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Previous Article | Next Article
Volume 17, Number 1, Issue of January 1, 1997 pp. 91-106
Copyright ©1997 Society for NeuroscienceThe Role of Synaptic and Voltage-Gated Currents in the Control of Purkinje Cell Spiking: A Modeling Study
Dieter Jaeger1, Erik De Schutter2, and James M. Bower1 1 Division of Biology, California Institute of Technology, Pasadena, California 91125, and 2 Born Bunge Foundation, University of Antwerp, 2610 Antwerp, Belgium
ABSTRACT
We have used a realistic computer model to examine interactions between synaptic and intrinsic voltage-gated currents during somatic spiking in cerebellar Purkinje cells. We have shown previously that this model generates realistic in vivo patterns of somatic spiking in the presence of continuous background excitatory and inhibitory input (De Schutter and Bower, 1994b
Key words: cerebellum; coding; dendrite; spiking; inhibition; synapse; balance; model; simulation; genesis). In the present study, we analyzed the flow of synaptic and intrinsic currents across the dendritic membrane and the interaction between the soma and dendrite underlying this spiking behavior. This analysis revealed that: (1) dendritic inward current flow was dominated by a noninactivating P-type calcium current, resulting in a continuous level of depolarization; (2) the mean level of this depolarization was controlled by the mean rate of excitatory and inhibitory synaptic input; (3) the synaptic control involved a voltage-clamping mechanism exerted by changes of synaptic driving force at different membrane potentials; (4) the resulting total current through excitatory and inhibitory synapses was near-zero, with a small outward bias opposing the P-type calcium current; (5) overall, the dendrite acted as a variable current sink with respect to the soma, slowing down intrinsic inward currents in the soma; (6) the somato-dendritic current showed important phasic changes during each spike cycle; and (7) the precise timing of somatic spikes was the result of complex interactions between somatic and dendritic currents that did not directly reflect the timing of synaptic input. These modeling results suggest that Purkinje cells act quite differently from simple summation devices, as has been assumed previously in most models of cerebellar function. Specific physiologically testable predictions are discussed.
INTRODUCTION
The dendritic trees of single cerebellar Purkinje cells receive about 175,000 excitatory glutamatergic inputs from granule cells in the rat (Napper and Harvey, 1988
The objective of the present study was to use the realistic Purkinje cell model to examine the pattern of synaptic and voltage-gated dendritic currents that produces ongoing somatic spiking. We applied several new analysis techniques to examine this issue. We find that in the model, intrinsic dendritic currents strongly influenced the time course of dendritic membrane potential. As a consequence, the timing of somatic spikes did not reflect the timing of particular synaptic inputs. The common assumption in cerebellar network models that Purkinje cell spiking reflects a simple summation of inputs (Marr, 1969
MATERIALS AND METHODS
Model construction The model studied in this report is identical to the model described previously by De Schutter and Bower (1994a
Morphology and voltage-gated conductances. The morphology of an HRP-injected Purkinje cell from a guinea pig cerebellum was reconstructed, and a passive membrane model of the cell was made (Rapp et al., 1994 80 mV, which provided for a stable resting membrane potential of the cell at
Synaptic input. After establishing the intrinsic conductances of the model, a set of synapses was added with the aim of replicating the expected natural input to the cell in vivo (De Schutter and Bower, 1994b
Stellate cell inhibition mediated by GABAA receptors was simulated by synaptic conductances with a reversal potential of Techniques for quantifying synaptic and voltage-gated currents The primary objective of the present study was to examine the differential contribution of synaptic and voltage-gated conductances to dendritic depolarization and ultimately the spiking output of the modeled Purkinje cell. To do this, we have developed several new approaches to analyze the contribution of different conductances to the behavior of the model, as described below. In general, these techniques have allowed us to somewhat simplify the analysis of this very complex model.
The pattern of synaptic activation used here was identical to that used in our previous work (De Schutter and Bower, 1994b
Collapsing the spatial complexity of the dendritic tree: total currents. The principle form of simplification we have adopted to analyze the model involves collapsing the spatial distribution of dendritic membrane currents into total currents flowing across the entire dendritic surface. These total dendritic currents were calculated separately for each type of synaptic or voltage-gated current by adding the current from all dendritic compartments at each simulation time step. We have analyzed membrane currents rather than conductances because currents are directly related to the time course of membrane potential, which can be measured experimentally.
