Abstract
The properties of neurotransmitter receptor channels are important for determining synaptic transmission in the nervous system. The presence of quantal variability complicates the use of conventional non-stationary noise analysis for determining the unitary conductance and number of channels involved in synaptic currents. Peak-scaled non-stationary noise analysis has been used to compensate for quantal variability, but there is evidence that the resulting variance versus mean relationships can be transformed from parabolic to skewed. We have used computer modelling based on experimentally derived kinetic schemes to investigate such relationships and demonstrate that their shape is a consequence of the temporal structure of the fluctuations during synaptic responses. Covariance analysis showed that peak-scaling generates a skewed relationship when the covariance function decays rapidly (compared to the average response waveform), corresponding to a low correlation between fluctuations at the peak and in neighbouring regions of the decay phase. A parabolic relationship is obtained when the covariance function decays more slowly, corresponding to a higher correlation. Irrespective of a skewed or parabolic relationship, we demonstrate that the unitary current can be reliably estimated, with a coefficient of variation (CV) as low as 0.05 and bias as low as ±2% under ideal conditions. While the shape of the variance versus mean curve after peak-scaled non-stationary noise analysis is ultimately a consequence of the kinetic properties of the channels, inadequate alignment of individual waveforms can transform the relationship from parabolic to skewed, and low-pass filtering can transform the relationship from skewed to parabolic. These findings have important implications for analysis of experimental data.
Estimating the conductance and number of receptor ion channels in chemical synapses in the central nervous system remains an important problem (Silver & Farrant, 1999; Benke et al. 2001; Momiyama et al. 2003). In general it is a difficult problem, mainly due to the experimental inaccessibility of synaptic receptors. An indirect approach involves the use of patch recordings of extrasynaptic receptors, either recording single-channel currents when the conductance can be directly resolved, or macroscopic currents where non-stationary noise analysis (Sigworth, 1980) can be used to estimate the single-channel conductance. In either case, the implicit assumption is that the properties of the extrasynaptic receptors are identical, or at least very similar, to those located in the synapse. Alternatively, one can apply non-stationary noise analysis directly to either spontaneous or evoked postsynaptic currents (PSCs).
The application of non-stationary noise analysis to PSCs presents a series of challenges (Traynelis & Jaramillo, 1998). The technique was originally developed to analyse responses where the ensemble of contributing ion channels is invariant from one response to the next, and it cannot be applied directly to PSCs arising from multiple release sites because quantal variability contributes to the variation in PSC amplitude. It is therefore necessary to isolate variations in current arising from the stochastic gating of the ion channels from variations arising from sources such as quantal variability. For conventional non-stationary noise analysis, fluctuations around the mean are isolated by directly subtracting the mean response waveform from each individual response waveform. Thus, for analysis of PSCs from more than one release site, scaling the mean PSC waveform to each individual PSC waveform before subtracting them from each other should, in principle, isolate fluctuations around the mean arising from ion channel gating. Robinson et al. (1991) described a method where the mean PSC waveform was fitted to each individual PSC, but this resulted in an underestimation of the single-channel conductance. Traynelis et al. (1993) (and see also De Koninck & Mody, 1994) scaled the mean PSC waveform to the peak amplitude of each individual PSC before subtracting the two waveforms, thereby isolating fluctuations arising from ion channel gating (reviewed by Silver & Farrant, 1999). With this method of peak-scaling, information about the total number of channels (N) exposed to transmitter is lost, and only the average number of channels open at the peak of the PSC can be estimated (Traynelis et al. 1993; Silver et al. 1996). Ideally, the method of peak-scaled non-stationary noise analysis should also be applicable to analysis of spontaneous PSCs (spPSCs) where there is minimal asynchrony in activation of receptor channels, but where the number and identity of channels activated changes from one PSC to the next. However, as discussed in a recent review on this topic (Silver & Farrant, 1999), a series of assumptions has to be fulfilled in order for the analysis to be valid. First, the contributing channels have to be identical. Second, the contributing channels have to be independent. Third, the mean PSC waveform should be the same at all contributing synapses. Fourth, contributing ion channels must have a single open state. Finally, it has been suggested that the shape of the theoretically parabolic relationship between variance and mean current will be distorted when channels open for the first time after the peak and contribute to the current decay without having been open at the peak (Traynelis et al. 1993). Momiyama et al. (2003) recently observed a skewed variance versus mean relationship for simulated responses based on a kinetic scheme proposed for AMPA receptors in cerebellar Purkinje cells, but did not further investigate the basis for this. Using a kinetic scheme developed by Geiger et al. (1999), we have observed a similarly skewed variance versus mean relationship from simulated responses of AMPA receptors in hippocampal interneurons. This suggests that a skewed variance versus mean relationship after peak-scaled non-stationary noise analysis might be a common property. Accordingly, we wanted to explore the consequences of applying peak-scaled non-stationary noise analysis to simulated, spPSC-like responses generated by a series of experimentally derived kinetic schemes, and to evaluate the reliability of estimates of channel conductances obtained from such measurements. In addition, we explored the consequences of low-pass filtering of spPSC-like waveforms by the combination of series resistance and membrane capacitance (RC filtering) for the estimate of single-channel conductance obtained with peak-scaled non-stationary noise analysis. Finally, we investigated the performance of different methods of aligning individual response waveforms, as this seems to be of critical importance for the analysis of experimental data (e.g. Traynelis et al. 1993). The major conclusions from our study are that a skewed or parabolic variance versus mean relationship follows from the kinetic properties of specific receptor channels, that the single-channel conductance can be reliably estimated irrespective of a skewed or parabolic relationship, and that analysis of covariance functions constitutes a useful tool for evaluating such relationships. However, a challenge for analysis of experimental data is that the shape of the variance versus mean relationship can be transformed by low-pass filtering or inadequate alignment of spPSCs.
Methods
Simulations
Simulations of ion channel current responses were based on Markov-type kinetic models, both experimentally derived models proposed for AMPA and glycine receptor gating, as well as more simplified, semi-realistic models. Stochastic (Monte Carlo or microscopic) simulations were performed with the software programs AxoGraph (version 4.6; Molecular Devices, Sunnyvale, CA, USA), AxoGraph X (version 1.0; Molecular Devices) and ChanneLab (version 2; Synaptosoft, Decatur, GA, USA). Initially, all stochastic simulations were run in duplicate under AxoGraph and ChanneLab to verify the consistency of results between simulators. The number of receptor channels (N) was set to 1 (for first latency measurements) or 50; the driving force membrane potential minus reversal potential; (Vm−Erev) to −60 mV; the time interval to 10 μs; and for each condition an ensemble of events was generated by repeating the simulation 1000 times (n). All channels occupied the closed unliganded state at the start of a simulation sweep. To mimic one source of quantal variability, we also ran simulations where the number of available channels was varied randomly between trials (Nmean= 50; Gaussian distribution with s.d.= 10). The single-channel conductance was specified separately for each kinetic model (see below). To generate current responses resembling spontaneous synaptic events, release of single synaptic vesicles was mimicked by modelling the agonist waveform as a 1 ms-duration square-wave pulse of agonist, instantaneously rising (from zero baseline) and falling within one simulation time-step (10 μs). For one set of experiments, we also varied the duration of the agonist pulse from 0.1 to 1.0 ms (cf. Clements, 1996). Agonist concentration was varied between 1 μm and 100 mm, and at least four different concentrations, spanning a range of peak open probability (Po,peak) values (see below), were used for each kinetic scheme. In another set of experiments we modelled the agonist waveform as instantaneously rising, followed by either an exponential decay (τdecay= 100 μs) or by an instantaneous fall to zero. Po,peak is the open probability at the maximum peak of the response, defined by the equation Po = I/iN, where i is the unitary current and N is the number of available channels. For some simulations, Gaussian noise was added with s.d.= 0.25 for multichannel simulations (N = 50) and s.d.= 0.025 for single-channel simulations (N = 1). Macroscopic simulations, e.g. to generate concentration–response relationships, were performed with Berkeley Madonna (version 8.1 or 8.3; R. I. Macey & G. F. Oster, USA). Responses were simulated for the following Markov kinetic models for glutamate receptors (henceforth referred to by the indicated abbreviations): Geiger et al. (1999; their Fig. 4; GluGei99), Momiyama et al. (2003; original scheme in Häusser & Roth, 1997; GluMom03), Robert & Howe (2003; their Fig.6a; GluRH03), and glycine receptors: Legendre (1998; his Fig. 8, but with rate constants taken from Legendre, 2001; GlyLeg98), Burzomato et al. (2004; their Fig.3 with rate constants from their Scheme 5; GlyBur04) and a model included in the standard distribution of AxoGraph version 4.6 (GlyAG).
Time constants for the burst length distributions of single-channel openings for specific kinetic schemes were calculated with the free software programs SCJUMP (http://www.ucl.ac.uk/pharmacology/dc.html) or GNU Octave (http://www.octave.org) as the reciprocals of the eigenvalues of −QEE, with QEE being the burst state submatrix of the Q matrix (Colquhoun & Hawkes, 1982; Wyllie et al. 1998). The mean Popen within bursts of single-channel openings was also calculated by SCJUMP, or, for a specific three-state kinetic scheme (see Results for definitions, including the rate constants α, β and k−1), directly as the total open time divided by the mean burst length:
 |
General data analysis
Analysis was performed with the following computer programs: AxoGraph, PulseFit and PulseTools (HEKA Elektronik, Lambrecht/Pfalz, Germany) and Igor Pro (WaveMetrics, Lake Oswego, OR, USA).
Concentration–response relationships were fitted with Hill-type equations of the following form:
 |
where R is the peak current (I) or Po,peak at a given concentration of agonist ([A]); Rmax is the maximum response, EC50 is the concentration of agonist giving rise to half-maximal response and nH is the Hill coefficient. For single-channel simulations (N = 1), latency to first opening and corresponding distributions were analysed with PulseTools. The bias of a parameter estimate (a′) relative to the nominal (true) parameter value (a) was calculated as: 100%× (a′−a)/a. Data are presented as mean ±s.d.
Bootstrap analysis
Statistical errors in the best-fit parameters were estimated by bootstrap analysis (Efron & Tibshirani, 1993) of the simulated events. Balanced resampling was done by generating 100 random lists of 1000 event numbers (from 1 to 1000) with the routine created in Mathematica by Roth & Häusser (2001). Each number corresponded to an event in the original ensemble, and each list of numbers was subsequently used to generate a synthetic data set with 1000 events.
Alignment of response waveforms
Three different methods of aligning individual current traces of an ensemble were used. The first method aligned traces on the point of steepest rise (calculated as the location of the minimum value of the first derivative) between onset and peak response. For the second method, each individual waveform was fitted with a function that combined two exponential functions and had an abrupt onset,
 |
where δ is the delay to onset of the response, A describes the peak amplitude and τrise and τdecay are the time constants of the rise and decay phases, respectively. The traces were aligned first by the delay to onset and second by the location of the peak of the fitted function. For the third method, each individual waveform was fitted with a function that combined an error function with an exponential function (modified from Borst & Sakmann, 1996) and had a gradual onset,
 |
where A describes the peak amplitude, erf is the error function, δ is the delay to onset of the response and τdecay is the time constant of the decay phase. As for the second method, the traces were aligned first by the delay to onset and second by the location of the peak of the fitted function. It should be noted that eqns (3) and (4) are not the correct equations to describe the time course of the responses (they would have to be calculated directly from the models), but simplified empirical curves that may (or may not) fit reasonably well.
Artificial de-alignment (jitter) of individual response waveforms of an ensemble (relative to the alignment after simulation) was performed by shifting each waveform left or right in time by an integer number of sample intervals (simulation time-steps). The number of sample intervals was varied randomly such that an infinite number of such values would be evenly distributed between −num and num, with num being a user-input value.
Non-stationary noise analysis
In order to estimate the single-channel current (i) and the available number of channels (N), we performed non-stationary noise analysis (Sigworth, 1980) on each ensemble of responses simulated with a constant number of channels (N = 50). The ensemble variance (σ2) was plotted against the mean current (I) and the data points were fitted with the following parabolic function (Sigworth, 1980), omitting the rising phase of the response:
 |
where σb2 is the variance of the background noise. In order to assign similar weights to all phases of the ensemble mean waveform, from the peak to the end of the decay, the amplitude interval from the peak to the baseline was divided into an equal number of intervals (30–100 bins), with each interval corresponding, on average, to the same number of channel closings (see Traynelis et al. 1993; Steffan & Heinemann, 1997). The amplitude intervals were then translated to the corresponding time intervals, and the variance and mean current were calculated for each interval over all the event waveforms before constructing the variance versus mean curves and fitting with eqn (5). The slope conductance value (γ) for each channel was then calculated by the equation i/(Vm−Erev), with Vm=− 60 mV and Erev= 0 mV. The open probability (Popen) at any given time was determined by the equation Popen = I/iN.
In order to estimate i and N from responses where the available number of channels varied between trials (see above), we performed peak-scaled non-stationary noise analysis. Briefly, the peak of the mean current response waveform was scaled to the response value at the corresponding point in time of each individual event before subtraction to generate the difference waveforms (Traynelis et al. 1993). In this case, when the resulting variance versus mean curve is fitted with eqn (5), N does not correspond to the number of available channels, but ideally to the average number of channels open at the peak. Thus, no information about Po,peak is available. We also analysed these responses with conventional non-stationary noise analysis (without peak-scaling). In this case, the variance versus mean curves did not have a parabolic shape, and the results were analysed by performing a linear fit to the initial part of the curve (10–15% of the response range) where the slope factor corresponds to the single-channel current (i).
