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The Journal of Neuroscience, January 15, 2001, 21(2):444-461
Combining Deconvolution and Noise Analysis for the Estimation of
Transmitter Release Rates at the Calyx of Held
Erwin
Neher and
Takeshi
Sakaba
Max-Planck-Institute for Biophysical Chemistry, Department of
Membrane Biophysics, D-37077, Göttingen, Germany
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ABSTRACT |
The deconvolution method has been used in the past to estimate
release rates of synaptic vesicles, but it cannot be applied to
synapses where nonlinear interactions of quanta occur. We have extended
this method to take into account a nonlinear current component
resulting from the delayed clearance of glutamate from the synaptic
cleft. We applied it to the calyx of Held and verified the important
assumption of constant miniature EPSC (mEPSC) size by combining
deconvolution with a variant of nonstationary fluctuation analysis. We
found that amplitudes of mEPSCs decreased strongly after
extended synaptic activity. Cyclothiazide (CTZ), an inhibitor of
glutamate receptor desensitization, eliminated this reduction, suggesting that postsynaptic receptor desensitization occurs during strong synaptic activity at the calyx of Held. Constant mEPSC sizes
could be obtained in the presence of CTZ and kynurenic acid (Kyn), a
low-affinity blocker of AMPA-receptor channels. CTZ and Kyn prevented
postsynaptic receptor desensitization and saturation and also minimized
voltage-clamp errors. Therefore, we conclude that in the presence of
these drugs, release rates at the calyx of Held can be reliably
estimated over a wide range of conditions. Moreover, the method
presented should provide a convenient way to study the kinetics of
transmitter release at other synapses.
Key words:
synaptic transmission; exocytosis; deconvolution; noise
analysis; calyx of Held; desensitization
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INTRODUCTION |
Quantitative analysis of the
kinetics of transmitter release is essential for elucidating the
mechanisms underlying synaptic vesicle fusion. Several methods have
been developed in the past to estimate the kinetics of transmitter
release from postsynaptic currents (PSCs). One useful method is to
directly count the number of quanta (Katz and Miledi,
1965 ; Barrett and Stevens, 1972a, 1972b; Issacson
and Walmsley, 1995; Stevens and Wang, 1995;
Borst and Sakmann, 1996). Unfortunately, this method can
be used only when release rates are low. Another method involves
deconvolution of the PSC with the miniature PSC (mPSC) (van der
Kloot, 1988a, 1988b; Aumann and Parnas, 1991;
Borges et al., 1995; Diamond and Jahr,
1995; Vorobieva et al., 1999). This method
assumes that the postsynaptic current consists of a linear summation of
quanta, an assumption that was experimentally verified at the
neuromuscular junction (Hartzell et al., 1975;
Magleby and Pallotta, 1981).
Recent studies, however, have shown that postsynaptic currents are not
only shaped by the kinetics of transmitter release but are also
influenced by desensitization (Trussell et al., 1993; Otis et al., 1996b) and saturation (Clements et
al., 1992; Jonas et al., 1993; Tang et
al., 1994) of the postsynaptic receptors. Additionally, delayed
clearance of transmitter from the synaptic cleft (Barbour et
al., 1994; Mennerick and Zorumski, 1995;
Otis et al., 1996a) and also asynchronous quantal
release (Borges et al., 1995) can cause a slow residual
current. These factors must be carefully taken into account when
estimating quantal release rates from postsynaptic currents.
For these reasons, the simple deconvolution method may not be an
adequate tool to analyze quantal release at the calyx of Held, a giant
glutamergic synapse in the rat auditory brainstem (Forsythe,
1994). Specifically, GluR-D AMPA receptor subunits with
flop splice variants are expressed in the postsynaptic principal neurons. This receptor type shows strong desensitization with fast time
constants (1 msec) (Geiger et al., 1995). Furthermore, EPSCs seem to be shaped at least partially by the delayed
clearance of glutamate, especially when desensitization of the
postsynaptic receptors is inhibited by cyclothiazide (CTZ)
(Schneggenburger et al., 1999; Wu and Borst,
1999).
Therefore, to apply the deconvolution method to the calyx of Held, we
first needed to estimate the size of the residual current component
attributable to the delayed clearance of glutamate. We did this by
developing a simple model of glutamate diffusion and then incorporated
this model into the deconvolution algorithm. We then needed to verify
that the mEPSC amplitudes were constant. For this purpose, we applied
nonstationary noise analysis to estimate the quantal size
(Haller et al., 1998; Silver et al.,
1998; Oleskevich et al., 2000). We found that
postsynaptic receptors were desensitized strongly during strong
synaptic activity. CTZ inhibited this desensitization. However, without
desensitization, postsynaptic AMPA receptors were more readily
saturated, and large currents caused serious voltage-clamp errors. The
additional application of kynurenic acid (Kyn), a low-affinity
competitive inhibitor of glutamate receptors, reduced the size of the
EPSC and prevented clamping problems, even after a strong bout of
exocytosis. Variance analysis confirmed that the size of the mEPSC was
constant in the presence of these two drugs. Thus, we conclude that the
deconvolution method works most reliably in the presence of CTZ and Kyn.
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MATERIALS AND METHODS |
Deconvolution. Deconvolution has been used in
a number of studies to calculate release rates,  (t), from
the measured postsynaptic current, I(t) (Cohen et
al., 1981; van der Kloot, 1988a; Aumann and Parnas, 1991). Thismethod assumes that I(t) is a
linear combination of elementary signals, h·F(t), given
by:
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(1)
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Here, the notation of Segal et al. (1985) has
been used, in which the elementary event (the mEPSC) is written as a
product of an amplitude, h, and a time function,
F(t). The latter is normalized to a peak amplitude of 1. As
discussed above, the basic assumption of the deconvolution method
(constant mEPSC amplitudes) may not be valid at the calyx of Held
synapse, where glutamate is likely to build up during prolonged
stimulation. In another calyx-type synapse, asynchronous current
generated by such "residual glutamate" has been documented
(Otis et al., 1996a). We call such current "residual
current" and expect it to be particularly prominent in the presence
of CTZ. As we show below, such current was observed experimentally and
most likely was not mediated by asynchronous transmitter release,
because there was very little associated noise. Also, the proportion of
the residual current increased as the rate of exocytosis increased.
These findings are consistent with reports from other experiments
(Trussell et al., 1993; Mennerick and Zorumski,
1995).
Therefore, we assumed that the residual current,
Ir(t), is mediated by residual
glutamate in the synaptic cleft and that it can be described by a power
function of the residual glutamate concentration
Cr(t), according to:
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(2)
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where  (the weighing factor) and n (the exponent
of the power law of glutamate channel activation) are two model
parameters. We formulate the residual glutamate concentration,
Cr(t), based on past release, as:
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(3)
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Here, c(t) is a simple diffusion type kernel
(Crank, 1975) with an exponent
nD, a mean diffusional distance
rD, and a diffusion coefficient D
given by:
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(4)
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This form of the kernel can be rationalized, if we assume the
glutamate contribution of a given release event to consist of two
parts: first, the contribution to the local active zone, which results
in the mEPSC elicited by the event, and second, the effect at all other
active zones, which results in Ir(t). To calculate the mean contribution of a given event to
Cr(t) at some time t after
the event, we have to integrate the concentration profile created by
this event over the whole synapse except for the local active zone.
(More precisely, we would have to sum up the contributions from all
active zones except for the local one.) If we assume (for simplicity)
that the diffusion takes place in a planar infinite synaptic cleft with
impermeable walls and integrate the solution for an instantaneous point
source within such a structure over the whole cleft except for a
central disc with radius rD, we obtain Equation 4 with nD = 0. If we do the same
calculation for diffusion into an open semi-infinite space (i.e.,
assuming that the presynaptic compartment does not present a
significant barrier for diffusion orthogonal to the postsynaptic
surface), the solution is Equation 4 with
nD = 0.5. The decay of the residual glutamate may be even faster, if active glutamate uptake mechanisms contribute significantly. This would result in apparent
nD >0.5. Because we do not know the exact
geometry and the contribution of glutamate uptake mechanisms, we
consider nD, as well as
rD, D (Eq. 4) and the parameters and n of Equation 2 as free parameters, which must be
determined empirically from fits to the data (see below). Unless
n is equal to 1, the total current will be a nonlinear function of the release process. This will prohibit ordinary
deconvolution. We therefore split the total current
I(t) into a sum of residual current and current
induced by direct release. We assume that Equation 1 adequately
describes the current induced by direct release and thus calculate the
total current to be:
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(5)
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A complication in this approach may be seen in the fact that
both the convolution integral in Equation 5 and
Ir(t) (which itself contains a
convolution integral) depend on (t), which is the
function to be found by deconvolution. However, the diffusional kernel
in Equation 3 is a function rising from zero with some delay. At any
new time point, t, the integral (Eq. 3) can be readily calculated from values of (t') where t' < t.
This observation suggests a simple way to deconvolve the following
function point-by-point:
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(6)
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Cohen et al. (1981) showed previously that a
signal y(t), composed of simple exponentially decaying
mEPSCs, can be deconvolved by calculating:
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(7)
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where  is the decay time constant of the mEPSC and the
subscript m denotes the assumption of a monoexponential
decay. This equation is valid for instantaneously rising mEPSCs. It can
be derived and generalized to the case of double exponentially decaying mEPSCs as shown in Appendix A, where the numerical algorithm of
deconvolution used in this work is explained in detail.
