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J Neurophysiol (November 1, 2002). 10.1152/jn.00839.2001
Submitted on 15 October 2001
Accepted on 24 July 2002
Institute National de la Santé et de la Recherche Médicale U.483, Université Pierre et Marie Curie, Boîte 23, 75252 Paris Cedex 05, France
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ABSTRACT |
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Genet, Stéphane and Bruno Delord. A Biophysical Model of Nonlinear Dynamics Underlying Plateau Potentials and Calcium Spikes in Purkinje Cell Dendrites. J. Neurophysiol. 88: 2430-2444, 2002. 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.
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INTRODUCTION |
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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 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.
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DENDRITIC MODEL |
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Electric properties of the membrane
The present study examines an isopotential,
single-compartment model of a dendrite with radius
Rd (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
(Kdr) 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 Kdr
channel (Gruol et al. 1991
), we have lumped these currents into a generic IKsub,
embedded with voltage activation at sub-threshold potentials.
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In the present model, dynamics of the membrane potential, V
(in millivolts), obeyed the differential equation
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(1) |
2) stands for
the specific membrane capacitance;
Ileak is a leakage current, and
I
and
Idc, 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
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(2a) |
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(2b) |
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(2c) |
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(2d) |
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
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(3) |
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
in Eq. 2a. The same assumption was made for
IKsub, whose activation variable
u was described by its equilibrium value
u
. As spiking requires delayed
activation of ICa and
IKdr, the latter current was assumed
to activate with a classical, bell-shaped time constant (Hille
1992
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(4) |
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
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(5) |
h) left as a freely
adjustable parameter. Other simulations were run with activation of
gKsub depending on
[Ca]i to mimic the BK-type K conductance of
Gruol et al. (1991)
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(6) |
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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
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(7) |
1M
1), T
is the absolute temperature (K), and F is the Faraday
constant (CM
1). 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
= 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
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(8) |
(2Rd
)/[2k(Rd
)],
where k corresponds to a one-dimensional diffusion constant (cms
1).
(µMs
1)
denotes a sink term accounting for the binding of Ca to the buffer.
This process was described by a first order reaction
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(9) |
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(10) |
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(11) |

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(12) |
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2, T = 298 K,
F = 96, 500 Cmol
1,
R = 8.32 JK
1mol
1,
Rd = 5 × 10
5 cm, [B]T = 150 µM, Kd = 1 µM,
[Ca]b = 50 nM, [Ca]o = 1.1 mM,
= 3 × 10
5 cm,
k = 0.01 cms
1,
gleak = 20 µScm
2, gCa = 600 µScm
2,
gKsub = 30 µScm
2, gKdr = 4,200 µScm
2,
ELeak =
60 mV,
EKsub =
95 mV,
EKdr =
95 mV,
Vs =
22 mV, Vu =
44.5 mV,
Vn =
25 mV,
ks = 4.53 mV,
ku = 3 mV,
kn = 11.5 mV,
n0 = 0.2 ms,
n1 = 4.15 ms, c
n = 0.6, k
n = 17 mV, V
n =
22.5 mV;
gKsub inactivation:
Vh =
50 mV,
kh = 8 mV; Ca-dependence of
gKsub:
KdCa = 10 nM,
kCa = 200 nM.
Analytical and numerical methods
To simplify the typography, we introduce the following notation
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(13) |
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(14) |
Let
(µ) denotes an hyperbolic equilibrium point of
system (Eq. 13), i.e., a point satisfying
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(15) |
![<B>F</B>[<OVL><B>X</B></OVL>(&mgr;)]](/content/vol88/issue5/fulltext/2430/img024.gif)



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(16) |
![<B>F</B>[<OVL><B>X</B></OVL>(&mgr;)]](/content/vol88/issue5/fulltext/2430/img024.gif)
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.
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RESULTS |
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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 nAcm
2) triggered a train of fast Ca spikes,
each of them accompanied by a distinct [Ca]i
transient. The amplitude of the spikes decreased slightly during the
first 300 ms of the pulse, but the model settled hereafter into a
regular firing mode with a frequency of approximately 10 Hz. The firing
abruptly ceased at the pulse offset and V recovered to its
resting value at
58.3 mV. [Ca]i did not fully
relax to its resting level between spikes, resulting in a slow increase of the baseline level that culminated at 2.5 µM within 0.3 s
after onset of the pulse. Spike-induced [Ca]i
transients developed with increasing amplitude as the envelope
progressively saturated the buffer (Kd = 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
).
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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 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
= 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. Increasing
I
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 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
Idc 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 (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. 2A.
The slope of the limit cycle
frequency/Idc curve decreased rapidly
with increasing currents (Fig. 2D), and the relation became
close to linear above 700 nAcm
2.
Idc = 103
nAcm
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
range computed above corresponded to a current domain where the
model exhibited bistability. However, the bifurcation parameter in Fig.
