance, as each of these ISIs were constructed from only 800 events. The computation times required to compute responses in the larger models (the 14,000 spine model had almost 30,000 compartments!) prevented us from collecting more events. For the same reason, relatively high excitatory input rates were used, resulting in the 111 Hz firing frequency. At this level of excitation, the model showed a bimodal ISI.
We conclude that the scaling of number of inputs versus firing frequency seemed to be linear for the range examined (i.e. for 10% of the inputs or less). Further, the sensitivity of the model to input frequencies was also extremely robust.
The actual conductance of a parallel fiber synapse is unknown. Therefore we also tested the
scaling of average frequency versus
for a linear scaling (Fig.
10B). We again expected that the model response would be identical in simulations were the
product of these 2 values is constant. We found a small systematic difference in the average
firing frequency (r2=0.96, n=8), with a 6% decrease in frequency for a change in
from 0.1 to 1.0 nS. However,
as demonstrated by the ISIs in Fig. 10B, these differences were much smaller than
biological variability and have therefore little significance.
Interaction of other synaptic inputs with asynchronous parallel fiber input
Climbing fiber inputs.
Each of the climbing fiber simulations described previously were made under conditions in which there were no background parallel fiber inputs, resulting in a constant waveform. This consistency is comparable to recordings obtained in slice preparations (Llinás and Sugimori 1980a). However, climbing responses in vivo are often somewhat more variable in their shape, even when recorded from the same cell (Campbell and Hesslow 1986).
Figure 11
Simulation of complex spikes in a Purkinje cell firing simple spikes.
A: 3 different evoked complex spikes recorded in the soma during asynchronous excitation at
25 Hz and inhibition at 1 Hz.
B: a complex spike recorded in the soma during asynchronous excitation at 22 Hz.
C: a complex spike in the soma during asynchronous excitation at 30 Hz.
D: same complex spikes as in A-C in the smooth dendrite, traces
superimposed.
We have used the model to examine whether some of the variability in climbing fiber waveform might be due to changes in background levels of dendritic synaptic input. Fig. 11 shows several examples of complex spikes in response to a climbing fiber input in a continuously firing Purkinje cell. If one compares the somatic recordings with and without background activity differences are, in fact, apparent. The complex spike by itself consistently had 3 somatic action potentials, while in the presence of background activity, the number of spikes varied between 1 or 2. Also, the number of these action potentials and their amplitude (particularly of the second one) was quite variable. In Fig. 11A this was caused by random variations in the asynchronous inputs to the cell, which changed the somatic recording of the complex spike considerably. At the same time the variability of the dendritic recordings of the complex spikes was quite small (Fig. 11D). In the main dendrite, the complex spike shape was also variable as the somatic spikes were still quite prominent at that location (not shown, see also Fig. 1A).
The only apparent effect of the different asynchronous inputs on climbing fiber responses in the dendrite was to produce variation in the length of the afterhyperpolarization after the complex spike. The resulting variable delays in occurrence of simple spikes after the complex spike have also been reported in extracellular recordings of cat Purkinje cells (Armstrong and Rawson 1979; Ebner and Bloedel 1981).
From these simulations, we conclude that the pattern of asynchronous inputs could change the somatic excitability and so affect the shape of the complex spike in the soma, but may have little effect on dendritic excitability directly related to the complex spike. This is the case even with respect to the changes in the duration of afterhyperpolarization as this is presumably unrelated to the occurrence of the next climbing fiber response. The dependence of the somatic response (and thus the output of the cell) is likely due to the fact that the ionic currents in the soma have faster time constants and would therefore be expected to be more sensitive to fast changes in the parallel fiber input.
Changes in the average frequency of the asynchronous excitation frequencies did not modify the complex spike beyond the variability noticed above (Fig. 11B, C). Even over a range of background synaptic input, climbing fiber responses in the soma changed, but remained remarkably constant in the dendrites.
Figure 12
Simulation of basket cell inhibition of a firing Purkinje cell neuron. At the time indicated, the
equivalent of 20 basket cell synapses were activated. Membrane potential as it would be recorded
in (from top to bottom) the soma, the main dendrite, a smooth dendrite and a spiny dendrite. The
model was excited by asynchronous inputs at 30 Hz and inhibited at an asynchronous rate of 1 Hz.
Basket cell inhibition.
