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J Physiol (2003), 547.3, pp. 665-689
© Copyright 2003 The Physiological Society
DOI: 10.1113/jphysiol.2002.032714

Local routes revisited: the space and time dependence of the Ca²⁺ signal for phasic transmitter release at the rat calyx of Held

Christoph J. Meinrenken, J. Gerard G. Borst* and Bert Sakmann

	ABSTRACT

Top Abstract Introduction References

During the last decade, advances in experimental techniques and quantitative modelling have resulted in the development of the calyx of Held as one of the best preparations in which to study synaptic transmission. Here we review some of these advances, including simultaneous recording of pre- and postsynaptic currents, measuring the Ca²⁺ sensitivity of transmitter release, reconstructing the 3-D anatomy at the electron microscope (EM) level, and modelling the buffered diffusion of Ca²⁺ in the nerve terminal. An important outcome of these studies is an improved understanding of the Ca²⁺ signal that controls phasic transmitter release. This article illustrates the spatial and temporal aspects of the three main steps in the presynaptic signalling cascade: Ca²⁺ influx through voltage-gated calcium channels, buffered Ca²⁺ diffusion from the channels to releasable vesicles, and activation of the Ca²⁺ sensor for release. Particular emphasis is placed on how presynaptic Ca²⁺ buffers affect the Ca²⁺ signal and thus the amplitude and time course of the release probability. Since many aspects of the signalling cascade were first conceived with reference to the squid giant presynaptic terminal, we include comparisons with the squid model and revisit some of its implications. Whilst the characteristics of buffered Ca²⁺ diffusion presented here are based on the calyx of Held, we demonstrate the circumstances under which they may be valid for other nerve terminals at mammalian CNS synapses.

(Received 19 September 2002; accepted after revision 10 January 2003; first published online 31 January 2003)
Corresponding author B. Sakmann: Max Planck Institute for Medical Research, Heidelberg, Germany. Email: zpsecr{at}mpimf-heidelberg.mpg.de

	INTRODUCTION

Top Abstract Introduction References

The work of Bernard Katz and his collaborators led to the formulation of the calcium hypothesis, which proposed that neurotransmitter release from presynaptic vesicles is triggered by elevations of the Ca²⁺ concentration ([Ca²⁺]) in the presynaptic terminal (Katz & Miledi, 1965; Dodge & Rahamimoff, 1967; Katz, 1969; Miledi, 1973). At the time, these Ca²⁺ signals were inaccessible by direct measurement. They remain so until this day. Considering the elegance of the calcium hypothesis, it is doubtful that its discoverers imagined just how difficult the study of the presynaptic Ca²⁺ signal would become. The quest gained urgency once it became widely accepted that Ca²⁺ does not merely trigger release, but that the exact amplitude and time course of the [Ca²⁺] govern the amplitude and time course of the release and thus the transmission characteristics of the synapse (Barrett & Stevens, 1972). During the next twenty years, a number of experimental and theoretical studies found that presynaptic Ca²⁺ signals were not spatially uniform, but 'localised', i.e. characterised by steep spatial gradients (across tens of nanometres) and a rapid rise and decay (in tens of microseconds) (for reviews, see Smith & Augustine, 1988; Augustine & Neher, 1992). This finding marked the debut of Ca²⁺ signalling studies in the rather in-elegant world of high-powered computing and numerical methods. To complicate matters, many parameters important for the calculations were not known, among them the distance between calcium channels and vesicles at release sites (Neher, 1998a). Finally, the inherent noise of the Ca²⁺ signal - mediated by the stochastic gating of voltage-sensitive ion channels (Hodgkin & Huxley, 1952; Fenwick et al. 1982) - posed a further challenge for a quantitative description.

Despite these difficulties, remarkable progress has been made, notably on the squid giant presynaptic terminal, on which most of the qualitative and quantitative aspects of presynaptic Ca²⁺ signalling were pioneered (Augustine et al. 1987; Llinás, 1999). Interpreting the relative effects of different Ca²⁺ buffers on the release probability of the squid terminal (Adler et al. 1991), Augustine et al. (1991) concluded that the action potential-evoked change in presynaptic [Ca²⁺] was of the order of 100 µM or higher. Whilst, quantitatively, this finding was based on calculations that assumed spatial equilibrium, it actually provided a further argument for the non-equilibrium, 'local' character of the Ca²⁺ signal, for the authors believed that such a large change in the [Ca²⁺] was generated by localised domains of elevated [Ca²⁺] in the immediate vicinity of open calcium channels (Augustine et al. 1991).

More recently, another giant terminal, the calyx of Held (henceforth 'calyx'; Held, 1898) joined the presynaptic terminal of the squid as a widely studied model system for neurotransmitter release (Forsythe et al. 1995; von Gersdorff & Borst, 2002). With the adaptation of patch clamp recordings to brain slices (Edwards et al. 1989), including the ability to image and record from subcellular structures (Stuart et al. 1993), pre- and postsynaptic currents at the calyx could be recorded simultaneously (Borst et al. 1995). We found that details of Ca²⁺ signalling at the calyx differ substantially from those at the squid presynaptic terminal. For example, we were surprised to find that 10 mM EGTA in the presynaptic solution, our standard saline at the time, blocks about 50 % of the release (Borst & Sakmann, 1996), whereas even higher concentrations do not affect release in the squid giant synapse (Adler et al. 1991). We also found that Ca²⁺ diffusion models intended to explain these differences were inconclusive because too many crucial parameters were not yet known. Consequently, much of our experimental effort in the past 7 years has been directed towards constraining these parameters, including the Ca²⁺ current, the volume-averaged Ca²⁺ dynamics, the Ca²⁺ sensitivity of transmitter release, and the 3-D structure of the calyx.

To our knowledge, the calyx today represents the only model system of action potential-evoked, phasic transmitter release that includes a numerical model consistent with most available experimental data (Meinrenken et al. 2002). Hence we use the calyx as the basis to review presynaptic Ca²⁺ signalling. The calyx model provides detailed predictions of the spatio-temporal Ca²⁺ signal, the function of endogenous buffers, and the time course and amplitude of release. Its main elements build on the vast number of previous studies of buffered Ca²⁺ diffusion in nerve cells (reviewed by Neher, 1998a; Smith, 2001). Still, the model offers new insights into the mechanisms controlling the Ca²⁺ signal that triggers phasic transmitter release, in particular into the effects of endogenous buffers. To better illustrate this, we compare the calyx model with earlier concepts of presynaptic Ca²⁺ signalling, particularly those developed for the presynaptic terminal of the squid. Finally, we explore the circumstances under which the results from the calyx may be applicable to presynaptic terminals in cortical synapses.

Advances in understanding presynaptic Ca²⁺ dynamics

Thanks to their comparatively large size (~15 µm across), calyx-type terminals have enjoyed broad popularity amongst physiologists. Intracellular recordings were made as early as 1963 (Martin & Pilar, 1963). Further facilitated by whole-cell recordings (Sakmann & Neher, 1983) and improved imaging methods, a large amount of detailed data has been collected from a number of calyceal preparations: chick ciliary ganglion (Stanley, 1992) with simultaneous pre- and postsynaptic recordings (Yawo & Momiyama, 1993); chick auditory calyx (Sivaramakrishnan & Laurent, 1995); and the calyx of Held (Fig. 1A and B; Forsythe, 1994; Borst et al. 1995).

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Figure 1. Membrane currents and volume-averaged [Ca2+] A, pseudocolour image of a calyx of Held (yellow) and the medial nucleus of the trapezoid body (MNTB) principal cell (blue) in a brain slice. The calyx was filled with Lucifer Yellow, the principal cell with Cascade Blue (after Borst et al. 1995). B, time course of the signalling cascade, showing (top to bottom) the presynaptic AP waveform and resulting Ca2+ current, (inferred) release rate, postsynaptic EPSC and postsynaptic AP (after Borst et al. 1995; Borst & Sakmann, 1998). C, fluorescence image of a calyx in a brain slice. The calyx was filled with 1 mM Fura-2. D, fluorescence of Fura-2 in calyx (at two different concentrations) to measure volume-averaged [Ca2+]. Note that the measured decay of the fluorescent signal is much slower than the decay of calculated, local [Ca2+] transients (see Fig. 4). E, inverted whole-cell [Ca2+] amplitude (A-1) in calyx as a function of exogenous buffer Ca2+-binding ratio (B). Regression line crosses x-axis at implied endogenous binding ratio ~40 (C-E reprinted after Helmchen et al. 1997).

The experimental results reviewed below represent average data of calyces of the rat at postnatal days (P) 8-10 and at room temperature. The effects of developmental changes and of temperature will be discussed in 'Conclusions and questions'. We focus only on those experimental results that were used to devise and calibrate a quantitative model of calcium-secretion coupling during a single action potential (AP) at the calyx (Table 1). Technical aspects of recording presynaptic calcium influx at the calyx were reviewed by Borst & Helmchen (1998). Advances in studying calyceal development (Friauf & Lohmann, 1999), changes in Ca²⁺ influx and synaptic transmission during multiple action potentials (von Gersdorff & Borst, 2002), and the regulation of different vesicle pools (Schneggenburger et al. 2002) have been reviewed elsewhere.

Membrane currents and presyaptic [Ca²⁺] dynamics

Whole-cell, presynaptic calcium current (I_Ca). By using an AP waveform as a voltage command, one can estimate the membrane currents during an AP (Borst & Helmchen, 1998). The calcium current shows an approximately Gaussian time course (full width at half maximum (fwhm), 360 ± 10 µs), with a peak of 2.5 ± 0.1 nA and an integral of 0.92 ± 0.05 pC (Borst & Sakmann, 1998). I_Ca begins approximately at the time of the peak of the AP and ends before the calyx is fully repolarised (Fig. 1B). During repolarisation, the calcium conductance is much smaller than the potassium conductance (Forsythe, 1994; Borst & Sakmann, 1996). Therefore, the shape of the AP does not depend measurably on the amplitude of the Ca²⁺ influx (Borst et al. 1995). Likewise, there is no evidence (yet) for an indirect effect of calcium channels via Ca²⁺-dependent potassium channels, in contrast to findings from the neuromuscular junction (Meir et al. 1999).