2C was Idc. With
Idc = 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
gKdr from the model, because this
conductance never activated more than 5% of its maximum value during
plateaus. However, in the range [
50,
40] mV,
gKdr and gKsub 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
gKsub suppressed
(gKdr left unchanged), the model lost
its capacity to sustain plateaus; but it could still fire Ca spikes,
showing that spikes arose from interaction between
ICa and
IKdr. When gKdr was suppressed
(gKsub 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.
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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 nAcm
2). 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
ICa 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.
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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-nAcm
2 pulse resulted in a 8-mV transient
depolarization, accompanied by a moderate [Ca]i
increase (peak approximately 150 nM). With I
= 130 nAcm
2, 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 nAcm
2 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 nAcm
2. 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,
Idc 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 Idc = 25 nAcm
2 (laying at the center of
) 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
= 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 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-nAcm
2 pulse.
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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. Idc 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
Idc 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-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
to
100 nAcm
2 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-nAcm
2 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.
|
Figure 6C summarizes characteristics of valley potentials as
a function of I
. 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
= 90 nAcm
2 represented a threshold value, above
which rectangular valley potentials adopted stereotyped characteristics
(mean level approximately
53 mV, 1 s duration;
[Ca]i peak at 250 nM and
[Ca]i deviation of 0.5 µMs).
Global behavior
The previous sections have shown that the plateaus with
spontaneous reset, which could be triggered by a brief depolarizing pulse (Figs. 1 and 3), were transformed into infinite-duration plateaus
by injecting a tonic current laying in the
range (Fig. 5). It is
apparent from this result that Idc 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 Idc. The pulse duration was
100 ms in all cases, and I
had the
following amplitudes (nAcm
2): 150 (×), 200 (+), 250 (
), and 300 (
). Figure 7A shows that hyperpolarizing tonic currents prevented the pulses from triggering plateau potentials, down to a critical
Idc value where plateaus emerged with
a triangular shape. This critical current value was more negative when
I
was large, ranging from about
100 nAcm
2 for
I
= 300 nAcm
2 to
25
nAcm
2 for
I
= 150 nAcm
2. 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
Idc values where the curves exhibit a
slope discontinuity. The curves for the 250 and 300 nAcm
2 had to be interrupted at
Idc =
30 and
70
nAcm
2, respectively, because the pulses
triggered Ca spikes with hyperpolarizing Idc 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
Idc between approximately
100
nAcm
2 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)
.
|
Figure 7A also illustrates the influence of
Idc on the duration of valley
potentials on the right of the
region. The 100-ms pulses used to
generate these valley duration/Idc
curves had the following magnitudes (nAcm
2):
-100 (
),
150 (
),
200 (
), and
300 (
).
The valley duration decreased as Idc
increased, and valleys with the longest duration were found near
like the longest plateaus. In contrast to plateaus, the duration of
valleys varied smoothly with Idc. In
summary, pulses with an appropriate magnitude could trigger plateaus
and valleys within a range of Idc
values spanning almost seven times the width of the
region.
Figure 7A shows, through variations in
Idc, 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
(Idc, V) plane as vertical
bars in Fig. 7B, 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
(Idc, 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 ICa and
IKsub (Fig. 7B). Due to
this distortion of the potential derivative curves, the one-dimensional model crossed regions of different dV/dt values in the
(Idc, V) plane during the
responses illustrated in Fig. 7, C and D. With the triangular plateau (Fig. 7C, 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. 7B) and this relatively
low dV/dt value entailed the initial slow rate of
repolarization of the one-dimensional model (Fig. 7C). As
the plateau decayed, however, the rate of repolarization accelerated
because the model crossed regions with growing values of
dV/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. 7C). The rectangular plateau
on the bottom of Fig. 7C had a longer duration
because, at the value of Idc = 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
1
during the triangular plateau. It is seen in Fig. 7C 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 and
D, 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
region, due to the activation of
ICa and
IKsub.
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
region, due to its critical role in setting sub-threshold responses of the model. Figure
8, A and B, shows that increasing gKsub narrowed
by
positively shifting the left endpoint of the excited stable branch in
the bifurcation diagram. Decreasing
gKsub widened
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 gKsub = 25.25 µScm
2 (marked by a vertical dashed line
through
). With further decrease in
gKsub, the excited branch eventually
lost stability at a lower Idc than the
right endpoint of the resting branch (Fig. 8B), turning
into a current domain where the model could still display bistability, but no longer hysteresis. To highlight the difference,
boundaries were plotted as dashed lines when
corresponded to a bistable region
(same symbols were used to locate Hopf/homoclinic and
hysteresis/bistability transitions throughout Fig. 8). Note that
vanished for gKsub = 15 µScm
2, due to the coalescence of the left and
right endpoints of the excited branch. Figure 7, C and
D, illustrates the effects of changing activation parameters
of gKsub.
rapidly vanished when the half-activation potential Vu
became more negative, while less negative
Vu widened
up to
Vu =
37.95 mV, where
vanished. Overall, Fig. 8C shows an approximate 10-mV-wide range of
Vu values in which the model had a
significant region of hysteresis/bistability. As seen in Fig.
