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 long-lasting, low-amplitude plateau potentials, but stronger synaptic activation can generate fast, large amplitude calcium spikes. Pharmacological data have suggested the involvement of only the P-type 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, Ca-imaging, and single ion channel data. A model is presented here incorporating a simplified description of [Ca]i regulation and three ionic currents: 1) the P-type Ca current,2) a delayed-rectifier K current, and 3) a generic class of K channels activating sharply in the sub-threshold voltage range. This model sustains fast spikes and long-lasting 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, finite-duration plateaus. Analyzing the slow subsystem obtained by assuming instantaneous activation of the delayed-rectifier reveals that the time course of plateaus and valleys is controlled by the slow [Ca]i dynamics, which arises from the high Ca-buffering capacity of PCs. A bifurcation analysis shows that tonic currents modulate sub-threshold dynamics by displacing the resting state along a hysteresis region edged by two saddle-node bifurcations; these bifurcations mark transitions from finite-duration plateaus to bistability and from bistability to valley potentials, respectively. This low-dimensionality model may be introduced into large-scale models to explore the role of PC dendrite computations in the functional capabilities of the cerebellum.
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 calcium-dependent responses: weak stimulation causes low-amplitude 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 so-called “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 low-amplitude (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 large-scale, multi-compartmental 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 finite-duration plateaus, but these are not all-or-none. 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 P-type Ca current, a delayed rectifier K current and a sub-threshold, generic K current that lumps together the set of low-voltage 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 saddle-node 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.
Electric properties of the membrane
The present study examines an isopotential, single-compartment model of a dendrite with radiusR d (centimeters) (Fig.1). In mature PCs, P-type 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 voltage-dependent Na channels (Llinas and Sugimori 1992; Stuart and Haüsser 1994). The model, therefore incorporated the P-type Ca conductance as the unique voltage-dependent inward conductance. The situation is less clear regarding outward conductances. In 1989, Gähwiler and Llano identified two types of K conductances with single-channel recordings from PCs. One had properties reminiscent of the delayed-rectifier, while the other was suggested to correspond to a large-conductance, Ca-dependent K channel (or “BK-type” 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 Ca-dependent K conductance presents significant sub-threshold 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 sub-threshold, 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 fast-inactivating (τ < 100 ms) A-type 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 sub-threshold 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 drchannel (Gruol et al. 1991), we have lumped these currents into a generic I Ksub, embedded with voltage activation at sub-threshold potentials.
In the present model, dynamics of the membrane potential, V(in millivolts), obeyed the differential equation Equation 1where C (μFcm−2) stands for the specific membrane capacitance;I leak is a leakage current, andI Φ andI dc, respectively, denote phasic and tonic currents injected into the model. The different ionic currents (expressed as nAcm−2 densities) were derived from Ohm's law according to the Hodgkin-Huxley (HH) formalism Equation 2a Equation 2b Equation 2c Equation 2dwhere E Ca,E Ksub, andE Kdr represent Nernst potentials andg Ca,g Ksub, andg Kdr correspond to maximum channel conductance (μScm−2). In the HH formulation, actual conductance are given by the product of these maximum conductance by voltage- (and possibly [Ca]i-) dependent gating variables, which are dimensionless functions defined on the range [0,1]; s stands for the activation variable of the Ca current and u and n for that of the sub-threshold and delayed-rectifier K currents, respectively. These variables obey the general differential equation Equation 3where τp(V) is the relaxation time and p ∞(V) is the equilibrium value of variable p. As P-type channels activate very fast (Regan 1991), we equated s to its equilibrium value s ∞ in Eq.2a . The same assumption was made forI Ksub, whose activation variableu was described by its equilibrium valueu ∞. As spiking requires delayed activation of I Ca andI Kdr, the latter current was assumed to activate with a classical, bell-shaped time constant (Hille 1992) Equation 4The different parameters appearing in this equation have the following units: τn0 and τn1 are in milliseconds,c τn is dimensionless, andV τn andk τn are in millivolts. Full HH description of membrane currents implicated unnecessarily complicated calculations given the experimental uncertainty on rates of (in)activation of these currents. Steady-state values of voltage-sensitive gates were therefore described with Boltzmann functions Equation 5where V p (in millivolts) andk p (in millivolts), respectively, stand for the half-activation potential and activation slope of gating variable p. Equation 2b deserves special comments. The body of results presented in this paper was obtained with the generic I Ksub described inEq. 2b . However, the various sub-threshold conductances lumped in I Ksub must display some heterogeneity in their activation function. We therefore investigated the robustness of the results to variations ing Ksub,V u, andk u. Moreover, some of these conductances exhibit Ca-dependence or inactivation as noted above, which led to the possibility that these properties may challenge conclusions derived from the crude description ofI Ksub. We therefore considered alternative schemes introducing these properties. To model A-type conductances (Midtgaard 1995; Wang et al. 1991), simulations were run withI Ksub multiplied by an inactivation variable h, whose dynamics obeyed Eq. 3 , with the time constant (τh) left as a freely adjustable parameter. Other simulations were run with activation ofg Ksub depending on [Ca]i to mimic the BK-type K conductance ofGruol et al. (1991). Sensitivity of BK channels to [Ca]i consists in a shift of their activation range toward more negative potentials with increasing concentrations of the cation (Hille 1992). Jacquin and Gruol's (1999) data on this shift were fitted by the following equation for the half-activation potential ofg Ksub Equation 6 which was substituted into Eq. 5 .