Calcium channel subtypes mediating I_Ca and release. Ca²⁺ enters the terminal through three subtypes of calcium channels, P/Q-, N-, and R-type channels (at P9-10). Through the use of specific pharmacological blockers, one can determine the contribution of each subtype to the total Ca²⁺ influx (P/Q: ~50 %, N: ~25 %, R: ~25 %; Wu et al. 1999). The composition of the calcium channels in the presynaptic membrane differs from that at the soma in the anteroventral cochlear nucleus (AVCN), which in addition expresses L- and T-type calcium channels, indicating that calcium channel expression can vary within the same neurone (Doughty et al. 1998).

Blocking the same portion of the total I_Ca at the calyx by partially blocking only a specific channel subtype blocks different portions of transmitter release, depending on which channel subtype is blocked. This is often referred to as different channel subtypes having different apparent co-operativity or efficacy ('m') in triggering transmitter release. Interpretations of these results are difficult; different hypotheses may lead to the same observed differences in m (Wu et al. 1999). Nevertheless, the results do indicate that different calcium channel subtypes probably differ in how they are located with respect to the releasable vesicles (the number of channels per vesicle and their distance from vesicles). In addition, part of the R-type, but not the P/Q-type, channels is located on the non-innervating side of the calyx (Wu et al. 1999). Here, the distance from the releasable vesicles at active zones (AZs) is most likely so large that, during single APs, these channels contribute only marginally to the local [Ca²⁺] transients that trigger transmitter release into the cleft (Meinrenken et al. 2002).

Finally, developmental studies have shown that the contributions of N- and R-type calcium channels to Ca²⁺ influx gradually decline (Iwasaki & Takahashi, 1998), leaving P/Q-type channels as the predominant subtype to trigger release from about P10 onwards (Iwasaki et al. 2000; but see discussion in Wu et al. 1999).

Hodgkin-Huxley (HH) Model for time course of single calcium channel currents. Even though I_Ca is mediated by different channel subtypes, with presumably different gating behaviours, no significant difference in the time courses of Ca²⁺ entry through these subtypes has been reported (except for a lower activation threshold of R-type channels; Wu et al. 1999). Hence the time course of the measured whole-cell I_Ca can be readily described by an HH Model that assumes a single channel subtype with two gates (Borst & Sakmann, 1998).

The HH Model offers the advantage that its parameters are easily compared to other published values. In comparison to somatic calcium channels, calcium channels at the calyx activate and deactivate rapidly. Because of their fast activation, they are efficiently opened even by the 'brief' AP; longer lasting activation potentials imposed on the calyx increase the Ca²⁺ current by only ~40 % (Borst & Sakmann, 1998). The peak of the open probability is reached at about 240 µs after the peak of the AP. With an increasing driving force during the repolarisation phase of the AP, the peak of the predicted Ca²⁺ influx is reached even later, when the predicted channel open probability has already decreased. This allows for a fast time course of I_Ca, because Ca²⁺ flux is restricted to the brief time window when the membrane potential is already repolarising but the calcium channels are still open. Consequently, the decay of I_Ca is more rapid than that of the AP (Fig. 1B).

Single channel current, open probability, and total number of calcium channels. Since single calcium channel recordings at the calyx have not yet been possible, the channel current can only be estimated. In chick ciliary ganglion neurones (Church & Stanley, 1996) and in smooth muscle cells (Gollasch et al. 1992), calcium current through L-type channels was ~0.1 pA (at 2 mM external [Ca²⁺] and a holding potential of -30 mV). The calyx whole-cell I_Ca of 2.5 nA (reached at a membrane potential of -30 mV) thus implies 25 000 open channels in the entire calyx membrane (during an AP) or, on average, ~10 open channels per µm² membrane (membrane area of the calyx ~2500 µm²; Sätzler et al. 2002).

The actual open probability of calcium channels at the calyx has not been measured. Based on single channel measurements in cloned calcium channels, even the maximum open probability (i.e. the time-average open probability during large depolarisations) may be as low as 10-20 % (P/Q-type channels; Colecraft et al. 2001). During APs, the open probability will be about 30 % lower still. This means that the total number of calcium channels at the calyx (open and closed) may be substantially larger than 25 000.

Glutamate release measured as excitatory postsynaptic current (EPSC). In voltage clamp, the EPSC in response to the presynaptic I_Ca can be measured. Its time of onset (5 % of peak) is at about 500 µs after the peak of I_Ca. The EPSC rises to a peak of 5.3 ± 0.4 nA (at a holding potential of -80 mV), with a 20-80 % risetime of 200-290 µs (Borst & Sakmann, 1996). It decays more slowly, within several milliseconds (Fig. 1B; for temperature effects and changes during early development, see Taschenberger & von Gersdorff, 2000). The measured EPSC is the combined effect of 210 ± 22 quantal EPSCs (the postsynaptic response to glutamate released from a single vesicle). Quantal EPSCs have an average amplitude of 32 ± 2 pA and a risetime of 130 µs (with cyclothiazide, an agent that reduces AMPA receptor desensitisation; Bollmann et al. 2000).

The contribution of individual quantal EPSCs to the composite EPSC is complicated, for different reasons. (1) Quantal EPSCs start at different times so that the first ones may already be decaying while later ones are just starting ('convolution' of quantal EPSCs into composite EPSC). This explains why the risetime of the composite EPSC is longer than that of the quantal EPSC and why the composite EPSC amplitude (5.3 nA) is smaller than the summed amplitude of the quantal EPSCs (210 32 pA = 6.7 nA). (2) Postsynaptic glutamate receptors can desensitise, thus rendering the postsynaptic current a non-linear measure of glutamate release, especially for large EPSCs and/or repetitive stimulation. (3) Glutamate released in response to APs may be insufficiently cleared from the cleft and 'spill over' to neighbouring AZs. As a result of (2) and (3), the incremental current elicited by consecutive quantal events may change in amplitude and time course, so that the composite EPSC is no longer a simple convolution of quantal EPSCs.

Inferring the absolute number of released vesicles as well as the time course of release from the recorded, composite EPSC is anything but trivial. Still, by 'linearising' the postsynaptic response with cyclothiazide and kynurenic acid and by employing mathematical techniques to deconvolve the EPSC, one can back-calculate the original release rate even for prolonged stimulation of the synapse (Neher & Sakaba, 2001). More recently, capacitance measurements at the calyx have been added as assays for exo- and endocytosis. They may provide an independent measure of the reliability of EPSC-based release measurements during prolonged stimulation, with some as yet unresolved discrepancies (Sun & Wu, 2001; Sun et al. 2002).

Stability of the time course of release. The amplitude of unitary EPSCs evoked by stimulation of a single presynaptic axon is often used as a relative measure of transmitter release. But is it justified to 'translate' a threefold reduction in EPSC amplitude into a threefold reduction in the number of released vesicles? The answer is 'yes' (for physiological and smaller EPSCs, i.e. when the postsynaptic response is sufficiently linear). The reason for this goes to the heart of a much debated issue of transmitter release: whereas the onset of the EPSC evoked with reduced external [Ca²⁺] is delayed (Taschenberger & von Gersdorff, 2000), the relative time course of the EPSC is remarkably stable, even if the number of released quanta is reduced more than 100-fold (van der Kloot, 1988; Isaacson & Walmesley, 1995; Borst & Sakmann, 1996). Since the time course of the probability of release is stable, the scaling factor between the summed quantal amplitudes and the actual amplitude of the composite EPSC remains largely the same (6.4 vs. 5.3 nA in the above example), even if the number of released vesicles is reduced. Therefore, the relative reduction in EPSC amplitude may be used as a measure of the relative reduction in the number of released vesicles.

Volume-averaged [Ca²⁺] dynamics during single APs. Using fluorescence Ca²⁺ imaging (Fig. 1C-E), Helmchen et al. (1997) have measured the resting [Ca²⁺] in the calyx to be ~50 nM. In the absence of any buffer(s), the 0.91 pC [Ca²⁺] entering during a single AP would raise the volume average [Ca²⁺] to 12 µM (volume ~400 µm³; Helmchen et al. 1997). However, once spatial and chemical equilibria are reached, 97.5 % of the entered Ca²⁺ is bound to buffer(s), so that the [Ca²⁺] reaches ~50 nM plus (1-0.975) 12 µM = 350 nM within a few milliseconds after I_Ca ends (the Ca²⁺ binding ratio (~buffer concentration/K_D) is 26-71, with an average of 40; Helmchen et al. 1997). After the AP, on a much slower time scale, the decay of [Ca²⁺] is well described by a single compartment model with a linear, non-saturable clearance mechanism operating at 400 s^-1. Since this clearance is accompanied by simultaneous unbinding of Ca²⁺ from the buffer(s), the effective decay time constant of [Ca²⁺] is not 1/(400 s^-1) = 2.5 ms, but is instead 106 ± 6 ms (Helmchen et al. 1997). Note that the binding ratio as well as the effective decay time constant apply only to single APs.

The study of Helmchen et al. (1997) further suggests that the endogenous Ca²⁺ binding ratio at the calyx is due mostly to a non-diffusible buffer (i.e. one not washed out during dialysis) and possibly to other, fast Ca²⁺ sequestration mechanisms. Diffusible buffers, notably ATP, are present as well, but with a binding ratio much below 40. The agreement of transmembrane I_Ca during a single AP with the change in presynaptic [Ca²⁺] (Borst & Helmchen, 1998) shows that any additional increase in [Ca²⁺] by release from intracellular stores is marginal. In agreement with this conclusion, depletion of intracellular Ca²⁺ stores with thapsigargin does not affect EPSC amplitudes (Chuhma & Ohmori, 2002). However, Ca²⁺ stores such as mitochondria, which have been shown to be preferentially located near release sites in the cat calyx of Held (Rowland et al. 2000), do participate in Ca²⁺ sequestration (Billups & Forsythe, 2002).

Release under non-physiological conditions

Since the calyx can be voltage clamped and the composition of its cytosol controlled, it is an ideal preparation for imposing non-physiological conditions to study the relationship between presynaptic Ca²⁺ signals and release, including reduced [Ca²⁺] of the external solution (Borst & Sakmann, 1996), changes in AP waveform (Borst & Sakmann, 1999), subtype-specific calcium channel toxins (Wu et al. 1999; see above), and intracellular Ca²⁺ buffering by the addition of BAPTA or EGTA to the presynaptic cytosol (see references below).

