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The Journal of Neuroscience, December 15, 2001, 21(24):9638-9654
Estimating Transmitter Release Rates from Postsynaptic
Current Fluctuations
Erwin
Neher and
Takeshi
Sakaba
Max Planck Institute for Biophysical Chemistry, Department of
Membrane Biophysics, D-37077 Göttingen, Germany
 |
ABSTRACT |
A method is presented that allows one to estimate transmitter
release rates from fluctuations of postsynaptic current records under
conditions of stationary or slowly varying release. For experimental
applications, we used the calyx of Held, a glutamatergic synapse, in which "residual current," i.e., current attributable to
residual glutamate in the synaptic cleft, is present. For a characterization of synaptic transmission, several postsynaptic parameters, such as the mean amplitude of the miniature postsynaptic current and an apparent single channel conductance, have to be known.
These were obtained by evaluating variance and two more higher moments
of the current fluctuations. In agreement with Fesce et al.
(1986) , we found both by simulations and by analyzing experimental records that high-pass filtering of postsynaptic currents
renders the estimates remarkably tolerant against nonstationarities. We
also found that release rates and postsynaptic parameters can be
reliably obtained when release rates are low (~10 events/msec). Furthermore, during a long-lasting stimulus, the transmitter release at
the calyx of Held was found to decay to a low, stationary rate of 10 events/msec after depletion of the "releasable pool" of synaptic
vesicles. This stationary release rate is compatible with the expected
rate of recruitment of new vesicles to the release-ready pool of
vesicles. MiniatureEPSC (mEPSC) size is estimated to be similar to the
value of spontaneously occurring mEPSC under this condition.
Key words:
synaptic transmission; exocytosis; noise analysis; skewness; cumulants; calyx of Held; vesicle pool
| |
INTRODUCTION |
Quantitative analysis of transmitter
release is crucial for the understanding of mechanisms underlying
various forms of plasticity, such as paired-pulse facilitation and
short-term depression. However, estimation of release rates
requires that the size and time course of the synaptic quantum [the
miniature EPSC (mEPSC) for the case of excitatory transmission] are
known. Furthermore, analysis of postsynaptic currents at glutamatergic
synapses is complicated by the fact that the transmitter is cleared
only slowly from the synaptic cleft, such that "residual glutamate"
accumulates and activates "residual current" (Trussell et
al., 1993; Barbour et al., 1994;
Mennerick and Zorumski, 1995; Otis et al.,
1996a; Kinney et al., 1997; Carter and
Regehr, 2000). Neher and Sakaba (2001) adapted
the method of deconvolution to allow for such residual current.
By combining this method with the analysis of the variance of
postsynaptic current records, they showed that the mEPSC amplitude decreases during prolonged stimulation attributable to desensitization of postsynaptic AMPA-type receptors. This confirmed the results of
previous studies at various glutamatergic synapses, demonstrating both
desensitization (Trussell et al., 1993; Otis et
al., 1996b) and saturation (Clements et al.,
1992; Jonas et al., 1993; Tang et al.,
1994; Liu et al., 1999) of postsynaptic
receptors during intense stimulation. Deconvolution was found to be
very useful for studying large and rapid changes in transmitter release
rate (Sakaba and Neher, 2001). However, the method is
not very accurate during low, continuous release, in the presence of
residual current. The analysis of EPSCs in such a situation is
necessary not only at the calyx of Held but also at other synapses at
which asynchronous release plays a major role in transmission
(Parsons et al., 1994; Rieke and Schwartz,
1994; von Gersdorff and Matthews, 1994;
Lu and Trussell, 2000). Therefore, it is
desirable to explore an alternative method that is reliable under such conditions.
Fluctuation analysis has been used extensively in electrophysiology to
study the properties of elementary events that underlie cellular
signals, either channel currents or synaptic quanta (Katz and Miledi, 1972; Anderson and Stevens,
1973; Neher and Stevens, 1977; Colquhoun
and Hawkes, 1981; Heinemann and Sigworth, 1993). However, the application of fluctuation analysis to the study of
synaptic processes has been primarily restricted to either the
statistical analysis of fluctuations in the amplitudes of evoked
responses (for review, see Zucker et al., 1999;
McLachlan, 1978) or the study of channel properties
(Traynelis and Jaramillo, 1998). A few studies, however,
have estimated release rates and mEPSC properties under stationary or
slowly varying conditions (Segal et al., 1985;
Fesce et al., 1986; Martin and Finger,
1988; Rossi et al., 1994). They
demonstrated that such an analysis, based on an extension of
Campbell's theorem, can be very powerful in elucidating mEPSC
properties, particularly if high-pass-filtered records are used.
Here, we adapt this method to the specific case of a large
glutamatergic synapse, by calculating the variance and cumulants of
higher order (skew and fourth cumulant) (Courtney,
1978; Segal et al., 1985; Fesce,
1990; Heinemann and Sigworth, 1991) of the postsynaptic current. By combining information from these quantities, we allow for the contribution of the residual current to noise and
obtain reliable estimates for the amplitudes and rates of mEPSCs under
moderately nonstationary conditions. In particular, we estimate the
release rate and mEPSC amplitude under the condition of deep depression
(long-lasting strong stimulation) in which releasable vesicle pools
should be empty and remaining release should be representing the
delivery of new vesicles to the release-ready pool. We find a release
rate that is compatible with the time course of recovery of the
release-ready pool after stimulation, and we find normal mEPSC size.