Autocovariance analysis
Autocovariance functions (for simplicity termed covariance functions) were calculated from an n-by-m event matrix (n events and m sample points for each event) in PulseTools. The resulting m-by-m covariance matrix is a measure of the linear strength between the m variables and can be stated formally as,
 |
where the yi are the current values from the ith event and the μ are the means of the nyi values (Sigworth, 1981, 1984). Each value in the covariance matrix, σ2ij, corresponds to the covariance between the corresponding columns i and j in the original event matrix. The diagonal of the covariance matrix (for i = j) corresponds to the variance values for the n columns of the event matrix. For a given centre point (t1 or tc), the decay of a covariance function, C(t1,t2), can be estimated by fitting with exponential functions such that each decay time constant corresponds to a correlation time (tcorr).
Digital filtering
A simple RC filter (1-poled), used to mimic the filtering effect of the combination of Rs (series resistance) and Cm (cell membrane capacitance), as well as a digital Gaussian filter, used to mimic instrument filtering, were applied from built-in routines of Igor Pro. Cut-off frequencies are −3 dB. Correction for RC filtering was performed by a procedure similar to that described by Traynelis (1998) and implemented in Igor Pro (adapted from software available at http://www.mpibpc.gwdg.de/abteilungen/140/software/index.html).
Results
In the first part of this report, we present results for conventional and peak-scaled non-stationary noise analysis of simulated spPSC-like events for a series of experimentally derived kinetic schemes for glycine and glutamate receptors. With a fixed number of receptor channels, non-stationary noise analysis of ideal data gave good estimates for the unitary current, i, and the number of available channels, N. With stochastic, trial-to-trial variation of the number of available channels, peak-scaled non-stationary noise analysis of ideal data could always be used to estimate i. For some kinetic schemes, the peak-scaled variance versus mean curves were parabolic, such that curve fitting also gave an estimate for the average number of channels open at the peak. For other kinetic schemes, however, the variance versus mean curves were skewed, and curve fitting only gave an estimate for i. In the second part of the report, we explore the critical differences between the kinetic schemes and identify parameters that give rise to one type of behaviour or the other. We further demonstrate that covariance analysis is a useful tool for understanding the results of peak-scaled non-stationary noise analysis. In the third part of the report, we explore simulation results based on a kinetic scheme with multiple, concentration-dependent open states. Finally, we explore the effects of low-pass filtering, the performance of procedures to correct for such filtering, and the performance of procedures for alignment of spPSCs with respect to the results of peak-scaled non-stationary noise analysis.
Glycine receptor kinetic schemes
The first kinetic scheme we used for glycine receptors was a relatively simple scheme included with the standard distribution of the software program AxoGraph (GlyAG; Fig. 1A). With 1 ms square-wave pulses of agonist, macroscopic simulations (Fig. 1B) gave a concentration–peak response curve with EC50 of ∼265 μm and a maximum response at saturating concentrations of ∼97 pA, corresponding to Po,peak= 0.65 (Fig. 1C). Here and later, we also illustrate the responses to long pulses of agonist (485 ms; Fig. 1B) and the corresponding concentration–response curve (Fig. 1C), in order to be able to compare the peak amplitude and location between responses to short and long pulses. With stochastic simulations at a series of agonist concentrations, the fraction of first openings that occurred after the peak location of the corresponding macroscopically simulated response at the same concentration, varied from 7.5% to 9.2% (Fig. 1D).
Figure 1. Non-stationary noise analysis of responses generated by the GlyAG kinetic scheme.
A, kinetic scheme (included in the standard distribution of the software program AxoGraph). U: unbound state; B1: singly liganded state; B2: doubly liganded state; O: open state; Dfast: fast desensitised state; Dslow: slow desensitised state. Here and elsewhere, a rate constant labelled kAB indicates the transition rate from state A to state B and when a rate constant has the unit of m−1 in addition to s−1, it indicates a ligand-binding step. The rate constants were as follows: kUB1= 10 × 106m−1 s−1, kB1U= 100 s−1, kB1B2= 5 × 106m−1 s−1, kB2B1= 200 s−1, kB2O= 1700 s−1, kOB2= 750 s−1, kB2Dfast= 60 s−1, kDfastB2= 10 s−1, kDfastDslow= 1 s−1, kDslowDfast= 1 s−1. B, macroscopically simulated responses evoked by short (1 ms; continuous lines) or long (485 ms; broken lines) pulses of agonist, N (number of channels) = 50, single-channel conductance (γ) = 50 pS, driving force −60 mV, time course of agonist application indicated above responses. C, concentration–response relationships for short (•) and long (○) agonist pulses, response measured as both peak current (left axis) and peak open probability (right axis), data points have been fitted with a Hill-type equation (eqn (2)). D, histograms (binwidth 20 μs; here and in all subsequent figures) of latency to first opening for stochastically simulated responses (N = 1 channel) at a series of agonist concentrations (1 ms pulses), time course of agonist application indicated by horizontal bar (top), n (number of repetitions) = 1000, vertical arrows under each histogram indicate peak location of mean response (macroscopically simulated) at the corresponding agonist concentration. Here and later, each histogram is mirrored across the horizontal line, with the scaling taking this into account. E, variance versus mean curves for stochastically simulated responses (N = 50; n = 1000) after conventional non-stationary noise analysis. Here and below, continuous lines represent results for the agonist concentrations indicated in panel D (1 ms pulses) and broken line represents theoretical curve according to eqn (5). F, variance versus mean curves for stochastically simulated responses with trial-to-trial variation (Gaussian) in the number of available channels (N = 50 ± 10 (s.d.); n = 1000) after conventional non-stationary noise analysis. G, variance versus mean curves for stochastically simulated responses (N = 50; n = 1000) after peak-scaled (PS) non-stationary noise analysis. H, variance versus mean curves for stochastically simulated responses with trial-to-trial variation (Gaussian) in the number of available channels (N = 50 ± 10; n = 1000) after peak-scaled (PS) non-stationary noise analysis.
We next used stochastic simulations and non-stationary noise analysis to estimate i and N (Fig. 1E; Table 1A). Here and later, the statistical errors were estimated both by repeating the stochastic simulations 10 times and by analysis of 100 bootstrap re-sampled synthetic data sets from one of the original data sets. As can be seen from Table 1A, both procedures generated very similar error estimates, both for i and N, with coefficient of variation (CV) typically around 0.05 and bias close to zero. For the estimates of N, however, both CV and bias were larger for the responses simulated at an agonist concentration of 500 μm. This is probably due to the fact that at this concentration, Po,peak did not exceed the value of 0.5, thus constraining the parameter N of eqn (5) less well.
Table 1.
GlyAG
| 500 μm | 1 mm | 3 mm | 5 mm | ||
|---|---|---|---|---|---|
| A | |||||
| Parabolic fit | |||||
| i | Mean ±s.d. (pA) | 2.93 ± 0.16 | 3.02 ± 0.15 | 2.96 ± 0.10 | 2.93 ± 0.16 |
| CV | 0.055 (0.050) | 0.051 (0.030) | 0.035 (0.030) | 0.056 (0.034) | |
| Bias (%) | −2.1 | 0.79 | −1.5 | −2.3 | |
| N | Mean ±s.d. | 53.9 ± 6.4 | 49.3 ± 4.9 | 51.2 ± 2.7 | 51.7 ± 4.6 |
| CV | 0.118 (0.120) | 0.099 (0.054) | 0.052 (0.047) | 0.089 (0.062) | |
| Bias (%) | 7.8 | −1.4 | 2.5 | 3.4 | |
| B | |||||
| Linear fit | |||||
| i | Mean ±s.d. (pA) | 2.90 ± 0.13 | 2.99 ± 0.06 | 3.00 ± 0.19 | 2.90 ± 0.13 |
| CV | 0.046 (0.049) | 0.021 (0.049) | 0.063 (0.043) | 0.046 (0.047) | |
| Bias (%) | −3.3 | −0.38 | 0.01 | −3.3 | |
| Parabolic fit (after peak-scaling) | |||||
| i | Mean ±s.d. (pA) | 3.00 ± 0.14 | 2.96 ± 0.15 | 3.02 ± 0.18 | 3.06 ± 0.13 |
| CV | 0.048 (0.049) | 0.050 (0.058) | 0.061 (0.044) | 0.043 (0.043) | |
| Bias (%) | −0.13 | −1.4 | 0.60 | 2.1 | |
| N | Mean ±s.d. | 56.8 ± 7.1 | 71.7 ± 10.5 | 73.0 ± 19.5 | 68.2 ± 9.1 |
| CV | 0.125 (0.158) | 0.147 (0.152) | 0.266 (0.145) | 0.133 (0.121) | |
A, GlyAG, simulations (1 ms agonist pulses with concentrations as indicated) with constant number of channels (γ= 50 pS; i= 3 pA; N= 50; n= 1000). Here and later, i (unitary current) and N (number of channels) were calculated with conventional non-stationary noise analysis, curve fit with parabolic function, eqn (5). Here and later, CV was calculated from analysis of 10 repetitions, CV in parentheses obtained from bootstrap re-sampling of 100 synthetic data sets from one of the original data sets. B, GlyAG, simulations with stochastic trial-to-trial variation of the number of channels (γ= 50 pS; i= 3 pA; N= 50; s.d.= 10; n= 1000). Here and later, upper row (i) indicates results from non-stationary noise analysis with no peak-scaling and linear fit to initial part of the variance versus mean curve, lower two rows (i and N) indicate results from peak-scaled non-stationary noise analysis and parabolic fit, eqn (5).
We next repeated the simulations with stochastic variation of the number of available channels from trial to trial, in order to mimic a potential source of quantal variability. The stochastic variation of available channels increased the variance over that contributed by the stochastic channel gating alone, and with conventional non-stationary noise analysis, the variance versus mean curves deviated strongly from a parabolic shape (Fig. 1F). By fitting the initial part of the curves with a straight line, however, we obtained good estimates of i, with CV typically around 0.05 and bias close to zero (Table 1B). By scaling the peak of the ensemble mean to each individual event before calculating the difference traces, it should be possible to compensate for the increased variability caused by the stochastic variation in the number of available channels. When peak-scaling was applied to non-stationary noise analysis of the responses simulated with either a constant or variable number of channels, it expectedly reduced the variance at the peak response, but it also generated a skewed instead of a parabolic variance versus mean relationship, with increased variance compared to the theoretical curve calculated according to eqn (5)(Fig. 1G and H). Importantly, the variance versus mean curves for responses simulated with a variable number of channels overlapped closely with those for responses simulated with a constant number of channels (Fig. 1G and H). This indicates that peak-scaling was effective in compensating for the increased variance caused by variations in the number of available channels. When a subregion of each variance versus mean curve (for responses simulated with a variable number of channels) was fitted with eqn (5), excluding the rightmost data points (from the maximum variance and to the right; cf. Traynelis et al. 1993; Momiyama et al. 2003), the estimates of i were very similar to those from the linear fits to the analysis results without peak-scaling, with CVs around 0.05 and bias close to zero (Table 1B). However, the estimates for N did not correspond to either the average number of channels open at the peak or the average number of available channels, and do not seem to have a meaningful physical interpretation. The shape of the curves did not constrain the parameter estimates for N, and the CVs were correspondingly large (Table 1B).
The second kinetic scheme we used for glycine receptors was developed for gating of receptors in zebrafish hindbrain (Legendre, 1998) (GlyLeg98; Fig. 2A). Macroscopic simulations (Fig. 2B) gave a concentration–peak response curve with EC50 of ∼306 μm and a maximum response of ∼137 pA, corresponding to Po,peak of 0.92 (Fig. 2C). With stochastic simulations at a series of agonist concentrations, the fraction of first openings that occurred after the peak location of the corresponding macroscopically simulated responses, varied from 0.3% to 2.8% (Fig. 2D). Non-stationary noise analysis of stochastic simulations reliably estimated both i and N with bias close to zero (Fig. 2E; Table 2A). The CV for estimates of both i and N was around 0.05, except for N at the two lowest concentrations (400 and 500 μm) where the Po,peak was lower (Table 2A).
Figure 2. Non-stationary noise analysis of responses generated by the GlyLeg98 kinetic scheme.
A, kinetic scheme for glycine receptor gating in Mauthner cells in zebrafish hindbrain (Legendre, 1998). U: unbound state; B1: singly liganded state; B2: doubly liganded state; O1: open state 1; B2rel: reluctant state; O2: open state 2. The rate constants were as follows (Legendre, 2001): kUB1= 12 × 106m−1 s−1, kB1U= 1452 s−1, kB1B2= 6 × 106m−1 s−1, kB2B1= 2904 s−1, kB2O1= 8938 s−1, kO1B2= 680 s−1, kB2B2rel= 536 s−1, kB2relB2= 136 s−1, kB2relO2= 3180 s−1, kO2B2rel= 1300 s−1. B, macroscopically simulated responses (as in Fig. 1B). C, concentration–response relationships (as in Fig. 1C). D, histograms of latency to first opening for stochastically simulated responses (as in Fig. 1D). E, variance versus mean curves for stochastically simulated responses after conventional non-stationary noise analysis (as in Fig. 1E). F, variance versus mean curves for stochastically simulated responses with trial-to-trial variation (Gaussian) in the number of available channels after conventional non-stationary noise analysis. G, variance versus mean curves for stochastically simulated responses after peak-scaled (PS) non-stationary noise analysis. H, variance versus mean curves for stochastically simulated responses with trial-to-trial variation in the number of available channels after peak-scaled (PS) non-stationary noise analysis.
Table 2.