Contribution of individual conductances to membrane potential. Most physiological experiments measure membrane potential over time. After having calculated the total current carried by each conductance we derived the contribution of individual conductances to membrane depolarization from the equation of a capacitor, Q = C × V (Q is charge in Coulomb, C is capacitance in Farad, V is potential in V). Because the dendritic capacitance of the Purkinje cell model is 4.2e 9 F (1.64 µF/cm2), a charge of 4.2e
Comparison of modeling results with experimental data A critical test in the performance of a single-cell model is its ability to replicate a whole range of physiological data. We have shown previously that the Purkinje cell model can replicate a wide range of intra- and extracellular data (De Schutter and Bower, 1994a
Interaction between the soma and the dendrite. In most mammalian neurons, the significance of dendritic current flow for neuronal processing ultimately lies in its effect on somatic spiking. In most Purkinje cells and in the model, the only connection between the dendrite and the soma is provided by a single dendritic trunk. Using the model, we have analyzed the axial current flowing through this junction because it provides the only means of electrical information transfer between the soma and the dendrite. The amplitude of this somato-dendritic current (I s-d) was calculated from the membrane potential of the soma (Vms), the membrane potential of the initial dendritic segment (Vmdi), and the axial resistance between the two using Ohm's law as I s-d = (Vmdi , even a small potential difference of 1 mV leads to a somato-dendritic current of 1.3 nA. The low capacitance of 4.6e
Intracellular Purkinje cell responses to current injection. To analyze the role of voltage-gated conductances in the response to somatic current injection, we describe the detailed behavior of the model during and after a current injection pulse. The modeling results are compared with intracellular Purkinje cell recordings. A detailed description of the in vitro guinea pig preparation used to acquire these data can be found in Jaeger and Bower (1994)
Interspike interval variability in neuronal spike trains. To compare the output spiking properties of the model with those of real Purkinje cells, extracellular spike data were obtained from crus IIa of rats under ketamine anesthesia. A detailed description of the preparation used to acquire these data can be found in Bower and Woolston (1983)
RESULTS
Although the present paper is focused on understanding the interaction between synaptic currents and voltage-gated currents in the Purkinje cell, we will first examine the activation of voltage-gated currents with direct somatic current injection. As we will show, the activation of intrinsic currents with direct current injection bears many similarities to the case of synaptic input. It is easier, however, to support the modeling results with physiological data in the case of current injection than in the case of synaptic input because the experimental conditions are much better controlled with current injection than with synaptic input.Purkinje cell responses to current injection in vitro
As in many neurons, when current is injected into the soma of a Purkinje cell at resting membrane potential, a fast train of somatic action potentials is elicited (Fig. 1A) (Llinás and Sugimori, 1980a
Fig. 1. Comparison of voltage traces during current injection into the soma between a physiological recording and the model. A, Intracellular recording of Purkinje cell soma in vitro. A current injection pulse (cip) of 1.5 sec duration and 0.24 nA amplitude was started at 100 msec into the recording. B, Somatic voltage trace of simulation of same current injection paradigm in the Purkinje cell model. In both cases, the current injection was started when the cell was in a quiescent state. The voltage response to current injection in the model and in vitro started with an initial period of slow depolarization, followed by fast regular somatic spiking. This spiking continued at a slightly reduced rate after offset of the current injection pulse (cip off).
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Membrane currents during current injection Purkinje cells in vitro often remain silent for considerable periods of time. In the model, such stable resting potentials (Fig. 2A;
Fig. 2. Current flow underlying voltage response to current injection in the model. Traces shown are taken from the same simulation for which the somatic voltage response is depicted in Figure 1B. The time around onset and offset of the current injection pulse (cip) was expanded for improved resolution of details, and the middle section of current injection was left out. A, Voltage response in the soma. B, Current flow between the soma and the main dendritic segment. Current depolarizing the dendrite is depicted upward. The large amplitude of I s-d during somatic spiking (peaks at +25 and
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If all membrane conductances in the Purkinje cell model are blocked, we can calculate from the capacitance of the total membrane that the charge carried by the injected current of 0.24 nA would depolarize the whole cell within 280 msec to
Similar to the period of initial depolarization with current injection, the characteristics of the ensuing phase of sustained spiking were governed primarily by the balance of large intrinsic dendritic currents. In particular, the mean dendritic depolarization of
As might be expected, during active spiking there was also a flow of current between the dendrite and the soma. It was surprising that, on average, this current was from the soma to the dendrite, providing a mean flow of 0.45 nA into the dendrite (Fig. 2B). Although the overall flow of current was from the soma into the dendrite, the amplitude and direction of this current was modulated strongly during the time course of somatic action potentials. This phasic current flow was much larger than the injected current of 0.24 nA and determined the time course of spiking. During the upswing of each action potential, most of the large inward somatic NaF current flowed into the dendrite (Fig. 3). Because the dendrite has a high capacitance, however, this current led only to a small dendritic depolarization with each somatic spike (Fig. 2C). Subsequent activation of Kdr led to somatic spike after-hyperpolarization, and somatic Na currents were deactivated (Fig. 3). In contrast, the dendrite remained depolarized, and a current with a peak amplitude of 2.5 nA flowed back into the soma from the dendrite (Fig. 3). Because of the small size and capacitance of the soma, this current effectively redepolarized the soma after each spike. Overall, the interaction between soma and dendrite is best viewed as a dynamic push-and-pull operation, in which the soma mostly pushed current into the dendrite, but a phasic pull of current from the dendrite led to redepolarization of the soma during spike after-hyperpolarzation. This depolarization then reactivated the NaF current in the soma, thus initiating an inward current flow across the somatic membrane. When the NaF current reached an amplitude of 0.5 nA, the somato-dendritic current reversed and again pushed current into the dendrite (Fig. 3, second vertical dashed line). 
