Abstract
Computational capabilities of Purkinje cells (PCs) are central to the cerebellum function. Information originating from the whole nervous system converges on their dendrites, and their axon is the sole output of the cerebellar cortex. PC dendrites respond to weak synaptic activation with longlasting, lowamplitude plateau potentials, but stronger synaptic activation can generate fast, large amplitude calcium spikes. Pharmacological data have suggested the involvement of only the Ptype of Ca channels in both of these electric responses. However, the mechanism allowing this Ca current to underlie responses with such different dynamics is still unclear. This mechanism was explored by constraining a biophysical model with electrophysiological, Caimaging, and single ion channel data. A model is presented here incorporating a simplified description of [Ca]_{i} regulation and three ionic currents: 1) the Ptype Ca current,2) a delayedrectifier K current, and 3) a generic class of K channels activating sharply in the subthreshold voltage range. This model sustains fast spikes and longlasting plateaus terminating spontaneously with recovery of the resting potential. Small depolarizing, tonic inputs turn plateaus into a stable membrane state and endow the dendrite with bistability properties. With larger tonic inputs, the plateau remains the unique equilibrium state, showing long traces of transient inhibitory inputs that are called “valley potentials” because their dynamics mirrors that of inverted, finiteduration plateaus. Analyzing the slow subsystem obtained by assuming instantaneous activation of the delayedrectifier reveals that the time course of plateaus and valleys is controlled by the slow [Ca]_{i} dynamics, which arises from the high Cabuffering capacity of PCs. A bifurcation analysis shows that tonic currents modulate subthreshold dynamics by displacing the resting state along a hysteresis region edged by two saddlenode bifurcations; these bifurcations mark transitions from finiteduration plateaus to bistability and from bistability to valley potentials, respectively. This lowdimensionality model may be introduced into largescale models to explore the role of PC dendrite computations in the functional capabilities of the cerebellum.
INTRODUCTION
The cerebellum is one of the principal regions of the brain implicated in adaptive control of movements (Ito 1984). The way this nervous structure operates during acquisition of motor skills remains, however, a matter of debate. Central to this issue are computational capabilities of Purkinje cells (PCs), because axons from these large neurons constitute the sole output of the cerebellar cortex. Information originating from nearly the entire nervous system converges onto PC dendrites in the form of two excitatory inputs—hundreds of thousands parallel fibers (PFs) contact the distal spiny dendrites while a single climbing fiber (CF) establishes a distributed, powerful synapse on the proximal smooth dendrites. The two inputs interact through the PC dendritic tree, which is endowed with highly nonlinear membrane properties (Llinas and Sugimori 1992).
Stimulation of PC dendrites can result in two very different types of nonlinear calciumdependent responses: weak stimulation causes lowamplitude plateau potentials, which can last up to several hundred milliseconds until the resting potential is spontaneously restored, and stronger stimulation can generate fast, large amplitude Ca spikes (Llinas and Sugimori 1980b). In vivo, Ca spikes underlie the socalled “complex spike” evoked by activation of the CF (Eccles et al. 1966), which results in a generalized [Ca]_{i} increase in the dendrites. Plateau potentials are lowamplitude (approximately 15 mV) depolarizations from resting potential. They exhibit a threshold behavior and display variable duration ranging from 100 ms to several seconds (Ekerot and Oscarsson 1981; Llinas and Sugimori 1980b,1992). Plateaus putatively participate in dendritic computations and synaptic plasticity, but these roles could not be explored thoroughly due to an uncertain mechanism underlying these electric signals. Several models have attempted to understand this mechanism, like the largescale, multicompartmental PC model ofDe Schutter and Bower (1994). This model sustains plateaus, but these are unconditionally stable and do not account for spontaneous reset of experimental plateaus. Miyasho et al. (2001) have recently proposed a modified version of this model, which produces finiteduration plateaus, but these are not allornone. Such discrepancies with experimental plateaus are difficult to interpret, due to the complexity of models incorporating numerous ion channel types. Yuen et al. (1995) have adopted an opposing viewpoint by building a simple model that sustains spikes and plateaus. However, their model displays plateaus at unrealistic depolarized potentials that do not spontaneously reset. Moreover, their model predicts transition from spiking to plateaus with increasing stimuli, whereas Llinas and Sugimori (1980b) have observed the opposite transition in intracellular recordings. Thus either simplistic or detailed descriptions of membrane properties have failed to interpret the dual electroresponsiveness of PC dendrites.
The objective of this study was to investigate the minimal biophysical properties required to produce the dual electroresponsiveness of PC dendrites. The strategy was to build up a computationally tractable model that may subsequently be introduced into network models of the cerebellum. For the model presented in this paper to be conclusive, several constraints were imposed: 1) the model had to reproduce characteristic features of plateaus, including shape, amplitude, duration, and the threshold behavior evidenced byLlinas and Sugimori (1980b); 2) the model also reproduced landmark electrophysiological properties such as passive membrane properties and Ca spiking; and 3) the model was based on a careful use of available ion channel data to avoid interpretations based on peculiar solutions of poorly constrained models.
Here we present a biophysical model that shows that plateau potentials and calcium spikes can both be generated by the same underlying currents: the Ptype Ca current, a delayed rectifier K current and a subthreshold, generic K current that lumps together the set of lowvoltage activated K currents described in PCs (Gruol et al. 1989, 1991; Jacquin and Gruol 1999;Midtgaard 1995; Midtgaard et al. 1993;Wang et al. 1991). The plateaus of the model give a correct quantitative fit for experimental plateaus. Besides, a yet unobserved form of inverted plateau, or “valley potential,” emerges as a natural property of the saddlenode bifurcation underlying the existence of plateaus. A robustness analysis proves that the results are not dependent on the particular set of parameters used in the simulations. Availability of this reliable, simplified model sets the stage for future studies on the role of PC dendrites computations in information processing in the cerebellum.