This latter finding indicates that, even after prolonged stimulation,
partially filled vesicles do not contribute appreciably to the EPSC, as
might be expected if vesicles recycle rapidly (Pyle et al.,
2000). It also indicates that, even under these strong
stimulation conditions, no vesicles contribute that are located more
remote from postsynaptic densities.
| |
THEORY |
Following Segal et al. (1985) and Fesce
(1990), we assume that the postsynaptic current is a sum of
randomly occurring mEPSCs, each of which has a mean time course,
h·F(t). Here, h is an amplitude factor, and
F(t) is the time course of the mEPSC scaled to a peak amplitude of 1. Additionally, current fluctuations around the mean,
attributable to the random opening and closing of channels, have to be
considered. For simplicity, we assume that these fluctuations are
independent of the mean mEPSC time course, have a Gaussian distribution, and are characterized by a variance,
Vc, that is proportional to the total current.
This assumption is problematic for the rising phase of the mEPSC, when
channel openings are highly synchronized; however, for any given mEPSC,
the variance attributable to the mean time course is large compared
with the channel variance around its peak. The assumption is
well warranted in the late decay phase of an mEPSC, when channels
flicker is uncorrelated, and also for the residual current generated by
slowly varying accumulations of neurotransmitter in the synaptic cleft.
Below, we will demonstrate by simulation that the assumption is valid for the purpose of this study.
Campbell's theorem (Campbell, 1909; Rice,
1944), which is the basis of this technique, states that mean,
variance, and the higher semi-invariants (or cumulants) of the noise
signal generated by a stream of randomly occurring elementary events
are proportional to the frequency of occurrence of elementary events,
 , and to the integral over the nth power of the
elementary signal waveform:
![&lgr;<SUB>n</SUB>=&xgr;h<SUP>n</SUP><LIM><OP>∫</OP><LL><UP>−</UP>x</LL><UL><UP>+</UP>x</UL></LIM>[F(t)]<SUP>n</SUP>dt=&xgr;h<SUP>n</SUP>I<SUB>n</SUB>.](/content/vol21/issue24/fulltext/9638/img001.gif)
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(1)
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Here  n is the nth cumulant
of the fluctuating signal, is the frequency of occurrence of
elementary events, h is the peak amplitude of the elementary
signal (mEPSC, see above), and In is a
short-hand notation for the integral over the nth power of the normalized mEPSC time course [F(t)]. Three
requirements have to be satisfied for Equation 1 to hold: (1) , the
rate of independently and randomly occurring mEPSCs, has to be
reasonably constant within the observation interval, (2)
F(t) can be different from zero only within an interval
short with respect to the observation interval, and (3) the total
current is a linear superposition of randomly occurring elementary
events (Rice, 1944; Segal et al.,
1985).
When these conditions are satisfied, Equation 1, applied to the cases
n = 2 (variance) and n = 1 (mean),
readily leads to the equation used for standard noise analysis
(h = 2/1·(I1/I2)). However, in the presence of a residual current, standard noise analysis
is not applicable, because the elementary event is very long lasting
(including a long "tail" of its contribution to the residual
current) and the ratio I2/I1
cannot be evaluated because of the ill-defined and variable nature of
the residual current. Also, residual current is most likely a nonlinear
function of residual glutamate, which violates the assumption of linear
superposition. However, a high-pass-filtered version of the original
current more readily satisfies the requirement of Campbell's theorem
because high-pass filtering turns the elementary events into short,
spike-like signals and eliminates the slowly-varying nonlinear
components. Additionally, high-pass filtering allows more accurate
measurement of the skew and higher moments (Segal et al.,
1985). After high-pass filtering, however, the mean of the
signal is zero and can no longer be used for calculation of . For
the case that the variance originating from channel flickering,
Vc, is small after high-pass filtering,
an estimate for can be obtained from the ratio of skew and variance
instead. Unfortunately, we found that, at the calyx of Held, channel
variance, Vc, very often makes a
substantial contribution to the total variance, especially when
cyclothiazide (CTZ) is used to block glutamate receptor desensitization
(Trussell et al., 1993; Yamada and Tang,
1993). However, Vc is expected to be
proportional to the mean current (as measured before high-pass filtering). If, as argued above, Vc is
statistically independent of mEPSC variance
'2, the total variance after filtering can be written as a sum of the two contributions:
|
(2)
|
where i' is a proportionality constant representing
the filtered single channel current, I'2
is the integral over the square of the filtered EPSC time course, and
Ip is the mean postsynaptic current before
high-pass filtering. The primes (') in this and the following equations
denote the fact that they refer to the filtered record.
The skew ('3) of the filtered postsynaptic
current does not contain contributions from the residual current (the
skew of a Gaussian distribution is zero) and is, therefore, given by
the following:
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(3)
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where I'3 is the integral of the
filtered mEPSC raised to the third power.
From Equations 2 and 3, we obtain the following:
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(4)
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and
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(5)
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Because '2,
'3, and Ip can
readily be measured and both I'2 and
I'3 can be calculated from the mEPSC time
course (see legend of Fig. 1), these equations allow the amplitude,
h, and the rate, , of mEPSCs to be calculated from
postsynaptic current traces. i', the proportionality
constant between channel variance and mean current, has to be
determined independently as described in Results. Alternatively, it can
be calculated from the cumulants (including information from the fourth
cumulant, 4; see below). The fourth cumulant
4 is given by the following:
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(6)
|
where µ4 is the fourth central moment. Like the
other cumulants, 4 has the property that, when derived
from a sum of two statistically independent signals, it is equal to the
sum of the two cumulants. Furthermore, both the skew and the fourth
cumulant of a random signal with Gaussian distribution are zero, such
that the channel variance does not contribute to either of them. We therefore obtain, in analogy to Equations 4 and 5, the following:
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(7)
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and
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(8)
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In practice, however, '4 can be determined
with reasonable accuracy only for long stretches of data; therefore, as
a rule, we determine h and from Equations 4 and 5.