GlyLeg98
| 400 μm | 500 μm | 1 mm | 3 mm | 5 mm | ||
|---|---|---|---|---|---|---|
| A | ||||||
| Parabolic fit | ||||||
| i | Mean ±s.d. (pA) | 2.98 ± 0.16 | 2.95 ± 0.20 | 2.96 ± 0.18 | 2.97 ± 0.11 | 2.97 ± 0.12 |
| CV | 0.053 (0.038) | 0.067 (0.059) | 0.061 (0.039) | 0.037 (0.039) | 0.039 (0.038) | |
| Bias (%) | −0.59 | −1.5 | −1.4 | −0.96 | −1.1 | |
| N | Mean ±s.d. | 50.0 ± 5.1 | 52.1 ± 6.9 | 50.8 ± 3.8 | 50.6 ± 2.4 | 50.4 ± 2.3 |
| CV | 0.103 (0.072) | 0.131 (0.100) | 0.074 (0.052) | 0.047 (0.043) | 0.045 (0.045) | |
| Bias (%) | 0.01 | 4.2 | 1.7 | 1.2 | 0.75 | |
| B | ||||||
| Linear fit | ||||||
| i | Mean ±s.d. (pA) | 2.90 ± 0.10 | 2.93 ± 0.09 | 3.00 ± 0.11 | 2.97 ± 0.16 | 3.01 ± 0.11 |
| CV | 0.033 (0.045) | 0.032 (0.039) | 0.036 (0.039) | 0.054 (0.037) | 0.036 (0.034) | |
| Bias (%) | −3.4 | −2.2 | 0.09 | − 1.1 | 0.22 | |
| Parabolic fit (after peak-scaling) | ||||||
| i | Mean ±s.d. (pA) | 2.98 ± 0.13 | 3.05 ± 0.15 | 3.07 ± 0.18 | 3.08 ± 0.08 | 3.07 ± 0.15 |
| CV | 0.043 (0.042) | 0.048 (0.039) | 0.059 (0.046) | 0.025 (0.044) | 0.047 (0.039) | |
| Bias (%) | −0.72 | 1.5 | 2.4 | 2.6 | 2.2 | |
| N | Mean ±s.d. | 32.2 ± 1.7 | 37.1 ± 2.3 | 47.2 ± 3.8 | 50.5 ± 1.9 | 51.4 ± 3.4 |
| CV | 0.054 (0.055) | 0.063 (0.052) | 0.081 (0.056) | 0.038 (0.061) | 0.067 (0.057) | |
A, GlyLeg98, simulations with constant number of channels (γ= 50 pS; i= 3 pA; N= 50; n= 1000). B, GlyLeg98, simulations with stochastic trial-to-trial variation of the number of channels (γ= 50 pS; i= 3 pA; N= 50; s.d.= 10; n= 1000).
We next repeated the simulations while varying the number of available channels from trial to trial. Without peak-scaling, the variance versus mean curves deviated strongly upwards (Fig. 2F), but curve fitting with a linear function to the initial part of the curve gave a good estimate for the unitary current (i; bias close to zero; CV around 0.05 or less; Table 2B). In contrast to the results obtained for the first glycine receptor scheme (GlyAG), peak-scaled non-stationary noise analysis resulted in parabolic variance versus mean curves (Fig. 2G and H). There was marked overlap of the curves for simulations with constant and variable numbers of available channels (Fig. 2G and H). Curve fitting with eqn (5), for responses simulated with a variable number of channels, gave good estimates for both i (bias close to zero) and N, with CVs around 0.05 (Table 2B). In this case N corresponded approximately to the average number of channels open at the peak, calculated by dividing the average peak response by the unitary current (i) obtained from curve fitting with eqn (5).
The third kinetic scheme we used for glycine receptors was developed for gating of heteromeric rat α1β receptors (heterologously expressed in HEK293 cells; Burzomato et al. 2004) (GlyBur04; Fig. 3A). In contrast to the two previous schemes, the rate constants of this scheme were estimated from single-channel current recordings as opposed to macroscopic current recordings. Macroscopic simulations (Fig. 3B) gave a concentration–peak response curve with EC50 of ∼1.28 mm and a maximum response of ∼144 pA, corresponding to Po,peak of 0.96 (Fig. 3C). With stochastic simulations at a series of agonist concentrations, the fraction of first openings that occurred after the peak location of the corresponding macroscopically simulated responses, varied from 0% to 21.5% (Fig. 3D). Non-stationary noise analysis of stochastic simulations reliably estimated both i and N (Fig. 3E; Table 3A). The CV for estimates of both i and N was around 0.05, except for N at the lowest concentration (1 mm) where the Po,peak was lower, and for i and N at the highest concentration (100 mm) where the Po,peak was very high (Table 3A). The bias was slightly larger than for the two first kinetic schemes (for both i and N), but for N it was considerably larger for the lowest concentration with lower Po,peak.
Figure 3. Non-stationary noise analysis of responses generated by the GlyBur04 kinetic scheme.
A, kinetic scheme for gating of heteromeric rat α1β glycine receptors (Burzomato et al. 2004). R: unbound state; AR, AF and AF*: singly liganded states; A2R, A2F and A2F*: doubly liganded states; A3R, A3F and A3F*: triply liganded states. Open channels indicated by an asterisk (F*). F indicates a distinct pre-opening conformation (‘flipped’; see Burzomato et al. 2004). The rate constants were as follows (Burzomato et al. 2004): α1= 3400 s−1, β1= 4200 s−1, α2= 2100 s−1, β2= 28000 s−1, α3= 6700 s−1, β3= 130000 s−1, γ1= 29266 s−1, δ1= 180 s−1, γ2= 18000 s−1, δ2= 6800 s−1, γ3= 948.1 s−1, δ3= 22000 s−1, k−= 302 s−1, k+= 0.59 × 106m−1 s−1, kF−= 1250 s−1, kF+= 150 × 106m−1 s−1. B, macroscopically simulated responses (as in Fig. 1B). Single-channel conductance (γ) identical for AF*, A2F* and A3F* (γ set to 50 pS for comparison with the other glycine receptor kinetic schemes). C, concentration–response relationships (as in Fig. 1C). D, histograms of latency to first opening for stochastically simulated responses (as in Fig. 1D). E, variance versus mean curves for stochastically simulated responses after conventional non-stationary noise analysis (as in Fig. 1E). F, variance versus mean curves for stochastically simulated responses with trial-to-trial variation (Gaussian) in the number of available channels after conventional non-stationary noise analysis. G, variance versus mean curves for stochastically simulated responses after peak-scaled (PS) non-stationary noise analysis. H, variance versus mean curves for stochastically simulated responses with trial-to-trial variation in the number of available channels after peak-scaled (PS) non-stationary noise analysis.
Table 3.
GlyBur04
| 1 mm | 3 mm | 5 mm | 100 mm | ||
|---|---|---|---|---|---|
| A | |||||
| Parabolic fit | |||||
| i | Mean ±s.d. (pA) | 2.87 ± 0.16 | 2.91 ± 0.15 | 2.92 ± 0.12 | 2.88 ± 0.02 |
| CV | 0.056 (0.052) | 0.052 (0.043) | 0.041 (0.032) | 0.009 (0.038) | |
| Bias (%) | −4.4 | −3.0 | −2.6 | −4.2 | |
| N | Mean ±s.d. | 61.7 ± 24.6 | 52.1 ± 3.1 | 51.7 ± 2.2 | 52.2 ± 0.6 |
| CV | 0.399 (0.177) | 0.059 (0.054) | 0.043 (0.036) | 0.011 (0.041) | |
| Bias (%) | 23.4 | 4.3 | 3.3 | 4.5 | |
| B | |||||
| Linear fit | |||||
| i | Mean ±s.d. (pA) | 3.08 ± 0.08 | 3.14 ± 0.11 | 3.12 ± 0.16 | 3.12 ± 0.14 |
| CV | 0.027 (0.041) | 0.034 (0.038) | 0.051 (0.041) | 0.046 (0.043) | |
| Bias (%) | 2.6 | 4.5 | 3.9 | 4.0 | |
| Parabolic fit (after peak-scaling) | |||||
| i | Mean ±s.d. (pA) | 2.84 ± 0.09 | 2.89 ± 0.11 | 2.83 ± 0.12 | 2.88 ± 0.10 |
| CV | 0.032 (0.046) | 0.039 (0.037) | 0.044 (0.042) | 0.036 (0.034) | |
| Bias (%) | −5.5 | −3.6 | −5.8 | −3.7 | |
| N | Mean ±s.d.) | 23.5 ± 1.0 | 49.9 ± 2.6 | 56.2 ± 3.0 | 55.2 ± 2.1 |
| CV | 0.041 (0.063) | 0.051 (0.048) | 0.053 (0.050) | 0.038 (0.040) | |
A, GlyBur04, simulations with constant number of channels (γ= 50 pS; i= 3 pA; N= 50; n= 1000). B, GlyBur04, simulations with stochastic trial-to-trial variation of the number of channels (γ= 50 pS; i= 3 pA; N= 50; s.d.= 10; n= 1000).
When the simulations were repeated with a varying number of available channels from trial to trial, non-stationary noise analysis resulted in variance versus mean curves that deviated strongly upwards (Fig. 3F), but curve fitting with a linear function to the initial part of the curve gave a good estimate for the unitary current (i; bias consistently above zero; CV around 0.05 or less; Table 3A). Similar to the GlyLeg98 kinetic scheme, peak-scaled non-stationary noise analysis resulted in parabolic variance versus mean curves (Fig. 3G and H). There was marked overlap of the curves for simulations with a constant (Fig. 3G) and variable (Fig. 3H) number of available channels. Curve fitting with eqn (5), for responses simulated with a variable number of channels, gave reasonably good estimates for both i (bias consistently below zero) and N, with CVs at 0.05 or less (Table 3B). In this case, N corresponded approximately to the average number of channels open at the peak, calculated by dividing the average peak response by the unitary current (from the parabolic fit). The reasons for the different shapes of the variance versus mean curves obtained with peak-scaled non-stationary noise analysis of simulations based on the different kinetic schemes for glycine receptors will be discussed below.
Glutamate receptor kinetic schemes
The first scheme used for simulation of glutamate receptors was that of Geiger et al. (1999) for AMPA receptor gating in hippocampal interneurons (GluGei99; Fig. 4A). It is a modification of a scheme for AMPA receptor gating in hippocampal pyramidal cells (Jonas et al. 1993). Macroscopic simulations (Fig. 4B) gave a concentration–peak response curve with EC50 of ∼1 mm and a maximum response of ∼18.6 pA (Po,peak= 0.73; Fig. 4C). Stochastic simulations with a series of different agonist concentrations yielded fractions of first openings occurring after the peak of the corresponding macroscopically simulated responses varying from 5.9% to 9.5% (Fig. 4D). Stochastic simulations followed by non-stationary noise analysis were used to obtain estimates for i and N (bias close to zero; Fig. 4E; Table 4A). The CVs were typically between 0.05 and 0.1, except for the estimate of N at the lowest agonist concentration (1 mm), where Po,peak was less than 0.5. With stochastic simulations and trial-to-trial variation in the number of available channels, conventional non-stationary noise analysis resulted in upwardly deviating, non-parabolic variance versus mean curves (Fig. 4F). A linear fit to the initial part of the curves gave estimates for the unitary current i, with bias consistently larger than zero and CVs less than 0.05 (Table 4B). With peak-scaled non-stationary noise analysis, we obtained skewed variance versus mean curves for all concentrations tested (Fig. 4G and H), very similar to those obtained with the GlyAG kinetic scheme. Curve fitting with eqn (5), for responses simulated with a variable number of channels, to an appropriate range of the curves yielded estimates for the unitary current (i; bias close to zero) and N (Table 4B), but the estimates for N had no obvious physical interpretation. For i, the CVs were around 0.05, but for N they were larger, particularly for the lower concentrations of agonist (Table 4B).
Figure 4. Non-stationary noise analysis of responses generated by the GluGei99 kinetic scheme.
A, kinetic scheme proposed for AMPA receptor gating in hippocampal interneurons (Geiger et al. 1999). C0: unbound state; C1: singly liganded state; C2: doubly liganded state; O: open state; C3: singly liganded, desensitised state; C4 and C5: doubly liganded, desensitised states. The rate constants were as follows: kC0C1= 17.1 × 106m−1 s−1, kC1C0= 157 s−1, kC1C2= 3.24 × 106 M−1 s−1, kC2C1= 3.76 × 103 s−1, kC2O= 14.9 × 103 s−1, kOC2= 4.00 × 103 s−1, kC1C3= 1.53 × 103 s−1, kC3C1= 408 s−1, kC2C4= 502 s−1, kC4C2= 0.377 s−1, kOC5= 121 s−1, kC5O= 191 s−1, kC3C4= 0.611 × 106m−1 s−1, kC4C3= 2 s−1, kC4C5= 1.59 × 103 s−1, kC5C4= 899 × 103 s−1. B, macroscopically simulated responses (as in Fig. 1B, except γ= 8.5 pS). C, concentration–response relationships (as in Fig. 1C). D, histograms of latency to first opening for stochastically simulated responses (as in Fig. 1D). E, variance versus mean curves for stochastically simulated responses after conventional non-stationary noise analysis (as in Fig. 1E). F, variance versus mean curves for stochastically simulated responses with trial-to-trial variation (Gaussian) in the number of available channels after conventional non-stationary noise analysis. G, variance versus mean curves for stochastically simulated responses after peak-scaled (PS) non-stationary noise analysis. H, variance versus mean curves for stochastically simulated responses with trial-to-trial variation in the number of available channels after peak-scaled (PS) non-stationary noise analysis.
Table 4.
