The Journal of Neurophysiology Vol. 78 No. 6 December 1997, pp. 2870-2880
Copyright ©1997 by the American Physiological Society

Estimating the Distribution of Synaptic Reliabilities

Emily P. Huang¹ and Charles F. Stevens²

¹ Neuroscience Department, University of California San Diego, La Jolla 92093; and ² Howard Hughes Medical Institute, The Salk Institute, La Jolla, California 92037

	ABSTRACT

Abstract Introduction Methods Results Discussion References

Huang, Emily P. and Charles F. Stevens. Estimating the distribution of synaptic reliabilities. J. Neurophysiol. 78: 2870-2880, 1997. Using whole cell recording from CA1 hippocampal pyramidal neurons in slices, we examined the progressive decrease of N-methyl-D-aspartate receptor-mediated synaptic responses in the presence of the open-channel blocker MK-801. Previous studies analyzing this decrease have proposed that hippocampal synapses fall into two distinct classes of release probabilities, whereas studies based on other methods indicate a broad distribution of synaptic reliabilities exists. Here we derive the theoretical relationship between the MK-801-mediated decrease in excitatory postsynaptic current amplitudes and the underlying distribution of synaptic reliabilities. We find that the MK-801 data are consistent with a continuous distribution of synaptic reliabilities, in agreement with studies examining individual synapses. In addition, changes in the MK-801-mediated decrease in response size as a consequence of altering release probability are consistent with this continuous distribution of synaptic reliabilities.

	INTRODUCTION

Abstract Introduction Methods Results Discussion References

Synapses release neurotransmitter probabilistically (Katz 1969). Release probabilitythe probability that at least one exocytotic event will occur when a nerve impulse invades a presynaptic terminalis regulated by the Ca²⁺-dependent processes governing synaptic vesicle exocytosis and is generally much less than one in central synapses (Allen and Stevens 1994; Hessler et al. 1993; Jack et al. 1981; Korn et al. 1986; Raastad 1995; Raastad et al. 1992; Rosenmund et al. 1993; Walmsley et al. 1988).

Synaptic release probability (P_r) plays an important role in determining the efficacy, or strength, of a given synapse: the average size of the postsynaptic response is proportional to P_r. Thus the strength of a synapse can be altered by modification of its release probability (Bekkers and Stevens 1990; Bolshakov and Siegelbaum 1995; Magleby 1987; Malinow and Tsien 1990; Manabe et al. 1993; Stevens and Wang 1994; Zucker 1989). Because synaptic strength is so important for the function of neuronal circuits, a significant question is how release probabilities are distributed in central synapses. In other words, we want to know the density function that describes the occurrence frequency of synapses with a given P_r.

Two studies (Hessler et al. 1993; Rosenmund at al. 1993) addressed this question in large populations of hippocampal synapses in culture and in brain slices. These studies analyzed the progressive decrease of N-methyl-D-aspartate (NMDA) receptor-mediated synaptic current amplitudes in the presence of the open-channel NMDA receptor blocker MK-801. MK-801 blocks receptors only when they are opened, and because receptors open only when transmitter is released, the progressive block of receptors in MK-801 provides an indirect measure of the transmitter release probability.

The conclusion from this analysis was that populations of hippocampal synapses are nonuniform in release probability. More specifically, it was proposed that hippocampal synapses fall into two classes with about sixfold difference in release probability; we call these classes unreliable and very unreliable. In this scenario, more than half of the synapses belong to the very unreliable class. This conclusion, however, is in conflict with the results of studies using other, more direct methods to examine P_r of synapses (Allen and Stevens 1994; Murthy et al. 1997); these suggest a wide range of release probabilities exists. Indeed, Rosenmund et al. (1993) note the possibility that more than two classes of synaptic P_r are present, although they do not pursue this analytically.

Murthy et al. (1997) measured the endocytotic uptake of the styryl dye FM-143 at stimulated synaptic terminals in culture. The amount of dye taken up at each synapse is directly proportional to its release probability (a synapse with high P_r releases synaptic vesicles more often in response to stimulation and therefore undergoes more vesicle endocytosis). Murthy et al. thus could calculate the P_r for each of a large number of synapses. When a frequency histogram of synaptic P_r was plotted, the distribution appeared to be a continuous function skewed heavily toward low P_r values; in contrast to the conclusions of the MK-801 studies, this directly observed P_r distribution cannot be fitted as a bimodal function. These results are corroborated by data from minimal stimulation experiments (Allen and Stevens 1994), which indicate synapses have P_rs ranging continuously from <0.1 to 1.0.

In the following analysis, we demonstrate that the progressive block of NMDA-mediated synaptic responses in the presence of MK-801 is in fact consistent with a continuous release probability distribution. First, we analyze the MK-801 method and derive the mathematical relationship between the progressive block of synaptic responses and the underlying release probability distribution. Then we show that a continuous distribution fits the data as well as the bimodal (two-class) distribution postulated in previous studies. Finally, we demonstrate that the same continuous distribution fits robustly the progressive block of excitatory postsynaptic current (EPSC) amplitudes in MK-801 under two conditions that alter the synaptic release probability: paired pulse facilitation and changing the external Ca²⁺ concentration.

	METHODS

Abstract Introduction Methods Results Discussion References

Slice preparation and physiology

Transverse hippocampal slices (350-µm thick) were prepared from 2- to 3-wk-old Long-Evans rats using standard procedures (Stevens and Wang 1993). Slices were stored at room temperature (24°C) in saline bubbled with 95% O₂-5% CO₂ for 1.5 h before recording; the saline contained (in mM) 120 NaCl, 3.5 KCl, 1.25 NaH₂PO₄, 26 NaHCO₃, 1.3 MgCl₂, and 2.5 CaCl₂. After incubation, slices were mounted in a recording chamber at room temperature and superfused with external solution identical to the storage solution described above with the addition of 50 µM picrotoxin (a cut was made between area CA3 and CA1 in the slices to prevent epileptiform activity). Flow rate was 2 ml/min. Whole cell recordings were obtained in CA1 pyramidal cells using standard procedures as described by Stevens and Wang 1993. Recording pipettes were 3-4 M and were filled with internal solution containing (in mM) 130 Cs gluconate, 5 CsCl, 5 NaCl, 10 N-2-hydroxyethylpiperazine-N-2-ethanesulfonic acid, 0.5 ethylene glycol-bis(-aminoethyl ether)-N,N,N,N-tetraacetic acid, 1 MgCl₂, 2 Mg-ATP, and 0.2 Li-guanosine 5-triphosphate.

