Characterization of the variability of glutamatergic synaptic responses

INSTITUTE OF PHYSICS PUBLISHING

NETWORK: COMPUTATION IN NEURAL SYSTEMS

Network: Comput. Neural Syst. 12 (2001) 175–198

www.iop.org/Journals/ne

PII: S0954-898X(01)22490-7

Characterization of the variability of glutamatergic synaptic responses to presynaptic trains in rat hippocampal pyramidal neurons Marco Canepari1,2 and Alessandro Treves Programme in Neuroscience and Istituto Nazionale Fisica della Materia (INFM unit), International School for Advanced Studies, Via Beirut 2-4, 34014 Trieste, Italy E-mail: [email protected]

Received 8 June 2000, in final form 30 January 2001 Abstract Excitatory postsynaptic currents from CA3 hippocampal neurons, elicited by trains of presynaptic action potentials either in mossy fibres or associative commissural fibres, have been analysed, by using a quantal analysis approach, in order to characterize their variability and the correlation among successive responses. As quantal parameters may change during the train according to the previous release events, correlation within consecutive EPSCs is expected. We tested simple hypotheses on how quantal parameters p and N may change on the basis of correlation detection in EPSCs. The statistical significance of these tests has been evaluated. The tests showed that, although simple binomial distributions can give a good description of synaptic responses at the level of single spikes, only stochastic chains can always account for correlations observed within the train. A systematic model fitting procedure has been developed and applied to extract information on the dynamics of synaptic transmission. As an application of this novel type of analysis, a measure of transmitted information to be associated with synaptic variability, a quantity that allows an estimate of the capability of the synapse to transmit reliable information in time, is proposed. We showed that this transmitted information depends on short-term plasticity and that the change in the type of short-term plasticity from facilitating to depressing obtained by increasing the extracellular calcium concentration results in a change of the related transmitted information.

1. Introduction The characterization of the variability of synaptic transmission and, more generally, of neural activity, is crucial to an understanding of information processing in the nervous system. Since the early studies on the neuromuscular junction showing that the variability of end plate 1 2

Present address: National Institute for Medical Research, The Ridgeway, Mill Hill, London NW7, UK. Corresponding author.

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potentials reflects the stochastic process of release of acetylcholine quanta (Del Castillo and Katz 1954), quantal theory (QT) of neurotransmitter release has been widely used to investigate synaptic responses also in the CNS (Redman 1990). The application of discrete statistical models and, in particular, of binomial models, reflecting the independence of the release of each quantum from the others, to fit experimental distributions of synaptic response amplitudes, is known as quantal analysis (QA) and allows synapses to be characterized in terms of release probability p, of number of available quanta N , and of quantal size of synaptic responses q (Van der Kloot 1991). Although QA methods have been proven to be efficient in characterizing the variability of synaptic responses (Larkman et al 1997a, Stratford et al 1997, Stricker and Redman 1994), classical QA has severe limitations when applied to central synapses (Silver et al 1998). In terms of extraction of statistical parameters with a precise biophysical meaning, classical QA cannot work well because the necessary simplifying assumptions are often not valid. For instance, in many systems (Hessler et al 1993, Murthy et al 1997) a compound binomial model (Stricker et al 1996) has been found to give a better statistical fit of experimental data than a simple binomial model. However, the limited number of available recordings and the possible variations in quantal size may still prevent discriminating among different hypotheses of synaptic transmission (Stricker et al 1994). Furthermore, in the case of synaptic responses to short trains of presynaptic action potentials, associated with short-term plasticity processes, quantal parameters may in general be subject to substantial changes, correlated to the previous synaptic responses in the train, which prevent the applicability of conventional QA methods (Larkman et al 1997b). These problems can be partially overcome by developing new approaches extending QA methods and requiring less stringent assumptions and a minimal number of extra parameters. In these extensions, each parameter may not have a precise biophysical meaning, but in general can still be interpreted in terms of the QT of neurotransmitter release. In particular, new relevant physiological information can be acquired by using statistical tools that allow a reliable interpretation of limited numbers of synaptic responses generated by synapses that are generally non-uniform in space and in time (short-term plasticity). An interesting example is the multiple-probability fluctuation analysis which has recently been used to analyse synaptic responses in the cerebellum and that allows to extract a mean release probability when quantal parameters are spatially non-uniform (Silver et al 1998). Similarly, new insight on the problem of short-term plasticity, i.e., on how quantal parameters can change in time during repetitive synaptic activation, can be gained by appropriate statistical techniques. This paper presents a new approach based on stochastic chain theory to analyse and interpret synaptic responses following repetitive activation of presynaptic fibres. The characterization of repetitive release, as already indicated by several extensions of the theory of neurotransmitter release at the neuromuscular junction (Elmquist and Quastel 1965, Magleby and Zengel 1982, Mallart and Martin 1967, Quastel 1997, Sen et al 1996), requires the extension of a traditional QT scheme based on static probability distributions to a dynamic quantal theory (DQT) scheme based on the theory of stochastic processes. In particular, if the statistics of the amplitude of the first synaptic responses in the train can be represented by simple or compound binomial distributions, the statistics of the amplitude of the other synaptic responses can be represented by distributions extracted from binomial chains where the quantal parameters p, N and q are also, in general, stochastic variables. Their distribution may depend on previous release, thus generating correlations within synaptic responses in the train. We addressed the question of whether the detection of correlations in hippocampal excitatory synaptic responses could discriminate among simple models of repetitive synaptic transmission, i.e., among simple binomial chains with different hypotheses as to how quantal

Dynamic characterization of synaptic responses

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parameters may change within the train. We analysed sets of hundreds of excitatory postsynaptic currents in the hippocampal CA3 region generated by the stimulation of either mossy fibres or associative commissural fibres. Within a general approach, that we termed dynamic quantal analysis (DQA), to determine the best fit of experimental data to a binomial chain, we developed a criterion to select the simplest binomial hypothesis that accounts for the statistics of repetitive synaptic transmission. At central synapses, the variability of synaptic responses evoked by trains of action potentials is particularly relevant to the time integration of synaptic signals. As proposed by Lisman (1997) the variability of synaptic responses can still allow synapses, which are unreliable at the level of a single action potential, to be reliable at the level of a burst. Synapses that are highly stochastic can, thus, still transmit reliable information to postsynaptic neurons. In this spirit, in the last part of this paper we have calculated a measure of transmitted information associated with synaptic transmission, to investigate how the variability of synaptic responses, associated with short-term plasticity processes during trains of presynaptic action potentials, can determine the properties of temporal integration of synaptic signals.