Fig. 3. Somatic voltage and currents for a spike cycle after the offset of current injection. Somatic Vm (upper trace) and somatic currents (lower trace) during a spike cycle with current injection. Inward currents are plotted downward. The dashed current trace represents the somato-dendritic current (I s-d). The NaF and I s-d currents are truncated at 10 nA maximal amplitude to give a better resolution of smaller currents. The first dashed vertical line marks the time of the most hyperpolarized somatic potential. Note that at this time I s-d flowed into the soma with an amplitude of 2 nA because of maintained dendritic depolarization. The second dashed line marks the time in the spike cycle at which the injected current provided a significant proportion of the total inward current responsible for continued somatic depolarization.
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Currents after the offset of current injection The activity of model conductances after the offset of current injection was surprisingly similar to the activity during the period of sustained spiking during current injection (Fig. 2). In fact, the mean level of dendritic depolarization changed only by 0.2 mV after the offset of current injection, and dendritic voltage-gated membrane currents showed only small changes (Fig. 2C-E). The continued depolarization was a result of the activation of inward P-type calcium conductance.
Analysis of the somato-dendritic current flow during somatic spiking after the offset of current injection showed the same pattern of current flow as during current injection. As described above, the dynamics of current flow during an action potential were governed by a dynamic push-and-pull operation between the soma and the dendrite (Fig. 3).
These findings show that the intrinsic properties of the Purkinje cell support fast regular spontaneous spiking at a depolarized membrane potential. In fact, the typical sequence of activity recorded in vitro leads to increasing depolarization during such spiking, and ultimately calcium spike bursting. We will now turn to the question of how synaptic input may interact with these intrinsic properties. Purkinje cell spiking in vivo
In Figure 4, we contrast the pattern of spontaneous somatic spiking with spiking during continuous synaptic input for real and modeled data. The typical ISI distribution in vitro obtained just after the offset of current injection was very narrow, indicating very regular spiking (Fig. 4A, solid line). This narrow ISI distribution is quite similar to the pattern generated by the model after the offset of current injection (Fig. 4B, solid line). The typical ISI distribution for Purkinje cell spiking recorded in vivo, however, is quite different (Fig. 4A, dashed line). In particular, the in vivo data typically show a pronounced tail of long intervals and a higher degree of variability in spike intervals.
Fig. 4. Comparison of ISI histograms for spiking with and without synaptic input. A, Physiological recording of extracellular activity in vitro (solid line) and in vivo (dashed line). The gc input is inactive under in vitro conditions, and inhibitory input through sc inputs was blocked with bicuculline. Spontaneous spiking under these conditions most likely reflects purely intrinsic mechanisms. The resulting fast regular spike pattern showed a strong modal interval at 7.5 msec. The in vivo recording was obtained from an anesthetized rat in the absence of external stimulation. In this case, the recorded cell is embedded in an intact cerebellar network and presumably receives a background of spontaneous synaptic inputs. The modal ISI of 9.5 msec was longer than in the in vitro case, and a pronounced tail of very long intervals was present. B, Spike interval distributions obtained with the model in the absence (solid line) and presence (dashed line) of synaptic input. The spike train in the absence of synaptic input was obtained after a current injection of 0.24 nA. Synaptic input consisted of the random activation of all excitatory synapses at the rate of 37 Hz and inhibitory synapses at 1.5 Hz. The obtained interval distributions closely resemble the physiological recordings under in vitro and in vivo conditions, respectively.
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The Purkinje cell model was capable of generating ISIs similar to those seen in vivo when background excitatory and inhibitory synaptic activation was added (De Schutter and Bower, 1994b Response after the onset of synaptic input in the model
Figure 5 shows a simulation in which excitatory and inhibitory random background synaptic inputs were initiated simultaneously after 100 msec of simulation at the resting potential of
Fig. 5. Voltage response and membrane currents resulting from asynchronous synaptic input. The simulation was started at the stable resting membrane potential (
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Synaptic and dendritic voltage-gated currents associated with synaptic input An important distinction between the direct current injection we examined above and the synaptic currents analyzed here lies in the fact that the amplitude of synaptic currents is dependent on the synaptic driving force, i.e., the distance of membrane potential from the synaptic reversal potential. For this reason, any change in membrane potential leads to an immediate change in driving force and, hence, of synaptic current.