In some of our records, however, we encounter suitable stretches of
data with relatively large residual current and little mEPSC activity.
From these, i' can be calculated, because the variance is
dominated by residual current. mEPSC-induced fluctuations, in this
case, constitute a minor contribution, which can be taken into account
by combining Equations 2, 3, and 6 for the calculation of
i':
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(9)
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It can be seen that i' is given by the ratio
'2/Ip (which is the
expression for pure channel-derived fluctuations) and a correction term
involving the ratio of third and fourth cumulants. We thus can
calculate i' on a suitable segment and determine
h and according to Equations 4 and 5 for the rest of the
trace unless there is reason to believe that i' varies along
that trace. It will be shown below that h can be calculated
at much better resolution than . Thus, it is advantageous to
calculate h from Equation 4 and insert this (averaged over
longer stretches of data) together with i' into Equation 2,
for calculation of .
An additional complication arises from the fact that, for most
synapses, the amplitude, h, is not a constant but rather a stochastic variable with a certain distribution. Fortunately, this
amplitude distribution of mEPSCs can be measured at the calyx of Held
(Borst and Sakmann, 1996; Chuhma and Ohmori,
1998; Schneggenburger et al., 1999). Assuming
that the different size classes of mEPSCs occur statistically
independently and that no correlation exists between size and release
probability, the quantities h, h2,
h3, and h4 in
Equations 2, 3, and 6 can be replaced by their expectation values
(Fesce, 1990), and we arrive at the following:
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(10)
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and
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(11)
|
Here, we have introduced the quantities
h's and
's, which, respectively,
represent estimates proportional to h and from the skew
of the filtered records. Both h's and
's have to be multiplied by
calibration factors H's and
Z's, which, according to Equations
10 and 11, are given by the following:
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(12)
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and
|
(13)
|
Equivalent equations and definitions
(H'4,
Z'4; see legend of Table 1) can be
introduced to replace Equations 7-9 for obtaining estimates from the
fourth cumulant and the skew:
|
(14)
|
|
(15)
|
|
(16)
|
Calibration factors H's,
Z's, H'4,
and Z'4 were calculated from the average
time course of the mEPSCs (Fig. 1B, Table 1) and from the
mEPSC amplitude distribution. The moments of the distribution were as
follows: h = 31.1 pA (mean);
h2 = 1182 pA2;
h3 = 5.4 · 10 32A3; and
h4 = 2.91 · 1042A4. The fact that the
calibration factors contain the quantity of interest,
h, is of little concern, because they do so in
combination with the higher moments in a way that they are
nondimensional and invariant, when an amplitude distribution uniformly
shrinks or expands. Because we are mainly interested in using Equation 9 to study possible postsynaptic desensitization, and, assuming that
desensitization scales all of the mEPSC amplitudes by a constant factor, we expect H's to be constant
during our recordings.
In practice, the calibration factors H's
and Z's are readily calculated from the
equations above, provided that the time course of an individual mEPSC
and its amplitude distribution is known (see legends of Fig. 1 and
Table 1). Alternatively, they can be determined from simulated data,
using the measured amplitude distribution for the simulation (see
below) and applying exactly the same routines, as used for the analysis
of experimental postsynaptic currents. In two previous papers
(Neher and Sakaba, 2001; Sakaba and Neher,
2001), we used this latter method to obtain relative values of
mEPSCs amplitudes. In the present work, we provide calculated values
throughout, using mainly Equations 10-16.
| |
MATERIALS AND METHODS |
Analysis. Postsynaptic, as well as presynaptic,
currents were sampled at rates of 20 kHz after low-pass filtering at
either 2.9 or 6 kHz and using a partial hardware series resistance
compensation (see below for recording conditions). The remaining series
resistance (typically 2-3 M ) was compensated by software using a
routine similar to that of Traynelis (1998). In this
manner, 100% compensation was readily obtained. Analysis was performed
with custom-written macros in the environment of IGOR-Pro (WaveMetrics
Inc., Lake Oswego, OR). The deconvolution method was performed as
described by Neher and Sakaba (2001). The Igor macros
used are available on our departmental homepage
(www.mpibpc.gwdg.de/abteilungen/140/software).
A first type of fluctuation analysis was performed on single traces of
postsynaptic currents by calculating sliding averages of the mean,
variance, skew, and fourth cumulant (averaged over a specified analysis
window). For the latter three quantities, the data were band-pass
filtered before calculating the moments. Variance and skew were set
equal to the second and third moment. The fourth cumulant was
calculated according to Equation 6 from the fourth moment and the
variance after smoothing these quantities by the sliding window. This
order of processing is important, because Equation 6 involves a
nonlinear operation. Likewise, the quantities
h's and
's (from Eqs. 10 and 11) or the
corresponding quantities h'4 and
'4 (estimates from the fourth cumulant and skew)
were calculated after smoothing. For this type of analysis, the sliding
windows had to be relatively large and the traces had to be reasonably
stationary within the length of a window (see Results for an analysis
of the effects of nonstationarity).