2. Methods 2.1. Patch clamp recordings Excitatory postsynaptic currents (EPSCs) from CA3 pyramidal neurons have been recorded in hippocampal slices as described in a previous report (Canepari and Cherubini 1998). Briefly, slices (200 µm thick) were prepared from 7 to 11 day old rats and neurons were patch clamped with pipettes (R = 2–5 M) containing an intracellular solution (in mM): 120 K-gluconate, 4 MgCl2 , 49 N-2-hydroethylpiperazine-N -2-ethanesulfonic acid (HEPES), 0.6 ethylene glycolbis (b-aminoethyl ether) N,N,N ,N -tetraacetic acid (EGTA), 2 Na2 ATP, 0.2 Na2 GTP, adjusted to pH 7.2 with KOH, 290 mOsm. Extracellular recording solution contained (in mM): 126 NaCl, 3.5 KCl, 2 CaCl2 , 1.2 NaH2 PO4 , 1.3 MgCl2 , 14 NaHCO3 , 11 glucose, gassed with 95% O2 and 5% CO2 at 24 ◦ C (pH 7.3), 305 mOsm. Bicuculline (10 µM, purchased from Sigma, St. Louis, USA) was added in order to block GABAA -mediated synaptic responses. EPSCs from CA3 hippocampal neurons were recorded at −70 mV in voltage clamp mode, with a standard patch clamp amplifier (EPC-7, List Medical Instruments). Series resistance (typically 10–15 M) was continuously checked for stability during the experiment. Currents were evoked by minimal extracellular stimulation of fibres in the stratum lucidum–radiatum (Canepari and Cherubini 1998). With our experimental protocol, either mossy fibres from dentate gyrus granule cells and/or associative commissural fibres from other CA3 pyramidal neurons could be stimulated. In comparison with double recording experimental protocols (Markram and Tsodyks 1996), extracellular stimulation does not allow the monitoring of presynaptic action potentials nor that of actual number of stimulated fibres. The very low probability of connectivity between excitatory neurons in the hippocampus (less than 5% in the adult hippocampus) (Amaral et al 1990) makes the use of the double recording technique difficult in this preparation. However, as shown in a previous report (Canepari and Cherubini 1998), the threshold for activation of one or more presynaptic fibres is basically the same for all the pulses in the train. Thus, possible phenomena such as K+ accumulation near the presynaptic fibres, leading to a change in the number of stimulated fibres, are likely to be negligible using this stimulating procedure.

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2.2. Data analysis and computer simulations Signals were low-pass filtered at 2 kHz and acquired at 10 kHz with a 12 bit laboratory interface (1401plus, Cambridge Electronic Design, Cambridge, UK). EPSC amplitudes were measured after the detection of stimulation artefacts. The amplitude was calculated as the absolute value of the mean of five points centred on the peak minus the baseline calculated as the mean of 100 points before the stimulus. A qualitative discrimination between mossy fibre EPSCs and associative commissural fibre EPSCs was obtained by testing the compound 2(2,3-dicarboxycyclopropyl)glycine (DCG-IV) (Canepari and Cherubini 1998), that selectively reduces the release probability at mossy fibre synapses. Both types of synaptic responses exhibited variability in quantal parameters and short-term plasticity behaviours and were analysed with the same procedures (see next paragraphs). In the case of an EPSC having a peak superimposing to the tail of the previous one, the amplitude was compensated by fitting the decay phase of the previous EPSC with a single exponential function (Canepari and Cherubini 1998). The simulations of the stochastic release process, the analysis of experimental synaptic responses and the calculation of the relevant statistical quantities in both theoretical and experimental cases have been done using programs developed with Matlab 5.1 for PC (The Math Works, Inc., Natick, USA). In particular the Matlab function rand has been used to generate random numbers in the simulation routines. 2.3. Statistical models The binomial QT has been used as a model of synaptic transmission (Fesce 1990). In the context of this theoretical framework, each synapse will then be characterized by a release probability p, by a number of quanta available for release N (i.e., releasing sites), and by a quantal size (i.e., by the size of the synaptic response corresponding to a single quantum) q. In a more general hypothesis, each quantum of the synapse has distinct release probability and quantal size leading to a multinomial statistics of synaptic responses (an even more general hypothesis would include trial-to-trial variability in the response associated with the same quantum, Stricker and Redman 1994). This synapse will then be characterized by 2N + 1 free parameters. The number of free parameters becomes independent of N and is reduced to 3 in the particular case of the simple binomial model in which p and q are uniform. As described in the results, by analysing sets of 100 EPSCs in CA3 pyramidal neurons we found that the high variability and the relatively small number of synaptic responses did not allow us to extract meaningful distributions by using statistical models with a large number of free parameters such as the multinomial model. This is quantitatively analysed in the results. Since we adopted the procedure of calculating the quantal size q directly as the ratio between the mean EPSC and the mean number of released quanta in the model (see appendix A2), the number of free parameters was reduced further to 2. In the simple binomial model the probability p(n) that n quanta are released after one action potential is N! p(n) = · p n · (1 − p)N −n . (1) n!(N − n)! The mean number of released quanta n, its standard deviations σn and the probability of a failure p(0) (release of 0 quanta) are n = p · N (2) 2 (3) σn = N · p · (1 − p) N p(0) = (1 − p) . (4)


