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The Journal of Neuroscience, February 15, 2000, 20(4):1374-1385
Interplay between Facilitation, Depression, and Residual Calcium
at Three Presynaptic Terminals
Jeremy S.
Dittman,
Anatol C.
Kreitzer, and
Wade G.
Regehr
Department of Neurobiology, Harvard Medical School, Boston,
Massachusetts 02115
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ABSTRACT |
Synapses display remarkable alterations in strength during
repetitive use. Different types of synapses exhibit distinctive synaptic plasticity, but the factors giving rise to such diversity are
not fully understood. To provide the experimental basis for a general
model of short-term plasticity, we studied three synapses in rat brain
slices at 34°C: the climbing fiber to Purkinje cell synapse, the
parallel fiber to Purkinje cell synapse, and the Schaffer collateral to
CA1 pyramidal cell synapse. These synapses exhibited a broad range of
responses to regular and Poisson stimulus trains. Depression dominated
at the climbing fiber synapse, facilitation was prominent at the
parallel fiber synapse, and both depression and facilitation were
apparent in the Schaffer collateral synapse. These synapses were
modeled by incorporating mechanisms of short-term plasticity that are
known to be driven by residual presynaptic calcium
(Cares). In our model, release is the product of two
factors: facilitation and refractory depression. Facilitation is caused by a calcium-dependent increase in the probability of release. Refractory depression is a consequence of release sites becoming transiently ineffective after release. These sites recover with a time
course that is accelerated by elevations of Cares.
Facilitation and refractory depression are coupled by their common
dependence on Cares and because increased transmitter
release leads to greater synaptic depression. This model captures the
behavior of three different synapses for various stimulus conditions.
The interplay of facilitation and depression dictates synaptic strength
and variability during repetitive activation. The resulting synaptic plasticity transforms the timing of presynaptic spikes into varying postsynaptic response amplitudes.
Key words:
short-term plasticity; residual calcium; cerebellar
granule cell; cerebellar Purkinje cell; climbing fiber; Schaffer
collateral; hippocampal CA1 pyramidal cell; synaptic model
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INTRODUCTION |
Fast, chemical synaptic transmission
provides the dominant means of information transfer between neurons,
but presynaptic action potentials do not all give rise to identical
postsynaptic responses. It has long been known that the history of
recent activity dynamically regulates the strength of most synapses
(Eccles et al., 1941 ; Feng, 1941 ; Magleby, 1987 ; Zucker, 1989 ). By
converting trains of action potentials into varying amplitudes of
postsynaptic responses, synapses perform a type of temporal filtering
(Lisman, 1997 ; Zador and Dobrunz, 1997 ). This synaptic plasticity
results from presynaptic changes in neurotransmitter release (Varela et al., 1997 ) and alterations in the responses of postsynaptic neurons to
a given amount of transmitter (Trussell and Fischbach, 1989 ; Magee et
al., 1998 ). In this study, we focus on how the temporal pattern of
activity influences neurotransmitter release on the time scale of
milliseconds to seconds.
Synapses show a wide range of responses to high-frequency stimulation
such as enhancement, depression, and more complex behaviors (Feng,
1941 ; Eccles et al., 1964 ; von Gersdorff et al., 1997 ; Selig et al.,
1999 ; Kreitzer and Regehr, 2000 ). This diversity has important
implications for the transmission of information in the CNS. To
demonstrate the effects of use-dependent plasticity on synaptic
transmission, we considered some possible responses to a typical spike
train recorded in vivo (Fig. 1A). We simulated responses to this spike train for two synapses with different presynaptic properties but identical postsynaptic properties (Fig. 1B,C). The temporal pattern of postsynaptic
depolarization in each example reflected the amount of facilitation and
depression present during periods of high- and low-frequency activity.
At one synapse, bursts of activity transiently boosted synaptic
strength (Fig. 1B). In contrast, the other synapse conveyed
the average rate of activity through EPSPs of relatively constant
amplitude (Fig. 1C). These simulations highlight the
importance of understanding presynaptic properties that determine the
influence of a neuron on the firing of its targets.
To understand the mechanisms underlying presynaptic plasticity, it is
necessary to consider presynaptic residual calcium
(Cares) (Zucker, 1999 ). In contrast to the high
calcium levels that trigger transmitter release after presynaptic
depolarization (>10 µM for ~1 msec), Cares
is the modest elevation in calcium levels (hundreds of nanomolar)
lasting for hundreds of milliseconds. During periods of high-frequency
presynaptic activity, Cares accumulates and is involved in
various short-term plasticities such as facilitation (Katz and Miledi,
1968 ; Zucker and Stockbridge, 1983 ; Kamiya and Zucker, 1994 ; Atluri and
Regehr, 1996 ), augmentation (Zengel et al., 1980 ; Swandulla et al.,
1991 ; Delaney and Tank, 1994 ), post-tetanic potentiation (Delaney et
al., 1989 ), and recovery from presynaptic depression (Dittman and
Regehr, 1998 ; Stevens and Wesseling, 1998 ; Wang and Kaczmarek, 1998 ).
It is likely that differences in the contribution of various
calcium-dependent mechanisms to synaptic strength will add to the
diversity of synaptic behavior during repetitive activation.
