Abstract
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 Nmethyldaspartate receptormediated synaptic responses in the presence of the openchannel blocker MK801. 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 MK801–mediated decrease in excitatory postsynaptic current amplitudes and the underlying distribution of synaptic reliabilities. We find that the MK801 data are consistent with a continuous distribution of synaptic reliabilities, in agreement with studies examining individual synapses. In addition, changes in the MK801–mediated decrease in response size as a consequence of altering release probability are consistent with this continuous distribution of synaptic reliabilities.
INTRODUCTION
Synapses release neurotransmitter probabilistically (Katz 1969). Release probability—the probability that at least one exocytotic event will occur when a nerve impulse invades a presynaptic terminal—is regulated by the Ca^{2+}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 Nmethyldaspartate (NMDA) receptormediated synaptic current amplitudes in the presence of the openchannel NMDA receptor blocker MK801. MK801 blocks receptors only when they are opened, and because receptors open only when transmitter is released, the progressive block of receptors in MK801 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 FM143 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 MK801 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_{r} s ranging continuously from <0.1 to 1.0.
In the following analysis, we demonstrate that the progressive block of NMDAmediated synaptic responses in the presence of MK801 is in fact consistent with a continuous release probability distribution. First, we analyze the MK801 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 (twoclass) 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 MK801 under two conditions that alter the synaptic release probability: paired pulse facilitation and changing the external Ca^{2+} concentration.
METHODS
Slice preparation and physiology
Transverse hippocampal slices (350μm thick) were prepared from 2 to 3wkold LongEvans rats using standard procedures (Stevens and Wang 1993). Slices were stored at room temperature (24°C) in saline bubbled with 95% O_{2}5% CO_{2} for ≥1.5 h before recording; the saline contained (in mM) 120 NaCl, 3.5 KCl, 1.25 NaH_{2}PO_{4}, 26 NaHCO_{3}, 1.3 MgCl_{2}, and 2.5 CaCl_{2}. 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 MΩ and were filled with internal solution containing (in mM) 130 Cs gluconate, 5 CsCl, 5 NaCl, 10 N2hydroxyethylpiperazineN′2ethanesulfonic acid, 0.5 ethylene glycolbis(βaminoethyl ether)N,N,N′,N′tetraacetic acid, 1 MgCl_{2}, 2 MgATP, and 0.2 Liguanosine 5′triphosphate.
MK801 experiments
Whole cell recordings were obtained as above in standard recording solution containing 5 μM 6,7dinitroquinoxaline2,3dione (DNQX). NMDAmediated 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 MK801 was added as specified in the text. DNQX and MK801 were obtained from Research Biochemicals.
Twopathway 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 50ms interval). In twopathway 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 MK801, as follows
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 pairedpulse 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 40ms 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 leastsquare fitting procedures. Differential equations describing the NMDAreceptor kinetic scheme also were solved in Mathcad.
RESULTS
Evaluation of the MK801 method
Whole cell recordings were obtained from CA1 pyramidal cells in hippocampal slices, and NMDA receptormediated 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 NMDAreceptor openchannel blocker MK801 was added to the bathing medium for 10 min. When stimulation was resumed in the presence of MK801, the NMDA receptormediated EPSCs decreased progressively in amplitude with each stimulus event (Fig. 1 A). The shape and time course of the amplitude decrease in 40 μm MK801 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. 1 B). In subsequent discussion, we shall call curves such as the one illustrated in Fig. 1 B, blocking functions.
Our goal is to determine the relationship between this EPSC amplitude decrease in MK801, the blocking function, and the underlying release probability distribution for a population of synapses. MK801 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 MK801 when a release event does occur. The MK801 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 MK801. Thus a proper evaluation of the MK801 blocking function must involve an analysis of channel block by MK801 under our particular experimental conditions. Toward this end, we obtained blocking functions for different concentrations of MK801 (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 receptormediated EPSCs so that their initial peaks match in the presence and absence of 40 μM MK801 (Fig. 2 A) confirmed that the decay of the EPSC is faster when MK801 is present; this faster decay represents the block of open channels by MK801 (Hestrin et al. 1990; Huettner and Bean 1988). EPSCs were fitted with a simple fourstate kinetic model (Fig. 2 B), 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 MK801 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 MK801 (Jahr 1992). When unblocking was used, the unblocking rate was at least an order of magnitude smaller than the blocking rate.