Intracellular Ca²⁺ buffering. As an assay for Ca²⁺ diffusion distances, the exogenous buffers BAPTA and EGTA provide information about the spatial organisation of calcium channels and releasable vesicles. In contrast to the squid giant synapse (Adams et al. 1985; Adler et al. 1991), 10 mM EGTA in the presynaptic cytosol of the calyx reduces the average release probability of vesicles to about half (Borst & Sakmann, 1996). This suggests that the average diffusion distance between vesicles and release-relevant calcium channels in the calyx is larger than that in the squid terminal. In this context, it was surprising that the much faster binding buffer BAPTA (1 mM) reduced release from the calyx with similar efficacy to that of 10 mM EGTA (Borst et al. 1995). One of the solutions to this conundrum has been a possible local saturation of BAPTA (but not EGTA) during presynaptic Ca²⁺ signalling (Naraghi & Neher, 1997). However, this mechanism now appears less likely, particularly since the local [Ca²⁺] levels near releasable vesicles at the calyx have been shown to be 10-30 µM rather than the previously assumed hundreds of micromolar (see below).

Changing the channel open probability. Regarding how many calcium channels contribute to the Ca²⁺ signal at any one vesicle, experiments with EGTA first suggested that, at the calyx, overlapping domains of multiple channels control transmitter release (Borst & Sakmann, 1996). This view has been confirmed by the observation that reducing Ca²⁺ influx by reducing the number of open channels (via variable AP waveform commands in voltage-clamped terminals) reduces the release probability with the same supralinear dependence as reducing the [Ca²⁺] in the extracellular solution (from 3rd to 4th power; Borst & Sakmann, 1999).

Photolysis of caged Ca²⁺ compounds

As soon as it appeared likely that the presynaptic Ca²⁺ signal was characterised by steep concentration gradients (Augustine & Neher, 1992), it was obvious that measurements of volume-averaged [Ca²⁺] during APs probably did not reflect the [Ca²⁺] sensed by the putative molecular sensor that triggers transmitter release. This has prompted several studies which used Ca²⁺ uncaged from photosensitive compounds to produce spatially homogeneous Ca²⁺ signals in fast synapses including the squid giant synapse (Delaney & Zucker, 1990), the goldfish retinal bipolar neurone (Heidelberger et al. 1994), the cochlear inner hair cell (Beutner et al. 2001), and the calyx of Held (Bollmann et al. 2000; Schneggenburger & Neher, 2000). By measuring the rate of release in response to step increases in presynaptic [Ca²⁺], these studies devised kinetic models which predict the time course and amplitude of the release rate in response to a given [Ca²⁺] transient (Fig. 2A and B). Although the models for the different preparations vary in detail, a common property is that the predicted maximum release rate varies supralinearly with [Ca²⁺] (with approximately 4th power dependence). This implies an intrinsic supralinearity of the putative sensor and explains the observed supralinear dependence of transmitter release on both the extracellular calcium concentration (Dodge & Rahamimoff, 1967) and the presynaptic Ca²⁺ currents (Augustine & Charlton, 1986).

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Figure 2. Ca2+ sensitivity and kinetic model for putative sensor A, maximum release rates of calyx in response to step-increases of presynaptic, intracellular [Ca2+] (after Bollmann et al. 2000). B, cartoon of kinetic release model (Bollmann et al. 2000). The corresponding putative sensor binds 5 Ca2+ (a and b; reversible), then switches to a release-promoting state (c; reversible), and finally triggers the fusion of the releasable vesicle (d; irreversible). C, simulation of release rate and EPSC (thin line) evoked by a brief increase of intracellular [Ca2+]. Rate and EPSC were shifted by 250 µs to the right, to allow comparison to a measured, AP-evoked EPSC (thick line) (after Bollmann et al. 2000). D, model-predicted vesicle release probability (Pr, vesicle) (Bollmann et al. 2000) in response to a hypothetical [Ca2+] transient with the same time course as that of ICa (~Gaussian, fwhm = 383 µs) and an amplitude of [Ca2+]vesicle (after Meinrenken et al. 2002).

Photolysis studies at the calyx did not provide direct evidence for differences in the intrinsic Ca²⁺ sensitivity across vesicles. Whilst such heterogeneity has been observed at the calyx during long, non-physiological Ca²⁺ signals and during recovery from short-term depression (Schneggenburger et al. 2002), intrinsic heterogeneity in those vesicles that contribute to phasic release during a single AP was not observed (in contrast to e.g. the sea urchin egg; Blank et al. 1998). However, some intrinsic heterogeneity at the calyx may nonetheless be present during AP-evoked release. A 'small' heterogeneity - we estimate a maximal 2-fold difference in release rate at a [Ca²⁺] of 10 µM - would be beyond the resolution of the experiments.

The calyx studies estimated the peak [Ca²⁺] sensed by a 'typical' readily releasable vesicle during an AP to be ~9 µM (Bollmann et al. 2000; Fig. 2C) and ~28 µM (Schneggenburger & Neher, 2000). Although the two models differ, in a simulation of physiological Ca²⁺ signalling and release at the calyx, the release model of either study produces results consistent with all of the above electrophysiological data (Meinrenken et al. 2002). The models differ particularly when inferring the release probability of single vesicles, and thus the intrinsic Ca²⁺ sensitivity of the putative, molecular release apparatus. With one model, a roughly Gaussian [Ca²⁺] transient of fwhm ~400 µs and 9 µM amplitude causes a 25 % release probability per vesicle (Bollmann et al. 2000). With the other model, a broader [Ca²⁺] transient (fwhm ~500 µs) with a higher amplitude (28 µM) causes only a 10 % release probability per vesicle (Schneggenburger & Neher, 2000). (For comparison, to obtain the same release probability, the less Ca²⁺-sensitive release model proposed for the goldfish retina bipolar neurone (Heidelberger et al. 1994) would require a [Ca²⁺] amplitude of 180 µM (assuming a Gaussian transient of fwhm ~400 µs). The corresponding EPSC (calculated by superposition of quantal EPSCs; Bollmann et al. 2000) has a risetime of 340 µs and an onset at 155 µs after the peak of the [Ca²⁺] transient.)

The discrepancy between the two models of release for the calyx stems partly from converting measurements of the release rate for aggregate release from the entire terminal to release rates per vesicle. This conversion depends on estimates of the total number of releasable vesicles ('pool size'; ~800 in Bollmann et al. 2000 vs. ~2000 in Schneggenburger & Neher, 2000; for review see Schneggenburger et al. 2002). Although both models were calibrated to describe the same molecular release apparatus of the calyx in young rats, they differ not only in the release-relevant steps downstream of Ca²⁺ binding, but also in the kinetic properties of the five Ca²⁺ binding sites (binding rate constants, affinity, and co-operativity). The fact that different models are able to describe transmitter release in the same preparation shows that a mere comparison of release models with models describing the Ca²⁺ binding proteins in vitro may be insufficient to unequivocally identify Ca²⁺ sensor molecules. Differences in the overall Ca²⁺ sensitivity of the release process across different preparations are clearly present. However, these may be due to several factors (Kasai, 1999), including differences in steps downstream of the actual Ca²⁺ binding itself. Currently, these steps are usually captured as one final, rate-limiting step ('').

Plotting the predicted release probability in response to an assumed, I_Ca-shaped [Ca²⁺] transient on a linear scale reveals one of the main functional advantages of the putative sensor's supralinearity: transmitter release at low [Ca²⁺] is negligible (Fig. 2D). Consequently, release is largely constrained to a brief period during which the [Ca²⁺] near vesicles is above 'threshold'.

3-D reconstruction of the calyx anatomy

Reconstruction of the morphology from EM micrographs of serially sectioned synapses has become a valuable tool to visualise the anatomy of synapses at nanometre resolution (e.g. Harris et al. 1992). This may include identification of the relative locations of calcium channels and docked vesicles at AZs (Harlow et al. 2001) as well as estimation of the number of readily releasable vesicles.

A recent full reconstruction of a serially sectioned calyx (P9, rat; Sätzler et al. 2002) reveals a large, cup-like structure which covers almost half of the spherical postsynaptic cell (diameter ~20 µm, Fig. 3A and B). The calyx has a volume of ~480 µm³, and a membrane area of ~2500 µm². The rims of the cup form finger-like stalks, which have been shown to grow more pronounced in maturing calyces past P9 (Kandler & Friauf, 1993). Two percent of the presynaptic area (or 5 % of the apposition area) consists of specialised AZs which harbour the release sites of the calyx and face the synaptic cleft. The 554 AZs cover approximately circular patches of membrane, of varying size (mean = 0.1 µm², CV = 0.9). About 6 % of the AZs, instead of being flat patches, cover spine-like protrusions of the postsynaptic membrane into the presynaptic membrane.

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Figure 3. 3-D reconstruction of calyx anatomy at the EM level A, 3-D EM reconstruction of a single calyx (yellow) and its medial nucleus of the trapezoid body (MNTB) principal cell (blue). B, contour-lined single-sectional cut through A, showing individual AZs (red) and puncta adhaerentia (magenta). C, 3-D view of a cluster of vesicles (green) at a single AZ (red). Scale: vesicle diameter, 45 nm. D, electron micrograph through single active zone, showing the pre- and postsynaptic (bottom) membrane, as well as anatomically docked and non-docked vesicles (A-C after Sätzler et al. 2002).

The centre of any AZ is separated from that of the nearest neighbour AZ by at least 0.15 µm (mean separation = 0.59 µm, CV = 0.5). Given the limited diffusion of Ca²⁺ in Ca²⁺-buffering cytosol, most AZs are placed sufficiently far from each other that local Ca²⁺ signals, and thus release, at each AZ can be modelled as independent events (Meinrenken, 2001). The calyx may thus be viewed as a structure containing hundreds of functional synapses with release-relevant Ca²⁺ signals acting independently but in parallel (Sätzler et al. 2002; Schneggenburger et al. 2002).