Here, we present a simple framework for understanding synaptic dynamics
in terms of underlying plasticities and Cares. We build on
different aspects of models that were developed previously (Magleby,
1987 ; Abbott et al., 1997 ; Markram et al., 1998 ) and take advantage of
recent clarification of the role of Cares in facilitation
and recovery from depression. We develop an analytical expression for
the use-dependent enhancement and depression of transmitter release
driven by Cares and apply the scheme to three synapses with
very different properties. This highly simplified model of presynaptic
plasticity captures much of the apparent complexity observed during
realistic Poisson train stimuli at all three synapses.

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Figure 1.
Effects of presynaptic plasticity on synaptic
transmission. A, Representative spike train recorded from
the basal ganglia of an awake, behaving macaque. B, C,
Simulated EPSPs resulting from the stimulus in A (see
Materials and Methods). The mean peak EPSPs were the same in both
examples (i.e., synaptic currents were normalized to give the same
average depolarization). The only difference between the presynaptic
terminals in B and C is the amount of
facilitation and initial release probability
(F1). Comparison of the traces reveals
the different temporal patterns of depolarization associated with each
type of presynaptic plasticity. The postsynaptic cell was simulated
using a passive single compartment model with parameters
m = 20 msec, Vrest = 70 mV, RN = 100 M . The FD model
parameters for B were = 3.1, F1 = 0.05, F = 100 msec, D = 50 msec, kmax = 30 sec 1, ko = 2 sec 1, KD = 2. Model
parameters for C were = 2.2, F1 = 0.24, F = 100 msec, D = 50 msec, kmax = 30 sec 1, ko = 2 sec 1, KD = 2. The
spike train in A was kindly provided by John Assad and Irwin
Lee.
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MATERIALS AND METHODS |
Synaptic physiology. For cerebellar recordings,
300-µm-thick transverse slices were cut from the cerebellar vermis of
9- to 14-d-old Sprague Dawley rats (Llano et al., 1991 ). For
hippocampal recordings, 300-µm-thick slices were cut from the
hippocampus of 16- to 19-d-old Sprague Dawley rats, and the CA3 region
was removed. Cerebellar slices were superfused with an external
solution containing (in mM): 125 NaCl, 2.5 KCl, 1.5 CaCl2, 1 MgCl2, 26 NaHCO3, 1.25 NaH2PO4,
and 25 glucose, bubbled with 95% O2/5%
CO2. For hippocampal slices, the external divalents were 2 CaCl2 and 3 MgCl2 to minimize multisynaptic
activity and CA1 population spikes during repetitive stimulation. Flow
rates were 4-6 ml/min at 34°C. Bicuculline (20 µM) was
added to the external solution to suppress synaptic currents mediated
by GABAA receptors. During parallel fiber trains, the
GABAB receptor antagonist CGP55845a (2 µM),
the adenosine A1 receptor antagonist DPCPX (5 µM), and the mGluRIII antagonist CPPG (30 µM) were included in the external saline. During studies
of the hippocampal Schaffer collateral synapse, (R)-CPP (5 µM) was present to suppress NMDA receptor-mediated currents.
Whole-cell recordings of Purkinje cells and CA1 pyramidal cells were
obtained as described previously (Mintz et al. 1995 ) with an
internal solution of (in mM): 35 CsF, 100 CsCl, 10 EGTA, 10 HEPES, and 0.2 D600, adjusted to pH 7.2 with CsOH. Synaptic currents
were monitored at a holding potential of 40 mV to inactivate voltage-gated Na channels, and D600 was included to block voltage-gated calcium channels. In some CA1 recordings, QX-314 (5 mM) was
also included in the internal solution. The access resistance and leak current ( 20 to 200 pA holding at 40 mV) were monitored
continuously. Experiments were rejected if either access resistance or
leak current increased significantly during recording. Presynaptic fibers were stimulated with two glass electrodes (tip diameter, 10-12
µm) filled with external saline solution placed in the molecular layer. Brief pulses (200 µsec) of current (5-15 µA) were passed between the two stimulating electrodes. This configuration greatly reduced the size of the stimulus artifact. The inter-stimulus interval
was 2 min for trains of 10 stimuli and 3 min for trains of >10
stimuli. Low stimulus intensities were used to keep initial synaptic
currents small ( 50 to 150 pA) thereby minimizing series resistance
errors during the train.
Detecting presynaptic calcium transients. Parallel fibers,
made up of granule cell axons and presynaptic terminals, were labeled with a high-pressure stream of the low-affinity calcium indicator magnesium green-AM (Molecular Probes, Eugene, OR) (Zhao et al., 1996 )
using techniques developed previously (Regehr and Atluri, 1995 ).
Parallel fiber tracts were stimulated extracellularly, and
epifluorescence was measured with a photodiode from a spot several
hundred micrometers from the loading site, where the vast majority of
the fluorescence signal arises from parallel fiber presynaptic boutons
that synapse onto Purkinje cells. The peak F/F change
produced by a single stimulus was used as a linear measure of
presynaptic calcium influx, as established previously (Regehr and
Atluri, 1995 ).
Data acquisition and analysis. Outputs of the Axopatch 200A
were filtered at 1 kHz and digitized at 20 kHz with a 16-bit D/A converter (Instrutech, Great Neck, NY) using Pulse Control software (Herrington and Bookman, 1995 ). Random train stimuli were generated off-line and sent through the DAC to the stimulus isolation unit. Both
on- and off-line analysis as well as computer simulations were
performed using Igor Pro software (Wavemetrics, Lake Oswego, OR).