GluGei99
| 1 mm | 3 mm | 5 mm | 100 mm | ||
|---|---|---|---|---|---|
| A | |||||
| Parabolic fit | |||||
| i | Mean ±s.d. (pA) | 0.51 ± 0.04 | 0.52 ± 0.03 | 0.50 ± 0.02 | 0.52 ± 0.02 |
| CV | 0.078 (0.093) | 0.062 (0.049) | 0.044 (0.057) | 0.046 (0.043) | |
| Bias (%) | −0.34 | 2.1 | −1.9 | 1.6 | |
| N | Mean ±s.d. | 52.7 ± 14.9 | 48.4 ± 4.6 | 51.2 ± 3.4 | 49.0 ± 3.3 |
| CV | 0.284 (0.339) | 0.094 (0.093) | 0.066 (0.092) | 0.067 (0.056) | |
| Bias (%) | 5.4 | −3.1 | 2.5 | −1.9 | |
| B | |||||
| Linear fit | |||||
| i | Mean ±s.d. (pA) | 0.52 ± 0.02 | 0.53 ± 0.01 | 0.55 ± 0.03 | 0.56 ± 0.02 |
| CV | 0.029 (0.038) | 0.026 (0.048) | 0.047 (0.042) | 0.036 (0.039) | |
| Bias (%) | 1.0 | 3.8 | 6.8 | 10.7 | |
| Parabolic fit (after peak-scaling) | |||||
| i | Mean ±s.d. (pA) | 0.52 ± 0.03 | 0.52 ± 0.04 | 0.51 ± 0.02 | 0.52 ± 0.03 |
| CV | 0.053 (0.084) | 0.074 (0.064) | 0.039 (0.052) | 0.049 (0.058) | |
| Bias (%) | 0.99 | 1.7 | −0.15 | 1.4 | |
| N | Mean ±s.d. | 42.2 ± 6.8 | 62.4 ± 12.8 | 61.4 ± 5.8 | 61.3 ± 6.3 |
| CV | 0.161 (0.417) | 0.205 (0.179) | 0.094 (0.118) | 0.102 (0.113) | |
A, GluGei99, simulations with constant number of channels (γ= 8.5 pS; i= 0.51 pA; N= 50; n= 1000). B, GluGei99, simulations with stochastic trial-to-trial variation of the number of channels (γ= 8.5 pS; i= 0.51 pA; N= 50; s.d.= 10; n= 1000).
The next kinetic scheme for AMPA receptors was used in the analysis of gating of receptors in cerebellar Purkinje cells by Momiyama et al. (2003) (GluMom03; Fig. 5A). It is identical to the scheme developed by Häusser & Roth (1997), except for a single-channel conductance of 5 pS instead of 8 pS. The concentration–peak response curve obtained with macroscopic simulations (Fig. 5B), displayed an EC50 of ∼456 μm and a maximum peak response of 11.3 pA (Po,peak= 0.75; Fig. 5C). Stochastic simulations with agonist pulses at a series of concentrations resulted in fractions of first openings occurring after the peak response of the corresponding macroscopically simulated responses ranging from 1.5% to 7.1% (Fig. 5D).
Figure 5. Non-stationary noise analysis of responses generated by the GluMom03 kinetic scheme.
A, kinetic scheme proposed for AMPA receptor gating in cerebellar Purkinje cells (Momiyama et al. 2003; original scheme by Häusser & Roth, 1997). C0: unbound state; C1: singly liganded state; C2: doubly liganded state; O: open state; C3: singly liganded, desensitised state; C4, C5, C6 and C7: doubly liganded, desensitised states. The rate constants were as follows: kC0C1= 13.66 × 106m−1 s−1, kC1C0= 2093 s−1, kC1C2= 6.019 × 106m−1 s−1, kC2C1= 4.719 × 103 s−1, kC2O= 17.23 × 103 s−1, kOC2= 3.734 × 103 s−1, kOC7= 114.1 s−1, kC7O= 90.47 s−1, kC1C3= 4.219 × 102 s−1, kC3C1= 31.15 s−1, kC2C4= 855.3 s−1, kC4C2= 46.65 s−1, kOC5= 3.108 s−1, kC5O= 0.6912 s−1, kC7C6= 18.78 s−1, kC6C7= 0.3242 s−1, kC3C4= 6.019 × 106m−1 s−1, kC4C3= 3.486 × 103 s−1, kC4C5= 476.4 s−1, kC5C4= 420.9 s−1, kC5C6= 1.034 × 104 s−1, kC6C5= 636.3 s−1. B, macroscopically simulated responses (as in Fig. 1B, except γ= 5 pS). C, concentration–response relationships (as in Fig. 1C). D, histograms of latency to first opening for stochastically simulated responses (as in Fig. 1D). E, variance versus mean curves for stochastically simulated responses after conventional non-stationary noise analysis (as in Fig. 1E). F, variance versus mean curves for stochastically simulated responses with trial-to-trial variation (Gaussian) in the number of available channels after conventional non-stationary noise analysis. G, variance versus mean curves for stochastically simulated responses after peak-scaled (PS) non-stationary noise analysis. H, variance versus mean curves for stochastically simulated responses with trial-to-trial variation in the number of available channels after peak-scaled (PS) non-stationary noise analysis.
Non-stationary noise analysis of stochastic simulations at the same series of agonist concentrations, resulted in variance versus mean curves displayed in Fig. 5E and reasonably good estimates for i and the number of available channels (N; Table 5A). The bias for i tended to be negative, while the bias for N tended to be positive. The CV was typically between 0.05 and 0.1, except at the lowest agonist concentration (300 μm) where Po,peak was less than 0.5. We next repeated the simulations with stochastic variation of the number of available channels from trial to trial. Without peak-scaling, a line fit to the initial part of the upwardly deviating variance versus mean curves (Fig. 5F) gave a reasonably good estimate for i, with bias consistently larger than zero and CVs around 0.05 (Table 5B). With peak-scaled non-stationary noise analysis, the variance versus mean curves had a skewed shape (Fig. 5G and H), similar to the results obtained with the GlyAG and GluGei99 kinetic schemes. Curve fitting with eqn (5), for responses simulated with a variable number of channels, to an appropriate range of the curves yielded estimates for i and N, but the estimates for N had no obvious physical interpretation. For i the bias tended to be negative and the CVs were around 0.05. For N the CVs were larger, particularly for the lower concentrations of agonist (Table 5B).
Table 5.
GlyMom03
| 300 μm | 1 mm | 3 mm | 5 mm | 100 mm | ||
|---|---|---|---|---|---|---|
| A | ||||||
| Parabolic fit | ||||||
| i | Mean ±s.d. (pA) | 0.30 ± 0.02 | 0.29 ± 0.01 | 0.27 ± 0.02 | 0.29 ± 0.01 | 0.28 ± 0.02 |
| CV | 0.068 (0.050) | 0.050 (0.054) | 0.072 (0.051) | 0.042 (0.047) | 0.054 (0.034) | |
| Bias (%) | 0.52 | −2.0 | −8.6 | −4.7 | −6.5 | |
| N | Mean ±s.d. | 53.6 ± 18.3 | 50.3 ± 4.2 | 57.9 ± 6.4 | 53.8 ± 3.1 | 55.6 ± 3.7 |
| CV | 0.341 (0.243) | 0.083 (0.091) | 0.111 (0.079) | 0.058 (0.069) | 0.066 (0.055) | |
| Bias (%) | 7.1 | 0.56 | 15.9 | 7.6 | 11.2 | |
| B | ||||||
| Linear fit | ||||||
| i | Mean ±s.d. (pA) | 0.31 ± 0.01 | 0.31 ± 0.02 | 0.31 ± 0.02 | 0.31 ± 0.01 | 0.32 ± 0.01 |
| CV | 0.042 (0.042) | 0.050 (0.039) | 0.052 (0.036) | 0.035 (0.043) | 0.031 (0.038) | |
| Bias (%) | 2.2 | 1.8 | 4.3 | 3.9 | 6.7 | |
| Parabolic fit (after peak-scaling) | ||||||
| i | Mean ±s.d. (pA) | 0.30 ± 0.02 | 0.29 ± 0.02 | 0.29 ± 0.02 | 0.29 ± 0.01 | 0.30 ± 0.02 |
| CV | 0.061 (0.043) | 0.076 (0.059) | 0.064 (0.061) | 0.046 (0.061) | 0.058 (0.052) | |
| Bias (%) | 0.06 | −4.3 | −2.5 | −2.5 | −0.32 | |
| N | Mean ±s.d. | 19.2 ± 2.2 | 66.6 ± 16.8 | 65.4 ± 7.1 | 63.5 ± 6.2 | 60.0 ± 5.5 |
| CV | 0.115 (0.075) | 0.252 (0.184) | 0.108 (0.124) | 0.097 (0.136) | 0.091 (0.098) | |
A, GlyMom03, simulations with constant number of channels (γ= 5 pS; i= 0.3 pA; N= 50; n= 1000). B, GlyMom03, simulations with stochastic trial-to-trial variation of the number of channels (γ= 5 pS; i= 0.3 pA; N= 50; s.d.= 10; n= 1000).
Peak-scaled non-stationary noise analysis and parabolic or skewed variance versus mean curves
As predicted by theory (Sigworth, 1980), all five kinetic schemes examined above gave parabolic variance versus mean curves following conventional non-stationary noise analysis when simulations were performed with a constant number of available channels. When the number of available channels was varied from trial to trial, in order to simulate one source of quantal variability, conventional non-stationary noise analysis gave non-parabolic, upwardly deviating variance versus mean curves. With peak-scaled non-stationary noise analysis, the variance versus mean curves for simulations with a constant number of channels overlapped closely with the corresponding curves for simulations with varying number of channels. This was the case for all kinetic schemes examined, and indicates that peak-scaling was consistently effective in reducing the increased variance caused by stochastically varying the number of channels. However, peak-scaled non-stationary noise analysis resulted in skewed variance versus mean curves for three of the kinetic schemes (GlyAG, GluGei99 and GluMom03), and parabolic curves for the other two kinetic schemes (GlyLeg98 and GlyBur04). Only with parabolic curves did curve fitting with eqn (5) give an estimate for N that corresponded (approximately) to the average number of channels open at the peak of the response. It was not immediately clear which properties of the kinetic schemes could account for these differences in behaviour. For the kinetic schemes we analysed, the differences with respect to the fraction of first openings occurring after the peak of the mean response (Traynelis et al. 1993; Momiyama et al. 2003) did not seem large or consistent enough to satisfactorily explain the presence of a skewed or parabolic variance versus mean relationship (Figs 1D, 2D, 3D, 4D, 5D).
We first speculated that the parabolic shape of the variance versus mean curves for GlyLeg98 and GlyBur04 was the consequence of low variability with respect to peak location of the individual events relative to the stimulus timing (low jitter). Indeed, in histograms of the peak location, both GlyLeg98 and GlyBur04 displayed relatively narrow distributions (Fig. 6B and C), whereas GlyAG displayed a relatively broad distribution (Fig. 6A). However, GluGei99 and GluMom03 also displayed narrow distributions (Fig. 6D and E) very similar to those of GlyLeg98 and GlyBur04, but peak-scaled non-stationary noise analysis produced skewed variance versus mean curves (Fig. 4G and H and 5G and H). Thus, low variability of peak location is not sufficient to generate a parabolic variance versus mean curve. Nevertheless, we found that introducing artificial jitter between the traces in a simulated dataset could result in a markedly skewed variance versus mean relationship (see below), for both conventional and peak-scaled non-stationary noise analysis.
Figure 6. Distribution of peak location for individual, stochastically simulated responses for different kinetic schemes.
A–E, histograms (binwidth 100 μs) of peak location for stochastically simulated responses (N = 50; n = 1000), 1 ms agonist pulses at the concentrations indicated. Horizontal bar in each panel indicates time course of agonist application and white vertical line in each histogram indicates peak location of mean response (macroscopically simulated) at the same agonist concentration. A, GlyAG kinetic scheme for glycine receptor gating. B, GlyLeg98 kinetic scheme for glycine receptor gating. C, GlyBur04 kinetic scheme for glycine receptor gating. D, GluGei99 kinetic scheme for glutamate receptor gating. E, GluMom03 kinetic scheme for glutamate receptor gating.
Relation between kinetic properties and results from peak-scaled non-stationary noise analysis for a simple three-state model
We next attempted to identify properties intrinsic to the kinetic schemes that could account for the different effects of peak-scaling on the resulting variance versus mean curves (skewed versus parabolic). We took advantage of the similarities between two of the glycine receptor kinetic schemes, GlyAG (Fig. 1A) and GlyLeg98 (Fig. 2A). We first simplified the kinetic schemes by eliminating the ‘branching’ states in each, leaving only the four states U (unbound), B1 (singly liganded), B2 (doubly liganded), and O (open), without changing the rate constants between the remaining states. This had no effect on the parabolic (GlyLeg98) or skewed (GlyAG) variance versus mean relationships after peak-scaled non-stationary noise analysis (not shown).