To estimate the release rate, (t), parameters of the
glutamate diffusion model, h, and F(t) must be
known. F(t) was obtained by sampling mEPSCs in the presence
of 100 µM CTZ. In four cells, the mean amplitude of the
mEPSC was 29.8 ± 2.03 pA (mean ± SEM). The decay phase of
the mEPSC could be fit by two exponentials with time constants of
2.38 ± 0.59 msec (relative amplitude 48.7 ± 4.8%) and
11.43 ± 1.99 msec. The time course of mEPSCs, especially during
the slow decay phase, was highly variable among mEPSCs (Wu and
Borst, 1999; our unpublished observations). For
deconvolution, we assumed that mEPSCs decayed either biexponentially
with time constants similar to those observed or monoexponentially with a time constant of 2-3 msec. In the latter case, the slow decay of the
mEPSC was considered to be part of the residual current. Then, the
parameter rD in Equation 4 had to be selected
smaller, and the contribution of the residual current to the total
current was larger than in the former case. Very similar estimates of the release rates were obtained for both the experimental data (as
shown in Results) and simulations (E. Neher, unpublished observations) regardless of the assumption applied. In the absence of CTZ, we assumed
that the mEPSC had a mean amplitude of 30 pA and decayed monoexponentially with a time constant of ~1 msec. These settings were similar to experimental findings of Borst and Sakmann
(1996) and Schneggenburger et al. (1999). In the
presence of CTZ and Kyn (1 mM), we were not able to sample
mEPSCs, because they were too small to be detected reliably. Therefore,
we assumed that mEPSC amplitudes in the presence of Kyn decreased by
the same factor as EPSC amplitudes. We determined this factor to be
0.50 ± 0.01 (n = 4 cell pairs) but found that the
macroscopic EPSCs decayed slightly faster in the presence of Kyn.
Parameters of the glutamate diffusion model were determined as follows
(see Figs. 6, 9, 12, 15). We depolarized the presynaptic terminal for 4 msec to +80 mV, where there was almost no Ca2+
influx. Then, the terminal was repolarized to 0 mV for several short
periods (1-5 msec) to elicit several episodes of transient Ca2+ influx into the presynaptic terminal. By
changing the duration of the pulses, EPSCs of various amplitudes and
durations could be evoked. We will refer to this protocol as "fitting
protocol" and the corresponding EPSC as "fitting EPSC." EPSCs,
evoked by current inflow, decayed rapidly each time the terminal
voltage was returned to +80 mV. Variance analysis (see below) indicated that there was very little release during pauses between the influx episodes such that current between depolarizing episodes can be considered as a sum of the decaying mEPSCs (from the immediately preceding release) and the residual current. Once mEPSCs have decayed,
the remaining measured postsynaptic current is residual current and can
be used for the fitting procedure.
We designed the fitting protocol so that different sections of the
resulting postsynaptic current are particularly sensitive to one or the
other parameters of the model. Thus, the first episode of
Ca2+ current inflow was chosen to be very short,
such that the resulting EPSC resembled a scaled version of a mEPSC.
This allowed a confirmation or fine adjustment of the mEPSC decay time
constant(s) by the criterion that the deconvolution rate should return
to zero immediately after the stimulus. The residual current was small
after the first pulse, because of its nonlinear nature. The next
stimulation episode was chosen to be larger, such that it elicited
larger currents and also a sizeable residual current (at least in the
presence of CTZ; see Fig. 9b). A good fit to the EPSC decay
after this episode depended critically on the setting of
rD (Eq. 4), which determines the speed of the
rise of Ir(t). The late decay that follows subsequent strong stimuli was used to adjust the remaining parameters. In practice, we performed deconvolutions of a given fitting
EPSC repetitively starting with a complete trial set of parameters
(including those of the mEPSC). We judged the quality of fits from a
display of Ir and from the outcome of the
resulting deconvolution rate (t) and applied corrections
to parameters by trial and error to optimize the fit. The criterion for
optimization was that (t) should be close to zero between
stimulation episodes. Whenever the variance between release episodes
was larger than that of expected AMPA-channel noise, the fitting target
for (t) was adjusted to match the release rate expected
for the extra noise (see below). This procedure will be detailed by the
examples presented in Results.
Once parameters had been determined for a given synapse, the same
parameters were used for other traces within the same cell pair. When
using monoexponential mEPSCs, typical parameters of the glutamate
diffusion model were n (the exponent of the power law of
glutamate channel activation) = 1.2, rD
(the diffusion distance) = 0.8 µm, nD
(the exponent of the diffusion law) = 0.8-0.9, and D
(the diffusion coefficient of the transmitter) = 30 µm2/sec.
Of course, this model is very simplified. For example, the power law of
channel activation might be influenced not only by the kinetics of
glutamate receptors but also by the densities and locations of
receptors at postsynaptic membranes. The synapse, in reality, has
complicated geometrical arrangements of release sites and postsynaptic
receptors, and the resulting structures may differ among active zones
(Walmsley et al., 1998). However, as described below,
this simple model fits quite well a wide range of EPSCs. The fitting
procedure might be readily extended to cover an even wider range of
EPSCs, by modeling a full dose-response curve of AMPA receptors.
However, we chose not to increase the number of parameters at the
present time and chose to explore the range of conditions under
which the above formulation is valid.
Noise analysis. To verify the new form of deconvolution (Eq. 6) and to test mEPSCs amplitudes under the experimental conditions examined, noise analysis was introduced. Because it is very
difficult to achieve a stationary state, especially during the EPSC,
simple stationary noise analysis (Katz and Miledi, 1972)
could not be applied. Thus, we had to extend variance analysis to the
nonstationary case. Ensemble noise analysis is a useful method for
analyzing nonstationary data (Sigworth, 1980) and has
already been applied to the analysis of the transmitter release process
(Silver et al., 1998; Oleskevich et al.,
2000). This method, however, requires a large number of traces
with very similar statistical properties. Unfortunately, obtaining such
data is difficult in the case of paired recordings at the calyx of
Held, because series resistances of both the presynaptic and
postsynaptic recordings usually deteriorate during experiments. For
this reason, we modified nonstationary variance analysis such that it
requires a reduced number of traces. This method can also be viewed as
an extension of the analysis used by Haller et al.
(1998).
Assuming that the observed EPSC, I(t), is a stationary
process that consists of a linear summation of uniformly sized mEPSCs, F(t), with an amplitude of h and rate
(t), we obtain the mean EPSC, I, from
Campbell's theorem (Rice, 1944; Segal et al.,
1985):
|
(8)
|
where  (t) is the mean rate. Its variance,
var, is given by:
|
(9)
|
if the duration of the mEPSC is short relative to the analysis
interval. Then, the value obtained by dividing the variance by the mean
current should be proportional to the square of the averaged amplitude
of the mEPSC (Katz and Miledi, 1972; Haller et
al., 1998).
In the nonstationary case, the expectation value of the power spectral
density S(f) can be expressed as (Rice,
1944):
|
(10)
|
where n(t) is the number of release-ready vesicles,
 (f) is the Fourier transform of the mEPSC,
 (f) is the Fourier transform of the
probability function p(t), which together with
n(t) is defined so that n(t)p(t) is the mean
number of occurrences of the mEPSC between the time interval of
t and t +  t. This equation shows that
S(f) is affected by both (f)
and (f). However, this Equation also shows
that S(f) can be dominated by
(f) within a certain frequency range in which
(f) makes only a small contribution. This
situation can be achieved by controlling the presynaptic holding
potential and adjusting it such that release rate changes slowly
compared with the time course of the mEPSC. Then,
S(f) is proportional to the mean rate of occurrence
of events. By choosing an appropriate spectral window, information on
the relative size of the mEPSC can be obtained by taking the ratio of
S(f) (or variance within that spectral window) over
the release rate, in analogy to Equations 9 and 10. The high-pass
filtering applied in this analysis eliminates trends and other
nonstationarities, and it has the additional advantage of shortening
the underlying elementary event, satisfying a prerequisite of
Campbell's theorem. Furthermore, narrowing the width of the elementary
event compared with the averaging interval improves the signal-to-noise
ratio of the variance estimate (see below). Instrumental and
AMPA-receptor channel noise dominate at higher frequencies compared
with the contribution from quantal release. The signal was therefore
low-pass-filtered to reduce the noise not directly associated with
quantal release.
In this study, differentiation was used to high-pass filter the signals
because this method can be easily implemented in the Igor program
(Wavemetrics, Lake Oswego, OR), and it yields coefficients of variation
(CV) of the variance estimates that compare favorably with other filter
methods tested (our unpublished observations). The CV of variance is
also influenced by the amplitude distribution of the mEPSCs
(Katz and Miledi, 1972) and, in principle, can be improved by averaging over data stretches as long as possible. Unfortunately, the amplitude distribution of mEPSCs is relatively broad
at the calyx of Held (Schneggenburger et al.,
1999). The relationship between the sampling length and
the CV of variance was examined by simulations and is shown in Results.
For low-pass filtering, the "box-smooth function" of Igor with a
window of 0.3 msec was used. Finally, variance was calculated and
smoothed by using a gliding window of 3 msec length. In experiments, we repeated a given protocol 5-10 times and averaged the variance records. Alternatively, consecutive traces were subtracted from one
another before filtering. This subtraction was done to eliminate long-term trends (Sigworth, 1980,
1981), if necessary.
Variance values were divided by 2 in such cases to account for the
subtraction procedure.