We have not shown any records of the response of our model to exclusive basket cell inhibition, because it has not been possible to activate only this input in experimental preparations. Fig. 12 shows an example of strong inhibition caused by the activation of 20 basket cell synapses on the model while it was firing somatic spikes due to background parallel fiber and stellate cell inputs. The basket cell input caused a membrane hyperpolarization that interrupted firing for about 60 ms. The amplitude and duration of this hyperpolarization was quite sensitive to the number of basket cell synapses activated and could therefore be easily changed to fit experimental data for a particular Purkinje cell.
Fig. 12 shows also how little the basket cell inhibition penetrated into the dendrites. In the main dendrite there was still a pronounced hyperpolarization, but in the smooth and spiny dendritic recordings this was hardly noticeable. The basket cell synapse could thus shunt the plateau currents causing the continuous depolarization in the soma and main dendrite, but not in the rest of the dendrite.
Discussion
Model assumptions
As with all detailed single cell modeling efforts, several relevant properties of Purkinje cells are not known or are still being debated in the literature. In each case, we have had to make parameter assumptions based on the best available data.
For example, the detailed kinetics of excitatory postsynaptic conductances are not yet clear. There is currently a vivid discussion going on among experimentalists about what the opening and closing time constants for AMPA receptor channels are (Silver et al. 1992), with a wide range of decay time constants being reported in different systems and under different conditions (from 0.3 ms to 8 ms; Finkel and Redman 1983; Hestrin et al. 1990). Recently, careful measurements of miniature AMPA currents in cerebellar granule cells, which are very small and thus electrotonically compact, gave a rise time of 0.2 ms and a fast decay time constant of 1.0 ms (Silver et al. 1992), comparable to the kinetics we used in the model. Whole-cell patch clamp measurements of Purkinje cell postsynaptic currents evoked by climbing fiber stimulation reported a much slower average decay constant of 8.3 ms (Konnerth et al. 1990; Llano et al. 1991b). However, our results with a simulated voltage clamp (De Schutter and Bower 1994) raise the question whether an adequate space clamp (Rall and Segev 1985) was maintained in these studies, even though the use of smaller, immature rat Purkinje cells may have resulted in better voltage clamp control. Nevertheless, this concern has led us not to use the slow decay times reported by these authors. Some of their findings, for example the slower kinetics of parallel fiber synaptic currents compared to climbing fiber currents and the voltage dependence of the decay time constant of these currents (Llano et al. 1991b), could be explained by space clamp problems.
Similarly, the kinetics of inhibitory synapses on Purkinje cells were also not clear when this model was developed. We used kinetic data from GABAA receptor channels in pyramidal neurons of the hippocampus (Ropert et al. 1990). Recently Vincent et al. (1992) reported whole-cell voltage clamp recordings of rat Purkinje cell inhibitory currents in slice. They measured rise times of the postsynaptic currents (1 to 3 ms) comparable to the values we used, but the decay time constants were faster (7 to 13 ms). These kinetics are considerably faster than those reported by Kaneda et al. (1989) in whole-cell clamp of freshly isolated rat Purkinje cells. The differences might be explained by the different methods used by the two groups to activate the synapse. Vincent et al. (1992) used molecular layer stimulation in the slice preparation, while Kaneda et al. (1989) used a less reliable "concentration-jump" method.
The current model also does not simulate all synaptic phenomena known to exist in the Purkinje cell. Because of our simple model of Ca2+ concentrations and the absence of second messengers in the model, properties like long term depression of the parallel fiber synapse (Ito 1989; Linden et al. 1991; Konnerth et al. 1992), metabotropic receptor activation (Crepel et al. 1991; Glaum et al. 1992; Staub et al. 1992) and the change in inhibitory synaptic currents caused by changes in internal Ca2+ concentration (Llano et al. 1991a; Kano et al. 1992; Vincent et al. 1992) were not modeled.
Model validation
Even with these limitations, we have been able to use the current model to explore many of the synaptic and dendritic events believed to underlie the kinds of Purkinje cell activity seen with in vivo extracellular recordings. The model was able to simulate both the complex dendritic spike evoked by climbing fiber inputs as well as its dual reversal potential under current clamp conditions (Llinás and Nicholson 1976). The model was also able to generate stochastic simple spike activity at the appropriate frequencies under asynchronous excitatory and inhibitory inputs.