To systematically explore the potential influence of specific rate constants on the shape of the variance versus mean curve generated by peak-scaled non-stationary noise analysis, we further simplified the kinetic scheme to a three-state scheme (S3) with only one binding step (Fig. 7A). This scheme is similar to that used previously by, e.g. del Castillo & Katz (1957) and Colquhoun & Hawkes (1977). It has four rate constants: binding (k+1), unbinding (k−1), opening (β) and closing (α). In order to compare simulation results for each modification of this scheme under equivalent conditions, we first simulated macroscopic responses to generate concentration–response curves and calculated the agonist concentration that resulted in a Po,peak of 0.5. For each modification and combination of rate constants, we then simulated, stochastically, responses to 1 ms agonist pulses at the appropriate concentration and performed non-stationary noise analysis. When the data were analysed without peak-scaling, all curves lined up along the theoretically predicted curve corresponding to eqn (5) (Fig. 7B). With peak-scaling, the resulting variance versus mean curves displayed large variability with respect to their shape, ranging from parabolic to markedly skewed (Fig. 7C–F). For each series of simulations, we varied one rate constant while keeping the other three fixed. After exploring a large number of combinations, we concluded that α, β and k−1 (Fig. 7C–E), but not k+1 (Fig. 7F), can influence the shape of the variance versus mean relationship after peak-scaling. Figure 7C (S3mod30–mod34) shows an example where increasing β incrementally from 1000 s−1 to 9000 s−1 changed the variance versus mean curves from skewed to increasingly parabolic. Figure 7D (S3mod24–mod29) shows an example where decreasing α incrementally from 2000 s−1 to 100 s−1 changed the variance versus mean curves from skewed to increasingly parabolic. Figure 7E (S3mod7–mod11) shows an example where increasing k−1 incrementally from 100 s−1 to 3000 s−1 changed the variance versus mean curves from skewed to increasingly parabolic. We were not able to find any combination of rate constants where varying k+1 had an effect on the shape (skewed versus parabolic) of the resulting variance versus mean curves. Figure 7F illustrates an example where k+1 was changed from the original value of 6 × 106m−1 s−1 to 2 × 104m−1 s−1, 1.2 × 107m−1 s−1 or 1.2 × 109m−1 s−1 for two different combinations of α, β and k−1. We also noted that the effect of a given change of a specific rate constant depended on the values of the other rate constants. For example, varying k−1 from 100 s−1 to 3000 s−1, as in Fig. 7E, but with α= 100 s−1 instead of 1000 s−1 (β= 1000 s−1 for both cases), resulted in a parabolic variance versus mean relationship for each case (not shown). When corresponding modifications were applied to a four-state kinetic scheme (with two identical binding steps), we observed similar results, indicating that the rate constants α, β and k−1 affected the shape of the variance versus mean relationship (not shown).
Figure 7. Relation between rate constants of three-state kinetic scheme and skewed or parabolic varianceversusmean relationship after peak-scaled non-stationary noise analysis.
A, simple three-state kinetic scheme for ligand-gated channel. U: unbound state; B: singly liganded state; O: open state. Rate constants (k+1= binding; k−1= unbinding; β= opening; α= closing) varied in order to examine influence on shape (skewed versus parabolic) of variance versus mean relationship after peak-scaled non-stationary noise analysis. B, variance versus mean curves for stochastically simulated responses after conventional non-stationary noise analysis (N = 50; n = 1000), 1 ms agonist pulses with concentration adjusted to give Po,peak= 0.5. C–F, variance versus mean curves for stochastically simulated responses after peak-scaled (PS) non-stationary noise analysis. C, S3mod30–mod34, β varied; mod30: 1000 s−1; mod31: 3000 s−1; mod32: 4000 s−1; mod33: 6000 s−1; mod34: 9000 s−1; α (750 s−1), k−1 (100 s−1) and k+1 (6 × 106m−1 s−1) constant. D, S3mod24–mod29, α varied; mod29: 2000 s−1; mod24: 1000 s−1; mod25: 750 s−1; mod26: 500 s−1; mod27: 250 s−1; mod28: 100 s−1; β (2000 s−1), k−1 (100 s−1) and k+1 (6 × 106m−1 s−1) constant. E, S3mod7–mod11 (agonist concentration was 100 mm as the maximum attainable Po,peak was less than 0.5), k−1 varied; mod7: 100 s−1; mod8: 500 s−1; mod9: 1000 s−1; mod10: 2000 s−1; mod11: 3000 s−1; α (1000 s−1), β (1000 s−1) and k+1 (6 × 106m−1 s−1) constant. F, S3mod43–mod48, for S3mod43–mod45, k+1 varied; mod43: 2 × 104m−1 s−1; mod44: 1.2 × 107m−1 s−1; mod45: 1.2 × 109m−1 s−1; α (5000 s−1), β (9000 s−1) and k−1 (100 s−1) constant. For S3mod46–mod48, k+1 varied; mod46: 2 × 104m−1 s−1; mod47: 1.2 × 107m−1 s−1; mod48: 1.2 × 109m−1 s−1; α (1000 s−1), β (9000 s−1) and k−1 (100 s−1) constant. G, plot of Popen within burst versus peak variance of variance versus mean relationship after peak-scaled non-stationary noise analysis for various modifications of the three-state kinetic scheme (C–F). Data points have been fitted with a straight line. Popen within burst calculated by eqn (1). Kinetic schemes with Popen (within burst) >0.9 generate increasingly parabolic variance versus mean relationships (e.g. S3mod28) and kinetic schemes with Popen (within burst) < 0.9 generate increasingly skewed variance versus mean relationships (e.g. S3mod7). H, two isosurfaces representing constant scalar values (iso-values) for Popen within bursts (top isosurface at iso-value of 0.9; bottom isosurface at iso-value of 0.95) in a three-dimensional scalar distribution of the rate constants α, β and k−1. Notice the large range (1–50000 s−1) included for each rate constant.
The observations detailed above could be analysed more systematically when we realised that the shape (skewed versus parabolic) of the variance versus mean relationship was correlated with the Popen within bursts of single-channel openings for each variant of the three-state kinetic scheme (Fig. 7G). For this scheme the Popen within bursts could be directly calculated as a function of the rate constants α, β and k−1 according to eqn (1). When Popen within bursts was less than ∼0.9, the variance versus mean relationship became increasingly skewed. When the Popen within bursts was higher (e.g. >0.9), the relationship became increasingly parabolic. The relationship between the three rate constants and Popen within bursts can be visualised by displaying two isosurfaces for Popen with isovalues of 0.9 (top surface) and 0.95 (bottom surface) in a three-dimensional scalar distribution with each rate constant spanning a range from 1 to 50000 s−1 (Fig. 7H). The complex relationship between Popen within bursts and the rate constants α, β and k−1 explains our observation of a context-dependent effect of varying a particular rate constant with respect to the effect on the shape of the variance versus mean relationship. The parameter combinations explored in Fig. 7C–F can be qualitatively visualised and confirmed from the shape of the isosurfaces in Fig. 7H. However, the shape of the isosurfaces indicates that a high Popen can also be obtained for very low values of β. We confirmed this by running simulations for the combination of rate constants used in Fig. 7C while reducing β to very low values. This condition increased Popen within bursts by reducing the mean burst length such that it approached the single-channel mean open time, thus its general significance is reduced.
Covariance analysis of responses obtained with a simple three-state kinetic model
In order for peak-scaling to generate a parabolic variance versus mean curve, it has to reduce the variance not only at the peak (given that Po,peak < 1), but also for response values close to the peak during the decay phase (e.g. S3mod34 in Fig. 7C). We hypothesised that the important property was not the magnitude of the variance itself, but the correlation between the fluctuations at the peak and the fluctuations during the decay phase. Depending on the degree of correlation, peak-scaling might result in a decrease or an increase in the variance compared to the theoretical variance versus mean curve obtained without peak-scaling according to eqn (5) (for no trial-to-trial variation in the number of available channels). With increasing extent of correlation, peak-scaling should make the variance versus mean relationship increasingly parabolic. Non-stationary autocovariance (or, simply covariance) functions have been analysed in considerable detail by Sigworth (1981, 1984), and can be considered as a generalisation of the variance that characterises not only the size of the fluctuations, but also their temporal structure. The covariance function (in the time domain) is equivalent to the power spectral density function in the frequency domain (e.g. Bendat & Piersol, 2000) and indicates how well the fluctuations are correlated at different times. Colquhoun & Hawkes (1977) derived analytical expressions demonstrating that for a kinetic scheme with k states, the covariance function, like the current relaxation, will decay exponentially with k−1 time constants (or correlation times). For a scheme with two states, e.g. C (closed) and O (open), the covariance function will decay as a single exponential function, and the correlation time will be equal to the mean open time (Colquhoun & Hawkes, 1977). For the three-state scheme we analysed above, the covariance function will decay as a double exponential function, and the two correlation times will be equal to the two time constants of the burst length distribution (Colquhoun & Hawkes, 1977). Based on these insights, we hypothesised that kinetic schemes resulting in skewed or parabolic variance versus mean relationships should display covariance functions (centred at the location of the peak of the mean response; t1=t2= time of peak of the ensemble mean current) with rapid and slow decay, respectively. The terms rapid and slow are, however, relative to the decay of the macroscopic responses. For example, if the covariance function displays a fast decay, the variance versus mean relationship might still be parabolic as long as the macroscopic response also displays a correspondingly fast decay. The same decay of the covariance function would give rise to a skewed variance versus mean relationship if the macroscopic response decays more slowly. In addition, for a given kinetic scheme the decay of the covariance function should also depend on Po,peak (and therefore on agonist concentration). One obvious reason for this conclusion is that peak-scaling of responses with high Po,peak (approaching 1) cannot reduce the variance below zero, corresponding to eqn (5)(e.g. Figure 2H). Accordingly, we calculated covariance functions for all the modifications of the S3 kinetic schemes in order to see how well the fluctuations were correlated at different times.
Figure 8 shows results for S3mod37 and S3mod39, two variants of the three-state kinetic scheme where the macroscopic responses displayed similar slow decays, but where S3mod39 generated a parabolic and S3mod37 a skewed variance versus mean relationship. For the condition with the agonist concentration adjusted to give Po,peak= 0.5, the figure shows the ensemble mean currents (Fig. 8A and B), the two-dimensional covariance functions (Fig. 8C and D), as well as the ensemble variance functions (σ2(t)) and individual covariance functions (C(tc,t)) with centre point corresponding to the location of the peak response of the ensemble mean current (Fig. 8G and H). It can be seen that for S3mod37, with a skewed variance versus mean curve after peak-scaling (Fig. 8N), the covariance function decayed rapidly (Fig. 8D and H). This contrasts with the slow decay of the covariance function resulting from S3mod39 (Fig. 8C and G) that gave rise to a parabolic variance versus mean curve after peak-scaling (Fig. 8M). In addition, for both kinetic schemes, we illustrate ensemble variance functions (σ2(t)) and individual covariance functions (C(tc,t)) with centre point corresponding to the location of the peak response of the ensemble mean current for three additional conditions, corresponding to higher (Fig. 8E and F) and lower (Fig. 8I–L) Po,peak.
Figure 8. Covariance functions for two modifications of three-state kinetic scheme resulting in macroscopic responses with similar, slow decays, but either parabolic or skewed varianceversusmean relationships after peak-scaled non-stationary noise analysis.
A, ensemble mean response for S3mod39 (k+1= 6 × 106m−1 s−1, k−1= 200 s−1, α= 1000 s−1, β= 9000 s−1), τdecay as indicated. Here and in B, 1 ms agonist pulses, concentration adjusted to give Po,peak= 0.5 (N = 50; n = 1000). Time scale as in C. B, ensemble mean response for S3mod37 (k+1= 6 × 106m−1 s−1, k−1= 50 s−1, α= 5000 s−1, β= 9000 s−1), τdecay as indicated. Time scale as in D. Notice similar decay of responses in A and B. C, two-dimensional covariance function for S3mod39 responses (A), variance coded according to colour bar (right), diagonal (top left to bottom right) corresponds to ensemble variance (G). Notice slow decay of covariance function. D, two-dimensional covariance function for S3mod37 responses (B), variance coded according to colour bar (left), diagonal (top left to bottom right) corresponds to ensemble variance (H). Notice rapid decay of covariance function. E,G,I,K, variance (black trace) for S3mod39 response ensemble, covariance function (red trace) with centre point (tc or t1) corresponding to location of peak response of ensemble mean. Double exponential fit (broken line) overlaid on each covariance function, the two time constants were 98 μs and 50.9 ms (calculated from the QEE submatrix of the Q matrix), with amplitude contributions depending on agonist concentration (amplitude-weighted decay time constant, τw, indicated in each panel). Here and below, agonist concentration indicated in the graph. F,H,J,L, variance (black trace) for S3mod37 response ensemble, covariance function (red trace) with centre point (tc or t1) corresponding to location of peak response of ensemble mean. Double exponential fit (broken line) overlaid on each covariance function, the two time constants were 71 μs and 56.1 ms, with amplitude contributions depending on agonist concentration (amplitude-weighted decay time constant, τw, indicated in each panel). M, variance versus mean curves after peak-scaled (PS) non-stationary noise analysis for S3mod39 (Po,peak as in E,G,I and K). N, variance versus mean curves after peak-scaled (PS) non-stationary noise analysis for S3mod37 (Po,peak as in F,H,J and L).
Figure 9 shows corresponding results for S3mod5 and S3mod7, two variants of the three-state kinetic scheme where the macroscopic responses displayed similar faster decays, but where S3mod5 generated a parabolic and S3mod7 a skewed variance versus mean relationship. For the condition with the agonist concentration adjusted to give Po,peak= 0.5, the figure shows the ensemble mean currents (Fig. 9A and B), the two-dimensional covariance functions (Fig. 9C and D), as well as the ensemble variance functions (σ2(t)) and individual covariance functions (C(tc,t)) with centre point corresponding to the location of the peak response of the ensemble mean current (Fig. 9G and F). It can be seen that for S3mod7, with a skewed variance versus mean curve after peak-scaling (Fig. 9N), the covariance function decayed rapidly (Fig. 9D and F). This contrasts with the slower decay of the covariance function resulting from S3mod5 (Fig. 9C and G) that gave a parabolic variance versus mean curve after peak-scaling (Fig. 9M). For both kinetic schemes, we illustrate ensemble variance functions (σ2(t)) and individual covariance functions (C(tc,t)) with centre point corresponding to the location of the peak response of the ensemble mean current for three additional conditions, corresponding to higher (Fig. 9E) and lower (Fig. 9H–L) Po,peak.
Figure 9. Covariance functions for two modifications of three-state kinetic scheme resulting in macroscopic responses with similar, fast decays, but either parabolic or skewed varianceversusmean relationships after peak-scaled non-stationary noise analysis.