Current fluctuations are caused not only by quantal release but also by
AMPA-receptor channel noise. To estimate this noise, the postsynaptic
cell was voltage-clamped at  80 mV. In the presence of CTZ (100 µM), S-AMPA (100 µM) was applied locally
from a glass pipette (Fig. 1). The
AMPA-evoked current was bandpass-filtered using the same filtering
protocol (high-pass filtering by differentiation followed by low-pass
filtering) as used for noise analysis. Then, variance was calculated
and smoothed by a gliding window. Variance was plotted against the mean
of the unfiltered current (Fig. 1B). The slope of a linear
fit was estimated by linear regression and should be a measure of the
amplitude of the filtered waveform of single-channel currents. We will
call this an effective single-channel current amplitude i'.
To correct for the variance arising from channel noise,
Vc, we assumed that channel noise and
mEPSC-induced noise are statistically independent. Therefore, the
contribution of Vc was calculated by forming the
product of the EPSC and the apparent single-channel current and simply
subtracting this value from the total variance. Of course, this
calculation is simplified, especially considering that the kinetics of
AMPA receptors is most likely much more complex (Jonas et
al., 1993; Rosenmund et al., 1998). However,
this effective channel amplitude was found to adequately explain the
variance associated with the late decay phase of EPSCs, where there is
very little quantal release. It may be argued that channel variance
contributed by mEPSCs is already represented in the amplitude
distribution of mEPSCs and therefore should not be subtracted. If this
were the case, an appropriate correction for Vc
would be to subtract the product of i' and
Ir(t). However, simulations
demonstrated (our unpublished observations) that the difference
between the two options is small and that the correction applied here
yields results that are indistinguishable (at the given level of
resolution) from the correct one.
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Figure 1.
Estimation of the AMPA-receptor channel noise.
A, The postsynaptic principal neuron was whole-cell-clamped
at 80 mV. Extracellular solution contained CTZ (100 µM), and S-AMPA (100 µM) was applied to the
cell locally by a puffer pipette, and the induced current was recorded
(top). After bandpass filtering, we calculated the variance
and smoothed it by a gliding window (bottom). B,
Variance was plotted against the amplitude of the (unfiltered)
AMPA-induced current. From this relationship, an apparent
single-channel AMPA current was estimated by linear regression. See
Figure 2 for the definition of variance units.
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To estimate the mEPSC size, the variance after correcting for
Vc was divided by the release rate that had been
estimated by deconvolution. Because the variance is proportional to the
release rate and to the square of the mEPSC size (Eq. 9), this ratio
should be proportional to the square of the mEPSC size. However, it is seen from Equation 7 that the release rate is itself a function of
the mEPSC amplitude. Thus we have to combine Equations 9 and 7, to see
that:
|
(11)
|
where "(t) is the product of h and
(t), which is the immediate output of the deconvolution
routine (see Appendix A and Eq. 7). We will refer to this ratio as the
mEPSC size estimated from variance and mean rate. However, because the
data have been filtered, this estimate does not represent the actual
amplitude of the mEPSCs (h in Eqs. 8 and 9) but rather a
filtered version, the amplitude of which we designate h'. We
established the relationship between h and h' by
performing simulations under conditions identical to those used
experimentally. For simplicity, the simulated value of h'
was defined to be equal to 1 relative amplitude unit (rel.u.). We
therefore divided the experimentally obtained values of h' by the value obtained from simulations to obtain the values of the
experimental h' in terms of rel.u.
When estimating relative errors in this analysis, one has to appreciate
that h' is a ratio and that relative errors in both the
numerator (variance) and denominator (rate) may contribute additively
to the total relative errors. Systematic errors in variance may
originate from erroneous estimation of channel variance, such as from
an incorrect estimate of i'. Another important source of
error relates to series resistance. Any series resistance error will be
very serious because it contributes to the variance estimate according
to its square and will strongly attenuate high-frequency components of
the signal. We found that Rs > 8 M
cannot be tolerated, even if series resistance compensation is applied.
Systematic errors in rate (as estimated from deconvolution) are most
likely to originate from y/ in Equation 7. Two cases
should then be distinguished. For sections of records in which
|dy/dt| > |y/|, the relative error is given by
(y/)/(dy/dt), where y is the systematic
error in the current (as it appears in Eq. 6) that might result from
inaccurate estimation of Ir(t). This
error should be relatively small [given |dy/dt| > |y/| and that Ir(t)
changes with some delay and more slowly than y(t)].
However, for more or less stationary records (|dy/dt|  |y/|), the relative error is given by y/y,
which means that the estimate of cannot be more accurate in
relative terms than the error in y, which again is sensitive
to inaccuracies of Ir(t). We will
discuss examples, in which such an error is critical, in Results.
Physiological recordings. Presynaptic and postsynaptic
recordings at the calyx of Held were performed in the slice preparation of the rat brainstem following the procedures described by Borst et al. (1995) and Schneggenburger et al. (1999).
Briefly, 8- to 10-d-old Wistar rats were decapitated without anesthesia
according to local guidelines. The brainstem was immersed in ice-cold,
low-calcium saline that contained (in mM): NaCl 125, KCl
2.5, CaCl2 0.1, MgCl2 3, glucose 25, NaHCO3 1.25, ascorbic acid 0.4, myo-inositol 3, and
Na-pyruvate 2, pH 7.3-7.4, 320 mOsm, and was bubbled with 95%
O2 and 5% CO2. Transverse slices of the
brainstem (150-200 µm thick) were cut using a vibratome. Slices were
incubated in the chamber for at least 30 min at 36°C in normal
extracellular solution while being continuously bubbled with 95%
O2 and 5% CO2. Normal extracellular solution
was the same as the low-Ca2+ saline except that 2.0 mM CaCl2 and 1.0 mM
MgCl2 were used. Experiments were performed within 5 h
after preparation of the slices.
All recordings were done at room temperature (~21-24°C). The slice
was transferred to the recording chamber and perfused continuously with
the normal saline at a rate of ~1 ml/min. Slices were visualized by
IR-DIC microscopy through a 40× water immersion objective on an
upright microscope (Axoscope, Zeiss, Oberkochen, Germany). A single,
large, calyx-type terminal and a postsynaptic principal neuron of the
medial nucleus of the trapezoid body were identified visually.
A presynaptic terminal and a postsynaptic target were simultaneously
clamped with patch pipettes. Both holding potentials were 80 mV. No
correction for liquid junction potentials was applied. EPSCs reversed
close to +10 mV (nominally), and the driving force for the postsynaptic
current was 90 mV. The presynaptic pipette (4-7 M) was filled with
a solution containing (in mM): Cs-gluconate 125-130, TEACl
20, HEPES 10, Na2 phosphocreatine 5, MgATP 4, GTP 0.3, EGTA
0.5, pH 7.2 with CsOH, 310 mOsm. The postsynaptic pipette (2-4 M)
was filled with the same solution as the presynaptic pipette except
that the concentration of EGTA was increased to 5 mM.
During recordings, 0.5-1 µM TTX, 10 mM
TEACl, and 50 µM D-AP5 were added to the normal
extracellular solution to isolate the presynaptic calcium current and
block NMDA receptors. AMPA receptor-mediated EPSCs were used to monitor
quantal release. Bicuculline (10 µM) and strychnine (2 µM) were also added in some recordings to block the
possible inhibitory input. In some experiments, 1 mM Kyn
was also added. TTX was purchased from Alamone Labs (Jerusalem,
Israel). D-AP5, NBQX, CTZ, and Kyn were from Tocris (Köln,
Germany). Other drugs were from Sigma (Deisenhofen, Germany).
Both presynaptic and postsynaptic cells were whole-cell-clamped with an
EPC9/2 amplifier controlled by the Pulse program (HEKA-Electronik, Lambrecht, Germany). Thirty to seventy percent of the presynaptic series resistance (Rs; 8-35 M, typically 15 M) was compensated. Presynaptic leak currents of >200 pA typically
caused run-down of the postsynaptic response, and cells showing such
leak were excluded from analysis. The postsynaptic series resistance
(3-10 M, typically 5 M) was compensated so that the
uncompensated series resistance was ~2-3 M. The remaining
deviation from holding potential (= Rs × IEPSC) was calculated off-line, and EPSC
amplitudes were corrected by multiplying by V/(V Rs·IEPSC). Postsynaptic leak
currents were typically between 50 and 100 pA. Currents were low-pass-filtered at 2.9 or 6 kHz and stored at 10 or 20 kHz. Recordings were discontinued once the uncompensated postsynaptic Rs became >10 M or the postsynaptic leak
current became >300 pA. Most of the data that had uncompensated
postsynaptic series resistances of >8 M were rejected from analysis
because such traces had reduced noise power density at higher frequencies.
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RESULTS |
Calibration of the mEPSC size estimate of noise
analysis (simulation)
Our method assumes that the variance of the filtered records is
proportional to the release rate and insignificantly affected by the
residual current (apart from channel variance
Vc). As described in Materials and Methods,
h', our estimate for mEPSC amplitude, should be independent
of the release rate, provided that adequate filters are used to remove
nonstationarities. However, h' does not directly correspond
to h. Therefore, we established the relationship between
h' and h using a set of simulations described in
Figure 2. Specifically, the release rate was changed from 0 to 5 msec 1, then to 10, 15, and finally to 20 msec1. EPSCs were simulated by convolving this
release rate with the mEPSC using the Monte-Carlo method. The
mEPSC used for the simulation had a rise time of 200 µsec and decayed
monoexponentially with a time constant of 3 msec. The amplitude
distribution of mEPSCs (with a mean amplitude of 30 pA) was obtained
from postsynaptic recordings (see Materials and Methods). During
simulations, mEPSC amplitudes were randomly selected according to the
experimental amplitude distribution. After the EPSC was generated, the
residual current caused by the delayed clearance of glutamate from the synaptic cleft was added. This residual current was generated using the
glutamate diffusion model according to Equations 2-5. To mimic
experimental data, we used parameters of the glutamate diffusion model,
which were similar to those used for fitting the experimental data. The
proportion of the residual current relative to the total EPSC was
adjusted so that it was similar to that observed in experiments.