For most types of experiments, however, we wanted to analyze records
with quite pronounced nonstationarities. Typically, we had 5-10 such
traces with very similar mean time courses available. It was then
possible to improve the analysis by a method similar to "ensemble
noise analysis," introduced by Sigworth (1980) for nonstationary records of voltage-dependent currents. This technique analyzes fluctuations around the mean of an ensemble. The most effective way of removing long-term trends is to subtract subsequent records from one another. The variance of such difference records is
twice the variance around the ensemble mean (Heinemann and Sigworth, 1993). For the skew, such simple subtraction is not possible, because the expected skew of a difference between two similar
records is zero. However, the difference between an individual record
and the sample mean over N such records is quite suitable for analysis. Such a difference record is given by the following:
|
(17)
|
and the expectation value for its skew,
3D,i (given statistical independence
among traces), is as follows:
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(18)
|
where 3 is the skew around the true ensemble
mean. When N is greater than 5, the multiplier of
3 is larger than 0.5, and, thus, a substantial fraction
of the skew is recovered. The corresponding equations for variance
(2D) and fourth cumulant
(4D) are as follows:
|
(19)
|
and
|
(20)
|
The advantage of this type of ensemble analysis is that it
tolerates substantial nonstationarities of the original records. To
efficiently reduce slow trends between records, we subtracted scaled
and shifted versions of the sample mean record. Scalers and shift
parameters for individual records were determined by minimizing the
mean square deviation between a given record and the ensemble mean over
a suitable time window. Thereby, trends and transient artifacts are
eliminated but results are not modified in any other way, provided that
scalers stay within the range of 0.8-1.2 (see below). The time
window for determining the scalers should be selected from a region of
the mean trace in which the largest nonstationarities occur (or else
the nonstationary region of interest), and it should be long enough
that it contains a large number of random events in any one trace. The
latter requirement derives from the fact that the least-square
optimization procedure reduces the number of degrees of freedom of the
fluctuations by two.
The combined effect of high-pass filtering of the current records and
of subtracting the weighted mean made the analysis quite independent of
nonstationarities, as will be documented by simulations below.
Filtering. Digital filtering was performed using
combinations of smoothing routines supplied by IGOR, as follows. The
rationale for the particular combinations will be given in Results. For low-pass filtering, a sliding average using the "Box-Smooth"
function with a specified averaging window T1
(usually 0.3 msec) was used. To remove the secondary maxima of the
resulting sinx/x-type filtering characteristics, the first smoothing
was followed by a second smoothing with a window 0.8 × T1, resulting in a filter function with a
 3dD point at fo = 0.5/T1 (= 1670 Hz for T1 = 0.3 msec), which dropped to values lower than 60 db at
~1.6·fo. Also, secondary maxima were lower
than 50 db. For high-pass filtering, a low-pass-filtered record (see
above) was subtracted from unfiltered data. Two passes of such
high-pass filtering were performed on all records, one using a window
Th and a second one using
8·Th. In the first pass, the low-pass-filtered
record was shifted by Th/2 before
subtraction (to preserve the asymmetry of the waveform), and, in the
second pass, it was shifted by 4Th. For
band-pass filtering, low-pass filtering and high-pass filtering were
applied sequentially. The choice of the smoothing windows
T1 and Th for low- and
high-pass filtering, respectively, will be explained in the context of
the simulations described below. The filter characteristics used for most of the variance and skew analyses is shown in Fig. 1. It was
calculated with T1 = Th = 0.3 msec. Note that a different filter procedure was used by
Neher and Sakaba (2001) and Sakaba and Neher
(2001).
Simulation. The purpose of the simulation was to provide
traces for a given set of model assumptions and a specified time course
of the release rate r(t). The corresponding IGOR macros were
written such that either experimental release time courses [as derived
from deconvolution (Neher and Sakaba, 2001)] or
appropriate test functions (this paper) could be specified. The macros
build the output trace as a sum of miniature currents progressing from beginning to end. For each sample interval,  t, the number
of release events, nR, was determined as
the output of a random number generator with a Poisson distribution and
the mean number of events equal to r(t)·t. If
nR was greater than 0, an mEPSC-type waveform (as specified by model parameters) was added to the output trace with
its origin at the given sample point. The amplitude of that addition
was determined as a sum of nR random numbers
drawn from a random number generator with the distribution of the mEPSC
amplitudes. In effect, this procedure merged the mEPSCs originating
from one sample interval into a single one. This simplification is
considered to be valid, if the sample interval (typically 50 µsec) is
much shorter than the mEPSC rise time (200-300 µsec). The simulation also tracked the mean residual glutamate concentration,
CD, in the synaptic cleft. This was
assumed to be a convolution of the released amount equal to the
following:
|
(21)
|
where r(t') is the release rate and
cr(t) is a diffusion-type kernel, given by
(Crank, 1975):
|
(22)
|
The residual current, ID, which
was added to the sum of mEPSC currents, was assumed to be as
follows:
|
(23)
|
In a previous publication (Neher and Sakaba,
2001), it has been shown that this formalism allows one to fit
glutamatergic postsynaptic currents over a wide range of experimental
situations. The choice was explained in that paper; here we just list
the parameters and give typical values. The peak amplitude of mEPSC was
calculated according to the measured distribution with a mean of 32.1 pA. Its time course was assumed to be a double or triple exponential
sum with the following:  r of 0.2 msec, rise time constant; 1 of 2 msec, decay time constant;  of 0 or
0.2, fraction of a second slow component of decay; and 2
of 10 msec time constant of slow decay phase. These values are typical
for recordings in the presence of 100 µM cyclothiazide
(which was used in most recordings to prevent glutamate receptor desensitization).
The residual current was described by an amplitude factor, selected
such that residual current correctly reproduced experimental currents
at long times after release. The parameters were as follows: n = 1.2 exponent of the power law of glutamate channel
activation; rD = 0.76 µm diffusional
distance; nD = 0.9 exponent of the
diffusion law; and D = 30 µm2/sec,
diffusion coefficient of the transmitter.
For the analysis of simulated records, the resulting sum of
mEPSC-derived and residual currents was filtered and processed in the
same way as experimental traces. It should be noted that most of the
simulation did not include noise from channel open-close fluctuations
and background recording. If required, such "channel noise" was
introduced by adding to the simulated trace a real current trace
obtained from an experiment in which glutamate-activated current was
elicited by bath application of 100 µM
S-AMPA.