The fact that the simulations could replicate these phenomena, increases confidence that the
model captures several basic features of the real electrophysiology of the Purkinje cell. This is
particularly the case, because the model was not optimized to replicate the synaptic responses
presented here. Instead, as described in the preceding paper (De Schutter and Bower 1994), the
active membrane properties of the model were tuned and optimized only to replicate cellular
responses to intrasomatic and intradendritic current injections. During the extensive parameter
searching involved in this process, responses to synaptic inputs were never tested. Because the
other parameters in the model were held constant, the only parameters manipulated in the current
simulations were those directly related to the synapses that were added to the model. Data for the
kinetics of the postsynaptic conductances were directly based on measured values obtained mainly
in hippocampal neurons (Forsythe and Westbrook 1988; Ropert et al. 1990), while synaptic receptor
channel distributions in Purkinje cells have been established by EM-studies (Palay and Chan-Palay
1974; Harvey and Napper 1991). This left the synaptic receptor density, expressed as
for the conductance, as the
only remaining free parameter.
The complex spike
Because the dendritic location of the synapses made by the climbing fiber are well known, a relatively small number of synapses is involved, and all synapses are made by the same axon (Ito 1984), more is known about the nature of this input than about the granule cell and stellate cell inputs also modeled. Further, excellent electrophysiological (Llinás and Sugimori 1980a, b; Ekerot and Oscarsson 1981; Campbell and Hesslow 1986) and imaging data (Knöpfel et al. 1991; Konnerth et al. 1992; Miyakawa et al. 1992) are now available for the climbing fiber response. This provided us with detailed experimental data to compare to model output. While other authors have modeled complex spike responses in a passive membrane model of the Purkinje cell (Llinás and Nicholson 1976), our results represent the first ever published active membrane model of the complex spike and replicate well the available data.
Dual reversal potential.
Under current clamp conditions, the climbing fiber response has a dual reversal potential. Llinás proposed that this was caused by the distribution of a single synaptic input over electrotonically distant parts of the cell and demonstrated a biphasic reversal potential of the complex spike in a small, passive membrane model (Llinás and Nicholson 1976). While the explanation for the dual reversal potential was biophysically correct, the model they used was not accurate. Because only 4 compartments were used to model the entire dendritic tree, the modelers simply selected the electrotonic length constants for each compartment to produce the correct result. Consequently, each passive compartment had a sizable electrotonic length (0.24 [[lambda]] and more), and was unlikely to be unipotential (Rall 1962, 1964).
Another passive membrane Purkinje cell model, this time with realistic cell morphology has recently been described (Rapp et al. 1994). In this model a synaptic input distributed like the climbing fiber synapse produces a single reversal potential (I. Segev, personal communication). When it is taken into account that the Rapp et al (1994) model uses the same cell morphology as the model described here, it is clear that the difference must be the active membrane properties. The electrotonic length of a passive Purkinje cell dendrite is simply too short to produce the reversal effect (Shelton 1985; Rapp et al. 1994). As described in the previous paper (De Schutter and Bower 1994), adding active channels to the dendritic membrane almost doubles the electrotonic length of the dendritic tree compared to what one would expect from the anatomy alone. The active dendritic properties alone, therefore, account for the dual complex spike reversal potential in the current model.
Extent of dendritic activation by climbing fiber input.
Climbing fiber input to this model resulted in extensive activation of the dendritic tree. Several aspects of this activation are worth noting. First, the spatial distributions of both depolarization and Ca2+ concentration during a complex spike are quite different from those during a dendritic spike evoked by current injection (compare Fig. 5 with De Schutter and Bower 1994, Fig 10). However, both activation by current injection or by a climbing fiber input resulted in a spread of activity throughout the dendritic tree. Thus, climbing fiber input activated not only the region of the smooth dendrite on which climbing fiber synaptic contacts are made, but also the region of the spiny branchlets that contain granule cell synapses. Substantial Ca2+ currents were associated with this depolarization in all regions. When we first simulated the complex spike, we were surprised by the large recruitment of the spiny dendritic branches in the generation of the Ca2+ spike. While it had been shown before that dendritic Ca2+ channels play an important role in generating the complex spike (Knöpfel et al. 1991), it has only recently been reported experimentally that the complex spike causes Ca2+ inflow in both the smooth and the spiny dendrites (Konnerth et al. 1992; Miyawaka et al. 1992).