When the synaptic input was turned on at the resting potential of Control of dendritic depolarization by synaptic conductances As described above, the mean synaptic current was close to zero in the dendritic plateau depolarization associated with continuous synaptic input (Fig. 5C). Nevertheless, the synaptic input still controlled the overall level of depolarization as any change in potential resulted in a change in driving forces of synaptic currents that led to a "control" current, bringing the overall balance back to its stable point. This mechanism by which a baseline of open synaptic conductances stabilizes dendritic Vm can be described as a partial voltage clamp (Staub et al., 1994
Similar to the effect on intrinsic conductances of dendritic depolarization caused by somatic current injection, the initial depolarization caused by inward synaptic current activated a large P-type calcium current and, at some delay, a potassium current (Fig. 5B,C). Even before the total synaptic current decreased as a result of the change in driving forces described above, the amplitude of intrinsic inward current surpassed the amplitude of synaptic current and became the main cause for continued depolarization (Fig. 5B,C). After a small peak in dendritic membrane potential caused by the delay between Ca and K current activation, the dendrite was kept depolarized at its plateau level because of a balance of inward and outward currents very similar to the current levels seen with somatic current injection. This balance consisted of a mean Ca current of 5.8 nA, a mean K current of where gex/in and Vex/in are the excitatory and inhibitory conductances and reversal potentials, respectively. For example, Vclamp in the model for the mean conductances associated with at a rate of 1.5 Hz inhibition and 33 Hz excitation (output spike rate 12 Hz) comes to
Fig. 6. Summary diagram of current amplitudes associated with various levels of gc and sc input. Simulations were run for a period of 2.0 sec for each level of synaptic input. The first 1.5 sec were treated as equilibration period, and only the final 0.5 sec was analyzed for spike rates and current amplitudes. In each panel of the figure, different levels of sc inhibition (0.5, 1.0, 1.5, 2.0 Hz) are depicted as separate curves. For each curve, the gc input rate was increased in small steps to study voltage responses and current amplitudes. A, Somatic spike rate as function of gc and sc input frequency. Note that the same spike rate could result from different total levels in synaptic input. B, Average membrane potential in soma and dendrite for different spike rates. Note that the soma was on average more depolarized than the dendrite and that the difference in potential is roughly proportional to the somato-dendritic current (E). C, Dendritic voltage-gated currents. Inward current (combined Ca currents) is down; outward current (combined K currents) is up. The sum of inward and outward currents (Ca+K) is inward for spike rates up to 150 Hz. The leak current counteracting this current to produce a stable membrane potential is not shown. Currents for the four different levels of inhibition are superimposed. D, Total inward (gc) and outward (sc) synaptic current as function of somatic spike rate. Inward current is down. The sum of inward and outward synaptic current (gc+sc) was outward for spike rates up to 120 Hz. The summed traces for different level of inhibition lie on top of each other. E, Temporal average of current flowing between soma and dendrite (I s-d). Although this current was strongly modulated during each spike cycle (Fig. 3), the mean direction of flow from soma to dendrite does indicate that overall the soma acted as a current source rather than as a current sink.
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Fig. 9. Spike-triggered averages of Vmd, synaptic currents, and channel currents. Spike-triggered averages were constructed for three sets of ISIs from 7 sec of simulation with 30 Hz gc and 1 Hz sc input. The ISI distribution for this level of input is shown in Figure 7C. Set 1 included 74 ISIs of 10-13 msec, set 2 included 32 ISIs of 16-20 msec, and set 3 included 39 ISIs of 30-50 msec. An upper and lower 95% confidence limit (1.7 SEs) is shown for each spike-triggered average as a pair of dashed traces. Vertical dashed lines denote the timing of dendritic traces with respect to the peak depolarization in the soma with each spike. Traces on the left are aligned to the initial spike in each ISI, and traces on the right are aligned to the terminating spike. A, Spike-triggered average of Vmd. B, Spike-triggered average of synaptic conductances summed for all gc inputs (G gc) and all sc inputs (G sc) in Siemens × 10e
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When different levels of excitatory and inhibitory input frequencies produced the same spike rate, the concomitant levels of somatic and dendritic membrane potential (Fig. 6B) were virtually identical. Any increase in spike rate was associated with an increase in the mean level of somatic and dendritic depolarization (Fig. 6B). Over the entire range of physiological spike rates between 10 and 140 Hz, the soma was on average more depolarized than the dendrite (Fig. 6B), meaning that the dendrite overall acted as a current sink at all somatic spike rates.
