INTRODUCTION

TOP ABSTRACT INTRODUCTION DENDRITIC MODEL RESULTS DISCUSSION REFERENCES

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 inputshundreds 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 of De 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 by Llinas 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.

	DENDRITIC MODEL

TOP ABSTRACT INTRODUCTION DENDRITIC MODEL RESULTS DISCUSSION REFERENCES

Electric properties of the membrane

The present study examines an isopotential, single-compartment model of a dendrite with radius R_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 ( < 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_dr channel (Gruol et al. 1991), we have lumped these currents into a generic I_Ksub, embedded with voltage activation at sub-threshold potentials.

View larger version (19K): [in this window] [in a new window]   Fig. 1. Membrane of the dendritic model is comprised of a constant leakage conductance and 3 voltage-dependent conductances: gCa is a high-threshold, P-type Ca conductance, gKdr is a classical delayed-rectifier, and gKsub is a low-threshold K conductance. As Ca channels open, Ca2+ ions entering the dendrite distribute uniformly into a shell of cytoplasm, inside which they combine with an endogenous buffer and are pumped into an inner core; the Ca concentration in the core is kept at a low basal value, [Ca]b. [Ca]i changes in the cytoplasm modify the value of the electromotive force on Ca2+ ions across the membrane.

In the present model, dynamics of the membrane potential, V (in millivolts), obeyed the differential equation

(1)

where C (µFcm²) stands for the specific membrane capacitance; I_leak is a leakage current, and I and I_dc, respectively, denote phasic and tonic currents injected into the model. The different ionic currents (expressed as nAcm² densities) were derived from Ohm's law according to the Hodgkin-Huxley (HH) formalism

(2a)

(2b)

(2c)

(2d)

where E_Ca, E_Ksub, and E_Kdr represent Nernst potentials and g_Ca, g_Ksub, and g_Kdr correspond to maximum channel conductance (µScm²). 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

(3)

where _p(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 for I_Ksub, whose activation variable u was described by its equilibrium value u. As spiking requires delayed activation of I_Ca and I_Kdr, the latter current was assumed to activate with a classical, bell-shaped time constant (Hille 1992)

(4)

The different parameters appearing in this equation have the following units: _n0 and _n1 are in milliseconds, c_n is dimensionless, and V_n and k_n 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

(5)

where V_p (in millivolts) and k_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 in Eq. 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 in g_Ksub, V_u, and k_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 of I_Ksub. We therefore considered alternative schemes introducing these properties. To model A-type conductances (Midtgaard 1995; Wang et al. 1991), simulations were run with I_Ksub multiplied by an inactivation variable h, whose dynamics obeyed Eq. 3, with the time constant (_h) left as a freely adjustable parameter. Other simulations were run with activation of g_Ksub depending on [Ca]_i to mimic the BK-type K conductance of Gruol 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 of g_Ksub

(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

(7)

and the magnitude of Ca currents at a given membrane potential (see Eq. 2a); R is the gas constant (JK¹M¹), T is the absolute temperature (K), and F is the Faraday constant (CM¹). Limited knowledge of the numerous processes involved in [Ca]_i regulation 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  = 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

(8)

The 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 (2R_d  )/[2k(R_d  )], where k corresponds to a one-dimensional diffusion constant (cms¹).  (µMs¹) 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 = k₂/k₁ (µM). Introducing the total buffer concentration [B]_T = [Ca-B] + [B] (µM), the sink term is written

(9)

Binding 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

(10)

Applying the chain rule of differentiation to Eq. 10 leads to

(11)

This expression was substituted for  into Eq. 8 to obtain

(12)

Basic parametric values were as follows: C = 1 µFcm², T = 298 K, F = 96, 500 Cmol¹, R = 8.32 JK¹mol¹, R_d = 5 × 10⁵ cm, [B]_T = 150 µM, K_d = 1 µM, [Ca]_b = 50 nM, [Ca]_o = 1.1 mM,  = 3 × 10⁵ cm, k = 0.01 cms¹, g_leak = 20 µScm², g_Ca = 600 µScm², g_Ksub = 30 µScm², g_Kdr = 4,200 µScm², 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, _n0 = 0.2 ms, _n1 = 4.15 ms, c_n = 0.6, k_n = 17 mV, V_n = 22.5 mV; g_Ksub inactivation: V_h = 50 mV, k_h = 8 mV; Ca-dependence of g_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) = [x₁(t), x₂(t), x₃(t)]^T, we can rewrite Eqs. 1, 12, and 3 written explicitly for n as

(13)

where

(14)

F₁, F₂, and F₃, 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 parameter I_dc was varied, were studied as follows.