MK-801 experiments

Whole cell recordings were obtained as above in standard recording solution containing 5 µM 6,7-dinitroquinoxaline-2,3-dione (DNQX). NMDA-mediated EPSCs were evoked by stimulating Schaffer collateral fibers with a bipolar tungsten electrode while clamping the recorded cell at 40 mV; the stimulus repetition interval was 8 s. Recording solution containing MK-801 was added as specified in the text. DNQX and MK-801 were obtained from Research Biochemicals.

Two-pathway experiments were performed as follows: two bipolar tungsten electrodes were placed in the Schaffer collaterals, each ~200 µm to either side of the recorded cell. The electrodes were stimulated alternately, evoking two independent populations of synapses on the cell. Pathway independence was ensured by confirming there was no paired pulse facilitation when one pathway was stimulated shortly after the other (at a 50-ms interval). In two-pathway paired pulse facilitation (PPF) experiments, the PPF interval was 50 ms; the stimulation patterns (repetition = 8 s) were arranged so that the same number of pulses were delivered to PPF and control pathways in the presence of MK-801, as follows In Ca²⁺ experiments, the external Ca²⁺ concentration was established by perfusing slices for 15 min at the specified [Ca²⁺] before beginning the experiment.

Minimal stimulation

Minimal stimulation experiments were performed using previously described methods (Dobrunz and Stevens 1997; Raastad et al. 1992; Stevens and Wang 1995). Whole cell recordings were obtained in standard recording solution, and EPSCs were evoked by stimulating the Schaffer collaterals with a tungsten bipolar electrode while clamping the cell at 70 mV. Stimulation rate was 0.1 Hz. Stimulation intensity was lowered until a significant number of failures were seen among the responses and the presence of a putative single synapse was confirmed as described in Dobrunz and Stevens 1997. The criteria for single synapse selection can be summarized as follows: the stimulus intensity is just high enough to elicit responses; small changes in stimulus intensity (± 5%) do not alter failure probability or average response amplitude; there is no variation in epsc shape or latency; and the response amplitude histogram does not change when the synapse is paired-pulse facilitated. Failure rate for the synapse was calculated as the fraction of responses that are failures; success rate (or P_r) was one minus the failure rate.

PPF at minimal stimulation was evoked by stimulating in pairs with interstimulus intervals of 19-25 ms (pairs were presented at a rate of 0.1 Hz). The PPF ratio was calculated as the success rate on the second pulse divided by success rate on the first pulse, whereas initial P_r was just the success rate on the first pulse.

Analysis

Signals were filtered at 2 kHz, digitized at 5 kHz, and stored for analysis. Analysis of epscs was performed using programs written in AXOBASIC; relative EPSC size was measured by integrating the current in a 40-ms window around the peak. For minimal stimulation experiments, responses were inspected visually to determine failures.

All fits of data with mathematical models were carried out in Mathcad by least-square fitting procedures. Differential equations describing the NMDA-receptor kinetic scheme also were solved in Mathcad.

	RESULTS

Abstract Introduction Methods Results Discussion References

Evaluation of the MK-801 method

Whole cell recordings were obtained from CA1 pyramidal cells in hippocampal slices, and NMDA receptor-mediated EPSCs were evoked by stimulating Schaffer collateral fibers in the presence of 5 mM DNQX while clamping the cells at 40 mV. After a stable baseline was obtained, stimulation was stopped and the NMDA-receptor open-channel blocker MK-801 was added to the bathing medium for 10 min. When stimulation was resumed in the presence of MK-801, the NMDA receptor-mediated EPSCs decreased progressively in amplitude with each stimulus event (Fig. 1A). The shape and time course of the amplitude decrease in 40 µm MK-801 plotted against stimulus number agrees generally with previous studies (Hessler et al. 1993; Kullmann et al. 1996; Manabe and Nicoll 1994; Rosenmund et al. 1993) and is fairly consistent from experiment to experiment (Fig. 1B). In subsequent discussion, we shall call curves such as the one illustrated in Fig. 1B, blocking functions.

View larger version (19K): [in this window] [in a new window]   FIG. 1. Decrease of N-methyl-D-aspartate (NMDA) receptor-mediated excitatory postsynaptic current (EPSC) amplitudes in MK-801. A: representative experiment showing EPSC amplitude plotted against stimulation number. After a stable baseline is obtained, stimulation is stopped and 40 µM MK-801 is perfused as indicated; stimulation is resumed after 10 min. EPSC amplitudes are normalized to the first response in the presence of MK-801. Insets: average EPSCs (n = 8 and 5, respectively) at indicated points in the experiment. B: summary graph of the decrease of EPSC amplitudes in the presence of 40 µM MK-801, 2.5 mM [Ca2+], and 1.3 mM [Mg2+]. EPSC amplitudes are plotted against stimulus number and are normalized to the first response. Error bars represent SE (n = 6).

Our goal is to determine the relationship between this EPSC amplitude decrease in MK-801, the blocking function, and the underlying release probability distribution for a population of synapses. MK-801 blocks NMDA receptors only when they are open; the fraction of total NMDA receptors that gets blocked at each stimulus, then, depends on synaptic P_r (the higher the P_r, the more likely a synapse will release and have open receptors to block) as well as the effectiveness of receptor block by MK-801 when a release event does occur. The MK-801 blocking function therefore provides us with a transform of the synaptic P_r distribution, but interpreting the underlying P_r distribution from these experiments presupposes an understanding of the nature of receptor block by MK-801. Thus a proper evaluation of the MK-801 blocking function must involve an analysis of channel block by MK-801 under our particular experimental conditions. Toward this end, we obtained blocking functions for different concentrations of MK-801 (10, 20, and 40 µM) and also examined the shape changes of the individual EPSCs as a function of concentration and/or time.