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We fitted the experimental data histograms (as described in appendices A2 and A3) with theoretical distributions extracted from binomial chains with different hypotheses on the variability of the parameters p and N within the train. In particular we selected sets of 100 EPSCs in which the computation of a temporal linear regression through mean EPCS amplitudes (n = 30–40 EPSCs) showed a change of less than 20% in the synaptic recording conditions. We used three levels of approximation. At the first level the fitting was done with binomial distributions in which the values p and N are independent of previous synaptic events (model 0), corresponding to a classical quantal approach where experimental distributions are associated with simple, and uncorrelated, binomial distributions. At the second level (model 1), the parameter N depended on the number of quanta released previously, and thus evolved in a stochastic manner, whereas the parameter p evolved in a simple deterministic manner. Finally, at the third level (model 2), the fitting was done with binomial chains in which both N and p were stochastic variables, dependent on previous release. The latter two approaches are referred to as dynamic quantal approaches. Each model is defined by the evolution of the release probabilities pj and of the numbers of available quanta Nj at the j th action potential of the train. Previous studies have shown that the balance between the effect of residual calcium accumulation and the depletion of vesicles accounts for the facilitating and depressing behaviours in the synaptic responses to a train of action potentials in CA3 hippocampal pyramidal neurons (Canepari and Cherubini 1998, Miles and Wong 1986). In keeping with this, we limited our investigation to binomial chains (models 1 and 2) in which the parameter Nj was changed by the phenomenon of depletion. Namely, at each action potential, Nj was decreased by the nj released quanta and, before the next action potential, increased by the number of previously released quanta that were made reavailable for the (j + 1)th release. Then, the number of quanta Nj available for release at the j th action potential of the train was N decreased by the number of quanta ν1 , . . . , νj −1 that have been released after the previous j − 1 action potentials and that were not yet reavailable for further release: j −1 Nj = N − νk . (5) k=1

The simple deterministic approximation of the process of reavailability, which would in general be itself stochastic (Melkonian 1993, Quastel 1997), has been used here. In this approximation, the number of available quanta Nj was a time function of the previously released quanta: Nj = N −

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νk (t) can be expressed as the integer number closer to the product f (t) · nk , where f (t) is a time function such that f (0) = 1 (no reavailability immediately after the release) and f (t → ∞) = 0 (complete reavailability after a long time). For synaptic release at a constant frequency of presynaptic firing 1/t, the reavailability has been simulated with a negative exponential behaviour f (tj − tj −k ) = f (k · t) = e−k·t/τ , where τ is the time constant of the deterministic approximation to reavailability. At the level of model 1, the release probability varied with a purely deterministic evolution, and was defined by a set of release probabilities pj at each spike j . In order to constrain the otherwise large number of parameters pj , particularly when describing processes with several presynaptic action potentials, one may introduce simple rules that determine the probabilities pj in terms of a minimal number of free parameters. A reasonable minimum is three, since one parameter is needed for the initial p value, one for its change after the first action potential, and one for its asymptotic value after many action potentials. A simple 3-parameter rule, that

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we used as a crude model of the pj series, is the generalized Dodge–Rahamimoff equation (Dodge and Rahamimoff 1967, Van der Kloot and Molgó 1994) pj =

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where determines the release probability at the first action potential and determines the release at the following action potentials. Although the original meaning of the parameter is related to the transient intracellular active calcium that triggers neurotransmitter release, the Dodge–Rahamimoff equation has been used here as a convenient simple model without any deep biophysical meaning. This model allowed to reduce the number of free parameters to three (pmax , and ) when the number of action potentials was higher than three. The method of analysis is not sensitive to the form of equation (7) and other simple rules may be used to account for changes in the release probability. At the level of model 2, the deterministic approximation for the evolution of the release probability must be replaced by a more general Markovian one in which also the change in the release probability depends on the previous release event. We used the simplest model that does not introduce further parameters and in which the application of the change in equation (7) is made conditional on the immediately preceding event not having been a failure, i.e., on there having been a genuine release. Such a model is a simple and extreme implementation of the hypothesis that failures may not be just normal presynaptic events followed by release of zero quanta, but may include failures in the cascade of events underlying exocytosis of quanta and, in particular, failures in the invasion of the presynaptic terminal by an action potential.

2.4. Correlation detection The statistics of cumulative responses Aj after j spikes Aj =

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3. Results 3.1. Discrimination among different hypotheses on the basis of correlation Experiments in which EPSCs were recorded in CA3 pyramidal neurons following trains of four action potentials have been analysed with the classical and with the dynamic quantal approaches. In particular, correlations among successive responses have been measured along with the statistical distribution of responses to individual spikes in the train.

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The ability of three successive simple approximations to describe the experimental data has been compared (model 0: independent binomial distributions, no correlation; model 1: stochastic release chain with predetermined changes in p; model 2: stochasticity also affecting changes in p). Figure 1(a) shows six consecutive recordings with EPSCs evoked by four pulses of stimulation at 60 ms interspike interval and the average of 100 recordings including those of (a); short-term plasticity was dominated, in this case, by facilitation. Figure 1(b) shows the histograms of EPSC amplitudes relative to the four action potentials (left column) and the distributions obtained by running 10 000 computer simulations for each of the three models with sets of parameters giving a good fit (always Pj (χ 2 ) > 10% ∀j ) to experimental distributions (see appendix A2). The distributions obtained with the three models are very similar at the level of single spikes and all of them could fit the experimental data. This is also true at the level of the mean response normalized to the mean of the first synaptic responses, and its standard deviation, figures 2(a) and (b). A schematic example of reshuffling for four recordings with three pulses is shown in C. The technique simply consists in randomly mixing the responses to the second, third and fourth (not shown in the example of C) pulses so that each recording loses the original chain of events. For uncorrelated chains (model 0), the difference P (Aj ) − P ∗ (Aj ) is close to zero at each bin used to measure A4 (not precisely zero because of finite sampling effects). For model 1, correlations among successive release events are entirely due to the variability in the number of released quanta and are negative, in the evident sense that higher release is more likely following a string of previous low release events, since a higher number of quanta is then available. Hence, the negative correlations generate a narrower distribution for A4 , which results in the expected central peak in the plot with two negative almost symmetrical flanks, figure (2(d) (2)). Positive correlations would result in a wider distribution and in a plot with an opposite shape. Model 2 embodies a mixture of positive and negative correlations. The positive contribution to correlations is stronger in facilitating behaviours with an initially high number of failures (because if pj can only increase after actual release, failures are slightly more likely to be followed by other failures). The result is a P (Aj ) with asymmetrical flanks as shown in figure (2(d) (2)). This is the same as that of experimental data, figure (2(d) (1)). Figure (2(d) (3)) shows the difference P (Aj ) in two other experiments with synaptic responses having a statistical behaviour similar to that of the experiment shown in figure (2(d)). Again, as A4 increases, first a positive difference, then a large negative one, are followed by a large positive one as predicted by model 2. A more quantitative criterion to evaluate the discrepancy between the P (Aj ) curves obtained from experimental data and that obtained with each model is to compute the integral of the absolute value of the difference of the two curves and to compare the result with the statistics obtained by running sets of computer simulations with the corresponding model parameters and with the number of simulations in each set equal to the number of experimental recordings. In this way, a model can be rejected on the basis of the value of the integral, i.e., whether and by how many SDs this value exceeds the mean of the distribution of the values obtained by running 100 sets of computer simulations. In the case of the experiment reported in figures 1 and 2, for models 0, 1 and 2 the statistics of the values obtained with the computer simulations were 0.20 ± 0.06, 0.18 ± 0.05 and 0.18 ± 0.06, while the values of the integral | experimental P (Aj )—model P (Aj )| were 0.29, 0.37 and 0.16. Thus, in this case, it was possible to reject model 0 and model 1 (P < 0.05), but not model 2 (P > 0.5). 3.2. Statistical significance of the correlation analysis The realistic synaptic dynamics are a function of a very large number of parameters. Indeed, any quantal analysis method is based on oversimplifying assumptions and models that, although