Model of use-dependent plasticity driven by presynaptic
calcium. We modeled the experimentally determined EPSC size as the product of the number of physical release sites
(NT) and the fraction of sites that undergo
release on arrival of an action potential. This fraction was divided
into two parts: a facilitation component (F) and a
depression component (D), both of which could range between
0 and 1:
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(1)
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Note that release is scaled by the average mEPSC amplitude ( )
to produce the final EPSC amplitude. The depression variable was set to
unity at rest, reflecting the assumption that all potential release
sites are available for release when no release activity has occurred
for some time. Therefore, the resting probability of release is
equivalent to the initial value of F
(F1).
Facilitation
Enhancement of release was assumed to be directly related to the
equilibrium occupancy of the release site by a calcium-bound molecule
CaXF with dissociation constant,
KF (see Fig. 3A for reaction
scheme):
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(2)
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where CaXF decays exponentially with time
constant F after a jump of size F after
an action potential at time to:
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(3)
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where (t) is the Dirac delta function and is
defined to have units of sec 1. Note that the
notation here differs from a previous publication in which we defined
(t) to be unitless (Dittman and Regehr, 1998 ). The
treatment of CaXF does not take into account the
underlying decay time course of free calcium, but instead serves as an
approximate description based on experimental evidence that the decay
of paired-pulse facilitation is roughly exponential in nature (Atluri
and Regehr, 1996 ). F represents the decay time constant,
which was measured to be ~100 msec at the granule cell to Purkinje
cell synapse at 34°C (Atluri and Regehr, 1996 ). We explored
variations of Equation 2 with a power law relationship between
facilitation and CaXF, but the performance of
the model was not improved with these modifications. We therefore chose
the linear relationship for the sake of simplicity. To create an
analytical expression for Equations 2 and 3 where CaXF is allowed to decay to 0, Equation 2 was
modified with the addition of a baseline release probability
F1 in the absence of CaXF:
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(2a)
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With this change, F(t) ranges from
F1 to 1 as CaXF increases
from 0. Given a quiescent presynaptic terminal with
CaXF = 0 and F = F1, then immediately after stimulation with a
single action potential, CaXF increases to
F and F increases to:
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(4)
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NTD1F1
sites have undergone release and entered a refractory state
while NT(1 D1F1) sites remain available. If a
second stimulus occurs just after the first stimulus such that no
recovery from the refractory state has occurred, then the second EPSC
will be determined by the increased release probability
F2 and the remaining number of release sites as
follows:
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(5)
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Therefore, the facilitation ratio ( ) can be expressed as:
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(6)
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if D1 = 1 (i.e., all release sites
are initially available). Because F2 cannot
exceed unity, one can establish an upper bound on the initial
probability of release: F1 1/(1 + ). By substituting Equation 4 into 6 for
F2 and solving for the constant
KF/ F, one arrives at:
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(7)
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so the value of KF/ F is
entirely determined by the experimentally observed value of
facilitation and the initial release probability.
During regular stimulus trains at rate r, the amplitude of
CaXF just before the ith stimulus is
given by:
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(8)
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where:
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(9)
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The value of release probability just before the ith
stimulus can be expressed as:
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(10)
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using Equation 2a. Release probability approaches a
steady-state value of:
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(11)
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Calcium dependence and kinetics of recovery from depression
Our model of recovery from depression has been previously
described for the climbing fiber [see Scheme II in Dittman and
Regehr (1998) ]. Here, we have adopted different symbols for some of
the parameters for clarity. We use CaXD in place
of Ca to refer to the calcium-bound site and
D instead of c for the decay time constant of this species. The model assumes three possible states of
the release apparatus (R, T, and N), where
R sites are in a refractory state, T are in a
transitional state, qj N sites are release-ready, and
there are a total of NT release sites
(R + T + N = NT).
Calcium dependence of the recovery rate (R N) was
captured by the equilibrium binding occupancy of a calcium-bound
molecule CaXD, which instantaneously
rises by D after an action potential at time
to and decays to 0 exponentially with time
constant D:
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(12)
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This treatment approximates the underlying kinetics of a
reaction between Cares and some site
XD in a manner similar to the facilitation
scheme described above. Experimentally, the value of D
was estimated from the amplitude and duration of the rapid phase of
recovery from paired-pulse depression at the climbing fiber synapse at
34°C [see Dittman and Regehr (1998) for a similar approach at
24°C]. Because F and D reflect
distinct binding reactions, there is no a priori reason to
assume that they will have equal values. As explained in Results, we
found that the experimental data could be better fit using a value of
D that was about half the value of
F, perhaps reflecting different underlying calcium-binding kinetics.
The probability that a release site is release-competent is then given
by the depression variable D = N/NT,
governed by the equation:
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(13)
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where
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(14)
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For Equations 12 and 13, rapid equilibration with
CaXD and the steady-state approximation
dT/dt 0 are assumed as described previously
(Dittman and Regehr, 1998 ). For CaXD = 0, krecov = ko, and
D recovers exponentially with time constant
recov = 1/ko. For values of
CaXD KD,
krecov = kmax, so
D recovers exponentially at a faster rate with
recov 1/kmax. For
intermediate, time-varying values of CaXD,
D recovers with both fast and slow kinetic components. The
form of Equation 14 was chosen for mathematical convenience as in
Equation 2a. With this form of rate dependence,
CaXD is allowed to decay to 0 while the
rate of recovery slows to a fixed, calcium-independent rate.