It was somewhat surprising that for all combinations of gc and sc input rates producing spike rates between 10 and 120 Hz, the mean synaptic current over the entire dendrite was close to zero, with a small outward bias (Fig. 6D). Maintained dendritic depolarization with a net synaptic outward current was possible as inward voltage-gated currents balanced the sum of outward dendritic currents (total synaptic current, potassium, and leak current) at membrane potentials supporting physiological spike rates (Fig. 6C). Even though the synaptic current was close to zero, the level of open synaptic conductances controlled dendritic depolarization through the partial voltage clamping mechanism described above. In fact, the small size of synaptic current indicates that dendritic membrane potential was close to the clamping potential given by the baseline level of open synaptic conductances with different rates of input. To achieve spike rates greater than 120 Hz, however, an increasing net inward synaptic current was necessary (Fig. 6D). This increase in inward synaptic current was needed because at the more depolarized membrane potential supporting such high spike rates, the balance between dendritic Ca and K currents shifted toward a net outward current (Fig. 6C). As a result, the dendrite was less depolarized than predicted from the synaptic clamping potential.
The mean somato-dendritic current was outward into the dendrite for the entire range of input frequencies tested (Fig. 6E). The amplitude of this mean somato-dendritic current increased with increasing spike rate because of the charge injected into the dendrite with each spike. Nevertheless, the phasic current from the dendrite into the soma after spike offset also increased with increased dendritic depolarization. In effect, this increased phasic current flow from the dendrite into the soma was responsible for an increase in the speed of somatic redepolarization and hence spike rate.
Although the preceding analysis demonstrates the pattern of current flow for different mean spike rates, it does not explain the processes underlying the variability in spike timing that shape the ISI distribution shown in Figure 4. The following sections address the issue of what mechanisms generate the irregularity in spiking and control the timing of each spike in the model. ISI variability in vivo
As we have already described, in vivo spike trains from Purkinje cells and spike trains in the model produced in the presence of background synaptic input typically show a pronounced mode in the ISI distribution around 10 msec and a tail of long ISIs (Fig. 4). A comparison between different Purkinje cells recorded in vivo indicates that the shape of the ISI tail varies from cell to cell (Fig. 7A,B). Below, we first examine what characteristics of the synaptic input may account for a similar range in ISI distributions in the model, and then we analyze the underlying membrane currents producing spike variability (Fig. 7C-F).
Fig. 7. ISI distributions for Purkinje cell spike trains. A, B, Baseline spike rate over 20 sec of recording from two Purkinje cells obtained in vivo in the anesthetized rat. The mean, mode, and coefficient of variation (cv = standard deviation/mean) of the spike interval distribution are printed above. These recordings presented two typical cases in the range of ISI distributions found in a sample of 15 recorded cells. Typically, the spontaneous spike rate was high and the peak in the ISI distribution around the modal interval was pronounced. The tail of the distribution was frequently larger than expected from an exponential decay. A pronounced tail was associated with a large cv. Values of cv in our sample ranged from 0.3 to 1.4 (mean = 0.7). The small number of very short intervals (<7 msec) was likely a result of spontaneous climbing fiber input that resulted in a brief burst of two to three spikes. C-F, ISI distributions obtained with different synaptic input rates in simulations of 20 sec duration. See Results for description.
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Source of ISI variability The ISI distribution in the in vivo recording shown in Figure 7A was closely matched by a simulation using 30 Hz gc and 1 Hz sc input frequencies (Fig. 7C). These frequencies are in the middle of the range used in the analysis of membrane currents described above. Under these conditions, the tail in the ISI distribution was quite pronounced. In further simulations, we examined how the observed ISI distribution depended on variability of the synaptic input. First, we tested whether spike variability could be produced without any variability in the synaptic input. To do this, we partly opened each synapse in the model at a constant level, such that the constant synaptic conductance matched the mean synaptic conductance level with 30 Hz gc and 1 Hz sc input. As output, the model produced regularly spaced spikes with an ISI of 14.5 msec, which was close to the modal interval seen with 30 Hz gc and 1 Hz sc input. This result indicates that the tail of long ISIs is a consequence of fluctuations in the input.
Dendritic currents underlying spike variability The relationship between synaptic input and the timing of spike output is a central question in the study of neuronal information coding. In our analysis above, we have demonstrated that irregularity in spike timing in the model was a consequence of variability in synaptic input. Before we showed that voltage-gated membrane currents in the model are much larger than synaptic currents. This poses the question of how these voltage-gated currents interact with fluctuations in synaptic input, and what processes determine spike timing. To address this issue, we first examine the fluctuations of dendritic membrane currents in relation to spike timing caused by random input through many synapses.