Simulation of mEPSCs as a superposition of single channel
currents. For some of the simulations, we constructed simple
mEPSCs with mean time courses rising with 0 and decaying
exponentially with 1 by activating channels at a rate
that has an initial value of A0 and decays
exponentially with time constant 0. After initial activation, channels fluctuate between an open state and a short-lived closed state, Csl, according to the
following scheme:
The rates and k1 were selected such
that bursts of openings were obtained with a mean burst duration of 2 msec, consisting of flickers with a mean duration of 0.5 msec and mean
gaps of 0.2 msec. The corresponding parameters are as follows:
k1 = 1590 sec1,
 = 3409 sec1, and = 2000 sec1 (Colquhoun and Hawkes, 1981).
We chose the amplitude of the single channel current such that
the mean current during the burst was 1.5 pA, and we chose
Ao such that the mean mEPSC amplitude was 32 pA.
This simple model is not intended to reproduce the complicated and
controversial open-close kinetics of AMPA-activated currents (Partin et al., 1996; Rosenmund et al.,
1998) but only to estimate the errors made by assuming
statistical independence of channel gating fluctuations and release
statistics (see Results).
Physiological recordings. Presynaptic and postsynaptic
recordings at the calyx of Held were performed in the slice preparation of the rat brainstem as has been described previously (Neher and Sakaba, 2001). Briefly, 8- to 10-d-old Wistar rats were
decapitated without anaesthesia according to local guidelines. The
brainstem was immersed in ice-cold, low-calcium saline, which contained (in mM): 125 NaCl, 2.5 KCl, 0.1 CaCl2, 3 MgCl2, 25 glucose, 1.25 NaHCO3,
0.4 ascorbic acid, 3 myo-inositol, and 2 Na-pyruvate, pH 7.3-7.4 (320 mOsm; 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 and 100 µM CTZ
and/or 1 mM kynurenic acid (Kyn) was added, as
indicated. Experiments were done within 4 hr after preparation of the
slices. All recordings were done at room temperature (~21-24°C). A
presynaptic terminal and a postsynaptic target were simultaneously
clamped at 80 mV with patch pipettes. The presynaptic pipette (4-6
M) was filled with a solution containing (in
mM): 125-130 Cs-gluconate, 20 tetraethylammonium (TEA)-Cl,
10 HEPES, 5 Na2-phosphocreatine, 4 MgATP, 0.3 GTP, 0.5 EGTA, pH 7.2 with CsOH (310 mOsm). The postsynaptic pipette (2-3.5
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
TEA-Cl, 0.1 mM 3,4-diaminopyridine, and 50 µM
D-AP-5 were added to the normal extracellular solution to isolate the presynaptic calcium current and to block NMDA
receptors. This way, AMPA receptor-mediated EPSCs were isolated to
monitor quantal release. TTX was purchased from Alomone Labs (Jerusalem, Israel). S-AMPA, D-AP-5, and CTZ
were from Tocris Cookson (Köln, Germany). Other drugs were from
Sigma (Deisenhofen, Germany).
Both presynaptic and postsynaptic cells were whole-cell clamped to a
holding potential of 80 mV, using EPC9/2 amplifiers controlled by the
Pulse program (Heka Elektronik, Lambrecht, Germany). No liquid junction
potential correction was applied. Thirty to 70% of the presynaptic
series resistance (Rs of 8-30 M, typically 15 M) was compensated. The postsynaptic series resistance (3-8 M, typically 5 M) was compensated so that the uncompensated series resistance was ~2-3 M. This remaining series resistance was software-compensated off-line, as described above.
| |
RESULTS |
Simulations
Optimizing the filtering procedure for high signal-to-noise ratio
and tolerance against nonstationaries
Optimal filtering of postsynaptic currents depends on a number of
criteria, such as suppression of background noise (including channel
noise), optimization of the ratio between the mean of the estimated
cumulants and their intrinsic statistical variation, suppression of
trends and nonstationarities in the data, and preservation of asymmetry
of the original signal (for skew). The dependence of noise estimates on
bandwidth can be derived analytically, and, for many reasons, high-pass
filtering with a high cutoff frequency is beneficial (Fesce,
1990). However, at very high frequencies, the noise power
originating from the mEPSC time course decreases rapidly, whereas
background noise, such as amplifier noise or ion channel noise,
increases or stays constant. Therefore, an optimal frequency band for
filtering has to be found. In the following, we will first discuss some
theoretical expectations and then search for optimal parameters by simulation.
We are interested in determining the relative accuracy of our
estimates of the cumulants and want to optimize the filtering procedure so that the coefficient of variation (CV),
  'v/'v, is as small as possible for a given observation interval. Neher and Sakaba (2001) pointed out that the coefficient of variation of the variance is independent of the frequency of elementary events
and is proportional to  , where n is
the number of independent samples that can be obtained during the
observation interval T. This, in turn, depends on the
filtering, given the requirement that the observation for obtaining
such a sample has to be longer than the duration of the filtered
elementary event. For this reason, the filtering should be done such
that the filtered elementary event has the shortest possible
half-width. This requirement for short half-width is even more serious
for the skew, as pointed out by Segal et al. (1985). The
argument calls for high-pass filtering (which eliminates the residual
current and reduces the slow decay phase of mEPSCs) with a corner
frequency as high as possible. Additionally, high-pass filtering
eliminates trends and other nonstationarities in the record. However,
when examining noise power spectra of stationary records derived from either a stream of mEPSCs or currents evoked by superfusion of AMPA, it
is noticed that channel and instrumentation noise may very well
dominate noise spectral density at frequencies above 1 kHz. Also, the
spectral density of the mEPSC-derived signal at such high frequencies
critically depends on the mEPSC rise time, which, in turn, may be
dependent on clamp speed and series resistance of the postsynaptic
recording. Therefore, we restricted the band pass on the high end to
~1.6 kHz, which corresponds to a smoothing window
T1 of 0.3 msec (see Materials and Methods), or
1.5-2 times the estimated mEPSC rise time (see paragraph on filtering,
above). With this restriction, a relatively narrow bandpass
(implemented by software, as shown in Fig.