Scaling of NMDA receptor-mediated EPSCs so that their initial peaks match in the presence and absence of 40 µM MK-801 (Fig. 2A) confirmed that the decay of the EPSC is faster when MK-801 is present; this faster decay represents the block of open channels by MK-801 (Hestrin et al. 1990; Huettner and Bean 1988). EPSCs were fitted with a simple four-state kinetic model (Fig. 2B), in which channels make the indicated transitions between the unbound, closed, open, or blocked states. An impulse of transmitter is assumed. For any given experiment, control EPSCs were first fitted with the blocking rate set to zero; EPSCs in the presence of MK-801 then were fitted using the same rate constants as control and allowing the blocking rate to vary. In general, an "unblocking" rate was unnecessary to fit the EPSCs except at low concentrations of MK-801 (Jahr 1992). When unblocking was used, the unblocking rate was at least an order of magnitude smaller than the blocking rate.

View larger version (15K): [in this window] [in a new window]   FIG. 2. Kinetic model of EPSCs in the presence and absence of MK-801. A: EPSCs in the presence (ii) and absence (i) of 40 µM MK-801 from a representative experiment. i: average of 8 EPSCs from the baseline before the perfusion of MK-801. ii: average of the 1st 5 EPSCs evoked after the perfusion of MK-801. EPSCs have been scaled so that their initial peaks match to show the faster decay of ii. B: EPSCs in the absence () and presence () of MK-801 fitted by a 4-state kinetic model (inset). On the left are EPSCs taken from an experiment using 10 µM MK-801, and on the right are EPSCs taken from an experiment using 40 µM MK-801. , average of 8 EPSCs from the baseline; , average of the 1st 5 EPSCs in the presence of MK-801. Inset: kinetic scheme; U, unbound state; C, bound and closed; O, bound and open; B, blocked; koff is unbinding rate; ko, opening rate; kc, closing rate; and kb, blocking rate. Average koff, ko, kc, and kb for 10 µM MK-801 experiments are 3.6, 14, 43.3, and 5.2 s1, respectively (n = 3); for 40 µM MK-801, they are 3.1, 8.8, 43.4, and 24.5 s1, respectively (n = 5). C: open-channel block fraction as a function of MK-801 concentration. Error bars represent SE (n = 3 for 10 µM; n = 4 for 20 µM; and n = 5 for 40 µM MK-801). Data are fitted by the blocking curve BF = ([MK]/k1/2)/(1 + ([MK]/k1/2), where BF is block fraction and [MK] is MK-801 concentration. Best-fit k1/2 by a least-squares fitting procedure is 28.6 µM MK-801.

The estimated blocking rate increased as the concentration of MK-801 was varied from 10 to 40 µm, indicating a dose dependence. To determine channel block fraction (the fraction of open channels that get blocked by MK-801 during a single synaptic response), the differential equations for the kinetic scheme (Fig. 2B) were solved with the fitted rate constants to calculate the time dependence of the blocked state; block fraction was calculated as the fraction of opened channels that end up in the blocked state after each synaptic current. This value is plotted against MK-801 concentration in Fig. 2C, and the data are fitted by an effective dose-response curve with k_1/2 = 28.63 µM MK-801. We emphasize that this constant k_1/2 is not the equilibrium dissociation constant of MK-801 binding to the NMDA receptor. Rather, it is the concentration of MK-801 at which half of the receptors that open during a stimulus get blocked. This value depends on the opening and closing kinetics of the receptor as well as the specific experimental conditions. In all subsequent experiments, a concentration of 40 µM MK-801 is used, with an estimated average channel block fraction of 0.611 ± 0.068 (mean ± SE; n = 5).

One might expect that block fraction, as determined above, would influence the rate at which NMDA receptor channels are blocked with successive stimuli. That is, an increase in the efficacy of MK-801 block would be expected to yield experimental blocking functions that decrease faster as a function of stimulus number. To compare the rate of decrease of two different blocking functions, we plotted the amplitude points of one as a function of the amplitude points of the other for each stimulus number. For instance, when we plotted the amplitudes of one 40 µM MK-801 blocking function against the amplitudes of another for each stimulus number (Fig. 3A), the points fell along a straight line of slope = 1, indicating the two functions have an identical rate of decrease. When we compared a 10 µM and a 40 µM MK-801 experiment in the same way (Fig. 3B), however, the points deviated greatly from the slope = 1 line, highlighting the slower decrease of EPSC amplitudes in 10 µM MK-801. Plots for decreases in 20 µM against those in 40 µM MK-801 fell in between (plots not shown).

View larger version (14K): [in this window] [in a new window]   FIG. 3. Block fraction affects rate of decrease of EPSC amplitudes in MK-801. A: normalized amplitudes of one 40 µM MK-801 blocking function are plotted against the normalized amplitudes of another for each stimulus number. Data fall on a straight line with slope = 1.0, indicating an identical rate of decrease. B: amplitudes of a 10 µM MK-801 blocking function plotted against the amplitudes of a 40 µM MK-801 blocking function for each stimulus number. Points deviate considerably from a line of slope = 1, indicating a far slower rate of amplitude decrease in 10 µM MK-801. C: representative EPSCs at early and late stimulus numbers in the presence of 40 µM MK-801. i: average of the 1st 5 responses and average of 6 responses beginning at stimulus number 50 in the presence of MK-801. In ii, the EPSCs are scaled so that their initial peaks match. There is no significant change in the decay course of the EPSCs, indicating that block fraction remains unchanged.

As the value of block fraction does clearly affect the blocking function, we must be sure the block fraction does not change during the course of the experiment. This point can be settled by scaling and comparing EPSCs after different number of stimuli in the presence of MK-801. Figure 3C shows typical results from an experiment in 40 µm MK-801; EPSCs at later stimulation numbers scale perfectly to EPSCs at early numbers. Thus block fraction does not change with either time or stimulus number in the course of our experiments.

