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1Computation and Neural Systems Program, California Institute of Technology 216-76, Pasadena, California; and 2Research Imaging Center, University of Texas Health Science Center at San Antonio and the Cajal Neuroscience Research Center at the University of Texas, San Antonio, Texas
Submitted 3 May 2004; accepted in final form 8 May 2004
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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; Euler and Denk 2001; Hausser and Mel 2003; Migliore and Shepherd 2002). Although neuronal dendrites have traditionally most often been modeled as spatial and temporal summing devices (see London and Segev 1999; Rall 1999), it is clear that the presence of active conductances in dendrites substantially affects how these structures integrate synaptic inputs (Cook and Johnston 1999; De Schutter and Bower 1994c; London and Segev 2001; Poirazi et al. 2003; Williams and Stuart 2002). In fact, model (Chance et al. 2002)- and experimental-based (Frick et al. 2004) studies of the functional consequences of active dendritic membranes are starting to raise questions about whether all synapses are fundamentally "driving" in nature. Specifically, it is now being suggested that some synapses might be more involved in regulating dendritic dynamics than in directly influencing the spiking output of neurons (Chance et al. 2002; De Schutter and Bower 1994b; Jaeger and Bower 1999; Jaeger et al. 1997; Rhodes and Llinas 2001; Santamaria et al. 2002; Sherman and Guillery 1998).
The cerebellar Purkinje cell was one of the first mammalian neurons in which experimental studies demonstrated substantial voltage-dependent dendritic conductances (Llinas and Sugimori 1980). Using an articulated set of modeling and experimental studies, we have engaged in a series of experiments intended to better understand the functional consequences of these highly active dendrites (De Schutter and Bower 1994b; Jaeger and Bower 1999; Jaeger et al. 1997; Santamaria et al. 2002). These studies have suggested that the distinction between synapses responsible for controlling dendritic dynamics (modulatory), and those directly influencing Purkinje cell output (driving) may have a specific anatomical correlate in the cerebellar cortex (Gundappa-Sulur et al. 1999). Specifically, we have proposed that the large number of parallel fiber synapses made by granule cells on Purkinje cells may be responsible for controlling the dynamics of the dendrite and not for directly driving the spiking output of the cell (Jaeger and Bower 1999; Jaeger et al. 1997). Previous physiological (Bower and Woolston 1983; Cohen and Yarom 1998; Jaeger and Bower 1994), anatomical (Gundappa-Sulur et al. 1999), and modeling studies (De Schutter and Bower 1994c; Jaeger et al. 1997; Santamaria et al. 2002) suggest that the timing of Purkinje cell spiking activity is more directly under the control of synapses associated with the ascending segment of the granule cell axon. This proposed differential functionality of synapses actually associated with the same axon is based on anatomical differences in the location of synapses (Gundappa-Sulur et al. 1999; Mugniani 1972), the dynamic properties of the large voltage-dependent conductances found in the Purkinje cell dendrite (Llinas and Sugimori 1980), and differences in the influences of inhibitory molecular layer interneurons on each type of input (Hausser and Clark 1997).
In this report, we describe the results of computer simulations intended to further predict interactions between parallel fiber and molecular layer interneuron synaptic inputs and those provided by the ascending granule cell axon segment. Specifically, we have focused on the consequences of the relatively long time constants associated with dendritic voltage-dependent conductances on the Purkinje cell model's response to two inputs from ascending segment synapses paired relatively closely in time. These responses have been evaluated in the context of different levels of background synaptic activity provided by parallel fiber synapses and inhibitory molecular layer interneurons. We demonstrate that the relatively long time constants of the intrinsic voltage-dependent dendritic conductances result in strong interactions between paired ascending segment inputs over periods of time as long as 120 ms. The model also predicts that the strength and duration of these interactions is modulated by the level of concurrent activity in parallel fibers and molecular layer interneurons. Taken together the results suggest a physiological mechanism for the dendritic interaction of modulatory and driving synapses that are particularly tuned to analyze sequences of afferent input.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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The model used in this study is based on an anatomically reconstructed Purkinje cell, discretized into 1,600 compartments based on a previous analysis of the passive electrical properties of these neurons (Rapp et al. 1994). The membrane capacitance was set as 1.64 mF/cm2; the membrane resistance was 10 k
