J. theor. Biol. (1979) 80, 441-443
LETTER
TO
THE EDITOR
On Estimating Parameters of the Binomial Model for Transmitter Release at Synapses The quantum hypothesis of transmitter release assumes that there are n identical and independent release sites, each having a probability p of releasing a quantum of chemical transmitter. If, for example, quanta1 size has a Gaussian distribution with mean v and variance s2, then the distribution of the amplitude of evoked potentials is given by the binomial model as f(x) =il
(~)prqn~‘(2nrs2)~1~2
exp [ - (x-ur)2/2rs2].
(1)
Using observed spontaneous potentials to estimate v and s, the parameters p and y1can be estimated with the method of Robinson (1976). However, sometimes measurements of spontaneous post-synaptic responses are impossible. Therefore, in a recent issue of this journal, Courtney (1978) proposed to use the first three moments of the observed distribution of N evoked synaptic responses in order to get estimates of n, p, v and the mean quantum content m = n . p. Applying the binomial model and the well-known asymptotic method (Kendall & Stuart, 1977), standard errors for these estimates were given for p = 0.1, 0.5, 0.8 and the case n = 4, v = 1, s = 0 with sample size N = 200. Based on these results, the method was found reliable. Closer inspection of this procedure nevertheless reveals that the moment estimators become inefficient for small p and/or larger numbers n of release sites. This is shown by simulating evoked potentials with the density function (1) and computing the moment estimates i, A, fi and 6. For v = 1, s = 0 and a given combination of it, p and N, sampling was continued until 500 estimates with 0 < fi < 1 were obtained, resulting in the sample means and standard deviations shown in Table 1. From this table it appears that for realistic values of n and p large sample sizes N are needed in order to get estimates of practical value. Especially the distribution of ri comes out to be heavily dispersed. The moment estimates become still more impaired if, as it has been observed repeatedly, a variance s2 > 0 must be taken into account. Courtney reports estimates for a single simulated realization from distribution (1) with 441 0022%5193/79/190441
f03
%02.00/O
0
1979 Academic
Press Inc. (London)
Ltd.
442
D. SCHENZLE
TABLE 1 Sample means and standard deviations of Courtney’s parameter estimates 0, A, IA and Qfor distribution (1) with v = 1, s = 0 and different values of n, p and N. Each
j?gure is based on 500 experiments with N simulated evoked potentials. N, denotes the number of failures (0 < 0 or fi 3 1) N=50
100
200
ml
Nr
05
0.47+0.13 8+27 2.2 + 0.7 1.0*0.3
0.8
0.77 k 0.06 5*3 3.8 + 1.6 a96kO.3
05
0.5 f 0.20 14+24 4.2+ 1.7 l.lkO.4
0.8
0.76 + 0.11 13+21 8.3+5.0 l.OkO.5
03
0.52kO.20 35+104 8.0 k 3.4 1.2*0.5
0.8
o-73*0.15 64,671 17.5+ 12.0 1.0+0.5
9
0.49 + 009 6+19 2.1) 0.5 l.OkO.2
1
0.49 & 0.06 4+1 2.0 & 0.3 1.0+0.1
0
0.49 * 004 4.1+ 0.6 2.0 f 0.2 1.0+0.1
4 0
55
0.79 f 0.04 4+1 3.4kO.8 0.99 f 0.2 0.49+0.14 14532 4.2+ 1.3 l.OkO.3
0
18
0.79 + 0.03 4*1 3.3 + 0.6 1.0+0.2 0.48 kO.12 11+13 4.2t 1.0 1.0,0.2
0
‘1
0.8 f 0.02 4.1 + 0.5 3.2kO.3 1.0+0.1 0.49 + 0.07 913 4.12 0.6 1.0+0.1
8 5
123
0.78 + 0.06 lOA 7.3 + 2.7 l.OkO.3 0.49+0.17 48 + 239 8.2k3.0 1.1+0.4
0
82
0.79 & 0.04 9+3 6.9+ 1.7 1.0+02 0.48+0.15 12Ok505 8.3 &- 2.6 1.0+0.3
0
0.8 + O-02 8&l 6.5 + la 1.0,@2
27
0.48+0.11 22+39 8.3 f 1.8 1.0+02
0
0.79 * 0.04 17*5 13.4k2.8 1.0+0.2
16 26
0.76kO.11 77 + 154 15.6k8.6 1.0+0.4
5
0.78 f 0.07 20*12 14.6k5.3 1.0+0.3