Let (µ) denotes an hyperbolic equilibrium point of system (Eq. 13), i.e., a point satisfying

(15)

at 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

(16)

Depending 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.

	RESULTS

TOP ABSTRACT INTRODUCTION DENDRITIC MODEL RESULTS DISCUSSION REFERENCES

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 2A shows how a large current step (I = 575 nAcm²) 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. Spike-induced [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 Ca-sensitive dyes (see e.g., Lev-Ram et al. 1992; Miakawa et al. 1992; Midtgaard et al. 1993).

View larger version (34K): [in this window] [in a new window]   Fig. 2. Spikes and plateaus in the dendritic model. A: V and [Ca]i time course during the train of Ca spikes triggered by a long depolarizing pulse (I = 575 nAcm2). B: sub-threshold responses to brief (100 ms) depolarizing pulses with varied amplitude. The passive response (*), triangular plateau (), and rectangular plateau (), respectively, correspond to I = 100, 115, and 130 nAcm2. Note the different [Ca]i scale compared with A. C: bifurcation diagram obtained by varying Idc. Steady and periodic solutions are respectively depicted as thin and thick lines, stable and unstable solutions as solid and dashed lines. Throughout the text,  denotes the current domain where the resting state coexists with an excited, plateau state. D: Ca spikes frequency/Idc curve reveals low frequency of discharge.

The range of voltages sub-threshold to Ca spikes was explored with small amplitude current pulses, which unraveled complex dynamical properties. Figure 2B illustrates three samples obtained with 100-ms-duration pulses of different amplitude. With I = 100 nAcm², 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 = 115 nAcm² brought the membrane potential to 49 mV, from which V recovered to its resting value after a triangular plateau response of 250 ms duration. Increasing I to 130 nAcm² 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 2B (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 2C illustrates the bifurcation diagram obtained by varying the intensity of a tonic current delivered to the model. From left to right, one first encounters a I_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  = [5.85,42.76] of tonic inputs (nAcm²). With larger currents, the excited state branch exchanged stability at 561.3 nAcm², 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 nAcm²), 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. 2A. The slope of the limit cycle frequency/I_dc curve decreased rapidly with increasing currents (Fig. 2D), and the relation became close to linear above 700 nAcm². I_dc = 10³ nAcm², 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  range computed above corresponded to a current domain where the model exhibited bistability. However, the bifurcation parameter in Fig. 2C was I_dc. With I_dc = 0, phasic inputs failed to switch the membrane to the excited state (Fig. 2B). 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. 2B) 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 remove g_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 and g_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). With g_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 between I_Ca and I_Kdr. When g_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.

View larger version (20K): [in this window] [in a new window]   Fig. 3. Mixed activation of IKsub and IKdr during finite-duration plateaus. A 100-ms pulse (I = 130 nAcm2) was used to trigger a plateau with spontaneous reset in the full model. The same pulse triggers a Ca spike in the model devoid of gKsub. Without gKdr, the pulse switches the model to a stable plateau state at a highly depolarized level (onset of the 3 pulses was shifted for clarity of the graph).

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. Figure 4A plots trajectories of the variables of this two-dimensional model in response to a 100-ms depolarizing pulse (I = 130 nAcm²). The V and [Ca]_i traces closely matched those illustrated for the full model with the same pulse in Fig. 2B. Figure 4B 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. 4A. 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. Three V-equilibria were found prior to the stimulus (Fig. 4B, 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 the I/V). Figure 4A 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 of I_Ca diminished progressively, resulting in a slow upward shift of the I/V (Fig. 4B). This shift forced the unstable and excited states to approach each other until they coalesced (Fig. 4B, ×) at the instant marked by a triangle in Fig. 4A. 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.