cm2 in the soma and 30 k cm2 in the dendrites; and the axial resistance was 250 cm. These passive properties result in a time constant of 46 ms and an input resistance of 19.6 M. The active dendritic compartments contained two types of Ca channels, a P-type, CaP (Llinas et al. 1989) and a T-type, CaT (Kaneda et al. 1990); two types of Ca-activated K+ channels, a BK-type, Kca (Latorre et al. 1989) and a K2-type, K2 (Gruol et al. 1991); and a persistent K+ channel. The soma includes two types of sodium channels, a fast current, NaF (Hirano and Hagiwara 1989) and a slow persistent current, NaP (French et al. 1990); one type of calcium current T-type; and four types of potassium channels, anomalous rectifier, Kh (Spain et al. 1987), delayed rectifier, Kdr (Yamada et al. 1989), persistent potassium, Km (Yamada et al. 1989), and an A-type, KA (Hirano and Hagiwara 1989). The ion channels had the following conductance function (De Schutter and Bower 1994a)
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). Previously published results have shown that the model's responses to current injection are robust to changes in densities of all of the ionic channels (De Schutter and Bower 1994a). In general, modifying densities by factors 
2 result in left or right shifts of the frequency-current curve. However, changes of >20% to the amount of P-type Ca channels of one of the Ca-activated K channels in the model either suppressed dendritic spikes or cause the model to always fire Ca spikes (De Schutter and Bower 1994a). Synaptic inputs
On the basis of previous physiological (Bower and Woolston 1983; Cohen and Yarom 1998) as well as anatomical studies (Gundappa-Sulur et al. 1999), granule cell synapses were divided into those made by the parallel fibers and those made by the ascending segments of the granule cell axon. Both types of inputs were modeled as AMPA-like synapses (Farrant and Cull-Candy 1991). These excitatory synapses were assumed to have a peak conductance of 0.7 nS, reversal potential of 0 mV, rising time constant of 0.5 ms, and closing time constant of 1.2 ms. The inhibitory synapses associated with molecular layer interneurons were modeled with a conductance of 1.4 mS/cm2 in smooth dendritic compartments and 7 mS/cm2 in spiny compartments with an opening time constant of 0.9 ms, a closing time constant of 26.5 ms, and a reversal potential of –80 mV. We did not implement basket cell synapses because the objective of this study was to understand dendritic dynamics, and previous analysis of our model has shown that the basket cell input to the soma has little physiological influence on the dendrite (De Schutter and Bower 1994b).
Parallel fiber inputs were restricted to spiny dendrites with diameters between 3.15 and 1.5 µm, while ascending segment synapses were made to contact dendrites with diameters <1.5 µm (Gundappa-Sulur et al. 1999; Palay and Chan-Palay 1974). Ascending segment input was distributed over eight different branchlets (De Schutter and Bower 1994c; Santamaria et al. 2002). Inhibitory interneuron synaptic inputs were distributed uniformly over the spiny branchlets (Fig. 1, A–C).
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Continuous random (Poisson) patterns of background parallel fiber and inhibitory interneuron synaptic input were provided to the Purkinje cell dendrite. To reduce the computational complexity of the model, this background activity was provided by a fraction (1,600 granule cells and the same number of passive spines) of the >160,000 parallel fiber inputs that actually project onto mammalian Purkinje cells (Harvey and Napper 1991). We compensated for this reduced number of inputs by increasing the firing rate of these synaptic inputs. Although the background firing rate of parallel fiber inputs in the cerebellar cortex is not yet known, we have previously shown that simulated interspike interval distributions for the Purkinje cell are relatively insensitive to proportional variations in the number of synapses and the background firing rate (De Schutter and Bower 1994b).
The number of modeled inhibitory inputs contacting the Purkinje cell dendrite in these simulations was 1,695, which is probably close to the real number of this type of input (Sultan and Bower 1998). Because in these studies the Purkinje cell model received random background levels of excitatory and inhibitory synaptic activity, these two synaptic inputs were not correlated.
Data analyses and simulations
To analyze Purkinje cell responses to synaptic input, peri-stimulus time histograms (PSTHs) were constructed using a fixed bin width of 2 ms bins. Histograms are the standard form for the presentation of Purkinje cell responses in the experimental and modeling literature. Response significance was determined using t-test by dividing the total number of trials (usually 300), into subsets of 10. A PSTH was generated from each subset (30 in total). Standard deviations (SDs) and standard errors (SEs) were determined based on these 30 samples. In this way, we were able to study the variability of the response to different stimuli. To quantify the change in response to the second stimulus, we normalized the height of the peak evoked by the second stimulus by the height to the first, see Fig. 3. To facilitate comparisons between different PSTHs, each histogram was normalized to the number of trials. In this case the y axis shows probability instead of instantaneous firing rate. We used the peak of the PSTH as opposed to other measurements of the response (e.g. area) because we have previously shown that synaptic input does not so much change the number of spikes but their distribution (Santamaria et al. 2002).