The estimated blocking rate increased as the concentration of MK801 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 MK801 during a single synaptic response), the differential equations for the kinetic scheme (Fig. 2 B) 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 MK801 concentration in Fig. 2 C, and the data are fitted by an effective doseresponse curve with k _{1/2} = 28.63 μM MK801. We emphasize that this constant k _{1/2} is not the equilibrium dissociation constant of MK801 binding to the NMDA receptor. Rather, it is the concentration of MK801 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 MK801 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 MK801 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 MK801 blocking function against the amplitudes of another for each stimulus number (Fig. 3 A), 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 MK801 experiment in the same way (Fig. 3 B), however, the points deviated greatly from the slope = 1 line, highlighting the slower decrease of EPSC amplitudes in 10 μM MK801. Plots for decreases in 20 μM against those in 40 μM MK801 fell in between (plots not shown).
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 MK801. Figure 3 C shows typical results from an experiment in 40 μm MK801; 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 MK801 depends on block efficacy and synaptic release probability
Now we can derive the quantitative relationship between synaptic release probability and the decrease of NMDAmediated EPSC amplitudes in the presence of MK801, 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 MK801. This fraction (a) depends on three factors: P_{r} , the release probability of the stimulated synapses; θ, the fraction of receptors participating in the current given a release event; and m, the fraction of these participating receptors that get blocked by MK801 (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 ρ(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
From Eq. 1 , we see that the predicted decrease of synaptic amplitudes in the presence of MK801 depends on mθ, the fraction of NMDA receptors blocked by MK801 when a synapse releases transmitter, as well as ρ(P_{r} ), the P_{r} weighting distribution. We now are prepared to determine what release probability distributions fit our MK801 data.
Decrease of EPSC amplitudes in MK801 is consistent with a continuous release probability distribution
Previous studies analyzing the decrease of EPSC amplitudes in MK801 (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 MK801 blocking function could be explained as well by a continuous distribution of release probabilities as by the twoclass (bimodal) P_{r}
distribution. An ideal bimodal weighting function, as used by Hessler et al. and Rosenmund et al., is represented mathematically as
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 MK801
Experimentally obtained blocking functions in the presence of 10, 20, and 40 μM MK801 were fitted with the predicted S_{n} functions incorporating the bimodal and continuous distributions by a leastsquares–fitting procedure (Fig. 4). For all cells (n = 15), the amplitude decreases were fitted by both models nearly indistinguishably; for 40 μm MK801 (n = 6), mean values of A, b _{1}, b _{2}, 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 MK801 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.
To obtain a quantitative description of the above P_{r} distributions, we must calculate P _{1} (=b _{1}/mθ) and P _{2} (=b _{2}/mθ), the P _{r} values of the two synaptic classes in the bimodal distribution, and 1/λ (λ = rmθ), the characteristic P _{r} for the continuous distribution. Calculation of these values requires an estimate of mθ, 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 MK801. θ is the fraction of receptors that participate in the current when transmitter is released; we take θ at the previously estimated value 0.5 (Rosenmund et al. 1995) and at 1.0. When θ is 0.5, P _{1}, P _{2}, and λ are 0.38, 0.05, and 2.99, respectively; when θ is 1, P _{1}, P _{2}, and λ 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/λ 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 Ca^{2+} 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 MK801, the EPSC amplitudes of one pathway could be plotted against the amplitudes of the other (for each stimulus number) as a straight line (Fig. 5 A), with a slope of 1.
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. 5
B). 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 φ(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
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_{r} s ranging continuously from 0.1 to 1.0; there was no evidence of a bimodal distribution of synaptic P_{r} (see Fig. 6).
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 φ(P_{r}
) as a straight line
We also calculated the S_{n}
for PPF synapses assuming a bimodal P_{r}
distribution
Ca^{2+} alters the P_{r} distribution according to the DodgeRahamimoff equation
Transmitter release is a Ca^{2+}dependent process, and changes in external Ca^{2+} profoundly affect synaptic release probability (Dodge and Rahamimoff 1967). We examined the MK801 blocking function at four different external Ca^{2+} concentrations: 1.0, 1.75, 2.5 (standard), and 6.0 mM. Figure 8 A shows blocking functions under these four conditions.