For regular stimulus trains given at frequency r, Equations
12, 13, and 14 can be used to generate an analytical expression for the
fraction of available release sites during the ith
stimulus:
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(15)
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where
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(16)
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CaXDi 1 is the level of calcium-bound
XD just after the (i 1)th
stimulus and is given by:
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(17)
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where:
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(18)
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For the ith stimulus in a regular train of rate
r, the normalized EPSCi can be expressed
analytically:
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(19)
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During a prolonged stimulus train, the number of available sites
reaches a steady-state value that can be expressed analytically using
Equations 9, 11, 15, 16, and 18:
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(20)
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Combining this equation with Equation 11, one can then generate
an analytical expression for the steady-state EPSC size normalized to
the initial EPSC:
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(21)
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Equations 19 and 21 were used to generate the simulations in
Figures 3-6. Simulated EPSC waveforms were produced using the
normalized alpha function,
(t*e/ E)*exp( t/ E)
where E was the decay time constant of the simulated
EPSC. The alpha function was then scaled in amplitude to the value
EPSCi for the ith EPSC in a train:
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(22)
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For the synapses modeled in this study, E was set
equal to 2 msec. For the Poisson train simulations in Figure 7,
Equations 10 and 15 were used with CaXFi and
CaXDi generated by the recursive equations:
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(23)
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(24)
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where ti is the ith inter-stimulus
interval in a Poisson train. A summary of the model parameters is
given in Table 1.
Simulated integrate-and-fire neuron. For Figure 8, we
implemented a single compartment model neuron with membrane time
constant m = 20 msec, input resistance
RN = 100 M , and resting potential Vrest = 70 mV. The EPSC waveforms
generated with Equation 22 were then scaled to conductance values,
Gsyn(t), chosen to produce a
particular EPSP magnitude (see Fig. 8) with synaptic reversal potential
Vsyn set equal to 0 mV. Membrane voltage
(Vm) was determined by solving the first-order
differential equation:
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(25)
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where if the voltage exceeded threshold
(Vthresh = 55 mV), its value was then
jumped to +40 for 1 msec and then hyperpolarized to 75 mV using an
idealized action potential waveform template. For Figure 8, Equation 25
was numerically integrated with a first-order Euler routine.
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RESULTS |
We recorded synaptic currents during various stimulus conditions
at three CNS synapses in rat brain slices. In addition, presynaptic calcium influx was measured in cerebellar granule cell presynaptic terminals for the same stimulus conditions. These experimental studies
formed the basis of a model of synaptic plasticity that was then
applied to each type of synapse.
Diversity of short-term synaptic plasticity in the CNS
The responses of three types of synapses to 50 Hz stimulation were
measured using whole-cell voltage clamp at 34°C (Fig.
2A). The inferior olive
climbing fiber to Purkinje cell synapse (CF) depressed during stimulus
trains as reported previously (Fig. 2A, top trace)
(Eccles et al., 1966 ). The cerebellar granule cell parallel fiber to
Purkinje cell synapse (PF) was activated under stimulus conditions
identical to those of the climbing fiber synapse (Fig. 2A, middle
trace) but synaptic strength enhanced markedly during the
stimulation. In contrast to the previous examples, the hippocampal CA3
to CA1 Schaffer collateral synapse (SC) briefly facilitated and then
depressed during the stimulus train. These three synapses demonstrate
the wide spectrum of short-term transmitter release properties
expressed during high-frequency stimulation.

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Figure 2.
Diversity of short-term plasticity in the CNS.
A, Top, Climbing fiber to Purkinje cell EPSCs
(CF); middle, parallel fiber to Purkinje
cell EPSCs (PF); bottom, CA3 to CA1
Schaffer collateral EPSCs (SC) recorded while stimulating
afferents at 50 Hz for 10 stimuli at 34°C. Traces are averages of
four to six trials each. Stimulus artifacts were suppressed for
clarity. Vertical scale bar is 2, 400, and 60 pA for the CF, PF, and SC
synapses, respectively. B, Average magnitude of the
8th-10th EPSC normalized by the first EPSC plotted as a function of
stimulus frequency for the climbing fiber (top), the
parallel fiber (middle), and the Schaffer collateral
(bottom) synapses. Data are shown as mean ± SEM
(n = 4-5). C, Measurement of parallel fiber
Cares during a 10 pulse, 50 Hz stimulus train using the
calcium-sensitive indicator magnesium green. Vertical scale bar is
percentage F/F. Parallel fiber data were adapted from
Kreitzer and Regehr (2000) .
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For all three synapses, the immediate effects of short-term
plasticity began to plateau by the eighth stimulus. Therefore, the
normalized average of EPSCs 8 through 10 (EPSC8-10/EPSC1) provided a useful
measure of steady-state behavior. The frequency-dependence of
steady-state transmitter release clearly differed at each type of
synaptic connection (Fig. 2B). The climbing fiber depressed and the parallel fiber facilitated at all frequencies. The Schaffer collateral synapse facilitated at low rates but depressed during higher-frequency stimulation. During prolonged stimuli, processes acting on the tens of seconds to minutes time scale, such as
post-tetanic potentiation and a slower form of presynaptic depression
(Galarreta and Hestrin, 1998 ), could potentially affect release. Thus,
stimulus trains were kept relatively short to isolate facilitation and shorter-lived forms of depression.
In addition to quantifying transmitter release, we recorded the
behavior of Cares during high-frequency stimulation (see
Materials and Methods). A representative measurement of granule cell
Cares during a 10 stimulus 50 Hz train using the
calcium-sensitive indicator magnesium green is shown in Figure
2C. Cares increased rapidly after each stimulus
and then decayed over hundreds of milliseconds with a time course that
is similar in many synapses in the CNS (Feller et al., 1996 ; Helmchen
et al., 1997 ; Sinha et al., 1997 ).