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). In some cases, we also studied the total dendritic conductance, which was calculated as the sum of the conductances in all dendritic compartments, Stot = KM + KA + KC + K2 + H + CaP + CaT + leak + excitatory synapses + inhibitory synapses. The effects of synchronous ascending segment inputs were isolated by subtracting total currents generated in two simulations with identical background synaptic inputs but differing in the presence or absence of the synchronous excitatory (ascending segment) input.
A second method for analyzing membrane currents used here involves the construction of phase planes comparing current levels in two different ionic currents. This data representation allows a better understanding of the dynamical influence of interacting conductances on responses to synaptic input (Rinzel and Ermentrout 1998; Santamaria et al. 2002).
All simulations were implemented in GENESIS (Bower and Beeman 1999) running on a Cray T3E operated by the San Diego Super Computer Center and Linux machines. Analyses were carried out using Matlab (Natick, MA).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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Purkinje cell somatic responses to paired ascending segment inputs
Figure 1D shows four somatic voltage traces of the Purkinje cell model when stimulated with a background parallel fiber activity of 15 Hz and a molecular layer interneuron of 0.5 Hz. At the time indicated (
), identical ascending segment stimuli were delivered (160 synapses). The raster plot shown in Fig. 1E indicates that the response to the Purkinje cell is more sensitive to the second stimulus than to the first. Note, however, that the average firing rate in the 100 ms after the onset of the first stimulus (85 Hz) does not differ from the background firing rate of the Purkinje cell without ascending segment stimulation (86 Hz). Therefore the primary effect of the stimulus is to change the temporal distribution of Purkinje cell output not the overall rate of spiking (Santamaria et al. 2002.)
Figure 2 shows responses of the Purkinje cell model to a series of paired inputs from 160 (left) or 80 (right) ascending segment synapses separated by different ISIs. Background inputs for these simulations were fixed at 15 Hz for parallel fibers and 0.5 Hz for molecular layer interneurons producing an average Purkinje cell firing rate of 86 Hz. Figure 2, A and E, shows control responses to a single synchronous activation of the ascending segment synapses. These are similar to those obtained physiologically in response to a single peripheral stimulus (Bower and Woolston 1983). As described in more detail in the following text, the multiple peaks in Fig. 2, C and D, after the second ascending segment stimulation are due to the activation of a series of interacting dendritic currents and are not due to a somatic bursting mechanism (see Fig. 1D). As also described in later parts of RESULTS, these multiple peaks only occur when low inhibitory background firing rates are present.
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Figure 3 shows summary data for a larger range of ISIs under the same background input conditions as seen in Fig. 2. This figure was constructed by plotting the ratio of the second to the first excitatory peak for ISIs ranging from 10 to 115 ms. As described in METHODS, we used the peak of the PSTH response because the average number of spike per trial remains very similar in the presence or absence of ascending segment stimulation. This "balance" of spikes per trial resulted in no change in the response if the window of integration was increased from 2 ms (the bin size of our PSTHs) to as little as 6 ms (data not shown). This particular graph indicates that ISIs of <30 ms result in a suppression of the response to the second input, whereas intervals between 30 and 40 ms result in an enhancement of the response to a second input. After 60 ms there is again a zone of depression that persists until ISIs are >100 ms. As already noted, the largest enhancement of the second response is 
40 ms.
Influence of background synaptic activity on the somatic response
The data reported in the previous section was obtained using a fix level of background parallel fiber and molecular interneuron synaptic input for all simulations. Figure 4 shows the ratio between amplitudes of the second and first response after activation of 160 ascending synapses for inputs separated by 10–60 ms over a wider range of background input frequencies. The simulation data shown in this figure clearly indicates that the relative levels of background synaptic input from parallel fiber and molecular layer interneurons can significantly influence the Purkinje cell model's response to paired inputs. For example, depending on the level of background inputs, the response to the second stimulus separated at a 40 ms ISI can be amplified (A, B, C, G, and K) or suppressed (E). The particular combination of background inputs also varies the duration of changes in response to the second stimulus. For example, Fig. 4, A–D shows that the region of amplification in response to the 2nd stimulus decreases as the parallel fiber frequency increases. A similar effect is shown in E–H where at low background parallel fiber input there is a slight depression of the response to the second stimulus (E) that becomes a wide amplification zone at higher background parallel fiber frequencies.
