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¹ University of Bergen, Department of Biomedicine, Bergen, Norway

	Abstract

Top Abstract Introduction Methods Results Discussion References

 
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.

(Received 19 April 2006; accepted after revision 18 May 2006; first published online 25 May 2006)
Corresponding author E. Hartveit: University of Bergen, Department of Biomedicine, Jonas Lies vei 91, N-5009 Bergen, Norway. Email: espen.hartveit{at}biomed.uib.no

	Introduction

Top Abstract Introduction Methods Results Discussion References

 
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

Top Abstract Introduction Methods Results Discussion References

 
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; (V_m – E_rev) 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 (N_mean = 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 (P_o,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 (_decay = 100 µs) or by an instantaneous fall to zero. P_o,peak is the open probability at the maximum peak of the response, defined by the equation P_o = 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 –Q_EE, with Q_EE being the burst state submatrix of the Q matrix (Colquhoun & Hawkes, 1982; Wyllie et al. 1998). The mean P_open 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 , ß and k_–1), directly as the total open time divided by the mean burst length:

(1)

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:

(2)

where R is the peak current (I) or P_o,peak at a given concentration of agonist ([A]); R_max is the maximum response, EC₅₀ is the concentration of agonist giving rise to half-maximal response and n_H 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% x (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,

(3)

where is the delay to onset of the response, A describes the peak amplitude and _rise and _decay 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,

(4)

where A describes the peak amplitude, erf is the error function, is the delay to onset of the response and _decay 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 (²) 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:

(5)

where _b² 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 () for each channel was then calculated by the equation i/(V_m – E_rev), with V_m = – 60 mV and E_rev = 0 mV. The open probability (P_open) at any given time was determined by the equation P_open = 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 P_o,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,

(6)

where the y_i are the current values from the ith event and the µ are the means of the ny_i values (Sigworth, 1981, 1984). Each value in the covariance matrix, ²_ij, 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 t_c), 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 (t_corr).

Digital filtering

A simple RC filter (1-poled), used to mimic the filtering effect of the combination of R_s (series resistance) and C_m (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

Top Abstract Introduction Methods Results Discussion References

 
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 P_o,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).

View larger version (29K): [in this window] [in a new window]   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 x 106M–1 s–1, kB1U = 100 s–1, kB1B2 = 5 x 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 () = 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.

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 P_o,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 P_o,peak was lower (Table 2A).

View larger version (31K): [in this window] [in a new window]   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 x 106M–1 s–1, kB1U = 1452 s–1, kB1B2 = 6 x 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.

The third kinetic scheme we used for glycine receptors was developed for gating of heteromeric rat 1ß 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 P_o,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 P_o,peak was lower, and for i and N at the highest concentration (100 mM) where the P_o,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 P_o,peak.

View larger version (31K): [in this window] [in a new window]   Figure 3.  Non-stationary noise analysis of responses generated by the GlyBur04 kinetic scheme A, kinetic scheme for gating of heteromeric rat 1ß 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): 1 = 3400 s–1, ß1 = 4200 s–1, 2 = 2100 s–1, ß2 = 28000 s–1, 3 = 6700 s–1, ß3 = 130000 s–1, 1 = 29266 s–1, 1 = 180 s–1, 2 = 18000 s–1, 2 = 6800 s–1, 3 = 948.1 s–1, 3 = 22000 s–1, k– = 302 s–1, k+ = 0.59 x 106M–1 s–1, kF– = 1250 s–1, kF+ = 150 x 106M–1 s–1. B, macroscopically simulated responses (as in Fig. 1B). Single-channel conductance () identical for AF*, A2F* and A3F* ( 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.

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 (P_o,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 P_o,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).

View larger version (28K): [in this window] [in a new window]   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 x 106M–1 s–1, kC1C0 = 157 s–1, kC1C2 = 3.24 x 106 M–1 s–1, kC2C1 = 3.76 x 103 s–1, kC2O = 14.9 x 103 s–1, kOC2 = 4.00 x 103 s–1, kC1C3 = 1.53 x 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 x 106M–1 s–1, kC4C3 = 2 s–1, kC4C5 = 1.59 x 103 s–1, kC5C4 = 899 x 103 s–1. B, macroscopically simulated responses (as in Fig. 1B, except = 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.

View larger version (27K): [in this window] [in a new window]   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 x 106M–1 s–1, kC1C0 = 2093 s–1, kC1C2 = 6.019 x 106M–1 s–1, kC2C1 = 4.719 x 103 s–1, kC2O = 17.23 x 103 s–1, kOC2 = 3.734 x 103 s–1, kOC7 = 114.1 s–1, kC7O = 90.47 s–1, kC1C3 = 4.219 x 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 x 106M–1 s–1, kC4C3 = 3.486 x 103 s–1, kC4C5 = 476.4 s–1, kC5C4 = 420.9 s–1, kC5C6 = 1.034 x 104 s–1, kC6C5 = 636.3 s–1. B, macroscopically simulated responses (as in Fig. 1B, except = 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.

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.

View larger version (22K): [in this window] [in a new window]   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.

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 (ß) and closing (). 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 P_o,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 , ß 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 ß 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 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 x 10⁶ M^–1 s^–1 to 2 x 10⁴ M^–1 s^–1, 1.2 x 10⁷ M^–1 s^–1 or 1.2 x 10⁹ M^–1 s^–1 for two different combinations of , ß 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 = 100 s^–1 instead of 1000 s^–1 (ß = 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 , ß and k_–1 affected the shape of the variance versus mean relationship (not shown).

View larger version (32K): [in this window] [in a new window]   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; ß = opening; = 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, ß varied; mod30: 1000 s–1; mod31: 3000 s–1; mod32: 4000 s–1; mod33: 6000 s–1; mod34: 9000 s–1; (750 s–1), k–1 (100 s–1) and k+1 (6 x 106M–1 s–1) constant. D, S3mod24–mod29, 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; ß (2000 s–1), k–1 (100 s–1) and k+1 (6 x 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; (1000 s–1), ß (1000 s–1) and k+1 (6 x 106M–1 s–1) constant. F, S3mod43–mod48, for S3mod43–mod45, k+1 varied; mod43: 2 x 104M–1 s–1; mod44: 1.2 x 107M–1 s–1; mod45: 1.2 x 109M–1 s–1; (5000 s–1), ß (9000 s–1) and k–1 (100 s–1) constant. For S3mod46–mod48, k+1 varied; mod46: 2 x 104M–1 s–1; mod47: 1.2 x 107M–1 s–1; mod48: 1.2 x 109M–1 s–1; (1000 s–1), ß (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 , ß and k–1. Notice the large range (1–50000 s–1) included for each rate constant.

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 P_o,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 P_o,peak (and therefore on agonist concentration). One obvious reason for this conclusion is that peak-scaling of responses with high P_o,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 P_o,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 (²(t)) and individual covariance functions (C(t_c,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 (²(t)) and individual covariance functions (C(t_c,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) P_o,peak.

View larger version (36K): [in this window] [in a new window]   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 x 106M–1 s–1, k–1 = 200 s–1, = 1000 s–1, ß = 9000 s–1), decay 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 x 106M–1 s–1, k–1 = 50 s–1, = 5000 s–1, ß = 9000 s–1), decay 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