Next we compared the relative contributions of excitatory and inhibitory input to spike variability in the model. Simulations were conducted in which either the gc conductance was constant in the presence of 1 Hz sc input, or the sc conductance was constant in the presence of 30 Hz gc input (Fig. 7D,E). We found that both simulations resulted in a realistic ISI distribution, but excitatory input variability resulted in a different ISI distribution than inhibitory input variability. In particular, a constant sc conductance with variable gc input resulted in an ISI distribution without a pronounced tail (Fig. 7D). This finding raised two possibilities, namely that long tails in the ISI distribution were a result of the kinetics of inhibitory input or were a result of the slow rate of sc input used. To dissociate between these two possibilities, we ran a simulation with 30 Hz sc and 30 Hz gc input and adjusted unitary inhibitory conductances such that the mean level of inhibitory synaptic conductance remained identical. This simulation did not have a tail in the ISI distribution (Fig. 7F), indicating that it was the low frequency of inhibitory input of previous simulations that was responsible for a pronounced tail in the ISI distribution rather than the kinetics of the sc conductance. In addition, we ran simulations in which we changed the long decay time constant of GABAA conductance from 26.5 msec in the standard model to 8 msec. When the reduced duration of IPSPs was compensated for by an increase in the peak conductance of single sc synapses, the resulting ISI distributions with a rate of 1 Hz sc input did show a pronounced tail. These results lead to the prediction that the tail of Purkinje cell ISI distributions in vivo is a result of a relatively low rate of inhibitory inputs.
Figure 8 shows the relative contribution of different dendritic currents to the time course of fluctuations in dendritic membrane potential. Figure 8A demonstrates that somatic spikes (spike times denoted by vertical dashed lines) were triggered only when the mean dendritic Vm was in a relatively depolarized state. It was surprising that synaptic currents contributed only a small part to the observed fluctuations in dendritic depolarization (Fig. 8B). Instead, fluctuations in voltage-gated currents were responsible for most of the increases in dendritic depolarization leading to somatic spiking (Fig. 8B). These fluctuations can be explained by the steep activation curve of the CaP conductance at this membrane potential and the ensuing secondary activation of calcium-dependent K conductances. Thus, a small fluctuation in membrane potential because of synaptic input can be amplified and changed in time course because of CaP and K conductance dynamics (De Schutter and Bower, 1994c 
Fig. 8. The contribution of individual currents to fluctuations in dendritic membrane potential. A segment of 650 msec for a simulation with 30 Hz gc and 1 Hz sc input is shown. A, Time course of Vmd for a period of 650 msec. B, Time course of the contribution to fluctuations in Vmd by synaptic and voltage-gated currents. C, Contribution of somato-dendritic and leak current. D, The sum of the contributions of the currents shown in B and C reconstructed the time course of Vmd. The contribution of each current was obtained by calculating the charge carried with the current (charge is integral of current) and then rescaling the charge to its equivalent change in membrane potential by Q = V × C (see Materials and Methods). The linear component of each current was removed before this procedure. Note that the Vmd trace starts and ends with the same potential and, therefore, that the sum of all linear current components was zero.
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Relationship among ISI duration, dendritic membrane potential, and dendritic current amplitudes Although Figure 8 clearly demonstrates the dominance of voltage-gated currents in de termining the trajectory of the mean dendritic Vm in the model, it does not directly address the issue of which currents determine the duration of ISIs. To examine how the level of dendritic depolarization and dendritic membrane currents were related to ISI duration in the model, we examined spike-triggered averages of these variables for spikes occurring before and after ISIs of different duration (Fig. 9). Spike-triggered averages were constructed for the mean voltage level of the dendrite (Fig. 9A), excitatory and inhibitory synaptic conductance (Fig. 9B), total synaptic current (Fig. 9C) and total voltage-gated current (Fig. 9D). Separate averages were triggered on the initial and the terminating spike of each ISI (Fig. 9, left and right columns, respectively). The data were further divided into spike-triggered averages for three different ranges of ISI duration, namely 10-13, 16-20, and 30-50 msec. Spike-triggered averages for these ranges are labeled 1, 2, and 3, respectively, in Figure 9.
As shown in Figure 9A, the dendritic membrane potential (Vmd) before the initial spike of each ISI was, on average, the same for all three sets of ISI durations. Thus, the initial state of dendritic depolarization was no indication of the subsequent ISI duration for these spike-triggered averages. Immediately after the first spike in an interval, however, the voltage traces for the three sets of ISI duration diverged. The trace in which the terminating spike followed most rapidly (10-13 msec) maintained a more depolarized state after the initial spike than the trace representing 30-50 msec intervals. The depolarization for 16-20 msec intervals was intermediate between the sets of short and long intervals. The 95% confidence limits for the population means of the different sets of intervals show that these differences were highly significant (Fig. 9A, dashed lines). The difference in the level of dendritic depolarization for short and long intervals remained significant until the terminating spike occurred (Fig. 9A, right panel).
The level of sc conductance was well related to ISI duration, whereas the level of gc conductance was not (Fig. 9B). In contrast to Vmd, the level of sc conductance showed a dependence on ISI duration already before the initial spike of the respective ISI (Fig. 9B). In particular, the net sc conductance in the 5 msec preceding an ISI was significantly larger before an ISI of 30-50 msec duration than before a 10-13 msec ISI. This difference in sc conductance for ISIs of different duration increased in the first 10 msec after the initial spike and decayed gradually before the terminating spike of the respective ISI. Therefore, a strong effect of inhibitory input on ISI duration occurred early during the ISI. In contrast, the gc conductance showed no difference for ISIs of different durations.