Blocking functions for all Ca^{2+} concentrations were well fitted by our model incorporating a continuous P _{r} distribution. The average bestfit rs for external [Ca^{2+}] 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 [Ca^{2+}] concentrations. r, however, also is affected by MK801 block efficacy (recall that r = λ/mθ); because Ca^{2+} ions may interact with the MK801 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 [Ca^{2+}]). Analysis of the EPSCs revealed that MK801 block fraction was not significantly affected by changing the external Ca^{2+} concentration except at 6.0 mM [Ca^{2+}], 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 [Ca^{2+}] was scaled by a factor of 0.330/0.611 = 0.54 for the purposes of quantitative comparison.
Figure 8 B shows normalized r ^{−1} plotted against external [Ca^{2+}]; the data are fitted by the DodgeRahamimoff equation (Dodge and Rahamimoff 1967). Best fit by a leastsquares fitting procedure yielded k _{Mg} and k _{Ca} of 3.9 and 0.57 mM, respectively.
Blocking functions at different [Ca^{2+}] 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, b _{1}, and b _{2}, changed in a manner that we were unable to relate to the DodgeRahamimoff equation: for external [Ca^{2+}] of 1.0 mM, average A, b _{1}, and b _{2} 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 [Ca^{2+}] 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
In the work presented here, we analyzed the relationship between synaptic release probability and the decrease of NMDAmediated synaptic current amplitudes in the presence of MK801. Previous studies exploiting this method (Hessler et al. 1993; Rosenmund et al. 1993) have argued that the simplest fit of the MK801 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 longterm potentiation and longterm 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 MK801 can be described with a continuous P_{r} distribution. One attractive feature of this distribution is that it describes the MK801 blocking function with a single parameter (r). In contrast, three parameters are needed with the bimodal distribution (A, b _{1}, b _{2}). Our distribution thus provides the simplest fit of the MK801 data. Furthermore, this distribution fits the MK801 blocking function robustly under conditions of altered release probability, such as PPF and changes in external [Ca^{2+}]. In our PPF experiments, we can account for the change in MK801 blocking function in terms of our distribution and the amount of observed PPF alone. In our Ca^{2+} experiments, the predicted change in our distribution at different Ca^{2+} concentrations follows the DodgeRahamimoff equation. In contrast, we cannot identify a consistent relationship between the parameters of the bimodal distribution and the [Ca^{2+}].
The equation we use to describe the continuous distribution, ρ(P_{r} ) = e ^{−λPr }, 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 MK801 method, although this method is more accurate at low P _{r} than minimal stimulation. We estimate the MK801 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 stimulationderived 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 MK801 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: θ, 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 θ = 0.54. Thus a real, albeit incomplete, distribution derived from minimal stimulation data are consistent with the MK801 blocking function.
Although we have confirmed here that the MK801 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 [ρ(P_{r} ) = e ^{−λPr }] to describe our data—this distribution is the lowest order approximation to a more general function—but we must emphasize that the MK801 method alone cannot be used to determine the exact shape of the function that describes the synaptic reliability distribution. The problem is that the MK801 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 ρ(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 ρ(P _{r}) that is consistent with the observed blocking function, but within that range, cannot decide whether one ρ(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 MK801 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 MK801 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_{r} s is provided by the characteristic release probability 1/λ of our fitted continuous distribution: twothirds of the synapses have a P_{r} value less than or equal to the characteristic P_{r} . Our estimation of 1/λ 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.
Acknowledgments
This work was supported by the Howard Hughes Medical Institute and a National Institute of Neurological Disorders and Stroke Grant NS12961 to C. F. Stevens and by a Department of Defense Graduate Fellowship from the Office of Naval Research to E. P. Huang.
Footnotes

Address for reprint requests: C. F. Stevens, The Salk Institute, 10010 N. Torrey Pines Rd., La Jolla, CA 92037.
 Copyright © 1997 the American Physiological Society
APPENDIX
MK801 blocking equation
Here we derive the relationship between synaptic P_{r}
and the decrease of EPSC amplitudes in the presence of MK801. 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 MK801 can be calculated by solving the difference equation
Solving the above difference equation yields
Because release probability is not constant, there must be a probability density function ω(P_{r}
) that describes the relative distribution of synaptic release probabilities in the population; thus F_{n}
can be weighted accordingly
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}
)
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 ρ(P_{r}) as a negative exponential function
The choice of an exponential weighting function can be motivated as follows. First, recall
MK801 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
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