Internal dendritic calcium regulation
Ca-imaging 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 Equation 7and the magnitude of Ca currents at a given membrane potential (see Eq. 2a ); R is the gas constant (JK−1M−1), Tis the absolute temperature (K), and F is the Faraday constant (CM−1). Limited knowledge of the numerous processes involved in [Ca]iregulation hindered elaboration of a faithful model of this concentration effect. A number of simplifying assumptions were thus made to derive a simple model of [Ca]i dynamics coherent with the low resting Ca level in neurons and calcium imaging data. First, lateral diffusion of Ca along the dendrite was neglected as the cation diffuses slowly within neurons (Hille 1992). Second, Terasaki et al. (1994) have shown that the endoplasmic reticulum extends in PCs from the soma up to the very tip of dendrites and even inside spines. Ca entering the dendrites is therefore compelled to distribute within a thin shell of cytoplasm beneath the membrane; its thickness was taken as δ = 0.3 μm. We assumed that Ca can exchange between this shell and the inner dendritic core (Fig. 1); [Ca]i in the core was fixed to value [Ca]b. This naive representation was aimed at providing a simple formalism for the complicated processes of calcium diffusion and pumping into the ER. Besides, PCs have a high Ca-buffering capacity (Fierro and Llano 1996), which must markedly slow down [Ca]i dynamics, according to the modeling work of Sala and Hernandez-Cruz (1990). In consequence, we have introduced, in the model, an immobile calcium buffer with fixed concentration [B]T. With these assumptions, the balance equation of Ca in the sub-membrane shell can be written Equation 8The rightmost term in Eq. 8 corresponds to the Ca exchange between the cytoplasm and the inner core of the dendrite, the calcium concentration being kept constant at value [Ca]b in the latter compartment; this process has a time constant δ(2R d − δ)/[2k(R d − δ)], where k corresponds to a one-dimensional diffusion constant (cms−1). Π (μMs−1) denotes a sink term accounting for the binding of Ca to the buffer. This process was described by a first order reaction with dissociation constant K d =k 2/k 1(μM). Introducing the total buffer concentration [B]T = [Ca-B] + [B] (μM), the sink term is written Equation 9Binding of Ca ions to the buffer was assumed to be fast with respect to the overall evolution time of [Ca]i. The free buffer concentration, [B], could therefore be equated to its equilibrium value at each point in time Equation 10Applying the chain rule of differentiation to Eq.10 leads to Equation 11This expression was substituted for Đ into Eq.8 to obtain Equation 12 Basic parametric values were as follows: C = 1 μFcm−2, T = 298 K,F = 96, 500 Cmol−1,R = 8.32 JK−1mol−1,R d = 5 × 10−5 cm, [B]T = 150 μM, K d = 1 μM, [Ca]b = 50 nM, [Ca]o = 1.1 mM, δ = 3 × 10−5 cm,k = 0.01 cms−1,g leak = 20 μScm−2, g Ca = 600 μScm−2,g Ksub = 30 μScm−2, g Kdr= 4,200 μScm−2,E Leak = −60 mV,E Ksub = −95 mV,E Kdr = −95 mV,V s = −22 mV,V u = −44.5 mV,V n = −25 mV,k s = 4.53 mV,k u = 3 mV,k n = 11.5 mV, τn0 = 0.2 ms, τn1 = 4.15 ms, c τn = 0.6,k τn = 17 mV,V τn = −22.5 mV;g Ksub inactivation:V h = −50 mV,k h = 8 mV; Ca-dependence ofg Ksub:K dCa = 10 nM,k Ca = 200 nM.