Based on anatomical estimates, the calyx contains at least 70 000 vesicles, which are clustered within hundreds of nanometres from an AZ, with about one-half of all vesicles located within 200 nm distance of the nearest AZ (Fig. 3C). At each AZ, an average of 2.0 vesicles are anatomically 'docked' (where no cytoplasmic space between vesicular and calyx membrane is visible at EM resolution, Fig. 3D). A docked vesicle has been proposed to be the anatomical correlate of a readily releasable vesicle (Schikorski & Stevens, 2001). In fact, the estimated 2 554 = 1108 docked vesicles, in the one calyx studied at the EM level, is consistent with earlier, electrophysiological measurements of the number of vesicles that are readily releasable by AP-evoked Ca²⁺ signals (an average of ~800 in several studies; Schneggenburger et al. 2002).

At the calyx, docked vesicles do not seem to form organised structures such as lines or rings, or to occupy a preferential location, such as the centre or the edge of AZs. Since the location of calcium channels at the calyx cannot (yet) be visualised at EM resolution, a possible topographical organisation between the calcium channels and individual vesicles cannot be measured. Such an organisation of the presynaptic cytomatrix was shown for the frog neuromuscular junction (Harlow et al. 2001) as well as for other terminals (Dresbach et al. 2001).

Single compartment modelling

The above measurements would provide an almost complete description of Ca²⁺ signalling and release at the calyx, if effects of spatial [Ca²⁺] gradients due to diffusion were ignored. The only missing parts are the concentration(s) and kinetic properties of the endogenous buffer(s). Before we face the conceptual complications of Ca²⁺ diffusion, we might ask whether the domain-like, spatial [Ca²⁺] gradients around calcium channels are in fact necessary to control the time course of phasic transmitter release, or whether they are simply an unavoidable side effect of limited diffusion coefficients.

To investigate this, we model the calyx as a single compartment, assuming that Ca²⁺ entering the calyx diffuses unrealistically fast, such that any spatial gradients dissipate instantaneously (spatial equilibrium model). To comply with the measured endogenous Ca²⁺ binding ratio (~40) we assume that the calyx contains 80 µM of an unidentified buffer, with a single binding site of K_D = 2 µM. The Ca²⁺ entering during the AP (not instantaneously but with a time course given by the above whole-cell I_Ca) perturbs the chemical equilibrium and results in a [Ca²⁺] of ~400 nM, once the Ca²⁺ influx stops and the new equilibrium is reached (assuming a resting [Ca²⁺] of 50 nM and a total Ca²⁺ influx of 12 µM per volume). We explicitly calculate the time course of [Ca²⁺] between the two equilibria by first order binding of Ca²⁺ to the buffer. In this model, the time course and amplitude of [Ca²⁺] are now completely determined by only a single free parameter, the forward binding rate constant of the buffer (K_on). In case of a very high K_on of 10⁹ M^-1 s^-1 (i.e. one approaching the diffusion limit), Ca²⁺ entering the calyx is buffered so fast that the [Ca²⁺] reaches a maximum of only ~700 nM, and returns to ~400 nM within 1 ms after the onset of I_Ca. The predicted release probability for this [Ca²⁺] transient is 0.013 % (using the release model from Bollmann et al. 2000), which is more than three orders of magnitude less than the 25 % observed in experiments. In the case of a low K_on of 10⁶ M^-1 s^-1, the [Ca²⁺] reaches a peak of 11.7 µM and returns to below 1 µM only after 40 ms. The predicted release probability reaches 100 % within 2 ms of the onset of I_Ca, resulting in a complete depletion of the vesicle pool.

At a moderate K_on of 3.4 10⁷ M^-1 s^-1, the [Ca²⁺] transient reaches a peak of 6.7 µM, has a fwhm of 630 µs, and returns to below 1 µM within 1.5 ms of the onset of I_Ca. The predicted release probability is 25 %. The resulting EPSC (calculated as described in Bollmann et al. 2000) has a risetime of ~500 µs (with an onset at 350 µs after peak of I_Ca). Given the simplicity of this model, the agreement with the experimental data is remarkable. In fact, returning to the above question, one might conclude that phasic transmitter release would be possible even if Ca²⁺ diffusion were much faster and therefore spatial gradients of [Ca²⁺] were negligible, or in other words, if presynaptic [Ca²⁺] signalling were not local at all.

Unfortunately, the simplicity of a spatial equilibrium-model comes at a cost. By implying that the amplitude and time course of the release-relevant [Ca²⁺] signal are controlled solely by I_Ca and the concentration and kinetics of the buffers, the model misrepresents the effect of buffers during transmitter release. Therefore, it may lead to false conclusions regarding the functional relevance of differences in endogenous buffers across different synapses (see also 'Comparison with other concepts - kinetic competition' in 'Conclusions and questions', below). Likewise, the above model fails to describe the observed effects of exogenous buffers. For example, adding 1 mM BAPTA to the compartment (K_D = 220 nM, K_on = 4 10⁸ M^-1 s^-1; Naraghi & Neher, 1997) reduces the peak of the predicted [Ca²⁺] transient from 6.7 to 0.15 µM. The resulting release probability (~10^-6 % in 5 ms) is many orders of magnitude smaller than that observed in experiments with 1 mM BAPTA (Borst et al. 1995). As expected, more realistic, spatially non-uniform models are required to describe presynaptic [Ca²⁺] signalling.

The local presynaptic Ca²⁺ signal - a minimum model

While we can determine the volume-averaged [Ca²⁺] before and a few milliseconds after an AP, any local processes in the meantime, i.e. before spatial and chemical equilibria of Ca²⁺ and its buffers are re-established, cannot (yet) be measured at sufficient spatio-temporal resolution. Instead, local diffusion and buffering of Ca²⁺ may be addressed by quantitative modelling only. Critics might warn that 'a sufficient number of equations and parameters will explain just about anything'. Indeed, stochastic channel gating, spatial heterogeneity, competitive binding to multiple buffers, and finally the supralinearity of the proposed release models make robust predictions difficult, and different parameter sets may produce similar, aggregate results.

While acknowledging this, we review here a numerical model that predicts the time course and amplitude of [Ca²⁺] at the vesicles and, consequently, the time course and amplitude of transmitter release. As a 'minimum' model, it does not claim to offer a comprehensive description of calcium-secretion coupling. Rather, it comprises a minimum biophysical cascade (Table 1) that can reproduce the available experimental data of the calyx.

Inferring a channel-vesicle topography at active zones

The 'local' Ca²⁺ signal sensed by a vesicle is affected by the spatial pattern of the calcium channels that are close enough to contribute to the local change in [Ca²⁺]. In fact, without knowledge of this spatial pattern (the topography of the release sites) the Ca²⁺ signal and the resulting release probability of the vesicle cannot be quantified, even if the channel current, the biochemical properties of the cytosol, and the sensitivity of the Ca²⁺ sensor for release are known (Neher, 1998a). At present, the release site topography at most synapses, including the calyx, cannot be measured. This is unfortunate since differences in the release site topography could be responsible for differences in release characteristics across fast synapses (Augustine, 2001).

Aided by advances in quantitative modelling of buffered Ca²⁺ diffusion and by faster computers, a number of theoretical studies have investigated the relevance of release site topography (e.g. Zucker & Fogelson, 1986; Yamada & Zucker, 1992; Chow et al. 1994; Cooper et al. 1996; Klingauf & Neher, 1997; Bennett et al. 2000; Gil et al. 2000). In a recent study, the calyx data on membrane currents, Ca²⁺ sensitivity, and geometry of AZs were used to infer the topography of release sites of the calyx (Meinrenken et al. 2002): at each AZ, the Ca²⁺ that controls phasic release is supplied by ten or more calcium channels. These are grouped into one or a few clusters, with a channel cluster covering an area of less than 50 nm across (Fig. 4A). While the simulation does not predict the exact distances between readily releasable vesicles and the channel clusters, it does imply that these distances vary widely across vesicles in the readily releasable pool (CV of distance ~0.5). An assumed random location of releasable vesicles and channel clusters anywhere within AZs reproduces the experimental data (cluster-to-vesicle distance 30-300 nm, average ~120 nm, S.D. ~60 nm). This topography was chosen because it was particularly suited to account for two sets of seemingly contradictory experimental results. Briefly, the heterogeneity in the channel-vesicle distances was essential for reproducing the relative efficacies of internal BAPTA and EGTA in reducing release. The clustering of the calcium channels was chosen to account - at the same time - for experiments suggesting that the variance in the [Ca²⁺] signals sensed by the releasable vesicles is low. The non-uniform topography results in a heterogeneous release probability of vesicles in the readily releasable pool, with release probabilities in response to a single AP ranging between < 1 % and 100 % (Fig. 4D). Heterogeneous release probability at the calyx (with channel-mediated Ca²⁺ transients) has been observed experimentally (Sakaba & Neher, 2001). This calyx model assumes that this heterogeneity is 'positional', i.e. of merely spatial rather than of intrinsic or biochemical origin. Clearly, an additional, intrinsic heterogeneity of the vesicles could be superimposed.

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Figure 4. A minimal model of local, non-uniform Ca2+ signalling at the calyx A, presynaptic membrane with active zone (AZ), calcium channel cluster, and three readily releasable vesicles. The location of cluster and vesicles, as well as the number of vesicles on the active zone, represent one example of random placement at multiple active zones. Drawings are to scale. B, as A but viewed from the top and with superimposed false colour coding of [Ca2+] on the membrane around the channel cluster. The concentrations reflect a simulation for the control case at time = 0.60 ms (see E). C, as B but viewed from the side, with an additional, vertical panel to show [Ca2+] in the plane perpendicular to the membrane (time = 1.15 ms). Dashed circle indicates a vesicle that has already fused. D, variable channel-to-vesicle distances result in [Ca2+] transients with variable peaks and thus heterogeneous release probability. A-D are excerpts from a video 'The Ca2+ signal controlling phasic transmitter release', available from the authors or at: http://sunny.mpimf-heidelberg.mpg.de. E, AP (trace 1, time course only) and whole-cell Ca2+ current (as predicted by the Hodgkin-Huxley Model; trace 2, left axis). Red traces show two simulated [Ca2+] transients (right axis, each normalised to the same amplitude) the average across all vesicle positions (trace 3) and at 200 nm from the channel cluster (trace 4). The first vertical dashed line from left indicates the time of peak AP (0.54 ms), and the second line indicates the time of peak Ca2+ influx (0.92 ms). F, predicted, average release rate of calyx (trace 1, right axis) and predicted EPSC (trace 2, left axis). For comparison, a measured, AP-evoked EPSC (Bollmann et al. 2000) is also shown (trace 3; shifted by ~400 µs to the left to allow comparison of the time course). The vertical dashed line indicates the time of the peak of the predicted release rate (1.03 ms) (E and F after Meinrenken et al. 2002).