v = 1, s = 0.3, p = O-3, n = 6 and N = 350: fi = O-3, ri = 5.7,5 = 1.06. These figures look favourably, but again some more extensive simulation results (Table 2) obtained in the same way as described above, show that in general the moment estimators undergo large fluctuations. The examples given here indicate that with sample sizes N reported in the
LETTER
2
TABLE
Rest&s
for s = 0.3 100
N=50
@41 k@17 0.5
443
TOTHEEDITOR
200
0.39kO.16
4.0+ 33+2151.4 1.1 kO.4
106
40+297 4.2 f 1.2 l.OkO.3
500
0.37 + 0.13 77
4.3il.O 19+42 1.010.2
0.37 + 0.1 28
4.3kO.7 15k23 1-o+ 0.2
4
8 0.8
0.60+0,17 23+41 9.Ok4.6 0.9 * 0.4
74
0.60+0.15 65+977 9.2k4.0 0.8 + 0.3
0.60FO.11 19$-40 8.9k3.0 0.8 + 0.2
2.
7
0.63 + 0.06 14*5 8.2k1.6 0.8 &- 0.2
’
literature (see e.g. McLachlan, 1975) it might be worthwhile to solve maximum likelihood equations. An efficient numerical method for doing this with distributions of type (1) is currently developed and will be described elsewhere. Inst. f. Med. Biometrie University of Tiibingen D 7400 Tiibingen, Germany (Received 19 February
1979,
D. SCHENZLE
and in revisedform
21
May 1979)
REFERENCES COURTNEY, K. R. (1978). J. theor. Biol. 73, 285. KENDALL, M. & STUART, A. (1977). The Advanced MCLACHLAN, E. M. (1975). J. Physiol. 245,447. ROBINSON, .I. (1976). Biomerrics 32, 61.
Theory
of Statistics.
London:
Griffin.
LETTER
TO
THE EDITOR
On Estimating Parameters of the Binomial Model for Transmitter Release at Synapses The quantum hypothesis of transmitter release assumes that there are n identical and independent release sites, each having a probability p of releasing a quantum of chemical transmitter. If, for example, quanta1 size has a Gaussian distribution with mean v and variance s2, then the distribution of the amplitude of evoked potentials is given by the binomial model as f(x) =il
(~)prqn~‘(2nrs2)~1~2
exp [ - (x-ur)2/2rs2].
(1)
Using observed spontaneous potentials to estimate v and s, the parameters p and y1can be estimated with the method of Robinson (1976). However, sometimes measurements of spontaneous post-synaptic responses are impossible. Therefore, in a recent issue of this journal, Courtney (1978) proposed to use the first three moments of the observed distribution of N evoked synaptic responses in order to get estimates of n, p, v and the mean quantum content m = n . p. Applying the binomial model and the well-known asymptotic method (Kendall & Stuart, 1977), standard errors for these estimates were given for p = 0.1, 0.5, 0.8 and the case n = 4, v = 1, s = 0 with sample size N = 200. Based on these results, the method was found reliable. Closer inspection of this procedure nevertheless reveals that the moment estimators become inefficient for small p and/or larger numbers n of release sites. This is shown by simulating evoked potentials with the density function (1) and computing the moment estimates i, A, fi and 6. For v = 1, s = 0 and a given combination of it, p and N, sampling was continued until 500 estimates with 0 < fi < 1 were obtained, resulting in the sample means and standard deviations shown in Table 1. From this table it appears that for realistic values of n and p large sample sizes N are needed in order to get estimates of practical value. Especially the distribution of ri comes out to be heavily dispersed. The moment estimates become still more impaired if, as it has been observed repeatedly, a variance s2 > 0 must be taken into account. Courtney reports estimates for a single simulated realization from distribution (1) with 441 0022%5193/79/190441
f03
%02.00/O
0
1979 Academic
Press Inc. (London)
Ltd.