1A), appeared to be optimal.
We also restricted the filter kernel to as few points as possible to
avoid contamination of those parts of the record, which are relatively
stationary, by large transients of current in neighboring segments.
This, too, is a property of the software filter described in Materials
and Methods, as shown in Fig. 1B. The filter converts an
mEPSC into a pulse with ~0.5 msec half-width, preceded and followed
by shallow depressions that extend over ~5 msec. Fig. 1B
also shows that the square and the cube of the filtered signal are
narrow pulses with hardly any components preceding and following. In
the following simulations, we explore the properties of fluctuation
analysis of records filtered in such a way.
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Figure 1.
The bandpass filter. A, Filter
characteristics used for most of the bandpass filtering in the
fluctuation analysis. Five thousand points of Gaussian white noise
(from Igor function gnoise) were interpreted as a current record with
50 µsec sample interval and bandpass filtered with a low-pass window
T1 of 0.3 msec and a high-pass window
Th of the same length. Spectral analysis was
performed using the Igor Routine PSD with a segment length of
1024 points and a Hanning-type window. The filter characteristic has
its maximum at 1074 Hz. The 3 db point is at 1670 Hz. The
broken line represents the curve f/1074 and indicates that
the filter curve is well fitted by a single pole high pass (3 db
point at 537 Hz) between 200 Hz and 1 kHz. B, Time course of
a typical mEPSC, with a 0.2 msec time constant and a 2 msec decay time
constant. For simulations, this standard mini was multiplied by an
amplitude, as drawn from the amplitude distribution. Also included in
this figure are the filtered mEPSC and the filtered mEPSC raised to the
second and third power. The latter two curves are expanded
five times for better visibility. By numerical integration, the
following values are obtained from Equations 1-3:
I'2 = 4.3 · 105,
I'3 = 1.06 · 105,
and I'4 = 3.156 · 106 sec.
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Averaging over 500 msec of a rather wavy record gives useful
information on the variance, skew, and fourth cumulant
Postsynaptic current traces were simulated as described in
Materials and Methods section. For the first series of simulations, we
used the mEPSC parameter set (described in Materials and Methods) without a second decay component ( of 0) and without a residual current ( of 0). These parameters are appropriate for recordings in
the presence of 100 µM CTZ.
First, we simulated stretches (500 msec each) of superimposed mEPSCs at
constant release rates of 0.5, 1, 2, 8, 12, and 24 events/msec. Example
excerpts from such simulated traces before and after filtering are
shown in Figure 2, A and
B. We calculated the variance, skew, and fourth moment of
the filtered records (mean of 0) and formed averages over individual
records (500 msec each). We repeated this procedure 50 times and
obtained mean values and SDs (given in Table
1, top seven rows). Also given in Table 1
are the estimates for mEPSC amplitude and release rates as derived from
the skew and variance (according to Eqs. 10 and 11) using quantities
H's and Z's
(given by Eqs. 12 and 13). The same estimates based on the fourth
cumulant and skew are also shown (Eqs. 14 and 15). Estimates of the
mean mEPSC amplitudes (32.1 pA), as well as the release rates, are
reproduced quite accurately by the analysis based on skew and variance.
In contrast, estimates deviate for rates above 2 events/msec when the
analysis is instead based on the fourth cumulant (Table 1, Fig.
3A). This is because at higher
frequency, according to the central limit theorem, the distribution of
current values approaches a Gaussian, which has a relatively small
fourth cumulant.
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Figure 2.
Simulated current traces. A, Excerpts
from simulated traces at rates of 0.5 (top), 2 (middle), and 8 (bottom) events/msec. mEPSCs had
rise time constants of 0.2 msec, decay time constants of 2 msec, and a
peak amplitude distribution. B, Same segments of traces as
in A after bandpass filtering. Traces are offset to match
the DC level of the corresponding traces in A.
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Figure 3.
Estimates for rates, mEPSC amplitudes, and the
coefficients of variation. Simulated records 500 msec in length
were analyzed as described in Materials and Methods. Estimates for the
rates and mEPSC amplitudes (A) and coefficients of variation
(SD divided by the absolute magnitude of the mean) of cumulants
(B) and mEPSC amplitudes and rates (C) are
plotted against the rate of occurrence of mEPSCs. Values are taken from
Table 1. For abbreviations used in the labeling, see the legend to
Table 1.
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Coefficients of variation, calculated from the means and SDs of Table
1, are plotted in Figure 3, B and C. As expected
from the consideration above, the estimates for the coefficient of variation of the variance are almost independent of the mEPSC frequency
for rates greater than 2 events/msec. The same is also observed for
estimates of the fourth moment. However, the accuracy of the fourth
cumulant and of estimates derived from it deteriorate at rates greater
than 2 events/msec. The skew is significantly less well resolved than
the variance, and its resolution decreases for rates greater than 5 events/msec (Segal et al., 1985). Both the variance and
(even more so) the fourth moment are less well resolved at very small
rates, probably because their distributions are far from Gaussian, even
if averaged over long time intervals. The coefficients of variation of
rates and amplitudes reflect the properties of the moments from which
they are derived (Fig. 3C) (see also the theoretical
predictions; Fesce, 1990). The rate estimate derived
from the fourth moment is without any significance for rates greater
than 2 events/msec. Amplitudes and rates estimated from the skew and
variance are optimally resolved at rates between 1 and 2 events/msec
and are quite accurate over the entire frequency range studied.