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quanta Figure 2. Correlation of EPSCs. (a) Mean ±SD of the EPSC amplitudes shown in figure 1. (b) Mean/mean(1st) ±SD of the experimental EPSCs and of the three models shown in figure 1. (c) Simple example of the reshuffling procedure applied to four recordings of responses to three pulses of stimulation. Values in the second and third column of the four recordings are randomly redistributed. (d) (1) Difference between the distribution of the total cumulative current of experimental responses and that obtained by reshuffling data as described in the methods. (2) Same as (1) but for sets of 10 000 computer simulations obtained with the parameters of the three models of figure 1. (3) Same as (1) with data from two other experiments with a similar statistics of synaptic responses. The x axis in (d) represents the total number of released quanta in the models, i.e., the corresponding bin number of the histogram of the experimental EPSC amplitudes.

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unable to capture the details of the process, can still provide a statistical fit of the data and therefore useful information on the physiology of the synapse. The level of simplification of a model that provides a statistical fit of a complicated stochastic process is a function of the number of events available for analysis, i.e., of the number of recordings. In those four experiments in which we succeeded to obtain more than 200 EPSCs, the computation of a temporal linear regression through mean EPCS amplitudes (n =30–40 EPSCs) typically showed a change (20% or more) in the synaptic recording conditions, possibly because of an EPSC run-down or other long-term plasticity effects. Indeed usually 100–200 stable responses are available during a patch clamp experiment. We have, therefore, set out to test the capability of an unrealistic model, in which the release probability p and the quantal size q are uniform in the synapse, to fit more realistic processes with non-uniform values of p and q, as a function of the number of events. We run sets of synaptic response computer simulations, with randomly chosen values of non-uniform p and q, for the three different models. We then calculated P (χ 2 ) (as described in appendix A2) for the original set of non-uniform parameters and for sets of uniform parameters that maximize P (χ 2 ) for each of the three models. In figure 3(a), the mean ±SD of P (χ 2 ) relative to 100 sets of 50–5000 computer simulations of non-uniform synapses following model 2 are reported. For sets of 50–250 responses, the computation of P (χ 2 ) (which is a logarithmic average of the Pj (χ 2 ) over the j action potentials) gives similar values for either multinomial or simple binomial statistics maximizing P (χ 2 ). In particular, with 250 responses, in about 90% of the


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cases the simple binomial statistics cannot be rejected with P < 0.1. It is clear from the figure, that at least 500 responses are necessary to start having a significant discrimination between simple or more complex binomial statistics, i.e., to be able to distinguish between non-uniform and uniform synapses, having a small number of quanta. In this respect, the interpretation of the quantal parameters can be that of a mean property of a set of synaptic responses, rather than a biophysical property of the synapse. It is important to point out that several sets of parameters differing up to 50% one from the other can be associated with P (χ 2 ) values higher than 0.1. Therefore, although the results presented here are relative to the sets of parameters that maximize P (χ 2 ), no unequivocal determination of quantal parameters can be obtained. This negative conclusion was reached, here, using a simple χ 2 statistics, but it is expected to hold also for other statistical procedures (Stricker and Redman 1994), at most with a slight decrease in the number of trials required for effective model discrimination. The effectiveness of the correlation analysis, on the other hand, based on the comparison between the cumulative statistics of experimental EPSC and that of different binomial chains, has been investigated, again, by running computer simulation of possible synaptic responses. We investigated whether the simplifying simple binomial statistics can still account for the correlation of sets of 100 synaptic responses even in synaptic responses generated by more complicated statistics. Thus we computed the integral of the absolute value of the difference between the P (A4 ) calculated with 100 computer simulations and the ‘exact’ one (i.e., calculated with 50 000 computer simulations) normalized to the exact |P ∗ (A4 )|. For this purpose we used sets of 100 computer simulations obtained, as described above, both by using uniform parameters (simple binomial) or by using non-uniform parameters. Figure 3(b) shows the mean ±SD obtained from 100 integrals with the different models. We evaluated the statistical significance of the correlation analysis by analysing by how many SDs the value of the difference described above exceeds the mean of the same difference obtained with the real non-uniform set of parameters. For the sets of computer simulation with uniform parameters, it was found that, in 82% of the cases, both model 0 and model 1 could be rejected with P < 0.05, whereas model 2 could not be rejected (P > 0.3). This is not surprising since model 2 corresponds to the exact simulated dynamics. However, when the same type of analysis was applied to the sets of computer simulation with non-uniform parameters, model 0 and model 1 could still be rejected with P < 0.05 in 76% of the cases, and model 2 could not be rejected (P > 0.3), although the fitting model referred to a simple set of binomial parameters not corresponding to the non-uniform set generating the computer simulations. It must be stressed that model 0, model 1 and model 2, giving significant differences in experimental data fitting, have eventually the same number of degrees of freedom. These results demonstrate that a dynamic quantal approach based on an accurate correlation analysis, but using simple binomial statistics is, in principle, able to account for the dynamic process of repetitive release, even if this occurs with a more complex non-uniform binomial statistics. 3.3. Dynamic quantal analysis of synaptic responses A DQA procedure has been developed, based on the exact calculation of the release probabilities (see appendix A1), on the evaluation of the agreement between the experimental and the theoretical distributions in terms of mean probability P (χ 2 ) (see appendix A2) and on a process of P (χ 2 ) maximization with a Monte Carlo Metropolis algorithm (see appendix A3). The DQA has been applied to analyse EPSCs recorded in CA3 pyramidal neurons from P7–P11 hippocampal slices and elicited by stimulation of synaptic fibres in the stratum lucidum– radiatum with trains of 3–6 pulses at 12.5–25 Hz (see methods). Both synaptic responses from