Twenty-five traces were simulated and filtered, and the variance was
calculated for each record and then averaged (Fig.
2A). The relationship
between the variance and release rates was plotted (Fig. 2B)
and could be fit with a straight line passing through the origin. A
linear relationship is expected according to Equation 9. We scaled the
variance, such that the slope of the fit, or else the variance at a
rate of one event per millisecond, was equal to 1, and we designate
this value as the arbitrary unit (a.u.) of variance. All plots of
variance in this study are given in these units. It should be noted
that this simulation did not include channel variance. Therefore, no correction regarding channel variance was performed in the simulations, contrary to the case of actual recordings.
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Figure 2.
Relationship between variance and release rate
derived from simulations. A, The release rate was changed
from 0 to 5, 10, 15, and finally 20 msec1. For
simulations, mEPSCs amplitudes were randomly chosen from an amplitude
distribution of mEPSCs, obtained experimentally, and superimposed. The
residual current caused by the delayed clearance of glutamate was added
to the simulated EPSCs. The resulting current is shown as a solid
line. Parameters of the diffusion model used to calculate the
residual current were n = 1.2, rd = 0.8 µm, nD = 0.8, D = 30
µm2/sec. Variance (dotted line) was
smoothed by a gliding window (3 msec) in each simulated trace and was
then averaged over 25 traces. The large spike of variance observed at
the end of the record was caused by a rapid drop in the release rate
from 20 to 0 msec1. B, The relationship
between variance and the release rate was plotted (error bars indicate
SD obtained from 25 repetitions of the protocol). Variance was scaled
such that the regression line in this plot has a slope of 1 a.u.
per one event per millisecond. All variance values of this study were
normalized with this same scaling factor (see Results).
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Robustness of nonstationary noise analysis under
various conditions
Next we examined whether the noise analysis performed after
optimal filtering of the signal is robust to various kinds of perturbations that potentially introduce nonstationarities. In these
simulations, the release rate was stepped from 0 to 5 msec1 then 15 and 5, and finally back to 0 msec1 (Fig.
3A). The parameters of the
glutamate diffusion model were set as described in Figure 2. Twenty
traces were simulated, and the variance was calculated for each record
over segments of 3 msec length and then averaged. Even when the release
rate was constant, the simulated EPSC traces changed gradually, because a residual current was present as a result of previous exocytosis. EPSCs changed more abruptly during sudden changes of the release rate
(Fig. 3A). Mean variance of filtered records, however,
stayed constant during episodes of constant release; its fluctuations were ~10% of the mean value. Variance fluctuations remained fairly low even when the release rate changed abruptly by 5 msec1 (Fig. 3A). When the release rate
was increased from 5 to 15 msec1 (or the reverse),
a spike in the variance fluctuation occurred. However, the amplitude of
the spike was only 20-30% of the mean value. More importantly, when
the release rate was 5 msec1, variance was
5.2 a.u. (corresponding to h' = 1.04 rel.u.). When release rate was 20 msec1, variance was
19.4 a.u. (corresponding to h' = 0.97 rel.u.). Thus, h' was relatively independent of quantal release
rates, which suggests that bandpass filtering and subsequent averaging effectively eliminated nonstationarities.
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Figure 3.
Robustness of variance to nonstationary
conditions. A, EPSCs (solid line) were simulated
by convolving the release rate (top) with mEPSCs. The
release rate was changed from 0 to 5, 15, 5, and finally 0 msec1. The parameters of simulation were the same
as those described in Figure 2. B, The same protocol as in
A, except that the proportion of the residual current to the
total current has been doubled. Note that the scale of the left
axis (EPSC) is different from that in A. C, The release
rate was changed from the 0 to 5, 30, 5, and finally returned to 0 msec1. Other settings were the same as described
in A.
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We also changed various parameters of the glutamate diffusion model to
examine the robustness of noise analysis. In the case of Figure
3B, the ratio of the residual current to the total EPSC (ß of Eq. 2) was double the value used in Figure 3A. Other
parameters remained unchanged. By increasing ß, the total size of
EPSCs was twice that observed in Figure 3A. Also, the slope
of the EPSC became steeper. Although the signal became more
nonstationary, variance did not fluctuate by >10% of the mean, and
the estimate was not distorted by the trend as long as the release rate
was constant. The amplitude of the spike observed during an abrupt change in the release rate was similar to that seen in Figure 3A. When the release rate was 5 msec1,
we calculated an h' value of 1.02 rel.u. When the release
rate was 20 msec1, h' was 0.97 rel.u.
We also changed other parameters of the glutamate diffusion model (for
example, rD) as well as the decay time constant of the mEPSCs. However, estimates of h' were not affected by
these manipulations (data not shown). Again, these results suggest that bandpass filtering and subsequent subtraction of consecutive traces effectively eliminated nonstationarities.
Despite the effectiveness of high-pass filtering in handling
nonstationarities, variance estimates may be distorted if changes of
the release rate are too rapid or too large. For example, changes of
the release rate from 5 to 30 msec1 and the
reverse (step sizes that were twice as large as in the case of Fig.
3A) resulted in large spikes in the variance (Fig. 3C). Nevertheless, variance was still stable except for the
short segments exactly before and after these rapid changes in the
release rate. Therefore, we analyzed only the regions outside the
transition zones, where h' was 0.96 rel.u. when the release
rate was 5 msec1 and 0.92 rel.u. when the release
rate was 30 msec1. This simulation suggests that
abrupt changes of release rate should be avoided to obtain reliable
estimates of the mEPSC size from noise analysis. In our experiments, we
achieved this by adjusting the Ca2+ influx so that
release rates and EPSCs did not change rapidly within certain regions
of interest (see below).
Accuracy of variance analysis
Because the coefficient of variation (CV) of the variance
estimate is quite large because of the random nature of the release process and the non-uniformity of the mEPSC amplitudes (Katz and Miledei, 1972), it is important to obtain multiple records to reduce the CV. At the calyx of Held, mEPSC amplitudes are variable (see
Materials and Methods); thus, sampling over sufficient lengths and averaging the signal are essential to reduce the CV. The effect of
averaging is demonstrated in Figure 4. In
all cases, the release rate was changed from 0 to 5, 20, and 5 msec1, and finally back to 0 msec1. EPSCs were simulated by convolution using
these release rates. Variance was smoothed by a gliding window and
averaged over an increasing number of traces (n = 1, 3, 5, 10) (Fig. 4). Averaging reduced variance fluctuations that were
observed while the release rate was kept constant. Therefore, averaging
records reduces the CV of variance substantially.
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Figure 4.
Effect of averaging on variance fluctuations. The
release rate was changed from 0 to 5, 20, 5, and finally 0 msec1. EPSCs (solid line) were
simulated using the same parameters as described in Figure 2. The EPSC
and variance (dotted line) were averaged over 1, 3, 5, or 10 traces. As the number of averages increased, the CV of variance
decreased.
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To establish the relationship between the CV of variance and the total
recording time, we performed a "thought experiment" in which we
subdivided the observation interval, T, into n
intervals of length T/n. During these subintervals, we
assumed that the transmitter release process was stationary and that
the subintervals would be long enough to ensure that the requirements
for Campbell's theorem were satisfied (that is, T/n must be
longer than the duration of the individual event). We considered the
n estimates of variance from these subintervals as a sample
with a sample mean of s, which for large n
has a SD of s [in the case
that the frequency of elementary events is high enough such that the
sample mean is normally distributed (Spiegel, 1975)]. Then, the coefficient of variation is , and therefore we expected that the relative accuracy of our estimate does not depend on the amplitude of the signal. This finding implies that the relative accuracy of our estimate does not depend on the
frequency of the mEPSC or the mEPSC amplitude. Instead, the relative
accuracy is determined by the number of independent samples that can be
taken. This in turn depends on the duration of the filtered mEPSCs,
which becomes smaller as more high-pass filtering is applied. A
plot of 1/CV2 against the number of averages,
N, should be linear. From such a plot, we obtained the
number of averages necessary for a desired CV. Simulations were used to
clarify this relationship. Specifically, we considered two different
release rates (5 and 20 msec1). For both cases,
the variance was calculated using the procedures described in Figure
2A while changing the number of averages (N) from
1 to 64. Then, s2/ variance (s)
was plotted against N (Fig.
5). The relationship was fitted by linear
regression, and the slope was 2.4 (Fig. 5) regardless of the release
rate (5 or 20 msec1). Thus, to obtain an accuracy
of 10%, which means that s2/variance
(s) equals 100, it is necessary to take 100/2.4 or 42 averages. Given that in the above analysis one measurement consists of
an average over 3 msec (the length of the smoothing window), a similar
accuracy can be obtained by averaging one record over 3 × 42 or
126 msec. In our experiment, we used protocols that were 40-50
msec long and repeated these protocols 5-10 times (for a total of
200-500 msec). Therefore, the expected accuracy is somewhat better
than 10%.
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Figure 5.
Relationship between variance fluctuations
and the number of averages. Variance was calculated for each simulated
EPSC and was averaged over various numbers (1, 2, 4, 8, 16, 32, or 64)
of EPSCs. In simulations, the release rate was either 5 msec1 ( ) or 20 msec1 (+).
Variance fluctuations were calculated from
s2/var(s), where s
denotes sample variance. This value was then plotted against the number
of averages. The slope of this relationship was 2.4 regardless of the
release rate.