Analytical and numerical methods
To simplify the typography, we introduce the following notation Defining vector X(t) = [x 1(t),x 2(t),x 3(t)]T, we can rewrite Eqs. 1, 12, and 3 written explicitly for n as Equation 13where Equation 14 F 1,F 2, andF 3, respectively, stand for the right-hand side of differential Eqs. 1, 12, and 3 ; μ is the vector of parameters of the model (μ dimension is not specified as alternative models had different numbers of parameters). One-dimensional bifurcations, when parameterI dc was varied, were studied as follows.
Let (μ) denotes an hyperbolic equilibrium point of system (Eq. 13 ), i.e., a point satisfying Equation 15at which location the linearization of vector field has no eigenvalue with zero real part. Bifurcations of arise when the Jacobean matrix of system Eq. 13 , , evaluated at is singular Equation 16Depending on the value of other parameters, limit cycles emerged at critical I dc values from Hopf or homoclinic bifurcation. Hopf bifurcation arose when a pair of complex eigenvalues of the linearization of vector field crossed the imaginary axis at nonzero speed. Homoclinic bifurcations were either at saddle-node (the point of coalescence of a stable and an unstable branch in the bifurcation diagram) or at regular saddle (on an unstable branch turning back from a stable branch).
Equation 13 was numerically studied with XPP, Matlab, and Maple V software. Numerical integration in the time-domain was carried out with the stiff-robust 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.
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]itransient. 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. Spike-induced [Ca]itransients 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 Ca-sensitive dyes (see e.g., Lev-Ram et al. 1992; Miakawa et al. 1992; Midtgaard et al. 1993).
The range of voltages sub-threshold to Ca spikes was explored with small amplitude current pulses, which unraveled complex dynamical properties. Figure 2 B illustrates three samples obtained with 100-ms-duration 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 above-mentioned 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 short-duration, 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 = 103nAcm−2, that is about twice the bifurcation current, led only to a 35-Hz 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 three-dimensionality. 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 sub-threshold 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 time-scale than V and [Ca]i during plateaus. This suggested that plateaus could be studied by considering the slow sub-system 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 two-dimensional model in response to a 100-ms 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 V-equilibria along the V axis evolved in time with [Ca]i changes. ThreeV-equilibria 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 V-equilibria, 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 V-equilibrium.
As the reduced model was two-dimensional, 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 100-nAcm−2 pulse resulted in a 8-mV transient depolarization, accompanied by a moderate [Ca]iincrease (peak approximately 150 nM). WithI Φ = 130 nAcm−2, membrane voltage was made to cross theV-nullcline, which it did nearly horizontally owing to the slow rate of [Ca]i evolution. From then on, the trajectory turned leftward to follow the right-most branch of theV-nullcline. 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 V-nullcline, crossed the [Ca]inullcline, 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(100-ms-duration 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]ipeak (∼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 Fitzhugh-Nagumo's model (in the parameter range where it has an excitable resting state, see Murray 1993). This analogy suggests that finite-duration plateaus triggered by phasic currents represent genuine action potentials, with a much slower time course and lower amplitude than fast Ca spikes. However, these spike-like 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.
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 35-nAcm−2, 100-ms 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 4-mV hyperpolarization triggered by a −35-nAcm−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 −100-nAcm−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 two-dimensional model. I dc had shifted the V-nullcline upward, resulting in a saddle-node 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 sub-threshold, consistent with experimental observations (Llinas and Sugimori 1980b).
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 de-excited state as they did previously. Figure6 A shows, however, that the plateau exhibited complex dynamical responses to such brief inputs. Thus a −50-nAcm−2, 100-ms 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 −150-nAcm−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 theV-nullcline, 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 U-shaped region of the V-nullcline 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 V-nullcline induced rectangular valleys; the slow phase of these valleys corresponded to the part of the trajectory that ran along the middle branch of the V-nullcline. With larger pulses, perturbations could even cross the left branch of the V-nullcline. These trajectories were quickly brought back toward the U-shaped region of the V-nullcline, 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).