Mosaic pattern of dendritic activation.
In addition to the extensive activation of the dendrite, our model also produced a mosaic pattern of small dendrite activation, i.e. there were large differences in local depolarization and Ca2+ concentration levels between neighboring dendritic branches (Fig. 5B, E). While such mosaic patterns of activation were a robust phenomenon in the model, direct imaging techniques do not yet have the spatial resolution to confirm this modeling result. However, the image of a complex spike shown by Miyawaka et al. (1992, their fig. 3) is more fractured than the image of a Ca2+ spike evoked by current injection in the same cell.
The model's generation of mosaic activity patterns is also relevant to a specific controversy in the Purkinje cell literature. It has previously been supposed that such differences in the Ca2+ concentration profiles at different locations in the Purkinje cell dendritic tree, are positive evidence for variable Ca2+ channel densities and therefore dendritic calcium "hot spots" (Tank et al. 1988; Llinás and Sugimori 1992). However, in the current model, channel densities were uniform (De Schutter and Bower 1994) and mosaic patterns were still generated. A closer look at the model reveals that this non-uniformity was directly related to the morphology of local dendritic regions (Fig. 5B, C), which effectively seemed to isolate some small spiny dendritic branches from their neighbors. Recently, Jaslove (1992) came to a similar conclusion, i.e. the geometrical properties of a highly simplified model of a spiny neuron could influence the shape of dendritic spikes in ways that have been thought to require hot spots. Accordingly, these results suggest that if similar non-uniformities are found with the more sophisticated imaging techniques now under development, that they do not necessarily prove the existence of inhomogeneities in dendritic Ca2+ channel densities. The model suggests that the only way to quantitatively determine Ca2+ channel locations is by making direct measurements of channel density, not by imaging of Ca2+ concentrations.
Ionic mechanisms responsible for the complex spike response.
Over the last several years there have been numerous debates about which of the 2 or more Ca2+ channels apparently present in Purkinje cells (Mintz et al. 1992) are responsible for the long depolarization during the complex spike event (Miyawaka et al. 1992). For example, Llinás and Sugimori (1992) have claimed that the CaP channel alone is responsible for the entire time course of the dendritic spike. However, it has also been suggested that the later phases of the potential are caused by activation of a separate Ca2+ channel in the distal dendrites (Ekerot and Oscarsson 1981).
The present modeling results suggest that Llinás and Sugimori (1992) may be correct in that the CaP channels were entirely responsible for the Ca2+ related events of the complex spike. However, one obvious alternative candidate for calcium entry is the CaT channel, which exists in Purkinje cells, although in low densities (Kaneda et al. 1991; Regan 1991). While our model includes these channels, they did not contribute to the complex spike response. Gruol et al. (1992) have recently claimed to demonstrate that the developmental appearance of complex spikes in Purkinje cells coincided with the appearance of CaT channels in the dendritic membrane. However, these authors did not examine true complex spikes in their Purkinje cell cultures, but Ca2+ spikes evoked by depolarizing current pulses from resting or hyperpolarized membrane potentials. Based on the kinetics of these channels, we would expect that this type of activity might be more sensitive to CaT channel activation than the complex spike. An interpretation of Gruol et al.'s data must also take into account that at the same time CaT channels appeared in their cell culture, the density of the CaP channels increased by a factor of 2 or more. This should, by itself, facilitate spike generation. Finally, data from Miyawaka et al. (1992) show that the Ca2+ inflow during the complex spike is caused by the same Ca2+ channels that are opened by Ca2+ action potentials, and that there is no significant change in the Ca2+ concentration during the prolonged phase of the complex spike. If correct, it seems unlikely that the late phase is caused by channels different from the CaP channel.
While confirming the speculations of Llinás and Sugimori (1992) with respect to the role of the CaP channels in dendritic spikes, our model is not consistent with the idea that these CaP currents have an inhomogenous distribution in different regions of the Purkinje cell dendrite. Thus, in the model, no distinction can be made between channels in one region of the dendrite that are involved in the initial activation of the complex spike and channels in some other region which generate its later prolonged phase. The same result has also been demonstrated experimentally (Miyawaka et al. 1992). Consequently, both modeling and experimental data suggest that the debate about whether a special "plateau current" causes the later phases of the complex spike is irrelevant, as there is no separate plateau phase in Purkinje cell responses to climbing fiber activation.