DENDRITIC MODEL
Electric properties of the membrane
The present study examines an isopotential, singlecompartment model of a dendrite with radiusR _{d} (centimeters) (Fig.1). In mature PCs, Ptype Ca channels sustain more than 90% of dendritic Ca currents (Kaneda et al. 1990; Usowicz et al. 1992), and the dendritic membrane is devoid of voltagedependent Na channels (Llinas and Sugimori 1992; Stuart and Haüsser 1994). The model, therefore incorporated the Ptype Ca conductance as the unique voltagedependent inward conductance. The situation is less clear regarding outward conductances. In 1989, Gähwiler and Llano identified two types of K conductances with singlechannel recordings from PCs. One had properties reminiscent of the delayedrectifier, while the other was suggested to correspond to a largeconductance, Cadependent K channel (or “BKtype” channel, see Hille 1992). Gruol and collaborators later extended these findings. On the one hand, they correlated activity of the delayed rectifier to the repolarization phase of spikes (Gruol et al. 1991); we therefore incorporated a delayed rectifier potassium conductance (K _{dr}) in our model, which was adapted from the model of Yuen et al. (1995). On the other hand,Jacquin and Gruol (1999) showed that the Cadependent K conductance presents significant subthreshold voltage activation at Ca concentrations as low as 100 nM. Gruol et al. (1991)found four more K channel types that still have not been clearly identified. However, Midtgaard (1995) has reviewed experimental evidence suggesting that several subthreshold, inactivating conductances may participate in synaptic integration in PCs dendrites (Midtgaard 1995; Midtgaard et al. 1993). Following the same direction, Wang et al. (1991) characterized a fastinactivating (τ < 100 ms) Atype conductance, but the existence of a conductance inactivating on the second time scale was suggested by Midtgaard (1995). All in all, a precise identification of subthreshold K conductances is still lacking. However, as they all activate in a critical voltage range between −50 and −30 mV, which is more negative than the activation threshold for the K _{dr}channel (Gruol et al. 1991), we have lumped these currents into a generic I _{Ksub}, embedded with voltage activation at subthreshold potentials.
In the present model, dynamics of the membrane potential, V(in millivolts), obeyed the differential equation
Internal dendritic calcium regulation
Caimaging techniques have revealed large [Ca]_{i} increases in PC dendrites on activation of their excitatory synapses (Callaway et al. 1995;Miyakawa et al. 1992). These concentration changes modify the Nernst potential (mV) of Ca ions
Analytical and numerical methods
To simplify the typography, we introduce the following notation
Let
Equation 13 was numerically studied with XPP, Matlab, and Maple V software. Numerical integration in the timedomain was carried out with the stiffrobust method CVODE implemented in XPP. Bifurcation diagrams were built with the AUTO part of XPP. Plateau and valley potentials were quantified to study the particular influence of the different parameters of the model. Duration of a plateau (or valley) was defined arbitrarily as the time elapsed between the end of its triggering stimulus and the inflection point in the potential decay at the end of plateaus (valleys); plateaus and valleys of duration <100 ms were discarded because they could not be distinguished by visual inspection from passive exponential relaxation to steady states.Potential of a plateau (or valley) was defined as the mean potential within its duration. Calcium variation for plateaus and valleys was calculated as the time integral of [Ca]_{i} changes from the resting concentration of the cation caused by the stimulus. The plateaus maximum and valleys minimum calcium reached after stimulation were also computed.
RESULTS
Dual electroresponsiveness of the model: plateaus and spikes
Figure 2 illustrates membrane voltage and [Ca]_{i} responses of the model to square pulses of depolarizing current that simulated activation of excitatory synapses on the dendrite. Figure 2 A shows how a large current step (I _{Φ} = 575 nAcm^{−2}) triggered a train of fast Ca spikes, each of them accompanied by a distinct [Ca]_{i}transient. The amplitude of the spikes decreased slightly during the first 300 ms of the pulse, but the model settled hereafter into a regular firing mode with a frequency of approximately 10 Hz. The firing abruptly ceased at the pulse offset and V recovered to its resting value at −58.3 mV. [Ca]_{i} did not fully relax to its resting level between spikes, resulting in a slow increase of the baseline level that culminated at 2.5 μM within 0.3 s after onset of the pulse. Spikeinduced [Ca]_{i}transients developed with increasing amplitude as the envelope progressively saturated the buffer (K _{d}= 1 μM). Calcium relaxation dynamics were accordingly much slower below 1 μM, compared with higher concentrations; thus [Ca]_{i} rapidly fell to 1 μM after the end of the stimulus pulse, but subsequently stayed elevated above its resting level (96 nM) for more than a second after the end of pulse. These features of calcium dynamics correspond very well with optical signals from PCs loaded with Casensitive dyes (see e.g., LevRam et al. 1992; Miakawa et al. 1992; Midtgaard et al. 1993).
The range of voltages subthreshold to Ca spikes was explored with small amplitude current pulses, which unraveled complex dynamical properties. Figure 2 B illustrates three samples obtained with 100msduration pulses of different amplitude. WithI _{Φ} = 100 nAcm^{−2}, the voltage response was dominated by passive properties of the membrane; V decayed exponentially at the end of pulse. This decay was profoundly modified when larger pulses activated nonlinear properties of the membrane.I _{Φ} = 115 nAcm^{−2} brought the membrane potential to −49 mV, from which V recovered to its resting value after a triangular plateau response of 250 ms duration. IncreasingI _{Φ} to 130 nAcm^{−2} caused a further (approximately 1.5 mV) depolarization at the end of the stimulus. From then on, and instead of repolarizing as before, V underwent a slow upward deflection to −45 mV, from which it produced a rectangular plateau of approximately 800 ms duration. V slowly drifted toward more negative values during the plateau, and below −49 mV, the model abruptly repolarized with kinetics similar to the triangular plateau. Close resemblance of this repolarizing phase between the two pulses as well as high sensitivity of the response to small current changes implied a voltage threshold in both the triggering and the spontaneous reset of plateaus.
Figure 2 B (bottom) displays the time course of [Ca]_{i} during the abovementioned voltage responses. [Ca]_{i} did not increase significantly from its resting value during the passive response. A very limited increase (peak approximately 150 nM) accompanied the triangular plateau, which can be related to the virtual absence of significant [Ca]_{i} changes reported during shortduration, triangular plateaus (Miyakawa et al. 1992). On the contrary, [Ca]_{i} increased up to five times its resting value during the rectangular plateau. This result corresponds well with data from Callaway et al. (1995; see Fig. 10) that show marked [Ca]_{i} increases during long duration, rectangular plateaus.