Calcium channel subtypes differ in whether they merely contribute to the total Ca²⁺ current or whether they actually control the release of transmitter (Wu et al. 1999). This is reflected in the topographic study (Meinrenken et al. 2002): clustered channels at AZs (P/Q-type and possibly N- and R-type) contribute 20-40 % of the total Ca²⁺ current, implying that the density of calcium channels at AZs is higher than that in the non-innervating membrane. The remainder of the total Ca²⁺ current is mediated by additional channels, most of which are located away from AZs and thus affect only the non-release-relevant, volume-averaged [Ca²⁺] (N- and R-type only). In particular, this includes channels located in the non-innervating membrane opposite the synaptic cleft. While one expects that some of the additional calcium channels are located closer to AZs, their contribution to the net release probability of the calyx is likely to be small. The simulated Ca²⁺ signal arising from such non-clustered channels does not reproduce the experimental results (Meinrenken et al. 2002).

Time course of signalling in the calyx

Figure 4E and F shows the time course of the presynaptic signalling cascade, superimposing those quantities that can be measured in experiments (AP, I_Ca, and EPSC; black traces) with those that can only be calculated ([Ca²⁺] transients and the release rate; red traces). The results are for a calyx containing (at rest) 50 nM Ca²⁺, 4 mM ATP (with 0.58 mM available for Ca²⁺ binding), and 80 µM of an unknown, non-diffusible buffer with an affinity of 2 µM and a forward binding rate constant of 5 10⁸ M^-1 s^-1. Ca²⁺ enters at each AZ through a cluster of calcium channels (Meinrenken et al. 2002).

Vesicles at different distances from the channel cluster are controlled by the same kinetic model for the Ca²⁺ sensor, with five Ca²⁺ binding sites, each with a K_D of 10 µM (Bollmann et al. 2000). Note that Fig. 4E shows a predicted, smooth Ca²⁺ current (left axis), which was averaged across > 100 APs and scaled to the amplitude of the whole-cell I_Ca. The actual AP-evoked I_Ca mediated by a single cluster of calcium channels is 'noisy', owing to the stochastic opening and closing of individual channels (average peak = 0.66 pA per cluster).

The Ca²⁺ influx through the channel cluster gives rise to [Ca²⁺] transients that vary in amplitude depending on the distance from the cluster (Fig. 4E right axis; transients are rescaled to the peak of I_Ca to allow comparison of the time courses). Transients shown in the figure were averaged across > 100 APs. The transients' peaks range from tens of micromolars (at 30 nm from the cluster centre) to ~1 µM (at 200 nm). Interestingly, the time courses of the transients are remarkably similar (having fwhm values between ~380 and ~480 µs for distances from 30 to 200 nm, respectively). The average [Ca²⁺] across all vesicles (30-300 nm distance) follows the time course of I_Ca with a lag of ~50 µs. This confirms the common finding that the diffusion times of Ca²⁺ contribute only a small fraction to the total synaptic delay (Yamada & Zucker, 1992; Borst & Sakmann, 1996).

The [Ca²⁺] transients sensed by the different vesicles result in an average release rate which varies in time (Fig. 4F). Convolving this release rate with the quantal EPSC (Bollmann et al. 2000) yields the model-predicted compound EPSC. For comparison, Fig. 4F includes the measured, AP-evoked EPSC of the calyx (with cyclothiazide; Bollmann et al. 2000). Its risetime (424 µs) is longer than that of the model-predicted EPSC (~300 µs). Its onset is later than that of the predicted EPSC (~90 µs after the peak of I_Ca ). To allow comparison of the time courses, in Fig. 4F the measured EPSC was shifted to the left, by about 400 µs.

Effect of release model parameters and release site topography on synaptic delays. The above mismatch between the predicted and the measured synaptic delays is expected. It is consistent with the finding that a large part of measured synaptic delay is caused by processes downstream of the Ca²⁺-dependent response of the molecular release apparatus (Borst & Sakmann, 1996). These processes, which are not (yet) included in the model, delay the onset of the EPSC and presumably increase its risetime, unless the additional delay of all quantal EPSCs is the same. Still, the discrepancy of approximately 400 µs may seem quite large. Therefore, we investigated whether the short delay time predicted by the calyx model is caused by: (1) the choice of the release model parameters; or (2) the release site topography.

(1) Using the calyx release model as described by Schneggenburger & Neher (2000) yields a predicted EPSC with a risetime of 295 µs and an onset only 20 µs later than that based on the release model described by Bollmann et al. (2000). While the forward Ca²⁺ binding rate constants in the former model (0.9 10⁸ M^-1 s^-1) are about 3 times lower than those in the latter model, this effect is counteracted by the about 2.5-times-higher local [Ca²⁺] transient required to achieve the release probability (10 %) as measured under control conditions (Meinrenken et al. 2002).

(2) During a single AP, the release predicted by the above topography comprises mostly vesicles that are located in the immediate vicinity of the calcium channels (within 30 nm of the cluster centre). Since these vesicles sense comparatively large [Ca²⁺] transients, this topography contributes to shorten synaptic delay. However, the effect is still small in comparison with the above discrepancy of 400 µs. Moving all releasable vesicles to a hypothetical, uniform distance of ~100 nm from the channel cluster centre yields a predicted EPSC with a risetime of 410 µs and an onset 150 µs later than that based on the non-uniform topography (both using the release model as described in Bollmann et al. 2000).

In summary, the current model of calcium-secretion coupling at the calyx lacks additional interactions that would account for an additional 200-400 µs in the synaptic delay. However, this discrepancy does not imply that the Ca²⁺ binding rate constants of the sensor are lower or that the release site topography is different. Likewise, our findings on the signalling cascade time course and its determinants are not restricted to only specific release site topographies (such as that proposed for the calyx) or narrow parameter ranges of putative Ca²⁺ sensors.

Time course of [Ca²⁺] transients

Upon inspection of Fig. 4E, perhaps the most striking prediction is how well the [Ca²⁺] transients 'follow' I_Ca . While the [Ca²⁺] transients at different distances from the channel cluster differ in amplitude, their time courses are very similar to that of I_Ca , even at 200 nm from the cluster. Since I_Ca is, in turn, controlled by the voltage-sensitive gates of the calcium channels and the electrochemical driving force, it is ultimately the AP itself (and its dependence on other presynaptic ion channels; Meir et al. 1999) that controls the time course of [Ca²⁺] at the vesicles.

Why are the time courses of the [Ca²⁺] transients so similar to that of I_Ca? There are two reasons. One relates to the times of onset, peak and offset of the [Ca²⁺] transient. The second relates to the overall shape of the transient.

(1) Rise and decay of the (noisy) I_Ca are sufficiently 'slow' to be followed by the buffered diffusion system that governs [Ca²⁺]. In a buffered diffusion system, Ca²⁺ and buffers reach steady state concentrations in response to a given Ca²⁺ influx (Neher, 1998b). Because diffusion and buffer equilibration are sufficiently fast, the time to reach this steady state is short enough (tens of microseconds, depending on buffers and distance from channels) that the steady state can follow the slowly changing I_Ca (fwhm ~400 µs), even at distances of hundreds of nanometres. This remains true for physiological temperatures at which Ca²⁺ currents at the calyx have been measured to be faster, with a fwhm value of about 200 µs (Borst & Sakmann, 1998). Note that the slow rise and decay of I_Ca are due to the fact that, according to the calyx model, I_Ca at each AZ is mediated by a cluster of ~10 or more calcium channels. Their aggregate current is sufficiently smooth, even though individual channels of the cluster open and close abruptly and thus cause steps in the Ca²⁺ flux. In other synaptic terminals, the Ca²⁺ signal reaching individual vesicles may instead be dominated by a single channel, and consequently by a more step-like Ca²⁺ current (e.g. in the squid giant synapse, Augustine et al. 1991). In such a case, the times of onset, peak, and offset of [Ca²⁺] could deviate from those of the individual channel current, particularly at distances of more than ~100 nm. However, at smaller distances of tens of nanometres, as was suggested for the squid giant synapse, a steady state [Ca²⁺] in response to a step-like Ca²⁺ influx will be established within tens of microseconds (assuming diffusion and buffering properties similar to those at the calyx; see example calculation in Meinrenken et al. 2002). Consequently, for channel open times of 100 µs or longer, the time course of [Ca²⁺] at the vesicle will still resemble that of the single channel current.

(2) Steady state [Ca²⁺] is (nearly) proportional to I_Ca. Even given the relatively short time necessary to reach steady state with I_Ca, it is still surprising that the overall shapes of the [Ca²⁺] transients are very similar to that of I_Ca. In other words, not only are onset, peak, and 'offset' of the Ca²⁺ signal not much delayed compared to those of I_Ca, but the steady state of [Ca²⁺], at any one distance, is nearly proportional to I_Ca (neglecting the resting [Ca²⁺]). This is due to the fact that the local changes in the concentration of Ca²⁺-bound ATP, particularly at large distances from the channels where the effect of the buffer is strong, are sufficiently small (based on linearised steady state approximation; Naraghi & Neher, 1997).

Spatial profile of [Ca²⁺] transients

The peak [Ca²⁺] amplitude reached in the vicinity of an open Ca²⁺ source (single channel or cluster) decreases steeply within tens of nanometres, with increasing distance from the source. To illustrate a typical spatial profile, we use the following approximation for the [Ca²⁺] at steady state: [Ca²⁺] above resting level is proportional to the channel current, and further proportional to exp(-d/)/d, where d is the distance from the calcium channel. The parameter depends on the diffusion coefficient of Ca²⁺ as well as on the concentration and forward binding rate constant of diffusible, unbound buffer(s) (Neher, 1998b). The decrease in [Ca²⁺] becomes less steep as the distance d increases. For example, with = 28 nm (for ATP, as used in the above calyx model), the [Ca²⁺] will decrease by 58 % when moving from d = 5 nm to d = 10 nm. But it will decrease by only 20 % when moving from d = 100 nm to d = 105 nm. At larger distances it will decrease even less, especially since here the peak of [Ca²⁺] is not much larger than the resting [Ca²⁺] itself. At such distances, the [Ca²⁺] will be a function of I_Ca, the cytosolic volume, the buffer concentrations, and the buffers' affinities, rather than of the rate constants of the buffers (Naraghi & Neher, 1997). It should be emphasised that the spatial dependence described above applies only to situations with discrete, point sources of single calcium channels or several closely clustered channels from which Ca²⁺ diffuses freely into all three dimensions. For vesicles located within or above a large, dense field of channels (i.e. with channel spacing of tens of nanometres) diffusion is effectively limited to only one dimension and the spatial profile of [Ca²⁺], as well as its dependence on buffers, is markedly different (Meinrenken et al. 2002).