Constancy of climbing fiber dendritic potentials.
It is a striking result of the model that the dendritic potentials evoked by climbing fiber
input were relatively constant even in the presence of background parallel fiber activity. In fact,
the only synaptic mechanism in our model that could change the dendritic component of the complex
spike were changes in of the
climbing fiber synapse. This is interesting, because changes in the amplitude of the dendritic
spike of the complex spike have been observed during Ca2+ imaging (Miyawaka et al. 1992). These
authors favored a postsynaptic cause of this variability, based on the fact that they could modify
the amplitude of the Ca2+ concentration changes by current injection. However, while there is at
present no experimental evidence for a large variability in the strength of climbing fiber
synaptic inputs, our results suggest that presynaptic factors resulting in variable amounts of
transmitter release (c.f. Faber et al. 1992) should also be considered.
Variability in simple spikes produced by climbing fiber input.
While climbing fiber responses in Purkinje cells are in some sense "all or none" (Llinás and Sugimori 1980a; Kimura et al. 1985), intracellular recordings of in vivo complex spikes show that these events can evoke a variable number of somatic spikes. Our own experimental data suggests that the number of somatic spikes usually varies from 2 or 3 in an irregular fashion (D. Jaeger, personal communication). We have shown that the model can produce similar variability through an interaction between background parallel fiber activity and the depolarization of the soma (Fig. 11).
It has previously been proposed that the secondary spikes evoked by climbing fiber activation might actually result from spikes generated in the Purkinje cell dendrite (Campbell and Hesslow 1986). This suggestion was based on differences between the amplitudes of the first spike and increases or decreases of secondary spikes after direct stimulation of the parallel fibers (Campbell and Hesslow 1986). However, these measurements were made with extracellular recordings techniques, which do not allow a direct measurement of the site of origin of these spike potentials. In our model, secondary somatic spikes had a variable amplitude under conditions of background parallel fiber activity (Fig. 11A-C) due entirely to effects on the soma of the cell.
Modeling granule and stellate cell synaptic inputs
Having demonstrated that the model was capable of replicating many features of the extensively described response of the Purkinje cells to climbing fiber inputs, we pursued the response of the model to the less well defined synaptic influences of granule and stellate cells. The ability of the model to replicate the climbing fiber response increases our confidence in the granule and stellate cell modeling results. However, the first section of this discussion considers several issues directly related to the fact that these inputs are less well defined and harder to simulate. The final sections consider our initial conclusions from this modeling study. Because many features of the Purkinje cell responses to these inputs have not yet been explored experimentally, the model has allowed us to generate several testablepredictions.
Modeling total parallel fiber input.
Our simulations of synaptic activity are based entirely on a single cell model of the cerebellar Purkinje cell. No attempt was made to model the network in which this cell is imbedded, despite the fact that we have explored the combined effect of many different individual parallel fibers on the modeled cell.
Ultimately we intend to construct a network model of the cerebellar cortex in order to provide more realistic patterns of parallel fiber input. However, we believe the nature of cerebellar cortical circuitry makes the absence of this network model less limiting in these initial modeling efforts than would be the case in other regions of the brain. For example, Purkinje cells do not directly influence the synaptic input they receive through feedback excitation or inhibition on a short time scale (i.e. from within the cerebellum itself), except by the very sparse Purkinje axon collaterals which are inhibitory (Ito 1984; Bishop 1988). This is in contrast with, for example, pyramidal cells in cerebral cortex, which receive massive feedback inhibition (Shepherd 1990). This absence of local synaptic feedback makes it possible to examine the response properties of an isolated Purkinje cell model over a range of input frequencies without modeling the rest of the cerebellar network.
Modeling the temporal pattern of input.
In these simulations, we were specifically interested in replicating the background "spontaneous" activity of the Purkinje cell that is recorded with in vivo experiments (Llinás 1981; Bower and Woolston 1983). Under these conditions, Purkinje cell simple spike activity occurs at frequencies of 30 to 100 Hz, even in the absence of obvious peripheral stimulation (Murphy and Sabah 1970). Our model replicated this background pattern of activity quite well under the assumption that parallel fibers provide low levels of continuous, asynchronous and random (uncorrelated) patterns of synaptic input.