8D,
was widened when
gKsub activated with steeper slopes
(i.e., smaller ku). On the other hand,
was continuously narrowed by decreasing
ku
5 mV, above which
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
(gCa) and F
(gKdr). A
region could be obtained
with deviations of the two conductances up to approximately 50% around
their reference value.
remained a true hysteresis region with all
tested values of gKdr, while it became
a bistability region with gCa'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 radius
Rd = 0.5 µm, corresponding to the thinnest spiny dendrites (Shelton 1985
). Figure
8G shows that increasing Rd
began by steeply increasing the
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 8H
displays essentially similar results when parameter k was
varied in a large range around its reference value (k sets the time constant that relaxation of [Ca]i
would exhibit without buffer in the cytoplasm). Finally, Fig. 8,
I and J, illustrates the effects of varying the
total buffer concentration, [B]T, and its
dissociation constant, Kd, on the
duration of a spontaneously resetting plateau (triggered by a
130-nAcm
2, 100-ms-duration pulse with
Idc = 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. Figure 8J shows that Kd's around
0.35 µm resulted in maximal duration plateaus. Decreasing
Kd 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 Kd'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 with
Iksub 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)
. Figure
9A displays boundaries of the
region in the model with inactivation versus
gKsub (boundaries did not depend on
h, which was taken as a constant). The graph is similar to that obtained without inactivation (Fig. 8A),
except
occurred in a gKsub range
above that found in the model without inactivation. This difference
stemmed from the voltage dependence of
h
, which resulted in significant
inactivation of gKsub 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 gKsub range
studied. As suggested by the
region in Fig. 9A, 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
h modulated
their duration as illustrated with plateaus in Fig. 9B. The
figure plots the length of a finite-duration plateau
(130-nAcm
2, 100-ms-duration pulse) versus
h for three values of
gKsub: 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
h
continuously increased the plateau duration, which became infinite at
h = 2 s; the model discharged a Ca spike
below this critical value. Plateaus were lengthened by the decay of
Iksub as it partly overcame the
decrease in ICa responsible for the
spontaneous resetting of plateaus. This effect was enhanced by
decreasing
h down to the critical value where Iksub decayed too fast to prevent
V from reaching the spike threshold. Critical
h could be decreased by using larger
gKsub values
(
h = 50 ms with
gKsub = 75 and
h = 1 ms with
gKsub = 100). With these larger
gKsub values, however, plateau
duration decreased in the neighborhood of the critical
h instead of becoming infinite. As critical
h was approached, this duration decreased due
to an early transient depolarization of growing amplitude (data not shown). This resulted in a larger initial [Ca]i
increase at the plateau onset that advanced the resetting effect.
|
The basic version of the model omitted Ca-dependent K conductances,
while it highlighted a critical role for [Ca]i
changes in plateau generation. We therefore introduced a Ca-dependence of Iksub 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 of
Iksub became controlled by the slow [Ca]i dynamics. In other words,
Iksub activated too slowly in front of
ICa for the two currents to produce
the balance required for plateau generation.
| |
DISCUSSION |
|---|
|
|
|---|
The model analyzed in this paper accounts for the major features
of the dual electroresponsiveness of PC dendrites. It sustains 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 by
Llinas 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, because
gKsub 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 gKsub 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 when
gKsub 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 idealized
gKsub endows the model with a strong
outward I/V rectification near
45 mV when
gCa 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 gKsub
represents 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 IKdr in
balancing ICa 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 the
Kd of these proteins. Neither
Kd 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 in
Kd and [B]T
values. Slow [Ca]i increases in the
sub-threshold voltage range decreased the Ca Nernst potential, thereby
reducing the magnitude of ICa on a
time scale of hundreds of milliseconds. This induced the reset of
plateaus in the model by breaking the balance between
ICa 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 Idc magnitude.
Spiking emerges at null frequency with the homoclinic bifurcation and only at 5 Hz with the Hopf bifurcation (Fig. 2D). 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 (see
Midtgaard 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
triggering nonlinear responses,
whose duration is modulated by Idc. 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 as
Idc 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 for
I
, 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, whereas Llinas 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 with
Idc values inside the hysteresis
region
, 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. 2C). But it suffices that
Idc decreases below the lower
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 Idc above the upper
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 Idc falls below
the upper
bound (see Fig. 2C). 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.
| |
ACKNOWLEDGMENTS |
|---|
We thank the two referees for sharp criticisms and clever suggestions.
| |
FOOTNOTES |
|---|
Address for reprint requests: Institute National de la Santé et de la Recherche Médicale U.483, Université Pierre et Marie Curie, Boîte 23, 75252 Paris Cedex 05, France (E-mail: stephane.genet{at}snv.jussieu.fr).
| |
REFERENCES |
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Purkinje synapses in rat cerebellar slices.
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