View larger version (38K): [in this window] [in a new window]   Fig. 4. Mechanism of spontaneously resetting plateaus in the model with Idc = 0. Graphs illustrate the dynamics of the 2-dimensional system derived from the model by assuming instantaneous activation of IKdr. A: time course of V and [Ca]i in response to a 100-ms depolarizing pulse (I = 130 nAcm2); traces faithfully reproduce those obtained with the full model with the same stimulation (Fig. 2C). B: instantaneous dendritic I/V relation at the 3 times marked by dashed vertical lines in A. Before the pulse (*), the I/V intersects the 0-current axis at 3 points, which represent pseudo-equilibrium voltages (or V-equilibrium) (, stable; , unstable). Voltages of the middle (dashed) and right (solid) V-equilibrium are superimposed on the V trace in A. Depolarization above the middle (saddle) point brings the system toward the right (plateau) equilibrium. As [Ca]i increases, ICa decreases, which shifts the I/V relation upward. This brings the 2 right equilibrium points closer, until they coalesce (), and V is then forced to return to its resting value. Note that, after reaching a maximum (), [Ca]i decreases and allows the saddle and plateau points to reappear; but V has already decayed under the saddle voltage at this time and the dendrite keeps repolarizing. C: phase plane analysis. Trajectory of the plateau illustrated in A (outer trajectory) and a passive response (inner trajectory) to a smaller, 100-nAcm2 pulse are displayed in the (V, [Ca]i) plane. Graph also displays the [Ca]i-nullcline (d[Ca]i/dt = 0) and V-nullcline (dV/dt = 0), which intersect at a stable state (solid dot) corresponding to the resting potential. From this point, a trajectory must be driven by an injected current across the unstable middle branch of the V-nullcline to form a rectangular plateau. D: quantitative analysis of resetting plateaus as a function of I (100-ms-duration pulse). Steep region on the left of curves corresponds to triangular plateaus, while the flat part corresponds to rectangular plateaus.

As the reduced model was two-dimensional, the above features were best captured in the ([Ca]_i, V) plane. Figure 4C illustrates two significant trajectories of the reduced model in this plane and depicts the nullclines. A 100-nAcm² pulse resulted in a 8-mV transient depolarization, accompanied by a moderate [Ca]_i increase (peak approximately 150 nM). With I = 130 nAcm², membrane voltage was made to cross the V-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 the V-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]_i nullcline, and finally returned to the resting state; this last phase of the trajectory corresponded to the rapid plateau decay.

Figure 4D quantifies properties of spontaneously resetting plateaus as a function of I (100-ms-duration pulses). Nonlinear membrane properties began to activate at 110 nAcm² and led to the triangular plateaus illustrated in Fig. 2B. Their mean potential, duration, [Ca]_i peak, and [Ca]_i deviation all increased steeply up to I = 125 nAcm². Beyond this value, plateau potentials became largely insensitive to I 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. 4C, 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. 2C). 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 nAcm² (laying at the center of ) modified responses to phasic inputs. A 35-nAcm², 100-ms duration pulse triggered a transient depolarization that decayed passively after the pulse. But I = 100 nAcm² switched the dendrite to 45 mV, this excited state being maintained indefinitely after the pulse. The transient 4-mV hyperpolarization triggered by a 35-nAcm² negative pulse, delivered at time t = 1.5 s, demonstrates the stability of this plateau. However, the model could be switched back to its unexcited state by a 100-nAcm² pulse.

View larger version (19K): [in this window] [in a new window]   Fig. 5. Stable plateau in the reduced model with Idc = 25 nAcm2. A: V and [Ca]i responses to brief (100 ms) depolarizing pulses. With I = 35 nAcm2 (*), the model returns to its resting state, while I = 100 nAcm2 (, ) switches the model to a plateau state. Stability of the plateau is illustrated by the transient hyperpolarization triggered by a small hyperpolarizing pulse (, I = 35 nAcm2). The dendrite can be, however, switched off to its resting state by a larger pulse (, I = 100 nAcm2). B: phase plane representation reveals the origin of this threshold behavior. Traces shown in A (thin lines) are displayed in the (V, [Ca]i) plane, together with the V- and [Ca]i-nullclines of the system (thick traces). Reduced model has 2 stable attractors (resting and plateau states, ) and an unstable equilibrium (saddle, ). Starting from 1 of the 2 stable points, a trajectory must cross the stable manifold of the saddle (dashed line) to converge to the other stable point.