A, ensemble mean response for S3mod5 (k+1= 6 × 106m−1 s−1, k−1= 1000 s−1, α= 100 s−1, β= 1000 s−1), τdecay as indicated. Here and in B, 1 ms agonist pulses, concentration adjusted to give Po,peak= 0.5 (N = 50; n = 1000). Time scale as in C. B, ensemble mean response for S3mod7 (k+1= 6 × 106m−1 s−1, k−1= 100 s−1, α= 1000 s−1, β= 1000 s−1), τdecay as indicated. Time scale as in D. Notice similar decay of responses in A and B. C, two-dimensional covariance function for S3mod5 responses (A), variance coded according to colour bar (right), diagonal (top left to bottom right) corresponds to ensemble variance (G). Notice slow decay of covariance function. D, two-dimensional covariance function for S3mod7 responses (B), variance coded according to colour bar (left), diagonal (top left to bottom right) corresponds to ensemble variance (F). Notice rapid decay of covariance function. E,G,I,K, variance (black trace) for S3mod5 response ensemble, covariance function (red trace) with centre point (tc or t1) corresponding to location of peak response of ensemble mean. Double exponential fit (broken line) overlaid on each covariance function, the two time constants were 488 μs and 20.5 ms (calculated from the QEE submatrix of the Q matrix), with amplitude contributions depending on agonist concentration (amplitude-weighted decay time constant, τw, indicated in each panel). Here and below, agonist concentration indicated in the graph. F,H,J,L, variance (black trace) for S3mod7 response ensemble, covariance function (red trace) with centre point (tc or t1) corresponding to location of peak response of ensemble mean. Double exponential fit (broken line) overlaid on each covariance function, the two time constants were 488 μs and 20.5 ms (identical to S3mod5), with amplitude contributions depending on agonist concentration (amplitude-weighted decay time constant, τw, indicated in each panel). M, variance versus mean curves after peak-scaled (PS) non-stationary noise analysis for S3mod5 (Po,peak as in E,G,I and K). N, variance versus mean curves after peak-scaled (PS) non-stationary noise analysis for S3mod7 (Po,peak as in F,H,J and L).
For each condition (with respect to Po,peak) tested, we verified that the decay of the non-stationary covariance function centred at the location of the peak of the mean response was well described by a double exponential function with time constants equal to the time constants of the burst length distribution (see above). The amplitude contributions of the time constants changed with increasing agonist concentration, corresponding to increasing Po,peak, resulting in an overall faster decay of the covariance function. This corresponded to an increase in the amplitude contribution of the faster of the two time constants and is shown by the traces overlaid on each covariance function in Figs 8E–L and 9E–L. We quantified this by calculating an amplitude-weighted time constant for each condition (τw). Overall, increasing agonist concentration was accompanied by a decrease in τw (Figs 8E–L and 9E–L). For all other modifications of the three-state kinetic scheme tested, we verified for two conditions (Po,peak= 0.5 and Po,peak∼ 0.001) that the decay of the non-stationary covariance function centred at the location of the peak of the mean response was well described by a double exponential function with time constants equal to the time constants of the burst length distribution. Irrespective of whether a variant of the kinetic scheme gave rise to skewed or parabolic variance versus mean relationships, the decay of the covariance function became faster with increasing agonist concentration, corresponding to increasing Po,peak.
For the special case of a concentration jump to zero concentration (as used here), the decay of the macroscopic response has kE, rather than k−1 (see above), time constants, with kE being the number of states in the subset of all states that can be occupied during a burst (Colquhoun et al. 1997; Wyllie et al. 1998). These time constants are the time constants of the burst length distribution and are determined by the rate constants of QEE (equal to the reciprocals of the eigenvalues of −QEE), with QEE being the burst state submatrix of the Q matrix. We assume that in general these time constants also determine the decay of the covariance function after a step to zero concentration. For the three-state kinetic scheme used here, the relevant rate constants are α, β and k−1. This verifies why these rate constants, and not k+1, had an influence on the shape of the variance versus mean relationship (see above). We assume that the changing amplitude contributions of the two time constants with increasing agonist concentration and Po,peak, corresponding to an overall faster decay of the covariance function, can be explained by a changing state of the system at the peak, resulting in a changing initialisation vector (see Appendix in Wyllie et al. 1998). For kinetic schemes with channel activations that include partially and fully liganded open and closed states, e.g. GlyBur04, the situation will be more complicated because the submatrix QEE and the corresponding burst length distribution will be concentration dependent (Wyllie et al. 1998). Nevertheless, the time constants for a jump to zero concentration will be the same as the time constants of the burst length distribution at ‘zero’ concentration, i.e. in practice the burst length distribution at a very low concentration (Colquhoun et al. 1997; Wyllie et al. 1998).
These observations also cast light over the consequences of peak-scaled non-stationary noise analysis of responses generated by a two-state kinetic scheme (see above; Traynelis et al. 1993), with ligand binding and channel opening occurring as a single step. In this case, the time constant of the decay of the macroscopic current after a jump to zero will be equal to the channel mean open time or the inverse of the closing rate constant (Colquhoun & Hawkes, 1977), and no first openings can occur after the location of the peak response. The time constant of decay of the macroscopic current will also be the time constant of decay of the covariance function as long as its centre point does not occur before the response starts to decay. Accordingly, for a two-state kinetic scheme the correlation time of the covariance function is inherently linked to the decay time of the macroscopic response and will always be equal to the channel mean open time, irrespective of the binding constant (k+1) or agonist concentration. Thus, after an agonist jump to zero concentration, responses generated by a two-state scheme will always generate parabolic variance versus mean relationships. We verified this by simulating a two-state kinetic scheme with a forward rate constant of 10 × 106m−1 s−1 and a backward (closing) rate constant of 1000 s−1 at a series of agonist concentrations between 1 μm and 5 mm (not shown). However, if the agonist waveform was modelled with an exponential decay, instead of a jump to zero, we were able to generate skewed variance versus mean curves, in agreement with the results of Traynelis et al. (1993). When the the closing rate constant was relatively large (e.g. 1000 s−1 as in the example above), the variance versus mean curve remained parabolic for small agonist decay time constants, but became markedly skewed when the decay time constant was larger than ∼500 μs. When the closing rate constant was relatively small (e.g. 100 s−1), the variance versus mean curve remained parabolic even when the agonist decay time constant was as large as 1.5 ms (data not shown). With an exponentially decaying agonist waveform, we observed first openings after the peak of the mean response.
Covariance analysis of responses obtained with multistate kinetic models
On the basis of our observations for the three-state model, we next verified that changing rate constants equivalent to α, β or k−1 had corresponding effects for the experimentally derived kinetic schemes examined earlier. For example, for the GlyLeg98 kinetic scheme (m0), increasing the closing rate constant kO1B2 (m1), or decreasing the opening rate constant kB2O1 (m2), changed the variance versus mean curve from parabolic to skewed (Fig. 10A; Po,peak= 0.5). Both changes of rate constants altered the decay time course of the ensemble mean response (Fig. 10C), but also altered the covariance function centred at the location of the peak of the ensemble mean response to more rapidly decaying (Fig. 10B). Both changes decreased the Popen within bursts (from 0.83 for the original scheme to 0.72 and 0.74, respectively). For the values of kO1B2 and kB2O1 in the original GlyLeg98 kinetic scheme, decreasing the unbinding rate constant (kB2B1) did not transform the variance versus mean relationship from parabolic to skewed.
Figure 10. Changing opening and closing rate constants of kinetic schemes GlyLeg98 and GluGei99 transforms the varianceversusmean relationships after peak-scaled non-stationary noise analysis.
A, variance versus mean curves from peak-scaled (PS) non-stationary noise analysis with modifications of the GlyLeg98 kinetic scheme: GlyLeg98 m0 (original; as in Fig. 2); GlyLeg98 m1 (kO1B2 increased from 680 s−1 to 5000 s−1) and GlyLeg98 m2 (kB2O1 decreased from 8938 s−1 to 1200 s−1). Here and in D, 1 ms agonist pulses, concentration adjusted to give Po,peak= 0.5 (N = 50; n = 1000). B, covariance functions for same simulations as in A (m0, m1, and m2), centre point (tc or t1) corresponding to location of peak response of ensemble mean. Time scale as in C. C, ensemble mean responses for same simulations as in A and B (m0, m1, and m2). D, variance versus mean curves from peak-scaled (PS) non-stationary noise analysis with modifications of the GluGei99 kinetic scheme. GluGei99 m0 (original; as in Fig. 4); GluGei99 m1 (kOC2 decreased from 4000 s−1 to 1000 s−1); GluGei99 m2 (kC2O increased from 14900 s−1 to 50000 s−1). E, covariance functions for same simulations as in D (m0, m1, and m2), centre point (tc or t1) corresponding to location of peak response of ensemble mean. Time scale as in F. F, ensemble mean responses for same simulations as in D and E (m0, m1, and m2).
Similarly, for the GluGei99 kinetic scheme (m0), decreasing the closing rate constant kOC2 (m1), or increasing the opening rate constant kC2O (m2), changed the variance versus mean curve from skewed to parabolic (Fig. 10D; Po,peak= 0.5). Both changes of rate constants altered the decay time course of the ensemble mean response (Fig. 10F), but also altered the covariance function centred at the location of the peak of the ensemble mean response to more slowly decaying (Fig. 10E). Furthermore, both changes increased the Popen within bursts (from 0.86 for the original scheme to 0.96 and 0.94, respectively). Contrary to our expectation, however, increasing the unbinding rate constant kC2C1 did not change the variance versus mean relationship from skewed to parabolic (Fig. 11A) even though Popen within bursts increased to 0.99. Closer examination revealed that the skewed relationship was a consequence of the peak response occurring before the end of the agonist pulse, combined with a biphasic relaxation phase (desensitisation followed by deactivation). This can be seen by comparing the very fast decay of the covariance function (centred at the location of the peak of the ensemble mean; Fig. 11B) with the much slower decay of the ensemble mean corresponding to the desensitisation phase during the agonist pulse (Fig. 11C). When we repeated the simulations for this modification of the GluGei99 kinetic scheme with reduced duration of the agonist pulse, such that it ended at the point in time when the ensemble mean response reached its peak, the resulting variance versus mean relationship was markedly parabolic (Fig. 11A), and the decay of the covariance function (Fig. 11B) was very similar to the decay of the ensemble mean (Fig. 11C).
Figure 11. Changing unbinding rate constant of kinetic scheme GluGei99 transforms the varianceversusmean relationship after peak-scaled non-stationary noise analysis. Influence of agonist pulse duration and amplitude on shape of varianceversusmean curves after peak-scaled non-stationary noise analysis.
A, variance versus mean curves from peak-scaled (PS) non-stationary noise analysis with a modification of the GluGei99 kinetic scheme (kC2C1 increased from 3760 s−1 to 50000 s−1). Pulse duration 1 ms (broken line) or 0.32 ms (continuous line), concentration adjusted to give Po,peak= 0.5 (N = 50; n = 1000). B, covariance functions for same simulations as in A, pulse duration 1 ms (broken line) or 0.32 ms (continuous line), centre point (tc or t1) corresponding to location of peak response of ensemble mean. Time scale as in C. C, ensemble mean responses for same simulations as in A and B, pulse duration 1 ms (broken line) or 0.32 ms (continuous line). D–F, variance versus mean curves after peak-scaled (PS) non-stationary noise analysis of simulated current traces based on the original GluGei99 kinetic scheme. Each panel shows results for six different pulse durations (continuous traces) at a given agonist concentration: 0.1 ms (leftmost curve), 0.3 ms, 0.5 ms, 0.7 ms, 0.9 ms, and 1.0 ms (rightmost curve). Agonist concentration (1, 5 and 100 mm) indicated in each panel.
With 1 ms agonist pulses, the original GluGei99 kinetic scheme also generated ensemble mean responses where the peak occurred before the end of the agonist pulse and where the decay was biphasic with initial desensitisation followed by deactivation (Figs 4B and 10F). When the duration of the agonist pulse was reduced such that it ended at the peak of the ensemble mean (0.56 ms), the resulting variance versus mean relationship became somewhat less skewed (not shown). We therefore extended our investigation by varying both the duration and the amplitude of the agonist pulse. Figure 11D–F shows results for the original GluGei99 kinetic scheme when we ran simulations with agonist pulses of 1, 5 and 100 mm (cf. Fig. 4G) and varied the duration from 0.1 to 1.0 ms for each concentration. Lowering the agonist concentration had qualitatively the same effect as shortening the pulse duration. For a given concentration, shortening the duration progressively reduced the maximum variance, but did not transform the variance versus mean curves from skewed to parabolic. For each simulation result, obtained with a given combination of agonist concentration and duration, the decay of the covariance function (with centre point corresponding to the location of the peak of the ensemble mean current) became faster with increasing agonist concentration and increasing pulse duration. In another set of experiments we modelled the agonist waveform as instantaneously rising, followed by either an exponential decay or an instantaneous fall to zero. In both cases, the shape of the variance versus mean curve remained skewed for results obtained with GluGei99 (data not shown). We observed similar results for the GluMom03 kinetic scheme. Similar simulations for the GlyLeg98 kinetic scheme resulted in variance versus mean relationships that remained parabolic, except for very slow exponential decays (τ larger than ∼1.5 ms), where the relationship started to become somewhat skewed (data not shown). Thus, in general, and within the limits of the parameter ranges explored, changing the agonist concentration or pulse duration does not transform a parabolic variance versus mean relationship to a skewed one. However, for receptor channels with very fast kinetics, the variance versus mean relationship can, under certain conditions, be transformed from parabolic to skewed when the duration of the agonist pulse exceeds the time to peak of the response itself, and the decay of the ensemble mean is characterised by relatively slow desensitisation followed by faster deactivation (Fig. 10A–C).