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Comparison between the somatic spiking and dendritic currents
After investigating the effects that different levels of background excitatory and inhibitory synaptic activity have on Purkinje cell somatic responses, we were interested in better understanding the interactions between dendritic currents responsible for the observed behavior. One possible explanation for the somatic response is a change in input resistance caused by synaptically induced changes in total dendritic conductance. Figure 5A shows the total dendritic conductance (KM + KA + KC + K2 + H + CaP + CaT + leak + excitatory synapses + inhibitory synapses) during the 25 ms prior to the first stimulation for several pairs of excitatory and inhibitory inputs. Interestingly, the total dendritic conductance is relatively constant for fixed background levels of inhibitory input and varying background levels of parallel fiber, and Purkinje cell, firing rates. For example, while the somatic responses of the cell when receiving 13- and 19-Hz excitatory input and the same 0.5-Hz inhibitory input are completely different (Fig. 4, A and D), the total dendritic conductance varies <20% (Fig. 5A). We also measured the maximum amplitude of the total dendritic conductance after the first and second stimuli for several levels of background stimulations. We chose an ISI of 40 ms because this interval often gave the highest amplification of the somatic response. For the cases shown in Fig. 5A, the dendritic conductance amplitude between the first and second stimuli is <20% and decreases as the inhibitory background input increases (Fig. 5, B and C). We also compared the absolute dendritic conductance just before the first and second stimuli and found a difference of at most 13%. In the examples shown in Fig. 5B, the value of the total dendritic conductance at 40 ms after the first stimulus relative to its steady level before the stimulus decreases as the excitatory input increases. However, this relative increase in input impedance is not reflected in an increase of the somatic response. A closer analysis of conductances changes under different conditions indicate that most are related to the P-Type calcium (CaP) and the two calcium-activated potassium (Kca) dendritic channels (Fig. 5D), whereas the other dendritic conductances essentially average out.
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) and then experimentally (Jaeger and Bower 1999), that the dendro-somatic current is responsible for regulating the spiking activity of the Purkinje cell. This current counter-balances the somatic currents when stimulated with background activity and directly drives the soma to fire when a direct stimulus is delivered (De Schutter and Bower 1994c; Jaeger and Bower 1999; Jaeger et al. 1997). We have also shown that the dendritic CaP and Kca currents in the model are the most activated under this kind of stimulation while the other dendritic currents are averaged out (Santamaria et al. 2002). For this reason, the analysis here examines the balance between CaP and Kca currents instead of dendritic conductances as the best measure of model behavior.
Figure 6 plots the relationship between the shape of PSTH responses and induced changes in the average dendritic CaP and Kca currents. These data are presented for a single input provided at t = 0 ms in A and C and for paired stimuli separated by 40 ms in B and D. The current data presented in C and E were obtained by subtracting steady state values for CaP and Kca (see METHODS). These figures therefore indicate net changes in CaP and Kca currents induced by the stimulus. Figure 6C indicates long-lasting changes in CaP and Kca after a single synchronous ascending segment input. Note that this figure shows the net change in CaP and Kca currents from steady state and therefore a negative deflection for CaP means that there is more current through the CaP channel than under baseline (i.e., prestimulus) conditions. Inversely, a positive deflection indicates less current flowing than in baseline conditions and not that Ca is flowing out of the neuron. Analyzed in this way, the response of these dendritic currents, on average, is characterized by a damped oscillatory alternation in relative changes in CaP and Kca currents compared with baseline. The reduction in Purkinje cell firing rate after the stimulus corresponds to a time when there is less net calcium current flow into the dendrite than under baseline conditions and when, accordingly, there are also less Kca channels open (Jaeger et al. 1997; Santamaria et al. 2002).
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Figure 6E represents the net result of the second stimulus on dendritic currents obtained by subtracting the currents in Fig. 6D from those in C. Comparing Fig. 6, C and E, indicates that the current dynamics for the second impulse are very similar to the response to a single stimulus alone. The net effect in D therefore is a result of current summation.
The conclusion from this analysis is that the enhancement of the PSTH response to the second stimulus is due to the state of the dendritic currents when the second stimulus is presented. This conclusion is also supported by the phase plane plot shown in Fig. 6F in which CaP is plotted against Kca currents during the entire paired stimulus condition. The small solid arrows mark the direction of the trajectory of the plot, and the thick solid arrow indicates the timing of the second stimulus. As already shown, with a 40 ms interval separating the two inputs, the second stimulus occurs when the net flow of Ca into the cell is the smallest with respect to steady state, and the net flow of K in Kca currents out of the cell is also the largest. The phase plane plot also shows that the relative level of CaP and Kca currents follow a trajectory in which both currents switch signs. These relative levels reflect a different degree of interaction between these currents. A dynamical system as the one studied here is further characterized by the instantaneous response of the currents to the first stimulus at steady state or to the second stimulus at different points along the CaP-Kca trajectory. This response will determine the direction and relative strength of each current (Morris and Lecar 1981; Rinzel and Ermentrout 1998). This interaction can be quantified by calculating the slope of the instantaneous response (m = 
Kca/Ca). The dashed arrows parallel to the trajectories indicate the sections where the slope was measured, between 0.5 and 2 ms after each stimulus. The slope after the first stimulus is m1 = –0.37, while the slope after the second is m2 = –0.29. The steeper, or more negative the slope, the faster the response of Kca compared with CaP. Thus the analysis shows that the Kca currents are in a state to compensate faster for the influx of CaP resulting from the first than the second stimulus. This difference results in a larger net CaP driven voltage change to the second stimulus. In the model, 20% more calcium enters the dendrite per K exiting in response to the second stimulus when compared with the first input.