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mossy fibre and associative commissural fibres are characterized by high unreliability within single recordings and by the presence of failures (Allen and Stevens 1994). Furthermore, very different facilitating or depressing behaviours (EPSC patterns) of mean EPSCs (n 30) can be observed by stimulating different synaptic fibres or even the same fibre at different extracellular calcium concentrations ([Ca]o ) (Canepari and Cherubini 1998). For all data analysed, the correlations were detected as described in the methods and, on this basis, in these experiments model 2 turned out to be the best model to account for the dynamics of repetitive synaptic transmission. In particular, in 14 out of 18 experiments in which 100 (stable) responses have been analysed P (χ 2 ) was never below 0.1 for any of the three models. However, with the correlation analysis, in none of these cases could model 2 be rejected with P < 0.05, whereas model 0 and model 1 could be rejected (P < 0.05) 11/14 times and 9/14 times, respectively. As shown in a previous report (Canepari and Cherubini 1998), most EPSC patterns observed in the presence of 2 mM [Ca]o are distinguished by the facilitation of the second mean EPSC and, in about 10% of the (n = 71) cases, also by the facilitation of third and fourth mean EPSC when stimulated with four pulses at 20 Hz. Figure 4(a) shows ten representative recordings of EPSC elicited by the stimulation of an associative commissural fibre characterized by the facilitation of the second, third and fourth mean EPSCs, figures 4(b)–(d). These recordings are dominated by a large number of failures and by a high variability (unreliability). The decreasing time course of the CV (i.e., of the ratio between the SD and the mean) indicate a typical behaviour of facilitation due to an increase in the release probability. Figure 4(e) shows P (A4 ), calculated from the same experimental data, superimposed to P (A4 ) of the model 2 that, at the level of single distributions, fits the experimental data. About 25% at 2 mM [Ca]o and more than 80% at 4 mM [Ca]o of the EPSC patterns in CA3 pyramidal neurons are dominated by the depression of the second, third and fourth mean EPSC when stimulated with four pulses at 20 Hz. Figure 5(a) shows ten representative recordings of EPSC elicited by the stimulation of an associative commissural fibre characterized by a depressing behaviour, figures 5(b)–(d), and an increasing time course of the CV. Figure 5(e) shows the experimental P (A4 ) and the corresponding P (A4 ) of the model 2 that, at the level of single distributions, fits the experimental data. The correlation quantified by P (A4 ) relative to synaptic responses was slightly negative, in this case, as predicted by model 2. The DQA also allows the estimation of the quantal current (18±15 pA, n = 15). Variations in the quantal current observed in different experiments can reflect the type of synaptic input stimulated (Henze et al 1997). In comparison to other studies on mossy fibre synaptic responses (Jonas et al 1993), done in maturer rat slices, these values were slightly higher. Typically, experiments with larger quantal currents are DCG-IV sensitive (Canepari and Cherubini 1998), whereas smaller quantal currents are more likely to be DCG-IV insensitive. Finally, DQA reduces to a classical QA when applied to a single action potential. 3.4. Transmitted information in synaptic signals The stochasticity characterizing synaptic events affects the fidelity with which synapses can transmit the occurrence of presynaptic action potentials. For example, a presynaptic high frequency train might be confused postsynaptically with one of lower frequency, because of failures in the transmission of individual spikes. Obviously, the discrimination between the two frequencies will be increasingly reliable with time, as successive spikes in the train arrive at the synapse and are integrated in the cumulative synaptic response A. One direct way to quantify this reliability is to measure the trans-information, i.e., the mutual information that the cumulative response at time t, A(t), conveys about a presynaptic variable such as the frequency


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(a)

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Model 2 0

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10 15 20 25 cum. resp. A4 (bin number)

Figure 4. DQA of synaptic responses: facilitating behaviour. (a) Ten typical recordings of EPSCs from a CA3 pyramidal neuron elicited by stimulation of a synaptic fibre with a facilitating behaviour. (b) Average of 100 recordings including those shown in (a). (c) Mean ±SD of the 100 EPSC amplitudes averaged in (b). (d) Comparison of the statistics (mean normalized to the first mean response ±SD) of the experimental EPSCs and of model 2 of the 100 synaptic responses used for the average shown in (b). (e) P (A4 ) of experimental EPSCs superimposed to that obtained from model 2 by maximizing the mean probability P (χ 2 ) with the procedure described in the appendix.

of presynaptic firing. The trans-information I (t) (Rolls and Treves 1998) for ℘ given different inputs (frequencies) occurring with the same probability can be defined as I (t) =

℘ N ·t·νh h=1 k=0

℘ · Pt (k, h) Pt (k, h) · log2 ℘ l=1 Pt (k, l)

(9)

where Pt (k, h) is the joint probability that a total number of k quanta are released by time t and that the input h is activated and νh is the frequency of input h. Interestingly, it turns out by running sets of computer simulations that the qualitative behaviour of I (t) does not change when the number of inputs (frequencies) is increased. Thus, the use of I (t) can give interesting insights on how the synapse behaves in terms of signal integration over ranges of different frequencies.

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20 pA 40 pA 50 ms

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15 20 25 cum. resp. A4 (bin number)

30

Figure 5. DQA of synaptic responses: depressing behaviour. (a) Ten typical recordings of EPSCs from a CA3 pyramidal neuron elicited by stimulation of a synaptic fibre with a depressing behaviour. (b) Average of 100 recordings including those shown in (a). (c) Mean ±SD of the 100 EPSC amplitudes averaged in (b). (d) Comparison of the statistics (mean normalized to the first mean response ±SD) of the experimental EPSCs and of model 2 of the 100 synaptic responses used for the average shown in (b). (e) P (A4 ) of experimental EPSCs superimposed to that obtained from model 2 by maximizing the mean probability P (χ 2 ) with the procedure described in the appendix.

An important question to be answered is whether and how I (t) is affected by short-term plasticity and by the correlation of synaptic responses. As an example, I (t) has been explicitly computed from experimental data in which two frequencies (12.5 and 25 Hz) were applied. The cumulative responses were evaluated at 80 ms (A1 and A2 , respectively), 160 ms (A2 and A4 ) and 240 ms (A3 and A6 ). The joint probability Pt (k, h)(h = 1, 2) was computed as the probability that the cumulative EPSCs occupies the bin k of the quantal distribution given by the quantal analysis of synaptic responses of the first pulse at the frequency h, times 0.5 which is the probability of each frequency.


