J. Math. Biology 7, 31-40 (1979) Jowmol of

by Springer-Verlag 1979

Neurotransmitter Release Statistics: Moment Estimates for Inhomogeneous Bernoulli Trials

Donald H. Perkel and Marcus W. Feldman

Department of Biological Sciences, Stanford University, Stanford, California 94305, USA

Summary. Evoked release of quanta of neurotransmitter is generally treated as a set of homogeneous, stationary Bernoulli trials, hence governed by the binomial distribution. Relaxing the assumptions of uniformity and stationarity leads to a more realistic physiological model of transmitter release but also introduces systematic biases in the moment estimates of the binomial param- eters. We derive probability generating functions for quantal release and expressions for the moment estimates of ~ and/~ for a generalized model that incorporates temporal variation and nonuniformity in individual release probabilities and in numbers of release sites.

1. Introduction

When an action potential invades a nerve terminal, it causes the release of a random number of individual packets or quanta of neurotransmitter. Experimentally, the distribution of quanta has been shown to follow a binomial distribution, which in many cases is adequately approximated by the Poisson limit; for a review, see Martin, 1977. In addition to measuring the mean quantal content (the expected number of quanta released at each stimulation), physiologists have estimated the two binomial parameters : n, the number of release sites, and p, the probability that a site will release a quantum at a single trial. The standard method for doing so is to equate the first and second sample moments with those of the population. Specifically, let X be the random variable corresponding to the number of quanta released at a trial; its mean and variance are/z and a2, and for the binomial dis- tribution they are related to n and p as follows:

p = 1 - (~ /~) (1)

n = / ~ l p . (2)

Replacing the population moments by the corresponding sample moments m and s 2 leads to the following moment estimates of binomial parameters

p = 1 - ( s2 /m) (3)

fi = m / p . (4)

0303-6812/79/0007/0031/$02.00

32 D.H. Perkel and M. W. Feldman

Brown, Perkel, and Feldman (1976) and Barton and Cohen (I 977) have shown that departures from the assumptions underlying the binomial distribution lead to moment estimates that are systematically biased from the corresponding population parameters. In particular, temporal variation in the number of release sites or in the probability of release causes the mean release probability to be underestimated and the mean number of sites to be overestimated, whereas spatial variation in the probability of release has the opposite effect.

Four specific equations for/~ ,~ 1 - (a2//~) are presented without proof by Brown et al. (1976); here we derive a general proposition from which those results and several others follow as special cases. In particular, we propose a general model of probabilistic transmitter release at nerve terminals. From the probability-generating function (p.g.f.) of the model we derive explicit expressions for the moment estimate of mean release probability and exhibit simple forms for certain specific cases.

2. The Basic Model for Transmitter Release

We assume first that there are L distinct categories of release site. At each trial (i.e., at each stimulation of the axon leading to the nerve terminal), two independent assignments are made: assignment A determines the numbers of release sites in the L categories, and assignment B determines the transmitter-release probabilities for each category of release site. Specifically, according to assignment A, each category takes on a random integer Nt of release sites. The total number of release sites at any trial is the random variable

L

N = y Nf. (5) t = 1

Let J r " - {N1, N z . . . . . NL} and k - {kl, k2 . . . . . kL}, where each kt is a non- negative integer. Also let

w(k) = erob ( ~ = k). (6)

The sample space of ~ is denumerable; the outcomes k are indexed by v. We write

~r v = Prob ( ~ = kv), ~ rrv = 1. (7) v

Then the expectation and variance of N are given by

L

- z ( N ) = 9 , k , . . (8) V i = 1

var N = ~ ~'~ k~,, _ ~2, (9)

where kv = {k l , , , k2 , , , . . ., kL.,}.

In the second assignment, B, each release site takes on a transmitter-release proba- bility. To do this we define a random variable P representing the transmitter-release probabilities over the population of sites in the ensemble of trials, through a

Moment Estimates for Inhomogeneous Release 33

weighted sampling procedure. All the release sites in a given category have the same release probability. These probabilities are designated P1, P z , . �9 PL ; the L random variables are not necessarily independent. They take on values rl, rz . . . . . rL, which are real numbers lying between 0 and 1. The probability of a particular assignment is

p(r) = Prob (P1 = rl,/~ = rz . . . . , PL = rL), (10)

where r = (rl, r2 . . . . . rL}. 1 The assignment first postulates that, given outcome v of the preceding assignment, the probability that P is drawn from the ith random variable is

Prob (P = P~ [ v) = k~.v k,,,. (11)

Now the probabilities ~r~ are weighted by the contributions that they would make to the number of release sites; this transforms the outcome probabilities from ~r~ to 7~, with

L

~v Y ki.v L ' (12) ~]V = L ~ TrY

Under assignment B the probability that P represents the ith release category is obtained by multiplying (11) and (12) and summing over v:

Prob (P = P,) = ~ ~r,ki.~/ti. (13) y

This assignment now defines a random variable expressing the transmitter-release probabilities. In fact, under this assignment,

L

erob (P = r) = ~. Prob (Pi = r) ~_~ - , k , .U~ t = 1 V

L

= ~. ~ p(r) ~ ~,,k,.,la. (14) t = 1 r ; r l = r v

The mean and variance of P defined in this way are

L

p = E(P) = ( l /n )~ ~ ~,,p(r)~ rA.da (15) v r t = 1

L

var P = (l/n) ~. ~. ~r~p(r) ~. r~k,,~ - fi2. (16) V It" l = 1

The moments (15) and (16) are used in the subsequent estimation procedure. It is a straightforward matter to write expressions for the mean and variance of the mean (random) number of quanta released per trial. For each category i, at a single trial, the number of quanta released, X~, is the outcome of the random number N~ of

1 The assumption that the set of random variables {P1, P2 . . . . . PL} takes values from a denumerable set is not essential but will be retained here.

34 D.H. Perkel and M. W. Feldman

Bernoulli trials. Each trial has the (random) probability of success P~. Thus, given Nt and P , X~ is a binomially distributed random variable. The total number of quanta released by all categories of release site is

L

X = ~ Xt. (17)

The L sets of Bernoulli trials are made independently. Define the random variable M, the expected number of quantal releases at each trial given the sets N, and P,, to be

L

M - ~ P,N,, (18) t = 1

which takes on values ~ = 1 r~k~. We have

L

= E ( M ) = ~ ~ ~rvp(r) ~. r,k , . , (19) Y F 1:I

and from (15)

/z = aft. (20)

The variance of M is

var M = ~. ~ Ir~o(r) r,k,., _ r (21)

The above assignments are now shown to enter into the moment-estimation procedure. We use a standard compounding approach to produce the generating function of X and hence its moments (see, e.g., Feller, 1968; Ch. 12).

3. Probability-Generating Functions and Moment Estimates

For any integer-valued random variable Y, the probability-generating function (p.g.f.) is defined by

f(O) -- E ( Y ~ = ~ O k Prob (Y = k) (22) k = 0

where 0 is a dummy variable. The derivatives of the p.g.f, are related to the mo- ments of the distribution of Y; the first two derivatives, evaluated at O = 1, yield the expressions

f ' (1) = E(Y) (23)

f"(1) = E(Y2) _ E(Y) (24)

from which the variance is obtained as

var Y = f"(1) + f ' (1) - [f'(1)] z. (25)

For the random variable X, having expectation t~ and variance ~2, it is easily seen that the expected value of the moment estimate for release probability is given by

E(,6) = 1 -- ((r2/tz) = f ' ( 1 ) - [f"(1)/f'(1)]. (26)

Moment Estimates for Inhomogeneous Release 35

For our model of a nerve terminal, this quantity can be obtained by writing the expression for the p.g.f, of X and performing the required differentiations and substitutions.

For a binomial distribution having parameters n and p the p.g.f, is given by

f(O) = (pO + q)~, (27)

where q - I - p. For independent random variables, the p.g.f, of their sum is given by the product of their individual p.g.f.'s. These two results enable us to write the p.g.f, of X as

r.

f(O) = ~ rr~ ~ p(r) I--I (riO + q~)k~,~, (28) V 1" ~ = 1

where q~ = 1 - r~. Its first two derivatives (with respect to O) are given by

f'(O) = ~, ~'v ~ p(r) (r,O + qs)~, 'v rtk,.~(r,O + qO -1 (29)

f " ( O ) : ~ r r , ~ P ( r ) [ ~ = l ( r , O + q , ) ~ " v ]

x r~kt,~(r,O +q,)-~ - ~ r~ki,~(r~O + q,)-2 . (30)

Setting 0 = 1 yields L

f'(1) = ~ ~ ~ p(r) r~k~,~ (31) r

which is equal to/z (Eq. 20), and

Eq. (32) may be seen to be reducible to certain of the expectations and variances defined earlier:

f"(1) = var M +/~z _ ~(varP + p2). (33)

By substitutingf'(1) = t~ and Eq. (33) into Eq. (26), we obtain, after some manipu- lation, the following expression for the expectation of the moment estimate for release probability:

E(/~) = - ( v a r M)/I~ + (~/t*)(var P + ffz). (34)

This expression may be simplified by combining Eqs. (10), (16), and (20) to obtain

ff = tz/~ from which we obtain

E(p) = fill + (var P)/1 ~2 - ~(var M)/tz2].