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Deconvolution of postsynaptic currents and determination of the
contribution of residual current
The presynaptic terminal and the postsynaptic target were
simultaneously voltage-clamped to a holding potential of 80 mV, as
described in Materials and Methods. The extracellular solution contained TTX (0.5 µM) and TEA (10 mM) to
block presynaptic Na+ and K+
currents. D-AP5 (50 µM) was also added to block NMDA
receptors and to isolate AMPA receptor-mediated EPSCs. To record traces suitable for estimation of the model parameters, the fitting protocol that was described in Materials and Methods was applied. This protocol
started with a short depolarization of +80 mV. The
Ca2+ influx was minimal as long as the terminal was
held at +80 mV, and hence no EPSC was evoked by this depolarization
(Fig. 6a). The terminal was
then repolarized to 0 mV several times for episodes of increasing
duration (1-5 msec). This evoked Ca2+ influx and
the duration of these episodes were adjusted so that various sizes of
EPSCs were obtained (Fig. 6b), which allowed the fitting of
the diffusion model at several levels of residual current, as described
in Materials and Methods. The residual current in this recording was
very small, unlike the cases with CTZ, as shown below. The time course
of the fit to the residual current is indicated in Fig. 6b
(broken line). Each trace was concluded by a stimulation
episode of 10 msec to completely deplete the readily releasable pool
(RRP) of synaptic vesicles (Schneggenburger et al.,
1999; Wu and Borst, 1999). Subsequently, the
holding potential was returned to 80 mV. We apply the depleting pulse
routinely to be able to measure the size of the RRP of synaptic
vesicles, as described in the companion paper (Sakaba and Neher,
2001).
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Figure 6.
Protocol for setting the parameters of the
glutamate diffusion model (the "fitting protocol"). The presynaptic
terminal was depolarized from 80 mV to +80 mV (a). The
presynaptic holding potential was repetitively repolarized to 0 mV
several times for short periods (1-5 msec) to evoke EPSCs
(b) rapidly. At the end of the protocol, the terminal was
held at 0 mV for 10 msec to completely deplete the RRP of synaptic
vesicles. The inset shows the EPSC evoked by the first
stimulus. Parameters of the glutamate diffusion model were set such
that the decay phases of the EPSCs could be explained by the residual
current (b, dotted line). After subtracting the
residual current component from the total postsynaptic current, the
release rate (c) was calculated by deconvolving the
remaining current with the mEPSC. Variance (d, dotted
line) was calculated after bandpass filtering the postsynaptic
current and smoothed by a gliding window of 3 msec. It is superimposed
on the plot of the release rate (d, solid line). Axes
of both traces were adjusted such that traces are expected to
superimpose for h' = 1 rel.u.
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It is likely that the small current between stimulation episodes is
attributed mainly to residual current based on the following observations. This small current was not mediated by NMDA receptors, because the extracellular solution contained 1 mM
Mg2+ and 50 µM D-AP5. Furthermore,
7-chloro-kynurenic acid (30 µM), a noncompetitive blocker
of NMDA receptors, did not eliminate this current (data not shown).
Therefore, this current must arise either from residual glutamate or
from asynchronous vesicular release.
To distinguish between these two cases, noise analysis was used as
described in Materials and Methods. The current trace was bandpass-filtered, and variance was calculated. Between stimulation episodes, the EPSC approached a sufficiently stationary state (Fig.
6b), and bandpass filtering was able to eliminate
efficiently the influence of slow trends. Variance measured between
individual stimuli was very low. Values were typical for
glutamate-induced current fluctuations that represent opening and
closing of channels. The channel noise was estimated as explained above
(Fig. 1) and subtracted from total variance, and then estimate for the
release rate during episodes between stimulations was determined by
dividing the remaining variance by the a.u. as determined by
simulations (see Materials and Methods). In Figure 6d,
variance observed between the individual stimuli was <2 a.u. (with
channel variance contributing <10%). Because the unit in this
calculation was mEPSC size (see above), the release rate during this
period should be <2 events msec1. We
adjusted the model parameters such that the estimated release rate
stayed within these limits. It should be pointed out that the influence
of 0.5 mM EGTA in the presynaptic pipette filling solution
(as routinely used in this study) is very helpful in limiting
asynchronous release, even after relatively strong stimuli. Because the
release rate is so small, postsynaptic current attributable to
asynchronous release is on the order of magnitude of <50 pA. Therefore, most postsynaptic current between the stimuli (typically 100-300 pA) arises from the residual current, which is caused by
delayed clearance of glutamate. Consequently, the release rate, as
returned from the deconvolution routine, gave correct results (i.e.,
rates low enough to agree with variance measurement) only, if the
residual current was modeled correctly. In practice, we adjusted the
parameters of the diffusion model by trial and error to achieve this
situation. Figure 6d shows an example of a fit that we
consider appropriate. Here, the release rate (solid line, as
returned by deconvolution), is superimposed onto variance. Variance is
scaled such that the two curves should agree for h' = 1
rel.u. It is seen that our choice of parameters fulfills this postulate
for all sections, during which a valid variance estimate is available
(i.e., when variance is not compromised by rapid transients in
EPSC). On the other hand, the results below will show that
h' is most likely <1 rel.u. late in the record, such that
the correction for asynchronous release is not accurate in this
particular case. Figure 6c shows the same deconvolution
result at lower gain, demonstrating that peak release rates in this
record are orders of magnitude larger than the inaccuracies in
the comparison between deconvolution- and variance-based estimates. The
parameters obtained from this modeling were used for deconvolution of
other traces within the same cell pair. Typical model parameters were n = 0.7, rD = 0.8 µm, and
nD = 0.9.
Noise analysis used to examine the constancy of mEPSC size
Importantly, our deconvolution method assumes that the mEPSC
amplitude is constant. Therefore, after fixing the parameters of the
glutamate diffusion model, we once more invoked noise analysis to
verify that the amplitudes of the mEPSCs were indeed constant. For
noise analysis, it is best to avoid nonstationarities (see above), and
thus records with only small changes in the release rate are preferred.
However, using release rates that are too small is also problematic,
because subtle over- or underestimation of the residual current
component will introduce serious errors during calculations. We found
that release rates between 20 and 40 msec1 were an
optimal compromise. To obtain such release rates, the presynaptic
terminal was held at +30-+50 mV for 50 msec (the voltage protocol for
"early noise" analysis) (Fig.
7Aa). During this episode, Ca2+ influx was relatively small, and the evoked
EPSC increased slowly (Fig. 7A, section between 0.009 and
0.059 sec). The release rate usually rose to values between 20 and 40 msec1. Variance was calculated for the whole
record but should be considered reliable only during the episode
intended for noise analysis. We repeated the same protocol 5-10 times
to get a sampling length long enough to obtain sufficiently low CVs of
variance.
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Figure 7.
Noise analysis for testing the constancy of mEPSCs
amplitudes. A, The early noise protocol. The presynaptic
terminal was depolarized to +80 mV, held at +45 mV for 50 msec, and
then held at 0 mV for 10 msec (a). A single trace of the
EPSC (b, solid line) is shown with residual current
(b, dotted line) superimposed. The release rate
(c) was estimated from deconvolution. Plots of the release
rate (d, solid line) and variance (d,
dotted line) calculated from the postsynaptic current of a
single trace were superimposed and scaled (as in Fig. 6) to allow
comparison of the relative time courses. B, The late noise
protocol. The presynaptic terminal was depolarized to +80 mV, held at 0 mV for 2 msec to evoke a strong bout of exocytosis, and then held at
+35 mV for 50 msec. The holding potential (+35 mV) was lower than used
in the control protocol (+45 mV) to obtain comparable release rates.
The terminal was then held at +80 mV for 30 msec, during which time
release rate decreased, to verify the correctness of the glutamate
diffusion model. Both protocols were concluded by holding the terminal
at 0 mV for 10 msec to completely deplete the RRP of synaptic vesicles.
The insets show the EPSC at the beginning of the episode
used for noise analysis. Data were obtained from the cell pair shown in
Figure 6.
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mEPSC amplitudes are expected to decrease when postsynaptic receptors
saturate (Clements et al., 1992; Jonas et al.,
1993; Tang et al., 1994; Tong and Jahr,
1994) or desensitize (Trussell et al., 1993;
Geiger et al., 1995; Otis et al., 1996b).
This may occur during synaptic activity or as a result of previous
bouts of exocytosis. Studies from other synapses that seem to share properties with the calyx of Held have shown that postsynaptic AMPA
receptors desensitize during synaptic transmission (Trussell et
al., 1993; Otis et al., 1996b). To
examine the possibilities of receptor saturation and desensitization as
well as the effects of previous release events, we used another test
protocol (Fig. 7B) (termed "late noise" protocol) in
which noise was analyzed after a strong bout of exocytosis. During this
stimulation protocol, a large influx of Ca2+ was
elicited by depolarizing the presynaptic terminal to 0 mV for 1-3
msec. An episode appropriate for noise analysis (similar to that of
Fig. 7A) followed directly after the stimulation. The amplitude of the strong EPSC elicited by large stimulation was adjusted
to be similar to or slightly larger than that of an action potential
evoked EPSC by selecting an appropriate level of polarization. Both the
early noise protocol (Fig. 7A) and the late noise protocol (Fig. 7B) were concluded by holding the presynaptic terminal
at 0 mV for 10 msec to deplete the RRP completely. We repeated each protocol 5-10 times with pauses of 10-15 sec between individual runs.