Given the complexity of this spatio-temporal dependence of [Ca²⁺], it may seem somewhat arbitrary to separate the space around a Ca²⁺ source into different regimes, such as 'close' and 'far'. Since the spatial profiles depend on the diffusion coefficients of Ca²⁺ and buffers as well as on the buffer concentrations and binding kinetics, such regimes may correspond to different absolute lengths in different synaptic terminals or may change during development. These difficulties notwithstanding, it is helpful to define such regimes, since this will facilitate the discussion of Ca²⁺ signalling in different synaptic terminals, for example in giant terminals vs. the much smaller synaptic boutons in mammalian CNS.

'Close' regime (within ~250 nm of the Ca²⁺source). In this spatial regime, the exact locations of the Ca²⁺ source(s) and releasable vesicles is crucial. Differences of only ~10 nm will result in functional differences of the terminal, for three reasons.

(1) The distance between a vesicle and its release-controlling channel(s) affects the release probability. For example, in the calyx model above, a vesicle placed at d = 80 nm (from the channel cluster) has a release probability of 26 %, vs. only 13 % if placed at a distance of 90 nm.

(2) The distance affects the synaptic delay. For example, in the above calyx model, 5 % of those vesicles at d = 30 nm that fuse during an AP will do so within 180 µs after the peak of the AP. The respective time for those vesicles at d = 60 nm is 425 µs. Note that the reason for the increased delay is not primarily that the Ca²⁺ signal reaches the farther vesicles later but instead that the peak amplitude of [Ca²⁺] reaching these vesicles is lower.

(3) The overall distribution of channel-to-vesicle distances across different release sites, as well as the number and spatial organisation of calcium channels controlling each vesicle, affects other properties of transmission, including the heterogeneity of the release probability, changes in the release probability during consecutive APs and the temporal fidelity of the postsynaptic response (Meinrenken et al. 2002).

'Far' regime (over ~250 nm from the Ca²⁺source). At distances of more than ~250 nm from the Ca²⁺ source, any changes in local [Ca²⁺] are comparatively small and slow, and they are largely governed by the change in the volume-averaged [Ca²⁺]. During single APs, this change is so small that it could not trigger phasic release, even if readily releasable vesicles were present in the 'far' regime. This has three important implications.

(1) The 500-600 AZs of a calyx may be considered multiple synapses which act independently and in parallel. At the calyx, individual AZs are separated by hundreds of nanometres so that [Ca²⁺] dynamics of individual AZs affect each other only via the 'far' regime, i.e. in the flat and slowly changing tails of the Ca²⁺ signals around AZs. Therefore, local release-relevant Ca²⁺ signals of individual AZs are largely independent, and Ca²⁺ signalling can be understood by focussing on individual compartments of (~500 nm)³ in size around each AZ (Meinrenken, 2001).

(2) Size and macroscopic shape of a terminal do not affect its fast Ca²⁺ signalling characteristics. The actual cup-like shape of the calyx at P8-10 is not relevant to fast Ca²⁺ signalling around AZs, provided a minimum core space for the 'close' regime around release sites is maintained (circa (500 nm)³, depending on diffusion coefficients and buffers). The shape of the terminal, of course, does affect the extrusion of Ca²⁺. A large surface to volume ratio, as in the case of the calyx allows for rapid extrusion of Ca²⁺.

(3) The 'far' regime affects only volume-averaged [Ca²⁺] during and after APs. Most details of the 'far' regime do not affect 'local' Ca²⁺ signals. For example, release-relevant Ca²⁺ signals can be modelled without specific knowledge of the properties of the 'far' calcium channels, of the identity and spatial distribution of the endogenous Ca²⁺ buffers, or of the terminal's geometry and total membrane area. However, while changes in [Ca²⁺] in the 'far' regime are too small to affect release during a single AP, they are important during consecutive APs: the gradual increase in volume-averaged, 'resting' [Ca²⁺] (~300 nM during a single AP at the calyx) will affect [Ca²⁺] during the following AP, in both the 'close' and the 'far' regime. This 'residual' Ca²⁺ affects short-term plasticity (Zucker, 1999). The determinants of the volume-averaged [Ca²⁺] - total Ca²⁺ influx, buffer capacity and cytosolic volume - are thus important parameters for the transmission characteristics of a synaptic terminal.

Effects of endogenous buffers on [Ca²⁺] signals

Endogenous buffers affect predicted Ca²⁺ signals and, thus, release. However, the effects differ substantially, depending on whether the buffer is diffusible (here ATP) or non-diffusible (endogenous fixed buffer, termed 'EFB'). The buffers' effects differ further with respect to [Ca²⁺] transients in the 'close' versus those in the 'far' regime. In the following, we demonstrate the effects of the buffers by removing ATP, EFB, or both buffers from the model cytosol. The predicted effects on [Ca²⁺] transients in the 'close' regime are demonstrated as changes in the average [Ca²⁺] across all vesicle positions (i.e. within 30-300 nm from the channel cluster, '[Ca²⁺]_{vesicle avg}'). The predicted effects in the 'far' regime are demonstrated as changes in the volume-averaged [Ca²⁺] (across entire calyx, '[Ca²⁺]_{volume avg}').

With both ATP and EFB in the model cytosol (control condition), [Ca²⁺]_{vesicle avg} reaches a peak of 8 µM and decays to ~400 nM (Fig. 5A, black traces). The predicted average release probability (after 5 ms) is 25 %. If both buffers are removed, [Ca²⁺]_{vesicle avg} reaches a peak of 26 µM and decays to 12 µM (not shown). The predicted release probability is 100 %, i.e. the vesicle pool would be fully depleted by a single AP.

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Figure 5. Effects of endogenous buffers on [Ca2+] transients Predicted [Ca2+] transients for different, hypothetical buffer conditions. Thick traces, [Ca2+] averaged across all vesicle positions (30-300 nm from channel cluster, left axes). Thin traces, [Ca2+] averaged across calyx volume (right axis). Black traces show control condition (ATP and EFB, in A-C). The vertical dashed line indicates the time of peak Ca2+ influx (0.92 ms). Red traces show hypothetical buffer conditions (different in A-C). A, no EFB (ATP only). B, as A but assuming the calyx to have infinite volume . C, no ATP (EFB only).

Endogenous fixed buffer. If only EFB is removed from the model cytosol (ATP remains), [Ca²⁺]_{vesicle avg} reaches a peak of 11 µM instead of 8 µM (Fig. 5A, red traces). This relatively small effect of EFB in reducing [Ca²⁺]_{vesicle avg} is expected, because EFB is non-diffusible. After it binds to Ca²⁺, EFB is not replenished by diffusion of fresh, unbound buffer (Naraghi & Neher, 1997). Instead, during the AP, EFB equilibrates with the local Ca²⁺ (if changes in I_Ca, relative to the kinetics of the buffer, are sufficiently slow; see above). Once locally equilibrated (for any location in the 'close' regime and for any given I_Ca), the rate of Ca²⁺ binding to the buffer equals that of unbinding. Thus the buffer no longer acts as a local Ca²⁺ sink. Despite the relatively small effect of non-diffusible buffers on [Ca²⁺] in the 'close' regime, the presence of EFB is nonetheless paramount: without EFB, the predicted release probability reaches 60 % after 5 ms and 100 % after 30 ms. In other words, release is not turned off before the vesicle pool is fully depleted. The reason is that, in the absence of EFB, the [Ca²⁺]_{vesicle avg} does not decay as effectively as in the control condition. The [Ca²⁺]_{vesicle avg} is broader (fwhm 435 vs. 391 µs in control) and does not decay below 3 µM, a concentration sufficient to trigger appreciable release (Bollmann et al. 2000). Why does [Ca²⁺]_{vesicle avg} remain too high? The problem is that, in the absence of EFB, the binding ratio of the model cytosol is reduced from 40 to ~3 (i.e. the binding ratio of ATP in the model cytosol). Consequently, a large portion of the Ca²⁺ that enters the calyx remains unbuffered, and [Ca²⁺]_{volume avg} itself reaches 3 µM during a single AP. Interestingly, the problem can be remedied by assuming that the volume of the calyx is infinite. In this case, [Ca²⁺]_{volume avg} remains at the resting concentration (50 nm), even in the absence of EFB (Fig. 5B, red traces). The predicted [Ca²⁺]_{vesicle avg} for this assumption reaches a peak of 10 µM (fwhm ~ 400 µs, as in control). The predicted release probability is 32 %.

In summary, the local effects of a non-diffusible buffer are marginal. The buffer does not strongly affect the [Ca²⁺] in the 'close' regime, either in its amplitude or in its time course. However, with a sufficiently large binding ratio, a non-diffusible buffer has an important global effect: it keeps [Ca²⁺]_{volume avg} low during the AP, so that [Ca²⁺]_{vesicle avg} can decay rapidly - by diffusion of Ca²⁺ into the 'far' regime - once calcium channels close.

The near absence of any 'close' regime effects of non-diffusible buffers has important implications for the sensitivity of fast Ca²⁺ signalling simulations to the often elusive parameters of non-diffusible endogenous buffers. If the binding ratio of the buffer is known, release-relevant [Ca²⁺] transients may be calculated fairly accurately, even if the total concentration, the forward binding rate constant, the number of binding sites, and the spatial distribution of the non-diffusible buffer(s) are not known exactly (Meinrenken et al. 2002). This includes the possibility that part of the apparent endogenous Ca²⁺ binding ratio in the calyx ( = 40) is in fact not mediated by an actual Ca²⁺ buffer, but instead by Ca²⁺ sequestering organelles such as mitochondria.