Faced with such different responses, we investigated the range of possible behaviors of the model by means of the bifurcation theory. Figure 2 C illustrates the bifurcation diagram obtained by varying the intensity of a tonic current delivered to the model. Fromleft to right, one first encounters aI _{dc} range where the resting state is a globally stable attractor (bottomsolid branch). Just above the zero current axis, a narrow current region is found, where this state coexists with another stable, depolarized state (topsolid branch). In this region of hysteresis, the resting state is separated from the excited one by an unstable (dashed) branch. We found that the hysteresis region laid in the range Ω = [5.85,42.76] of tonic inputs (nAcm^{−2}). With larger currents, the excited state branch exchanged stability at 561.3 nAcm^{−2}, where a limit cycle appeared. Classical algebraic criteria allowed us to show that this limit cycle arose from a Hopf bifurcation (see Mattheij and Molenaar 1996). The new oscillatory branch was unstable and became stable at a turning point (555.6 nAcm^{−2}), demonstrating the subcritical nature of the bifurcation. Thus stable oscillations of the membrane potential started with finite amplitude and corresponded to the regular firing of fast Ca spikes illustrated in Fig. 2 A. The slope of the limit cycle frequency/I _{dc} curve decreased rapidly with increasing currents (Fig. 2 D), and the relation became close to linear above 700 nAcm^{−2}.I _{dc} = 10^{3}nAcm^{−2}, that is about twice the bifurcation current, led only to a 35Hz frequency, indicating that the model predicted low frequency firing. Compilation of published traces of Ca spike discharge in intracellular recording gives a frequency range of approximately 5–30 Hz (see e.g., Llinas and Sugimori 1980b), which corresponds well with this result.
The Ω range computed above corresponded to a current domain where the model exhibited bistability. However, the bifurcation parameter in Fig.2 C was I _{dc}. WithI _{dc} = 0, phasic inputs failed to switch the membrane to the excited state (Fig. 2 B). This proved that the origin of the spontaneous reset of plateaus was not to be found in the bistability of the model. The following section investigates the mechanism of this reset.
Mechanism of spontaneously resetting plateaus
The ionic basis of spontaneously resetting plateaus (Fig.2 B) was difficult to determine from the full equation system, owing to its threedimensionality. We therefore attempted to simplify this system. It was tempting to removeg _{Kdr} from the model, because this conductance never activated more than 5% of its maximum value during plateaus. However, in the range [−50, −40] mV,g _{Kdr} andg _{Ksub} were of the same order of magnitude, suggesting that plateau generation involved the two K conductances. This was confirmed by zeroing either of the two K conductances (Fig. 3). Withg _{Ksub} suppressed (g _{Kdr} left unchanged), the model lost its capacity to sustain plateaus; but it could still fire Ca spikes, showing that spikes arose from interaction betweenI _{Ca} andI _{Kdr}. Wheng _{Kdr} was suppressed (g _{Ksub} left unchanged), the model lost its ability to sustain either Ca spikes or subthreshold plateaus. Instead, current pulses switched the membrane to a highly depolarized stable potential (approximately 52 mV). This result reproduced the large plateau at 55 mV observed by Llinas and Sugimori (1980a) after blocking K conductances with tetraethylammonium remarkably well.
The above results show that the three active currents interacted strongly during plateaus. Nevertheless, the n variable evolved on a much faster timescale than V and [Ca]_{i} during plateaus. This suggested that plateaus could be studied by considering the slow subsystem formed by the two latter variables. We therefore set n =n _{∞}(V) in the full system to obtain a degenerate system with V and [Ca]_{i} as variables. Figure4 A plots trajectories of the variables of this twodimensional model in response to a 100ms depolarizing pulse (I _{Φ} = 130 nAcm^{−2}). The V and [Ca]_{i} traces closely matched those illustrated for the full model with the same pulse in Fig. 2 B. Figure4 B displays the instantaneous I/V relation of the reduced model (i.e., at fixed [Ca]_{i}) at three different times, before and during the plateau, marked by vertical dashed lines in Fig. 4 A. Intersections of these curves with the V axis were not true equilibria of the reduced model, but equilibria of the V differential equation for given instantaneous values of [Ca]_{i}. As such, location of Vequilibria along the V axis evolved in time with [Ca]_{i} changes. ThreeVequilibria were found prior to the stimulus (Fig.4 B, curve labeled with an asterisk). The left (approximately −58.3 mV) and right ones (−42.1 mV) were stable (●) and corresponded, respectively, to the resting state and to an excited state of the model. The middle equilibrium point (○) was unstable (stability can be assessed from the sign of the local slope of theI/V). Figure 4 A plots the time evolution of the voltage of the unstable and excited Vequilibria, superimposed on the membrane voltage trace. The current pulse depolarized the membrane beyond the unstable equilibrium and the model switched toward the excited state. However, due to the slow [Ca]_{i} increase occurring during the early part of the plateau, the driving force ofI _{Ca} diminished progressively, resulting in a slow upward shift of the I/V (Fig.4 B). This shift forced the unstable and excited states to approach each other until they coalesced (Fig. 4 B, ×) at the instant marked by a triangle in Fig. 4 A. The model was then forced to recover its resting state, as it was the only equilibrium point left. Full recovery of the resting state was granted by the fact that, as [Ca]_{i} rediminished, the excited and unstable states reappeared only after V had decayed under the unstable Vequilibrium.
As the reduced model was twodimensional, the above features were best captured in the ([Ca]_{i}, V) plane. Figure 4 C illustrates two significant trajectories of the reduced model in this plane and depicts the nullclines. A 100nAcm^{−2} pulse resulted in a 8mV transient depolarization, accompanied by a moderate [Ca]_{i}increase (peak approximately 150 nM). WithI _{Φ} = 130 nAcm^{−2}, membrane voltage was made to cross theVnullcline, which it did nearly horizontally owing to the slow rate of [Ca]_{i} evolution. From then on, the trajectory turned leftward to follow the rightmost branch of theVnullcline. In this region, the overall dynamics of the model were governed by the slow [Ca]_{i} dynamics; this part of the trajectory corresponded to the plateau part of the response. The trajectory eventually went beyond the local maximum of the Vnullcline, crossed the [Ca]_{i}nullcline, and finally returned to the resting state; this last phase of the trajectory corresponded to the rapid plateau decay.
