INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY RESULTS DISCUSSION REFERENCES

The complexity of synaptic transmission between central neurons can pose fundamental problems to its investigators. The number of synaptic contacts between given sets of pre and postsynaptic neurons is often unknown, as are the average probability of release upon activation at each site and the average size of the postsynaptic event generated by the release of a potentially variable quantum of transmitter packaged in presynaptic vesicles of potentially variable sizes. Furthermore, there is no a priori reason that average properties should be constant and uniform from synapse to synapse. The overall measured evoked postsynaptic current (ePSC) distribution will be a convolution of the distribution of unitary evoked synaptic currents (uePSCs) evoked at each synapse. By itself, the ePSC distribution provides minimal information about actual quantal properties of the underlying system of synapses. Here we examine whether methods that assume a simpler and more uniform underlying synaptic system provide physiologically meaningful approximations of the true, complex behavior of the native system; we conclude that they can. We place our analysis within a physiological context by considering fast glutamatergic synaptic transmission between mammalian CNS neurons mediated by punctate unitary synapses, as occurs on hippocampal pyramidal neurons in situ and in tissue culture. Although hippocampal synapses are not necessarily representative of all types of CNS synapses in terms of morphology and biophysical characteristics, the main steps in synaptic transmission and the methods of statistical analysis should be comparable.

Presynaptic interactions at individual glutamatergic contacts between hippocampal neurons appear to be small or absent (Murphy et al., 1995; Diamond and Jahr, 1995; Asztely et al., 1997) such that each synaptic site is "binary" or binomial; it either releases a single quantum of transmitter or fails to respond when invaded by a presynaptic action potential (Jack et al., 1981; Redman, 1990; Faber and Korn, 1991; Raastad et al., 1992; Kullmann and Siegelbaum, 1995; Stevens and Wang, 1995; Rosenmund and Stevens, 1996; Walmley, 1995). Although a small proportion of terminals associated with punctate synapses may release more than one vesicle in response to an action potential (see Prange and Murphy, 1999) these terminals may contain more than one active zone/postsynaptic density complex (see Edwards, 1995) and hence might be considered to represent a cluster of independent unitary synapses, especially if release is determined by highly localized signals of elevated calcium.

There is evidence for two broad groups of synapses. Some synapses are high-output with probabilities of release >0.5, while others are low-output with release probabilities <0.2 (Rosenmund et al., 1993; Hessler et al., 1993). This distribution may well be a continuous one and is certainly influenced by activity (Markram et al., 1997). Within a given synapse, variability in release probability can arise due to interactions between the stimulation pattern and the state of depletion/refilling of the presynaptic vesicular pools (Murthy et al., 1997; Dobrunz and Stevens, 1997; Markram et al., 1997). Although the presynaptic active zone and postsynaptic density are closely matched in size and dimensions, these excitatory glutamate synapses do vary in shape. Moreover, release probability seems to be directly correlated with synaptic area (Schikorski and Stevens, 1997). It is also possible that the mean amplitude of uePSCs will, up to a point, be correlated with synaptic area, while the variance of uePSCs at a given synapse will be inversely correlated with synaptic area (see Uteshev and Pennefather, 1996, 1997). This, in turn, implies that variance and mean amplitude will be inversely correlated. In addition, because of intrinsic variability in vesicle size and content, the number of glutamate molecules released during a given fusion event can vary (see Bekkers et al., 1990). Thus, the observed variability in quantal size observed within and between these synapses may arise from a number of causes. This variance clearly complicates quantal analysis, but because of its structural origin may be tuned and exploited biologically.

A deconvolution theory for nonuniform synaptic systems, where all intrinsic synaptic parameters simultaneously exhibit intra (type I), inter (type II), and temporal synaptic variability, is the most general but the least developed to date. To deal with the problem of nonuniform synaptic systems, theoretical studies typically consider partial nonuniformity, i.e., type I, type II, or temporal variability (Brown et al., 1976; McLachlan, 1975; Bennett and Lavidis, 1979; Walmsley, 1995; Stricker et al., 1994; Wahl et al., 1995; Frerking and Wilson, 1996; Quastel, 1997). Experimental conditions are therefore sought to separate these types of variabilities from each other and to study one type at a time (Bekkers et al., 1990; Raastad et al., 1992; Jack et al., 1994; Liu and Tsien, 1995a,b; Forti et al., 1997; Murthy et al., 1997; Dobrunz and Stevens, 1997; Liu et al., 1999). Here we consider in detail the general case where all quantal parameters can vary and demonstrate useful relationships between parameters derived for an equivalent uniform system and those describing the actual underlying synaptic system containing mixtures of binomial synapses of variable "loudness" (see Kullmann, 1999).