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Figure 6(a) shows the averages of 100 recordings from three different experiments. Mean EPSCs shown in figure 6(a) (1) are dominated by a strong facilitation at both frequencies, while facilitation is weaker in the mean EPSCs shown in figures (6(a) (2)) and (6(a) (3)) and figures (6(b) (1)) and (6(b) (2)). The time course of I (t) depends on short-term plasticity, figure (6(b) (3)). In the case of strong facilitation, I (t) has a rather slow onset at the beginning (80 ms) that accelerates at 160 ms; in contrast I (t) has the largest increase already at 80 ms for less facilitating behaviours; in the case shown in figure 6((a) (3)), where EPSCs are almost depressing at 25 Hz stimulation, I (t) is only very slightly increasing after 80 ms. These results show that the time scale for integrating reliable synaptic signals increases with facilitation, as expected. This evidence does not depend on the bin used to calculate Pt (k, h). Interestingly, it turns out not to depend much, either, on the correlation in the data, as similar results have been obtained by running computer simulations with the three different models (data not shown). Furthermore the qualitative behaviour of I (t) does not change when the number of inputs (frequencies) is increased. Since short-term plasticity is not an invariant property of synapses, but can be modulated by activity (e.g., via long-term plasticity) or by changing the composition of the extracellular solution, in another series of experiments I (t) has been calculated at two different [Ca]o (2 and 4 mM) in the same cell. Figure (7) shows the result of a typical experiment in which the first response was potentiated by increasing [Ca]o . With this protocol, facilitation was reversed to depression for stimulations at 25 Hz, figures 7(a) and (b), in the presence of 4 mM [Ca]o ; I (t) reaches its maximum already at 80 ms, while at 2 mM [Ca]o it has a longer lasting increase, up to about 1 bit (fully reliable discrimination) after 240 ms, figure 7(c). This result suggests that the time scale at which synapses can integrate and transmit reliable information is itself not constant, but may be modulated by external influences.

4. Discussion This paper was aimed at investigating how the variability of synaptic responses following trains of presynaptic action potentials can be studied and characterized. We showed that correlations can be detected among EPSCs elicited by trains of stimulation pulses. On this basis, we developed a dynamic quantal approach based on stochastic chains. At the level of analysis of individual synaptic responses, synaptic transmission can be characterized in terms of facilitation or depression. These behaviours may be interpreted as increases in the release probability and decreases in the number of available quanta or releasing sites (synaptic depletion) (Miles and Wong 1986). At this level, the statistics of synaptic responses can be modelled either with simple binomial distributions by changing p and N at each action potential or with simple (but correlated) binomial chains. For instance the CV and the percentage of failures (Faber and Korn 1991, Kullmann 1989, 1994b) are indicators that do not depend on how quantal parameters evolve during a train and, at the level of cumulative responses, equation (8), individual (uncorrelated) binomial distributions are an unsatisfactory description of synaptic activity. For sets of 100 stable responses, the simple technique of reshuffling was found to be an efficient way to discriminate different models in terms of their correlation, even by assuming an oversimplifying uniform binomial statistics. Furthermore, this level of characterization in terms of cumulative responses allows investigation of how synaptic signals can be integrated in time: in particular, a calculation of the information transmitted by synaptic responses can be obtained from experiments with different frequencies of presynaptic firing.

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Figure 6. Information transmitted by synaptic signals. (a) Averages of 100 recordings with trains of three pulses of stimulation at 12.5 and with trains of six pulses of stimulation at 25 Hz in three different experiments. (b) (1) and (2) Mean/mean(1st) of EPSCs for the stimulation at 12.5 Hz (1) and for the stimulation at 25 Hz (2). (3) Information transmitted by the synapse, equation (9), in the three different experiments.


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Figure 7. Variability of synaptic responses at different extracellular calcium concentrations. (a) Averages of 100 recordings with trains of three pulses of stimulation at 12.5 and with trains of six pulses of stimulation at 25 Hz in the presence of 2 mM (left column) and 4 mM extracellular calcium concentration. (b) Mean normalized to the first mean EPSC ±SD of the experimental EPSCs at the two different frequencies of stimulation and at the two different calcium concentrations. (c) Information transmitted by the synapse, equation (9), in the two different conditions relative to the two frequencies of stimulation.

4.1. The dynamic quantal approach The necessity for a dynamic approach to the problem of synaptic transmission has been suggested by several lines of experimental evidence, in particular by the frequency-dependent behaviour of synaptic responses in the neocortex (Markram and Tsodyks 1996), in the

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brainstem Calyx synapse (von Gersdorff et al 1997) and in the hippocampus (Canepari and Cherubini 1998). An interesting stimulus-dependent mobilization model, aimed at characterizing the frequency-dependent changes of quantal parameters at the lobster neuromuscular junction (Worden et al 1997), has been recently found efficient in predicting synaptic depression at the Calyx of Held (Neher et al 1998). A simple model proposed by Tsodyks and Markram can account for the time course of mean synaptic responses in the CNS (Markram et al 1998, Tsodyks and Markram 1997). Tsodyks and Markram do not consider the variability of synaptic responses around their mean, but in terms of mean responses their formulation is equivalent to that described in this paper, once their ‘fraction of restored resources’ Rj is identified with Nj /N and their ‘utilization’ uj with pj . Two minor differences are that (i) since their formulation is not in terms of explicit quanta, but only of continuous variables, the restored resources are not discretized (this makes no practical difference as long as only means are considered); and (ii) their model for the evolution of uj (pj ) follows a different equation from the Dodge–Rahamimoff equation, although also used without reference to specific biophysical mechanisms (Markram and Tsodyks 1996). Another more recent model has been proposed by Dittman et al (2000) to account for facilitation and depression at several central synapses. Also in this model, the interplay between residual calcium and synaptic depletion (‘refractory depression’) accounts for shortterm plasticity behaviours. All these models, although capturing the time course of short-term plasticity, do not use any information on the stochasticity of synaptic transmission, i.e., on the correlation within consecutive release events. In contrast, the dynamic quantal approach used here accounts for the statistics of cumulative synaptic responses, i.e., for the correlation which is an intrinsic property of synaptic dynamics, and this allows the extraction of unavailable physiological information. It has been found that repetitive synaptic responses (particularly in facilitating behaviours) are almost always positively correlated. This property can be accounted for by setting the change in the release probability dependent on the previous release events, i.e., by requiring that the changes in pj are dependent on the number of quanta actually released: in the simplified scheme that is proposed here, the change in the release probability occurs only when the previous release event is not a failure. The important ingredient is the additional element of stochasticity, i.e., the variability in the changes of release probability. This is not in accord with models that attribute facilitation to the binding of low-affinity sites (Tang et al 2000), as this depends only on the influx of calcium and not on the release events. Indeed, if only N is a stochastic variable, a purely negative correlation would be expected. Instead, correlation was found to be more negative for depressing synapses, suggesting that the positive source of correlation is in the (increasing) behaviour of the parameter p. In this respect, since the release process is now understood as a complex cascade of events starting from activation of the presynaptic terminal, terminating with vesicle exocytosis and involving a series of protein– protein interactions, the release probability can be considered the product of the probabilities of each step of the cascade. Thus, a failure in any of these steps determines an EPSC failure and may result also in a failure of the process that, keeping the memory of the previous release (for example the accumulation of residual calcium), provides a change in the following release probability. A more trivial possible source of positive correlation among successive synaptic responses could be the presence of failures of propagation of action potentials in the terminal, or failures in stimulation, which in our case cannot be fully controlled. In this situation, the release probability would be conditional on this type of failure. More accurate descriptions of the variability may be used as well as the introduction of additional elements of stochasticity, e.g., in the reavailability process. In particular, a