A model of presynaptic plasticity
We developed a model to account for the behavior of these synapses
during repetitive stimulation. Our aim
was to keep the model as simple as possible while including a level of
detail matched to the constraints provided by our experiments. A number of simplifying assumptions allowed us to obtain an analytical solution
while retaining important aspects of facilitation and recovery from
depression. According to our scheme, sustained presynaptic activity
depletes the number of functional sites but also elevates Cares, thereby facilitating release and accelerating
recovery from depression (Fig.
3A). The model is based on the
following principles:

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Figure 3.
FD model for Ca-dependence of short-term
plasticity. A, Left, Residual presynaptic calcium
binds to site XF, and the complex
CaXF then binds to the release site causing an
enhancement of release probability. Right, Schematic of
residual presynaptic calcium binding to site XD,
which then binds with the refractory release site driving a transition
back to the release-ready state. B, Left, F
plotted as a function of CaXF ranging from a
minimal probability of F1 (no residual calcium)
to a maximum of 1. The dissociation constant for
CaXF is KF.
Right, the recovery rate for depression is plotted as a
function of CaXD with a minimum rate of
ko, a maximum rate of
kmax, and CaXD
dissociation constant KD. C,
Presynaptic levels of CaXF (thin
line) and CaXD (thick line),
fraction of available synapses that undergo release (F),
fraction of release-ready synapses (D), and normalized EPSC
during a train of 10 stimuli at 100 Hz. Model parameters for this
simulation were = 3.4, F1 = 0.15,
F = 100 msec, D = 50 msec,
kmax = 30 sec 1,
ko = 2 sec 1,
KD = 2. Note that
CaXF and CaXD were
normalized to their respective dissociation constants.
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Assumption I
Release = NT·pR where
NT is the number of physical release sites and
pR is the probability of transmitter release
(Del Castillo and Katz, 1954a ).
Assumption II
Release probability is composed of two independent variables
(probabilities): pR = F·D
where F and D both range between 0 and 1. Therefore, the average EPSC is given by
·NT·F·D where is the
average mEPSC amplitude (Eq. 1). The values of F and D determine the state of the presynaptic terminal. Although
F and D are thought to be stochastic variables at
individual release sites, we treated them as deterministic averages in
this study for the purpose of simplicity (Del Castillo and Katz, 1954b ;
Stevens and Wang, 1995 ). In this way, "release" represents the
average of many trials at a particular synaptic connection, or
equivalently the summation of many simultaneously activated synapses.
Assumption III
Facilitation is attributable to the calcium-dependent increase in
the value of F from an initial value
F1 according to the fractional occupancy of some
calcium-bound molecule CaXF at the release site
(Fig. 3A, B, left). CaXF
increases by a constant amount ( F) with each
action potential and decays exponentially toward 0 with time constant
F (Eqs. 2, 2a, and 3). A constant increment in
CaXF is equivalent to assuming that
XF remains unsaturated during a stimulus train.
Assumption IV
Depression is determined by the fraction of sites that undergoes
transmitter release in such a way that D = (number of
available sites)/(total number of sites). According to Assumptions I
and II,
NT*F1*D1
sites have undergone transmitter release and enter a refractory state,
leaving NT*(1 F1*D1) sites available to release
immediately afterward. Therefore D2 = 1 F1*D1 just after an action potential and
recovers back to D1 at a specified rate (see
assumption V). For maximal simplicity in the model we explore here, it
was assumed that D1 = 1 (i.e., after long
quiescent intervals, all sites are available to undergo release; see
Eqs. 5 and 6). We refer to this form of depression as refractory
depression [see also Dittman and Regehr (1998) ].
Assumption V
Recovery from refractory depression is dependent on
Cares in the following manner. In the absence of
Cares, recovery proceeds at some minimal rate
ko. After a stimulus, the rate is accelerated by
the interaction of a calcium-bound molecule CaXD
with the release site (Fig. 3A, right, and Eqs. 12,
13, and 14). CaXD increases by a constant amount
D and decays exponentially to 0 with a time constant
D. We refer to this calcium-dependent acceleration of recovery from refractory depression as CDR. This treatment is based on
our studies at the climbing fiber synapse (Dittman and Regehr, 1998 ).
As in the case of XF, we assume that
XD is not saturated during stimulus trains (see
assumption III).
Assumption VI
Facilitation and depression are inherently coupled through
assumptions I, II, and III. As facilitation increases the number of
sites that undergo release, a larger fraction of the total number of
sites enters the refractory state, thereby increasing depression.
F was estimated by the decay of paired-pulse
facilitation at the parallel fiber to Purkinje cell synapse (Fig.
3B) (Atluri and Regehr, 1996 ). D was
estimated as in Dittman and Regehr (1998) . The values of
F and D are set in part by the decay of
Cares, so changes in Cares will perturb
the recovery kinetics of both facilitation and depression (Eqs. 15 and
16).
An example of the proposed model is shown in Figure 3C
during a 10 pulse 100 Hz stimulus train. The details of the model are described in Materials and Methods. During the train, Cares
and the calcium-bound species CaXF and
CaXD build up and then decay with their
characteristic time constants. F is initially 0.15 and
increases approximately fivefold during the train. Because the
CaXF binding sites are nearly saturated after 10 stimuli, F increases toward unity and remains elevated after
free CaXF decays toward 0. D quickly
declines during the train, but recovery from the refractory state
accelerates as CaXD reaches a high
concentration. As a result of the increase in F and the
decrease in D, transmitter release first enhances and then
declines during the stimulus train. The EPSCs shown in Figure
3C are scaled to reflect the product F·D
normalized to the initial value F1 (see
Materials and Methods).