Figure 4 D quantifies properties of spontaneously resetting plateaus as a function of I _{Φ}(100msduration pulses). Nonlinear membrane properties began to activate at 110 nAcm^{−2} and led to the triangular plateaus illustrated in Fig. 2 B. Their mean potential, duration, [Ca]_{i} peak, and [Ca]_{i} deviation all increased steeply up toI _{Φ} = 125 nAcm^{−2}. Beyond this value, plateau potentials became largely insensitive toI _{Φ} and adopted a stereotyped rectangular form. Thus rectangular plateaus were characterized by a uniform amplitude (approximately −46 mV) and duration (∼800 ms), which translated into nearly constant [Ca]_{i}peak (∼550 nM) and deviation (∼450 nMs).
According to the above analysis, spontaneously resetting plateaus reflected the excitability of the resting state of the model. Looking at Fig. 4 C, one recognizes an homology between geometrical properties of this point and that of the resting state in FitzhughNagumo's model (in the parameter range where it has an excitable resting state, see Murray 1993). This analogy suggests that finiteduration plateaus triggered by phasic currents represent genuine action potentials, with a much slower time course and lower amplitude than fast Ca spikes. However, these spikelike plateaus were obtained without tonic currents. In our model,I _{dc} was able to shift the operating regime of the dendrite in response to phasic inputs with respect to the hysteresis region (Fig. 2 C). We therefore investigated model properties at different levels of depolarizing tonic currents.
Stable plateaus
Figure 5 illustrates how feeding the model with I _{dc} = 25 nAcm^{−2} (laying at the center of Ω) modified responses to phasic inputs. A 35nAcm^{−2}, 100ms duration pulse triggered a transient depolarization that decayed passively after the pulse. But I _{Φ} = 100 nAcm^{−2} switched the dendrite to −45 mV, this excited state being maintained indefinitely after the pulse. The transient 4mV hyperpolarization triggered by a −35nAcm^{−2} negative pulse, delivered at timet = 1.5 s, demonstrates the stability of this plateau. However, the model could be switched back to its unexcited state by a −100nAcm^{−2} pulse.
The origin of these features is more evident in Fig. 5 B, which illustrates how the tonic input modified nullclines of the reduced twodimensional model. I _{dc} had shifted the Vnullcline upward, resulting in a saddlenode bifurcation. This led to the appearance of two additional equilibrium points in contrast with the zero tonic input diagram (Fig.4 C). The right most equilibrium corresponded to a plateau state of the membrane and was stable, like the resting state. The central point was a saddle, whose stable manifold separated the basins of attraction of the resting and plateau states and therefore acted as a threshold between the two stable states (dashed curve in Fig.5 B). Thus perturbations of the resting state that stayed to the left of the stable manifold eventually died away; perturbations of the plateau state that stayed to the right of the stable manifold also died away. But any perturbation from one of the stable states, large enough to cross the stable manifold, brought the model over to the other state. In these conditions, the dendrite behaved like a switch between the resting and plateau states, as was suggested by Yuen et al. (1995). However, the model of these authors predicted very depolarized potentials for stable plateaus (∼0 mV), whereas the ones obtained in our reduced (Fig. 5 A) and full models (Fig.2 C) were clearly subthreshold, consistent with experimental observations (Llinas and Sugimori 1980b).
Valley potentials
A third kind of dynamical behavior was obtained withI _{dc} larger than the upper bound of the Ω range. With such tonic inputs, the model had a globally stable attractor, corresponding to a stable plateau state. Whatever initial conditions, the dendrite eventually converged to this state because the lower stable branch in the bifurcation diagram had vanished. Thus short inhibitory inputs could not switch off the dendrite to a deexcited state as they did previously. Figure6 A shows, however, that the plateau exhibited complex dynamical responses to such brief inputs. Thus a −50nAcm^{−2}, 100ms duration pulse resulted in a transient passive hyperpolarization. Increasing the intensity to −75 nAcm^{−2} turned this passive response into a triangular, inverted plateau of approximately 150 ms duration. Further increase of I _{Φ} to −100 nAcm^{−2} lengthened this response to 1 s. Comparison with Fig. 2 B shows how the time course ofV and [Ca]_{i} during these responses mirrored dynamics of the variables during spontaneously resetting plateaus. These inverted plateaus were therefore termed “valley potentials.” The shape of rectangular valleys was robust to increases in I _{Φ} as can be seen from the trace with the −150nAcm^{−2} pulse. This shows that the model could produce a stereotyped trace of past inhibitory inputs. However, Fig. 6 A suggests that such traces could take place only following inhibitory inputs with a magnitude sufficient to bringV under a threshold located around −50 mV. This threshold behavior can be understood in Fig. 6 B, which plots trajectories of the reduced model in the ([Ca]_{i}, V) plane. The resting and saddle points had coalesced, leaving the plateau as the unique steady state. As V evolved faster than [Ca]_{i}, the vector field was nearly horizontal in the portion of the plane considered, except near branches of theVnullcline, where the field was tilted vertically. This implicates that model dynamics were controlled by the [Ca]_{i} differential equation in these regions. Thus with growing pulse amplitude, perturbations approached the Ushaped region of the Vnullcline where their relaxation was slowed down by [Ca]_{i} dynamics (i.e., triangular valleys). Perturbations that were just large enough to cross the middle branch of the Vnullcline induced rectangular valleys; the slow phase of these valleys corresponded to the part of the trajectory that ran along the middle branch of the Vnullcline. With larger pulses, perturbations could even cross the left branch of the Vnullcline. These trajectories were quickly brought back toward the Ushaped region of the Vnullcline, thereby producing a peak hyperpolarization followed by a stereotyped valley potential.