The separate assumptions that parallel fiber activity is continuous, and asynchronous each have their own justifications. First, the assumption that parallel fibers are continuously active at low levels is related to the observation that some mossy fiber inputs are also spontaneously active (Ito 1984), making it seem likely that parallel fibers would be as well even though this has not been measured directly (see below). The assumption that parallel fiber synaptic inputs, in the absence of a specific peripheral stimulus, are asynchronous is based on the expectation that the non-uniform conduction velocity of parallel fibers will desynchronize any initially synchronous signal (Bernard and Axelrad 1991).
The model assumed that stellate cells also fire asynchronously. This assumption is based on two observations. First, these inhibitory interneurons receive input mainly from parallel fibers, which in our simulations were asynchronous. Second, the probable distribution of inputs from a single stellate cell on a particular Purkinje cell make it unlikely that it would produce synchronous effects. This can be deduced from the Purkinje cell-to-stellate cell ratio, estimated to be about 1:16 (Palkovits et al. 1971), and the short axons with a simple terminal arborizations of stellate cells (Ito 1984), so that the synaptic input from a single stellate cell to a particular Purkinje cell would be expected to be quite small. The effective strength of a stellate cell synapse in our model seems comparable to that of a real synapse as measured by dual recordings in vitro in the turtle cerebellum (Midtgaard 1992, his fig. 4). In these experiments a single inhibitory synapse, even excited at an extremely high frequency, could not inhibit Purkinje cell simple spike firing if the cell was depolarized. At less depolarized levels it would interrupt simple spike firing, but no IPSP was measurable in the soma. This is comparable to the effects seen in the responses to combined asynchronous excitatory and inhibitory inputs in the model (Fig. 7), which were simulated with a single inhibitory input occurring on average every 0.6 ms..
Parameters for individual parallel fiber synapses.
Of the three parameters that govern the influence of parallel fibers on Purkinje cells in this model, values for only one are known. Thus, of the number of inputs, the maximum conductance of the synapse and the average firing frequency, only the number of inputs is known with any accuracy (Harvey and Napper 1991). While, the single channel conductance has been measured, and multiple conductance levels have been reported (Cull-Candy and Usowicz 1989), the number of channels at postsynaptic sites is not known, making it impossible to determine the total conductance of a single postsynaptic event. The normal frequency of parallel fiber inputs is also not known. There are not even any reports about the firing properties of individual granule cells in vivo, presumably because their small size makes intracellular recording very difficult and their high density has made it difficult to isolate single cells in extracellular recordings (Ito 1984).
Scaling parallel fiber inputs.
Because of the massive synaptic convergence on Purkinje cells, it was not possible to simulate all the synaptic inputs. As described in the Methods section, the number of parallel fiber inputs was scaled by approximately a factor of 100 in the standard model (Harvey and Napper 1991). However, we also ran a number of simulations to investigate if this scaling was linear in our model and found that the total excitatory input to the Purkinje cell was determined by the linear product of the 3 model variables just described, i.e. the number of inputs, the maximum conductance of the synapse and the average firing frequency. While this finding may seem mainly of theoretical interest to modelers, it also has important consequences for the generality of our model and the strength of its predictions.
The 3 variables which determine total excitatory input on the model were linearly related to
each other for asynchronous inputs (within the range that we could examine). This scaling had a
practical value as it allowed us to model only about 1% of the inputs, which kept the model to a
reasonable size (4550 compartments in total). Because we know approximately the number of granule
cell inputs to the Purkinje cell (Harvey and Napper 1991), the values for
and parallel fiber firing
frequencies we used can be related to real electrophysiological parameters.
This was not true for the
of the stellate cell synapse and the corresponding average frequency. There are no data on the
number of stellate cell inputs, or on their exact distribution, so it was impossible to determine
whether a single stellate cell synapse in our model would correspond to a few high
real synapses or a lot of synapses with a low
.
Response to granule and stellate cell synaptic inputs
The role stellate cell of inhibition in cell bursting.