Categories of synaptic behavior arising from the
facilitation-depression model
We next explored the contributions of calcium-dependent processes
to synaptic behavior simulated by the facilitation-depression (FD)
model (Fig. 4). The four simulations
shown here describe the same synapse in the presence or absence of
either facilitation (±Fac) or calcium-dependent recovery (±CDR)
during a 50 Hz stimulus train. In the absence of facilitation and CDR,
sustained activity gradually depressed synaptic strength (Fig.
4A1). The steady-state level of
depression was more pronounced at higher stimulus frequencies (Fig.
4A2). Thus, the synapse behaves as a
low-pass filter during regular stimulus trains with little attenuation
at frequencies below 1 Hz, a frequency that corresponds to the slow
recovery rate (ko). Stimulation at higher
frequencies eventually depletes a large fraction of available release
sites because there is not sufficient time for recovery between
stimuli.

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Figure 4.
Effects of facilitation and CDR on presynaptic
dynamics. A1, Simulation of 20 EPSCs
generated at 50 Hz in a reduced FD model with no facilitation or CDR.
Dashed line represents the steady-state EPSC size.
A2, Steady-state EPSC magnitude
(normalized to the first EPSC) as a function of stimulus frequency.
B, Same as A for a simulation with facilitation
only. C, Same as A for a simulation model with
CDR only. D, Same as A for a simulation with both
facilitation and CDR. Equations 19 and 21 were used with model
parameters from Figure 3. E1, EPSC peak
amplitudes versus stimulus number for the four model synapses.
E2, Steady-state EPSC versus frequency
curves from A2 to D2 are
superimposed. For A and C, F was held constant at
F1. For A and B, the
recovery rate was held constant at ko.
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When facilitation was added to this model synapse (+Fac, CDR), a
strikingly different behavior emerged during the first few stimuli
(Fig. 4B1). The EPSC amplitude
transiently increased, reflecting an increase in F. However,
the increasingly larger fraction of sites undergoing release rapidly
depleted the number of available sites (decrease in D).
After only five stimuli, transmitter release fell to similar levels as
in the previous example, despite the presence of facilitation (Fig.
4E1, traces A and
B). In the absence of facilitation, steady-state levels of
transmitter release were reached after more than 10 stimuli. Thus,
facilitation reduces the time to reach steady state according to the FD
model. The similarity in steady-state behavior between these two cases
can be seen more explicitly by comparing the superimposed frequency response curves (Fig. 4E2, traces
A and B). The only difference between these two
synapses at steady state is a boosting at intermediate frequencies
(~3-10 Hz).
For a nonfacilitating synapse, the inclusion of CDR ( Fac, +CDR)
allows the synapse to remain much more effective over a wide range of
stimulus frequencies (Fig. 4C). As the stimulus
frequency increases, Cares increases proportionally,
boosting the recovery rate at high frequencies and maintaining synaptic
efficacy even during prolonged trains (Fig.
4E2, traces A and
C).
When facilitation and CDR are both present (+Fac, +CDR), two features
become apparent (Fig. 4D). First, a transient
increase in release is observed during the first few stimuli because of increases in F, similar to the (+Fac, CDR) synapse.
Second, steady-state release was maximal at a stimulus frequency of
~12 Hz. This tuning curve arises from an interplay between
facilitation and depression. At low frequencies (<1 Hz), there is
little Cares accumulation, and neither facilitation nor
refractory depression influences transmitter release. At intermediate
frequencies (5-20 Hz), release probability begins to increase, and the
recovery rate from depression accelerates in concert leading to a net
enhancement. Finally, at high stimulus frequencies (>50 Hz), release
probability and recovery rate have saturated at their maximal values
but the inter-stimulus interval is too short for a significant amount
of recovery to occur, so synaptic strength falls off precipitously.
The transient and steady-state responses of all four simulated synapses
are superimposed in Figure 4E. Steady-state synaptic strength is identical for these synapses at stimulus rates <1 Hz
because little if any Cares accumulates for long
inter-stimulus intervals (>1 sec). The divergence in steady-state
behavior observed at higher frequencies arises from recruitment of the
calcium-dependent processes described above. Thus, the decay kinetics
of Cares determine a frequency threshold, below which
facilitation and CDR do not influence synaptic strength.
Fitting the FD model to three synapses
After characterizing some of the basic features of the FD model,
we determined whether it could account for the observed behavior of the
climbing fiber, parallel fiber, and Schaffer collateral synapses
studied above. We began by establishing experimentally based values for
many of the FD model parameters (Table
2). At the climbing fiber synapse,
F1 was set by the amount of paired-pulse depression (35%) for two closely spaced stimuli.
ko was estimated from the recovery time course
at long inter-stimulus intervals (Dittman and Regehr, 1998 ). For the
parallel fiber synapse, the decay of CaXF
( F) was constrained to match the decay of
paired-pulse facilitation, and the amplitude of facilitation determined
the value of (Atluri and Regehr, 1996 ). For the Schaffer collateral synapse, only the amount of facilitation ( ) was experimentally determined. To constrain the SC model further, we assigned all parameters ( F, D,
ko, kmax, and
KD) to the values used in the parallel fiber
simulations except for the initial release probability, F1. Table 2 summarizes the FD model parameter
values used to characterize all three synapses. The values in bold were
determined by experiments at 34°C. Note that the parameter values
differ from those reported in a previous study conducted at 24°C
(Dittman and Regehr, 1998 ). The other values were chosen both "by
eye" and with a least-squares minimization procedure. The predicted EPSC peaks and steady-state frequency curves shown in Figure
5 were generated by analytical solutions
to approximations of Equations 3, 13, and 14 (see Materials and
Methods, Eqs. 15-21 for analytical solutions).