Figure 6 C summarizes characteristics of valley potentials as a function of I _{Φ}. From 75 to 90 nAcm^{−2}, phasic inputs triggered triangular valleys at more hyperpolarized levels and with growing duration. The minimum [Ca]_{i} reached and integrated [Ca]_{i} diminution continuously decreased with increasing hyperpolarizing phasic inputs.I _{Φ} = 90 nAcm^{−2} represented a threshold value, above which rectangular valley potentials adopted stereotyped characteristics (mean level approximately −53 mV, 1 s duration; [Ca]_{i} peak at 250 nM and [Ca]_{i} deviation of 0.5 μMs).
Global behavior
The previous sections have shown that the plateaus with spontaneous reset, which could be triggered by a brief depolarizing pulse (Figs. 1 and 3), were transformed into infiniteduration plateaus by injecting a tonic current laying in the Ω range (Fig. 5). It is apparent from this result that I _{dc} was able to modulate the length of plateaus in the model. This property of tonic currents was analyzed in details. In Fig.7 A, on the left of the Ω region, are displayed four examples of curves relating the duration of a plateau triggered by a depolarizing pulse to the magnitude of the applied I _{dc}. The pulse duration was 100 ms in all cases, and I _{Φ} had the following amplitudes (nAcm^{−2}): 150 (×), 200 (+), 250 (⋄), and 300 (■). Figure 7 A shows that hyperpolarizing tonic currents prevented the pulses from triggering plateau potentials, down to a criticalI _{dc} value where plateaus emerged with a triangular shape. This critical current value was more negative whenI _{Φ} was large, ranging from about −100 nAcm^{−2} forI _{Φ} = 300 nAcm^{−2} to −25 nAcm^{−2} forI _{Φ} = 150 nAcm^{−2}. Whatever the value ofI _{Φ}, reducing the magnitude of the tonic hyperpolarizing current from the critical value increased sharply the duration of triangular plateaus, up to a point where plateaus adopted a rectangular shape; this change of shape occurred at theI _{dc} values where the curves exhibit a slope discontinuity. The curves for the 250 and 300 nAcm^{−2} had to be interrupted atI _{dc} = −30 and −70 nAcm^{−2}, respectively, because the pulses triggered Ca spikes with hyperpolarizingI _{dc} below these values. The curves for the two smaller pulses could be extended up to the lower bound of the Ω region, where the plateau duration became infinite. Overall, Fig.7 A shows that the duration of plateaus in the model could be made to cover an infinite range by varyingI _{dc} between approximately −100 nAcm^{−2} and the lower bound of Ω. Experimental plateaus have been reported to range from close to zero duration plateaus (nearly passive responses) to plateaus lasting for several seconds (Ekerot and Oscarsson 1981; Llinas and Sugimori 1992). These latter, longlasting plateaus may reflect the reset of otherwise stable plateaus by the spontaneous activation of inhibitory synapses; however, no experimental evidence of our knowledge can support or refute this interpretation at the current time. Interestingly, the onset of Ca spikes from plateau potentials predicted from Fig. 7 A as the phasic input magnitude increases is clearly evident in the voltage traces illustrated by Llinas and Sugimori (1992).
Figure 7 A also illustrates the influence ofI _{dc} on the duration of valley potentials on the right of the Ω region. The 100ms pulses used to generate these valley duration/I _{dc}curves had the following magnitudes (nAcm^{−2}): –100 (◃), −150 (▿), −200 (▵), and −300 (▹). The valley duration decreased as I _{dc}increased, and valleys with the longest duration were found near Ω like the longest plateaus. In contrast to plateaus, the duration of valleys varied smoothly with I _{dc}. In summary, pulses with an appropriate magnitude could trigger plateaus and valleys within a range of I _{dc}values spanning almost seven times the width of the Ω region.
Figure 7 A shows, through variations inI _{dc}, that the shape of plateaus and valleys was sensitive to the membrane potential. We devised a qualitative understanding of this influence from a geometrical representation of the model dynamics similar to the method of isoclines (see, e.g., Mattheij and Molenaar 1996). This approach consisted of generating a onedimensional approximation of the full model by extending to the [Ca]_{i} variable the rapid equilibrium approximation made for the n variable (notice that zeroing the time derivative in Eq.12 leaves an algebraic equation that cannot be solved explicitly for [Ca]_{i} in terms of V; strictly speaking, the onedimensional model was actually a differential algebraic system of equations). Stimulation parameters were adjusted for the full model to produce typical subthreshold responses (thick curves in Fig. 7, C and D): a triangular plateau (C,top), a rectangular plateau (C, bottom), a rectangular valley (D,top), and a triangular valley (D, bottom). Trajectories were also computed for the onedimensional model with the same stimulus parameters and are illustrated as thin curves in Fig. 7, C and D. These trajectories are also represented in the (I _{dc}, V) plane as vertical bars in Fig. 7 B, together with a subset of constant potential derivative curves of the onedimensional model. Each of these curves indicates the locus of points in the (I _{dc}, V) plane where the derivative of state variable, V, of the onedimensional model has a given value (see labels).
The onedimensional model reproduced qualitatively the shape of either triangular or rectangular plateaus and valleys in the full model. The only difference between the two models was that all active subthreshold responses had a shorter duration in the onedimensional model than in the full one, due to the neglecting of slow [Ca]_{i} dynamics. This result proved that the onedimensional model could be used to understand the qualitative dynamics of plateaus and valleys.