It seems somewhat paradoxical that the unique dendritic excitability of the Purkinje cell is very conspicuous in slice, while in vivo the cell fires a Ca2+ spike only during a complex spike (Crepel 1972; Armstrong and Rawson 1979; Llinás 1981; Montaloro et al. 1982; Ito 1984). When only excitatory inputs were given, even fairly low levels of parallel fiber activity resulted in the generation of dendritic Ca2+ spikes. But if stellate cell inhibition was provided also, this bursting did not occur. This results in the prediction that removal of stellate cell inhibition by pharmacological block should cause Purkinje cells to fire spontaneous dendritic bursts in vivo.
While this experiment has not been done specifically in vivo, data from the in vitro preparation would seem to support this result (D. Jaeger, unpublished results). Midtgaard (1992) has shown in the turtle cerebellum that whether a single strong IPSP can abolish Ca2+ spiking depends on the level of depolarization of the dendrite. Our modeling results suggest that if enough stellate cells are inhibiting the Purkinje cell continuously, Ca2+ bursting is completely suppressed at any level of excitation.
Difference between Purkinje cells in vitro and in vivo.
An interesting emergent property of the model, was the difference between what can be described as its "in vitro" and "in vivo" states. With no synaptic input, the model had a stable resting membrane potential of -68 mV (De Schutter and Bower 1994) and was silent (i.e. produced no simple spikes or dendritic events). We refer to this as the in vitro state of the model. In this state, the ionic channels were mostly inactive, outward currents were fully deinactivated and inward currents were fully (CaP) or partially (NaF and CaT) deinactivated. Under these conditions, there was a variable delay before the model could fire a spike during current injections (De Schutter and Bower 1994, Fig. 6B), because the cell first needed to depolarize up to the firing threshold. Also, strong enough inputs cause the cell to fire dendritic Ca2+ spikes in this state. Finally, transient perturbations, like a climbing fiber input or short duration current injection, did not produce prolonged activity but instead resulted in the model's return to a non firing state. However, if a prolonged depolarization activated the CaP current sufficiently, then the cell would sometimes switch by itself to an active state in which it fired action potentials spontaneously (for example De Schutter and Bower 1994, Fig. 3). Similar behavior is also sometimes observed in in vitro slice preparations (Llinás and Sugimori 1980a)
In contrast to the in vitro state, the addition of continuous excitatory and inhibitory input alone made the model behave more like Purkinje cells recorded in vivo. Under these conditions, the model showed a second semi-stable membrane potential between -50 an -55 mV. In this case, the modeled cell could fire an action potential without delay, because the membrane potential was very close to the activation threshold. Also, as in vivo, the model did not fire spontaneous dendritic bursts. The in vivo state was stable, as the model did not return to the -68 mV resting membrane potential spontaneously, even if all synaptic input was stopped. This behavior was based on an interaction of ionic currents that is discussed in detail in the preceding paper (De Schutter and Bower 1994, Fig. 11). In brief, the membrane potential under conditions of continuous input was determined in the soma mainly by a balance between the window current of the NaF channel and the Kdr current and in the dendrites by a feedback loop between the CaP channel and the Ca2+-activated K+ channels.
Recently several reports concerning the changes in passive membrane properties of neuron models under conditions of background synaptic activity have appeared (Holmes and Woody 1989; Bernander et al. 1991; Rapp et al. 1992) and it has been suggested that this could have a profound effect on synaptic integration. In our model, background excitatory and inhibitory inputs also appeared to quite profoundly change the basic firing pattern of the neuron as a result of an interaction with active channels in the membrane.
The role stellate cell of inhibition in stochastic firing.
We have shown that in the absence of inhibitory input, parallel fiber input alone causes a very regular pattern of cell firing (Fig. 6). The firing pattern of our Purkinje cell model was stochastic only when asynchronous inhibition was added to the excitation (Figs. 7, 8). Recently, Softky and Koch (1993) also reported that a detailed pyramidal cell model could not fire irregularly under conditions of asynchronous excitatory inputs. However, they proposed that fast Na+ channels on the dendrites might be responsible for stochastic firing. Our model suggests that they might also want to consider asynchronous inhibition as an alternative mechanism to create random firing patterns.
The regulation of Purkinje cell ongoing activity.