Decrease of EPSC amplitudes in MK-801 depends on block efficacy and synaptic release probability

Now we can derive the quantitative relationship between synaptic release probability and the decrease of NMDA-mediated EPSC amplitudes in the presence of MK-801, that is, between the synaptic reliability distribution and the blocking function. First, let us outline the assumptions used in our derivation. We assume we repetitively stimulated a fixed number of synapses (with a fixed number of postsynaptic NMDA receptors) the release probabilities of which remained constant throughout the experiment [Allen and Stevens (1994) found this to be the case for the synapses in their sample]. With each stimulus, a fraction of the available postsynaptic NMDA receptors are blocked by MK-801. This fraction (a) depends on three factors: P_r, the release probability of the stimulated synapses; , the fraction of receptors participating in the current given a release event; and m, the fraction of these participating receptors that get blocked by MK-801 (block fraction). If P_r varies across synapses (Allen and Stevens 1994; Hessler et al. 1993; Murthy et al. 1997; Rosenmund et al. 1993), a also will depend on (P_r), the function that describes the relative occurrence frequency of synapses with a given P_r.

With these initial assumptions, we write and solve a difference equation describing the fraction of synaptic receptors remaining unblocked at nth stimulus, F_n (see APPENDIX for full derivation). Our experimental blocking functions measure synaptic current, however, not receptor fraction, so we then calculate from F_n the predicted synaptic current on the nth stimulus, S_n. When this is done (see APPENDIX), we get

(1)

S_n is normalized so that S₀ = 1

From Eq. 1, we see that the predicted decrease of synaptic amplitudes in the presence of MK-801 depends on m, the fraction of NMDA receptors blocked by MK-801 when a synapse releases transmitter, as well as (P_r), the P_r weighting distribution. We now are prepared to determine what release probability distributions fit our MK-801 data.

Decrease of EPSC amplitudes in MK-801 is consistent with a continuous release probability distribution

Previous studies analyzing the decrease of EPSC amplitudes in MK-801 (Hessler et al. 1993; Rosenmund et al. 1993) have suggested that synapses are distributed between two classes of release probabilities (termed high and low). As noted earlier, the conclusion that there is a bimodal distribution of release probabilities, however, contradicts more direct estimates of the P_r weighting function from studies using other methods (Allen and Stevens 1994; Murthy et al. 1997), in which release probabilities seem to follow a continuous distribution.

To resolve this conflict, we next examined whether the MK-801 blocking function could be explained as well by a continuous distribution of release probabilities as by the two-class (bimodal) P_r distribution. An ideal bimodal weighting function, as used by Hessler et al. and Rosenmund et al., is represented mathematically as where P₁ and P₂ are the P_r values of the two synaptic classes, respectively, and A is the proportion of synapses with P_r = P₁. On the other hand, a continuous distribution can be characterized as a function with a negative exponential dependence on P_r (see APPENDIX), such as where  is a constant; 1/ is the "characteristic P_r" of the distribution.

When Eq. 1 is evaluated with the above bimodal and continuous distributions, we get two predictions for S_n, the decrease in EPSC amplitudes in the presence of MK-801 where b₁ = P₁m, b₂ = P₂m, 0  A  1, and where r = /m. We see that S_n for the bimodal distribution is the sum of two exponentials (the form used by Hessler et al. 1993; Rosenmund et al. 1993), whereas S_n for the continuous distribution is the sum of an infinite number of exponentials. Note that the parameter r is inversely proportional to the characteristic release probability 1/.

Experimentally obtained blocking functions in the presence of 10, 20, and 40 µM MK-801 were fitted with the predicted S_n functions incorporating the bimodal and continuous distributions by a least-squares-fitting procedure (Fig. 4). For all cells (n = 15), the amplitude decreases were fitted by both models nearly indistinguishably; for 40 µm MK-801 (n = 6), mean values of A, b₁, b₂, and r were 0.677 ± 0.012, 0.116 ± 0.007, 0.014 ± 0.0006, and 9.78 ± 0.502, respectively. We conclude that the data from MK-801 experiments are perfectly consistent with at least one type of continuous distribution of synaptic release probabilities as well as with a bimodal P_r distribution.

View larger version (18K): [in this window] [in a new window]   FIG. 4. MK-801 data are fitted by a continuous release probability distribution. Representative blocking function in 40 µM MK-801 fitted by Sn incorporating a bimodal distribution (- - -) and a continuous distribution (). To reduce variability and for the purposes of quantitative fitting, the EPSC amplitudes are normalized to the y intercept of a line fitted to the 1st 10 points in the decrease. ···, fit of the data by a Sn incorporating a distribution derived from minimal stimulation experiments, as described in the DISCUSSION.

To obtain a quantitative description of the above P_r distributions, we must calculate P₁ (=b₁/m) and P₂ (=b₂/m), the P_r values of the two synaptic classes in the bimodal distribution, and 1/ ( = rm), the characteristic P_r for the continuous distribution. Calculation of these values requires an estimate of m, the fraction of receptors blocked per stimulus. m is the fraction of participating receptors on a given trial that get blocked and was estimated earlier as 0.611 for 40 µM MK-801. is the fraction of receptors that participate in the current when transmitter is released; we take  at the previously estimated value 0.5 (Rosenmund et al. 1995) and at 1.0. When  is 0.5, P₁, P₂, and  are 0.38, 0.05, and 2.99, respectively; when  is 1, P₁, P₂, and  are 0.19, 0.025, and 5.98, respectively. Thus for the bimodal distribution, the "high" synaptic P_r may range from 0.19 to 0.38 and the low P_r from 0.025 to 0.05. For the continuous distribution, the characteristic release probability 1/ ranges between 0.17 and 0.33.

PPF of a continuous P_r distribution

PPF is an increase in presynaptic release probability that occurs when a synapse is stimulated to release within a few hundred milliseconds of a previous stimulation event; the increased P_r is thought to reflect residual Ca²⁺ in the axon terminal (see Zucker 1989). To examine the effect of PPF on the blocking function, we performed experiments in which two separate and independent pathways onto the same recorded CA1 cell were stimulated alternately. This was accomplished by stimulating the Schaffer collaterals on either side of the cell. When such pathways were alternately given single stimuli in the presence of MK-801, the EPSC amplitudes of one pathway could be plotted against the amplitudes of the other (for each stimulus number) as a straight line (Fig. 5A), with a slope of 1. 