In a passive model, potential derivative would be straight lines, but in the onedimensional model these curves were Sshaped due to the activation of I _{Ca} andI _{Ksub} (Fig. 7 B). Due to this distortion of the potential derivative curves, the onedimensional model crossed regions of different dV/dt values in the (I _{dc}, V) plane during the responses illustrated in Fig. 7, C and D. With the triangular plateau (Fig. 7 C, top), the membrane potential of the onedimensional model was −49.7 mV at the end of the pulse. At this voltage, dV/dt was −55 mVs^{−1} (Fig. 7 B) and this relatively low dV/dt value entailed the initial slow rate of repolarization of the onedimensional model (Fig. 7 C). As the plateau decayed, however, the rate of repolarization accelerated because the model crossed regions with growing values ofdV/dt until it approached its resting state. Then, the model encountered again a region with low dV/dt value, which was responsible for the slow phase terminating the triangular plateau in the onedimensional model (Fig. 7 C). The rectangular plateau on the bottom of Fig. 7 C had a longer duration because, at the value of I _{dc} = 0, the trajectory of the onedimensional model intersected constant potential derivative curves with globally smaller dV/dt values than with the triangular plateau. Thus the smallest dV/dt value met during the rectangular plateau was −10 mVs^{−1} versus −50 mVs^{−1}during the triangular plateau. It is seen in Fig. 7 C how the sequence of slow/fast/slow potential derivatives in the onedimensional model accounted well for the qualitative shape of plateaus in the full model. Similar conclusions can be derived from Fig. 7, B andD, as regard to the valley potentials. Thus the simplified onedimensional model shows that plateaus and valleys stood as a direct consequence of the distortion of the voltage/current relationship around the Ω region, due to the activation ofI _{Ca} andI _{Ksub}.
Robustness of results and alternative schemes
The parameter sensitiveness of the results was examined systematically. Particular emphasis was placed on the influence of the parameters on the Ω region, due to its critical role in setting subthreshold responses of the model. Figure8, A and B, shows that increasing g _{Ksub} narrowed Ω by positively shifting the left endpoint of the excited stable branch in the bifurcation diagram. Decreasingg _{Ksub} widened Ω with the opposite effect and also shifted the right endpoint of the excited branch (the Hopf bifurcation) toward the left of the diagram. First, this shift resulted in transition from the Hopf to a homoclinic bifurcation at regular saddle at g _{Ksub} = 25.25 μScm^{−2} (marked by a vertical dashed line through Ω). With further decrease ing _{Ksub}, the excited branch eventually lost stability at a lower I _{dc} than the right endpoint of the resting branch (Fig. 8 B), turning Ω into a current domain where the model could still display bistability, but no longer hysteresis. To highlight the difference, Ω boundaries were plotted as dashed lines when Ω corresponded to a bistable region (same symbols were used to locate Hopf/homoclinic and hysteresis/bistability transitions throughout Fig. 8). Note that Ω vanished for g _{Ksub} = 15 μScm^{−2}, due to the coalescence of the left and right endpoints of the excited branch. Figure 7, C andD, illustrates the effects of changing activation parameters of g _{Ksub}. Ω rapidly vanished when the halfactivation potential V _{u}became more negative, while less negativeV _{u} widened Ω up toV _{u} = −37.95 mV, where Ω vanished. Overall, Fig. 8 C shows an approximate 10mVwide range ofV _{u} values in which the model had a significant region of hysteresis/bistability. As seen in Fig.8 D, Ω was widened wheng _{Ksub} activated with steeper slopes (i.e., smaller k _{u}). On the other hand, Ω was continuously narrowed by decreasingk _{u} ≤ 5 mV, above which Ω vanished. Together, these results show that dynamical behaviors described in previous sections were robust to significant deviations in the subthreshold K current parameters, but that an overall steep slope of activation was required for the model to reproduce experimental plateaus.
The effects of changing the density of the two other active conductances are illustrated in Fig. 8, E(g _{Ca}) and F(g _{Kdr}). A Ω region could be obtained with deviations of the two conductances up to approximately 50% around their reference value. Ω remained a true hysteresis region with all tested values of g _{Kdr}, while it became a bistability region with g _{Ca}'s larger than 800 μS cm^{−2} due to lower thresholds for spiking at these high Ca conductance values.
The bottom of Fig. 8 illustrates the influence of several key parameters of [Ca]_{i} regulation. All results displayed above were obtained with a radiusR _{d} = 0.5 μm, corresponding to the thinnest spiny dendrites (Shelton 1985). Figure8 G shows that increasing R _{d}began by steeply increasing the Ω width, which became nearly constant between 1 and 5 μm (corresponding to the primary dendritic trunk). According to our model, all parts of PC dendrites should thus be able to sustain dynamical behaviors described above. Figure 8 Hdisplays essentially similar results when parameter k was varied in a large range around its reference value (k sets the time constant that relaxation of [Ca]_{i}would exhibit without buffer in the cytoplasm). Finally, Fig. 8,I and J, illustrates the effects of varying the total buffer concentration, [B]_{T}, and its dissociation constant, K _{d}, on the duration of a spontaneously resetting plateau (triggered by a 130nAcm^{−2}, 100msduration pulse withI _{dc} = 0). With increasing [B]_{T}, triangular plateaus of growing duration were first encountered. [B]_{T} = 0.3 μM marked appearance of rectangular plateaus; their duration increased linearly with the buffer concentration. This feature reflects the ability of the buffer to slow down [Ca]_{i} dynamics in the range of the cation concentration corresponding to the plateaus. Figure8 J shows that K _{d}'s around 0.35 μm resulted in maximal duration plateaus. DecreasingK _{d} from this value shortened the plateaus because it allowed the buffer to saturate at lower [Ca]_{i} levels, thereby reducing its efficiency at slowing down [Ca]_{i} dynamics during plateaus. With K _{d}'s > 0.35 μM, plateaus were also shortened because the buffer slowed down [Ca]_{i} dynamics at higher [Ca]_{i} levels than those reached during plateaus.