In Fig. 7, the phase plane analysis just described is extended to ISIs of 20 (A) and 50 ms (C), respectively. For comparison, the plot shown in Fig. 6F is also reproduced here (B). In Fig. 7A, the suppression of the response to the second input with an ISI of 20 ms results from the high conductance of the Kca currents at the time of input onset. The Kca channels are thus primed to limit the growth in CaP that would normally result from synaptic depolarization. In the case of a 50 ms ISI (Fig. 7C), the current in both the CaP and Kca channels are close to 0 resulting in a response to the second stimulus that is similar to the response to the first stimulus alone. Nevertheless as seen in the phase plot, the trajectories still diverge after the second stimulus, resulting in a slight increased excitability.
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In the analysis of Fig. 4, we showed that the differential response of the model to a second stimulus was dependent both on the ISI and the background state of parallel fiber and molecular layer inhibitory inputs. Figure 8 shows a phase plane analysis of the Kca and CaP currents for a range of ISIs for four different levels of background synaptic input. Note that these background inputs all result in ongoing spiking activity in the relatively narrow range of 80–88 Hz. As suggested by the results in Fig. 4, this analysis indicates that Purkinje cell current behavior after the second stimulus has a similar trajectory as the current response after the first stimulus as the frequency of background input, especially inhibitory input, increases. Second, with increased background activity, the phase plot trajectory diminishes, reducing the distance between the steady state (Kca = 0, CaP = 0) values and other states of the dendrite. Thus a second stimulus whenever it is presented sees current more similar to those at steady state and thus the condition of the first stimulus. In the plots, a small arrow indicates the point where the second stimulus was delivered, no arrow means that this point was indistinguishable from the background activity around the center.
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Time constant of Kca to changes in [Ca2+]i is critical in controlling the paired-pulse interaction
As just described, the analysis of the currents underlying the model's differential responses to paired-pulse stimulation suggest that the key parameter in the behavior of the model is the interaction of Kca with CaP and therefore the time constant of the Kca to changes in [Ca2+]i. Other possible sources for this interaction involve a combination of the membrane time constant, and the voltage time constant of the CaP and Kca currents. However, the membrane time constant of the PC = 46 ms; the voltage activation time constant for the CaP is between 0.05 and 1 ms, and the inactivation is 1,000 ms; the voltage activation time constant for the Kca channels are <0.1 ms. Therefore the Kca time constant to [Ca2+]i is the only parameter within the order of magnitude of the described dynamics. Based on our previous analysis of the currents in the model, we would predict that if the Kca time constant to [Ca2+]i is shorter than the Kca-CaP trajectories will be shorter and that there would be a reduced period of time after the first stimulus in which the response to the second stimulus would be amplified. If the Kca time constant is larger, we would predict a wider time window for amplification. To test these predictions, we ran a series of simulations in which we varied the [Ca2+]i time constants of both Kca channels from 5 to 25 ms (originally 10 ms in both cases). Figure 9A shows the firing rate of the Purkinje cell to different pairs of excitatory and inhibitory inputs for these different Kca time constants. The figure shows that the robustness of the model's firing rate to changes in this time constant is related to the level of inhibitory input. Only in one of the cases studied did the firing rate became oscillatory (). To study variations in channel parameters on dendritic and somatic responses to single and paired ascending segment stimulation, we chose simulations in which the firing rate of the Purkinje cell did not change significantly from 80 Hz (10%<). Figure 9B shows phase plane plots after a single stimulus for Kca time constants ranging from 5 to 25 ms for a Purkinje cell stimulated with 56 Hz parallel fiber and 2 Hz molecular interneuron background activity. As expected, as the time constant increases the trajectories become larger, and longer (note the different scale in the phase plot corresponding to a Kca time constant of 25 ms—compare with phase plot in Fig. 8). Our prediction from previous analysis is that in this case somatic responses to a second input would be further amplified with respect to the response to the first. To test this prediction, we calculated the amplification or suppression of the somatic response to the second ascending segment stimulus for several of the modified Kca time constants shown in Fig. 9A, choosing the ISIs that give the largest difference from control. This analysis demonstrated that changing the Kca time constant to 5 ms when the Purkinje cell is receiving a background excitatory frequency of 13- and 0.5-Hz (as in Fig. 4A) inhibitory results in almost no amplification of the second response compared with the first (Fig. 9C) while increasing the Kca time constant to 15 ms in the case of background excitatory 56- and 2.0-Hz inhibitory (as in Fig. 4O) increases this amplification (Fig. 9D).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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Parallel fiber and ascending segment synapses
With upward of 160,000 excitatory synaptic inputs, the cerebellar Purkinje cell is believed to receive a greater synaptic convergence than any other neuron in the brain. Except for a few hundred synapses associated with the climbing fiber input, all the remaining excitatory synapses originate from granule cells (Llinas 1990) with the majority associated with the parallel fiber segment of the granule cell axon (Harvey and Napper 1991). Each parallel fiber extends for several millimeters in a direction orthogonal to the planar structure of the Purkinje cell dendrite (Harvey and Napper 1991; Pichitpornchai et al. 1994). The size of this synaptic input and the strict geometrical relationship between the associated axon and the Purkinje cell dendrite has been a dominate influence on all structural theories of cerebellar function (for review, see Bower 2002).