For both protocols, the residual current component was estimated using
the glutamate diffusion model after the parameters had been determined
as shown in Figure 6. The residual current component was subtracted
from the total EPSC, and the release rate was estimated by deconvolving
the remaining current with the mEPSC, as described in Materials and
Methods. Then, the release rate was averaged over the whole group of
traces obtained with the same protocol. Variance was calculated as
described in Materials and Methods. Finally, h' was
calculated in rel.u. as explained in the context of Equation 11 and
plotted against time (Fig. 8). These
h' values should be reliable during the episodes intended for noise analysis (0.015-0.059 sec in the early noise protocol and
0.016-0.066 sec in the late noise protocol) (Fig. 7A,B).
They should be constant and close to 1, if all the underlying
assumptions hold. In contrast to our expectations, h' was
found to decrease in the early noise protocol from >2.0 to 0.3 rel.u.
During the late noise protocol, h' was lower than expected
but stayed constant (~0.3 rel.u.); (Fig. 8). The change in
h' was also apparent in a single trace when variance and
release rate traces were superimposed (Fig. 7, bottom). In
this Figure, variance and release rate were scaled so that both traces
should match if h' was 1 rel.u. In the early noise protocol,
h' was >1 rel.u. at the very beginning of the stimulation
episode. This was observed frequently and may represent an inaccuracy
caused by the fact that release rates are very small initially.
Alternatively, the large values of h' at the beginning of
the stimulation may actually represent the release of extra large
quanta, or synchronized multiple release. This might occur in case
there is some correlation between release probability and quantal size
or if a small number of quanta can be pairwise-triggered by the opening
of a single Ca2+ channel. h' was close to
1 rel.u. (Fig. 7A) when the release rate first approached a
value of 20 msec1 but became smaller by the end of
the protocol. In the late noise protocol (Fig. 7B), h' was
significantly <1 rel.u. throughout the episode used for noise
analysis. Thus, Figures 7 and 8 demonstrate that mEPSC amplitudes
decrease during continuous release and are very small after a strong
bout of exocytosis.
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Figure 8.
mEPSC size estimated from fluctuation analysis.
Data were obtained from the same cell pair as shown in Figures 6 and 7.
The early noise and late noise protocols were repeated five times.
h' values estimated from average traces of the early noise
protocol (solid line) and the late noise protocol
(dotted line) were plotted against time. The regions of
interest, during which the analysis is most reliable, are marked by
bars (0.015-0.059 sec for the early noise protocol and
0.016-0.066 sec for the late noise protocol). In the early noise
protocol, release rates are low during the first 5-10 msec of the
episode used for noise analysis, and these regions were excluded from
analysis.
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It is interesting to note that h' seems to be stable during
the depleting pulse (at t > 0.07 sec in the early
noise protocol and t = 0.1 sec in the late noise
protocol) and does not change abruptly, despite rapid changes in the
release rate that occur at the onset of the depleting pulse (Fig. 8).
Such findings indicate that subtraction of consecutive records and the
filtering protocol used in our study efficiently remove
nonstationarities that can distort the variance. Even so, we did not
use episodes with serious nonstationarities for later analysis (Fig.
3), and we restricted the analysis to the episodes designed for noise
analysis (Fig. 8, bars). The estimates of h' were
pooled among different cell pairs. h' was 0.82 ± 0.13 rel.u. in the early noise protocol and 0.32 ± 0.02 rel.u. in the
late noise protocols (four cell pairs; see Fig. 18). h' in
the late noise protocol was 0.42 ± 0.05 of the early noise
protocol, indicating that peak mEPSC amplitude in the late episode was
42% of that in the early episode. From deconvolution, we estimated
that 207 ± 59 vesicles were released during the strong release
period in the test condition. This value is slightly larger than the
number of vesicles that are evoked by an action potential (~150)
(Borst and Sakmann, 1996; von Gersdorff et al.,
1997; Schneggenburger et al., 1999).
The residual current was quite small throughout the relevant parts of
the records of Figure 7, such that neither the variance estimates nor
the deconvolution rates should be compromised by systematic errors (see
Results). Thus, our results indicate that the mEPSC amplitude was
indeed affected by previous or ongoing release. Therefore,
deconvolution will lead to incorrect results in conditions similar to
those used in Figure 7. Indeed, the release rates given in Figure 7,
A and B, should be divided by "the respective h' values," to provide self-consistent values. This in
turn would influence the predictions of the residual current (which
fortunately is only a small correction in the data of Fig. 7).
The reduction in mEPSC amplitude is most likely mediated by
postsynaptic receptor desensitization. We therefore examine this possibility in the following experiments by adding CTZ, which blocks or
at least minimizes desensitization of AMPA receptors (Trussell
et al., 1993; Yamada and Tang, 1993).
Cyclothiazide stabilizes mEPSCs amplitudes
To prevent postsynaptic receptor desensitization that may cause a
reduction in the mEPSC amplitudes, we performed deconvolution and
nonstationary noise analyses in the presence of CTZ (100 µM). At the start of each experiment, a fitting protocol
was applied to determine parameters of the glutamate diffusion model
(Fig. 9) under the conditions of the
given experiment. In the presence of CTZ, EPSCs decayed much more
slowly than in the absence of CTZ (compare Figs. 6 and 9)
(Trussell et al., 1993; Yamada and Tang,
1993; Barnes-Davis and Forsythe, 1995).
Thus the decay phase of EPSCs might be regulated by desensitization of
AMPA receptors. Alternatively, it is also possible that the apparent
affinity of AMPA receptors for glutamate is increased by CTZ or that
channel closure after removal of glutamate is slowed (Yamada and
Tang, 1993; Partin et al., 1994) (but see
Trussell et al., 1993). We also observed that EPSCs
decayed more slowly as the EPSC amplitudes got larger (Fig. 9). Such
findings are consistent with observations from other synapses, which
showed that delayed clearance of glutamate occurs during periods of
high quantal output (Trussell et al., 1993;
Barbour et al., 1994; Mennerick and Zorumski,
1995).
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Figure 9.
Protocol used for setting parameters of the
glutamate diffusion model in the presence of CTZ. The protocols and
calculations were performed as described in Figure 6 except that 100 µM CTZ was present. A monoexponential decay was assumed
for the mEPSC.
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Once parameters of the diffusion model were established by using the
fitting protocol (Fig. 9), which was described in Materials and Methods
and Figure 6, the release rate was estimated by deconvolution. Then we
performed noise analysis to test the constancy of the mEPSC amplitude
in the presence of CTZ (Fig. 10, in
analogy to Fig. 7) using the same protocols that were used previously.
In both the early and late noise protocols, h' remained
constant (0.5-0.7 rel.u.) during the episodes intended for noise
analysis (Fig. 11, 0.009-0.059 sec in A and 0.016-0.066
sec in B). When averaged over six cell pairs, h'
was 0.71 ± 0.09 rel.u. in the early noise protocol and 0.59 ± 0.09 rel.u. in the late noise protocol (six cell pairs). Thus
h' in the late noise protocol was 0.83 ± 0.03 of that
during the early noise protocol (see Fig. 18), indicating that peak
mEPSC amplitude in the late episode was 83% of that in the early
episode. Before the episode for noise analysis, a strong
Ca2+ influx as part of the test protocol evoked
secretion of 295 ± 24 quanta. This release was twice that evoked
by a single action potential (Borst and Sakmann,
1996).
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Figure 10.
Protocols for noise analysis in the presence of
CTZ. The protocols and calculations were performed in the presence of
100 µM CTZ as described in Figure 7 using the cell pair
and the parameters obtained in Figure 9.
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In Figures 9-11, we assumed
monoexponential decay (with a time constant of 2-3 msec) of the
mEPSCs. In the presence of CTZ, many mEPSCs decayed biexponentially
with a second time constant of ~10 msec (Wu and Borst,
1999; our unpublished observations). As described in
Materials and Methods, we can perform deconvolution analysis assuming
either mono- or biexponential mEPSCs. In the former case, the slow
component of mEPSC is considered to be part of the residual current. To
validate this claim, we compared the two variants of analysis using the
same data. We performed deconvolution and noise analysis of the records
used in Figures 9-11 assuming a biexponential decay of mEPSCs (Figs.
12-14).
As expected, the proportion of the residual current to the total
current decreased (Figs. 9b, 12b). However, the release
rates estimated were not different between the two conditions
(compare Figs. 10, 13). Also, the estimated h' was almost
the same: in the case of Fig. 14, h' was only slightly smaller (~10%) when biexponential decays were assumed as compared with the analysis using monoexponential decay (Fig. 11).
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Figure 11.
The mEPSC size estimated from fluctuation
analysis in the presence of CTZ. The protocols and calculations were
performed as described in Figure 8 except for the presence of 100 µM CTZ. Same cell pair as in Figure 9.
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Figure 12.
Protocol used for setting the parameters of the
glutamate diffusion model, assuming a biexponential decay of the mEPSC.
The protocols and calculations are as described in Figure 6 using the
cell pair shown in Figure 9. Decay time constants of the mEPSC were 3 msec (50% of the total) and 10 msec.
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Figure 13.
Protocols for noise analysis assuming a
biexponential decay of the mEPSCs. The protocols and calculations were
performed as described in Figure 10 using the cell pair shown in Figure
9.
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Figure 14.
The mEPSC size estimated from fluctuation
analysis assuming a biexponential decay of the mEPSC. The protocols and
calculations were performed as described in Figure 11 using the cell
pair shown in Figure 9.
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|
In the presence of CTZ, the residual current is quite large, and
therefore a discussion of possible systematic errors attributable to
inaccuracies in this quantity is necessary. Taking the case of Figure
10, we can see that Ir has approximately the
same amplitude as y (according to the definition of Eq. 6).