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 infinite-duration 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, long-lasting 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 100-ms pulses used to generate these valley duration/I dccurves had the following magnitudes (nAcm−2): –100 (◃), −150 (▿), −200 (▵), and −300 (▹). The valley duration decreased as I dcincreased, 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 dcvalues 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 one-dimensional 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 one-dimensional model was actually a differential algebraic system of equations). Stimulation parameters were adjusted for the full model to produce typical sub-threshold 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 one-dimensional 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 one-dimensional 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 one-dimensional model has a given value (see labels).
The one-dimensional 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 sub-threshold responses had a shorter duration in the one-dimensional model than in the full one, due to the neglecting of slow [Ca]i dynamics. This result proved that the one-dimensional 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 one-dimensional model these curves were S-shaped due to the activation of I Ca andI Ksub (Fig. 7 B). Due to this distortion of the potential derivative curves, the one-dimensional 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 one-dimensional 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 one-dimensional 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 one-dimensional 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 one-dimensional 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−1during the triangular plateau. It is seen in Fig. 7 C how the sequence of slow/fast/slow potential derivatives in the one-dimensional 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 one-dimensional 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 sub-threshold 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 half-activation potential V ubecame more negative, while less negativeV u widened Ω up toV u = −37.95 mV, where Ω vanished. Overall, Fig. 8 C shows an approximate 10-mV-wide 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 sub-threshold 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 dbegan 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]iwould 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 130-nAcm−2, 100-ms-duration 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 A-current 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 τh, 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 saddle-node 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 τh modulated their duration as illustrated with plateaus in Fig. 9 B. The figure plots the length of a finite-duration plateau (130-nAcm−2, 100-ms-duration pulse) versus τh 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 sub-threshold K channels at rest as in its basic formulation. With this value, decreasing τhcontinuously increased the plateau duration, which became infinite at τh = 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 τh down to the critical value whereI ksub decayed too fast to preventV from reaching the spike threshold. Critical τh could be decreased by using largerg Ksub values (τh = 50 ms withg Ksub = 75 and τh = 1 ms withg Ksub = 100). With these largerg Ksub values, however, plateau duration decreased in the neighborhood of the critical τh instead of becoming infinite. As critical τh was approached, this duration decreased due to an early transient depolarization of growing amplitude (data not shown). This resulted in a larger initial [Ca]iincrease at the plateau onset that advanced the resetting effect.
The basic version of the model omitted Ca-dependent K conductances, while it highlighted a critical role for [Ca]ichanges in plateau generation. We therefore introduced a Ca-dependence of I ksub based on Jacquin and Gruol's (1999) data. With this modification, the model failed to produce any long-lasting 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.
The model analyzed in this paper accounts for the major features of the dual electroresponsiveness of PC dendrites. It sustains finite-duration 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 low-dimensionality, provides a valuable account of the input-output 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 D-type Ca currents, which inactivate with time constants of several tens of milliseconds. In this model, brief depolarizing currents trigger long after-depolarizations that resemble experimental plateaus. But Miyasho et al.'s model comprises low-threshold Ca conductances with densities of the same order (or even larger) than P-type 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 P-type 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 voltage-activated K channels. The delayed-rectifier 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 sub-threshold K channels identified in PCs (Gruol et al. 1991), which are not understood well enough to be modeled individually. Among these conductances, the Purkinje BK-type conductance does not seem critical for plateau generation because endowing g Ksub with a quantitative model of its Ca-dependence (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 sub-threshold 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 sub-threshold 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 Ksubrepresents a key property for the generation of sub-threshold 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 Ca-binding 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]Tvalues. Slow [Ca]i increases in the sub-threshold 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 (Lev-Ram 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 mossy-fiber 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 long-term depression (LTD). The CF-induced [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 synaptic-evoked plateau responses with the stimulus intensity, whereasLlinas and Sugimori (1980b) obtained all-or-none plateaus by direct electric stimulation. Our model supports the idea that plateau potentials are indeed all-or-none 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 all-or-none plateaus. If so, plateaus would endow PCs with multistability properties in regard to their input-output 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 multi-compartmental 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 quasi-stable 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 long-duration 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 short-term 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 vestibulo-ocular reflex.
We thank the two referees for sharp criticisms and clever suggestions.
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 (E-mail:).
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