Although there is no doubt that synapses associated with the parallel fibers provide a massive input to Purkinje cells, physiological results going back >35 yr have suggested that this input may not directly drive somatic spiking. Thus for example, Bell and Grimm (1969) reported no spiking correlations in Purkinje cells >100 µm apart even though the recorded neurons presumably shared a common set of parallel fiber inputs. In 1972, Eccles and colleagues specifically proposed that parallel fiber synaptic effects were subthreshold when they failed to find elongated beams of activated Purkinje cells after peripheral afferent activation (Eccles et al. 1972). This result was confirmed and extended by our own report in 1983 that a focal excitation of the granule cell layer resulted in a restricted activation of only immediately overlying Purkinje cells (Bower and Woolston 1983). Since then several other authors have reported similar results (Cohen and Yarom 1998; Kolb et al. 1997).
In response to our first report of restricted Purkinje cell activation (Bower et al. 1980), Llinas proposed that the somatic output of the Purkinje cell was actually driven by synapses associated with the ascending segment of the granule cell axon and that the parallel fibers provided a modulatory influence (Llinas 1982). Subsequent physiological (Cohen and Yarom 1998; Jaeger and Bower 1999; Kolb et al. 1997) and anatomical (Gundappa-Sulur et al. 1999) studies have supported a driving role for the ascending segment synapses; however, difficulties in physiologically controlling and manipulating parallel fiber activity have limited experimental attempts to better define the nature of parallel fiber modulation. Better defining the possible interactions between ascending segment and parallel fiber inputs was a principle motivation for the development of the Purkinje cell model described here.
Long-duration modulatory effects
Each of our previous modeling studies (De Schutter and Bower 1994a–c; Jaeger et al. 1997; Santamaria et al. 2002) has supported a role for parallel fibers in modulating the responsive properties of the Purkinje cell dendrite and has associated the spiking behavior of the neuron as a whole with the dynamics of the large voltage-dependent currents found in the Purkinje cell dendrite (Llinas and Sugimori 1980). Not only are these currents large, but they also have long time constants, which once activated, influence membrane dynamics as well as somatic output over several tens of milliseconds. As a consequence, for example, our models predict that even a single synchronous activation of ascending segment inputs results in fluctuations in somatic spiking for considerable periods of time (Santamaria et al. 2002). These fluctuations resemble variations seen in PSTHs, in vivo, in response to single peripheral stimuli (Bower and Woolston 1983; Garwicz et al. 1998; Jorntell and Ekerot 2002).
In the study reported here, we have specifically explored the consequence of these long duration effects for the predicted response of Purkinje cells to inputs paired at short latency. Our modeling results predict interactions between responses to paired stimuli can also occur over tens of milliseconds, but that the duration and nature (e.g., increased or decreased amplitude) of this effect is dependent on the rates of background synaptic input mediated through the large voltage-dependent Ca and Kca currents. In fact, the proposed modulatory influence of background rates of parallel fiber and molecular layer inhibitory synaptic inputs is greater on the response of the model to paired pulses than it is in response to single inputs (Santamaria et al. 2002) or to control of background spontaneous firing activity (De Schutter and Bower 1994a, b; Jaeger et al. 1997). It would therefore appear that Purkinje cells might be particularly tuned or sensitive to different temporal afferent activation patterns. Unfortunately, to date, most experimental studies of cerebellar response properties to tactile stimulation have been conducted using isolated single stimuli (Bower and Woolston 1983; Chadderton et al. 2004; Morrissette and Bower 1996).