Performing the same analysis on segments of different lengths or
averaging values from several similar traces readily demonstrates that
the coefficient of variation changes with 1/ (Neher and Sakaba, 2001), where T is the
total recording time over which the cumulants are averaged. Thus, for
each estimate, the total recording time necessary for a desired
accuracy can be calculated. This recording time is 70 msec for variance
when 20% accuracy is desired (with release rates of 2 events/msec) and
much longer for the other estimates. For example, an estimation of the
mEPSC amplitude with 20% accuracy requires only 110 msec, significantly less than a similar estimate for the release rate (280 msec). This difference is expected, realizing that higher powers of
cumulants enter the equation for release rate (Eq. 11) than that for
amplitude (Eq. 10).
Second, we wanted to know how sensitive such estimates would be to slow
variations in the release rate. To this end, we simulated traces with a
mean rate of 2 events/msec that increased and decreased sinusoidally by
±50% with a period of 50 msec. The result for the mean cumulants are
shown (Table 1) for comparison with the values for a constant release
process. Within the statistical error it is seen that the values agree
with those corresponding to a constant rate of 2 events/msec. Thus, we
confirm that bandpass filtering effectively eliminates the effects of
nonstationarities (Fesce, 1990), if the
nonstationarities are sufficiently slower than the low frequency cutoff
of the filter. Below, we will present in more detail the effects of
more rapid transitions in release rates.
Third, we simulated traces with a constant rate of 2 events/msec and a
fourfold-reduced cutoff frequency of the high-pass branch of the
filter. Variance and higher cumulants were increased sevenfold to
30-fold (Table 1, compare row two, row nine), whereas the estimates of
rate and amplitude were still correct, although significantly more
noisy. This analysis shows that high-pass filtering should be done with
the highest corner frequency that is compatible with other constraints.
Contributions by slow decay components, residual current, and
channel noise
So far, the simulations were performed with a single component
mEPSC, in which fluctuations attributable to opening and closing of
channels were not considered. We now proceed to simulations in which we
include a slow component of mEPSC decay, as well as channel
fluctuations. We demonstrate that these additions lead only to small
changes in the mean values, provided that their contributions are slow
relative to the bandpass of the filter.
When a slow decay component of 20% amplitude and 10 msec time constant
was added in simulations at a release rate of two per millisecond, the
mean current at steady state was about twice as large as that for a
single component mEPSC. If we included a residual current component,
using parameters typical for the calyx of Held (see Materials and
Methods) (Neher and Sakaba, 2001), the steady-state
currents were increased approximately fivefold. Nevertheless, the
variance was unchanged and the skew increased only insignificantly (see
Table 1 for simulations at 0.5, 2, 8, and 24 events/msec).
Correspondingly, amplitudes and rates estimated from the skew and
variance were correct. A similar result was obtained when we added a
slow negative component, producing an undershoot.
We did two kinds of simulations to examine the influence of channel
noise. In the first simulation, we simply added current traces recorded
during S-AMPA bath application (at 100 µM) from a
postsynaptic neuron to the simulated traces at 0.5, 2, 8, and 24 events/msec. To do so, we selected experimental traces that had the
same mean current as the simulated ones. As expected, the variance of
the summed records was increased (Table 1), and the skew and fourth
cumulant were changed only insignificantly. More importantly, the SD of
all estimates changed very little. The contribution of channel noise to
most of the traces was ~15% of the total variance. When the variance
of the AMPA-induced current was measured directly (after bandpass
filtering) and plotted against mean current, a straight line was
usually obtained. However, the slopes of such lines (corresponding to
the parameter i' in Eqs. 2-10) varied between 12 and 24 fA
for different cells (Neher and Sakaba, 2001). When
AMPA-induced currents were added to simulated ones, the variance of the
superposition was very close to the sum of the individual variances of
the AMPA currents and the EPSC simulations. Thus, accurate estimates
for amplitudes and rates were obtained when channel variance was
accounted for by Equations 10 and 11 (Table 1, last four rows).
In the second simulation, we examined the error that might arise by the
assumption of statistical independence between channel noise and the
noise originating from the random superposition of mEPSCs.
Specifically, we simulated simple mEPSCs explicitly as superpositions
of channels (see Materials and Methods), superimposed such mEPSCs
randomly at a mean rate of 2 events/msec, and compared the resulting
total noise with that of a random superposition of mEPSCs with a fixed
time course. Time courses of the latter mEPSCs were similar to that of
the mean of the simulated ones. For this simulation, we used a constant
mEPSC amplitude, unlike all other simulations. If our assumption about
the independence of channel variance, Vc,
and mEPSCs variance (Eq. 2) were correct, we would expect the
difference in variance between the two cases to be given by
Vc, according to Equation 2. This is
exactly what we found. Variance values in the two simulations differed
only by ~20%. This difference (1.9·1023
A2) was very close to the variance of another
simulation (1.7·1023 A2), in
which the same type of channels was superimposed randomly at a rate
that resulted in the same mean current (176 pA). In the control
simulation, the total variance (9·1023
A2) was, however, smaller than that of the standard
simulation at the same mEPSC rate (Table 1, row three) because, in the
latter case, variance is increased by the dispersion of mEPSC amplitudes.