View larger version (14K): [in this window] [in a new window]   FIG. 5. Paired pulse facilitation (PPF) changes rate of EPSC amplitude decrease. A: normalized EPSC amplitudes of 1 pathway is plotted against the normalized amplitudes of the second pathway for each stimulus number in a 2-pathway experiment. Points fall on a straight axis with slope = 1.0. B: normalized amplitudes of the first pulse of every pair in the paired pulse facilitated pathway plotted against the normalized amplitudes of every other pulse in the control pathway, for a 2-pathway PPF experiment. The points fall under the straight axis, indicating that the PPF pathway decreases more quickly than control in MK-801.

When one of the pathways was paired pulse facilitated, however, a plot of the first pulse of each pair in the PPF pathway against every other pulse in the control pathway (with stimulus number as the plot parameter) deviated considerably from a line of slope equal to one (Fig. 5B). This deviation reflects a more rapid decrease of the blocking function for the paired pulse pathway, presumably due to an increase in release probability. To see whether we could relate directly the amount of PPF enhancement to the change in amplitude decrease, we modified our continuous distribution model (see APPENDIX) to accommodate an increase in P_r for every second pulse. This increase in P_r on the second pulse is given by the function (P_r), defined as the amount of PPF (or PPF ratio) as a function of initial release probability. The result of this simple modification is

(2)

All the parameters of this equation, except for (P_r), are fixed by values derived from fits of data from the control pathway (, m, ). The problem, of course, is to determine (P_r) because the MK-801 experiments do not provide a direct measure of the changes in release probabilities due to PPF. To estimate (P_r), we turned to the technique of minimal stimulation, which provides a means of activating putative single synapses on a neuron recorded in slices (Dobrunz and Stevens 1997; Raastad 1995; Stevens and Wang 1995). Synapses stimulated in this manner exhibit frequent response failures that are due to the probabilistic nature of transmitter release (Allen and Stevens 1994).

In minimal stimulation, the P_r of a given synapse is calculated as one minus the failure rate of responses to the minimal stimulus (Allen and Stevens 1994; Raastad et al. 1992; Stevens and Wang 1994). When closely spaced pairs of the minimal stimulus are given, the P_r of the second pulse is increased relative to the P_r of the first pulse, as expected with PPF. Thus we were able to measure PPF ratio and initial P_r for a number of synapses (n = 25, see METHODS). In these experiments, we found synapses with P_rs ranging continuously from 0.1 to 1.0; there was no evidence of a bimodal distribution of synaptic P_r (see Fig. 6).

View larger version (9K): [in this window] [in a new window]   FIG. 6. PPF as a function of initial release probability. PPF ratio, calculated by dividing release probability of the 2nd pulse by the release probability of the 1st pulse, plotted as a function of the release probability of the 1st pulse. Data are from 25 minimal stimulation experiments with paired pulse intervals from 19 to 25 ms.

When the PPF ratio is plotted as a function of initial P_r (Fig. 6), we see a decrease in PPF as P_r goes from zero to one. We thus can approximate (P_r) as a straight line

(3)

PPF_max is the PPF ratio at very low P_r values; (P_r) goes to one at P_r = 1.0. We assessed PPF_max in our MK-801 experiments by calculating average PPF ratio at late stimulus numbers; then, for ease of calculation, we further approximated (P_r) as a staircase function decreasing regularly from PPF_max to one in 10 steps. When this (P_r) was incorporated into Eq. 2, we obtained an expression describing S_n for PPF synapses that had no free parameters. For four experiments, this expression fitted the MK-801 mediated decrease of EPSC amplitudes in the PPF pathway very well (Fig. 7, A and B). The model incorporating the continuous distribution (P_r) = e^P_r is therefore robust when the release probabilities are modified, such as with PPF.

View larger version (13K): [in this window] [in a new window]   FIG. 7. Decrease of EPSC amplitudes in MK-801 for PPF synapses is fitted by a continuous Pr distribution. A: blocking function of a representative control pathway fitted by a continuous distribution model (Scn). Amplitudes are plotted against every other stimulus number; fitted r = 9.92. B: blocking function of the PPF pathway from the same experiment as A is fitted by a Sn incorporating the same Pr distribution as in A. Amplitudes of the 1st pulse of each stimulus pair are plotted against the stimulus pair number. Estimated (Pr) = 1.86-0.86 Pr was used to calculate the fitting function Sn.

We also calculated the S_n for PPF synapses assuming a bimodal P_r distribution

(4)

Again, the parameters A, P₁, P₂, m, and  are taken from fits of data from the control pathway, and (P_r) is taken from the minimal stimulation data by selecting the values of Eq. 3 at P₁ and P₂. The PPF blocking functions are fitted acceptably with these calculated S_n, so we cannot exclude a bimodal P_r distribution from consideration on the basis of fitting the PPF data. As we noted before, however, that the (P_r) obtained from minimal stimulation data are itself a continuous function; in Eq. 4, it is treated artificially as a bimodal function because only two of its points are selected in the calculation of S_n. Ignoring this inherent contradiction yields reasonable fits but indicates a bimodal P_r distribution is a less appropriate choice than a continuous one.

Ca²⁺ alters the P_r distribution according to the Dodge-Rahamimoff equation

Transmitter release is a Ca²⁺-dependent process, and changes in external Ca²⁺ profoundly affect synaptic release probability (Dodge and Rahamimoff 1967). We examined the MK-801 blocking function at four different external Ca²⁺ concentrations: 1.0, 1.75, 2.5 (standard), and 6.0 mM. Figure 8A shows blocking functions under these four conditions.

View larger version (17K): [in this window] [in a new window]   FIG. 8. Blocking functions in varying external [Ca2+] follow the Dodge-Rahamimoff equation. A: representative blocking functions for 1.0 mM (), 1.75 mM (), 2.5 mM (, standard), and 6.0 mM () external [Ca2+]. B: average r1 plotted against [Ca2+]. Error bars represent SE; data are normalized to r1 for [Ca2+] = 1.75 mM. Data are fitted by the Dodge-Rahamimoff equation: normalized r1 = {[1 + (1.3/kMg) + (2.5/kCa)][Ca]/[2.5][1 + ([Mg]/kMg) + ([Ca]/kCa)]}4, where [Mg] and [Ca] are Mg2+ and Ca2+ concentrations, respectively. Best-fit kMg and kCa are 3.9 and 0.57 mM, respectively.