442
D. SCHENZLE
TABLE 1 Sample means and standard deviations of Courtney’s parameter estimates 0, A, IA and Qfor distribution (1) with v = 1, s = 0 and different values of n, p and N. Each
j?gure is based on 500 experiments with N simulated evoked potentials. N, denotes the number of failures (0 < 0 or fi 3 1) N=50
100
200
ml
Nr
05
0.47+0.13 8+27 2.2 + 0.7 1.0*0.3
0.8
0.77 k 0.06 5*3 3.8 + 1.6 a96kO.3
05
0.5 f 0.20 14+24 4.2+ 1.7 l.lkO.4
0.8
0.76 + 0.11 13+21 8.3+5.0 l.OkO.5
03
0.52kO.20 35+104 8.0 k 3.4 1.2*0.5
0.8
o-73*0.15 64,671 17.5+ 12.0 1.0+0.5
9
0.49 + 009 6+19 2.1) 0.5 l.OkO.2
1
0.49 & 0.06 4+1 2.0 & 0.3 1.0+0.1
0
0.49 * 004 4.1+ 0.6 2.0 f 0.2 1.0+0.1
4 0
55
0.79 f 0.04 4+1 3.4kO.8 0.99 f 0.2 0.49+0.14 14532 4.2+ 1.3 l.OkO.3
0
18
0.79 + 0.03 4*1 3.3 + 0.6 1.0+0.2 0.48 kO.12 11+13 4.2t 1.0 1.0,0.2
0
‘1
0.8 f 0.02 4.1 + 0.5 3.2kO.3 1.0+0.1 0.49 + 0.07 913 4.12 0.6 1.0+0.1
8 5
123
0.78 + 0.06 lOA 7.3 + 2.7 l.OkO.3 0.49+0.17 48 + 239 8.2k3.0 1.1+0.4
0
82
0.79 & 0.04 9+3 6.9+ 1.7 1.0+02 0.48+0.15 12Ok505 8.3 &- 2.6 1.0+0.3
0
0.8 + O-02 8&l 6.5 + la 1.0,@2
27
0.48+0.11 22+39 8.3 f 1.8 1.0+02
0
0.79 * 0.04 17*5 13.4k2.8 1.0+0.2
16 26
0.76kO.11 77 + 154 15.6k8.6 1.0+0.4
5
0.78 f 0.07 20*12 14.6k5.3 1.0+0.3
v = 1, s = 0.3, p = O-3, n = 6 and N = 350: fi = O-3, ri = 5.7,5 = 1.06. These figures look favourably, but again some more extensive simulation results (Table 2) obtained in the same way as described above, show that in general the moment estimators undergo large fluctuations. The examples given here indicate that with sample sizes N reported in the
LETTER
2
TABLE
Rest&s
for s = 0.3 100
N=50
@41 k@17 0.5
443
TOTHEEDITOR
200
0.39kO.16
4.0+ 33+2151.4 1.1 kO.4
106
40+297 4.2 f 1.2 l.OkO.3
500
0.37 + 0.13 77
4.3il.O 19+42 1.010.2
0.37 + 0.1 28
4.3kO.7 15k23 1-o+ 0.2
4
8 0.8
0.60+0,17 23+41 9.Ok4.6 0.9 * 0.4
74
0.60+0.15 65+977 9.2k4.0 0.8 + 0.3
0.60FO.11 19$-40 8.9k3.0 0.8 + 0.2
2.
7
0.63 + 0.06 14*5 8.2k1.6 0.8 &- 0.2
’
literature (see e.g. McLachlan, 1975) it might be worthwhile to solve maximum likelihood equations. An efficient numerical method for doing this with distributions of type (1) is currently developed and will be described elsewhere. Inst. f. Med. Biometrie University of Tiibingen D 7400 Tiibingen, Germany (Received 19 February
1979,
D. SCHENZLE
and in revisedform
21
May 1979)
REFERENCES COURTNEY, K. R. (1978). J. theor. Biol. 73, 285. KENDALL, M. & STUART, A. (1977). The Advanced MCLACHLAN, E. M. (1975). J. Physiol. 245,447. ROBINSON, .I. (1976). Biomerrics 32, 61.
Theory
of Statistics.
London:
Griffin.