Perhaps the functionally most significant result of this modeling study is the suggestion that the ongoing spontaneous simple spike activity of Purkinje cells is regulated by a combination of parallel fiber excitatory and stellate cell inhibitory inputs. Using a combination of these two inputs, we have demonstrated that the model was exquisitely sensitive to small changes in input frequencies. Such sensitivity to small changes in input patterns was not expected a priori, considering the large number of granule cell inputs (Harvey and Napper 1991). Because of the scaling factor involved in our simulations of parallel fiber inputs, this result implies that Purkinje cells may be able to detect both large changes in the activity of a small number of inputs or small changes in the frequency of most inputs.
Interpreted in another way, our modeling results demonstrate that stellate cell inhibition could set the threshold at which a Purkinje cell will fire somatic spikes in response to parallel fiber activity. In Fig. 9 it is clearly shown that increasing frequencies of stellate cell inhibition shifted the firing threshold of the model toward higher excitatory input frequencies, without significantly affecting the shape of the frequency response curve. Similar effects have been reported in a compartmental model of a spinal motoneuron, but in this case parallel shifts of the frequency response curve occurred only if synaptic inputs were confined to the soma, while dendritic inputs caused a large change in slope of the curve (Kernell 1971).
Marr (1969) speculated in his influential paper, based on the organization of cerebellar circuitry, that stellate and basket cells could be "threshold-setting cells". In our model, stellate cell inhibition in the physiological firing range not only set the threshold for firing, but also resulted in a very sharp response curve to asynchronous excitatory inputs. Under these conditions, a steady small change in excitatory input frequency resulted in a large change in Purkinje cell firing frequency.
While Marr's proposed threshold setting function is consistent with our results, simulations suggest that stellate cells may do more than just set a threshold and establish a response curve. In the model, changes in stellate cell activity actually shifted the sensitivity of Purkinje cells to parallel fiber input (Fig. 9). Further, if it is assumed that a given stellate cell receives its input mainly from the same parallel fibers as the Purkinje cell it inhibits (Ito 1984), our model suggests that this cell would be positioned to assure that Purkinje cells stay maximally responsive over a wide range of background parallel fiber firing rates. In other words, as the activity of the parallel fibers goes up, the inhibitory synaptic input to the Purkinje cell would also go up, in effect keeping the Purkinje cell within its normal range of firing frequencies.
We believe that this last effect may be a particularly important mechanism with respect to the response of the Purkinje cell to peripherally activated inputs arising from the ascending branch of the granule cell axon (Llinás 1982; Bower and Woolston 1983). While a full discussion of this issue is beyond the scope of this paper, we have reason to suspect that the ascending branch of the granule cell axon produces a significant response in Purkinje cells following peripheral stimulation (Bower and Woolston 1983). If stellate cells are providing the normalizing function just described, Purkinje cells may be able to respond to small changes in excitatory input from the ascending branches, at any mean frequency of parallel fiber input. As a result a small synchronous stimulus from ascending branch inputs could still be sufficient to generate a significant response in a peristimulus histogram (Bower and Woolston 1983). These interactions are the subject of current modeling work in the laboratory (De Schutter and Bower 1992).
Conclusion
The putative role of stellate cell inhibition in the response properties of Purkinje cells is the most unexpected result of this modeling effort. However, while the several predicted effects of stellate cell inhibition are relatively robust and therefore insensitive to changes in the model (Fig. 9B), the modeling results also suggest that experimental confirmation of these predictions will require considerable care. For example, stellate cell inhibition did not seem to affect other properties of the modeled Purkinje cell firing response, like the ISI (Fig. 9C). Therefore, the average firing frequencies of either excitatory or inhibitory inputs cannot be deduced easily from recorded Purkinje cell output. Instead, Purkinje cell activity results from an intimate and coordinated interaction of its excitatory and inhibitory inputs within the smallest regions of the Purkinje cell dendrite. Nevertheless, as is often the case with modeling of this type (Bower 1990), simulation results have served to focus our attention on the importance of obtaining additional experimental data on the currently understudied inhibitory cells within cerebellar cortex.
Acknowledgements
We thank D. Jaeger for useful, often critical, comments. J. H. Thompson provided the experimental data shown in Fig. 8. Discussions with A. Marty, T. Knöpfel and A. Konnerth inspired some of the simulations presented here. We are also extremely grateful to Rapp, Segev, and Yarom for the anatomical data on which the modeled cell was based.
This work was supported by Fogarty fellowship F05 TW04368 to EDS and NINDS, NS31378 and NSF, BIR-9017153 to JMB.
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