Because y(t) is quite stationary, this indicates that the
relative systematic error of the rate estimate is approximately the
same as the relative error in Ir (see Materials
and Methods). Therefore, an accurate estimate of
Ir is very important in this case. We consider
our diffusion model prediction to be accurate at the 10-20% level. For the case of Figure 10B, this is confirmed by the episode
immediately after the episode of noise analysis. Here (between 0.08 and
0.095 sec), the release rate drops to low values, and the residual
current closely matches the total current. The error resulting from the inaccuracy of the channel variance should be small throughout traces in
Figure 10, because it is in the order of 0.5 a.u. per nA. Thus, it
should be ~3 a.u., maximally, in episodes for noise analysis of
Figure 10. Even if it was accurate to only 30%, the error would be
<5%, given the fact that the total variance in the relevant sections
is 20 a.u. or larger.
Comparison of the data of Figures 9-14 with those of Figures 6-8
strongly suggests that the reduction of mEPSCs amplitudes during synaptic activity was mediated by postsynaptic AMPA receptor
desensitization, because mEPSC amplitudes remained relatively constant
in the presence of CTZ. Compared with the early noise protocol, mEPSCs
amplitudes decreased by only 20% after a strong bout of exocytosis
(corresponding to a release of ~300 synaptic vesicles).
We also found that in some cells EPSC amplitudes were extremely large
in the presence of CTZ (>10 nA) (Fig. 9). Under these conditions, the
possibility of voltage clamp errors must be examined seriously. If
postsynaptic series resistance is 2 M and the EPSC amplitude is 15 nA, the voltage deviation from the holding potential will be 30 mV. We
were forced to discard several cell pairs because EPSC amplitudes were
>15 nA. This selection might cause some sampling bias of the data
toward cells with smaller than average EPSCs. To circumvent potential
problems of postsynaptic receptor saturation and clamp errors, we
examined release rates in the presence of Kyn.
Effect of Kyn on the constancy of mEPSCs amplitudes
Kyn, a competitive antagonist in rapid equilibrium
with the glutamate binding sites of the AMPA receptors (Diamond
and Jahr, 1997), was used to prevent saturation of postsynaptic
receptors. At the same time, clamp errors could be minimized by
reducing the EPSC amplitudes. First, we examined the effects of Kyn (1 mM) on macroscopic EPSCs. CTZ was applied, and the EPSC
were evoked by depolarizing the terminal from 80 mV to 0 mV. Kyn and
CTZ were then both applied, and the same depolarization protocol was performed once more. The EPSC amplitude was reduced to 0.50 ± 0.01 by Kyn (four cell pairs), and the EPSC decayed slightly faster (data not shown). This acceleration might indicate that AMPA receptors were saturated in the presence of CTZ alone.
Noise analysis was performed on the data recorded in the presence of
CTZ and Kyn to examine the constancy of the mEPSC amplitudes. As in the
cases described so far, we determined parameters of the diffusion model
using a fitting protocol (Fig. 15), and
we performed deconvolution and noise analysis (Fig.
16). h' values estimated
from the episode for noise analysis were compared between the early
noise and the late noise protocols (Fig.
17) and were found to be almost the
same. Also, the estimated h' values were constant during the
episodes for noise analysis. The size of h' (0.4 rel.u. in
this particular experiment) was much smaller than in the absence of
Kyn. From five similar experiments, the average value of h'
was found to be 0.47 ± 0.06 rel.u. in the early noise protocol
and 0.46 ± 0.07 rel.u. in the late noise protocol (five cell
pairs). Thus values are very close to what we expect (0.5 rel.u.) if
Kyn reduces h by 50%. h' in the late noise
protocol was 0.98 ± 0.03 of the early noise protocol (Fig.
18). We conclude that mEPSC amplitudes
were constant in the presence of CTZ and Kyn. Regarding systematic
errors in this analysis, similar arguments apply, as discussed in the
context of Fig. 10. It should be noted that the 50% reduction of mEPSC
amplitude, which we obtained here in the presence of Kyn, is expected
to be the same as the macroscopic current reduction, only if there is
negligible receptor saturation or the same degree of receptor
saturation for macroscopic and miniature events. Because mEPSCs cannot
be resolved adequately under Kyn, this point will need further
investigation.
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Figure 15.
Protocol for setting parameters of the glutamate
diffusion model in the presence of CTZ and Kyn. The protocols and
calculations were performed in the presence of 100 µM CTZ
and 1 mM Kyn as described in Figure 6. For the plots of
release rates in c, we assumed that Kyn decreased mEPSCs
amplitudes to 50% and therefore scaled the values by a factor of 2. Such a correction was not applied to traces of release rate, other than
those under Kyn.
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Figure 16.
Protocols for noise analysis in the presence of
CTZ and Kyn. The protocols and calculations were performed in the
presence of 100 µM CTZ and 1 mM Kyn as
described in Figure 7, using the cell pair shown in Figure 15. For the
plots of release rates in Ac and Bc, we assumed
that Kyn decreased mEPSCs amplitudes to 50% and therefore scaled the
values by a factor of 2.
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Figure 17.
mEPSC size estimated from fluctuation analysis in
the presence of CTZ and Kyn. The protocols and calculations were
performed in the presence of 100 µM CTZ and 1 mM Kyn as described in Figure 8, using the cell pair shown
in Figure 15.
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Figure 18.
Summary of the noise analysis.
A, Comparison of the h' estimates determined from
the early noise protocol (left) and the late noise protocol
(right) under three conditions: without CTZ or Kyn (4 cell
pairs), in the presence of CTZ (6 cell pairs), and in the presence of
CTZ and Kyn (5 cell pairs). The data were averaged among all cell pairs
for each condition. Error bars indicate SEM. B, Ratios of
h' determined from the late noise protocol to h'
determined from the early noise protocol under the corresponding
condition. Although the value of h' varied among different
cell pairs (A), the ratio of h' was less variable
(B).
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DISCUSSION |
In the present study, we extend the suitability of
deconvolution methods to estimate the rate of quantal release at the
calyx of Held. The simple form of the deconvolution method
(Cohen et al., 1981; van der Kloot,
1988a; Aumann and Parnas, 1991) is not readily
applicable to this synapse and possibly also not to other glutamatergic
synapses, because of a residual current component caused by delayed
clearance of glutamate from the synaptic cleft (Barbour et al.,
1994; Mennerick and Zorumski, 1995; Otis
et al., 1996a). Also, postsynaptic factors such as
desensitization (Trussell et al., 1993; Otis et
al., 1996b) and saturation (Clements et al.,
1992; Jonas et al., 1993; Tang et al.,
1994) of AMPA receptors complicate the analysis. To take into
account the residual current component, we developed a simple model of
glutamate diffusion and incorporated it into the deconvolution
algorithm. To identify postsynaptic factors that shape the EPSCs, we
combined deconvolution with a novel type of nonstationary variance
analysis. We found that postsynaptic AMPA receptors were desensitized
significantly by a strong bout of exocytosis. CTZ prevented this
desensitization. The mEPSC amplitude was almost constant in the
presence of CTZ and Kyn. Kyn seemed to prevent saturation of
postsynaptic receptors and also reduced possible voltage clamp errors.
A contribution of desensitization to synaptic depression at the
calyx of Held synapse
Trussell et al. (1993) showed that synaptic
depression was reduced by CTZ, suggesting that postsynaptic receptor
desensitization may play a role in the synaptic depression at
calyx-type synapses (Otis et al., 1996b). In another set
of experiments, the mEPSC amplitude was used to assay desensitization
of postsynaptic receptors (Magleby and Pallotta,
1981; Otis et al., 1996b). Using a
different approach, we now show that desensitization occurs during
synaptic activity at the calyx of Held. We observed that the amplitude of mEPSCs decreased after a large EPSC, which was comparable in amplitude to an action potential-evoked EPSC. This reduction was eliminated by CTZ. Therefore, desensitization of postsynaptic receptors
has an important role in shaping postsynaptic responses.
At the calyx of Held, postsynaptic responses showed pronounced
paired-pulse depression when two stimuli were applied with a short
interpulse interval and if the first stimulus released >100-200
quanta (Borst et al., 1995; von Gersdorff et al.,
1997; Schneggenburger et al., 1999). Also,
high-frequency stimulation caused strong synaptic depression
(Borst et al., 1995). We show that release equivalent to
that of a single action potential causes significant desensitization
and affects subsequent EPSCs, although the EPSC evoked by a single
action potential itself is shaped not by desensitization but rather by
asynchronous release of synaptic vesicles (Issacson and
Walmsley, 1995; Borst and Sakmann, 1996). Desensitization should build up during trains of action potentials, especially if intervals between successive stimuli is comparable or
shorter than the time constant of desensitization. The latter was found
to be 10-20 msec in another calyx-type synapse (Raman and
Trussell, 1995). In addition, there may be desensitization between stimuli if residual glutamate accumulates. Thus desensitization will contribute to depression for frequencies above 50 Hz and possibly
at lower frequencies, in the case that accumulation occurs. This
conclusion is quite reasonable, considering that postsynaptic AMPA
receptors at the calyx show rapid and strong desensitization when
activated by application of glutamate (Geiger et al.,
1995).
Validity of the use of deconvolution at the calyx of
Held synapse
From noise analysis, we showed that the amplitude of mEPSCs was
constant during synaptic activity in the presence of CTZ and Kyn. CTZ
alone seemed to reduce receptor desensitization very effectively,
because in its presence the mEPSC amplitude decreased only slightly
(20%; maximally 40% in our data set) after large amounts of
exocytosis. This reduction can be explained in several ways.