In the present study we ignored synaptic noise; however, the approach will be valid in the case of "noisy" synaptic systems as well. Addition of noise would have contributed one or more Gaussian amplitude distributions that would have convoluted with the response amplitude distributions to produce an overall "noisy" uePSC and/or ePSC amplitude distributions. Accordingly, this Gaussian noise amplitude distribution could have been deconvoluted from the overall distribution to reveal noise-free uePSC and ePSC amplitude distributions (for a review, see Redman, 1990).

	THEORY

TOP ABSTRACT INTRODUCTION THEORY RESULTS DISCUSSION REFERENCES

Equivalent uniform distributions: basic principles

It will be shown that an infinite variety of nonuniform synaptic systems can give rise to any given unimodal ePSC distribution. This impossibility of a unique solution does not, however, mean that such systems cannot be meaningfully quantified provided additional information is considered. We postulate that the simplest, most convenient, and most meaningful approximation to such nonuniform synaptic systems is the unique uniform synaptic system that best describes the experimentally recorded distribution. We then demonstrate how such an equivalent uniform system can be formally defined in terms of the underlying, intrinsic parameters of the nonuniform system. A specific set of apparent parameters can then be estimated from observation of such a system. Although the question of compound binomial systems has been discussed extensively (Brown et al., 1976; Mclachlan, 1975; Bennett and Lavadis, 1979; Korn and Faber, 1991; Quastel, 1997) our present analysis is more general and not only provides a useful extension of the previous work, but also gives rise to new insights into the stochastic nature of synaptic transmission.

Equivalent uePSC distributions

(1)

(2)

(3)

(4)

(5)

View larger version (14K): [in this window] [in a new window]   FIGURE 1   Definition of amplitude distributions and their components. (A) The complete amplitude distribution function of a unitary synapse. A complete amplitude distribution function, f(j)(x), is the sum of the delta function describing the amplitude distribution of release failures, qj(x), and the partial amplitude distribution function, fj(x). The latter contains no information regarding failures. These two components reflect presynaptic and postsynaptic influences, respectively, and are of a fundamentally different nature. Note that the partial amplitude distribution function is not determined at zero amplitude, owing to the assumption that the source of transmission failures is strictly presynaptic and not due to any failure in detection. (B) An overall partial uePSC amplitude distribution function of unitary evoked events. A compound partial uePSC amplitude distribution function, f, is normalized to M and obtained as a sum of components, fj, each normalized to Mj (long dashes), that correspond to amplitude distributions of unitary evoked synapses involved in transmission. An amplitude distribution function for an equivalent uniform system can be defined as,  = f/ñ and is normalized to .

(6)

Equivalent ePSC distributions

An ePSC consists of the summed responses of n independent but synchronous unitary synaptic events, each of which can vary in amplitude from one event to the next, or fail altogether. Therefore, the responses of each synaptic contact j measured over many stimulations can be described by an amplitude distribution consisting of the sum of two components (Eq. 7; Fig. 1 A): 1) the response failures, q_j(x), where q_j = 1  p_j and (x) is the Dirac delta function; and 2) the normalized failure-free uePSC amplitude distribution, f_j(x) = (1  q_j)f^*_j(x). The normalized complete amplitude distribution for the synapse j (containing both failures and responses and designated by the bracketed subscript) can thus be defined as,

(7)

If a synaptic system contains n independent synaptic units, the resulting complete ePSC amplitude distribution E_(z) will be expressed as a convolution of the n individual complete uePSC amplitude distributions, such that

(8)

The characteristic function, _(E)(t) for Eq. 8, assuming that each failure-free distribution f_j is Gaussian, is given by,

(9)

We solve Eq. 9 using the Fourier convolution theorem and considering a logarithmic transformation. Hence,

(10)

Therefore,

(11)

(12)

Where µ_(E) is the mean and _(E)² is the variance of the complete ePSC amplitude distribution. Equation 12 is the same as that derived by Quastel (1997) using a different approach.