Blocking functions for all Ca²⁺ concentrations were well fitted by our model incorporating a continuous P_r distribution. The average best-fit rs for external [Ca²⁺] of 1.0 mM (n = 2), 1.75 mM (n = 5), 2.5 mM (n = 6), and 6.0 mM (n = 4) were 22.51 ± 4.5, 12.67 ± 0.868, 9.78 ± 0.502, and 8.81 ± 0.607, respectively.

The change in r reflects changes in the P_r distribution at different [Ca²⁺] concentrations. r, however, also is affected by MK-801 block efficacy (recall that r = /m); because Ca²⁺ ions may interact with the MK-801 block site (Reynolds and Miller 1988), we examined the EPSCs from each condition to determine whether block fraction had been significantly changed from standard conditions (2.5 mM [Ca²⁺]). Analysis of the EPSCs revealed that MK-801 block fraction was not significantly affected by changing the external Ca²⁺ concentration except at 6.0 mM [Ca²⁺], where the average block fraction was reduced from 0.611 ± 0.068 to 0.330 ± 0.029. Because r is inversely proportional to block fraction, the fitted r for the condition of 6.0 mM [Ca²⁺] was scaled by a factor of 0.330/0.611 = 0.54 for the purposes of quantitative comparison.

Figure 8B shows normalized r¹ plotted against external [Ca²⁺]; the data are fitted by the Dodge-Rahamimoff equation (Dodge and Rahamimoff 1967). Best fit by a least-squares fitting procedure yielded k_Mg and k_Ca of 3.9 and 0.57 mM, respectively.

Blocking functions at different [Ca²⁺] also were fitted reasonably with a S_n incorporating a bimodal P_r distribution by varying the three required parameters for a best fit. However, the three parameters of the fitting function, A, b₁, and b₂, changed in a manner that we were unable to relate to the Dodge-Rahamimoff equation: for external [Ca²⁺] of 1.0 mM, average A, b₁, and b₂ were 0.649 ± 0.008, 0.039 ± 0.007, and 0.007 ± 0.002 (n = 2); for 1.75 mM, they were 0.655 ± 0.019, 0.089 ± 0.008, and 0.012 ± 0.001 (n = 5); for 2.5 mM, they were 0.677 ± 0.012, 0.116 ± 0.007, and 0.014 ± 0.0006 (n = 6); and for 6.0 mM, they were 0.749 ± 0.012, 0.224 ± 0.03, and 0.019 ± 0.004 (n = 4). The weighting factor A changed as [Ca²⁺] changed, implying some synapses move from low to high P_r in this model. Thus we know of no straightforward way to relate the changes of the three parameters of the bimodal distribution to the changes in external calcium concentration.

	DISCUSSION

Abstract Introduction Methods Results Discussion References

In the work presented here, we analyzed the relationship between synaptic release probability and the decrease of NMDA-mediated synaptic current amplitudes in the presence of MK-801. Previous studies exploiting this method (Hessler et al. 1993; Rosenmund et al. 1993) have argued that the simplest fit of the MK-801 blocking function points to two classes of synapses with a sixfold difference in release probability (although Rosenmund et al. did not exclude the possibility of multiple P_r classes). The proposal of two synaptic P_r classes, however, is inconsistent with the results of studies in which synaptic P_r is directly measured (Allen and Stevens 1994; Murthy et al. 1997) (see also Fig. 6); such studies indicate synapses range over all possible release probabilities. Furthermore, studies have shown that individual synapses are capable of both increasing and decreasing their P_r by varying amounts under conditions such as long-term potentiation and long-term depression (Bolshakov and Siegelbaum 1995; Stevens and Wang 1994), implying that two P_r classes may be inadequate to describe the available P_r landscape.

We have demonstrated that the data from MK-801 can be described with a continuous P_r distribution. One attractive feature of this distribution is that it describes the MK-801 blocking function with a single parameter (r). In contrast, three parameters are needed with the bimodal distribution (A, b₁, b₂). Our distribution thus provides the simplest fit of the MK-801 data. Furthermore, this distribution fits the MK-801 blocking function robustly under conditions of altered release probability, such as PPF and changes in external [Ca²⁺]. In our PPF experiments, we can account for the change in MK-801 blocking function in terms of our distribution and the amount of observed PPF alone. In our Ca²⁺ experiments, the predicted change in our distribution at different Ca²⁺ concentrations follows the Dodge-Rahamimoff equation. In contrast, we cannot identify a consistent relationship between the parameters of the bimodal distribution and the [Ca²⁺].

The equation we use to describe the continuous distribution, (P_r) = e^P_r, is, of course, a mathematical idealization. For instance, we actually expect the distribution to go to 0 at P_r = 0, as described in Murthy et al. (1997). Our idealization is acceptable because very low values of P_r are not detectable by the MK-801 method, although this method is more accurate at low P_r than minimal stimulation. We estimate the MK-801 method cannot detect synaptic P_r < 0.05; with minimal stimulation, synapses begin to be heavily underrepresented at P_r of 0.1-0.2. The FM1-43 method used by Murthy et al. (1997) is also inaccurate for low probability synapses and underrepresents synapses with release probabilities less than ~0.05-0.1.

Instead of idealizing the minimal stimulation-derived distribution as a negative exponential, we can express it exactly as a histogram (Allen and Stevens 1994). This "real" distribution is problematic in that it assuredly underestimates the relative number of synapses at the lower end of the P_r range (<0.2, as discussed earlier), so it is of limited use for analysis. Nonetheless, to see whether this distribution is consistent with the MK-801 data, we calculated its S_n and fitted it to the experimental blocking function. The calculated S_n had only two free parameters, which were varied for best fit: , the number of receptors that participate in the current, and y, a small constant added to the fitting equation to compensate for the underrepresentation of low P_r synapses. When these parameters were adjusted, the data were fitted quite well (Fig. 4, ···); the best fit yielded  = 0.54. Thus a real, albeit incomplete, distribution derived from minimal stimulation data are consistent with the MK-801 blocking function.