Kinetic schemes with more than one open state, and multiple conductance levels
We next wanted to investigate the consequences of applying non-stationary noise analysis to ion channels with concentration-dependent, multiple conductance levels, described by kinetic schemes with multiple, non-identical open states. The kinetic scheme for recombinant GluR4 AMPA receptors developed by Robert & Howe (2003) has three open states with single-channel chord conductances of 8, 16 and 24 pS for the doubly (O2), triply (O3) and quadruply (O4) liganded states, respectively (GluRH03; Fig. 12A). Macroscopic simulations (Fig. 12B) yielded an EC50 of ∼1.6 mm and a maximum response of ∼60 pA (Fig. 12C). Because there is no unique correspondence between a given current amplitude and the open probability, we calculated the open probability as the sum of the individual probabilities of occupancy of the open states (O2, O3 and O4) for each point in time (Fig. 12C). Stochastic simulations with a series of different agonist concentrations yielded fractions of first openings occurring after the peak response of macroscopically simulated responses at the same agonist concentrations between 0.1% and 27% (Fig. 12D). With non-stationary noise analysis, there was a clear effect of the agonist concentration on the shape of the variance versus mean curves (Fig. 12E) and the estimate for the unitary current i increased with increasing agonist concentration (Table 6A). N, the number of available channels, was markedly overestimated (varying from 66 to 88; Table 6A). When the simulations were repeated with the number of channels varying stochastically from trial to trial, the variance versus mean curves deviated strongly upwards (Fig. 12F). Linear fits to the initial part of the curves resulted in estimates for the unitary current that increased with increasing agonist concentration (Fig. 12F; Table 6B). For each concentration, the estimates were moderately higher than the results obtained for simulations with a constant number of available channels. With peak-scaled non-stationary noise analysis, the variance versus mean curves were skewed for all agonist concentrations tested (Fig. 12G and H). The estimates for the unitary current again increased with increasing agonist concentration (Fig. 12H; Table 6B) and were similar to those obtained for a constant number of available channels. Over the range of concentrations tested that resulted in a Po,peak > 0.5, the estimates clustered around the value corresponding to the middle conductance state (16 pS; triply liganded state). The estimates for N had no obvious physical interpretation. As for the kinetic schemes analysed above, the CVs for the estimates of i were around 0.05, but for N they could be considerably higher (Table 6).
Figure 12. Non-stationary noise analysis of responses generated by the GluRH03 kinetic scheme with multiple conductance levels.
A, kinetic scheme proposed for recombinant GluR4 AMPA receptors (Robert & Howe, 2003). R0: unbound state; R1: singly liganded, closed state; R2: doubly liganded, closed state; R3: triply liganded, closed state; R4: quadruply liganded, closed state; O2: doubly liganded, open state; O3: triply liganded, open state; O4: quadruply liganded, open state; D0-D4 and D22–D24: desensitised, closed states, liganded as indicated. The rate constants were as follows: α= 8000 s−1, β= 20000 s−1, k1= 107m−1 s−1, k−1= 104 s−1, k2= 103m−1 s−1, k−2= 1 s−1, δ0= 3.5 × 10−3 s−1, γ0= 6 s−1, δ1= 800 s−1, γ1= 45 s−1, δ2= 4000 s−1, γ2= 220 s−1. B, macroscopically simulated responses (as in Fig. 1B, except γ= 8 pS (O2), 16 pS (O3) and 24 pS (O4)). C, concentration–response relationships for short agonist pulses with response measured as peak current (•, left axis) and peak open probability (□, right axis), data points for peak current have been fitted with a Hill-type equation (eqn (2)). D, histograms of latency to first opening for stochastically simulated responses (as in Fig. 1D). E, variance versus mean curves for stochastically simulated responses after conventional non-stationary noise analysis, here and below, continuous lines represent results for four different agonist concentrations (panel D; 1 ms pulses), here and below, broken lines represent theoretical curves according to eqn (5), calculated separately for the three different single-channel conductance levels (8, 16 and 24 pS). F, variance versus mean curves for stochastically simulated responses with trial-to-trial variation (Gaussian) in the number of available channels after conventional non-stationary noise analysis. G, variance versus mean curves for stochastically simulated responses after peak-scaled (PS) non-stationary noise analysis. H, variance versus mean curves for stochastically simulated responses with trial-to-trial variation in the number of available channels after peak-scaled (PS) non-stationary noise analysis.
Table 6.
GluRH03
| 1 mm | 3 mm | 5 mm | 100 mm | ||
|---|---|---|---|---|---|
| A | |||||
| Parabolic fit | |||||
| i | Mean ±s.d. (pA) | 0.82 ± 0.056 | 0.91 ± 0.050 | 0.90 ± 0.081 | 1.05 ± 0.040 |
| CV | 0.068 (0.062) | 0.055 (0.053) | 0.090 (0.047) | 0.038 (0.046) | |
| N | Mean ±s.d. | 65.7 ± 10.3 | 80.1 ± 7.6 | 88.4 ± 8.5 | 77.8 ± 4.3 |
| CV | 0.156 (0.167) | 0.095 (0.107) | 0.097 (0.091) | 0.055 (0.066) | |
| B | |||||
| Linear fit | |||||
| i | Mean ±s.d. (pA) | 0.88 ± 0.036 | 1.02 ± 0.052 | 1.09 ± 0.049 | 1.20 ± 0.050 |
| CV | 0.041 (0.045) | 0.051 (0.046) | 0.045 (0.043) | 0.042 (0.046) | |
| Parabolic fit (after peak-scaling) | |||||
| i | Mean ±s.d. (pA) | 0.80 ± 0.043 | 0.88 ± 0.047 | 0.90 ± 0.066 | 0.98 ± 0.041 |
| CV | 0.054 (0.058) | 0.053 (0.058) | 0.073 (0.075) | 0.041 (0.052) | |
| N | Mean ±s.d. | 105.2 ± 25.6 | 119.0 ± 23.5 | 112.4 ± 19.1 | 97.6 ± 7.9 |
| CV | 0.243 (0.220) | 0.198 (0.192) | 0.169 (0.233) | 0.081 (0.093) | |
A, GluRH03, simulations with constant number of channels (γ= 8, 16 and 24 pS; i= 0.48, 0.96 and 1.44 pA; N= 50; n= 1000). B, GluRH03, simulations with stochastic trial-to-trial variation of the number of channels (γ= 8, 16 and 24 pS; i= 0.48, 0.96 and 1.44 pA; N= 50, s.d.= 10; n= 1000).
Effect of low-pass filtering and correction procedures
The results presented so far indicate that a skewed or parabolic variance versus mean relationship for peak-scaled non-stationary noise analysis can be determined by the kinetic properties of the receptor channels involved. Because the effect of peak-scaling is to reduce the variance (at the peak), this raises the question whether, and to what extent, low-pass filtering (both RC filtering, by the combination of Rs and Cm, as well as instrument filtering) of experimental data can influence the shape of variance versus mean relationships (cf. Momiyama et al. 2003).
We examined this for two different kinetic schemes, GlyAG and GluGei99, both of which gave rise to skewed variance versus mean curves with peak-scaled non-stationary noise analysis (see above). In order to mimic the effects of realistic RC filtering and instrument filtering, simulated data were digitally filtered with an RC filter at 1000, 750, 500 or 250 Hz, cascaded with a Gaussian filter at 5 kHz (see Methods). Data were then subjected to conventional and peak-scaled non-stationary noise analysis with various degrees of correction for the RC filtering (0, 25, 50, 75 and 100%: see Methods). Figure 13A–D shows results for the GluGei99 kinetic scheme after peak-scaled non-stationary noise analysis with various degrees of correction for the RC filtering (compare to Fig. 4G). Low-pass filtering caused an underestimation of i, and the error increased with the degree of filtering (Fig. 13E). For the GluGei99 kinetic scheme, the error ranged from approximately 17% with an RC filter at 1000 Hz, to approximately 45% at 250 Hz. Importantly, we observed that with an increasing degree of RC filtering, the variance versus mean curves were changed from a skewed to an increasingly parabolic shape and appeared parabolic already with an RC filter at 750 Hz (Fig. 13B). This indicates that low-pass filtering of experimental spPSCs can mask a genuinely skewed variance versus mean relationship and that the presence of a parabolic curve after peak-scaled non-stationary noise analysis must be treated with considerable caution.
Figure 13. Low-pass filtering of current traces transforms skewed varianceversusmean relationships to parabolic.
Variance versus mean curves after peak-scaled (PS) non-stationary noise analysis of simulated current traces based on the GluGei99 kinetic scheme, γ= 8.5 pS, 1 ms agonist pulses, concentration adjusted to give Po,peak= 0.5 (N = 50; n = 1000). Current responses digitally low-pass filtered with Gaussian filter (cut-off frequency at 5 kHz) and RC filter with variable cut-off frequency as indicated (fc; A–D). In panels A–D, the bottommost trace (thicker, continuous line) corresponds to variance versus mean curve after uncorrected low-pass filtering. The upper traces (thinner, continuous lines) correspond to curves after correcting 25, 50, 75 and 100% of the effect of RC filtering, respectively. The top trace (broken line) corresponds to variance versus mean curve after analysis of unfiltered data. A, RC filtering at 1000 Hz. B, RC filtering at 750 Hz. C, RC filtering at 500 Hz. D, RC filtering at 250 Hz. E, single-channel conductance (γ), estimated by peak-scaled non-stationary noise analysis of data illustrated in A–D, as a function of cut-off frequency of RC filter (1000 Hz (○), 750 Hz □, 500 Hz ▵, and 250 Hz (▿) and fractional correction of RC filtering (0, 25, 50, 75 and 100%). Horizontal arrows (right) indicate estimated conductances for the same dataset with no filtering (1) or with Gaussian filtering only (2). Data points have been fitted with straight lines.
We next considered the extent to which software-based procedures for correcting the effects of single compartment RC filtering (Traynelis, 1998) are able to correct the results obtained with peak-scaled non-stationary noise analysis. After correction, the variance versus mean curve was transformed from parabolic to its originally skewed shape (Fig. 13A–D) and the error in estimated i was reduced (Fig. 13E). With 100% correction, the analysis results were very similar to those obtained with Gaussian filtering alone. Interestingly, the estimates of single-channel conductance after different degrees of correction were linearly related (Fig. 13E). While 100% correction is unrealistic with real experimental data, because of an unacceptable amplification of noise, these results suggest that for the estimate of i it might be possible to perform a series of low to moderate fractional corrections and use the corrected estimates to extrapolate the correction to 100%.
We observed similar results with the kinetic scheme GlyAG, both with respect to underestimation of i and transformation of variance versus mean curves from skewed to parabolic (data not shown). The error was smaller, both with uncorrected RC filtering and with Gaussian filtering alone, probably due to the longer mean open time for the GlyAG kinetic scheme (1.33 ms) compared to the GluGei99 kinetic scheme (0.24 ms; see Silberberg & Magleby, 1993; Benke et al. 2001).
Effect of variable alignment
For both conventional and peak-scaled non-stationary noise analysis, the individual traces were aligned as they were generated during simulation, i.e. synchronised to the stimulus. If the individual current traces are artificially desynchronised by introducing jitter, a parabolic variance versus mean curve can be converted to a skewed one, and with increasing degree of jitter, the degree of skewness increases. Figure 14 shows an example of the effect of adding increasing degrees of jitter (from ±20 to ±100 sample intervals; see Methods) to responses simulated with a variant of the three-state kinetic scheme (Fig. 14, legend; Po,peak∼0.5) and analysed with conventional (Fig. 14A) and peak-scaled (Fig. 14B) non-stationary noise analysis. Accordingly, if non-stationary noise analysis of experimental spPSCs gives rise to a skewed variance versus mean relationship, this could be due to inadequate alignment instead of being related to the kinetic properties of the receptors. The question concerning which method of alignment is optimal becomes important because there is no stimulus with which to synchronise spPSCs. We examined this by subjecting simulated data to artificial desynchronisation, followed by re-alignment. The resulting data sets were analysed with both conventional and peak-scaled non-stationary noise analysis. An adequate method of re-alignment should be able to align individual current traces such that subsequent noise analysis results are as similar as possible to those obtained directly from the original data. Because we wanted to evaluate different alignment procedures for use with real experimental data, we first digitally low-pass filtered the simulated data (RC filter at 1 kHz cascaded with Gaussian filter at 5 kHz). The following methods of alignment were examined: alignment by the point of steepest rise of the ascending phase of the response; alignment by the onset or peak of a fitted function, eqn (3); or alignment by the onset or peak of another fitted function, eqn (4) (see Methods).
Figure 14. Jitter between individual current traces distorts varianceversusmean curves.
A,B, variance versus mean curves for stochastically simulated responses for a variant of the three-state kinetic scheme (S3mod49; k+1= 40 × 106m−1 s−1, k−1= 10000 s−1, α= 1000 s−1, β= 10000 s−1), 1 ms agonist pulses, concentration adjusted to give Po,peak= 0.5 (N = 50; n = 1000). The different curves in each panel correspond to introducing increasing degrees of jitter before conventional (A) or peak-scaled (PS; B) non-stationary noise analysis. The degree of introduced jitter varied from displacement by ±0 (bottommost curve; corresponds to original data set) to ±100 sample intervals (topmost curve; see Methods). C,D, variance versus mean curves for stochastically simulated responses for the same kinetic scheme as in A and B after conventional (C) and peak-scaled (PS; D) non-stationary noise analysis, 1 ms agonist pulses, concentration adjusted to give Po,peak= 0.75 (N = 50; n = 1000), the six different curves in each panel (C,D) correspond to different manipulations of original data as indicated in the Fig. (‘jitter’ refers to displacement by ±30 sample intervals, ‘filter’ refers to low-pass filtering with RC filter at 1 kHz and Gaussian filter at 5 kHz).