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realistic QT of neurotransmitter release at hippocampal synapses must account for details in the release at each site leading to multinomial models in which p and q are generally different for each quantum (Bekkers et al 1990, Bekkers and Stevens 1995, Frerking and Wilson 1996). However, with the information that can be derived at present from the analysis of relatively small sets of experimental synaptic responses, any approach using a larger number of parameters seems to be redundant. Indeed, the number of recordings needed to overcome the simple binomial framework is 500–1000 as evaluated in this paper, whereas the number of ‘statistically stable’ responses never exceeded 200. At the level of the estimation of the quantal parameters characterizing synaptic transmission, the novelty in this paper is the attempt to describe chains of distributions rather than individual distributions. Previous investigations have applied quantal analysis to pair-pulse facilitation in hippocampal synapses (Bolshakov and Siegelbaum 1995, Kuhnt and Voronin 1994) pointing at an increase in release probability as underlying this phenomenon. More recent work (Turner et al 1997) succeeded, in half of the analysed experiments, to reveal heterogeneity at the level of release sites with respect to this change, by applying a Bayesian analysis. Still, in these investigations, the changes in the parameter p were found, by the analysis of the second distribution of synaptic events, to contain no information on the preceding event. In contrast, in this paper, the changes in the parameters p and N were extracted from the statistics of chains of consecutive events, in a more realistic framework of repetitive release. Our results also confirm a wide range of values for the parameter p (Murthy et al 1997), from less than 0.1 to more than 0.5. In this respect, although in the simple binomial framework the quantal parameters p and q lack a precise biophysical meaning, the sequence of values that can be extracted by finding the best chain that accounts for correlations is also useful to understand the dynamics of repetitive synaptic transmission. 4.2. Reliability in synaptic signals The characterization of stochastic synaptic transmission is extremely relevant to the study of neural network activity (Liaw and Berger 1996). In synapses in which a large number of quanta are released by each action potential fluctuations of synaptic signals from the mean value are relatively small and can be, as a first approximation, neglected. In this case, once short-term synaptic plasticity is accounted for as activity-dependent changes of mean synaptic responses, the network activity can be considered deterministic at the level of single connections. In contrast, in synapses with a low number of quanta available for release as in the case of the hippocampus, the marked unreliability (Allen and Stevens 1994, Jonas et al 1993) of synaptic responses, although possibly exaggerated by the recording conditions (Hardingham and Larkman 1998), does not allow the approximation of deterministic synapses. Still, synaptic signals can be integrated both in space (the activity of several presynaptic neurons) and in time (integration of synaptic signals during a burst of action potentials) and the problem of understanding the space and time scales of the neural code, i.e., the space and time scales over which the activity becomes deterministic, is particularly relevant. The statistical analysis of synaptic responses evoked by trains of action potentials can be useful for investigating the time integration of synaptic signals. In particular, information quantities such as the one defined in equation (9) are indicative of the capability of a system to discriminate between different inputs (Rolls and Treves 1998) and quantify what is transmitted by spiking neurons (Zador 1998). In this paper an amount of information transmitted has been measured in hippocampal synaptic responses by using two frequencies of stimulation. The information rate transmitted by the cumulative synaptic responses was slower in facilitating behaviours, but could reach higher values at the end of the train. This indicates that the time

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scale for signal integration is longer in facilitating synapses, in which case bursts of action potentials could indeed be said to make unreliable synapses reliable, as proposed by Lisman (1997). Since short-term plasticity is not an invariant property of synapses, the information rate that can be transmitted by synaptic signals can change in the same synapses, as the present experiments with different [Ca]o have demonstrated. This evidence suggests how long term changes of synaptic efficacy may affect the coding expressed by neuronal networks. 5. Conclusions The approach described here has methodological advantages compared to traditional quantal techniques. It allows to extrapolate information on the correlations of synaptic responses to trains of presynaptic action potentials. This information is useful for understanding the biophysics of synaptic dynamics and to approach general problems related to neuronal information processing. In this paper, a simple binomial framework, which allows the development of fast algorithms for data interpretation, has been used, but the same type of analysis can be performed with different statistical models. The present method is statistically reliable with reasonably low numbers (100–200) of responses, which are those that are normally available in electrophysiology experiments. Acknowledgments We are grateful to Professor Tim Bliss, Professor Alan Fine, Professor Stephen Redman and to Dr Dmitri Rusakov for their valuable comments on the manuscript and to Professor Enrico Cherubini for helpful discussions during the course of this work. This work was supported by a Grant from Ministero Università e Ricerca Scientifica (MURST, PNR) and partially by Human Frontier grant RG0110/1998-B. Appendix A1. Calculation of the release probabilities in a binomial chain At the first action potential, the probability p1 (n) that n out of N quanta are released is given by equation (1). At the second action potential, the release probabilities p2 (n), n = 1, . . . , N are determined by p2 (n) =

N

p(n/N2 ) · P2 (N2 )

(10)

N2 =0

where p(n/N2 ) is the probability that n out of N2 are released according to the rule for the evolution of p, and the probability P2 (N2 ) that N2 are available at the second action potential is given by {nS : N −int (f (t) · nS ) = N2 } P2 (N2 ) = (11) p1 (nS ) nS

where N is the initial number of available quanta. At the j th spike, the probability Pj (Nj ) that Nj quanta are available is obtained by summing over all possible combinations C = {n1 , n2 , . . . , nj −1 } that lead to Nj available quanta. In practice, it is sufficient to iterate Pj (Nj ) = pj −1 (nj −1 ) (12) nj −1


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since pj −1 (nj −1 ) is itself determined by the recursive equation pj (n) =

N

p(n/Nj ) · Pj (Nj ).