The total synaptic current for the three sets of ISIs reflected the increase in sc conductance in that it was significantly more outward for the set of long ISIs than the set of short ISIs. The increase in current was less pronounced than the increase in conductance, however, because the driving force was reduced at the less depolarized Vm associated with long ISIs. The charge carried by the increased outward synaptic current for long ISIs in the period between 5 msec before and 10 msec after the initial spike accounted for a decrease in dendritic depolarization of 0.87 mV. Somatic currents controlling the timing of individual spikes Ultimately the timing of somatic sodium spikes is dependent on the control of NaF activation in the soma. As we showed in Figure 3, spontaneous somatic spiking in the absence of synaptic input was the result of a dynamic push-and-pull operation between the soma and the dendrite. In Figure 10, we examine this process in the presence of synaptic input for single examples of a short and of a long ISI. As in the case without synaptic input, we find that spike after-hyperpolarization was overcome through a current boost from the dendrite, which remained depolarized throughout the spike cycle. For the short ISI (Fig. 10A), this current boost depolarized the soma sufficiently to lead to a monotonically increasing activation of the NaF current, which directly resulted in a second spike. In contrast, for the long interval (Fig. 10B), the NaF current was activated less strongly during the current boost from the dendrite, and a tug of war between the subsequent outward current into the dendrite and the inward NaF current ensued. A second spike was generated only when the NaF current escaped the dendritic current sink.
Figure 9D shows that the net inward level of voltage-gated dendritic currents was significantly decreased for long ISIs. As with synaptic current, this difference started before the first spike in the respective ISI. The amplitude of the mean difference for voltage-gated currents between short and long ISIs for consecutive windows of 5 msec duration starting at 5 msec before the first spike was 0.24 nA, 0.39nA, and 0.4 nA, respectively. Overall, the decrease in inward voltage-gated current with long ISIs accounted for 1.2 mV of dendritic hyperpolarization over 15 msec. Voltage-gated current, therefore, contributed more to the relative dendritic hyperpolarization seen with long ISIs than did synaptic current. It was of interest that the influence of voltage-gated currents was reversed just before the terminating spike of an ISI (Fig. 9D, right panel). In effect, the increase in inward voltage-gated current before the terminating spike of a long ISI contributed to the depolarization of the dendrite necessary to trigger a somatic spike. This time course again indicates that the dendritic currents around the onset of an ISI determined ISI duration in the case of random input through many synapses. 
Fig. 10. A, Somatic currents during a 15 msec ISI. B, Same currents during a 23 msec ISI. The black arrows denote the time at which the current from the dendrite into the soma (I s-d) reversed, turning the dendrite from a current source into a current sink. As described in the text, the amplitude of the NaF current around the time of I s-d reversal was critical for obtaining a short or long ISI.
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Figure 10 suggests that the state of the somatic NaF current at the time when the phasic boost of current from the dendrite ends (I s-d reversal: Fig. 10, black arrows) may be important in determining ISI duration. This idea is examined more closely in Figure 11A, which shows a scatter plot of the NaF current at the time of I s-d reversal for 261 ISIs of varying duration. In fact, for intervals shorter than 20 msec the amplitude of NaF at I s-d reversal showed a high degree of correlation with ISI duration (r2 = 71%). For longer intervals this correlation broke down, suggesting that events subsequent to I s-d reversal determined the precise duration of long intervals. Because the current flow from the dendrite into the soma is proportional to the voltage difference between the two compartments, it seemed likely that the level of dendritic depolarization at the time of I s-d reversal might be an important factor in controlling NaF activation. As we saw above in Figure 9A, dendritic depolarization was significantly different for short than for long ISIs at the time of somatic spike after-hyperpolarization. Figure 11B shows that the activation of NaF at the time of I s-d reversal was indeed highly correlated (r2 = 80%) with the level of dendritic depolarization at this time. This correlation indicates that somatic redepolarization after the initial spike of an ISI was under tight control of dendritic depolarization and that NaF was largely activated as a function of this process. Given that the level of NaF activation at the time of I s-d reversal correlated well with Vmd, it is not surprising that Vmd showed a similar correlation to ISI duration at this time point, as did NaF (Fig. 11C). This result extends the observation that Vmd determines spike timing obtained from spike-triggered averaging to the level of single ISIs (Fig. 9A). As we saw in Figure 8, voltage-gated current contributed more to dendritic depolarization than did synaptic current. In fact, we failed to find any significant correlation between the amplitude of excitatory synaptic conductance at any point in time during an ISI and the duration of ISIs. The level of inhibitory conductance at the time of I s-d reversal on the other hand did show a significant correlation with ISI duration (Fig. 11D; r2 = 24% for ISIs shorter than 20 msec). Nevertheless, this correlation was much weaker than the correlation of Vmd itself with ISI duration, supporting the findings above that intrinsic properties of the cell have a large influence on ISI duration. In distinction to NaF and Vmd, the level of sc conductance showed also a small but significant correlation with the duration of long ISIs (r2 = 14%), suggesting that the duration of long ISIs was partly predetermined at the early time of I s-d reversal through the level of inhibition. The precise events that led to the termination of long ISIs were not examined. 