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Figure 5.
Application of the FD model to real synapses.
A1, F, D, and EPSC size during a 10 pulse
50 Hz stimulus train for a model climbing fiber to Purkinje cell
synapse. A2, Amplitude of the 8th-10th
EPSC (taken from Fig. 2B) plotted against stimulus
frequency. Solid line is the analytical solution given in
Equation 21. Model parameters were F1 = 0.35 for A1, D = 50 msec, kmax = 20 sec 1, ko = 0.7 sec 1, KD = 2. B1, B2, Same as
A for the parallel fiber to Purkinje cell synapse. Model
parameters for both B1 and
B2 were = 3.1, F1 = 0.05, F = 100 msec, D = 50 msec, kmax = 30 sec 1, ko = 2 sec 1, KD = 2. C1, C2, Same as
A for the CA3 to CA1 Schaffer collateral synapse. Model
parameters were = 2.2, F1 = 0.24
for C, F = 100 msec,
D = 50 msec, kmax = 30 sec 1, ko = 2 sec 1, KD = 2. D1, Normalized average EPSC magnitude
during a 50 Hz stimulus versus stimulus number. Open circles
represent mean EPSC amplitudes during 50 Hz trains plotted against
stimulus number for the climbing fiber (CF), parallel
fiber (PF), and Schaffer collateral (SC)
synapses. Data are mean ± SEM with n = 5-7.
D2, FD model fits to the steady-state data in
A2-C2 superimposed for
comparison.
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At the climbing fiber synapse, this model provided a reasonably good
description of both the time course and frequency-dependence of
depression (Fig. 5A2,
D1). Because no facilitation was observed at
this synapse, F was held constant for these simulations at ~0.35. This fixed value of F implies that 35% of
available sites release neurotransmitter when stimulated. The data were
well approximated with only three free parameters (Table 2).
The model also captured the basic features of synaptic plasticity at
the parallel fiber synapse (Fig. 5B2,
D1). According to this simulation, the
enhancement reflected a large increase in F from an initial
value of ~0.05 that was partially countered by a decrease in
D. By the end of the train, the fourfold enhancement in
release was the result of an eightfold increase in F and a twofold reduction in D. Simulations of the hippocampal
Schaffer collateral synapse were similar to the parallel fiber synapse (Fig. 5C2, D1), but
the initial release probability was about five times as large (0.24).
During the train, F was close to unity (saturation of
CaXF binding sites), leading to a large
reduction in D. The model could not account entirely for the
enhancement of release observed at 3 Hz at either the parallel fiber or
Schaffer collateral synapses (see Discussion). The model fits to
steady-state release are superimposed in Figure
5D2 for comparison. Despite the many simplifying
assumptions that we made, the model accounts for synaptic dynamics
during high-frequency stimulation fairly well at these three synapses.
Possible role for CDR at "low P" synapses
Calcium-dependent recovery from depression can play an important
role in sustaining transmitter release during high rates of presynaptic
activity. The importance of CDR has been demonstrated for synapses with
high release probabilities where depression is prominent (Dittman and
Regehr, 1998 ; Wang and Kaczmarek, 1998 ). However, CDR is more difficult
to assess at low probability synapses such as the parallel fiber
synapse, where facilitation tends to obscure any underlying depression.
We used the FD model to explore a possible role of CDR in transmission
at parallel fiber synapses (Fig. 6).
During 50 Hz 25-pulse stimulation, the parallel fiber synapse reaches a
steady-state level of facilitation with no sign of depression. In the
absence of CDR with only slow calcium-independent recovery from
depression present, the FD model did not predict a significant amount
of steady-state facilitation (Fig. 6A, ).

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Figure 6.
Importance of CDR at "low P" synapses.
A, Parallel fiber to Purkinje cell EPSCs recorded during 25 stimuli at 50 Hz. Trace represents a single trial. Open
circles are the predicted FD model EPSC magnitudes using Equation 19. Filled circles represent the same FD model without CDR
(CaXD = 0). B, Steady-state EPSC
size plotted against stimulus frequency for the parallel fiber synapse
with (thin line) and without (thick line) CDR.
Open circles are parallel fiber data from Figure
2B. Model parameters were = 3.8, F1 = 0.038, F = 100 msec, D = 50 msec, kmax = 30 sec 1, ko = 2 sec 1, KD = 2.
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We investigated several possible explanations for the sustained
enhancement observed in Figure 6. One possibility is that an extremely
low initial release probability (F1) can
account for the apparent lack of depression during a train. However, we were unable to fit the data in Figure 5 with values of
F1 smaller than ~0.05 (data not shown),
suggesting that simply lowering F1 cannot
explain the enhancement. Alternatively, recovery from depression could
be fast enough to prevent most of the release sites from accumulating
in the refractory state. This may reflect a fast basal rate of recovery
(ko) at the parallel fiber synapse (fit not
shown) or a calcium-dependent acceleration of recovery from depression,
as observed at the climbing fiber synapse. The FD model provided a good
approximation to synaptic strength when the properties of recovery from
depression were similar to those observed at the climbing fiber
synapse, in that a slow component of recovery from depression and
CDR were both present (Fig. 6A, ).