Finally, if we define the coefficients of variation as

% - (vat p)1~2//~

(35)

(36)

(37)

36

and

am -- (var g)l l2/ l~

then Eq. (36) reduces to the simple form

E(ff) = 1 - (~2//~) = if(1 + a~ - ~ ) .

D. H. Perkel and M. W. Feldman

(38)

(39)

4. Special Cases

Three special cases may be derived readily from Eq. (39); each of these eases involves just one kind of variation in the nerve terminal.

If the variation in P is spatial only (i.e., if release probability varies from site to site), and if these probabilities r~ and the numbers of sites k~ in each category remain constant from trial to trial, then a~ = a~, where the subscript designates spatial variation, and a~ = 0. This leads to

E(fi) = if(1 + a~), (40)

which is Eq. (5) of Brown et al. (1976), going back to Kendall (1948, Eq. 5.26). If N is constant and the variation in P is temporal only, so that the kt are fixed but the rt vary from trial to trial, we may write a~ = a~ and it also holds that a~ = a~, so that Eq. (39) becomes

E ( p ) = p [ l - ( n -

which is Eq. (6) of Brown et al. (1976).

(41)

Finally, if there is no spatial variation in P, there is only one category (so that L = 1). I f N varies from trial to trial, then a~ = 0 and a~ = ~ , so that

E(fi) =/~(1 - ~a.2), (42)

which is Eq. (7) of Brown et al. (1976).

5. Independence Among Release-site Categories

In order to extract further tractable special cases from Eq. (39), we make two further assumptions: The first is that the release-site categories are mutually independent. This means that the P~ are independently distributed with

E(Pt ) =/z~ (43)

and similarly the Nt are mutually independent. We also assume that the N~ are identically distributed, so that for all i,

e ( N , ) = +,. (44)

var Nt = (l /L) var N - ~2. (45)

Moment Estimates for Inhomogeneous Release 37

The P~ and the N~ remain independent of each other. It is clear that/z = / z , ~ = 1/zp~ and that the expectations and variances of N and P have the following simplified forms:

L

= ~ t~ = L/~, (46) t = 1

L

var N = ~. var N4 = Lo~ (47) t = l

L

ff = (I/L) ~ / z , , =/~/~ (48) 4 = 1

L L

vat P = ( l /L) ~ E(P~) - /~2 = (I/L) ~ E(P, -p)2. (49) i=l 4=1

Under these assumptions, it is possible to partition var P into temporal and spatial terms as follows:

L L

v a r P = (I/L) ~. E(P4-/~,t) 2 + (l /L) ~ (/z,~-/~)2 i=1 i=1

L L

= (I/L) ~. varP4 + (l /L) ~ / ~ 2 _ fi2 (50a) i=1 i=l

= vart P + var, P, (50b)

where vart P and vars P are defined by the corresponding terms in Eq. (50a). Temporal variation occurs from trial to trial, while spatial variation affects mean values across categories. It follows that

L

~./~v 2, = L vars P + ~z. (51) t = 1

Furthermore, we have the following expression for the expectation of M:

L L

i = l i=i

We may also decompose the variance of M. First, because the categories of release site are independent, we may write

L L

var M = var ~. P,N~ = ~. var P,N~. (53) f = l t = 1

It is easily shown that, in general, if a random variable is the product of two other, independent random variables

Y = WZ, (54)

then the variances and means are related as follows:

1 + (var Y)IIz~ = 1 + (var Y)l(tx~lx~) = [1 + (var W)//~2w][1 + (var Z)/tx~]. (55)

38 D . H . Perkel and M. W. Feldman

Since the P~ and the N~ are themselves independent, this result may be applied, to yield

1 + (var P~Nt)/(~, t/~) = [1 + (var P~)/tz~][1 + cr,2/t~,2], (56)

from which it follows that

var PiNt =/z~ var P~ + a~(var P~ + t~3. (57)

Summing over categories, we obtain

L L

vat M = t~, 2 ~ var P, + ~2 ~ (var Pl + tz~,). (58) ~1 i=1

Finally, by substituting Eqs. (46), (47), (50), and (51) into Eq. (58), we obtain

var M = Ltz~ vart P + La~(var P +/52)

= (KS/L) vart P + var N(vart P + var~ P +/52). (59)

We may now evaluate the moment estimate for/5, first defining ~ = (vars p)//52 and ~ = (vart p)//52 and noting that Eq. (50) implies

r = ~ + r = (1//52)(var~ P + vart P) (60)

and, with a~ = (var M)/tz 2 = (var M)/(ff~/52), that Eq. (59) implies

~2m = (1/L)u~ + =~(1 + =~ + u~). (61)

When Eqs. (60) and (61), together with Eq. (46), are substituted into the general formula (Eq. 39), they yield the expression for independent categories having a common distribution for N~:

E(/~) =/5[(1 + u] + a~)(1 - K~2) _ ~,c,~]. (62)