Additional computations were carried out withI _{ksub} inactivating with time constants ranging from that of the rapid Acurrent of Wang et al. (1991) to that of the slowly inactivating current hypothesized by Midtgaard (1995). Figure9 A displays boundaries of the Ω region in the model with inactivation versusg _{Ksub} (boundaries did not depend on τ_{h}, which was taken as a constant). The graph is similar to that obtained without inactivation (Fig. 8 A), except Ω occurred in a g _{Ksub} range above that found in the model without inactivation. This difference stemmed from the voltage dependence ofh _{∞}, which resulted in significant inactivation of g _{Ksub} at rest. Introduction of the inactivation scheme also changed the bifurcation from which the limit cycle emerged into a homoclinic bifurcation at saddlenode in the entire g _{Ksub} range studied. As suggested by the Ω region in Fig. 9 A, the model with inactivation also produced finite duration plateaus and valleys with appropriate levels of tonic currents (not shown). However, the time constant of inactivation τ_{h} modulated their duration as illustrated with plateaus in Fig. 9 B. The figure plots the length of a finiteduration plateau (130nAcm^{−2}, 100msduration pulse) versus τ_{h} for three values ofg _{Ksub}: 40.66, 75, and 100 μS cm^{−2}. With the value of 40.66, the model had the same density of active subthreshold K channels at rest as in its basic formulation. With this value, decreasing τ_{h}continuously increased the plateau duration, which became infinite at τ_{h} = 2 s; the model discharged a Ca spike below this critical value. Plateaus were lengthened by the decay ofI _{ksub} as it partly overcame the decrease in I _{Ca} responsible for the spontaneous resetting of plateaus. This effect was enhanced by decreasing τ_{h} down to the critical value whereI _{ksub} decayed too fast to preventV from reaching the spike threshold. Critical τ_{h} could be decreased by using largerg _{Ksub} values (τ_{h} = 50 ms withg _{Ksub} = 75 and τ_{h} = 1 ms withg _{Ksub} = 100). With these largerg _{Ksub} values, however, plateau duration decreased in the neighborhood of the critical τ_{h} instead of becoming infinite. As critical τ_{h} was approached, this duration decreased due to an early transient depolarization of growing amplitude (data not shown). This resulted in a larger initial [Ca]_{i}increase at the plateau onset that advanced the resetting effect.
The basic version of the model omitted Cadependent K conductances, while it highlighted a critical role for [Ca]_{i}changes in plateau generation. We therefore introduced a Cadependence of I _{ksub} based on Jacquin and Gruol's (1999) data. With this modification, the model failed to produce any longlasting responses to phasic inputs, either plateaus, bistability, or valley potentials (data not shown). Inability of the model to sustain this kind of responses arose because activation ofI _{ksub} became controlled by the slow [Ca]_{i} dynamics. In other words,I _{ksub} activated too slowly in front ofI _{Ca} for the two currents to produce the balance required for plateau generation.
DISCUSSION
The model analyzed in this paper accounts for the major features of the dual electroresponsiveness of PC dendrites. It sustains finiteduration plateaus with the various shapes reported in response to parallel fiber volleys (Campbell et al. 1983), activation of the climbing fiber (Ekerot and Oscarsson 1981), or direct electric stimulation (Llinas and Sugimori 1980b, 1992). The model also reproduces the transition from plateaus to spikes with increasing stimulation, as reported byLlinas and Sugimori (1980b, 1992). The robustness of these results relative to large deviations in key parameters around their standard value suggests that our model, despite its lowdimensionality, provides a valuable account of the inputoutput relation of PC dendrites. As the model predicts occurrence of valley potentials in response to inhibitory inputs during stable depolarized states, which have not been observed yet, we will first discuss consistency of the model in relation to synaptic and membrane intrinsic properties of PCs. We will then discuss the significance of the model regarding computations of PC dendrites.
We cannot exclude that other models may reproduce electroresponsiveness of PC dendrites equally well. However, models that have so far attempted to reproduce the dual electroresponsiveness of PC dendrites contain inconsistencies. In the introduction, we discussed that De Schutter and Bower's (1994) model fails to produce dendritic plateaus with spontaneous reset. Miyasho et al. (2001) recently introduced into this model E and Dtype Ca currents, which inactivate with time constants of several tens of milliseconds. In this model, brief depolarizing currents trigger long afterdepolarizations that resemble experimental plateaus. But Miyasho et al.'s model comprises lowthreshold Ca conductances with densities of the same order (or even larger) than Ptype Ca channels, whereas the latter channels sustain the major part of Ca currents into PC dendrites (Usowicz et al. 1992). In fact, Llinas et al. (1989) have shown that FTX toxin, a selective blocker of Ptype channels, abolishes spikes and plateaus, which support the idea that a unique type of Ca current underlies both electric signals. We have found that, to realistically reproduce salient features of experimental plateaus with the sole noninactivating P channel, a minimal model needs to contain two kinds of voltageactivated K channels. The delayedrectifier introduced into our model was clearly identified in PCs, where it serves to repolarize spikes (Gähwiler and Llano 1989; Gruol et al. 1991). The second channel is more conjectural, becauseg _{Ksub} lumps together several subthreshold K channels identified in PCs (Gruol et al. 1991), which are not understood well enough to be modeled individually. Among these conductances, the Purkinje BKtype conductance does not seem critical for plateau generation because endowing g _{Ksub} with a quantitative model of its Cadependence (Jacquin and Gruol 1999) completely abolished plateaus. On the other hand, overall properties of the model remained unchanged wheng _{Ksub} was endowed with inactivations similar to those exhibited by several PC's subthreshold K channels (Midtgaard 1995; Wang et al. 1991). Moreover, Yuen et al.'s (1995) model of a PC dendrite, which is devoid of such a subthreshold K current, produces unrealistic plateaus near 0 mV. Due to its steep slope of activation, the idealizedg _{Ksub} endows the model with a strong outward I/V rectification near −45 mV wheng _{Ca} is zero (data not shown), as can clearly be observed in PCs after blocking their Ca conductances (Genet and Kado 1997). Together, these results suggest that the steep activation of g _{Ksub}represents a key property for the generation of subthreshold plateaus into PCs. However, it must be noted that in the range [−50, −40] mV, the involvement of I _{Kdr} in balancing I _{Ca} to produce plateaus is quantitatively significant (Fig. 3).
Due to the high levels of Cabinding proteins in PCs, [Ca]_{i} dynamics must be largely slowed down in the concentration range corresponding to theK _{d} of these proteins. NeitherK _{d} nor the concentration of these Ca buffers are currently known with precision. The results illustrated in this paper were, however, computed with a buffer concentration, [B]_{T}, falling at the center of the range of estimated parvalbumin and calbindin concentrations in PCs (100–210 μM, Fierro and Llano 1996). Moreover, Fig. 8 shows that plateau responses in the model withstood large deviations inK _{d} and [B]_{T}values. Slow [Ca]_{i} increases in the subthreshold voltage range decreased the Ca Nernst potential, thereby reducing the magnitude of I _{Ca} on a time scale of hundreds of milliseconds. This induced the reset of plateaus in the model by breaking the balance betweenI _{Ca} and the two K currents. According to our model, the large [Ca]_{i} transients seen in PCs following their synaptic activation (Miakawa et al. 1992) or direct electric stimulation (LevRam et al. 1992) are therefore responsible for the spontaneous reset of experimental plateaus.