These simulations emphasize that rapid recovery from depression may be
important even at synapses with low initial release probabilities where
depression is not apparent. The model suggests that fast recovery from
depression is required to preserve synaptic strength during periods of
prolonged activity. Without rapid recovery from depression,
facilitating synapses cannot maintain significant enhancement at steady
state. As shown in Figure 6, inclusion of CDR can account for the
behavior of parallel fibers during long stimulus trains. However,
further experiments are required to determine whether CDR is the
mechanism responsible for fast recovery from depression at these
synapses. This will be difficult to establish until the underlying
molecular correlates of facilitation and CDR can be identified and
independently manipulated.
Synaptic plasticity during irregular stimulus trains
As a further test of the FD model, synaptic currents were recorded
at the climbing fiber, parallel fiber, and Schaffer collateral synapses
during stimulation with irregular stimulus trains (Fig. 7). The climbing fiber synapse showed
sustained depression during the train, with significant recovery after
inter-spike intervals greater than a few hundred milliseconds (Fig.
7A, top). In contrast, the parallel fiber synapse
facilitated during periods of high-frequency activity (Fig. 7A,
middle). The Schaffer collateral synapse enhanced to a
similar degree but the temporal pattern of EPSC peaks differed from the
parallel fiber synapse (Fig. 7A, bottom). At all three synapses, there was a high degree of peak-to-peak variability attributable in large part to the short-term plasticities described above.

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Figure 7.
Presynaptic dynamics during Poisson stimulus
trains. A, Examples of EPSCs recorded in response to an
irregular stimulus train with average rate 20 Hz at the climbing fiber
(CF), parallel fiber (PF), and
Schaffer collateral (SC) synapases. Stimulus artifacts were
suppressed for clarity. B, FD model simulations for the
three synapses. Vertical scale bar is 1, 400, and 200 pA for the CF,
PF, and SC synapses, respectively. Model parameters for CF were
F1 = 0.57, D = 50 msec,
kmax = 30 sec 1,
ko = 2 sec 1,
KD = 3.6. Model parameters for PF were
= 2.7, F1 = 0.05, F = 100 msec, D = 50 msec,
kmax = 30 sec 1,
ko = 2 sec 1,
KD = 2. Model parameters for SC were
= 3.2, F1 = 0.1, F = 100 msec, D = 50 msec,
kmax = 18 sec 1,
ko = 2 sec 1,
KD = 1.8. Parallel fiber data adapted from
Kreitzer and Regehr (2000) .
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The analytical FD model used in Figure 5 accounted for features of this
EPSC variability as shown in Figure 7B. The significant peak-to-peak variation in EPSC magnitude arose from the interplay between facilitation and depression. As facilitation increased the
fraction of activated sites, there were fewer available sites for the
next stimulus. During periods of low-frequency activation, CDR boosted
synaptic strength by increasing the number of available sites. Thus,
the variation in synaptic strength directly reflects the temporal
intervals of preceding stimuli according to the FD model. We noted some
systematic deviations between the FD model and the data at all three
synapses (for example, see the second to last EPSC in the burst of
stimuli in Fig. 7). A few possible explanations for the limitations of
the FD model are discussed below.
Simulated postsynaptic responses to the FD model
How does short-term synaptic plasticity contribute to a neuron's
ability to influence the firing of its targets? We examined this
question using the FD model for presynaptic transmitter release and a
single-compartment integrate-and-fire model for spike initiation in the
postsynaptic neuron (Fig. 8). The initial
EPSP amplitude was adjusted to a range in which individual synapses
could fire the postsynaptic cell. This is a greatly simplified
situation in that single synaptic inputs do not usually have such a
large impact on the postsynaptic neuron, although this situation is akin to "all-or-none" synapses such as brainstem auditory synapses (von Gersdorff et al., 1997 ), cerebellar climbing fiber synapses (Eccles et al., 1966 ), and the neuromuscular junction (Betz, 1970 ). Furthermore, the model for spike initiation did not take into account
many features of action potential threshold found in realistic neurons
(Llinas, 1988 ; McCormick, 1990 ). Despite its obvious limitations, this
approach allowed us to explore the manner in which facilitation and CDR
might contribute to neuronal firing.

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Figure 8.
Potential postsynaptic effects of facilitation and
CDR. A1, A2,
Postsynaptic responses of a model integrate-and-fire neuron using a
single nonfacilitating presynaptic input stimulated at 3 Hz with a 100 Hz burst. CDR is absent in A1 ( Fac, CDR) and
present in A2 ( Fac, +CDR). Synapse A had an
initial peak conductance of 15 nS (suprathreshold).
A3, Postsynaptic potentials for synapse A
with (thin lines) and without CDR (thick lines).
B1, B2, Postsynaptic
responses given the same presynaptic stimulus but facilitation was
included. Synapse B had an initial conductance of 6 nS (subthreshold).
CDR was absent in B1 (+Fac, CDR) and present
in B2 (+Fac, +CDR).
B3, Postsynaptic potentials for synapse B
with (thin lines) and without CDR (thick lines).
Model parameters for synapse A were F1 = 0.24, D = 50 msec,
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