Equivalent ePSC amplitude distributions with uniform synapses

Equation 8 can be further simplified if n identical synapses are considered ("uniform synapses"). The function E_(z) will simplify to,

(13a)

where q is the probability of failure for all the synapses, f  f_j is their failure-free amplitude distribution, and C_n^k is the binomial coefficient (i.e., C_n^k = n!/k!(n  k)!). Equation 13a can be rewritten as follows for Gaussian and gamma distributions (see McLachlan, 1975, 1978):

(13b)

(13c)

Both gamma and Gaussian distributions are members of a class of unimodal distribution functions that convolute regularly and thus can be used for defining the uniform equivalent synapse systems. The failure-free ePSC amplitude distribution function of members of this class can be introduced as E_z(z) = _k=0ⁿ¹ C_n^kq^k(1  q)^nkV_n^k(z). Based on the physical understanding of the failure-free distribution function E_z(z), the kernel, V_n^k(z), must be determined as the kth component of this distribution, i.e., as a convolution of k equivalent uePSC amplitude distributions with each other selected from a total of n distributions.

Because ePSC amplitude distributions represent sums of different components, i.e., a convolution of uePSC amplitude distributions, appropriate distribution functions for an equivalent uniform system should convolute analytically with each other, giving rise to a new function of the same class. Therefore, useful distribution functions must be stable with respect to convolution. This will minimize the number of parameters that is involved in describing the response distribution. Furthermore, the statistical moments generated by these equivalent distribution functions will be expressed in simplistic mathematical forms, readily permitting comparisons and physical interpretations.

To explicitly define the class of suitable equivalent distribution functions, let us consider an arbitrary (unimodal or multimodal) failure-free ePSC amplitude distribution consisting of n components, where n is the number of uniform synapses involved in transmission. An immediate consequence of the synaptic uniformity is that the mean of the (n  k)th individual component of the failure-free ePSC amplitude distribution must be equal to (n  k)µ, where µ is the mean of the unitary component, which will be the same as the uePSC amplitude distribution (see above). Thus,

(14a)

(14b)

The kernel V_n^k(z) denotes a class of failure-free distribution functions such that the first two statistical moments of E_z(z) about the origin are given by Eqs. 15a and b, respectively.

(15a)

(15b)

In Eqs. 15a and b we used the equalities
and
which we provide here without proofs. Although potentially the kernel V_n^k(z) can have arbitrary shapes, we will consider only unimodal kernels.

Equations 15a and b or 14a and b comprise a formal definition for the class of distribution functions that are appropriate for defining equivalent uniform systems. In particular, it can be readily verified that Gaussian and gamma distributions belong to the above class.

The relationships between parameters describing complete and failure-free portions of the ePSC distribution function (identified by the subscripts (E) and E, respectively) for uniform systems are known, but can be obtained from Eqs. 11 and 12 by making all synaptic parameters identical, and combining with Eqs. 15a and b. Thus,

(16)

(17a)

or

(17b)

Unique modeling of unimodal ePSC amplitude distributions

Using Eqs. 16 and 17a it is easily shown that there are an infinite number of sets of uniform quantal parameters that can describe a given unimodal ePSC amplitude distribution [i.e., (µ₁, ₁, q₁, n₁), (µ₂, ₂, q₂, n₂), ... , (µ_i, _i, q_i, n_i), ...]. Each such sets of parameters lead to identical values of µ_(E) and _(E)². Combining Eqs. 16 and 17a we obtain, for arbitrary i and j, µ_(E)i = µ_(E)j and _(E)i² = _(E)j². The latter leads to Eq. 18.