The origin of these features is more evident in Fig. 5B, 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. 4C). 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. 5B). 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. 5A) and full models (Fig. 2C) were clearly sub-threshold, consistent with experimental observations (Llinas and Sugimori 1980b).

Valley potentials

A third kind of dynamical behavior was obtained with I_dc larger than the upper bound of the  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. Figure 6A shows, however, that the plateau exhibited complex dynamical responses to such brief inputs. Thus a 50-nAcm², 100-ms duration pulse resulted in a transient passive hyperpolarization. Increasing the intensity to 75 nAcm² turned this passive response into a triangular, inverted plateau of approximately 150 ms duration. Further increase of I to 100 nAcm² lengthened this response to 1 s. Comparison with Fig. 2B shows how the time course of V 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 as can be seen from the trace with the 150-nAcm² pulse. This shows that the model could produce a stereotyped trace of past inhibitory inputs. However, Fig. 6A suggests that such traces could take place only following inhibitory inputs with a magnitude sufficient to bring V under a threshold located around 50 mV. This threshold behavior can be understood in Fig. 6B, 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 the V-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.

View larger version (35K): [in this window] [in a new window]   Fig. 6. Valley potentials in the reduced model with Idc = 50 nAcm2. A: responses to 100-ms hyperpolarizing pulses from the stable plateau potential (I = *, 50; , 75; , 100; and , 150 nAcm2, see bottom traces). Mirroring plateaus (compare to Fig. 2A), increasing magnitude of hyperpolarizing pulses turn a passive response into triangular and rectangular responses. B: representation in the phase plane illustrates changes in the nullclines intersections. Single equilibrium point left corresponds to the plateau state (). This point is excitable, and trajectories that cross the middle branch of the V-nullcline form rectangular valleys ( and ). C: quantitative analysis of resetting valley potentials as a function of I.

Figure 6C summarizes characteristics of valley potentials as a function of I. From 75 to 90 nAcm², 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 = 90 nAcm² 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 infinite-duration plateaus by injecting a tonic current laying in the  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. 7A, on the left of the  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 had the following amplitudes (nAcm²): 150 (×), 200 (+), 250 (), and 300 (). Figure 7A shows that hyperpolarizing tonic currents prevented the pulses from triggering plateau potentials, down to a critical I_dc value where plateaus emerged with a triangular shape. This critical current value was more negative when I was large, ranging from about 100 nAcm² for I = 300 nAcm² to 25 nAcm² for I = 150 nAcm². Whatever the value of I, 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 the I_dc values where the curves exhibit a slope discontinuity. The curves for the 250 and 300 nAcm² had to be interrupted at I_dc = 30 and 70 nAcm², respectively, because the pulses triggered Ca spikes with hyperpolarizing I_dc below these values. The curves for the two smaller pulses could be extended up to the lower bound of the  region, where the plateau duration became infinite. Overall, Fig. 7A shows that the duration of plateaus in the model could be made to cover an infinite range by varying I_dc between approximately 100 nAcm² and the lower bound of . 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. 7A as the phasic input magnitude increases is clearly evident in the voltage traces illustrated by Llinas and Sugimori (1992).

View larger version (46K): [in this window] [in a new window]   Fig. 7. Global behavior of the model. A: duration of plateaus and valleys as a function of Idc. Plateaus and valleys were triggered by 100-ms pulses with the following I value (nAcm2). Plateaus: 300 (), 250 (), 200 (+), 150 (×); valleys: -100 (), 150 (), 200 (), 300 (). B-D: origin of the influence of Idc on the duration of sub-threshold responses. By assuming that [Ca]i and n are at equilibrium at each point in time, the full model was reduced to a 1-dimensional model with variable V. B: subset of constant dV/dt curves of the 1-dimensional model in the (Idc, V) plane; the value of the potential derivative (mVs1) is indicated by a label on each curve. C and D: compare the time course of typical sub-threshold responses in the full model (thick lines) and in the 1-dimensional model (thin lines): triangular plateau (C, top, Idc = 25 and I = 155), rectangular plateau (C, bottom, Idc = 0 and I = 155), triangular valley (D, top, Idc = 75 and I