With its set of reference parameters, our model is very close to a transition between Hopf and homoclinic bifurcations for emergence of Ca spiking (Fig. 8). With these two bifurcations, oscillations become stable at a turning point, where they have a finite amplitude of low sensitivity to the I _{dc} magnitude. Spiking emerges at null frequency with the homoclinic bifurcation and only at 5 Hz with the Hopf bifurcation (Fig. 2 D). This difference would be difficult to observe experimentally, and the model was not designed to faithfully reproduce Ca spiking, which probably involves other conductances than those introduced into the model (seeMidtgaard 1995). The precise nature of the bifurcation, therefore, appears meaningless within the context of this study.
Our model assigns distinct roles to phasic and tonic inputs,I _{Φ} triggering nonlinear responses, whose duration is modulated by I _{dc}. A physiological counterpart for these two kinds of excitatory inputs can be found in actual inputs to PC dendrites. Thus spiny dendrites of PCs are bombarded with several thousands of PFs synapses, whose precise pattern of activation is still unknown. Contextual information in the mossy fiber system without specific correlation may activate PFs asynchronously. According to Rapp et al.'s (1992)simulations, individual fibers probably lose any individual functional meaning in these conditions and provide a tonic depolarizing input to PC dendrites, which can be identified asI _{dc} in our model. In addition to this tonic input, synchronous activation of a subset of PFs during a motor task [see the theory of Marr (1969)] may result in a phasic depolarizing input to the dendritic tree. Ekerot and Oscarsson (1981) were indeed able to trigger plateau potentials by stimulating bundles of PFs. The large number of PFs that must be activated synchronously to trigger plateaus initially led to the conclusion that granule cells do not evoke these prolonged responses under physiological conditions (Campbell et al. 1983).Jaeger and Bower (1994) later proved, however, that granule cells can actually do so via the ascending part of their axon, which runs along PC dendrites and provides more powerful excitatory synapses than PFs (Llinas and Sugimori 1999); according to Cohen and Yarom (1998), granule cell ascending axons provide the main source of excitation to the cerebellar cortex when it is activated via the natural mossyfiber system. The climbing fiber represents another attractive counterpart forI _{Φ}, as Ekerot and Oscarsson (1981) have observed that plateau potentials terminate the complex spike in vivo. Interestingly, Ekerot and Kano (1985) showed that activation of stellate cells, which make inhibitory synapses on PC dendrites, abolishes the plateau part of the complex spike and result in the failure of the CF to induce longterm depression (LTD). The CFinduced [Ca]_{i} signal is believed to constitute the initial stimulus for the LTD of parallel fibers (Daniel et al. 1998; Sakurai 1987), so that the result of Ekerot and Kano (1985) suggests a possible role of plateau potentials in the induction of LTD at PF synapses.
Jaeger and Bower (1994) have observed a gradation of synapticevoked plateau responses with the stimulus intensity, whereasLlinas and Sugimori (1980b) obtained allornone plateaus by direct electric stimulation. Our model supports the idea that plateau potentials are indeed allornone events. This feature could be reconciled with the data of Jaeger and Bower (1994) if plateaus can be triggered independently in different branches of PC dendrites, as suggested by Campbell et al. (1983). Jaeger and Bower's (1994) graded responses would thus reflect summation of individual allornone plateaus. If so, plateaus would endow PCs with multistability properties in regard to their inputoutput relation; computational perspectives of networks comprising such multistable units have been recently illustrated by Barto et al. (1999). This hypothesis on multiple plateaus originating in different dendritic branches could be explored by introducing our local model within a multicompartmental model of the PC.
An important prediction of our model is that the various plateau shapes reported by Ekerot and Oscarsson (1981) only reflect a part of the PC dendrites operating capabilities. Thus withI _{dc} values inside the hysteresis region Ω, the model can be switched to a stable plateau by brief depolarizing inputs. This feature could explain the quasistable plateaus observed by Llinas and Sugimori (1980b), even if a clear evidence for a bistability of PC dendrites with large depolarizing DC input is currently lacking. Brief hyperpolarizing currents with sufficient magnitude can actively reset the plateau by making V cross the unstable branch in the bifurcation diagram (Fig. 2 C). But it suffices thatI _{dc} decreases below the lower Ω bound to recover the resting state automatically; this overcomes limitation of theories of bistable dendrites (Baginskas et al. 1993), into which plateaus can only be reset by activation of inhibitory synapses. Transient inhibitory inputs are unable to reset the plateau with I _{dc} above the upper Ω bound. However, such inputs can trigger longduration valley potentials at approximately −52.5 mV (Fig. 6), from which the resting state is recovered if I _{dc} falls below the upper Ω bound (see Fig. 2 C). Valleys have not been observed, but these potentials represent a testable prediction of the model that could be used to experimentally validate the proposed membrane mechanisms underlying plateau potentials.
The present model of PC dendrite provides a modeling framework linking detailed cellular experimental data and large scale computational models of the cerebellum. Thus our model suggests that plateaus and valleys constitute shortterm memories of phasic inputs and that the control contextual tonic inputs exert on their duration enlarge the computational properties attributable to PC dendrites. These properties, together with plastic changes at PF synapses, LTD (Daniel et al. 1998), and potentiation (Hansel et al. 2001; Hirano 1991), may contribute to the temporal specificity of cerebellar learning, that has been revealed by the Pavlovian conditioning of eyelid responses (Medina et al. 2000) and adaptation of the vestibuloocular reflex.
Acknowledgments
We thank the two referees for sharp criticisms and clever suggestions.
Footnotes

Address for reprint requests: Institute National de la Santé et de la Recherche Médicale U.483, Université Pierre et Marie Curie, Boı̂te 23, 75252 Paris Cedex 05, France (Email:stephane.genet{at}snv.jussieu.fr).
 Copyright © 2002 The American Physiological Society