Endogenous diffusible buffers (in this case ATP). If only ATP is removed from the model cytosol (i.e. EFB remains), the [Ca²⁺]_{vesicle avg} reaches a peak of 16 µM instead of 8 µM (fwhm ~400 µs; Fig. 5C, red traces). The predicted release probability is 48 %. The strong effect of ATP in reducing [Ca²⁺] amplitudes in the 'close' regime is expected. ATP (0.58 mM is available for Ca²⁺ binding) has a buffer product ([ATP]_unbound forward binding rate constant) similar to that of 1 mM BAPTA. ATP intercepts part of the Ca²⁺ that diffuses from the Ca²⁺ source to the vesicles. In contrast to a non-diffusible buffer, however, ATP in any location is constantly replenished by 'fresh', unbound ATP. As a result, ATP does not reach equilibrium with local [Ca²⁺] anywhere in the 'close' regime. Hence ATP acts as a local Ca²⁺ sink throughout the course of I_Ca.

Focusing on the [Ca²⁺]_{volume avg}, we find that it remains almost as low as in the presence of ATP (Fig. 5C). This, of course, is not due to the fact that ATP is diffusible, but rather to the fact that its binding ratio (~3) is small in comparison to that of EFB.

In summary, a diffusible buffer can substantially affect [Ca²⁺] transients, in both the 'close' as well as the 'far' regimes. In the 'close' regime, depending on the buffer's concentration and kinetics as well as on the distance from the calcium source, the amplitudes of the [Ca²⁺] transients sensed by the vesicle are reduced. However, in the absence of equilibration of the buffer, the time course of the [Ca²⁺] transients in the 'close' regime remains largely unaffected (Fig. 5C) (except in the case of brief, pulse-like Ca²⁺ influx). In the 'far' regime, a diffusible buffer affects the [Ca²⁺] via the [Ca²⁺]_{volume avg}, depending on the binding ratio of the buffer.

Finally, one might ask how release-relevant [Ca²⁺] signals in a buffered diffusion model differ from those in an unbuffered diffusion model. We illustrate this by again removing both ATP and EFB from the model cytosol, whilst also assuming infinite volume (to keep the [Ca²⁺]_{volume avg} low). The predicted [Ca²⁺]_{vesicle avg} reaches a peak of 22 µM and then decays to 50 nM (not shown). Despite an increase in amplitude, the predicted time course of [Ca²⁺]_{vesicle avg} (fwhm = 396 µs) in unbuffered diffusion is still governed by the time course of I_Ca and is virtually the same as in the presence of ATP and EFB (fwhm = 391 µs).

But what if the rise and decay of I_Ca were more abrupt than those shown in the above example calculations, as would be the case if the [Ca²⁺] in the 'close' regime were controlled by a single calcium channel? In such a case, the rise and decay phases of the [Ca²⁺]_{vesicle avg}, i.e. while not at steady state, would indeed be affected by the presence of buffers (e.g. Naraghi & Neher, 1997). However, for channel open times of more than ~100 µs and distances shorter than ~100 nm (above), the overall duration of the [Ca²⁺] transients would still be largely determined by the duration of I_Ca itself, not by the buffers. As discussed in detail below, this conclusion differs from previous Ca²⁺ signalling concepts.

Probing local and volume-averaged [Ca²⁺] with fluorescent Ca²⁺ indicators

The [Ca²⁺] transients in presynaptic terminals can be measured with fluorescent, Ca²⁺-sensitive dyes. The virtues and limitations of such techniques have been discussed in several reviews (e.g. Augustine & Neher, 1992). One important consideration is that the dye, in order to faithfully report the actual [Ca²⁺] in the cytosol, must respond sufficiently rapidly to [Ca²⁺] transients. Therefore, dyes with a Ca²⁺ dissociation constant (K_D) in the range of the expected [Ca²⁺] (as well as sufficiently fast binding kinetics) are more suitable than high-affinity dyes. Whilst these are not yet available at the calyx, experiments using evanescent wave and multi-photon spot detection may, in the future, provide sufficient spatio-temporal resolution to measure local, fast fluorescence transients. But will the measured fluorescence signal in such experiments report the actual [Ca²⁺], even for smaller, more localised optical control volumes?

To investigate this, we added a low concentration (1 nM) of the fast, low-affinity Ca²⁺ indicator MagFura-2 to the model cytosol (K_on = 4 10⁸ M^-1 s^-1, K_D = 31 µM (Bollmann et al. 2000); diffusion coefficient 220 µm² s^-1 (the same as that of ATP and Ca²⁺)). All other parameters were the same as for the simulation shown in Fig. 4 and Fig. 5 (control case). The unrealistically low concentration of MagFura-2 (1 nM) was chosen such that the dye, although binding and thus reporting Ca²⁺, did not change the predicted [Ca²⁺] transients. To predict the [Ca²⁺] reported by the dye (for a given optical control volume and at any given time), we calculated the volume-averaged concentration of Ca²⁺-bound MagFura-2 in that control volume, as predicted by the buffered diffusion model. This concentration was then converted into an apparent [Ca²⁺] by:

[Ca²⁺]_apparent = K_D[bound dye]/[unbound dye].

If the optical control volume was set equal to the calyx itself, the dye reports the volume-averaged Ca²⁺ signal of the calyx ([Ca²⁺]_{volume avg}) fairly accurately, including its peak of 1 µM (Fig. 6, traces 2 and 3). The presence of a clear peak is in contrast to other studies, which predict a monotonically increasing fluorescence signal and use the signal's first derivative to infer the time course of the Ca²⁺ currents (Sabatini & Regehr, 1998). In the calyx model, even if the concentration of MagFura-2 was increased to 0.4 mM (a concentration suitable to measure the [Ca²⁺]_{volume avg} at the calyx; Helmchen et al. 1997) the peak in [Ca²⁺]_{volume avg} is reduced to half its amplitude, but still present (not shown). However, as illustrated in Fig. 5, the time course and amplitude of [Ca²⁺]_{volume avg}, including whether or not it shows a peak, critically depend on - amongst others - the binding kinetics and total capacity of endogenous buffers. These may differ across synaptic terminals of different types.

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Figure 6. Measuring [Ca2+] transients via fluorescent Ca2+ buffers Simulation of local and volume-averaged [Ca2+] dynamics (red traces), probed with a low concentration (1 nM) of the low-affinity Ca2+ dye MagFura-2 (black traces). The vertical dashed line indicates the peak time of the Ca2+ current. Except for the presence of the dye, all conditions were the same as in Fig. 4, including the Ca2+ current (trace 1, right axis). Trace 2 shows the predicted volume-averaged [Ca2+] (same as control case in Fig. 5). For comparison, trace 3 shows the predicted, volume-averaged concentration of Ca2+-bound MagFura-2 (after conversion into [Ca2+], see text), thus reporting an apparent [Ca2+] as it would be measured in experiments. For the same simulation, trace 4 shows the predicted average [Ca2+] in a more localised, hypothetical control volume, a circular disk of 250 nm diameter and 5 nm height centred around the calcium channel cluster (at 10 nm above the AZ). For the same control volume, trace 5 shows the apparent [Ca2+] reported by the dye. Since MagFura-2 is low-affinity, it reports the volume-averaged [Ca2+] in the terminal fairly accurately. For the smaller control volume, however, the reported [Ca2+] is much lower than the actual [Ca2+] and thus would underestimate the domain-like, local Ca2+ signals sensed by the releasable vesicles.

So what can the measured, apparent [Ca²⁺]_{volume avg} reveal about the Ca²⁺ current or the local [Ca²⁺] signal sensed by the releasable vesicles? As seen in Fig. 6, the onset of the measured signal marks the onset of the Ca²⁺ current, albeit with a small delay. In addition, its peak amplitude provides a lower limit for the average [Ca²⁺] signal sensed by the releasable vesicles (see Fig. 4E). Still, the actual amplitude as well as the exact time course of the release-relevant [Ca²⁺] remain largely unknown.

A potential remedy is to measure the fluorescence from a smaller optical control volume, preferably nearer the calcium channels on the AZ. Using a spot detection method, DiGregorio et al. (1999) have indeed measured highly localised, rapid transients in terminals of cultured nerve-muscle preparations. Even more ideally, one would restrict the fluorescence measurement to a thin region immediately below the AZ. Near-field optical techniques such as evanescent wave microscopy could be used to achieve this (Haydon et al. 1996; Cleeman et al. 1997). However, if the Ca²⁺ indicator is diffusible, even the localised detection of its signal from immediately below the AZ does not report the local Ca²⁺ signal. To illustrate this, we have restricted the optical control volume to a circular disk of 250 nm diameter and 5 nm height, centred around the calcium channel cluster (at 10 nm above the AZ). As expected, the predicted apparent [Ca²⁺] in this control volume (Fig. 6, trace 5) has a higher peak (3.8 µM) than [Ca²⁺]_{volume avg} and thus better reflects the average [Ca²⁺] signal sensed by the vesicles. However, the true [Ca²⁺] in this control volume is about three times higher still (12 µM; Fig. 6, trace 4). The reason for the discrepancy is that the diffusible MagFura-2, after binding Ca²⁺, escapes the optical control volume by diffusional exchange into the remaining cytosol and thus no longer contributes to the collected fluorescence signal. The extent of this discrepancy depends, among other factors, on the diffusion coefficient of the dye. A less mobile dye would report the local Ca²⁺ signal more faithfully. Therefore, an ideal strategy to measure local [Ca²⁺] signals near AZs would be to target genetically encoded, non-diffusible Ca²⁺ indicators to the AZ itself.

In summary, low affinity Ca²⁺ indicators such as MagFura-2 can be used to accurately measure whole-cell, volume-averaged Ca²⁺ signals. However, measuring [Ca²⁺] in more localised control volumes inside a synaptic terminal is much more difficult, with the mobility of the indicator being one of the limiting factors.

Summary - properties of local Ca²⁺ signals in a minimum model

Arguably, one of the reasons for the very existence of partial differential equations is the difficulty of describing processes such as buffered Ca²⁺ diffusion in a language other than mathematics. To capture the process in words means to compete with its true complexity and thus to risk oversimplification. Thus warned, we propose the following summary of our main findings.