Specifically, postsynaptic receptors may become partially saturated
during synaptic activity, or clamp errors may become relevant because
of large EPSCs. Unfortunately, we were not able to distinguish between
the two possibilities, although both are equally possible. The
possibility of receptor saturation is supported by the observation that
the EPSC decayed slightly faster after additional application of Kyn
(data not shown) [see Wu and Borst (1999), their Fig.
1]. Also, in some recordings, we saw a gradual decrease in the mEPSC
amplitudes during prolonged stimulation. This decrease could be
explained by saturation of postsynaptic receptors, because it was shown
that postsynaptic receptors saturate during synaptic transmission in
some synapses (Jonas et al., 1993; Tang et al.,
1994). However, clamp errors may also be serious, because EPSC
amplitudes are quite large under CTZ. Although we corrected for this
off-line, the data should be considered unreliable if the correction
factor exceeds 50%.
It is also possible that postsynaptic receptors are desensitized by
transmitter even in the presence of CTZ. Partin et al. (1994) showed that CTZ slowed down but did not completely
remove desensitization of the AMPA receptor flop variants. Because
GluR-D flop variants are expressed in the postsynaptic principal
neurons (Geiger et al., 1995), it might be possible that
receptors still desensitize in the presence of CTZ. However, we think
this scenario is unlikely, because mEPSCs amplitudes were constant in
the presence of CTZ and Kyn.
The mEPSC size estimate derived from the ratio of variance and release
rate was compared with the value expected from simulations. However,
simulations depended critically on parameter settings, especially on
the rise time of mEPSCs. If much slower rise times of mEPSCs (>200
µsec) were assumed for simulations, the mEPSC size expected from
simulations became smaller because of decreased noise power in the
bandpass chosen for analysis. The opposite happened if faster rise
times were assumed, although such changes were limited by the low-pass
filtering, which we applied to both experimental and simulated records.
In the presence of CTZ and Kyn, the estimate from experimental results
and the values obtained in simulations were exactly the same. This
suggests that the parameter settings of simulations are most likely
quite reasonable. Thus, discrepancies between the experimental results
and simulations in other conditions are not caused by the settings of
simulations, unless mEPSC rise times are different.
CTZ slowed the decay of EPSCs (Figs. 6, 9), making the estimates of
release rates more robust. As shown in Equation 7, the estimates of the
release rates will be proportional to the time derivative of the EPSCs,
when the decay of mEPSCs is very slow. If this is the case, the
deconvolution method becomes less sensitive to other factors, such as
the exact time course of mEPSCs or slowly varying residual currents. In
the study of Wu and Borst (1999), CTZ was added so that
the EPSC amplitudes approximated the cumulative transmitter release,
when the calyx of Held was stimulated with short current pulses.
Apart from these considerations, the deconvolution method described
here may be compromised by inadequate estimation of the residual
current. We show that the error under conditions of stationary release
cannot be better than the error of the residual current. During rapid
transients the same applies with respect to rates of change (i.e., the
relative error is at least as large as that of
dIr/dt). The estimate of the residual
current, on the other hand, depends on an appropriate choice of the
fitting protocol. We attempted to cover a wide range of possible
waveforms of the residual current in the fitting protocol of this
study. In general, however, it is advisable to design fitting protocols
that result in residual currents similar (in amplitude and time) to
those expected during the most relevant sections of the test run.
Ideally, the fitting protocol should be exactly the same as the test
protocol, except that it does not elicit release in the interval of
interest. Also, it should be noted that the estimate of the individual
current is not accurate if the mEPSC amplitude changes, because it
depends on the deconvolution rate, which is calculated assuming
constant mEPSC amplitude.
Applicability of this method to other preparations
We have developed a novel type of deconvolution that, in
combination with variance analysis, allows estimation of the kinetics of transmitter release. To apply this method to other synapses, the
rate of quantal release must be precisely controlled to obtain EPSCs
with low release rates that can be used for noise analysis. It is also
essential to voltage-clamp the postsynaptic cell reliably. If the EPSC
is distorted by voltage-clamp errors, the mean current and also
variance will be distorted. In fact, the error of the variance will be
significantly more distorted, because it is the square of the error of
the mean current. Finally, the conditions under which the mEPSC
amplitude is constant must be verified. Otherwise, correction factors
must be introduced to compensate for changes in the mEPSC amplitude.
The calyx of Held satisfied these criteria only in the presence of CTZ
and Kyn. In other preparations, it is also possible to voltage-clamp
the presynaptic and postsynaptic cells simultaneously (Yawo and
Momiyama, 1993; Yazejian et al., 1997). It will
be interesting to apply the method presented here to these synapses.
Furthermore, the nonstationary noise analysis presented here seems to
be a potent method to identify presynaptic and postsynaptic factors
shaping EPSCs. Specifically, we were able to separate the residual
current component from that arising from the asynchronous release of
synaptic vesicles, a task that has proven difficult in the past
(Barbour and Häusser, 1997). If the trace is
stationary for a long enough period, release rates and mEPSC amplitude
can be estimated from a single trace. We hope that this analysis is helpful for studies at other synapses. An application of this type of
analysis in which the dependence of release rate on calcium current is
established is described in the companion article (Sakaba and
Neher, 2001).
| |
FOOTNOTES |
Received July 20, 2000; revised Sept. 14, 2000; accepted Oct. 18, 2000.
This work was supported in part by a grant of the Deutsche
Forschungsgemeinschaft (SFB 406) to E.N. and by an Alexander von Humboldt fellowship (T.S.). We thank Ralf Schneggenburger for very
helpful advice during the course of this study, and Alexander Meyer,
Ralf Schneggenburger, and Henrique von Gersdorff for help with the
initial part of the experiments. We also thank Ruth Heidelberger, Isabel Llano, Alain Marty, and Sonja Pyott for critical comments on
this manuscript.
Correspondence should be addressed to Erwin Neher, Max-Planck-Institute
for Biophysical Chemistry, Department of Membrane Biophysics, D-37077,
Göttingen, Germany. E-mail:
eneher{at}gwdg.de.
| |
APPENDIX A |
The point-by-point deconvolution algorithm, used here, can be
derived by considering the general rule for differentiating an integral
of the form of Equation 6 (Madelung, 1964), which is:
|
(A1)
|
We consider the case of an mEPSC starting from zero at time zero
and rising to a value of h (its peak value) with a time constant of rise, 0, from where it decays in a
double exponential fashion according to:
|
(A2)
|
Here,  is the relative contribution of a second slow
mEPSC component and 1 and 2 are the time
constants of the rapid and the slow component, respectively.
A0 is a normalization factor, which is the
inverse of the peak amplitude of the expression within the parentheses
of Equation A2. This insures that the peak amplitude of
F(t") is 1. For evaluation of Equation A1, we note that the first term in the sum is zero, because F(0) = 0, and we
split the integration into two time intervals:
|
(A3)
|
with t representing the time to peak of
F(t").
For evaluating the second part we assume that the release rate
(t') is relatively constant over the short time interval
t and can be replaced by its value
 (t*) at an appropriately chosen time t*  [t t, t]. We consider (t*) as an
average in the sense that small rapid fluctuations in (t)
are averaged over the integration interval. Then this term will be
given by:
|
(A4)
|
where we make use of the normalization F(t) = 1.
A3 together with A2 and A4 results in the following equation:
|
(A5)
|
where:
|
(A6)
|
Calculation of (t*), therefore, requires the
evaluation of the derivative dy/dt at time t and
of three integrals, involving the function (t) at values
smaller than t. This, however, can be simplified by
considering that (Eqs. 6, A2, and A6):
|
(A7)
|
The second term in this equation is given by:
|
(A8)
|
where t** is an appropriately chosen time t** [t t, t]. Int0 can be simplified by considering
that this integration is performed over a very rapidly decaying
function, which is significantly different from zero in the integration
interval only at time t and slightly larger. Therefore,
Int0 can be evaluated to:
|
(A9)
|
Here, again, t*** denotes an appropriately chosen time
t*** [t t 0, t t]. Combining Equations A5 and A6-A9 and eliminating Int1, we arrive at:
|
(A10)
|
When written in this form one can readily see that the
deconvolution can be calculated according to the simple Equation 7 (Cohen et al., 1981), if the mEPSC is monoexponential
( = 0) and the rise time ( 0) is very
short. The additional terms in Equation A10 are corrections for
biexponential decay (the term proportional to ) and finite rise
time. For the actual calculation we evaluated:
|
(A11)
|
numerically, and chose t* to be at the half-point of
the mEPSC rising phase. Numerical simulations showed that the
deconvolution result was not very sensitive toward the exact choice of
t** and t***. We therefore set t** = t* and t*** tt and evaluated the quantity
"(t*), defined as the product of h and
(t*):
with
|
(A12)
|
"(t*) as well as all the terms within the
curled brackets of Equation A12 are in units of amperes per second, and
the release rate in events per second is obtained by division through
h. Note that this is a mean release rate, referred to the
mean mEPSC amplitude h, and when evaluated at time
t, it is an average over the interval [t t, t]. The quantity "(t**), appearing on
the right side of A12, is the same quantity as evaluated at an earlier
time point. Therefore, A12 can be evaluated point by point, when
starting values in the interval [0, t] are known. We
take these as zero, because our experimental traces always start with
episodes without stimulation. The mean current of the starting episode
is subtracted from the whole trace before deconvolution, to correct for
holding currents.
| |
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TOP
ABSTRACT
INTRODUCTION
Compound via MeSH