(18)

and reduce the number of parameters involved to three. Thus, two of the four parameters need to be specified before a unique set of equivalent parameters can be estimated. For example, if µ and ² are determined from the uePSC distribution and fixed at predicted values, such that µ_i = µ_j = µ and _i² = _j² = ² for any i and j, then Eq. 18 will give q_i = q_j = q and Eq. 16 will give n_i = n_j = n. In fact, a thorough analysis of Eqs. 16-18 determines that the uniqueness of the deconvolution will hold when the following pairs of parameters are known and fixed at correct values during the deconvolution: (µ and ²) or (q and n), or (µ and n), or (µ and q). In contrast, the knowledge of the pairs (² and q) or (² and n) is insufficient for a unique deconvolution.

Relation between actual and equivalent synaptic systems

For an equivalent complete ePSC amplitude distribution to approximate a nonuniform ePSC distribution, the equivalent complete mean, _(E), and variance, _(E)², must by definition be identical to values defined in Eqs. 11 and 12, respectively (because Gaussian functions are defined completely by their means and variances). Therefore, using Eqs. 11-17 we arrive at Eqs. 19 and 20.

(19)

(20)

Solving Eqs. 2, 4, 5, 19, and 20 simultaneously we arrive at Eqs. 21 and 22.

(21)

(22)

Equation 22 is somewhat similar to the equation derived by Bennett and Lavadis (1979) and Quastel (1997), but more general, as it now includes variability in p_jµ_j.

Thus, all binomial quantal parameters needed to define an equivalent uniform system can be expressed in terms of weighted averages of the intrinsic quantal parameters of a nonuniform system. These relations are summarized in Table 1 and represent generalizations of some of the equations derived by Brown et al. (1976).

The meaning of CV_pµ

If we define the following means:
and
then Eq. 22 can be rewritten as

(23)

where CV_pµ is a coefficient of variation of the values p_jµ_j. In contrast to the case considered by Brown et al. (1976), in the present case of continuous distributions, the means and variances are also variable. Therefore, the value CV_p used by Brown et al. (1976) is upgraded in our theory by a more general value of CV_pµ. Similarly, Eq. 2 can be combined with Eq. 23 to give,

(24)

Eqs. 23 and 24 lead to Eq. 25a,

(25a)

The approximation p²µ²  p²µ² and the approximations pµ = p  pµ and pµ²  pµ² hold relatively well for sufficiently smooth distributions of the parameters p² and µ². Moreover, these equalities do not require synaptic parameters to be independent from each other under our definitions of means (the derivation and the computer simulations are not shown). Therefore, under these conditions,

(25b)

(25c)

If synapses are uniform such that µ_j =  and _j = , then CV_µ = 0. Under this condition CV_pµ becomes CV_p, such that Eq. 24 reduces to that derived by Brown et al. (1976).

Correlation between parameters of inter-synaptic (Type II) variability

In Eq. 25a we have separated the two components of the CV_pµ. Here we take one further step to establish dependence among p-, µ-, and pµ-variabilities. Consider three definitions: _pµ² = p²µ²  pµ²; _p² = p²  p²; _µ² = µ²  µ². From Table 1 we have
and thus, ² p² = pµ². Solving all the above equations together, we obtain ² p² = ²(p²  _p²)  p²(µ²  _µ²). Therefore, for the relationship among _pµ², _p², and _µ², we obtain:

(26a)

Recalling that p²µ²  p²µ² for even distributions of p and µ, we further obtain

(26b)

Equations 26a and b provide an additional link among parameters of p- µ-, and pµ-variability and the averaged synaptic parameters.

Coefficient of variation and variance to mean analysis of ePSCs

Using the definition of _(E)², given in Eq. 17a, one can further show that for a fully equivalent system,

(27)

where ñ = np = m is the mean quantal content and C is the equivalent coefficient of variation for the uePSC amplitude distribution, which by definition approximates the actual CV. The form of this equation is well known for compound binomial systems; however, the contribution of CV_pµ has not previously been explicitly included and analyzed. Equation 27 shows that the inverse relation between CV_(E)² and quantal content allows CV_(E)² to give an accurate indication of presynaptic changes, even for nonuniform systems where values of p, µ, and ² vary between synapses (see also Quastel, 1997) provided p is low and C can be estimated from experimental observations.

Recently, there has been renewed interest in analyzing the expected parabolic relation between ePSC variance and ePSC means. From Eqs. 17a and 27, we obtain:

(28a)

or
Using the equations from Table 1, we can rewrite Eq. 28a in the form of Eq. 28b,