(1) Basis of findings. The presented properties of Ca²⁺ signals driving fast transmitter release are based on a minimum model. The model may be considered 'realistic' only in the sense that it explains and reproduces a large amount of experimental data available for the calyx.

(2) 'Close' regime. In the close regime (within ~250 nm around a Ca²⁺ source) [Ca²⁺] transients that reach vesicles are determined by local parameters: the open times, conductance, number, and location of individual calcium channels, as well as the concentration and kinetics of buffers.

(3) 'Far' regime. In the far regime around a Ca²⁺ source (beyond ~250 nm), [Ca²⁺] transients are most likely too low to trigger phasic release. These transients are determined by 'global' parameters: the cytosolic volume, total Ca²⁺ influx and binding ratios of Ca²⁺ buffers and/or intracellular stores, but not by the exact buffer kinetics or the anatomic shape of the terminal.

(4) Diffusion times. Given a Ca²⁺ diffusion coefficient of 220 µm² s^-1, the [Ca²⁺] in the 'close' regime reaches steady state with any given Ca²⁺ influx within ~100 µs (assuming only fast buffers such as ATP and EFB). Hence, diffusion times constitute only a small fraction of synaptic delay.

(5) Time course of the [Ca²⁺]. In the absence of appreciable local equilibration of diffusible buffer(s), the time course of the [Ca²⁺] at any given location in the 'close' regime is governed by that of the influx through the Ca²⁺ source (for the calyx, the fwhm is ~ 400 µs). This time course is controlled by the gating of the calcium channels and the AP.

(6) Amplitude(s) of the [Ca²⁺]. The amplitudes of the Ca²⁺ signals reaching vesicles in the 'close' regime decay steeply with increasing distance (tens of nanometres) from the Ca²⁺ source (for the calyx, it decays from 40 to 0.5 µM as the distance increases from 30 to 300 nm).

(7) Diffusible buffer. Depending on its Ca²⁺ binding rate constant, concentration, and local equilibration level, a diffusible buffer reduces the amplitudes of Ca²⁺ signals in the 'close' regime but does not (strongly) affect their time course. The (relative) reduction is stronger, the larger the distance from the Ca²⁺ source. A diffusible buffer thus helps to spatially restrict Ca²⁺ signals capable of triggering release to the 'close' regime. In the 'far' regime, a diffusible buffer reduces [Ca²⁺] via the volume-averaged [Ca²⁺], depending on its binding ratio (given by ~concentration/K_D).

(8) Non-diffusible buffer. Depending on its binding ratio, a non-diffusible buffer reduces the volume-averaged [Ca²⁺] (provided it equilibrates sufficiently fast). A non-diffusible buffer, however, has only small direct effects on [Ca²⁺] transients in the 'close' regime.

(9) Termination of local [Ca²⁺] and release. At the calyx, interestingly, volume-averaged Ca²⁺ dynamics are such that release would not terminate if the non-diffusible buffer were absent. However, the effect of the non-diffusible buffer in terminating release is only an indirect one; since the buffer keeps the [Ca²⁺] low in the 'far' regime, the Ca²⁺ signal in the 'close' regime can collapse rapidly as soon as calcium channels close, by diffusion of Ca²⁺ (free and buffer-bound) into the 'far' regime. (A diffusible buffer would have the same effect if its binding ratio at the calyx were larger.)

(10) Applicability. The above properties are applicable to synaptic terminals other than the calyx, given that: (i) the terminal had a minimum, unrestricted diffusion volume of about (500 nm)³ around AZs; (ii) Ca²⁺ sources (single channels or clusters/fields of channels) are concentrated at AZs, such that Ca²⁺ and buffers could diffuse in three dimensions away from the Ca²⁺ source; and (iii) characteristic open times of individual Ca²⁺ sources are long (~hundreds of microseconds) relative to relaxation times of the buffered diffusion system (~tens of microseconds for distances up to hundreds of nanometres in physiological, buffered cytosols).

Conclusions and questions

The experiments and modelling studies on the calyx provide answers to important questions of synaptic physiology, such as: how high does the [Ca²⁺] rise during an AP and which mechanisms terminate the Ca²⁺ signal and thus the release of transmitter? In addition, one might ask whether the spatio-temporal characteristics of local Ca²⁺ signalling in the calyx are comparable to those in boutons of mammalian cortical synapses, and, finally, what are the current methodological limitations for more realistic simulations of local Ca²⁺ transients?

How large is the [Ca²⁺] transient during an AP at the calyx?

According to the calyx model, the peaks of the [Ca²⁺] transients reaching releasable vesicles vary depending upon the position of the vesicles, from ~40 to ~0.5 µM (for distances of 30 to 300 nm from the calcium channels, respectively). The average transient at all vesicles of the releasable pool has a peak of ~10 µM. Note, however, that 10 µM should not be interpreted as a typical or effective transient. Considering only the average transient, rather than a heterogeneous ensemble of transients, cannot fully explain the transmission characteristics of the calyx (Meinrenken et al. 2002).

The non-uniformity of [Ca²⁺] transients at the calyx is the direct result of a non-uniformity of vesicle locations relative to release-controlling calcium channels. The distance to the channel is an important determinant of the [Ca²⁺] amplitude that is sensed by the Ca²⁺ sensor. Additional factors are the calcium channel conductance, the number of channels contributing to the Ca²⁺ signal at each vesicle, and the properties of the cytosol (diffusion coefficients and buffer properties). The time course of the [Ca²⁺] transients closely follows the time course of the Ca²⁺ influx through the channel clusters (fwhm ~400 µs), even at diffusional distances of up to hundreds of nanometres. As expected, the transient of volume-averaged [Ca²⁺] is smaller and broader than the local transients (with a peak ~1 µM and a fwhm ~500 µs).

Functional relevance. As the Ca²⁺ sensor has relatively 'high' affinity (i.e. a half maximum release rate at tens of micromolar [Ca²⁺]) and operates below saturation, even vesicles located further than tens of nanometres away from the channel may contribute to release, and even small changes of the local [Ca²⁺] will result in adjustments of the release probability (Schneggenburger & Neher, 2000). This permits effective fine-tuning of the release probability of the terminal.

The non-uniform channel-to-vesicle distances result in a 'positional heterogeneity' of the vesicles' release probability during a single AP (estimated range from < 1 % to 100 % per vesicle (Meinrenken et al. 2002), average 10-25 % (Bollmann et al. 2000; Schneggenburger & Neher, 2000). During consecutive APs, the heterogeneity creates an immediate backup pool of gradually 'facilitated' vesicles and thus seems suited to support transmission at frequencies greater than 10 Hz (Meinrenken et al. 2002). It could help to explain the observed decrease in release fraction at the beginning of an AP train at the calyx, as well as its surprising synaptic strength during steady state transmission (Schneggenburger et al. 2002). Note that the exo- and endocytosis of vesicular membrane during transmission will, presumably, affect the size of AZs (Sätzler et al. 2002) and thus may affect the channel-to-vesicle distances.

[Ca²⁺] signals at physiological temperatures and different developmental stages. The above simulations describe the Ca²⁺ signalling and transmitter release of the calyx at room temperature and age P8-10 only. At physiological temperatures, the Ca²⁺ influx will be about twice as large (~5 nA) and half as wide (fwhm ~200 µs; Borst & Sakmann, 1998). Since the local [Ca²⁺] transients will follow even the higher and faster Ca²⁺ influx, we speculate that release-relevant [Ca²⁺] signals at physiological temperatures at the calyx are likewise higher (~80 µM to ~1.0 µM) and faster (fwhm ~200 µs). Furthermore, the time course of the signalling cascade changes during development of the calyx. This includes changes in AP time course and calcium channel subtypes (Iwasaki & Takahashi, 1998; Taschenberger & von Gersdorff, 2000; reviewed in von Gersdorff & Borst, 2002). In addition, we expect mature synapses to differ with respect to the channel-vesicle topography of the release sites (discussed in Meinrenken et al. 2002). However, the main findings on the space and time dependence of the [Ca²⁺] signal, particularly its primary dependence on the time course of the Ca²⁺ current itself, are quantitatively robust and thus will remain valid if the changes during development are within limits (i.e. if the Ca²⁺ current is at the most ten times faster than that at P8-10, and the channel-to-vesicle distance is still within hundreds of nanometres, assuming similar endogenous buffer kinetics and diffusion coefficients for the mature synapse).

Comparison with squid giant presynaptic terminal. In contrast to the situation in the calyx, the release-relevant [Ca²⁺] signal at the squid giant presynaptic terminal has been estimated to be of the order of 100 µM or higher (Augustine et al. 1991). However, this estimate was based on calculations that assumed that during AP-evoked release Ca² is at spatial equilibrium, as in a single compartment model. Such calculations predict quantitative effects of varying buffer affinities (in this case BAPTA derivatives) that are not valid for diffusion-governed, 'local' Ca²⁺ signalling. Direct measurements of [Ca²⁺] at the squid terminal have also reported local concentrations as high as 200 µM (Llinás et al. 1992). However, the accuracy of such aequorin-based measurements has since been questioned (Augustine & Neher, 1992). Whatever the experimental evidence, a difference in Ca²⁺ sensitivity at the calyx vs. the squid nerve terminal should be almost expected. After all, the squid terminal is in an environment of ~10 mM external Ca²⁺ concentration.

Again in contrast to the calyx, most releasable vesicles at the squid nerve terminal appear to be controlled by the Ca²⁺ signal arising from a single calcium channel per vesicle. Possibly the strongest evidence for this one-to-one coupling has been that the amount of transmitter release in the squid is supralinearly dependent on the [Ca²⁺] of the external solution but apparently only linearly dependent on the number of calcium channels opened by APs of different widths (the channel co-operativity equals unity, Augustine et al. 1991). However, similar experiments did cite evidence for some domain overlap even in the squid (Zucker et al. 1991).

Which mechanism determines the duration of the Ca²⁺ signal and release?

In our model of local [Ca²⁺] signalling at the calyx, the deactivation of the Ca²⁺ sensor, and thus the termination of transmitter release, is controlled only by the decay of local [Ca²⁺] transients. This decay, even at distances of hundreds of nanometres from the calcium channels, is governed primarily by diffusion of Ca²⁺ away from the calcium channels, once calcium channels close during the late repolarisation phase of the AP. The importance of calcium channel gating and Ca²⁺ diffu