Predicted critical parameters
Any realistic neuronal modeling effort is, of course, intrinsically limited by the information currently available on the anatomy and physiology of the cell being modeled. Viewed the other way around, modeling studies can serve to highlight those parameters most in need of further experimental refinement and provide a context for the results once obtained. In the case of the current model, our results once again indicate how critical the model's behavior is on interactions between voltage-dependent dendritic Ca and Kca channel currents. This is particularly the case with respect to responses to paired afferent inputs as we have shown, using parameter variations, that changes in the relative time constants of Kca to [Ca2+]i result in relatively small effects on the Purkinje cell firing rate but significant effects on the response of the model to paired-pulse stimulation.
Fortunately, a growing number of experimental groups have begun to interpret their results in the context of this model. For example, complementary to our modeling studies, other groups have recently explored experimentally the modulatory effects of background activity on synchronous inhibitory input on Purkinje cells (Kreiner and Jaeger 2004). With respect to the kinetics of the Kca channels, Miyakawa and his colleagues (Miyasho et al. 2001; Watanabe et al. 1998) have published several papers suggesting that the amplitude of the principal calcium-sensitive K conductance in our model may be too large (cf. De Schutter and Bower 1994a, Table 1; and Miyasho et al. 2001 Table 2). On the basis of our parameter variations around conductance time constants, we would expect that reducing this conductance would result in less Ca and Kca coupling, extending the range of voltage-sensitive interactions and enhancing the modulatory effects of background inputs on paired inputs. Such a change in the model would also widen the range of background input frequencies producing differential responses to paired stimuli. The same authors have also proposed that CaP channels do not have an inactivation function, as assumed here, and that there are separable voltage- and calcium-sensitive potassium currents in the dendrite. Any additional complexity in the voltage dependence of dendritic currents would also be expected to enhance and extend the influence of both prior activity in the ascending branch inputs as well as the voltage-dependent modulation provided by background parallel fiber and inhibitory inputs. Accordingly, changing these particular parameters will certainly not qualitatively alter the interaction between the different synaptic influences in the model but would only change the quantitative input output relationships of the results. Being able to make specific quantitative predictions concerning Purkinje cell responses using the model will require making many more modeling enhancements and a considerable extension of the model's complexity. For example, a recent study also based on the current model (Chono et al. 2003) suggests that interactions between the CaP and Kca channels may also control the diffusion of calcium signals in the dendrite of Purkinje cells.
Another important set of parameters governing the behavior of the current model are those related to the properties of the background synaptic input to Purkinje cells. These include biophysical synaptic properties (strength, temporal response) as well as the actual rates and range of rates of synaptic activity. Recently, Barbour and colleagues have published kinetic data for single granule cell inputs to Purkinje cells in vitro (Isope and Barbour 2002). They report an opening time constant for parallel fibers in the range of 0.2–2 ms and a closing time constant of 4–21 ms at 32°C (cf. Isope and Barbour 2002, Fig. 11). The opening times reported are within the range of those used in this model (0.5 ms). The reported closing time constant of 1.5 ms, however, is between 2.6 and 15 times slower than assumed here. However, given that the dynamics of the Purkinje cell dendrite is under control of voltage-dependent conductances with much longer time constants (hundreds of milliseconds), it is unlikely that this difference in parameters will have a significant effect on the behavior of the model. These authors have also recently published results suggesting that, in vitro and with molecular layer inhibition blocked, individual ascending and parallel fiber synapses may have the same strength (Isope and Barbour 2002). As described in METHODS, computational limitations preclude us from simulating each of the 160,000 individual granule cell synapses terminating on each Purkinje cell. Accordingly, each simulated synapse is intended to represent many individual synapses. Thus there is not a direct mapping between synaptic strengths in vitro and those in the model. For this reason, the rate of activity in the average parallel fiber and ascending branch synapse is as, or perhaps more important than the strength of each modeled synapse.
In fact, the relative levels of activity in the parallel fibers and molecular layer interneurons and ascending segment synapses is perhaps the most important unknown parameter related to these modeling results. Unfortunately, the very small size of the granule cell axon and the enormity of the synaptic input to the Purkinje cell conspire to make these values very difficult to obtain experimentally and even harder to manipulate. Recent work by Hausser and his group are starting to provide important new information about the temporal response properties of individual granule cells (Chadderton et al. 2004), but it is still not clear how somatic responses are translated into parallel fiber activity (Mocanu et al. 2000) or how individual responses reflect the responses of large populations of granule cells to afferent input. The results presented in this paper were intended to begin to provide the basis for a new experimental approach to determining or at least constraining relative levels of cortical synaptic activity. In particular, the results reported here predict that Purkinje cell responses to paired afferent input are more sensitive to patterns of background synaptic input than is, for example, the overall spontaneous firing behavior of the Purkinje cell or its responses to single afferent inputs. Comparing Purkinje cell responses to paired inputs at different intervals during periods of the same or different rates of spontaneous spiking activity may allow us to discriminate between periods of heightened and lowered background synaptic input. Experiments of this nature are currently underway in our laboratory.