Six special cases result when one or two of the coefficients of variation are zero: c,~ for spatial variation in release probability, c~t for temporal variation in release probability, and c,, for temporal variation in number of release sites. The three cases corresponding to two vanishing coefficients of variation are shown above (Eqs. 40-42). I f cet = 0, we obtain

E(p') =/5(1 + ~])(1 - ~a~), (63)

which is Eq. (8) of Brown et al. (1976). The two remaining cases give new formulas: for ~ = 0, we have

E(p') =/5[(1 + r - ~az) _ t~,~] (64)

and for a, = 0, we have

E(p) =/511 + ~ - (tz, - 1)c~]. (65)

6. Discussion

It is easy to show that, in the model of Section 5, the results reduce to the binomial limit when/z, ~< 1, Prob (N~ > 1) = 0, and there is no spatial variation in P (i.e., if all tz~ are equal).

Moment Estimates for Inhomogeneous Release 39

It is possible to generalize slightly, so that we assume the N~ to be independently but not identically distributed, and so to arrive at similar results, but a covariance term between N and P renders the result corresponding to Eq. (62) rather cumbersome.

A generation ago, Del Castillo and Katz (1954) observed that 'some individual units [may] have a high probability and respond almost every time, while others have a low probability and contribute to the e.p.p. [end-plate potential] only occasionally . . . . In general, the coefficient of variation for this case is less than that expected for a binomial distribution . . . . Th i s . . . should not be confused with the case in which probabilities of response vary during the set of observations . . . . In this case the standard deviation of the e.p.p, amplitude would become greater, not less.' These qualitative effects on the variance of quanta1 release lead directly to the quantitative effects on estimates of the binomial parameters predicted by the results derived here and illustrated in Brown, Perkel, and Feldman (1976).

More recently, a number of workers have been investigating the quantitative effects of nonuniformity of release sites, e.g., Zucker (1973), Hatt and Smith (1976), Barton and Cohen (1977). Nonstationarity in the number of effective release sites would be expected from one of the transmitter-release models of Vere-Jones (1966). More systematic nonstationarities (i.e., periodic fluctuations) in evoked transmitter release are described by Meiri and Rahamimoff (1978). The systematic and longer- term nonstationarities associated with the well-known phenomena of synaptic depression and facilitation also give rise to the biases in the binomial estimates described above, but for these phenomena the nonstationarity is readily detected and measured.

The biases in the estimation of the binomial parameters are likely to be most crucial when applied to the interpretation of quantal statistics in the investigation of facilitation and depression; these and related physiological issues are discussed by Brown, Perkel, and Feldman 0976) and in forthcoming papers. It is becoming clear that the purely binomial model of transmitter release is no longer adequate for the detailed description of the process, and a more flexible, realistic model, such as that described here, will become increasingly necessary to account for the fine- scale morphological and physiological data furnished by contemporary investigative techniques.

Acknowledgements. Supported by U.S.P.H.S. Grants GM 10452 (M.W.F.) and NS 09744 (D.H.P.) and by NSF Grant GB 37835 (M.W.F.). We thank Thomas H. Brown for suggesting the problem and providing invaluable discussion and collaboration.

References

Barton, S. B., Cohen, I. S. : Are transmitter release statistics meaningful ? Nature 268, 267-268 (1977)

Brown, T. H., Perkel, D. H., Feldman, M. W. : Evoked neurotransmitter release: statistical effects of nonuniformity and nonstationarity. Proc. Natl. Acad. Sci. USA 73, 2913-2917 (1976)

Del Castillo, J., Katz, B. : Quantal components of the end-plate potential. J. Physiol. (London) 124, 560-573 (1954)

40 D . H . Perkel and M. W. Feldman

Feller, W.: An Introduction to Probability Theory and Its Applications. (Third Edition, Vol. 1) New York: Wiley 1968

Hatt, H., Smith, D. O.: Nonuniform probabilities of quantal release at the crayfish neuro- muscular junction. J. Physiol. (London) 259, 395-404 (1976)

Kendall, M. G.: The Advanced Theory of Statistics, pp. 122-123. (Fourth Edition, Vol. 1.) London: Griffin 1948

Martin, A. R.: Junctional transmission II. Presynaptic mechanisms. Pp. 329-355 in E. R. Kandel, Ed., The Nervous System, Vol. I, Cellular Biology of Neurons, Part I (Handbook of Physiology, Section 1.) Bethesda, Maryland: Amer. Physiol. Soc. 1977

Meiri, H., Rahamimoff, R.: Clumping and oscillations in evoked transmitter release at the frog neuromuscular junction. J. Physiol. (London); in press

Zucker, R. S.: Changes in the statistics of transmitter release during facilitation. J. Physiol. (London) 229, 787-810 (1973)

Received May 16, 1978