Influence of the dispersion of mEPSC amplitudes
The heterogeneity of mEPSC amplitudes has a pronounced influence
on the estimates for size and rate. The effect on mean values are taken
care of by the correction factors described in Equations 10-13. To
investigate whether this dispersion influences the coefficients of
variation of the estimates and to test the overall correctness of the
analysis (with regard to the expectation values), we compared simulations based on the experimental amplitude distribution (described above) with two additional simulations in which the mean amplitudes were close to the mean of the experimental distribution. In one case,
the amplitude was fixed at this value (31.1 pA), and, in the other, it
had a very large dispersion, with half of the mEPSCs having an
amplitude of 12.5 pA and the other half 52.5 pA. In both cases,
amplitudes and rates were estimated accurately and with the same
coefficients of variation as in control cases (analyses performed at 2 events/msec).
Time resolution of noise estimates
So far, we have shown that high-pass filtering of the records
renders noise estimates relatively insensitive to slow variations in
release rate. When we analyze experiments in which the release process
changes slowly (such as during slowly decaying, asynchronous release
after a strong stimulus), the question arises what the "time
resolution" of the estimates will be. We already showed above that a
total recording time of 70 msec is required to estimate the variance
with an accuracy of 20%. Likewise, we can ask what recording time is
required to achieve an SD of the variance estimate that is equal to its
mean. It is reasonable to define this time as the time resolution of
the variance estimate, because this duration is the shortest time
interval, T, for which the estimate bears any meaning.
Because the SD of the noise estimates at high enough mEPSC rates varies
with 1/ (when the estimates have a Gaussian
distribution), we can calculate the time resolution,
Tr, from values found in Table 1 from the following equation:
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(24)
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where CV0 is the coefficient of variation
(SD/mean), and T0 is the time window over which
the values in Table 1 were obtained (500 msec). Table
2 gives some of the calculated values for
the following quantities: variance, AmpS (amplitude, as estimated from
skew and variance), RateS (rate from skew, and variance), and the
corresponding values for the estimates based on the fourth cumulant and
skew (AmpF and RateF). We see that the resolution for variance is as
short as 1-3 msec, depending on the mEPSC rate. This demonstrates that
an order of magnitude estimate for variance can be obtained within
surprising short times. The time resolutions for amplitude and rate
estimates are within 10 and 100 msec, with better values at lower mEPSC
rates. It should be noted that the values given in Table 2 are the
total recording times necessary. Thus, if there is an ensemble of
n similar traces available, the time resolution for the
average estimate will be approximately Tr/n or else the coefficient of variation
for the time interval Tr will be
1/ . We performed simulations at 0.5 events/msec
over time windows equal to the T values predicted from
Equation 24. In approximately half of the cases shown in Table 2, these
calculations confirmed Equation 24. However, for estimates involving
the fourth cumulant, particularly at large release rates, Tr values turned out to be up to two times
larger than those extrapolated from the numbers obtained with a 500 msec analysis window (Eq. 24 and Table 1). The reason for the
discrepancy is most likely a violation of the assumption (underlying
Eq. 24) that the estimates have a Gaussian distribution.
Tolerance towards transient changes in release rate
The above consideration addressed only the question of how long a
recording interval must be to yield meaningful estimates. In addition,
we wanted to find out how the estimates are influenced by trends and
transient changes in release rates. To this end, we simulated
postsynaptic current traces with abruptly changing release rates,
subjected these to the same analysis as used above, and tested how well
the resulting rate estimates reproduce such transient changes. We
averaged estimates over time windows of 5 msec, which is expected to
result in a coefficient of variation of ~0.6 for the variance trace.
The parameters for the mEPSCs were selected to resemble those found in
recordings from the calyx of Held, and a residual current was included.
In agreement with a previous study, in which variance was analysed only
(Neher and Sakaba, 2001), we found that variance and
skew are capable of tracking changes in release rate, provided that
those are not excessive. In fact, the resolution (both in time and
amplitude) is quite remarkable, especially if a few similar traces of a
given protocol are available for averaging. Single traces provide order of magnitude estimates for variance and release rates with a time resolution of 5-10 msec. Correspondingly, for averages over 10 similar
traces time resolution approaches 1.5-3 msec. As expected from Figure
3, the coefficient of variation of the skew was larger than that of
variance, and that of the amplitude estimate (as derived from skew and
variance) was in between. The estimate of the release rate, on the
other hand, was not as well resolved (as expected from Fig. 3 and on
the basis of theory; Fesce, 1990). Thus, the best
strategy might be to analyze variance and skew together only for mEPSC
amplitude. In case that mEPSC amplitude is found to be constant or only
slowly varying with time, the rate can be inferred from variance alone
by inverting Equation 2 and inserting the known h:
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(25)
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Averaged simulation results were quite insensitive to variations
between individual traces. Thus, results did not change when release
rates used for simulations fluctuated by up to 20% between traces.
Transients in the variance or skew during abrupt changes in release
rate were greatly reduced, when mean subtraction was applied in an
ensemble of records (described in Materials and Methods). Figure
4 compares simulation results with and
without mean subtraction. A series of release episodes (each with a
square pulse-like release rate at increasing amplitudes between 5 and 20 events/msec) was simulated, and averages over 10 records are displayed: first without (Fig. 4A) and then after mean
subtraction (Fig. 4C). The mean trace (average of 10 records), as well as an example of a difference trace, is shown in
Figure 4B. The skew is particularly sensitive to transients.
In Figure 4A, it is seen that its estimate is out of range
for two of the larger transitions, which is not the case after mean
subtraction (Fig. 4C). Variance estimates, on the other
hand, were more tolerant to transient changes. An example is given in
Figure 5, in which the variance estimate
for an abrupt transition from 2 to 100 events/msec and back again is
shown. It is free of artifacts, although the rates of individual traces
were allowed to fluctuate by up to 20%.
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ABSTRACT
INTRODUCTION