Functional consequences for the cerebellum
The emerging picture from our models is that the CaP and Kca currents dynamically interact to integrate the dendritic state of the Purkinje cell set by the background parallel fiber and molecular interneuron synaptic input, and it is against this background state that input from the ascending segment is, in effect, evaluated. The dendritic current then modulates the somatic firing through the dendro-somatic current (Jaeger and Bower 1999; Jaeger et al. 1997). With these relationships in mind, one could try to build similar interactions into a model of less complexity than the one studied here, for example the simplified models proposed by Genet and Delord (2002), Mandelblat and collaborators (2001), and Yuen and collaborators (1995). However, it is important to keep two things in mind: first, the purpose of using detail models is to discover new relationships, not represent relationships already believed or understood. Second, our anatomically realistic model allows the ascending segment synapses to be placed in the correct spatial relationship to those of the parallel fibers. In fact, while comparing full reconstructed models and simplified ones, Roth and Hausser found that although the full compartmental models are electrotonically compact, they also heavily filter voltage transients such as synaptic potentials (Roth and Hausser 2001). In their position on the tiniest distal most Purkinje cell dendrites [diameters <1.5 µm (Gundappa-Sulur et al. 1999)], ascending inputs are in an ideal position to be modulated by parallel fiber and molecular layer inhibitory inputs affecting dendritic CaP and Kca currents (De Schutter 1998; De Schutter and Bower 1994c; Santamaria et al. 2002). This spatial relationship is likely to be important as we start to construct network models of the cerebellar cortex.
The results presented here suggest that the complex dynamics of the Purkinje cell dendrite may be particularly manifest in the response of this cell to more complex patterns of afferent input. Unfortunately, most physiological studies to date have used single distinct stimuli (Bower and Woolston 1983; Garwicz et al. 1998; Jorntell and Ekerot 2002; Shambes et al. 1978). Although for the sake of control and analysis (and eventual comparison to experimental data), we have only extended the current study to paired-pulse responses, recent recordings have demonstrated that activity in the granule cell layer consists of complex ongoing temporal patterns in awake behaving rats (Hartmann and Bower 2001). Given the complex interaction between parallel fiber and molecular layer interneuron synapses with even paired activation of ascending synapses used here, it may very well be that the cerebellar cortex is computationally designed to respond differentially to complex temporally varying sensory input patterns.
General distinctions between modulatory and driving synapses
Finally, as discussed in the preceding text, we have been exploring the distinction between modulatory synapses (parallel fibers and molecular layer interneurons) and those that provide direct drive on Purkinje cells (the ascending segment synapses) for >20 yr (Bower and Woolston 1983). Recently, however, it has been suggested that this distinction may also apply to synapses on cerebral pyramidal cells. Specifically, several authors have used models and experiments to suggest that some synaptic inputs to these cerebral cortical neurons may also be more involved in modulating dendritic excitability than in producing the classical spatial and temporal summation traditionally assumed to influence cell spiking (Chance et al. 2002; Pare et al. 1998; Rhodes and Llinas 2001). These authors have suggested a distinction between "driving" and "modulatory" synapses (Sherman and Guillery 1998), demonstrating in pyramidal cells that responses to identical stimuli can be modulated by different levels of background activity (cf. Fig. 3A in Chance et al. 2002), although only one group has specifically examined the potential contribution of voltage-dependent currents to these effects (Rhodes and Llinas 2001). This raises the possibility that in the cerebral cortex as well, different sets of synapses may serve to directly influence the output of the cell, whereas others modulate that output. Unlike our proposal for the cerebellum, the anatomical distinction between these types of synapses in the cerebral cortex is not as clear. Nevertheless, the phenomena we have been studying in the cerebellum may have broad application to the computational behavior of the neurons and circuits in the rest of the nervous system.
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Present address and address for reprint requests and other correspondence: F. Santamaria, Duke University Medical Center, P. O. Box 3209, Durham, NC 27710 (E-mail: santamaria{at}neuro.duke.edu)
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in each case. The value of slope measurements taken between 0.5 and 2 ms after the 1st (m1) and 2nd (m2) stimuli are shown in the top right of each graph.