Fig. 11. Scatter plots relating current levels and dendritic depolarization at the time of I s-d reversal to ISI duration. Each circle denotes the values for a single ISI out of 7 sec of simulated data for 30 Hz gc and 1 Hz sc input. Because short intervals (<20 msec) generally had a higher degree of correlation with different variables at the time of I s-d reversal, two linear regressions were calculated for short and long ISIs, respectively. The correlation coefficients (r2) for short and long ISIs are printed above each plot. A, Activation of NaF at the time of I s-d reversal versus ISI duration. B, NaF activation versus Vmd. Values associated with short ISIs (<20 msec) are denoted by circles; values for long ISIs are denoted by asterisks. C, Vmd at the time of I s-d reversal versus ISI duration. D, Total sc conductances at the time of I s-d reversal versus ISI duration.
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Overall, the relatively small control of synaptic input over the timing of somatic spikes was shown multiple times in the preceding analyses. This result ensued from dendritic voltage-gated conductances, which caused a strong indirection in the influence of synaptic input on spike timing. As discussed below, these findings have important implications for neuronal coding by Purkinje cells in the cerebellar cortical network.
DISCUSSION
We examined in a realistic Purkinje cell model how intrinsic currents interact with synaptic input to control somatic spiking. Although we supported model behavior with physiological data, the validity of any modeling results depends on the choice of model parameters and analysis procedures. We will first discuss our modeling assumptions. We will then consider the significance of the present results with respect to the input-output function of the Purkinje cell and implications for cerebellar function. Finally, we will propose several experimentally testable predictions based on our analysis.Validity of modeling results
Limitations because of disregarding spatial aspects of dendritic function Although summing currents across the whole dendritic tree reduced the complexity of our analysis significantly, this technique neglects all spatial aspects of dendritic processing. Several factors mitigate this problem in the present study. First, dendritic conductances were distributed uniformly in the model, thus avoiding spatially restricted modes in current activation. Recent calcium imaging studies support the concept of uniformly distributed conductances in the Purkinje cell dendrite (Lev-Ram et al., 1992
Recent studies using whole-cell recording techniques (Llano et al., 1991
The value of specific membrane capacitance (Cm) of 1.64 µF/cm2 used in the model is considerably higher than the estimate of Cm of 0.8 µF/cm2, which has been obtained recently in a careful study of hippocampal pyramidal cells (Major et al., 1994 Control of spiking in the Purkinje cell model
The general question of what information about synaptic input is coded in spike has recently received a lot of attention (Shadlen and Newsome, 1994Functional implications for cerebellar processing
Theorists have traditionally focused on the near-crystalline anatomical layout of cerebellar cortex to construct models of cerebellar function (Marr, 1969Functionality of the Purkinje cell in the cerebellar network Our modeling results indicate that the baseline level of open excitatory and inhibitory synaptic conductances exerted a partial voltage clamp on the dendrite, and that actual dendritic membrane potential stayed close to the clamping voltage determined by the balance of excitatory and inhibitory synaptic conductances. As a consequence, changes in the activity of any set of synaptic inputs were effective in changing the dendritic membrane potential. In turn, somatic spiking was very sensitive to small changes in dendritic potential (Fig. 6). This description of Purkinje cell behavior is quite different from that of a summation device of excitatory inputs used in most established cerebellar theories (Marr, 1969
Spike timing was also much more sensitive to rapid changes in inhibitory inputs than in excitatory inputs (Figs. 9B, 11D). The essential role of inhibition in determining spike rate and timing has not been recognized in established theories of cerebellar cortical function (Marr, 1969
Although not examined in the present study, synchronous inputs from many excitatory synapses produce time-locked spike responses in the Purkinje cell model (De Schutter, 1994 Model-based experiments and predictions
The present analysis identifies several model parameters that are critical for the described behavior. The results also lead to specific predictions about the role of synaptic input in controlling Purkinje cell spiking. Because of the direct correspondence between data traces generated by the model and physiological traces, we can propose specific experiments that test the validity of critical model assumptions, as well as our functional predictions. Voltage-gated membrane currents The P-type calcium current was crucial in the model to keep the dendrite depolarized, and it was strongly modulated with changes in synaptic input. Experimental findings support the presence of sustained dendritic plateau depolarizations because of the CaP conductance (Llinás and Sugimori, 1980b
ABSTRACT