(13)

Nj =0

It must be said that the evaluation of the j terms Pj (Nj ) implies the calculation of (N + 1)j −1 terms for the summation over all combinations. This means that the calculation of the release probabilities at the 10th action potential for a synapse with five initial available quanta requires the computation of 69 –107 products. Therefore, when the number of spikes in the train and/or the number of available quanta is relatively high, the estimation of the release probabilities by using computer simulations becomes convenient. A2. Evaluation of the significance of a model The significance of the agreement between the distributions of EPSCs with those predicted by a model can be evaluated by using the quantity 2 exp Q N (n) − 1/2 'j (n) − 'model j . (14) χ2 = model ' (n) j j =1 n=0 Q is the number of spikes in the train and N is the number of available quanta in the binomial chain. 'model (n) is the predicted number of events with n released quanta calculated as the j probability pj (n) to have n released quanta at the j th action potential times the number of experimental recordings. In the case of binomial chains (models 1 and 2), the first mean number of released quanta n1 and the first mean EPSC amplitude A1 are used to normalize exp the two distributions and to calculate the observed frequency of events 'j (n), while in the case of simple binomial distributions (model 0) each distribution is normalized separately by using the ratio A/n. Namely the ratio A/n(A1 /n1 for binomial chains) is utilized exp as a bin for the experimental distribution to evaluate 'j (n):  number of failures for n = 0    A    number of events > 0 and < 1.5 · for n = 1   n    A  number of events > (n − 0.5) · exp 'j (n) = n   A   and < (n + 0.5) · for 1 < n < N    n      number of events > (N − 0.5) · A for n = N . n (15) Finally the term 1/2 in equation (14) is the Yates correction to take into account the fact that exp (n) < 1 have been grouped together. 'j (n) takes only integer values, and bins with 'model j Equation (15) groups responses together in a deterministic way, i.e., by assigning to each EPSC the closest multiple of the quantal amplitude. In order to account for the fluctuations due to exp recording noise and the variability in quantal amplitude, an evaluation of 'j (n) more realistic than that given by equation (15) is provided by a stochastic assignment for each EPSC using realistic distributions for the quantal amplitude and noise fluctuations, which is equivalent to a noise deconvolution approach (Kullmann 1994a). Noise distribution can be measured directly from the recordings whereas values of the quantal amplitude variability can be obtained from the literature and we estimated an upper limit for the CV associated with a quantal EPSC to

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M Canepari and A Treves exp

be 1. For four experiments, we compared the evaluation of 'j (n) using equation (15) with that obtained with a stochastic assignment assuming Gaussian distributions for the quantal exp amplitudes. As expected theoretically, the difference in the evaluation of 'j (n) depended on the width (SD) of the Gaussian distribution and, in particular, the two methods could give considerably different results (more than 30% difference in the χ 2 ) when the CV associated with a quantal EPSC was more than 1. In contrast, in our cases, the simple (and fast) evaluation of exp 'j (n) with equation (15) can be considered reasonable (typically less than 20% difference in the χ 2 calculated using (15) with respect to that obtained with a stochastic assignment assuming Gaussian distributions for the quantal amplitudes). In keeping with this, the EPSC amplitude CV obtained directly from the experimental data is overestimated with respect to the correspondent binomial one; in the case of a synapse with N = 4, p = 0.2 and a CV associated with a quantal EPSC of 1, this overestimation is 60%. An alternative way to minimize the discrepancy between the statistics of experimental data and the binomial chain is to use the maximum likelihood L (Stricker et al 1994) instead of χ 2 which, in the case of a single distribution, has advantages, in that it allows to determine confidence limits and significance of the model used. Having to deal with multiple distributions, however, we have evaluated the significance of a model in the following way, without resorting to more time-consuming maximum likelihood calculations: for each distribution j the validity of a model is assessed by the probability of χ 2 Pj (χ 2 ) = 1 − )inc (χ 2 , µ − η + 1)

(16)

where )inc is the incomplete gamma function, µ is the number of utilized bins and η is the number of free parameters, which is 2 in the case of model 0 and is 5/(number of spikes) in the case of models 1 and 2. The mean (log) probability of χ 2 is then calculated as P (χ 2 ) = e

j

(log(Pj (χ 2 )))/4

.

(17)

The P (χ ) has no direct statistical meaning because the distributions for each j are not independent, but is still useful to quantify the significance of a model for the whole train. For a model to be rejected by experimental data, the individual Pj (χ 2 ) should be lower than a preset criterion, say 5%. 2

A3. Algorithm to determine the best fit of experimental synaptic responses with a binomial chain To perform a DQA of synaptic responses (i.e., to automatically find the best binomial chain that accounts for the variability of EPSCs), the following algorithm has been used. The maximization of P (χ 2 ) in the space of the parameters , , pmax and τ at fixed N is performed through a Monte Carlo Metropolis algorithm (Metropolis et al 1953) in which P (χ 2 ) is considered the energy of the system. Briefly, the algorithm consists of an iteration in which each parameter is alternatively changed by a random value (dependent on P (χ 2 )) and the new set of parameters is always accepted if the new value of P (χ 2 ) is bigger than the 2 older one or accepted with a probability e−β·P (χ ) otherwise. β is a parameter (representing the inverse temperature of the system) to be chosen at the beginning of each iteration. The iteration is stopped when P (χ 2 ) fails to increase further over many (∼100) parameter changes. References Allen C and Stevens C F 1994 An evaluation of causes for unreliability of synaptic transmission Proc. Natl Acad. Sci. USA 91 10383


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