Although we have confirmed here that the MK-801 blocking functions reflect the distribution of synaptic reliabilities, the limitations of this method for estimating synaptic reliabilities must be stressed. We consider these limitations in the following text.

We have used a particular continuous weighting function of synaptic reliabilities [(P_r) = e^P_r] to describe our datathis distribution is the lowest order approximation to a more general functionbut we must emphasize that the MK-801 method alone cannot be used to determine the exact shape of the function that describes the synaptic reliability distribution. The problem is that the MK-801 blocking function can be fitted with sums of anywhere between two (bimodal) and an infinite (continuous) number of exponential functions. Distinguishing whether a function with even a small amount of noise is the sum of 2, 10, or more exponentials is difficult because exponential functions with similar rate constants are very similar in shape. On the basis of our analysis, the reliability distribution (P_r) could be a continuous exponential function, a sum of two delta functions, or any function in between (such as sums of multiple delta or Gaussian functions); with correctly chosen parameters, the predicted S_n are nearly indistinguishable. We can determine the range of (P_r) that is consistent with the observed blocking function, but within that range, cannot decide whether one (P_r) is better based on fit. To make that decision, one must use other criteria, as we have done in the preceding text.

Whatever the precise problems with the MK-801 method for determining the distribution of synaptic reliabilities, we have resolved the conflict between the observed continuous distribution of release probabilities and the original interpretation of the MK-801 blocking data as indicating the presence of two or a few classes of synapses. A firm conclusion of the present work is that synapses are generally quite unreliable and that a broad distribution of reliabilities that is weighted to the less reliable synapses is present in slices. A measure of how much the distribution is weighted to low P_rs is provided by the characteristic release probability 1/ of our fitted continuous distribution: two-thirds of the synapses have a P_r value less than or equal to the characteristic P_r. Our estimation of 1/ is 0.17-0.33. We believe that any theory of brain function will have to take account of this broad distribution of synaptic reliabilities.

	ACKNOWLEDGEMENTS

  This work was supported by the Howard Hughes Medical Institute and a National Institute of Neurological Disorders and Stroke Grant NS-12961 to C. F. Stevens and by a Department of Defense Graduate Fellowship from the Office of Naval Research to E. P. Huang.

	APPENDIX

MK-801 blocking equation

Here we derive the relationship between synaptic P_r and the decrease of EPSC amplitudes in the presence of MK-801. Let us start with the simplest case where a hypothetical population of synapses all have the same release probability. In that case, the fraction F_n of receptors remaining unblocked after n stimulation trials in the presence of MK-801 can be calculated by solving the difference equation a, the fraction of receptors blocked for any given stimulus, is the product of three factors: P_r, synaptic release probability; , the fraction of receptors participating in the current; and m, the fraction of these participating receptors that get blocked by MK-801 (block fraction). In the text, we found m = 0.611 for an MK-801 concentration of 40 µM.

Solving the above difference equation yields For small a, such as when release probability P_r is small, this equation becomes Thus if release probability is uniform across a population of synapses and m and  are constants, the number of unblocked receptors at trial n will decline as a single exponential. Actual EPSC amplitude at trial n, S_n, will be proportional to the product of the fraction of unblocked receptors (F_n), the probability of release (P_r), and the average current per receptor (z) The actual value of S_n can be obtained by scaling this quantity by the total number of receptors. From the preceding text, we see that the assumption of uniform P_r yields a measured quantity S_n that declines exponentially; so for this hypothetical case, the blocking function is an exponential. However, as has been noted previously (Hessler et al. 1993; Rosenmund et al. 1993), the decline of synaptic current amplitudes in the presence of MK-801 cannot be fitted with a single exponential, indicating that release probability is not constant across the population of synapses responsible for the measured synaptic currents.

Because release probability is not constant, there must be a probability density function (P_r) that describes the relative distribution of synaptic release probabilities in the population; thus F_n can be weighted accordingly The upper limit of the integral can be taken as infinity if (P_r) approaches zero rapidly as P_r approaches one, so that higher values of P_r do not contribute; in practical terms, this means there are many more low P_r synapses than high P_r synapses. Several studies (Allen and Stevens 1994; Hessler et al. 1993; Rosenmund et al. 1993) have shown this to be the case.

Again, synaptic current amplitude for trial n will be proportional to the product of F_n, P_r, and the size of the average current as a function of P_r, z(P_r) where (P_r)  (P_r)z(P_r)P_r.

The actual magnitude of S_n depends on the total number of receptors, but this factor is unimportant in our analysis because we will normalize S_n to its initial value.

Motivation for (P_r) as a negative exponential function The choice of an exponential weighting function can be motivated as follows. First, recall which also can be written as Expanding the function ln(P_r) to first order about _r, we find the equation Generally, (P_r) decreases with P_r, so k is negative. Taking the first order approximation, we get Thus the continuous distribution e^P_r can be thought of as the lowest order approximation to the general function (P_r). This distribution reasonably describes the dominant tail portion of P_r histograms derived from the study of individual synapses (Allen and Stevens 1994; Murthy et al. 1997).

MK-801 blocking equation for PPF synapses A modified S_n for paired pulse facilitated synapses is derived next. The assumptions and analysis are similar to those used above to derive S_n for normal synapses (see Eq. 1). We write the difference equation that describes the fraction of unblocked receptors at stimulus trial k for the case in which every second pulse is facilitated where r_k = a if k is even (first pulse) and r_k =  if k is odd (second pulse).

Take n = k/2, so that n counts the first stimulus of each pair. The fraction of unblocked receptors on the first pulse of the nth stimulus pair is Using the same reasoning as in the text, the actual synaptic response size for the first stimulus of the nth pair is given by Remember that the fraction blocked a on the first pulse equals P_rm. For the second (facilitated) pulse, the fraction blocked  is related to a by where (P_r) is the PPF ratio at a given P_r. Thus With a continuous distribution (P_r) = e^P_r With a bimodal distribution (P_r) = A(P_r  P₁) + (1  A)(P_r  P₂)

	FOOTNOTES

  Address for reprint requests: C. F. Stevens, The Salk Institute, 10010 N. Torrey Pines Rd., La Jolla, CA 92037.

  Received 24 March 1997; accepted in final form 13 August 1997.