An example of the effects of the different methods of alignment, after introducing jitter and low-pass filtering of the responses, is shown in Fig. 14C and D. The data were simulated with the same values for the rate constants as above (Fig. 14, legend), but with the agonist concentration increased to give Po,peak∼ 0.75 which is more realistic for putative synaptic responses. The results from conventional non-stationary noise analysis are shown in Fig. 14C, and the results with peak-scaling are shown in Fig. 14D. As expected, low-pass filtering alone reduced the variance. Introducing a moderate amount of jitter (±30 sample intervals) markedly increased the variance and changed the variance versus mean curves for both conventional and peak-scaled noise analysis. Re-alignment by the point of steepest rise of the current trace (Traynelis et al. 1993) proved effective in restoring the original non-stationary noise analysis results, both with conventional and peak-scaled noise analysis (Fig. 14C and D). Re-alignment by onset after optimally fitting with eqn (3) was equally effective in restoring the original results (Fig. 14C and D). Re-alignment by onset after optimally fitting with eqn (4) was less effective (not shown). Finally, re-alignment by the peak of an optimally fitted function, either eqn (3) or eqn (4), proved inadequate (result for eqn (3) shown in Fig. 14CD).
In all cases, curve fitting with a parabolic function, eqn (5), to an appropriate range of the variance versus mean curves, gave a reasonably good estimate for the unitary current (cf. Traynelis et al. 1993). This was the case for filtered data without jitter (probably due to the relatively long mean open time, see above), for filtered data with jitter, and for data re-aligned by any of the methods tested. However, N, the number of available channels or the average number of channels open at the peak, was incorrectly estimated after introducing jitter and after inadequate re-alignment. While re-alignment of waveforms after introducing artificial jitter was able to restore the original variance versus mean curve reasonably well, we have never observed that any method of re-alignment of stimulus-synchronised simulation data was able to transform an originally skewed variance versus mean curve into a parabolic one. Furthermore, for responses generated by receptor ion channels with slower kinetics, the effect of introducing jitter was much less pronounced than described above and is unlikely to represent a problem for experimental investigations.
Discussion
In this study, we have evaluated the performance of conventional and peak-scaled non-stationary noise analysis with respect to estimating the single-channel conductance and the number of channels involved in spontaneous postsynaptic currents. A motivating factor for this study was to understand the basis of skewed or parabolic variance versus mean curves after peak-scaled non-stationary noise analysis. We explored the results obtained with several different experimentally derived kinetic schemes of varying complexity and varying conditions relevant for real synaptic transmission. Our results demonstrate that a parabolic variance versus mean curve should not necessarily be expected following peak-scaled non-stationary noise analysis.
With a constant number of available channels, conventional non-stationary noise analysis provides estimates of i, the single-channel current and N, the total number of available channels. For both parameters, the CV could be as low as 0.03–0.05, but for lower values of Po,peak (<0.5), the CV for estimates of N increased. The CV was estimated both with repeated analysis of 10 original data sets and with analysis of 100 synthetic data sets generated by balanced re-sampling (bootstrap) of individual waveforms from one of the original data sets. Both procedures gave very similar results. Previous results with balanced re-sampling of spPSCs from real and simulated data (Momiyama et al. 2003) indicated that the CV falls with increasing number of spPSCs in the ensemble. Our results are likely to be close to an asymptotic limit reached at high values of n, and agree well with the results of Momiyama et al. (2003).
In order to mimic one source of quantal variability, we stochastically varied the number of available channels from trial to trial. With conventional non-stationary noise analysis, the variance versus mean curves were upwardly concave with no reduction in variance at high Po,peak. Nevertheless, the initial slope of the curves could still be used to give a good estimate of the single-channel conductance. To compensate for the increased variance caused by the simulated quantal variability, we employed peak-scaled non-stationary noise analysis on a series of different kinetic schemes by scaling the peak of the average response to each individual response before calculating the fluctuations around the mean (Traynelis et al. 1993). Without exception, peak-scaling proved highly effective in compensating for the increased variance caused by quantal variability. For some kinetic schemes, the variance versus mean curves after peak-scaled non-stationary noise analysis were parabolic, and the estimated value of the parameter N corresponded approximately to the average number of channels open at the peak. These results are consistent with the predictions of earlier studies by Traynelis et al. (1993) and De Koninck & Mody (1994). However, for other kinetic schemes, the variance versus mean curves were markedly skewed and the estimated value of the parameter N did not have a physical interpretation. While the parameter estimates of conventional non-stationary noise analysis are model-independent (Traynelis & Wahl, 1997), our results indicate that this is not the case for peak-scaled non-stationary noise analysis. However, for both parabolic and skewed curves, fitting with eqn (5) to an appropriate portion of the curve provided adequate values for the single-channel current (i). The CVs for i after peak-scaled non-stationary noise analysis were very similar to those obtained after conventional analysis.
These results seem to question the value of peak-scaling when applying non-stationary noise analysis to real, experimental spPSCs. However, we can see two reasons for preferring peak-scaled non-stationary noise analysis and curve fitting with eqn (5) to an appropriate part of the variance versus mean curve over conventional non-stationary noise analysis and curve fitting with a straight line to the initial part of the curve. First, for real, non-ideal data, fitting with the parabolic function of eqn (5) offers the opportunity to take into account a larger number of data points and thereby better constrain the parameter estimates. Second, the shape of the variance versus mean curve obtained by peak-scaled non-stationary noise analysis (skewed versus parabolic) can potentially reveal important information about the kinetic properties of the underlying channels.
Parabolic or skewed variance versus mean curves after peak-scaled non-stationary noise analysis
By systematically varying either the opening, closing or unbinding rate constant of a simplified three-state kinetic scheme, we were able to transform the shape of the variance versus mean curve obtained by peak-scaled non-stationary noise analysis along a continuum between skewed and parabolic. The crucial property that determined the shape of the curve was the degree of correlation between the fluctuations at the peak and at neighbouring points in time during the decay phase. We analysed this quantitatively by calculating non-stationary covariance functions. The covariance takes into account not only the magnitude of the variance, but also its temporal structure (Sigworth, 1981, 1984). We studied the decay of the covariance function with centre point corresponding to the location of the peak of the average waveform. Responses that gave rise to skewed variance versus mean relationships displayed fast decay of the covariance function compared to the decay of the ensemble mean response, i.e. the degree of correlation of the fluctuations at the peak with those at neighbouring regions fell off more rapidly as one moved away from the peak. In contrast, responses that gave rise to parabolic variance versus mean relationships displayed a decay of the covariance function that was more similar to the decay of the ensemble mean response. When the covariance function decayed relatively slowly, peak-scaling reduced the variance below that of the theoretical curve predicted for a given pair of values for i and N according to eqn (5). This was not the case with a faster decay of the covariance function, and in such cases the variance could even be increased above the level of the theoretical curve, thus creating an increasingly skewed variance versus mean relationship. For all variants of the simple three-state kinetic scheme analysed in detail, we also verified that the decay of the covariance function centred at the location of the peak of the mean response was bi-exponential, with time constants equal to the time constants of the burst length distribution.
Skewed variance versus mean relationships have been observed for peak-scaled analysis of both experimental and simulated data (Traynelis et al. 1993; De Koninck & Mody, 1994; Momiyama et al. 2003). It has been proposed that a skewed variance versus mean relationship after peak-scaled analysis results when a substantial fraction of first channel openings occurs after the peak of the average response (Traynelis et al. 1993). In that case, channel openings and closings would contribute to the variance during the decay of the response, but not to the variance at the peak. In our analysis and comparison of simulation results from several different kinetic schemes, there was a tendency for the fraction of first openings to be somewhat larger for kinetic schemes that resulted in skewed variance versus mean relationships. However, there was overlap between different kinetic schemes, and we also observed cases where there was no difference with respect to the fraction of first openings after the peak in spite of marked differences with respect to skewed or parabolic variance versus mean curves. We therefore conclude, that differentiating responses with respect to the fraction of first openings after the peak, insufficiently characterises the temporal structure of the channel fluctuations that is the basis for the shape of the variance versus mean curve after peak-scaling. Instead, the covariance function and its properties, as analysed by the decay of the function with centre point corresponding to the location of the peak of the average waveform, explained all the examples of skewed versus parabolic relationships for the various kinetic schemes included in our study. The results of Traynelis et al. (1993) can most likely be explained by the use of a two-state model in their simulations. After a jump of agonist to zero concentration, the decay of the covariance function will always be identical to the decay of the ensemble mean response, there will be no first openings after the peak of the mean response, and the variance versus mean relationship will be parabolic. However, with an exponentially decaying agonist waveform, the variance versus mean relationship can become skewed (depending on the relation between the rate constants and the agonist decay time time constant), and there will be first openings after the peak of the mean response.
For a given kinetic scheme, among the examples we analysed, a high value of Popen within bursts of single-channel openings corresponded to a more parabolic variance versus mean relationship, and a lower value corresponded to a more skewed relationship. For the simple three-state scheme, the value for Popen within bursts could be directly calculated as a function of the opening, closing and unbinding rate constants according to eqn (1). The relationship between these rate constants and Popen within bursts is complex (Fig. 7H). If we ignore the cases where the mean burst length approached the mean channel open time, an increase in the Popen within bursts and a corresponding parabolic variance versus mean relationship could be obtained by decreasing the closing rate constant, increasing the opening rate constant or increasing the unbinding rate constant. The effect of changing a given rate constant, however, depended on the specific values of the other two. Accordingly, the results of peak-scaled non-stationary noise analysis of real, experimental data can reveal important information with respect to the kinetic properties of the underlying receptors. This is the case whether one analyses responses with a variable (e.g. spPSCs) or constant (e.g. outside-out patches) number of available channels. It therefore becomes important to identify conditions that can transform a skewed variance versus mean relationship into a parabolic one or vice versa. We identified two such conditions, inadequate alignment (jitter) and low-pass filtering. The issue of alignment of experimental data is critical for spPSCs, as there is no stimulus available with which to synchronise them. For responses generated by ion channels with fast kinetics, desynchronisation of simulated waveforms can transform the resulting variance versus mean curves from parabolic to skewed (after both conventional and peak-scaled non-stationary noise analysis). We examined the performance of different procedures for alignment by the degree to which they were able to restore the analysis results obtained from the original data sets aligned by the stimulus waveform. Alignment by either the point of steepest rise (Traynelis et al. 1993) or by the onset of an optimally fitted function, based on the product of two exponential functions, proved highly effective in restoring the original analysis results. This contrasted with the poor performance obtained with alignment by the peak of an optimally fitted function.
Low-pass filtering, both Gaussian filtering, intended to mimic instrumental filtering, and RC filtering, designed to mimic filtering imposed by the combination of series resistance and cell membrane capacitance, transformed a skewed variance versus mean curve into a parabolic curve. In addition, low-pass filtering also reduced the estimated single-channel current, consistent with previous reports (Silberberg & Magleby, 1993; Traynelis et al. 1993; Traynelis & Wahl, 1997; Benke et al. 2001; Mørkve et al. 2002; Momiyama et al. 2003). Importantly, off-line procedures designed to correct for the effects of low-pass RC filtering were able to fully correct for the effects of filtering on the variance versus mean relationships, as well as on the estimates of single-channel currents. These results suggest that attempts to correct for RC filtering should be made whenever possible. Low-pass filtering caused by electrotonic filtering has qualitatively similar effects (Benke et al. 2001), but will be more difficult to correct for.
Non-stationary noise analysis of channels with multiple, agonist concentration-dependent conductance levels
There is strong evidence that AMPA-type non-NMDA receptors can open to multiple, agonist concentration-dependent conductance levels (Rosenmund et al. 1998; Smith & Howe, 2000; Smith et al. 2000), and recently a kinetic scheme that accounts for these properties was developed by Robert & Howe (2003). In our simulations, we used their model for homomeric GluR4 receptors with three open states at 8, 16 and 24 pS, respectively. Previous work has indicated that when channels open to multiple, conductance levels, non-stationary noise analysis will yield an estimate of a mean channel conductance weighted towards larger conductance levels, provided that the probability of a channel being in any of the conductance states is small (Sigworth, 1980; Cull-Candy et al. 1988). Accordingly, it has been proposed (Traynelis et al. 1993; Momiyama et al. 2003) that estimates should be obtained from the initial slope of the variance versus mean curves, where the channel open probability is low. In our analysis of responses simulated with the kinetic scheme of Robert & Howe (2003; GluRH03), we found marked differences with respect to other kinetic schemes with single or multiple homogeneous conductance levels. With a constant number of channels from trial to trial, and conventional non-stationary noise analysis, increasing agonist concentration did not yield variance versus mean curves that followed each other along a theoretical curve predicted by eqn (5). With both conventional and peak-scaled non-stationary noise analysis, we found an increase in the estimated single-channel conductance with increasing agonist concentration, clustered around the value corresponding to the middle conductance state (16 pS; triply liganded state). Although these types of results from peak-scaled non-stationary noise analysis of spPSCs or conventional non-stationary noise analysis of responses in excised patches are not sufficient conditions for concluding that receptor channels with multiple conductance levels are present, it might be a necessary condition for the presence of such receptors. For excised patches, it should be experimentally feasible to test the receptors with a range of concentrations. For synaptic responses, it is an open question to what extent intra- and intersynaptic variability, with respect to the spatial and temporal concentration profiles of transmitter in the synaptic cleft (Kruk et al. 1997; Nusser et al. 2001), will generate results with peak-scaled non-stationary noise analysis where the presence of multiple, agonist concentration-dependent conductance levels can be identified.
Acknowledgments
Financial support from the Norwegian Research Council (NFR 161217/V40), the Meltzer fund (University of Bergen) and the Faculty of Medicine at the University of Bergen (fellowship for MLV) is gratefully acknowledged.
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