	REFERENCES

Abstract Introduction Methods Results Discussion References

• ALLEN, C., STEVENS, C. F. An evaluation of causes for unreliability of synaptic transmission. Proc. Natl. Acad. Sci. USA 91: 10380-10383, 1994.[Abstract/Free Full Text]
• BEKKERS, J. M., STEVENS, C. F. Presynaptic mechanism for long-term potentiation in the hippocampus. Nature 346: 724-729, 1990.[Medline]
• BEKKERS, J. M., STEVENS, C. F. Cable properties of cultured hippocampal neurons determined from sucrose-evoked miniature EPSCs. J. Neurophysiol. 75: 1250-1255, 1996.[Abstract/Free Full Text]
• BOLSHAKOV, V. Y., SIEGELBAUM, S. A. Regulation of hippocampal transmitter release during development and long-term potentiation. Science 269: 1730-1734, 1995.[Abstract/Free Full Text]
• DOBRUNZ, L. E., STEVENS, C. F. Heterogeneity of release probability, facilitation, and depletion at central synapses. Neuron 18: 995-1008, 1997.[Medline]
• DODGE, F. A. JR, RAHAMIMOFF, R. Co-operative action of calcium ions in transmitter release at the neuromuscular junction. J. Physiol. (Lond.) 193: 419-432, 1967.[Abstract/Free Full Text]
• HESSLER, N. A., SHIRKE, A. M., MALINOW, R. The probability of transmitter release at a mammalian central synapse. Nature 366: 569-572, 1993.[Medline]
• HESTRIN, S., SAH, P., NICOLL, R. A. Mechanisms generating the time course of dual component excitatory synaptic currents recorded in hippocampal slices. Neuron 5: 247-253, 1990.[Medline]
• HUETTNER, J. E., BEAN, B. P. Block of N-methyl-D-aspartate-activated current by the anticonvulsant MK-801: selective binding to open channels. Proc. Natl. Acad. Sci. USA 85: 1307-1311, 1988.[Abstract/Free Full Text]
• JACK, J. J., REDMAN, S. J., WONG, K. The components of synaptic potentials evoked in cat spinal motoneurones by impulses in single group Ia afferents. J. Neurophysiol. 321: 65-96, 1981.
• JAHR, C. E. High probability opening of NMDA receptor channels by L-glutamate. Science 255: 470-472, 1992.[Abstract/Free Full Text]
• KATZ, B. In: The Release of Neural Transmitter Substances., . Liverpool: Liverpool University Press, 1969
• KORN, H., FABER, D. S., TRILLER, A. Probabilistic determination of synaptic strength. J. Neurophysiol. 55: 402-421, 1986.[Abstract/Free Full Text]
• KULLMANN, D. M., ERDEMLI, G., ASZTELY, F. LTP of AMPA and NMDA receptor-mediated signals: evidence for presynaptic expression and extrasynaptic glutamate spill-over. Neuron 17: 461-474, 1996.[Medline]
• MAGLEBY, K. L. Short-term changes in synaptic efficacy. In: Synaptic Function, edited by G. M. Edelman, V. E. Gall, and K. M. Cowan . New York: John Wiley, 1987, p. 21-56
• MALINOW, R., TSIEN, R. W. Presynaptic enhancement shown by whole-cell recordings of long-term potentiation in hippocampal slices. Nature 346: 177-80, 1990.[Medline]
• MANABE, T., NICOLL, R. A. Long-term potentiation: evidence against an increase in transmitter release probability in the CA1 region of the hippocampus. Science 265: 1888-1892, 1994.[Abstract/Free Full Text]
• MANABE, T., WYLLIE, D. J., PERKEL, D. J., NICOLL, R. A. Modulation of synaptic transmission and long-term potentiation: effects on paired pulse facilitation and EPSC variance in the CA1 region of the hippocampus. J. Neurophysiol. 70: 1451-1459, 1993.[Abstract/Free Full Text]
• MURTHY, V. N., SEJNOWSKI, T. J., STEVENS, C. F. Heterogenous release properties of visualized individual hippocampal synapses. Neuron 18: 599-612, 1997.[Medline]
• RAASTAD, M. Extracellular activation of unitary excitatory synapses between hippocampal CA3 and CA1 pyramidal cells. Eur. J. Neurosci. 7: 1882-1888, 1995.[Medline]
• RAASTAD, M., STORM, J. F., ANDERSEN, P. Putative single quantum and single fibre excitatory postsynaptic currents show similar amplitude range and variability in rat hippocampal slices. Eur. J. Neurosci. 4: 113-117, 1992.[Medline]
• REYNOLDS, I. J., MILLER, R. J. ³[H] MK801 binding to the NMDA receptor/ionophore complex is regulated by divalent cations: evidence for multiple regulatory sites. Eur. J. Pharm. 151: 103-112, 1988.[Medline]
• ROSENMUND, C., CLEMENTS, J. D., WESTBROOK, G. L. Nonuniform probability of glutamate release at a hippocampal synapse. Science 262: 754-757, 1993.[Abstract/Free Full Text]
• ROSENMUND, C., FELTZ, A., WESTBROOK, G. L. Synaptic NMDA receptor channels have a low open probability. J. Neurosci. 15: 2788-2795, 1995.[Abstract]
• STEVENS, C. F., WANG, Y. Reversal of long-term potentiation by inhibitors of haem oxygenase. Nature 364: 147-149, 1993.[Medline]
• STEVENS, C. F., WANG, Y. Changes in reliability of synaptic function as a mechanism for plasticity. Nature 371: 704-707, 1994.[Medline]
• STEVENS, C. F., WANG, Y. Facilitation and depression at single central synapses. Neuron 14: 795-802, 1995.[Medline]
• WALMSLEY, B., EDWARDS, F. R., TRACEY, D. J. Nonuniform release probabilities underlie quantal synaptic transmission at a mammalian excitatory central synapse. J. Neurophysiol. 60: 889-900, 1988.[Abstract/Free Full Text]
• ZUCKER, R. S. Short-term synaptic plasticity. Annu. Rev. Neurosci. 12: 13-31, 1989.[Medline]

0022-3077/97 $5.00 Copyright ©1997 The American Physiological Society